Essays on Political Economy Contests
D I S S E R T A T I O N
of the University of St. Gallen,
School of Management,
Economics, Law, Social Sciences
and International Affairs
to obtain the title of
Doctor of Philosophy in Economics and Finance
submitted by
Philipp Georg Denter
from
Germany
Approved on the application of
Prof. Dr. Martin Kolmar
and
Prof. Stergios Skaperdas, PhD
Dissertation no. 4061
epubli GmbH, Berlin 2012
The University of St. Gallen, School of Management, Economics, Law,
Social Sciences and International Affairs hereby consents to the print-
ing of the present dissertation, without hereby expressing any opinion
on the views herein expressed.
St. Gallen, May 15, 2012
The President:
Prof. Dr. Thomas Bieger
Acknowledgements
During the process of writing this dissertation I have met so many people to
whom I now owe gratitude for their guidance and support, for distraction and
encouragement. Without them finishing this project would probably have been
much harder, maybe even impossible, and definitively not such a memorable
time.
First, I would like to thank my supervisors Martin Kolmar and Stergios
Skaperdas for their guidance and support. I have learned from both so much
more than just the art of conducting research. Learning from and working
with them was not only educational and enlightening but also inspirational
and entertaining. I am really grateful for having had this great opportunity.
A special thanks to Dana, who is not only my former office mate co-author
of Chapters 1, 3, and 4, but who is also my good friend for almost a decade
now. Discussing ideas and projects with her during day and night was always
very enjoyable and fruitful.
Another special thanks to John Morgan, who co-authored Chapter 3 and
who gave so many insightful comments on the other chapters.
Many thanks go to my friends and colleagues from St. Gallen, Irvine, and
Mainz, who made studying and conducting research such a pleasurable thing
to do. Fortunately, it is not possible to name them all now and I apologize to
everybody I am forgetting at this point. Thanks to Darjusch, Philip, Giovanni,
Mirco, Roland, Manuel, Kerstin, Vally, Alex, Sara, Martin, Hendrik, Ermira,
Marco, and Dirk.
A very special thanks goes to Sebastian, Christian, and Stefan for being
great friends for so many years, for always keeping me grounded, and for
always managing to put things in perspective for me. I have known them for
almost 30 years now and I am looking forward to the next 30.
Special thanks also to Barbara. Without her encouragement six years ago I
probably would never have started my PhD studies. Throughout the last years
she gave me encouragement and support and frequently reminded me that
there is more in life than just academia.
iv
Last but not least I would like to thank my family, Papa, Anne, Christoph,
Maria, Florian, Stepha, and my grandparents, for their unconditional support
throughout the years. The greatest thanks, however, goes to the only person
from this list that is not here anymore, Mama. Since it is not possible to express
in words all the things you did for us until you had to leave two years ago, and
how grateful I am and will remain, I keep it simple: Danke!
July 2012 Philipp Denter
Contents
Summary 1
Zusammenfassung 3
Introduction 5
1 The Effect of Polls in Political Campaigns 11
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 The Effect of Polls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.1 A Campaign Without a Poll: The Benchmark . . . . . . . 19
1.3.2 A Campaign with a Poll . . . . . . . . . . . . . . . . . . . . 21
1.4 Polls and the intensity of political competition . . . . . . . . . . . 28
1.5 Polls and candidates’ spending profiles . . . . . . . . . . . . . . . 29
1.6 Extending the basic model . . . . . . . . . . . . . . . . . . . . . . . 30
1.6.1 Functional Assumptions . . . . . . . . . . . . . . . . . . . . 30
1.6.2 Proportional Representation . . . . . . . . . . . . . . . . . 33
1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Political campaigns with specialized candidates 63
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.3 When Salience has no Effect: A Benchmark . . . . . . . . . . . . . 71
2.4 When Salience Matters . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3 Transparency in Contests 87
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
vi CONTENTS
3.3 Information Acquisition . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4 Information Disclosure . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5 Information Transmission . . . . . . . . . . . . . . . . . . . . . . . 98
3.6 More General Contest Success Function . . . . . . . . . . . . . . . 101
3.7 Mandatory Disclosure Policy . . . . . . . . . . . . . . . . . . . . . 102
3.8 Noisiness of the Contest and the Scope for Agreement . . . . . . 106
3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4 Imperfect Property Rights 131
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2.1 Basic Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2.2 Concerning Property Rights . . . . . . . . . . . . . . . . . 137
4.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.3.1 Complete Information: A Benchmark . . . . . . . . . . . . 138
4.3.2 Incomplete Information . . . . . . . . . . . . . . . . . . . . 138
4.4 Generalizing the Model . . . . . . . . . . . . . . . . . . . . . . . . 143
4.5 When There is a State . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Bibliography 153
Curriculum Vitae 165
List of Figures
1.1 Reactions functions and stage 1 equilibria. . . . . . . . . . . . . . 25
1.2 Parameter values that generate multiple equilibria. . . . . . . . . 26
1.3 Expected spending profiles, theory and data. . . . . . . . . . . . . 31
1.4 Anti-momentum in a polarized populace. . . . . . . . . . . . . . . 38
1.5 Appendix: Possibility of multiple equilibria in stage 1. . . . . . . 50
1.6 Appendix: Second derivatives in stage 1. . . . . . . . . . . . . . . 52
1.7 Appendix: Proof that polls decrease wastefulness of competition. 54
1.8 Appendix: Stage 2 mixed strategy equilibrium in the all-pay auc-
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.9 Appendix: Stage 1 pure strategy equilibrium in the all-pay auction. 59
2.1 Specialization and κ. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1 Best response functions of A against both opponent types. . . . . 95
3.2 Sequence of moves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.3 Welfare comparison in the all-pay auction. . . . . . . . . . . . . . 108
3.4 Appendix: Utility differences and zone of agreement in case of a
continuous uniform distribution. . . . . . . . . . . . . . . . . . . . 129
3.5 Appendix: Welfare implications of mandatory transparency in
case of a continuous uniform distribution. . . . . . . . . . . . . . 130
4.1 Marginal utility of effort. . . . . . . . . . . . . . . . . . . . . . . . . 141
4.2 Degree of property rights protection and expected relative valu-
ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
List of Tables
2.1 Equilibria of a parameterized version of the campaign game. . . 78
Summary
My dissertation consists of four chapters, all of which are devoted to the anal-
ysis of contests in a political economy context.
Chapter 1 studies the impact public opinion polls have on candidates’ incen-
tives to compete in a political campaign. Under a plurality rule, polls always
can cause situations in which the more popular candidate campaigns more
heavily. In close campaigns also the opposite may happen if having an influ-
ence is ceteris paribus not very expensive. Under proportional representation
no clear statement about the effect is possible unless there is more information
about the electorate available.
Chapter 2 studies communication strategies of candidates for a political
office during the campaign contest, when campaigning has both a policy ad-
vertising and an issue priming component. If candidates have comparative
advantages, they specialize in different issues. However, unlike predicted by
Riker’s dominance principle, specialization is usually imperfect, meaning can-
didates address all issue during the campaign. If one candidate has a large
marginal cost advantage, he may spend more than his opponent also on issue
in which he is disadvantaged. A candidate may also spend more on an issue
he does not specialize in if voters ex-ante perceive this issue to be relatively
important.
Chapter 3 studies situations where two lobbying groups compete for a po-
litical favor. When the goal of a policy maker is to keep the wastefulness of the
competition low, under mild conditions a policy that mandates the disclosure
of private information is not optimal and dominated by a policy of decentral-
ized information sharing. The same holds true if the goal of a politician is to
increase allocative efficiency.
Chapter 4 studies self-enforcement of property rights in a contest game.
While under complete information the defender may deter the appropriator
and so achieve secure property rights, this cannot happen when there is un-
certainty about the appropriator’s type, independent of risk preferences and
relative ability in the contest. Hence, we give a new rationale for imperfectly
2
enforced property that is not based on power but on heterogeneity and the
thereby caused uncertainty faced by the defender.
Zusammenfassung
Meine Dissertation hat vier Kapitel. In allen Kapiteln benutze ich Wettkampf-
modelle um politökonomische Fragestellungen zu untersuchen.
In Kapitel 1 untersuchen ich und meine Koautorin Dana Sisak welchen
Effekt öffentliche Meinungsumfragen für das Verhalten von konkurrierenden
Parteien in einem politischen Wahlkampf hat. Ohne Umfragen haben die
Parteien unabhängig vom Wahlsystem identische Anreize im Wahlkampf. Falls
es Umfragen gibt, existiert in Demokratien mit einfachem Mehrheitswahlrecht
immer ein Gleichgewicht in welchem die bislang beliebtere Partei stärker in
den Wahlkampf investiert. Falls der Wahlkampf einen starken Einfluss auf den
Wahlausgang hat, kann auch der gegenteilige Effekt auftreten. Handel es sich
statt um Mehrheitswahlrecht um Verhältniswahlrecht, dann kann keine allge-
meingültige Aussage bezüglich des Effektes von Umfragen gemacht werden,
da die Ergebnisse explizit von den Wählerpräferenzen abhängen.
In Kapitel 2 untersuche ich wie Parteien in Wahlkämpfen verschiedene The-
men hervorheben, wenn die Sichtbarkeit eines Themas die wahrgenommene
Wichtigkeit beeinflusst. Haben Parteien komparative Vorteile, werden sie sich
im Wahlkampf spezialisieren, solange kein Kandidat einen signifikanten Vorteil
durch niedrigere Grenzkosten hat.
In Kapitel 3 untersuchen ich und meine Koautoren John Morgan und Dana
Sisak den freiwilligen und dezentralen Austausch von Informationen in einem
Wettkampf. Die Ergebnisse stelle ich einer Situation mit verpflichtender Offen-
legung von Informationen gegenüber. Verpflichtende Offenlegung senkt häu-
fig die allokative Effizienz sowie steigert die ineffiziente Ressourcennutzung
im Vergleich zur dezentralen Lösung. Transparenz hat somit oftmals negative
Nebeneffekte.
In Kapitel 4 untersuchen meine Koautorin Dana Sisak und ich die endogene
Durchsetzung von Eigentumsrechten in Wettkämpfen mit sequentieller Strate-
gienwahl und unvollständigen Informationen. Zieht die Verteidigung zuerst
und ist hinreichend stark, können bei vollständigen Informationen Eigentums-
rechte perfekt gesichert werden, was in der Literatur häufig gezeigt wurde.
4 ZUSAMMENFASSUNG
Dieses Ergebnis bricht jedoch zusammen, sobald unvollständige Informationen
berücksichtigt werden. Somit wird eine neue Erklärung für unsichere Eigen-
tumsrechte gegeben.
Introduction
In this dissertation I employ contest models to analyze political economy phe-
nomena. But what exactly is a contest? While most people do not think about
contests in their lives frequently, contests are almost omnipresent. When we
compete for jobs or promotions we participate in a contest. We frequently
watch and participate in sporting contests. Firms compete in advertising con-
tests for market shares or engage in R&D contests to get a patent. Politicians
compete in campaign contests for voters’ support and lobbyists compete in rent-
seeking contest for political favors. And throughout history, mankind engaged
in violent contests such as war or social conflict. Although all these activities
are obviously very different in many respects, they share an important com-
mon characteristic: Different actors or players spend time, money and effort in
order to get or to win something they value, and if they are not successful they
nevertheless have to pay their outlays. These are the defining characteristics of
a contest.
I will now give a short and therefore naturally non-exhaustive account of
past research in the field of contest theory and its applications. For recent
surveys of this literature see Corchón (2007), Konrad (2009), or Sisak (2009).
Although the probably first paper using a contest model to analyze an
economic problem, due to Friedman (1958), was concerned with the study of
advertising, the classic application is lobbying and rent-seeking competition.
This literature dates back at least to the seminal paper of Tullock (1967), in
which he emphasizes that the actual costs of monopolies and tariffs are likely
to be much larger than suggested by the Harberger triangle, measuring just
the distorted market equilibrium. The reason is that apart from the distortion
of the market allocation, there is also a redistribution of rents from consumers
to producers. While this redistribution of rents is in itself not harmful from a
welfare perspective, it creates incentives to engage in socially wasteful activities:
Generally governments do not impose protective tariffs on their own. They
have to be lobbied or pressured into doing so by the expenditure of resources
in political activity. One would anticipate that the domestic producers
6 INTRODUCTION
would invest resources in lobbying for the tariff until the marginal return
on the last dollar so spent was equal to its likely return producing the trans-
fer. There might also be other interests trying to prevent the transfer and
putting resources into influencing the government in the other direction.
These expenditures, which may simply offset each other to some extent, are
purely wasteful from the standpoint of society as a whole; they are spent not
in increasing wealth, but in attempts to transfer or resist transfer of wealth.
Tullock (1967), page 228.
In other words, whenever there are transfers from one party to another to be
granted by a politician, there is likely to emerge a contest. Tullock’s analy-
sis sparked a whole literature analyzing the topic of rent-seeking and lobby-
ing, with notable contributions being Krueger (1974), Tullock (1980), Bhagwati
(1982), Hillman and Riley (1989), Ellingsen (1991), Baye et al. (1993), or Epstein
and Nitzan (2004). Surveys of this literature can be found in Nitzan (1994) and
Congelton et al. (2008). The focus of this literature is mostly the inefficiency
of the political process and the wastefulness of the competition. An important
question posed is to what extent the contested rents are dissipated in the com-
petition, because this is then a natural measure of the additional social costs
caused.
Another classic application of contest models is the study of violent con-
flict or war. Classical contributions include Hirshleifer (1991, 1995), Skaperdas
(1992), or Grossman and Kim (1995). A common denominator in this literature
is that property rights are not well defined or enforced and hence a contest in
which opposing players fight for possession might emerge. Hence, among the
central questions of this literature are when will there be peace or war, what are
the implications of imperfect property rights for economic efficiency under au-
tarky and trade, and under which conditions can secure property rights emerge.
Among the most important contributions to this literature is to show that (mar-
ket) income not necessarily reflects marginal productivity, what follows for ex-
ample from Hirshleifer (1991) and Skaperdas (1992), or that absent well defined
and enforced property rights trade may lead to (expected) efficiency losses rel-
ative to autarky, as has been shown by for example Skaperdas and Syropoulos
(2002) and Garfinkel et al. (2008). Other authors are concerned with identifying
conditions for the breakout of violent conflict as opposed to peace, for exam-
ple Skaperdas (1992), Grossman and Kim (1995), Slantchev (2010), or McBride
et al. (2011), and the emergence of political institutions and elites, for example
Grossman (2001, 2002), Hafer (2006), Kolmar (2008), or Hoffmann (2010). A
INTRODUCTION 7
recent and comprehensive survey of this literature can be found in Garfinkel
and Skaperdas (2007).
Related is also a strand of literature using contests to model political cam-
paigns. Two of the classics in the field, Brams and Davis (1973) and Snyder
(1989), study how candidates that compete in a campaign for political office al-
locate funds to different electoral districts under different electoral rules. Brams
and Davis (1973) thereby develop a ’3/2’ rule, that states that if districts are
of different sizes, larger districts receive a more than proportional share of
campaign resources. Klumpp and Polborn (2006) study the allocation of cam-
paign funds during the Primaries and Caucuses determining the candidates
for the presidential race. They thereby need to take into account the sequen-
tiality of the campaigns and find a strategic explanation for the so called ’New-
Hampshire effect’, which states that early states receive a more than propor-
tional share of funds, because success in an early state creates momentum.
Skaperdas and Grofman (1995) study conditions for candidates to engage in
negative campaigning. They show, for example, that in two candidate cam-
paigns the more popular candidate engages in less negative campaigning than
his opponent. Other more recent papers in this field are for example Konrad
(2004), Iaryczower and Mattozzi (2009, 2011), or Amoros and Puy (2011).
While the aforementioned strands of literature usually take the contest
structure as given and analyze the implications for behavior and outcomes,
in the tournament literature researchers study how a principal can design an
optimal contest to further his goals. For example, a principal may have the dis-
cretionary power to change the allocation of a prize purse to different prizes,
the number and characteristics of competing players, or the informational feed-
back rule during the tournament. Among the seminal contributions in this
field are Lazear and Rosen (1981), Nalebuff and Stiglitz (1983), O’Keeffe et al.
(1984), and Rosen (1986). See Lazear (1995) and Prendergast (1999) for sur-
veys. More recent contribution to this literature include Gradstein and Konrad
(1999), Moldovanu and Sela (2001, 2006), Gershkov and Perry (2009), Gürtler
and Münster (2010), or Ederer (2010).
This thesis has four chapters on contests in a political economy framework.
The first two chapters add to the literature analyzing political campaigns. The
first paper is concerned with the effect of public opinion polls for candidates’
campaigning incentives. Given the increased availability of polls and other in-
formation sources, it is interesting to ask in what way this kind of information
influences political outcomes. When studying political outcomes (at least) two
8 INTRODUCTION
sorts of actors are of importance: Voters and politicians. Since the early 1980s a
large literature has developed that analyzes the impact of public opinion polls
on voting decisions. To mutually exclusive hypotheses have been put forward
by scientists, namely that there are bandwagon effects and that there are under-
dog effects. If there is a bandwagon effect, voters’ propensity to vote for a can-
didate increases in the candidate’s past popularity. Justifications put forward
for the existence of bandwagon effects are various and include information cas-
cades (e.g. Bikhchandani et al. (1992)), and voters’ power to influence policy
(e.g. Straffin Jr. (1977)) to uncertainty aversion (e.g. Hong and Konrad (1998))
or the desire to vote for the winner (e.g. Callander (2007)). To the contrary, if
there is an underdog effect, there is to some degree a tendency that prevents
a party or candidate to become too dominant. A recent paper explaining why
there should be underdog effects is Goeree and Grosser (2007). In their model
a group that was in the majority in a pre-election poll has stronger incentives
to free-ride than the minority group, and hence voter turnout in the stronger
group is likely to be lower than in the minority group. While there exists a
literature analyzing candidate behavior in campaigns, and another literature
analyzing the effect of polls on voting behavior, there is no paper I am aware
of studying the effect polls have on candidate behavior. However, this topic is
interesting for various reasons. First, the availability of information through
polls increased significantly in recent years, and it is unlikely that the availabil-
ity of information does not have an effect on incentives. Second, due to increas-
ing party detachment in recent years, the group of voters that is targeted by
candidates in their campaigns – the swing voters – increased and hence cam-
paigning has become more important, too. In Chapter 1 my co-author Dana
Sisak and me develop a dynamic contest model to systematically study how
polls influence political campaigns. To keep the focus of the analysis on the
effect of polls we assume candidates have identical campaign technologies, but
may have different popularity levels before the campaign starts. We compare
two different electoral systems, a simple plurality rule and proportional repre-
sentation. Without a poll, candidates always invest identically in the campaign.
If there is a poll, we show that under a plurality rule there always exist equi-
libria in which the more popular candidate invests more in the campaign. In
close campaigns also the less popular candidate may invest more if the cam-
paign is important in determining election outcomes or – equivalently – if it
is not too expensive to have a significant influence on the campaign outcome.
Under proportional representation no generally valid conclusion can be drawn
unless there is more information concerning the electorate. If the electorate is
INTRODUCTION 9
relatively centrist results resemble those from a plurality rule. However, if the
electorate is polarized, in close campaigns there exist equilibria in which the
less popular candidate invests more. Therefore, the electoral system has an
important influence on candidates’ campaign incentives.
Chapter 2 also analyzes candidate behavior in a political campaign. In all
the papers mentioned above a campaign has only one dimension. However, in
reality we see candidates competing in a variety of issues and the winner is
the candidate that is perceived best in a weighted average of those issues. I
model a campaign between two candidates that compete in two issues, and the
issues’ weights for the aggregation are also endogenous. The reason for the lat-
ter assumption is that by talking a lot about an issue a candidate increases the
issue’s salience. Endogenous salience has been discussed in a variety of papers,
but I am aware of no paper except Amoros and Puy (2011) where this has an
effect on relative weights of the discussed issues in the campaign contest. In
their paper, however, candidates can only influence the issues’ weights by mak-
ing them salient. In contrast, I develop a model of three intertwined contests,
where one contest determines the relative importance of the other two contests
for competence in an issue. I show that when candidates have comparative ad-
vantages in the campaign, they must partially specialize in equilibrium; that is,
one candidate talks more about one issue and his opponent talks more about
the other issue. This is true unless one candidate has a significant advantage in
the marginal costs of funding. The model is helpful to explain and interpret for
example the presidential campaign between Barrack Obama and John McCain
2008 in the U.S.
Chapter 3 adds to the literature on lobbying or rent-seeking competition. In
many countries lobbyists are legally obliged to frequently disclose information
about their business. For example, in the U.S. the Lobbying and Disclosure Act
of 1995 and the Honest Leadership and Open Government Act of 2007 have
the goal to increase transparency of lobbying practices and thereby make lob-
byists accountable. In this paper I analyze together with my co-authors John
Morgan and Dana Sisak the effect of mandatory disclosure policies on different
measures of efficiency. To do so we study endogenous and voluntary transmis-
sion of information in a contest and contrast this to mandatory transmission.
It is shown that under mild conditions mandating transparency is detrimental
since it increases the wastefulness of the competition and decreases allocative
efficiency in the political process. Hence, we identify two negative side-effects
of transparency policy.
Finally, Chapter 4 adds to the literature on conflict and property rights.
10 INTRODUCTION
In this literature many authors have shown that without well-defined and en-
forced property rights, that is when an outside enforcer (Hafer (2006)) is miss-
ing, there are important inefficiencies and trade may actually decrease welfare,
see for example Skaperdas and Syropoulos (2002) or Garfinkel et al. (2008).
Other authors have claimed that even when no outside enforcer defines and
enforces property rights, incumbency may give rise to secure property rights
because it enables individuals to deter appropriative activities, see for exam-
ple Grossman and Kim (1995), Grossman (2001), Hafer (2006), or Gintis (2007).
It then seems that incumbency – or, equivalently, the opportunity to commit
resources to defend property rights – might overcome some of the discussed
problems and that some of the inefficiencies related to the absence of property
rights, for example distorted production incentives, disappear. However, as
shown by Kolmar (2008) this is not true, since deterrence is at the margin and
so production incentives remain distorted. In Chapter 4 my co-author Dana
Sisak and me go into a similar direction but show that incumbency does under
mild conditions not lead to deterrence, once informational asymmetries and
uncertainty are taken into account. Hence, contrary to the above cited papers,
we show that incumbency usually does not lead to secure property rights.
Chapter 1
The Effect of Polls in PoliticalCampaigns: A Dynamic Contest Model
Philipp Denter and Dana Sisak†
†We would like to thank Johann Bauer, Stefan Bühler, Patrick Button, Benoît Crutzen, Oliver Gürtler,Martin Kolmar, Michael McBride, Giovanni Mellace, John Morgan, Hendrik Rommeswinkel, PhilipSchuster, Stergios Skaperdas, participants of seminars at UC Irvine, University of St.Gallen, Universityof Tuebingen, the 2009 SSES meeting in Geneva, the 2011 APET meeting in Bloomington, the YoungResearchers Workshop on Contests and Tournaments 2011 in Berlin as well as the Erasmus PoliticalEconomy Workshop 2012 at Erasmus University Rotterdam. The kind hospitality of UC Berkeley andUC Irvine is gratefully acknowledged. All errors remain our own. Both authors gratefully acknowledgethe financial support of the Swiss National Science Foundation.
12 CHAPTER 1. THE EFFECT OF POLLS
1.1 Introduction
While an informed electorate is generally considered essential for the well-
functioning of a democracy, one exception concerns polls of candidates’ rela-
tive standing. Critics claim that polls undermine both the incentive to vote
as well as the vote itself; thus distorting voting decisions. As a consequence
the preferences of the populace are warped by the echo chamber of opinion
polls. For this reason, many countries have imposed a ban on the publication
of pre-election polling results. This ban can range from one day before the
election, as in France, to a whole month before the election, as for example
in Luxembourg (Spangenberg, 2003). Candidates, however, are still allowed to
commission opinion polls, even if the general public is not allowed access to
the results. In this paper we study the effect of opinion polls on candidates’
incentives to invest in their campaign. These investments in turn influence the
voters’ ballot choice on election day (Erikson and Palfrey, 2000) and thus the
final election outcome.
Polls are ubiquitous. In addition to traditional providers like Gallup and
Rasmussen Reports, newspapers and TV stations conduct their own polls. Poll
aggregator websites, such as realclearpolitics.com collect the plethora of poll re-
sults and offer a structured overview to the public. In addition, prediction mar-
kets such as politicalbetting.com and oddschecker.com offer alternative sources
of voters’ approval (Berg et al., 2008). Given the relevance of polls, numerous
studies have analyzed the various channels through which information about
candidates’ relative standing might affect voters’ behavior. A prominent hy-
pothesis is the existence of a bandwagon effect, which posits that voters in-
creasingly cast their vote for candidates doing well in the polls. This suggests
that polls create momentum in the sense that the front runner improves his
position over time. However, there are also studies predicting anti-momentum,
for example due to a mobilization effect, with the consequence that the front
runner in the polls experiences a loss of support in the election.
While scholars show a strong interest in the analysis of voters’ reaction to
polls, the other side of the political market – the politicians and parties – has
been virtually neglected. During political campaigns candidates spend large
amounts of resources to increase their chances of coming out ahead. For ex-
ample, total campaign spending during the 2008 Presidential Campaign in the
United States amounted to more than USD 1.6 billion, according to the Federal
Election Commission.1 Because the fraction of partisan voters has been shrink-
1Data on individual spending can be found on the commission’s home page: www.fec.gov/
1.1. INTRODUCTION 13
ing in many countries (e.g. Dalton and Wattenberg (2001)), there are more
swing voters to be swayed in a campaign, making campaigning increasingly
important in determining the election outcome.
The availability of reliable public information about candidates’ relative
standing has undoubtedly an effect on their incentives to compete. In this chap-
ter we analyze how this information influences the incentives of candidates to
spend in the course of a political campaign. In particular, we are interested
in how feedback about relative standing changes the candidates’ relative incen-
tives to compete and consequently whether polls affect the likelihood of an
incumbent coming out ahead. We construct a model in which candidates for
political office may spend resources early and late in the campaign to gain vot-
ers’ support. If there is no poll, candidates a priori have a common belief about
the median voter’s candidate ranking and cannot update their beliefs as the
election day comes closer. If there is a poll, candidates know the median’s can-
didate ranking when making their investment decisions. Our main results are
as follows:
• Under a plurality rule, polls always give the front runner an incentive to
campaign harder than his opponent and thereby create momentum. Thus
there always exists an equilibrium where the front-runner increases his
lead in expectation.
• In an environment where candidates have similar popularity and cam-
paign expenditures are very effective in influencing voters, polls can also
create anti-momentum. In other words, the trailing candidate may run
a more costly campaign than his adversary. In this case equilibria with
momentum and anti-momentum exist.
• Polls tend to make the campaign less "wasteful", i.e. expected aggregate
expenditures decrease.
To illustrate the intuition behind our results consider two ex-ante equally
popular candidates. They campaign over a certain period of time and during
this time random (unobservable) shocks to their popularity occur. Without a
poll, candidates never learn about their current standing with the voter and
thus at any point in time incentives are completely symmetric. With polls on
the other hand, candidates receive updates about their current relative stand-
ing. This alters their incentives in the following way. A candidate who receives
DisclosureSearch/MapAppRefreshCandList.do?d-16544-p=1&d-16544-s=4&d-16544-o=1.
14 CHAPTER 1. THE EFFECT OF POLLS
the information that he is ahead, now has an additional incentive to invest. The
reason is that any additional investment now will afford him an even greater
lead in expectation in the next poll. This in turn will decrease the expected
intensity and thus expected costs of future competition. A trailing candidate
on the other hand will have a weaker incentive to invest. Any additional unit of
investment now will bring him closer in expectation to his opponent in the next
poll and thus it will make future competition more fierce and costly in expec-
tation. Consequently polls drive a wedge between the investment incentives
of the trailing and leading candidate and thereby create momentum. When
campaigning is relatively effective and the candidates are relatively close, this
intuition is valid for the trailing candidate as well. A large investment will help
him overtake his opponent and at the same time defuse future competition in
expectation. In these situations both candidates have an incentive to preempt
the other with the objective to save costs in the future.
The chapter is organized as follows. In the remainder of this section we
review the relevant literature. Section 1.2 sets up and explains the basic model.
In Section 1.3 we explore the effect of providing information through polls in
a system of pure majority voting (FPTP). We discuss the effect of polls on the
intensity of political competition in Section 1.4. In Section 1.5 we study the
expected spending profile of candidates. Section 1.6 contains two important
extensions of the basic model. First we show that our technical assumptions
are not too restrictive and that in a more general model our findings will be
qualitatively preserved. In the second part of the section we study the effects
of polls under an alternative electoral system, proportional representation (PR).
Section 1.7 concludes. We relegate all formal proofs to the appendix.
Related Literature. Many scholars have directed their attention to the effect of
polls on election outcomes. The incentives to conduct polls about voters’ policy
preferences to choose a favorable campaign position are analyzed for example
in Bernhardt et al. (2009) and Jacobs and Shapiro (1994). In contrast, in our pa-
per polls inform about candidates’ relative standing and we study the effect of
polls on campaign spending. Earlier models of how candidates use their money
and time during an electoral contest were studied for example by Brams and
Davis (1973, 1974), Snyder (1989), Skaperdas and Grofman (1995), Stromberg
(2008), Iaryczower and Mattozzi (2009, 2011), and Meirowitz (2008). To be able
to study polls and the associated repercussions for electoral outcomes, we study
candidate incentives in a dynamic campaigning model. The dynamic nature of
the model is essential, since the effects we are interested in can only emerge if
1.1. INTRODUCTION 15
candidates can learn over time. The only other paper we are aware of study-
ing dynamics in campaign spending is Klumpp and Polborn (2006). While the
authors study incentives in sequential electoral contests during the primaries,
we consider spending dynamics and the role of informational feedback within
a single contest.
Among others Straffin Jr. (1977), Bikhchandani et al. (1992), Callander (2007),
and Hong and Konrad (1998) show how polls can distort voters’ decisions at
the ballot in favor of the more popular candidate. Rationales for this bandwagon
effect include information cascades, the desire to vote for the winner, and vot-
ers’ aversion to uncertainty. Other authors, e.g. Goeree and Grosser (2007),
argue in the opposite direction, claiming that leading in the polls might actu-
ally be harmful, due to a negative mobilization effect. A number of empirical
papers have tried to falsify one or the other hypothesis, but by now evidence is
at best inconclusive (see for example Blais et al. (2006), McAllister and Studlar
(1991), Nadeau et al. (1993), or Vowles (2002)). Our paper also addresses the
effect of polls on voting decisions and election outcomes, but through an indi-
rect channel: the strategies of political candidates. Candidates vie for voters
by spending time and money on their campaign. We study how candidates’
incentives to engage in such persuasive efforts are affected by polls.
Given that in our setting we find an advantage for more popular candidates
through polls, our paper also adds to the literature identifying and explaining
an incumbency advantage in political competition. For some early empirical
evidence on the existence of an incumbency advantage see for example Erikson
(1971) or Gelman and King (1990). A textbook justification for this phenomenon
are political business cycles as studied by Nordhaus (1975) or Drazen (2001). A
consequence of these theories is that incumbents are likely to have popularity
advantages at the outset of a campaign. Our paper shows that this advantage
is likely to be amplified through opinion polls.
Finally, a literature that is related because of the class of models employed
analyzes workers’ incentives in labor market tournaments. Gershkov and Perry
(2009), Aoyagi (2010), and Ederer (2010) study the optimal feedback policy in
a dynamic promotion tournament, when the firm’s goal is the maximization of
aggregate effort. For their purposes a two-period model with ex-ante identical
individuals is sufficient. We show that asymmetries in popularity only lead
to asymmetric behavior when a future period exists where the frontrunner can
reap the benefits of reduced competition caused by his earlier investment. Thus
to study momentum in a campaign we either need a model with at least three
periods or ex-ante asymmetric candidates.
16 CHAPTER 1. THE EFFECT OF POLLS
1.2 The Model
The model has three stages, t ∈ {1,2,3}, in which two candidates i ∈ {A, B}campaign to sway voters.2 In stage 1 and 2 candidates choose non-negative
campaign efforts xti ≥ 0. We denote the cumulative effort a candidate spent
in stages prior to t by Xti . Hence, X1
i = 0, X2i = x1
i , and X3i = x1
i + x2i . Effort
in each stage is costly and costs are quadratic. In particular, C(xti ) =
c2(xt
i )2,
c > 0. Convex costs may reflect both increasing marginal costs of effort or
decreasing marginal products. Similarly, c may both reflect marginal costs and
the marginal product of effort.3 In stage 3 the election takes place, in which
each voter casts a ballot for her preferred candidate. Each voter ι assigns to
each candidate utility utι,i(Xt
i , Eti ,αι,i) if he wins, where
utι,i(Xt
i , Eti ,αι,i) = Xt
i − Eti + αι,i. (1.1)
In each stage t = 1,2, after candidates chose effort, voters’ utility is muted
by a random shock ǫti , and the realization is observable not before the next
stage. Eti is the cumulative shock until t: E1
i = 0, E2i = ǫ1
i , and E3i = ǫ1
i + ǫ2i . ǫ1
i
and ǫ2i are i.i.d. random variables and, for simplicity, identical for all voters.
Hence we may interpret them as ’macro’ shocks.4 They represent changes in
the ranking that are not under the control of either candidate. For specificity
and simplicity, let ǫti be normally distributed with mean zero and variance σ/2.
This implies that the difference ǫt := ǫtA − ǫt
B is also normally distributed with
mean zero and variance σ. Throughout the paper, we shall use the symbols φ
and Φ to denote the PDF and CDF of the standard normal distribution where
needed. αι,i is the a priori value voter ι gains if i gets elected. As an example,
assume each candidate stands for a policy platform vi and voter ι has quadratic
preferences over policies of the form −(bι − pi)2. Moreover, each candidate may
2We assume campaigning is persuasive, rather than informative. Mueller and Stratmann (1994) arguedthat purely persuasive campaigning actually seems to be more in accord with most real world observa-tions than informative campaigning, that is, campaigning that reveals or discloses to the electorate theplatform of a candidate. Persuasive campaigning can, to the contrary, be defined as ’convincing a voterto vote for a particular candidate regardless of her position (p. 59)’. Hence, persuasive campaigningis very similar to persuasive advertising, see Chapter 2.2 in Bagwell (2007) for a recent survey of thisliterature.
3To see the analogy of decreasing marginal products and increasing marginal costs consider thefollowing: Assume the effective effort is f (x), which is weakly concave and increasing in effort x, andthat spending effort has costs C(x), which is weakly convex and increasing. We hence have increasingmarginal costs and decreasing marginal products. By defining effective effort x := f (x)⇔ x = f−1(x),we can also write costs in terms of effective effort, C(x) = C( f−1(x)), which is weakly convex in x.
4The assumption of such a ’macro’ shock is not necessary, but significantly simplifies the exposition.However, in general the shocks could as well have any other form that allows inferring the expectedranking of candidates in the next stage.
1.2. THE MODEL 17
have valence vi. Then, for example, αι,i =−(bι − pi)2 + vi. We do not model αι,i
explicitly but assume it is exogenously given. The example gives you an idea
what it may represent.
It is useful to define the difference of the utilities voter ι expects to get after
the election:
dtι := ut
ι,A − utι,B.
dtι is distributed according to Pt(dt
ι) on P t, with the differentiable density pt(dtι)
and median dt. The distribution may, for example, represent the distribution
of bliss points in the electorate. If d1> 0, then the median a priori prefers
candidate A over B, and vice versa if d1< 0. If the median is indifferent so
is the electorate as a whole, and both candidates receive a vote share of 50
percent.5
If there is no poll, candidates do not know the exact position of the median.
In this case they share the prior belief that d1 is uniformly distributed on [d,d],
d > d. Moreover, they do not learn the realization of ǫ1 before they campaign
in stage 2. To the contrary, if there is a poll, candidates learn d1 and, after
spending effort in t = 1, they learn d2. Note that it would only be possible
to correctly infer the median’s position from the result of a poll if pt had full
support on the real line, which follows from the assumption of normal shocks
ǫ. In particular, for all poll results stating one of the two candidate gets a
vote share of zero, both candidates are not able to tell which of the continuum
of points actually is correct. Such a situation is not extremely realistic, but
may happen in the model, which is due to the assumption of normal shocks,
which have full support. We keep this assumption in most of the paper for
analytical convenience, but note that is does not have any important qualitative
implications for our results (see Section 1.6.1). Hence, we assume a poll gives
candidates a precise estimate of the distribution of voters’ candidate ranking,
respectively of d1.6
The random shock deserves some more interpretation. A possible inter-
pretation is the one of random shocks to a voters’ candidate ranking. It is
well established how random events influence human decision making, also
in elections. Gassebner et al. (2008) show the influence of terrorist attacks on
5You may see that the assumed candidate ranking of a voter resembles a tug-of-war, as it has beenstudied by – for example – Harris and Vickers (1987), Konrad and Kovenock (2005) or Moscarini andSmith (2011). Where the analogy breaks down is how the game is decided. In the tug-of-war it is theplayer who first gets a given advantage over his opponent that wins, while in our model you need tohave the advantage exactly at t = 3.
6Morgan and Stocken (2008), Burke and Taylor (2008), and Meirowitz (2005) explore the informationcontent of polls when voters have an incentive to act strategically.
18 CHAPTER 1. THE EFFECT OF POLLS
politicians’ popularity. While this may still be related to policy or qualification,
other studies show the impact of pure random events – at east from the point
of view of a politician – influence election outcomes. For example, Healy et al.
(2010) study the impact of local college football games just before an election
takes place. If the local team wins, the incumbent’s vote share on average in-
creases by more than 1.6 percentage points. Similarly, natural disasters may
have a direct influence of voters’ candidate ranking.7 Finally, even weather con-
ditions influence our decision making and ranking of alternatives, as has been
shown by for example Saunders (1993) or Kamstra et al. (2003) in the context
of stock market behavior. Apart from shocks to candidate ranking, ǫ may as
well be interpreted as a random shock muting candidates’ campaign efforts in
a given stage, so that effective effort is not xti but xt
i − ǫti . This is a standard
assumption in the literature on labor tournaments that has been pioneered by
Lazear and Rosen (1981) and Nalebuff and Stiglitz (1983). In the context of po-
litical competition it may represent uncertainty about the effects of campaign
spending, and that campaigning might backfire as well (e.g. Dukakis in the
tank).
Finally, a word concerning the cost function. In our setting we naturally
need strictly convex cost functions to guarantee the existence of a pure strategy
equilibrium, since otherwise the second order conditions cannot hold for both
candidates simultaneously. However, strict convexity is not yet sufficient, but
costs need to be sufficiently convex. What this means will be discussed in the
relevant appendices. Throughout the paper, with the exception of our discus-
sion of the All-pay auction in Section 1.6.1, we will assume costs are sufficiently
convex.
1.3 The Effect of Polls
We now start our analysis of the campaign by studying candidate behavior in
simple plurality electoral systems (first past the post / FPTP). In a FPTP system,
the goal of a politician, a party, or a candidate is to win the simple majority of
7A recent example for such a shock is the disaster caused by the damaged Fukushima Daiichinuclear power plant in Japan that followed the earthquake and the thereby caused the tsunami ofMarch 11, 2011. In the following, in many countries around the world a shift of voters’ preferencesin favor of green or anti-nuclear movements was accounted for. In Germany, two important stateelections took place just one week after the disaster. In political polls a large increase of anti nuclearmovements and green party support was recognized. In the traditionally conservative state of Baden-Württemberg the Green Party won the election after 66 consecutive years of a conservative ChristianDemocratic government and the Green Party’s candidate, Winfried Kretschmann, is now the first greenstate governor (Ministerpräsident) in Germany. The Green Party’s success was attributed, among otherthings, to the revived anti-nuclear movement.
1.3. THE EFFECT OF POLLS 19
votes. That is, as long as a plurality is achieved, the candidate is indifferent
whether he wins by a small or by a large margin; because he is declared the
winner anyway. As an example, take the electoral college in the U.S. Candidates
for presidency compete in several states for voters’ support, and the candidate
who wins the plurality in a state wins all presidential electors, too, no matter
how close the election outcome was.8 In a second stage, it is again the abso-
lute majority in the Electoral College that is the majority of electors, which is
needed to elect the president and vice-president. Other countries with FPTP
systems are for example the United Kingdom and most of the countries that
were historically under significant British influence, such as India, or Canada.
With more than 2 billion people in 47 countries living with a FPTP electoral sys-
tem in 2005, it is together with list proportional representation (PR) the most
important electoral system worldwide (see Reynolds et al. (2005)).
Let us now start with the analysis of conducting a poll. First, we establish
a benchmark by looking at a situation without polls.
1.3.1 A Campaign Without a Poll: The Benchmark
If there is no poll the campaign protocol is as follows. In t = 1 and t = 2 both
candidates have the opportunity to spend effort, and they do not receive a
signal as to their current popularity with the electorate in either stage. Hence,
they have to base their decision on their common prior beliefs F(d1) in both
stages.
Let us conjecture what happens in equilibrium. First, because a player does
not receive a signal as to the realization of ǫ both candidates’ beliefs after t = 1
are unaltered. Given an identical technology to produce effective campaign
effort in both stages, we should expect each candidate to choose the same action
in each stage. But how does F(d1) influence effort choices? First, if d1 = 0, both
candidates are identical in expectation, and hence it is intuitive to assume there
is a symmetric equilibrium, too.9 However, what happens if one candidate
has a competitive edge over his opponent? The answer is not obvious because
there are many different effects at work and there are three different sources
8Exceptions are Nebraska and Maine, in which the winner of the popular vote receives twoelectoral votes, and the winner of each of the congressional districts, there are two in Maineand three in Nebraska, receives one electoral vote in addition. Just recently, a discussion,started by newly in power republican politicians, centered on the question whether or not tochange the electoral system also in Pennsylvania from FPTP to a mixture of winning the popu-lar vote and the single congressional districts. http://www.nytimes.com/2011/09/19/us/politics/
pennsylvania-republicans-weigh-electoral-vote-changes.html.9By a symmetric equilibrium we mean an equilibrium in which both candidates choose identical
strategies.
20 CHAPTER 1. THE EFFECT OF POLLS
of uncertainty. One source is the lack of knowledge about the current state of
the world, and finds representation in F(d1). The other two sources are related
to uncertainty about future shifts in the distribution of preferences, given by ǫ1
and ǫ2. Since candidates are different it seems intuitive that they also engage
in different behavior. The leading candidate might have an incentive to be
aggressive and secure his lead by spending a lot. But since effort is costly he
may as well take a soft position to safe on costs and hence spend relatively little
compared to his opponent. In analogy to an markets it is intuitive to believe in
the first course of things, since in such a setting the laggard often loses market
share in expectation (see for example Athey and Schmutzler (2001) or Cabral
(2002)). In our case this intuition is, however, flawed:
Proposition 1.3.1. The equilibrium is unique. Both candidates choose in each stage
identical effort, and efforts are also identical across stages. Hence, absent a poll the
ex-ante expected popular vote of a candidate in t = 3 is identical to his popularity in
t = 1.
Proof. See appendix.
To see why efforts are identical in equilibrium it is instructive to look at
candidates’ marginal incentives. For a fixed median position d1, note that the
marginal increase in popular vote / probability to win of A is exactly the share
(density) of currently indifferent voters. Of course, this also holds for B. Taking
into account uncertainty about d1 this must hold also in expectation. Moreover,
since candidates have identical technologies to produce effort, they also have
identical marginal products and marginal costs of effort. Consequently, for
any prior distribution F(d1) both candidates have identical marginal incentives,
and hence the equilibrium must be symmetric. This is similar to a result by
Lazear and Rosen (1981) in their analysis of labor market tournaments, when a
candidate enjoys a head start.
A direct implication of the proposition is, that the campaign does not have
any effect on the expected winner of the election, if there is no poll. Candidates
always spend identical effort, the net effect is zero, and the winning probability
will be unaffected in equilibrium. Here we clearly see how political campaigns,
or contests in general, are similar to the Prisoner’s Dilemma: both candidates
would be better of by reducing effort by some k > 0, or by spending nothing at
all, because effort is costly. However, this is not an equilibrium.
1.3. THE EFFECT OF POLLS 21
1.3.2 A Campaign with a Poll
Now turn to the case of a campaign with a poll. Unlike before, in t = 1 both
candidates learn d1 and d2. In most modern democracies polls are an almost
pervasive element of the political landscape and this kind of information is, by
now, accessible in abundance. Apart from polls, politicians may as well learn
something about their popularity from prediction markets in the internet. This
information makes it possible for a candidate to react to changes in his pop-
ularity, and, at the same time, even if the current popularity does not change
(d1 = d2), this gives additional information to the candidates; after all, learning
about the popularity in t = 2 is confounded by less noise than it was in t = 1,
and for sure by less than absent a poll.
Look at stage 2 first. After learning d2, both candidates spend effort. This
situation is strategically similar to the situation of the candidates absent a poll;
the structure and amount of noise is, however, different. But we have seen that
structure and amount of noise were not decisive for relative incentives. Hence,
although efforts probably differ in absolute levels, the relative level of effort is
as it was before:
Proposition 1.3.2. A unique equilibrium exists in stage 2. In this equilibrium both
candidates always choose identical effort.
Proof. See appendix.
The intuition for identical efforts is identical to before: by marginally in-
creasing effort, the additional probability to win equals exactly his opponent’s
loss in winning probability; the marginal benefit of effort is identical for both
candidates independent of their popularity. Because marginal costs are also
identical, both must choose identical effort. Hence, we can conclude that the
expected popular vote a candidate gets in t = 3 equals his popularity in stage
2.
Now turn to the comparative statics of the equilibrium. Most importantly,
consider the effect on efforts of varying d2. Those comparative statics are of
great importance, because from the point of view of t = 1, the relative popular-
ity in stage 2 is endogenous. Since the equilibrium is symmetric, the difference
in efforts, x2A − x2
B, is zero, independent of d2. Hence, the marginal increase
in the winning probability in equilibrium is the density of the shock / noise,
evaluated at d2. The normal density is strictly quasi-concave and symmetric at
zero and the shock distribution determines the marginal product of campaign-
ing effort. By spending more, the probability that the realization of the shock
22 CHAPTER 1. THE EFFECT OF POLLS
is such that the candidate wins, gets larger. The marginal product of effort is
exactly the marginal increase of this probability, and hence in the symmetric
equilibrium it must be φ(d2). We know now that φ(z) is strictly decreasing
in |z|, implying marginal effort decreases as |d2| gets larger. Put differently,
equilibrium effort in t = 2 is an inversely U-shaped function of d2 and has its
maximum at d2 = 0. More plastically, the intensity of the campaign in t = 2
increases as the popularity difference vanishes. What happens if we vary σ?
Increasing it implies φ goes up at the tails and goes down in the center, hence
becomes flatter. Therefore, if the competition is lopsided with one candidate
enjoying a big lead, increasing the variance of the shock increases equilibrium
spending. Intuitively, if the variance is large the trailing candidate has a real-
istic chance to catch up without spending overly in the campaign. This also
gives him an incentive to increase effort, since, as we have discussed before,
φ determines the marginal product of effort. To the contrary, if σ is relatively
low, catching up due to luck is relatively unlikely, and hence the trailing candi-
date has no incentive to compete anymore. As a result both candidates’ efforts
decrease. Finally, efforts strictly decrease in c.
Next turn to stage 1. Candidates take into account the expected effect their
campaign spending in t = 1 has on stage 2 through d2 = d1 + ∆1 − ǫ1. So far
we have seen that candidates choose identical efforts, no matter what is the
relative popularity. What should we expect now given the equilibrium in stage
2? If d1 = 0 and hence candidates are perfectly symmetric, it seems intuitive
that there also exists a symmetric equilibrium. It is, however, not completely
clear, intuitively, what happens if d1 6= 0. Has the trailing player an incentive
to try extra hard to “gamble for resurrection”? Or has the leading candidate
stronger incentives? As it turns out the results depend on both c and σ, more
specifically, on ρ := cσ2. We can interpret ρ as a measure of how expensive it is
to gain a substantial advantage due to effort in a close campaign. The higher
ρ, the more expensive it is. To see this note that a high variance σ implies the
density in the center of the shock distribution is low, and hence there is not
much mass. Spending a given level of effort hence has a lower impact on the
probability to win the campaign than it would have if σ was small. Hence, we
can interpret ρ as a measure of the expensiveness to influence the campaign’s
outcome. In particular, there exists a threshold, ρ := (334√
π)−1, that determines
two regimes of equilibrium outcomes. We will discuss both regimes separately.
ρ > ρ. If ρ is relatively large it is relatively expensive to influence the expected
outcome of the election when the race is close. Now what effect does this have
1.3. THE EFFECT OF POLLS 23
on the equilibrium of the campaign game in t = 1? When it is relatively costly
to influence the expected outcome in a close campaign, the leading candidate
(if d1 6=0) will probably not be able to make a decisive step forward, but the
high level of noise makes it more likely that the trailing candidate catches up.
Hence, it seems that the trailing candidate benefits from higher risk and thus
that he has stronger incentives to invest, too. However, deeper inspection of the
situation reveals that this logic is missing an important point. To see that con-
sider the decision each candidate has to make in t = 1 in more detail. There are
three different channels through which effort influences a candidate’s expected
utility. First, effort increases the probability to win the election. This effect is,
as in the previous section, identical for both, and hence cannot be the reason
for differences in campaign disbursements. Second, effort has immediate costs.
But candidates have identical cost functions, and hence marginal costs are also
identical. Therefore, this cannot be the reason for different behavior, either.
There is, however, a third channel. By increasing effort in t = 1 a candidate
changes the expected state of the campaign in t = 2 a bit in his favor, because
d2 = d1 + ∆1 − ǫ1; the marginal impact of effort in t = 1 on d2 is equal to one.
Changing d2 has consequences for spending, and hence for costs, too. From the
discussion of the comparative statics in t = 2 we know that costs are highest
when the race is tied, d2 = 0, and that costs decrease monotonically if we let |d2|grow, because spending goes down then. This implies that the leading candi-
date, by exerting effort in stage 1, locally increases |d2| in expectation and hence
decreases expected costs in stage 2 for both, while the opposite holds for the
trailing candidate. In more technical terms, this implies the leading candidate’s
efforts in stage 1 and 2 are strategic substitutes, while the trailing candidates
efforts are strategic complements (Bulow et al. (1985)). This implies the leader
has lower expected marginal costs than his opponent. As a consequence, he
will spend more in equilibrium and thereby in expectation increase the differ-
ence to his opponent; we have a situation in which the leader acts tough, while
his opponent takes a softer stance. Hence, unlike the situation without a poll,
the analogy to the increasing dominance results of Athey and Schmutzler (2001)
and Cabral (2002) is valid, although the mechanism is a different one.
Where now does the expensiveness to have an influence come into play? A
large ρ is due to either a high risk, or high marginal costs, or both. The degree
of risk or noise in the electoral contest implies the effect stage 1 effort has on
the expected stage 2 campaign is somewhat limited. Because both high risk
implies efforts are relatively low in equilibrium, so are costs. Hence, there is
not much to save in stage 2 due to spending more in stage 1. The expected
24 CHAPTER 1. THE EFFECT OF POLLS
marginal effect of effort today on costs tomorrow is low. High marginal costs,
on the other hand, make it directly unattractive to spend a lot. While this is
true for both candidates, it is even more relevant for the candidate trailing. For
him it is prohibitively expensive to overinvest in stage 1 so as to gamble for
resurrection; he would have to spend a lot to safe a bit only. This, however, is
not true for the leader. Hence, if the costs of influence are large there cannot
be an equilibrium in which the trailing candidate acts tough and tries to force
a change. Rather, he adopts a ’wait and see’ posture. Before discussing the
implication of a relatively low ρ we summarize the results our results in the
following proposition:
Proposition 1.3.3. If it is relatively expensive to change the outcome in expectation,
ρ > ρ, in the unique equilibrium the leading candidate chooses greater effort than his
opponent. If the race is tied both candidates spend identical efforts.
Proof. See appendix.
ρ ≤ ρ. Now consider the case that ρ is relatively low, which means it is ceteris
paribus relatively cheap to increase the probability to win in expectation if the
race is close. This is so because either marginal costs are low, there is little noise
/ σ is small, or both. Then the aforementioned logic, that the trailing candidate
cannot outspend the leader in equilibrium, does not apply anymore, and hence
’gambling for resurrection’ may actually be an equilibrium. However, there is
another condition that needs to hold for the leader taking a soft stance, and his
opponent acting aggressively: the difference between candidates must not be
too large. To see why this is the case recall the aforementioned logic of strategic
substitutes and complements. A candidate’s efforts are strategic substitutes if
and only if in equilibrium he is in expectation more popular than his opponent
in stage 2. If this does not hold, by spending marginally more a candidate
decreases the expected gap and hence increases expected costs, while the oppo-
site holds for his opponent. Hence, the opponent has lower marginal costs of
effort and hence it cannot be an equilibrium to spend more than the opponent.
Consequently, the trailing candidate may spend more in equilibrium, but if he
does this he needs to spend enough to turn the state in his favor in expectation.
Of course, turning the state becomes increasingly expensive as the difference
between candidates increases, and there exists a threshold gap determining the
maximum lead that the trailing candidate may try to turn by investing heavily
in campaigning. Obviously, if the trailing candidate can take charge and act
aggressively in the campaign in equilibrium, this is also possible for the leader.
1.3. THE EFFECT OF POLLS 25
0.40 0.45 0.50 0.55 0.60 0.65 0.70
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.40 0.45 0.50 0.55 0.60 0.65 0.70
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.40 0.45 0.50 0.55 0.60 0.65 0.70
0.40
0.45
0.50
0.55
0.60
0.65
0.70
x1Ax1
Ax1A
x1Bx1
Bx1B
Figure 1.1: Reaction functions for A (gray) and B (brown, dashed) and stage 1 equi-libria for ρ = 0.242 and d1 ∈ {0,0.04,0.1}.
After all, for him it is cheaper to stay in the lead than for his opponent to turn
the electorate’s sentiment. Hence, there are multiple equilibria. It remains to
show what happens when the race is tied. As before, to conjecture that there ex-
ists a symmetric equilibrium is appealing. But we also have seen that there are
multiple equilibria if the game is not tied and no candidate dominates. Those
asymmetric equilibria also exist in the tied race. In addition, a symmetric equi-
librium exists. Hence, there are three equilibria for d1 = 0, or more generally,
when |d1| is small. However, note that the symmetric equilibrium at d1 = 0 is
asymptotically unstable, while the asymmetric equilibria are stable. In Figure
1.1 we simulated the equilibria using the best response functions of the can-
didates for ρ < ρ and d1 ∈ {0,0.01,0.1}. We summarize the results formally:
Proposition 1.3.4. If ρ is relatively small, there exist multiple equilibria, and, if we
confine ourselves to asymptotically stable equilibria, there does not exist a symmetric
equilibrium. In close races either candidate may spend more, while if one candidate has
a sufficiently large advantage, this candidate spends more than his opponent.
Proof. See appendix.
A final word concerning the equilibria when ρ is low is in order. A low
value of ρ implies that either marginal costs are low, the campaign is – to
borrow from the theory of contests – relatively discriminating (low σ), or both.
Either of those makes it less likely that an interior pure strategy equilibrium
actually exists, since the second order conditions are then likely to be violated.
However, there exists a small range of values for ρ for which the second order
conditions hold and ρ ≤ ρ. In Figure 1.2 you can see combinations of c and σ
for which equilibria as discussed above actually exist.
26 CHAPTER 1. THE EFFECT OF POLLS
0 2 4 6 8 10
0.2
0.4
0.6
0.8
1.0
Figure 1.2: Combinations of c (ordinate) and σ (abscissa) for which multiple andasymmetric interior pure strategy equilibria in stage 1 exist.
Having discussed campaigning incentives, we now explore the expected
impact a poll has on election outcomes in more detail. We are in particular
interested in the probability that the more popular candidate increases his lead.
Because the campaign is noisy, spending more is not sufficient to improve one’s
position, but makes it more likely. Note that in a campaign without any noise,
this would not change because than there is no pure strategy equilibrium and
candidates hence create noise endogenously by randomizing (see for example
Nalebuff and Stiglitz (1983), Hillman and Riley (1989), or Baye et al. (1993)).
We can now determine the probability distribution of changes in the relative
standing. Absent a poll we have seen that efforts are always identical, and
hence (i) campaigning has no effect on the outcome in equilibrium, and (ii), each
candidate improves his standing with 50 percent probability. Conducting a poll
has no effect on relative incentives in stage 2, but alters incentives significantly
in stage 1.
Assume without loss of generality that d1 ≥ 0. The probability that A in-
creases his lead is
q(ρ,d1) := Pr[d2 ≥ d1|(ρ,d1)] = Pr[∆1 ≥ ǫ1|(ρ,d1)
]= Θ
(∆1(ρ,d1)
σ
).
Not surprisingly, this probability is strictly increasing in the difference of
efforts spent in equilibrium, ∆1. We hence need to explore the incentive dif-
ferential of the two candidates, as a function of the median’s ranking d1. To
develop an intuition for the underlying mechanism, consider first the two po-
lar case, d1 = 0 and d1 → ∞. In the first case candidates are identical, and as
we have seen before, the equilibrium is then symmetric. Hence in this case we
have q(ρ,0) = 12 . In the latter case A de facto already won the campaign. As a
result no candidate has an incentive to spend effort at all, and hence we find
1.3. THE EFFECT OF POLLS 27
also q(ρ,∞) = 12 . If we increase d1 starting at zero and let it converge to infin-
ity, the incentive differential is increasing first, and so is ∆1. At a given point,
however, this stops and incentives become more and more identical again. The
differential incentives first increase because it becomes more likely for A to ac-
tually improve his position. However, as his popularity grows this becomes
less and less interesting, because spending more means incurring higher costs;
but if A is already very popular he starts to care more about the costs he can
save by spending less, because the probability that he wins is already large, and
he cannot safe much on efforts either, because in a lopsided campaign in t = 2
both candidates will not spend much effort in equilibrium. Therefore, there
exists an inversely U-shaped relation between d1 ∈ [0,∞] and q(ρ,d1).
Proposition 1.3.5. Assume ρ < ρ, that is the campaign is expensive to decide. q(ρ,d1)
is an inversely U-shaped function of d1 on [0,∞]. It attains its maximal value of
Θ(
112√
2eπρ2
)>
12 when d1 =
√32σ.
Proof. See appendix.
Note that the maximum is strictly decreasing in ρ, what is intuitive. If
having a significant influence on campaign outcomes is relatively expensive
we should not expect candidates to try investing a lot to ’decide’ it. It also
follows that the maximum of the maxima will be attained when ρ = ρ and is
q(ρ,√
32σ) = 57.37. This does not seem bo be a very large number, but note that
absent a poll it would have been 50 percent instead, and that the infinite sup-
port of the normal distribution naturally decreases the number. For different
distributions it will certainly be larger.
If it is not expensive to influence the expected campaign outcome, the tra-
jectory of q changes for small values of d1, and is not equal to zero for d1 = 0.
However, assuming equilibria are stable, the results are qualitatively unaltered
– the trajectory is inversely U-shaped – and hence we relegate the discussion of
this case to the appendix.
What are now the effects of conducting a poll? One relates to the role of
campaigning to decide the race. We have seen in the previous section that ab-
sent a poll campaigning has no influence whatsoever on the outcome of the
election and candidates are basically trapped in a Prisoner’s dilemma. Even
when candidates are asymmetric in the sense that one has a popularity edge,
incentives to campaign are identical and hence campaigning is only costly with-
out helping anybody to increase his expected share. This changes as soon as we
28 CHAPTER 1. THE EFFECT OF POLLS
introduce a poll. The knowledge of getting feedback in a future stage and hence
having the ability to adjust campaigning efforts accordingly creates incentives
to behave differently in the earlier stage of the campaign. Hence, campaign-
ing has an effect on the expected outcome. Our second result relates to the
direction of this effect. In general there always is an equilibrium in which the
leading candidate assumes a tough posture by spending more and so increases
his expected lead. However, in a close campaign in which it is relatively cheap
to influence the probability to win, also the trailing candidate may act tough.
1.4 Polls and the intensity of political competition
So far we have focused on the effects of polls on the outcome of the election,
which is determined through relative candidate spending in the campaign. We
have neglected the actual level of expenditures. Campaign expenditures can
be quite substantial. For example, they typically exceed one billion U.S. dol-
lars in the case of presidential elections in the U.S. Thus the "wastefulness of
competition" is also an important aspect to study and relate to the existence of
polls. We are aware that not all campaign efforts are necessarily wasteful since
campaigns also inform voters about the candidates’ positions. Nevertheless,
we define wastefulness as expected aggregate spending over the course of the
campaign. Thus we implicitly assume that polls do not influence the level of
informative campaigning and candidates on the margin invest for example in
image building activities. The following proposition shows the result:
Proposition 1.4.1. When there are polls, aggregate campaign expenditures decrease in
expectation. Thus, ceteris paribus, polls make competition less wasteful.
Proof. See appendix.
The intuition for the result is the following. When polls create momen-
tum, initial asymmetries are amplified. This decreases the expected intensity of
competition in subsequent rounds, and hence expected wastefulness decreases,
too.
The effect of increased information on the intensity of competition has been
a subject of interest in other studies as well. A recent example is Chapter 3 of
this dissertation, where we analyze the effect of mandatory disclosure require-
ments in competitive environments such as lobbying competition and political
campaigns. They study the competitors’ incentives to share private information
about their characteristics, for example their valuation of winning, and how
mandated disclosure affects the outcome of competition. Their main result is
1.5. POLLS AND CANDIDATES’ SPENDING PROFILES 29
that mandatory disclosure increases the intensity and decreases allocative effi-
ciency of competition in expectation. In their setting, an uninformed player can
use the chance of facing a weak opponent to appease a strong opponent, and
at the same time the threat to face a strong opponent to discourage a possible
weak opponent. The lack of information about the opponent helps to commit
to strategies which would not be credible under complete information. In con-
trast, in this paper, candidates are identical except for their current popularity.
There is no private information held by the candidates, and hence they can not
use his lack of knowledge as a commitment.
1.5 Polls and candidates’ spending profiles
In this section we now take a closer look at the spending profiles of the candi-
dates when there is a poll. If there is no poll the result is straightforward, spend-
ing decreases as one candidate becomes more and more advantaged. Hence,
here the focus is on spending profiles when there are polls. We will then com-
pare our results with empirical findings and show that the model performs
relatively well in predicting spending profiles.
We know already that in a campaign that is costly to decide polls create
momentum and the more popular candidate invests more heavily in the cam-
paign. We are now interested how the two candidates’ spending varies in
absolute terms as we increase |d1| from zero. Conjecturing about how efforts
change one is tempted to say efforts unambiguously decrease, as the campaign
becomes more lopsided and hence the need to spend decreases. This also what
we observed in stage 2, where spending decreases monotonically in |d2|, and is
also what would happen in the models of for example Snyder (1989), Erikson
and Palfrey (2000), or Klumpp and Polborn (2006). However, as we have seen
already before, introducing dynamics may change results significantly and it
also does in this case:
Proposition 1.5.1. Assume ρ > ρ. Candidate A’s expected total effort increases in d1
when d1 = 0, while candidate B’s expected total effort decreases.
Proof. See appendix.
The trailing candidate’s campaign effort decreases monotonically as |d1|grows larger. His more popular adversary, however, has an incentive to first in-
crease effort when his lead grows larger, before he also cuts down on spending
when he becomes more and more advantaged in the campaign. Hence we can
now describe the two candidates spending profiles completely:
30 CHAPTER 1. THE EFFECT OF POLLS
Corollary 1.5.1. As |d1| increases from zero the more popular candidate first increases
effort in the campaign and his spending declines only after he reaches a certain popu-
larity advantage. The trailing candidate monotonically decreases spending and spends
less than his opponent for all |d1| > 0.
Proof. See appendix.
In the right panel of Figure 1.3 we plotted expected total campaign spend-
ing for both candidates. As stated in the corollary, the more popular candi-
date’s spending first increase until it reaches a maximum, and monotonically
decreases thereafter. The trailing candidate’s spending directly decreases mono-
tonically in d1. An interesting question is how well the model performs in pre-
dicting candidate spending in a campaign. For a comparison we show in the
left panel of Figure 1.3 the vote-on-spending effects estimated by Erikson and
Palfrey (2000). Although our model is relatively simple it predicts the spending
profile surprisingly well. At d1 = 0, which represents a predicted incumbent
vote of 50 percent, both candidates choose identical spending. The incumbent’s
spending increases first and decreases after peaking at around 55 percent un-
til the predicted incumbent vote reaches some 80-85 percent. Then, somewhat
surprisingly, spending goes up again. The challenger’s spending almost mono-
tonically decreases as the incumbents predicted vote increases over 50 percent
and also remains below the incumbent’s spending.
1.6 Extending the basic model
In this section we discuss two extensions to the basic model. We start off show-
ing that our results from the previous section are qualitatively robust to more
general functional specifications. Next we depart from the assumption of pure
maximization of the probability to win and consider candidates that are also
interested in the popular vote per se, as it is the case in proportional represen-
tation (PR) electoral systems.
1.6.1 Functional Assumptions
So far we assumed the distribution of shocks, respectively the noise, is normally
distributed. Also, costs were quadratic. In this section we show that similar
results will emerge if we relax these assumptions.
Consider the distribution of ǫ first. While, as we argued above, the nor-
mality assumption can be justified – for example by invoking the central limit
1.6. EXTENDING THE BASIC MODEL 31
50 55 60 65 70 75 80
0.0
0.2
0.4
0.6
0.8
Figure 1.3: Total spending per candidate, depending on popularity. The left panelshows Figure 3 from Erikson and Palfrey (2000), which shows candidatespending in the U.S. The right panel shows the predictions of our model(expected total spending) when there is a unique equilibrium under FPTP.Our predictions resemble the data pretty well. When popularity is around50 percent for each candidate, spending is more or less equal. If weincrease the popularity of one candidate this candidate’s total effort in-creases first and decreases then again. For the less popular candidatespending monotonically declines. Generally, the more popular candidatespends more than his opponent.
theorem – it is nevertheless of interest to explore the robustness of our results in
more detail. Hence we relax this assumption and assume the following instead:
Assumption 1.6.1. ǫ1 and ǫ2 are independently distributed on S1 ⊆ R and S2 ⊆ R
with densities g1(ǫ) and g2(ǫ). gt(ǫ) is differentiable on S t, symmetric around zero
and g′(|ǫ|) ≤ 0.
By Assumption 1.6.1 each shock has a quasi-concave density, implying that
a large shock is less likely to occur than a small shock. Equivalently, it is less
likely to have a series of random events only in support of one of the two can-
didates than a series that is relatively balanced, i.e. each candidate gains from
some events and loses from others. The identical logic applies to stochastic
productivity of effort. A specific distribution fulfilling the assumption is for
example the uniform distribution or a symmetrically truncated normal distri-
bution. At the end of this section we also explore what happens if the variance
of ǫ converges to zero and hence there is no noise in the game.
Now turn to the cost of effort function. Instead of quadratic costs we as-
sume the following:
Assumption 1.6.2. Spending effort x implies costs C(x), where C(0) = 0, C′(0) = 0,
C′> 0 for all x > 0, C′′
> 0 and |C′′′| is finite.
32 CHAPTER 1. THE EFFECT OF POLLS
This cost function is strictly convex and hence may reflect both increasing
marginal costs of funding and other campaign efforts, as well as diminishing
marginal returns of effort. Apart from this convex costs are also necessary to
guarantee the existence of a pure strategy equilibrium. That is also the purpose
of the bounded third derivative.
The protocol of the campaign game is as before. If the variance of ǫ is
positive we find the following:
Proposition 1.6.1. Absent a poll both candidates choose identical effort in stage t =
1,2. If there is a poll, both candidates choose identical effort in t = 2, but there may
only be a stable symmetric equilibrium in t = 1 if d1 = 0 and if there exists a unique
equilibrium (for all d1). If the equilibrium is unique, in all campaigns with asymmetric
candidates the leader takes a tough position and spends more in equilibrium. If there
are multiple equilibria, in a close race either candidate may spend more in equilibrium,
while if the race is sufficiently lopsided the leader spends more. If there is momentum,
polls decrease expected aggregate spending.
Proof. See appendix.
Now let the variance of ǫ converge to zero and hence the noisiness disap-
pears. The campaign is then a perfectly discriminating contest or – using the
language of auction theory – an All-pay auction (see Konrad (2009)). This form
of contests has been studied extensively with applications ranging from lobby-
ing contest (Hillman and Riley, 1989; Baye et al., 1993) to political campaigns
(Meirowitz, 2008), revolution or war (Polborn, 2006) and internal labor markets
(Nalebuff and Stiglitz, 1983; Moldovanu and Sela, 2001). Siegel (2009) studies
general equilibrium characteristics of All-pay auctions. An All-pay auction de-
scribes a world in which effective effort is completely decisive for the contest
outcome. Hence, we may interpret it as a limiting case in which contestants
have perfect control over their effective effort. Clearly, when E[(ǫ)2] = 0, in our
basic model we would have σ = 0 and hence ρ = 0, too. In this case there we
found stage 2 behavior is identical and in stage 1 there may be multiple equilib-
ria in close races. An equilibrium in which the more popular candidate invests
more always exists. However, we had the problem that for ρ too small the
second order conditions did not hold and hence our analysis does not directly
extend to this case. However, the following proposition shows that qualitatively
all our results are preserved:
Proposition 1.6.2. Absent a poll both candidates choose identical effort in stage t = 1,2
also if noise vanishes. If there is a poll, in stage 2 there is a unique equilibrium. If
1.6. EXTENDING THE BASIC MODEL 33
|d2| < C−1(1) this equilibrium is in mixed strategies and both candidates choose in
expectation identical effort. Otherwise, the campaign is decided already and both candi-
dates spend zero equilibrium. In stage 1 there always exists a pure strategy equilibrium
in which the leading candidate spends positive effort while the trailing candidate stays
passive, if costs are quadratic. In close races also the opposite – an equilibrium in which
the trailing candidate acts tough and spends a positive amount, while the leader remains
passive – exists. Hence, there are multiple pure strategy equilibria if the campaign is
close in t = 1. Expected aggregate spending is weakly smaller when there are polls.
Proof. See appendix.
The proposition shows that our findings for ρ < ρ are robust, too. Absent a
poll campaigning has no influence on the expected outcome in equilibrium. This
changes once there is informational feedback before stage 2, and campaigning
again matters in equilibrium. There may be multiple equilibria in t = 1 if there
is no clear leader. In general, it is likely to see the leader spending more in stage
1 and so the leader is more likely to improve his position than his contender.
However, in close campaigns both candidates may adopt a tough stance. In
stage 2 we see both candidates choosing the same expected effort.
1.6.2 Proportional Representation
So far we assumed throughout the analysis that candidates compete in a FPTP
system and hence care only about the probability to come out ahead, not the
actual share of votes won. This is so because the winner of the campaign gets
all rents as long as he gets a majority; candidates’ campaign in a winner-take-all
competition. Having a relatively large share in t = 1 is a good thing, but only
because this makes it more likely to win a majority. The additional voters in
excess of the 50 percent give no additional benefit. But this is actually not a very
good description of political competition in many democratic countries, for
example when the political system is one of proportional representation (PR)
instead of FPTP. Looking at the number of countries using different electoral
systems, with 70 countries PR was the most popular system worldwide in 2005
(see Reynolds et al. (2005)). PR systems (and its close derivatives) can be found
in most European countries, and are also dominant in Latin America and Africa.
Examples include Brazil, Indonesia, or South Africa. Under PR campaigning
incentives differ from FPTP, because at least a share of the benefits is allocated
according to vote shares in the popular vote. Hence, while winning a majority
is under PR also valuable in its own right, this is not the only goal a candidate or
party pursues. One reason is that under PR the composition of the parliament is
34 CHAPTER 1. THE EFFECT OF POLLS
proportional to the popular vote, and hence a party’s influence increases in the
gained vote share. Another reason is that with only a marginal majority it often
is relatively hard to pass laws and bills, since then all members of government
would have to tie the party line, which is often not the case. In many cases a
government with a slight majority only turned out to be not able to effectively
shape and pass its own policies. Thus, with a comfortable majority it is easier
to govern and lead a country.
Assume a candidate or party values winning the majority by λ ∈ [0,1]. We
can interpret this as a plurality premium as in Iaryczower and Mattozzi (2011).
In addition, candidates get utility si(1 − λ) from gaining a share of si of the
total popular vote. (1 − λ) measures the relative importance of the vote share.
For example, in many countries political parties receive financial subsidies in
proportion to their vote share. Increasing those subsidies would imply a higher
(1 − λ). Similarly, the influence of a normal member of parliament that is not
part of the government may determine (1 − λ), too. The benefit of candidate i
in the election is hence
bi =
{λ + si(1 − λ) if si >
12 ,
si(1 − λ) else.
As before, candidates maximize their expected benefit subject to costs of cam-
paigning effort. We assume the shocks and costs have the same shape as in
Section 1.3.
Now let us analyze the effect of polls, and hence we start with a situa-
tion without informational feedback. The protocol of the campaign game is as
above, both candidates spend effort in stage 1 and 2 and then the election takes
place. As under FPTP, absent a poll the information a candidate holds in stages
1 and 2 is identical. Moreover, in any stage, both candidates have the same
marginal incentives, because both the marginal increase in probability and the
marginal increase in the vote share must be identical for both. Hence, absent a
poll we find results that are qualitatively the same as they were under FPTP:
Proposition 1.6.3. The equilibrium is unique. Both candidates choose in each stage
identical effort, and efforts are also identical across stages. Hence, also under PR,
absent a poll the ex-ante expected popular vote of a candidate in t = 3 is identical to his
popularity in t = 1.
Proof. See appendix.
This shows that the result from before, that campaigning does not matter in
equilibrium for the election outcome, remains valid also in PR. The intuition for
1.6. EXTENDING THE BASIC MODEL 35
this result is also straightforward and similar to the one under FPTP. Looking
at the plurality premium, as before, the marginal increase in the probability
to win is identical for both at any effort profile, and hence, so are spending
incentives. Looking at the other part of interest, the vote share, the exactly
same is also true: the marginal vote share gained from spending is identical for
both. Consequently, efforts must be identical in equilibrium.
Now we introduce polls and look at stage 2 first. As in a campaign without
a poll and following the identical intuition as under FPTP, the equilibrium must
again be unique and symmetric.
Proposition 1.6.4. Candidates choose identical efforts in stage 2 and the equilibrium
is unique.
Proof. See appendix.
Given the analysis to far this result is not really surprising. Also, the im-
portance of the result does not lie in the symmetry of the stage 2 equilibrium,
but in the comparative statics with respect to d2 of this equilibrium. Because,
as we have seen in the previous section, these comparative statics determine
incentives in the first stage decisively.
There are two different effects at work that may lead to differential incen-
tives: the effect of a marginal change in effort on the winner-take-all part and
the effect on the vote share part. We already know how the first effect looks
like: efforts are decreasing in |d2| and thus there always exist momentum equi-
libria. Hence, turn to the latter effect, caused by the proportionality of the
obtained rent and the vote share. And here we are immediately in trouble.
Without assuming a particular shape of the distribution Pt(dtι) – the distribu-
tion of bliss points – it is not possible to make any general statement about the
effect of varying d2. To see this, consider the logic of campaigning for vote
shares in more detail. The marginal benefit of effort is identical to the marginal
vote share gained, and this vote share is equal to the density of the bliss point
distribution. But so far we did not specify this distribution at all, what was
not a problem under FPTP. Because there all we needed to know was the ’dis-
tance’ to the median voter d1, and not how the exact distribution of bliss points
looks like. Also, it is not obvious how we should expect the distribution of
bliss points to look like. For example, a centrist distribution would probably
have a quasi-concave shape where the bulk of voters are centered around the
median and the frequency of bliss points decreases when we move away from
the median. For example, we could use a normal distribution with zero mean
36 CHAPTER 1. THE EFFECT OF POLLS
to model such a case. But it is not in any way obvious that the distribution of
voters is centrist. In many countries it seems to be the case that the electorate
is rather polarized with respect to their tastes, implying when we move away
from the median the frequency (density) of bliss points is likely to increase
first, before it must at some point decrease again. This would be a reasonable
distributional assumption if the fraction of partisan voters is relatively large. It
is also not obvious that preferences are symmetrically distributed. And all this
has important consequences for comparative statics. In particular, it may be the
case that efforts increase in |d2| over a given range or that comparative statics
are not at all related to |d2| in some regions. But these comparative statics are
important for stage 1 behavior. Hence, depending on the distribution of bliss
points, the results under PR might differ significantly from those under FPTP.
To show that this is true we now look more closely at how the shape of the
distribution of bliss points determines stage 2 behavior and then show by way
of an example how this alters stage 1 incentives. To be able to do all this we
make the following assumption:
Assumption 1.6.3. The distribution of bliss points has density
pt(dt) =φ(−µp+dt
σp ) + φ(µp+dt
σp )
2,
where µp ≥ 0 and σp> 0.
pt(dt) is a mixture distribution of two normals with identical variances σp
and means −µp and µp. Hence, it is symmetric around dt = 0. We restrict
ourselves to symmetric distributions for analytical purposes and because it is
sufficient to make the general point that incentives in the different electoral
system are often different. It is easy to generalize the model to more flexible
distributions, but the additional insights gained from such a generalization are
only marginal.10
We now look at the comparative statics with respect to d2 of the equilibrium
in stage 2:
Lemma 1.6.1. If µp>
√σ2 + (σp)2, candidates’ stage 2 spending increases in |d2|
when |d2| and λ are small. An upper bound for λ is λ ≡ 2
2+e32≈ 0.308. Otherwise
spending decreases in |d2|.10For example, if we look at mixture distributions of two normals with different means and variances,
the resulting distribution may be asymmetric and skewed to the left or right. As a consequence,whether or not there is momentum can be completely independent of |d2| over a given range.
1.6. EXTENDING THE BASIC MODEL 37
Proof. See appendix.
If the condition in the lemma is fulfilled, the expected stage 3 distribution of
bliss points is bimodal, and hence we could say the electorate is polarized in
expectation. Note that µp> σp is sufficient for pt to be bimodal. However, if it
is only ’marginally bimodal’, taking the expectation makes it unimodal again.
Hence, the distribution needs to be sufficiently bimodal. If λ becomes larger
the whole expected benefit distribution again becomes more unimodal, since
the plurality premium becomes more and more important. Hence, λ needs to
be sufficiently small and µp needs to be sufficiently large.
If we look now at stage 1 behavior, a similar argument as before demands
that the expected stage 3 distribution of bliss point be bimodal. This implies
the conditions from stage 2 are not yet sufficient to generate any qualitatively
different behavior in stage 1. The following proposition proves the main result
in this section:
Proposition 1.6.5. If λ is small and µp is large, there are anti-momentum equilibria
in stage 1 in close races.
Proof. See appendix.
What is the difference now to the anti-momentum equilibrium we have
seen under FPTP? First, the mechanism is obviously a different one, since the
result is driven by the polarized electorate. Second, unlike before, the result
does not relate to the expensiveness of influence in the campaign. The result
holds whether or not it is costly to influence the election outcome and the
equilibrium might be unique as well. But why does the result only hold in close
races? To see this note that ’close’ in Proposition 1.6.5 is not to be interpreted as
an absolute measure of tightness of competition, but relative to the competitive
environment (the parameters of the game). Hence, it might well be the case
that a candidate currently gathers two-thirds of all voters, but there is still
anti-momentum. The general point is that once the campaign is extremely
lopsided at the outset, the trailing candidate must in expectation increase future
intensity of competition. This is seen most easily by looking at a candidate who
currently does not have any voter support. His campaign spending in stage 1
decreases the gap and thereby increases his expected share in stage 2, what in
turn increases his expected costs. In our example there is never a candidate
with a current share of zero because pt has full support, but the intuition is still
right. In Figure 1.4 you can find an example where there is anti-momentum in
close races.
38 CHAPTER 1. THE EFFECT OF POLLS
-4 -2 2 4
0.05
0.10
0.15
0.20
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.105
0.110
0.115
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.10
0.11
0.12
0.13
0.14
dt
pt(dt)
d1 d2
x2ix1
i
Figure 1.4: Distribution of voters’ bliss points and equilibrium efforts in stage 1 and2 of candidate A (black and solid line) and 2 (brown and dashed line),depending on dt, t = 1,2. The parameter values are mp =
√6, σp = σ = c =
1, λ = 1/10. The equilibrium is unique.
While we looked at symmetric distributions of bliss points in this section,
the results are easily generalized to more general distributions. However, by
looking at skewed distributions an additional effect has to be considered: rela-
tive stage 1 incentives may over some range be independent of |d1|. In partic-
ular, there might be a neighborhood of d1 ∈ (−o,o), o > 0, in which the same
candidate always has an incentive to spend more on campaigning than his op-
ponent, no matter if he is leading or trailing. For example, if pt is unimodal but
skewed, candidates spending incentives in stage will always differ. In particu-
lar, if the distribution is negatively skewed, candidate B has over a large range
of parameters stronger incentives to invest than candidate A, independent of d1,
and similar effects benefiting B are true if the distribution is positively skewed.
1.7 Conclusion
In this paper we explore the effect public opinion polls have on candidates’
incentives to campaign. We found that in FPTP systems there always exists
an equilibrium in which the leading candidate gains momentum, but in close
campaigns also the trailing candidate may adopt a tough stance and outspend
the leader. Hence, anti-momentum may also be an equilibrium. For this to be
the case it needs to hold that having an influence on the election outcome in the
campaign is relatively cheap. We have shown that those results are relatively
robust to functional form assumptions or the extent to which random events
influence the election outcome. Quite generally, polls lead to a decrease in
expected spending in the campaign and so decrease the expected intensity of
the electoral competition. However, these results are not robust if we look at PR
systems instead. There the distribution of voters’ candidate ranking determines
relative campaigning incentives. If the electorate is relatively polarized, in close
1.7. CONCLUSION 39
campaigns the trailing candidate has stronger incentives to campaign. If the
electorate is relatively moderate, for example if the density of bliss points is
normal, than incentives are similar to those under FPTP.
An implication of the model is that if there is an incumbency advantage at
the outset of the campaign, polls are likely to foster this advantage and hence
decrease the rate of turnover in political offices in countries with FPTP electoral
systems. Indeed, the spending profile predicted by our model fits well to the
estimated profiles in Erikson and Palfrey (2000), who show that incumbents
with an early popularity edge tend to improve their chances by spending more
than their opponents in the competition.
For future research it is interesting to empirically validate the results pro-
posed by the theory. We are currently starting this project, however, do not
have results yet. Another interesting direction for future research is to study
how polls influence not only campaigning incentives, but also candidates’ plat-
form choices.
40 CHAPTER 1. THE EFFECT OF POLLS
Appendix
Proof of Proposition 1.3.1
Proof. We prove the proposition assuming the more general model discussed
in Section 1.6.1, which as a special case nests the model discussed in Section
1.3.1. If there is no poll, candidates do not observe the realization of ǫ1 and
hence their information does not change between periods 1 and 2. Hence, they
maximize
max(x1
A,x2A)∈R2
+
1d−d
∫ dd G(d1 + x1
A + x2A − x1
B − x2B)d d1 − C(x1
A)− C(x2A),
max(x1
B,x2B)∈R
2+
1 − 1d−d
∫ dd G(d1 + x1
A + x2A − x1
B − x2B)d d1 − C(x1
B)− C(x2B),
where G is the cdf of the convolution g1 ∗ g2. The system of first order condi-
tions is1
d − d
∫ d
dg(d1 + x1
A + x2A − x1
B − x2B)d d1 − C′(xt
i ) = 0,
i = A, B and t = 1,2. It is easily observed that in any interior pure strategy
equilibrium it must hold that x1A = x2
A = x1B = x2
B = x∗. In particular,
x∗ = C′−1
(1
d − d
∫ d
dg(d1)d d1
).
The second order conditions are
1
d − d
∫ d
dg′(d1)d d1 − C′′
(C′−1
(1
d − d
∫ d
dg(d1)d d1
))< 0
for A and
− 1
d − d
∫ d
dg′(d1)d d1 − C′′
(C′−1
(1
d − d
∫ d
dg(d1)d d1
))< 0
for B. If C′′(.) is sufficiently large relative to g′(.) the second order conditions
hold for all d2 and hence this is an equilibrium. We assume this to be the case.
Note that if the variance of g1 and g2 vanishes, the second order conditions
always hold, as we show later in the proof of Proposition 1.6.2. If the variance
becomes very large, the slope of g1 and g2 decreases and hence the second
order conditions also always hold.
For the parameterized model in Section 1.3.1 the second order conditions
APPENDIX 41
hold if
c > max
−
φ(
d√2σ
)− φ
(d√2σ
)
d − d,φ(
d√2σ
)− φ
(d√2σ
)
d − d
.
For example, if σ = 1, d = 2, and d = −1, it must hold that c > 0.03864. For the
remainder of the paper we assume C′′ to be sufficiently large to guarantee the
second order conditions. Hence, the proof is complete and this also proves the
first part of Proposition 1.6.1.
Proofs of Proposition 1.3.2 and 1.6.1
Proof. We again proof the proposition directly in the general version of Propo-
sition 1.6.1.
First look at stage 2 when there is a poll. In this case both candidates know
the median’s exact position in stage 2, d2. Candidates maximize
maxx2
A∈R+
G2(d2 + x2A − x2
B)− C(x2A),
maxx2
B∈R+
1 − G2(d2 + x2A − x2
B)− C(x2B).
First order conditions read
g2(d2 + x2A − x2
B)− C′(x2i ) = 0,
where i = A, B, and the assumption on C′ guarantee that there exists an x
fulfilling these. It is immediately observed that – as without a poll – both
candidates choose identical effort, x2A = x2
B. In particular,
x2A = x2
B = x∗∗(d2) = C′−1(g2(d2)).
The second order condition for A is
g2′(d2)− C′′(C′−1(g2(d2))) < 0.
Since it must hold for all d2, if this holds for A it also holds for B. We assume
this holds for all d2. The problem each candidate faces is then continuous and
concave and hence a pure strategy equilibrium exists. For example, if σ = 1 it
must hold that c > 0.242. For the remainder of the paper we assume C′′ to be
sufficiently large to guarantee the second order conditions are fulfilled.
This proves Proposition 1.3.2 and the first part of Proposition 1.6.1.
42 CHAPTER 1. THE EFFECT OF POLLS
Proofs of Propositions 1.3.3, 1.3.4, and 1.6.1
Proof. Expected utility of A and B, conditional on being in state d2 in the second
stage, is
EU∗A(d
2) = G2(d2) − C(
C′−1(
g2(d2)))
,
EU∗B(d
2) = (1 − G2(d2)) − C(
C′−1(
g2(d2)))
.
Note that d2 = d1 + x1A − x1
B − ǫ1. Then we can write the optimization problemof the candidates as:
maxx1
A≥0
∫
S1G2(d1 + x1
A − x1B − ǫ1)− C
(C′−1
(g2(d1 + x1
A − x1B − ǫ1)
))g1(ǫ1)dǫ1 − C(x1
A),
maxx1
B≥0
∫
S1(1 − G2(d1 + x1
A − x1B − ǫ1))− C
(C′−1
(g2(d1 + x1
A − x1B − ǫ1)
))g1(ǫ1)dǫ1 − C(x1
B).
Taking the derivative with respect to the respective own strategy yields
∫
S1
(g2(·) −
[C′(C′−1(·))∂C′−1(·)
∂g2(·)∂g2(·)∂x1
A
])g1(ǫ1)dǫ1 − C′(x1
A),
∫
S1
(g2(·) −
[C′(C′−1(·))∂C′−1(·)
∂g2(·)∂g2(·)
∂x1B
])g1(ǫ1)dǫ1 − C′(x1
B).
Using C′(C′−1(z)) = z and ∂g2(·)/∂x1A = ∂g2(·)/∂d1 =−∂g2(·)/∂x1
B these equa-
tions simplify to
∫
S1
(g2(·) −
[g2(·)∂C′−1(·)
∂g2(·)∂g2(·)
∂d1
])g1(ǫ1)dǫ1 − C′(x1
A)
∫
S1
(g2(·) +
[g2(·)∂C′−1(·)
∂g2(·)∂g2(·)
∂d1
])g1(ǫ1)dǫ1 − C′(x1
B)
Using ∂C′−1(g2(·))/∂g2(·) = 1/(C′′(C′−1(g2(·))) and ∂g2(·)/∂d1 = g2′(·), and
letting d1 + x1A − x1
B =: κ we get
∫
S1
(g2(κ − ǫ1) −
[g2(κ − ǫ1)g2′(κ − ǫ1)
C′′(C′−1(g2(κ − ǫ1)))
])g1(ǫ1)dǫ1 − C′(x1
A), (1.2)
∫
S1
(g2(κ − ǫ1) +
[g2(κ − ǫ1)g2′(κ − ǫ1)
C′′(C′−1(g2(κ − ǫ1)))
])g1(ǫ1)dǫ1 − C′(x1
B). (1.3)
Before we analyze the equilibrium in detail we now need to show that an
equilibrium exists.
APPENDIX 43
Lemma 1.7.1. If C′′(x) > Γ(g1, g2) (defined below) a pure strategy equilibrium exists
for all d1.
Proof. The second derivative of A’s payoff function is
∫
S1
[g2 ′(κ − ǫ1)−
(g2 ′(κ − ǫ1)
C′′(x∗∗(·))
)2(C′′(x∗∗(·))− g2 ′(κ − ǫ1)C′′′(x∗∗(·))
C′′(x∗∗(·))
)− g2(κ − ǫ1)g2 ′′(κ − ǫ1)
C′′(x∗∗(·))
]dG(ǫ1)− C′′(x1
A).
To show strict concavity of the payoff function we need to show that this is
strictly negative for all κ and x1A. Consider
(g2′(κ − ǫ1)
C′′(x∗∗(ǫ1))
)2(C′′(x∗∗(κ − ǫ1))− g2′(κ − ǫ1)C′′′(x∗∗(ǫ1))
C′′(x∗∗(κ − ǫ1))
). (1.4)
Let the γ1 > 0 be the smallest value of C′′ for which
γ21 > g2′(κ − ǫ1)C′′′(x∗∗(κ − ǫ1))
for all κ − ǫ1. If the infimum of C′′ is at least γ1 (1.4) is non-negative. Since C′′′
is bounded by Assumption 1.6.2 and because it follows from differentiability
and quasi-concavity that g2′(κ − ǫ1) is bounded, too, there exists finite γ1 for
which this is the case. Hence assume (1.4) is zero. If this is the case and we can
show that the second derivative is negative, this is even more so the case when
(1.4) is positive. Therefore we are left with
∫
S1
[g2′(κ − ǫ1)− g2(κ − ǫ1)g2′′(κ − ǫ1)
C′′(x∗∗(·))
]dG(ǫ1)− C′′(x1
A). (1.5)
From strict quasi-concavity of g2 it follows that the expectation is bounded.
Hence, there exists γ2 > 0 such that if C′′(x) > γ2 for all x, (1.5) is strictly neg-
ative. It follows that if C′′(x) > Γ(g1, g2) := max{γ1,γ2} the second derivative
is strictly negative and hence the problem is strictly concave. Thus, assuming
this holds, and since payoffs are continuous in x1A and x2
B, existence of a pure
strategy Nash equilibrium follows from Theorem 1.2 in Fudenberg and Tirole
(1991), which is due to Debreu (1952), Fan (1952), and Glicksberg (1952).
Now let us go more into the details of the first order conditions to deter-
mine the properties an equilibrium must have. For this purpose we need the
following lemma:
Lemma 1.7.2. Let
ξ(κ) := Eǫ1
[g2(κ − ǫ1)g2′(κ − ǫ1)
C′′(C′−1(g2(κ − ǫ1)))
].
44 CHAPTER 1. THE EFFECT OF POLLS
Then ξ(0) = 0, ξ(+) > 0 and ξ(−) < 0.
Proof. Define
ω(κ) :=g2(κ − ǫ1)g2(κ − ǫ1)
C′′(C′−1(g2(κ − ǫ1))),
which is the function we want to take the expectation of. Now remember
that by adding an arbitrary constant a to the argument of a function, the
graph of the function is shifted horizontally by −a. Therefore, if g2(ǫ) is
axis-symmetric across zero, g2(κ − ǫ) is axis-symmetric across −κ. As a conse-
quence, the same holds true also for functions of this function, like C′−1(g2(κ −ǫ)), C′′(C′−1(g2(κ − ǫ))), and
g2(κ−ǫ1)C′′(C′−1(g2(κ−ǫ)))
. Because g2(ǫ) is axis-symmetric
across zero, its derivative is point-symmetric across zero. By a similar argument
as above it also holds that g2′(κ − ǫ) is point-symmetric across −κ. Therefore,
we have that also the product of an axis-symmetric function across −κ and a
point-symmetric function across this point, in our case this function is ω(κ), is
point-symmetric across −κ.
We first show that ξ(0) = 0. Let f (z) be a function which is axis-symmetric
across zero and let h(z) be another function which is point-symmetric across
zero. Both functions share the same support K. Then, if we want to find∫K f (z)h(z)dz, we can split the integral into two parts:
∫
Kf (z)h(z)dz =
∫
{z∈K:z≤0}f (z)h(z)dz +
∫
{z∈K:z>0}f (z)h(z)dz.
Because of the symmetry properties f (z) = f (−z) and h(z) = −h(−z) we can
rewrite the second term as∫
{z∈K:z>0}f (z)h(z)dz = −
∫
{z∈K:z≤0}f (z)h(z)dz.
Using this substitution it is easily verified that
∫
Kf (z)h(z)dz =
∫
{z∈K:z≤0}f (z)h(z)dz −
∫
{z∈K:z≤0}f (z)h(z)dz = 0.
Now let f (z) = ω(ǫ) and h(z) = g2(ǫ) and observe that the integral we want to
calculate is the expectation of ω(ǫ) and therefore equal to ξ(0) to complete this
part of the proof.
Next, consider κ > 0. ω is shifted to the left and is point-symmetric across
−κ. Now note two things: First, to the left of −κ the values of ω are positive,
to the rights the values are negative. Second, for any shock φ leading to a real-
ization ω(κ + φ) = m there exists exactly one other shock φ′, which leads to a
APPENDIX 45
realization ω(κ + φ′) = −m and is an inversion of the former point at (−κ,0).
Moreover, this holds true for any point in the graph of ω. Accordingly, we
can define the whole graph as pairs of inversion points. Now observe, that the
probability of an outcome −m is always weakly larger than the probability of
outcome m for all m ≥ 0. To see this note that a shock generating m must be
of size −κ − c, while the shock generating −m must be −κ + c, for some con-
stant c ≥ 0. But then the shock φ that produces outcome m is in absolute value
weakly larger than φ′. As a consequence, because shocks are distributed sym-
metrically around zero, the density of φ′ is weakly larger than the density of φ,
g2(φ′) ≥ g2(φ). Note that this must hold for all m, φ and φ′, and accordingly
the expectation of ω must be negative. From a similar argument it follows that
the converse must hold if we assume κ < 0. Hence the proof is complete.
Knowing now that a pure strategy equilibrium exists we focus on interior
equilibria henceforth. In Section 1.6.1 we also discuss the case when noise van-
ishes and the contest takes the form of a fully discriminating All-pay auction.
In this case there are only corner equilibria and the results are qualitatively
identical.
From (1.11) and (1.3) it follows that the first order conditions in an interior
equilibrium are
∫
S1
(g2(κ − ǫ1) −
[g2(κ − ǫ1)g2′(κ − ǫ1)
C′′(C′−1(g2(κ − ǫ1)))
])g1(ǫ1)dǫ1 − C′(x1
A) = 0
and
∫
S1
(g2(κ − ǫ1) +
[g2(κ − ǫ1)g2′(κ − ǫ1)
C′′(C′−1(g2(κ − ǫ1)))
])g1(ǫ1)dǫ1 − C′(x1
B) = 0.
Simple manipulations reveal that equivalently the following must hold:
x1A = C′−1
(∫
S1
(g2(κ − ǫ1) −
[g2(κ − ǫ1)g2′(κ − ǫ1)
C′′(C′−1(g2(κ − ǫ1)))
])g1(ǫ1)dǫ1
)
x1B = C′−1
(∫
S1
(g2(κ − ǫ1) +
[g2(κ − ǫ1)g2′(κ − ǫ1)
C′′(C′−1(g2(κ − ǫ1)))
])g1(ǫ1)dǫ1
).
Using κ = ∆1 + d1, it follows that in equilibrium it must hold that
46 CHAPTER 1. THE EFFECT OF POLLS
∆1 = Σ(κ)
:= C′−1(∫
S1
(g2(κ − ǫ1) −
[g2(κ − ǫ1)g2 ′(κ − ǫ1)
C′′(C′−1(g2(κ − ǫ1)))
])g1(ǫ1)dǫ1
)(1.6)
− C′−1(∫
S1
(g2(κ − ǫ1) +
[g2(κ − ǫ1)g2 ′(κ − ǫ1)
C′′(C′−1(g2(κ − ǫ1)))
])g1(ǫ1)dǫ1
).
The shape of this function is now important to determine equilibrium behavior.
We now establish a few lemmata that help us to characterize equilibria.
Lemma 1.7.3. Sign[Σ(∆1 + d1)
]= Sign[∆1 + d1]. Moreover, Σ(∆1 + d1) is contin-
uous, bounded, point symmetric at −d1 in ∆1, and lim|∆1|→∞ Σ(∆1 + d1) = 0.
Proof. From C′′> 0 it follows that the inverse C′−1
is increasing. This together
with Lemma 1.7.2 directly implies Σ(0) = 0, Σ(+) = (+), and Σ(−) = (−). For
the symmetry properties look at d1 = 0 first. Then we have that
C′−1(∫
S1
(g2(∆1 − ǫ1) −
[g2(∆1 − ǫ1)g2′(∆1 − ǫ1)
C′′(C′−1(g2(∆1 − ǫ1)))
])g1(ǫ1)dǫ1
)
and
C′−1(∫
S1
(g2(∆1 − ǫ1) +
[g2(∆1 − ǫ1)g2′(∆1 − ǫ1)
C′′(C′−1(g2(∆1 − ǫ1)))
])g1(ǫ1)dǫ1
)
are mirror images of each other (in ∆1) with the reflection axis being the vertical
through zero. This follows from the first argument of C′−1being axis symmet-
ric at zero and the second being point symmetric at zero. Hence, the difference
must be point symmetric at zero. Now note that by adding an arbitrary con-
stant – for example d1 – to the argument of a function, the function is shifted
horizontally by −d1. Hence, Σ must be point symmetric at −d1. Continuity
and boundedness follow directly from all terms and C′−1being continuous
and bounded. That the limit vanishes follows from lim|x|→∞ g2(x)g2′(x) = 0,
which follows from quasi-concavity of g2. This proves the lemma.
Lemma 1.7.4. Assume d1 6= 0 and let the effort of the more popular candidate in stage
1 be xL and the effort of his opponent be xT. In any equilibrium we have that
xT /∈ (xL, xL + |d1|).
APPENDIX 47
Proof. To see this look at the first order conditions. Without loss of generality
assume d1> 0 and also assume x1
B ∈ (x1A, x1
A + |d1|). This implies κ > 0 and thus,
by Lemma 1.7.2, ξ(κ) < 0. Hence, B’s efforts are strategic complements, and
A’s strategic substitutes. If A’s first order condition holds, B’s must be strictly
negative and he hence would like to decrease effort. If B’s first order condition
holds, A’s must be strictly positive and he would like to increase effort. Hence,
this cannot be an equilibrium.
Lemma 1.7.5. There exists d ≥ 0 such that if |d1| > d the equilibrium in stage 1 is
unique and xL ≥ xT.
Proof. This follows from Lemma 1.7.4 and the fact, that any effort greater than
x := C−1(1) is strictly dominated. Thus, when |d1| becomes larger and larger,
outspending the leading candidate becomes too expensive. The inequality is
weak because we did not assume ǫ2 has full support and hence the race might
be decided if |d1| is sufficiently large. If we assume ǫ2 has full support – S2 = R
– the inequality is strict because the leading candidate always spends positive
effort.
The derivative of Σ with respect to ∆1 is
Eǫ1
[g2′(·)
]− Eǫ1
[(g2′(·))2
C′′(w)+ g2(·)g2′′(·)
C′′(w)− g2(·)(g2 ′(·))2C′′′(w)
(C′′(w))3
]
C′′(
C′−1(
Eǫ1
[(g2(·)−
[g2(·)g2′(·)
C′′(C′−1(g2(·)))
])])) (1.7)
−Eǫ1
[g2′(·)
]+ Eǫ1
[(g2′(·))2
C′′(w)+
g2(·)g2′′(·)C′′(w)
− g2(·)(g2 ′(·))2C′′′(w)(C′′(w))3
]
C′′(
C′−1(
Eǫ1
[(g2(·) +
[g2(·)g2′(·)
C′′(w)
])])) (1.8)
where w = C′−1(g(.)). The proposition we want to prove states that in close
games there might be both equilibria in which the leading candidate spends
more and the some in which the trailing candidate spends more, depending
on the distributions of ǫ1 and ǫ2 and the shape of the cost function. If one
candidate has a sufficiently large advantage, in all equilibria this candidate will
spend weakly more. If the equilibrium is unique for all d1, in this equilibrium
also the leading candidate will spend weakly more. A necessary and sufficient
condition for a unique equilibrium for all d1 is that Σ′(∆1) < 1 for all ∆1. To se
this note that if there are to be multiple equilibria, that is ∆1 = Σ(∆1) intersect
more than once, then Σ must be steeper than ∆1 somewhere. Starting from
an intersection of the two functions, if the slope is strictly smaller than 1 to
the right of the intersection Σ is strictly smaller than ∆1, and to the left strictly
48 CHAPTER 1. THE EFFECT OF POLLS
larger, and hence there cannot be another equilibrium. If, however, there is
some region in which the slope is larger than 1, there exists d1 shifting Σ in a
way such that there are multiple equilibria. Hence, if and only if
Σ′(∆1) =Eǫ1
[g2′(·)
]− Eǫ1
[(g2′(·))2
C′′(w))+ g2(·)g2′′(·)
C′′(w))− g2(·)(g2′(·))2C′′′(w))
(C′′(w)))3
]
C′′(
C′−1(
Eǫ1
[(g2(·)−
[g2(·)g2 ′(·)
C′′(w)
])])) (1.9)
−Eǫ1
[g2′(·)
]+ Eǫ1
[(g2′(·))2
C′′(w))+ g2(·)g2′′(·)
C′′(w))− g2(·)(g2′(·))2C′′′(w))
(C′′(w)))3
]
C′′(
C′−1(
Eǫ1
[(g2(·) +
[g2(·)g2 ′(·)
C′′(C′−1(g2(·)))
])])) < 1
where w = C′−1(g(.)), for all ∆1, there is a unique equilibrium. It is easy to see
that the absolute value of the slope is strictly decreasing in C′′ (evaluated at
the equilibrium). Hence, if C′′(x) is sufficiently large for all x the equilibrium
is unique for all d1. For d1 = 0 the equilibrium is symmetric and ∆1 = 0. If
we now increase d1 we thereby shift Σ to the left, what because of the fact that
Σ(+) = (+) (see Lemma 1.7.3), implies the intersection is now where ∆1> 0.
This remains true for all d1> 0, and the opposite is similarly true for d1
< 0.
If
Σ′(∆1) > 1
for some ∆1 there are multiple equilibria for some d1. This follows from the
discussion above. If this is true for small |d1| it is likely that either candidate
might spend more in equilibrium. A sufficient condition for such equilibria is
that
Σ′(0) =−2Eǫ1
[(g2′(−ǫ1))2
C′′(C′−1(g(−ǫ1)))+ g2(−ǫ1)g2′′(−ǫ1)
C′′(C′−1(g(−ǫ1)))
]
C′′(
C′−1 (Eǫ1 [(g2(−ǫ1))])) > 1.
This derivative is strictly positive (follows from Lemma 1.7.2). If C′′ is suffi-
ciently small the derivative gets larger than 1. Then there is one equilibrium
∆1 = 0. Moreover, because Σ vanishes as |∆1| → ∞ (see Lemma 1.7.3) and Σ
is continuous, it follows from the intermediate value theorem that there are at
least two more equilibria, one with ∆1> 0 and one with ∆1
> 0. Because Σ
is point symmetric in ∆1 at zero (see Lemma 1.7.3) the asymmetric equilibria
are symmetric to each other. Note that for the analysis here hinges on the ap-
plicability of the first order conditions, and hence when C′′ gets too small the
second order conditions are violated. In the parameterized version below we
show that those equilibria can exist.
APPENDIX 49
From Lemma 1.7.5 it follows that in races with one dominant candidate this
candidate will always spend more effort in the campaign, or both spend zero.
Now go the specific example of a normal distribution with variance σ2 and
zero mean and a cost function C(x) = c2 x2. The marginal cost function is then
linear and the second derivative of the cost function is c. Hence, Σ simplifies
significantly:
Σ(∆1 + d1) = −Eǫ1
[2φ(∆1+d1−ǫ1
σ )φ′(∆1+d1−ǫ1
σ )
c2
]=
(∆1 + d1)e−(∆1+d1)2
3σ2
√27πc2σ4
.
Now look at the shape of Σ. Since d1 only shifts the function horizontally, we
assume d1 = 0 for now. The derivative with respect to ∆1 is
(3σ2 − 2(∆1)2)e−(∆1)2
3σ2
9√
3πc2σ6. (1.10)
This is strictly positive for |∆1| <√
32σ, negative for |∆1| >
√32σ, and zero for
|∆1| =√
32σ. The maximum of Σ = 1
3√
2eπc2σ3is attained at ∆1 + d1 =
√32σ, and
the minimum of Σ = − 13√
2eπc2σ3is attained at ∆1 = −
√32σ. Now look at the
second derivative of Σ,
2∆1(2∆1 − 9σ2))e− (∆1)2
3σ2
27√
3πc2σ8.
This is strictly negative on [−∞,− 3√2σ) ∪ (0, 3√
2σ), and hence the function is
strictly concave in this region, which also must include (and does) the max-
imum. Hence, Σ is strictly concave between zero and the maximum, and
decreases monotonically thereafter. Hence, if Σ′(∆1) ≤ 1, the slope is strictly
smaller than 1 (the slope of ∆1) for all ∆ ≥ 0. Hence, there exists a unique ∆1
fulfilling ∆1 = Σ(∆) not only for d1 = 0, but for all d1 ∈ R. Note that for given
σ the derivative becomes arbitrarily small at ∆1 = 0 if we increase c, and hence
there exists c such that for all c > c the slope is less than 1, and larger than 1
else. In Figure 1.5 we show two examples. If the derivative at D1 = 0 is larger
than 1 there exist multiple equilibria. Because Σ is strictly concave on [0,√
32σ],
and the derivative is zero at the end of this interval, it must become equal to 1
at some ∆1 ∈ [0,√
32σ]. Denote this by ∆. If we increase d1 now from zero we
thereby shift Σ to the left by d1. Hence, the two outer intersections of ∆1 and Σ
move to the right (∆1 increases), while the inner intersection moves to the left.
50 CHAPTER 1. THE EFFECT OF POLLS
-1.5 -1.0 -0.5 0.5 1.0 1.5
-1.0
-0.5
0.5
1.0
-1.5 -1.0 -0.5 0.5 1.0 1.5
-1.0
-0.5
0.5
1.0
∆1∆1
Figure 1.5: ∆1 (gray) and Σ(∆1 + d1) (brown) for σ = .5 and c ∈ {1.1,0.97}.
Hence, there are two intersections converging to each other, the ones where
∆1< 0. At d they converge to ∆, and hence there are only two equilibria left. If
we increase d1 now further this equilibrium vanishes and only one equilibrium
remains, in which d1> 0.
To complete the proof we now show by example that the second order
condition can hold in both stages when there are multiple equilibria in stage 1.
The second order condition in stage 2 for A is
φ′(d2
σ)− c < 0,
and this must hold for all d2. The second derivative has a maximum if d2 =−σ,
and if c > 1√2eπσ2
⇔ ρ >1√2eπ
this maximum is strictly negative and A’s second
order condition holds for all d2. Note that this is then also guarantees B’s
second order condition since candidates are symmetric in d2. Moreover, note
that 1√2eπ
< ρ = 133/4
√π
. Hence, if the second order condition holds marginally
in stage 2, there are multiple equilibria in stage 1.
Now look at stage 1. Assuming the second order condition in stage 2 holds
marginally, the second order condition in stage 1 is
− κe− κ2
4σ2
4√
πσ3− 1√
2eπσ2+
(3σ2 − 2κ2)e12− κ2
3σ2
9√
6πσ4,
where, as above, κ = ∆1 + d1. This seems to be strictly negative for all (σ,κ),
and is strictly negative in all our numerical calculation, see Figure 1.6.
What is left to prove Proposition 1.3.4 is to show that no stable symmetric
equilibrium exists when ρ < ρ. In the following we show that the slope of each
candidate’s best response function for d1 = 0, evaluated at the symmetric inter-
APPENDIX 51
section, is less than minus one. Being notationally a little sloppy we drop the
argument of g2(∆1 − ǫ1). Each candidate’s best response is implicitly defined
by
BR1A(x1
B) = max
{{x1
A : Eǫ1
[g2(.) −
[g2(.)g2′(.)
C′′(C′−1(g2(.)))
]]= C′(x1
A)},0
},
BR1B(x1
A) = max
{{x1
B : Eǫ1
[g2(.) +
[g2(.)g2′(.)
C′′(C′−1(g2(.)))
]]= C′(x1
B)},0
}.
Assuming a symmetric equilibrium with x1A = x1
B, it follows from Lemma 1.7.2
that the indirect effect is zero for both and hence
x1A = x1
B = C′−1(Eǫ1 g2(ǫ1))
is indeed an equilibrium. From the implicit function theorem it follows that the
slope of the best responses is
∂BR1i (x1
j )
∂x1j
=
Eǫ1
[(g2 ′(ǫ1)
C′′(x∗∗(ǫ1))
)2(
C′′(x∗∗(ǫ1))− g2(ǫ1)C′′′(x∗∗(ǫ1))
C′′(x∗∗(ǫ1))
)+
g2(ǫ1)g2′′(ǫ1)
C′′(x∗∗(ǫ1))
]
Eǫ1
[(g2 ′(ǫ1)
C′′(x∗∗(ǫ1))
)2 (C′′(x∗∗(ǫ1))− g2(ǫ1)C′′′(x∗∗(ǫ1))
C′′(x∗∗(ǫ1))
)+
g2(ǫ1)g2 ′′(ǫ1)
C′′(x∗∗(ǫ1))
]+ C′′(C′−1(Eǫ1 g2(ǫ1)))
.
If this is smaller than minus 1 the equilibrium is unstable. Note that the de-
nominator must be positive, because it is the negative of the second derivative
in equilibrium, and this has to be negative in equilibrium. This is the case
whenever
−2Eǫ1
[(g2 ′(ǫ1)
C′′(x∗∗(ǫ1))
)2(
C′′(x∗∗(ǫ1))− g2(ǫ1)C′′ ′(x∗∗(ǫ1))
C′′(x∗∗(ǫ1))
)+
g2(ǫ1)g2 ′′(ǫ1)
C′′(x∗∗(ǫ1))
]> C′′(C′−1
(Eǫ1 g2(ǫ1))).
Hence, the cost function must be sufficiently convex but not too convex. Using
the functional forms assumed in the main part of the paper, this expression
simplifies to
−2Eǫ1
[(φ′(
ǫ1
σ
))2
+ φ
(ǫ1
σ
)φ′′(
ǫ1
σ
)]> c2,
which yields
ρ := cσ2< (33/4
√π)−1 = ρ.
This is the condition for the existence of multiple equilibria in the game with
quadratic costs and normal shocks. Hence, whenever there are multiple equi-
libria in this game, there is no stable symmetric equilibrium. This completes
the proof.
52 CHAPTER 1. THE EFFECT OF POLLS
0
1
2
3
-10-5
05
10
-0.2
-0.1
0.0
-4 -2 2 4
-6
-5
-4
-3
-2
-1
κ
κ
σ
Figure 1.6: Numerical calculations of A’s second derivative in stage 1, assuming sec-ond order conditions in stage 2 hold marginally, c > 1√
2eπσ2 . Right panel:
σ ∈ {1/4,1/2,1}.
Proof of Proposition 1.3.5
Proof. After observing that the probability to increase the lead is a function
only of ∆1 and the distribution of ǫ2, the proposition follows immediately from
Lemma 1.7.3 and Equation 1.10 and the discussion thereafter.
Proof of Proposition 1.4.1
Proof. From the proof of Proposition 1.3.1 we know that aggregate expenditures
without a poll are equal to
4
c
(1
d − d
∫ d
dg(d1)d d1
).
With a poll, second period expected aggregate expenditures are also symmetric
and equal to
2
c
1
d − d
∫ d
d
(∫
S1(g2(d2))g1(ǫ1)dǫ1
)d d1.
as the proof of Proposition 1.3.2 shows. In period 1 we unfortunately cannot
generally get a closed form solution for equilibrium efforts with poll. So we
need to take an indirect approach. Recall from the proof of Proposition 1.3.3
that the first-order conditions are given by
APPENDIX 53
∫
S1
(g2(·) −
[g2(·)∂C′−1(·)
∂g2(·)∂g2(·)
∂d1
])g1(ǫ1)dǫ1 − cx1
A = 0,
∫
S1
(g2(·) +
[g2(·)∂C′−1(·)
∂g2(·)∂g2(·)
∂d1
])g1(ǫ1)dǫ1 − cx1
B = 0.
After simple manipulations, it follows from this that aggregate effort in period
1 is2
c
∫
S1g2(d1 + ∆1 − ǫ1) g1(ǫ1)dǫ1,
and expected aggregate stage 1 spending is
1
d − d
∫ d
d
(2
c
∫
S1g2(d1 + ∆1 − ǫ1) g1(ǫ1)dǫ1
)d d1.
Thus, the difference between expected aggregate effort with and without poll
is equal to
4
c
1
d − d
∫ d
d
(∫
S1g2(κ − ǫ1) g1(ǫ1)dǫ1 − g(d1)
)d d1.
By definition of g as the convolution of g1 and g2 we can simplify the expres-
sion:4
c
1
d − d
∫ d
d
(g(d1 + ∆1)− g(d1)
)d d1.
Because g is strictly quasi-concave and symmetric around zero, and because in
a momentum equilibrium Sign[d1 ] = Sign[∆1 ], this difference is weakly nega-
tive whenever there is momentum. Hence, in this case polls decrease expected
aggregate spending.
Now consider a situation where there is anti-momentum. To prove that
polls also in this case decrease intensity of competition, it is sufficient to show
that in all anti-momentum equilibria ∆1> −2d1. Because if that is the case,
|d1 +∆1|> d1 as before and therefore spending must decrease, too. Since as we
decrease |d1| to zero ∆1 increases, it suffices to show that for the maximum |d1|that still admits multiple equilibria, d, the relation holds as we need it.
We prove that this is the case with the help of Figure 1.7. A marks the
anti-momentum equilibrium with the highest value of d1 allowing for such an
equilibrium. AB is the effort difference in this equilibrium, ∆1. CD corresponds
to d. Hence, we need to show that AB > 2CD. Note that CD = AB − BC. Hence,
AB > 2(AB − BC) ⇔ 2BC > AB ⇔ ABBC < 2. Now note that AB
BC is exactly the
54 CHAPTER 1. THE EFFECT OF POLLS
D
CB
A
-0.4 -0.2 0.2 0.4
a
-0.4
-0.2
0.2
0.4
b
Figure 1.7: Appendix: Proof that polls decrease wastefulness of competition.
slope of line through A and C and we also know that the slope of the function
through both point is increasing from one as we approach C. Hence, for the
average slope to be smaller than two, it is sufficient that the function’s slope
in C is smaller than 2. It is straightforward to show that this is the case for all
ρ >1
33/4√
2π= ρ/
√2. Since the second order conditions are not guaranteed to
hold whenever ρ ≤ 1√2eπ
< ρ/√
2, we know that in all interior anti-momentum
equilibria ∆1> −2d1 when d1 = d. Whenever d1
< d this relation remains true.
As d1 decreases to zero, ∆1 increase and hence for all d1 admitting multiple
equilibria, ∆1> −2d1. This directly implies that the expected lead in stage 2
must increase, E[|d2|] > E[|d1|]. That is sufficient to complete the proof.
Proof of Proposition 1.6.2
Proof. Also consider the situation without a poll first. In this situation the
candidates do not know the median’s exact ranking but have beliefs F(d1). For
candidate A the probability to win the election is
Pr[x1A + x2
A + d1 − x1B − x2
B > 0] = 1 − F(x1B + x2
B − x1A − x2
A),
and similarly for B we get F(x1B + x2
B − x1A − x2
A). Hence, we may write the
candidates objectives as
APPENDIX 55
max(x1A,x2
A)∈R2+
1 − F(x1B + x2
B − x1A − x2
A)− C(x1A)− C(x2
A),
max(x1B,x2
B)∈R2+
F(x1B + x2
B − x1A − x2
A)− C(x1B)− C(x2
B).
The corresponding first order condition for i in t reads
f (x1B + x2
B − x1A − x2
A)− C′(xti )
!= 0.
This is identical for all t = 1,2 and i = A, B. Hence,
xtA = C′−1( f (0)).
For this to be an equilibrium the second order conditions need to hold:
f ′(0)− C′′(C′−1( f (0))) ≤ 0.
If costs are sufficiently convex this inequality holds generally. For the assumed
uniform prior we get f ′(0) = 0 and hence the inequality always holds.
Now look at the case of a campaign with polls. Candidates have perfect
knowledge of the median’s ranking, and since there is no shock, there is no
exogenous noise left. Consider stage 2 first, and assume without loss of gener-
ality d2 ≥ 0. It is easily shown that there cannot be a pure strategy equilibrium
in this stage (see e.g. Nalebuff and Stiglitz (1983) or Hillman and Riley (1989)).
The stage game is similar to the game analyzed in Meirowitz (2008) with the
difference that we have strictly convex costs. Let x := C−1(1), the maximum ef-
fort that is not strictly dominated. Moreover, let Qi(x2i ) be candidate i’s mixed
strategy with support Si, i = A, B. The following proposition summarizes the
stage 2 equilibrium:
Proposition 1.7.1. Without loss of generality, let d2 ≥ 0. There is a unique equilibrium
in mixed strategies if d2< x with SA = [0, x − d2] and SB = {0} ∪ [d2, x]. Candidate
A randomizes according to
QA(x2A) =
0 if xA < 0,
C(d2 + xA) if xA ∈ [0, x − d2],
1 if xA > x − d2,
while B’s mixed strategy is given by
56 CHAPTER 1. THE EFFECT OF POLLS
0.5 1.0 1.5 2.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.0
0.2
0.4
0.6
0.8
1.0
x2Ax2
A
QA(x2A)QA(x
2A)
Figure 1.8: Effort distribution functions for the leader (dashed) and his opponent(solid). We assumed d2 = 3
10 and in the left panel c = 1, while in theright c = 7.
QB(x2B) =
0 if xB < 0,
1 − C(x − d2) if xB ∈ [0,d2],
1 − C(x − d2) + C(xB − d2) if xB ∈ [d2, x],
1 if xB > x.
Expected utilities are 1− C(x − d2) for A and 0 for B. If d2 ≥ x both candidates spend
zero effort, x2A = x2
B = 0, and expected utilities are 1 and 0 respectively.
Proof. First we show that this is an equilibrium. Consider A choosing effort
x ∈ [0, x − d2]. The associated payoff is then
Pr[x + d2> x2
B]− C(x) = QB(x + d2)− C(x)
= 1 − C(x − d2) + C(x + d2 − d2)− C(x)
= 1 − C(x − d2) + C(x)− C(x) = 1 − C(x − d2).
He is hence indifferent between all pure strategies in SA given QB. Note also,
that x > x − d2 is strictly dominated since lowering effort to x − d2 saves costs
without altering the probability to win. Hence, given QB A is indifferent be-
tween all pure strategies in SA and cannot improve on the expected payoff by
deviating. Now consider B. Given QA, choosing x ∈ SB gives the following
payoff:
Pr[x > x2A + d2]− C(x) = Pr[x − d2
> x2A]− C(x) = QA(x − d2)− C(x)
= C(d2 + x − d2)− C(x) = C(x)− C(x) = 0.
Effort x = 0 never wins but does not imply costs, either. All efforts x ∈ [0,d2]
APPENDIX 57
are strictly dominated since they cannot win but imply costs. Choosing effort
x ∈ [d2, x] also yields zero payoff, since the probability to win exactly equals the
associated effort costs. Hence, all x ∈ SB yield zero expected payoff. Note that
deviating to x > x is strictly dominated, since it would win for sure but imply
costs that are greater than the value of winning. Hence, there is no beneficial
deviation for B as well and thus QA and QB are an equilibrium.
Now consider the support SA. By elimination of strictly dominated strate-
gies it is apparent that A will never choose effort greater than x − d2, because B
will never choose effort greater than x. The maximum bid neither can be lower,
because than B would have a winning strategy and could simply overbid A
and win for sure. That A’s minimum bid must be zero and that the support
cannot have holes, or that it must be convex, follows directly from the proof in
Hillman and Riley (1989). Hence, SA is the only support a mixed strategy of
A can have in equilibrium. Now consider SB. It is apparent that all x ∈ (0,d2)
can never win but imply positive costs, and are hence strictly dominated. The
maximum possible bid is naturally x, and must also be x. Otherwise A could
overbid x2B and win for sure. Convexity of SB follows again from Hillman and
Riley (1989).
It is now easy to see that given SA and SB there do not exist other effort
distribution making the opponent indifferent between all effort in his mixed
strategy. Hence, the proof is complete.
To see which candidate has stronger incentives to invest we now need to
calculate the expected effort of each. This is, unfortunately, not possible for our
general cost function. We hence resort to the quadratic form employed before,
where C(x) = c2 x2. Candidate A’s expected effort in the campaign is
Ex2A =
∫
SA
x C′(x + d2)dx =∫
SA
xc(x + d2)dx =
[c
3x3 +
cd2
2x2
]x−d2
0
=c
3
(√2
c− d2
)3
+cd2
2
(√2
c− d2
)2
=
√8
9c− d2 +
c
6(d2)3
Similarly, for candidate B we get
Ex2B =
∫
SB
x C′(x − d2)dx =∫
SB
xc(x − d2)dx =
[c
3x3 − cd2
2x2
]x
d2
=c
3(x)3 − cd2
2(x)2 − c
3(d2)3 +
c
2(d2)3 =
√8
9c− d2 +
c
6(d2)3,
58 CHAPTER 1. THE EFFECT OF POLLS
which is identical and hence the expected difference of efforts is zero. Formally:
Corollary 1.7.1. The difference in expected efforts in stage 2 is equal to zero.
This resembles our findings from a campaign with exogenous noise.
Now look at stage 1. We focus on pure strategy equilibria. Assume without
loss of generality d1 ≥ 0 and x1B = 0. A chooses effort to maximize
1 − C(x − d2)− C(x1A) = 1 − C(x − d1 − x1
A)− C(x1A).
His first order condition reads
C′(x − d1 − x1A)− C′(x1
A) = 0 ⇔ x − d1 = 2x1A.
The equivalence in the last step follows from the strict convexity of C(.), due
to which C′(.) is strictly increasing. Because the second derivative is strictly
negative,
−C′′(x − d1 − x1A)− C′′(x1
A) < 0,
we can use the first order condition to find the global maximum. A’s optimal
reaction to x1B = 0 is
x∗A =x − d1
2.
Now turn to B. He either spends zero and gets zero in expectation or maxi-
mizes
1 − C
(3
2x +
1
2d1 − x1
B
)− C(x1
B).
The first order condition reads
C′(3
2x +
1
2d1 − x1
B) = C′(x1B)⇔ 2xB =
3
2x +
1
2d1,
and hence
xB =3
4x +
d1
4.
The second order condition holds, but note that we only looked at positive
effort. For this to be optimal it needs to hold in addition that expected utility
is weakly positive:
1 − 2C
(3
4x +
d1
4
)≥ 0.
This inequality could in principle hold for small d1, depending on the curvature
of the costs function and when d1 is relatively small. For concreteness, however,
APPENDIX 59
0.0 0.5 1.0 1.5 2.0
-2
-1
0
1
2
Figure 1.9: Combinations of c (horizontal) and d1 (vertical) admitting a pure strategyNash equilibrium in which A spends more.
we stick to the functional employed mostly in the paper, C(x) = c2 x2. Then
1 − 2C
(3
4x +
d1
4
)= 1 − c
(3
4
√2
c+
d1
4
)2
< 0.
To see this note that the second term is strictly increasing in d1. Hence, if utility
is negative for d1 = 0 this is true also for all d1> 0. Using d1 = 0 we get
1 − c
(3
4
√2
c+
d1
4
)2
= 1 − c
(3
4
√2
c
)2
= 1 − 9
8= −1
8< 0.
Hence, xB cannot be an optimal choice and B chooses zero effort instead. There-
fore, we established that this is an equilibrium. Moreover, note that as long as
the utility from spending x1B is negative, (x∗A,0) is an equilibrium. This is the
case for all
d1 ≥ 4 − 3√
2√c
< 0.
Hence, as before when E[(ǫ)2] 6= 0, it may also be the case that the trailing
candidate spends more. In Figure 1.9 we show combinations of c and d1 for
which there exists a pure strategy Nash equilibrium in which A spends x1A > 0
while B stays passive.
Note that by symmetry the whole analysis also applies for B if d1 ≤ 0.
Hence, there always exists a pure strategy Nash equilibrium in stage 1 if we
assume costs to be quadratic, and there are multiple pure strategy equilibria in
close enough games:
Proposition 1.7.2. In stage 1 of the competition with quadratic costs there always
exists a pure strategy Nash equilibrium. In a close race there may also exist multiple
equilibria, and either candidate may adopt a tough stance and spend more. If one
60 CHAPTER 1. THE EFFECT OF POLLS
candidate has a relatively large advantage, this candidate will also choose larger effort.
Proof. This follows immediately from the discussion above.
It follows from Corollary 1.7.1 and Proposition 1.7.2 that the campaign
game qualitatively perfectly resembles what we have seen before in the case
of a noisy competition. This proves Proposition 1.6.2.
Proof of Proposition 1.6.3
Proof. Candidates maximize
Ed1
[λΦ
(d1 + ∆1 + ∆2
√2σ
)+ (1 − λ)Eǫ1,ǫ2
[1 − P(d1 + ∆1 + ∆2 − ǫ1 − ǫ2)
]]
− c
2(x1
A)2 − c
2(x2
A)2
Ed1
[λ
(1 − Φ
(d1 + ∆1 + ∆2
√2σ
))+ (1 − λ)Eǫ1,ǫ2
[P(d1 + ∆1 + ∆2 − ǫ1 − ǫ2)
]]
− c
2(x1
B)2 − c
2(x1
B)2(x2
B)2
respectively. Taking first order conditions with respect to spending in t leadsto
Ed1
[λ√2σ
φ
(d1 + ∆1 + ∆2
√2σ
)+ (1 − λ)Eǫ1,ǫ2
[p(d1 + ∆1 + ∆2 − ǫ1 − ǫ2)
]]− cxt
A!= 0
Ed1
[λ√2σ
φ
(d1 + ∆1 + ∆2
√2σ
)+ (1 − λ)Eǫ1,ǫ2
[p(d1 + ∆1 + ∆2 − ǫ1 − ǫ2)
]]− cxt
B!= 0
It is apparent that in any interior equilibrium both spend identical effort in
both stages. Moreover, since efforts perfectly cancel out, the equilibrium must
be unique.
Proof of Proposition 1.6.4
Proof. This follows immediately from the proof of Proposition 1.6.3. For the
stage 1 analysis we need an expression of the effort. Noticing that the noise
structure changes since only one random term is remaining, and that candi-
dates now know d2 precisely, from the FOCs is follows that
x2A = x2
B = x2 =
λσ φ(
d2
σ
)+ (1 − λ)Eǫ2
[p(d2 − ǫ2)
]
c.
APPENDIX 61
Making use of Assumption 1.6.3, we find
x2 =
λσ φ(
d2
σ
)+ (1−λ)√
σ2+(σP)2
φ
(−µP+d2√σ2+(σP)2
)+φ
(µP+d2√σ2+(σP)2
)
2
c.
Proof of Lemma 1.6.1
Proof. Stage 2 spending is
x2 =
λσ φ(
d2
σ
)+ (1−λ)√
σ2+(σP)2
φ
(−µP+d2√σ2+(σP)2
)+φ
(µP+d2√σ2+(σP)2
)
2
c.
Differentiation with respect to d2 gives
∂x2
∂d2=
λσ2 φ′
(d2
σ
)+ (1−λ)
σ2+(σP)2
φ′(
−µP+d2√σ2+(σP)2
)+φ′
(µP+d2√σ2+(σP)2
)
2
c.
At d2 = 0 the derivative is clearly zero. We hence take the second derivative to
see the curvature:
∂2x2
∂(d2)2=
λσ3 φ′′
(d2
σ
)+ (1−λ)
(σ2+(σP)2)3/2
φ′′(
−µP+d2√σ2+(σP)2
)+φ′′
(µP+d2√σ2+(σP)2
)
2
c.
At d2 = 0 we get
∂2x2
∂(d2)2|d2=0 =
λσ3 φ′′ (0) + (1−λ)
(σ2+(σP)2)3/2
φ′′(
−µP√σ2+(σP)2
)+φ′′
(µP√
σ2+(σP)2
)
2
c
=
λσ3 φ′′ (0) + (1−λ)
(σ2+(σP)2)3/2 φ′′(
µP√σ2+(σP)2
)
c
The first term is strictly negative, implying the derivative ∂x2
∂d2 decreases and so
does effort as we increase |d2| from zero. Hence look at the second term. It is
62 CHAPTER 1. THE EFFECT OF POLLS
straightforward to verify that
Sign
[φ′′(
µP
√σ2 + (σP)2
)]= Sign
[µP −
√σ2 + (σP)2
].
It follows that µP>
√σ2 + (σP)2 is necessary for effort to be increasing in d2. It
is, however, not sufficient. It is clear that when λ becomes large, the first term
dominates and the sign of the second derivative must be positive. However, if λ
is small, that is if the plurality premium is not very important, the second term
dominates. It is straightforward to show that for any λ ≥ 22+e3/2 the first term
always dominates and hence efforts decrease in |d2| (available upon request).
Proof of Proposition 1.6.5
Proof. It is apparent from the discussion that for too small µP incentives are as
under FPTP. However, if µP>
√σ2 + (σP)2 stage 2 efforts increase over some
range in |d2|. It is easily verified that when µP = µP =√
3√
σ2 + (σP)2 the
second derivative of second stage effort with respect to d2 has its maximum.
It is easily shown that comparative statics ∂x1A/∂d1 at d1 = 0, in case a unique
equilibrium exists, are negative if µP = µP and if λ is small (available upon
request). Hence, in this region there is always anti-momentum in close games.
Chapter 2
Political campaigns with specializedcandidates
Philipp Denter†
†I would like to thank Micael Castanheira, Martin Kolmar, Michael McBride, Stergios Skaperdas,seminar participants at UC Irvine, Tuebingen as well as participants of EPEW Rotterdam 2012 forinsightful comments and suggestions. The kind hospitality of UC Irvine is gratefully acknowledged. Ialso gratefully acknowledge the financial support of the Swiss National Science Foundation.
64 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
2.1 Introduction
“[...] there is no shortage of explanations for why issue convergence is such a rare
commodity in American campaigns. Perhaps surprisingly, though, there is a shortage
of convincing evidence that issue convergence really is a rare commodity.”
Sigelman and Buell (2004)
Obama has devoted 68 percent of his total TV advertising this year to
ads that include health care themes, and McCain has devoted 13 percent.
[...] In October, McCain spent 1.5 percent of his TV ads on health
care, while Obama upped the ante to 86 percent of his total budget.
Chris Frates on Politico.com (October 2008)1
Political competition determines which party or politician gets elected, and
therefore also which policies will be carried out during a term. Candidates
or parties develop a policy platform, which states more or less vaguely what
policies they plan to carry out if they should get elected. Then candidates
enter the campaigning stage, during which they spend considerable amounts
of time and money to convince voters that they are the right choice at the ballot.
But how does communication between politicians and the electorate look like?
Which policy issues will be important during the campaign? Which issues will
be neglected by one or both candidates? When will candidates communication
strategies converge, that is when will both emphasize the same issues during
the campaign? And under which conditions do candidates diverge? To answer
these questions is the purpose of this paper.
The campaigning literature so far is dominated by the seminal works of
Petrocik (1996) and Riker (1996). Petrocik (1996) argued that each candidate
owns certain issues, i.e. she is perceived to be more competent in this issue
than her political opponent. Such an advantage in perceived competence can
have many different sources, such as a party’s history, personal professional
experience of candidates and the like. For example, in the current campaign
contest with Barrack Obama, the Republican candidate Mitt Romney tries to
use his experience as a leader of a big business as a proof of his leadership abil-
ities. Riker (1996) developed two principals of campaigning rhetoric from issue
ownership. “When one side has an advantage on an issue, the other side ignores it;
but when neither side has an advantage, both seek new and advantageous issues” (page
106). He calls the former the dominance principle and the latter the dispersion prin-
1The article is available here: http://www.politico.com/news/stories/1008/14887.html.
2.1. INTRODUCTION 65
ciple. The prediction of his theory is that there is a strict form of divergence in
communication strategies and that some issues will be completely neglected.
How is the predictive power of these two principles in practice? As a first
example look at the 2008 campaign between Obama and John McCain.2 Obama
published some 23 percent of all his TV ads on the issue ’the economy’, which
was almost matched by McCain who published more than 20 percent of all
his TV ads on that issue. This issue was perceived the most important during
the campaign, with the US economy struck hard by the financial crisis and
many people fearing about their jobs. 91 percent of respondents of a Gallup
/ USA Today poll said this issue is either extremely or very important. Also,
48 percent of respondents believed Obama to be the right person to tackle eco-
nomic problems, while only 32 percent believed McCain to be more competent.
Although Obama was considered more competent by the average voter, candi-
dates roughly converged in their communication strategies, that is, the fraction
of TV ads they bought on that issue. Now look at the issue ’healthcare’, which
was considered less important than ’the economy’ but still 80 percent of poll
respondents classified it as either extremely or very important. In this issue,
Obama’s supremacy over McCain was even more extreme, with 51 percent
of respondents believing him to be more competent in this issue, and merely
26 percent believing in the opposite ranking. The above quotation by Chris
Frates shows that here candidates’ communication strategies diverged signifi-
cantly, and Obama was taking the lead. Finally, look at the issue ’terrorism’.
76 percent of poll respondents classified the issue as either extremely or very
important, and now Obama was perceived to be less competent in solving en-
ergy related problems, with 33 percent believing he was more competent vs.
52 percent believing McCain to be more competent. Also in this issue there
was significant divergence in communication strategies, but now McCain was
taking the lead, since Obama remained completely mute on that issue, while
McCain published some 5 percent of all his TV ads on that issue. On most other
issues both candidates’ strategies converged again, for example ’Iraq’ or ’taxa-
tion’. Hence, there is often some degree of divergence, and in some issues this
divergence can become quite extreme, but mostly there is only a moderate di-
vergence and most important issues were addressed by both candidates. Many
empirical studies confirm this conclusion, e.g. Damore (2004) and Sigelman
and Buell (2004). Therefore, the predictive power of the dominance principle is
2All numbers on TV ads are calculated by me using data from an online archive of the Wash-ington Post: http://projects.washingtonpost.com/politicalads/issues/. Data on issues’ im-portance and candidates’ competence are taken from a Gallup / USA Today poll from June 2008:http://www.gallup.com/poll/108331/obama-has-edge-key-election-issues.aspx.
66 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
rather weak, as has been put forward by a number of authors already.
In this paper I develop a theoretical model of political communication. Can-
didates can buy TV ads to sway voters. Buying TV ads has two effects: it ad-
vertises a candidate’s policy platform in the issue and it primes the issue. The
first effect increases a candidate’s perceived competence in the issue, or makes
voters like the platform better. The second increases the issue’s salience by di-
recting attention to it, and thereby increases the perceived importance of the
issue. This simple structure is sufficient to generate the following main results:
• In a pure advertising campaign candidates perfectly converge in their com-
munication strategies if they have identical marginal costs of funding. If
one candidate has a marginal cost advantage, he spends more on both
issues.
• In an advertising and priming campaign in which candidates do not have
comparative advantages, there exists an equilibrium in which they converge
completely in their communication strategy.
• In an advertising and priming campaign in which candidates have compar-
ative advantages, there will be some form of divergence, but never com-
plete divergence.
• A candidate might publish the biggest number of TV ads on an issue in
which he is disadvantaged, if this issue is sufficiently important and the
disadvantage not too big.
The intuition for the main result is as follows. If candidates have compar-
ative advantages, it is beneficial to highlight this advantage for two reasons.
First, advertising policy strengthens the advantage. Second, priming the issue
of the comparative advantage draws attention to one’s strength and away from
one’s weakness. This is also beneficial. While the advertising effect is also ben-
eficial in issues in which a candidate does not have an advantage, the priming
effect is now detrimental. However, as long as the first effect dominates the
candidate will still publish some TV ads on that issue. However, due to the ad-
ditional cost as which we can interpret the negative priming effect, candidates
publish more ads on their comparative advantages than on the other, unless the
issue in which there is a disadvantage becomes extremely important. If this is
the case a marginal increase in perceived quality of one’s policy platform in an
issue has a relatively strong effect on a candidate overall assessment, and hence
in that issue the advertising effect becomes dominating. To the contrary, in less
2.1. INTRODUCTION 67
important issues the priming effect is more important and there we should
hence expect more divergence.
The model can explain most features of the 2008 campaign such as the
strong form of convergence in communication strategies in the issue ’the econ-
omy’ and most other issues, and the strong form of divergence in ’healthcare’
and ’terrorism’. The issues in which candidates’ strategies diverged signifi-
cantly were not among the most important issues and in both issues the per-
ceived competence advantage of one candidate was relatively large. The model
predicts that both of these facts make divergence more likely. In issues that are
more important and in which comparative advantages are smaller, there is a
tendency to converge.
The chapter is organized as follows. Next I discuss the relevant related
literature. In Section 2.2 I present the formal model. In Section 2.3 I discuss
the benchmark of a pure policy advertising campaign. Section 2.4 studies the
general model with both policy advertising and issue priming. Section 2.5
concludes.
Related literature. Many scholars have directed their attention to the study of
political campaigns. An important question thereby is and was how candidates
allocate their time and money during an electoral contest to different states and
electoral districts, e.g. Brams and Davis (1973, 1974), Snyder (1989), Klumpp
and Polborn (2006), and Stromberg (2008), or to different forms of campaigning,
e.g. Skaperdas and Grofman (1995). Meirowitz (2008), Iaryczower and Mattozzi
(2009), and Sahuguet and Persico (2006) consider different effects of campaign
spending regulations. In contrast to all those papers, the focus of my paper is
to study candidates’ communication strategies during a campaign, that is how
different issues are addressed during a campaign when there are advertising
and priming elements of campaigning.
Petrocik (1996) argued in his influential paper that candidates and parties
have a reputation to own certain issues, i.e. that they are in the perception
of the populace more competent in this issue than their political opponent. A
consequence of owning an issue is then that candidates will focus in their cam-
paign communication on issues that they own, and mute others. This strategy
has been called the ’dominance principle’ by Riker (1996) and subsequent au-
thors, e.g. Amoros and Puy (2011). The predictions of the dominance principle
are that there is some extreme form of divergence in communication strate-
gies. However, empirical scholarship so far tends to reject the hypotheses and
finds that there is moderate divergence in some issues, but often candidates
68 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
also tend to converge in their communication, see for example Bélanger and
Meguid (2008), Damore (2004, 2005), Green and Hobolt (2008), Petrocik et al.
(2003), or Sigelman and Buell (2004).
Recently, some authors have developed theories that want to explain where
issue ownership might come from, for example Krasa and Polborn (2010) and
Aragonès et al. (2012). In contrast to this literature the focus here is not the
emergence of issue ownership, but the consequences for campaigning. Moen
and Riis (2010) develop a theoretical justification for issue trespassing, which
is based on a signaling argument. In the current paper issue trespassing can
also happen, but the reason is that policy advertising has an effect on voters’
policy assessment. Egan (2009) shows how candidates can use their perceived
advantage in the issues they own to choose policy platforms that are not as
conform with voters’ preferences. In contrast, I am not concerned with how
candidates’ choose policy platforms, but how issue ownership and policy plat-
forms influence candidates’ communication strategies during a campaign.
2.2 The Model
I model a multi issue campaign between two candidates i ∈ {1,2} in two is-
sues j ∈ {H,S}, which are mnemonics for health care and homeland security.
A candidate spends effort xji ≥ 0 to increase his competence in this issue as
perceived by the voters, and thereby his probability to win the election. This
can include positive campaigning in which the candidate highlights parts of his
policy plans to convince voters that his plan is good, or it could mean negative
campaigning, which targets the opponent’s weaknesses in the issue. Marginal
costs of effort are constant and equal to mi > for candidate i and reflect costs
to raise and spend funds as well as time. Differences in marginal costs may re-
flect, for example, different budgets. I assume that campaigning is effective in
the sense that spending increases a candidate’s perceived competence. Letting
θ j ∈ (0,1) and 1− θ j denote the electorate’s common prior concerning candidate
1’s and 2’s competence, after campaign beliefs are
cj1 =
θ j f (xj1)
θ j f (xj1) + (1 − θ j) f (x
j2)
(2.1)
and cj2 = 1 − c
j1. Competence is Hence defined as relative competence. (2.1) is a
competence ’production function’ and depends on both candidates’ efforts or
inputs. This form of contest has been used to analyze various problems such as
2.2. THE MODEL 69
lobbying, litigation, or war. Skaperdas (1996) provides an axiomatic foundation
and Skaperdas and Vaidya (2009) provide an inferential micro foundation for
this function to model persuasion, i.e. the transformation of ex-ante beliefs into
ex-post beliefs due to persuasive effort. Accordingly, I interpret θ j and 1− θ j as
the respective prior beliefs regarding competence in issue j.
The function f (.) transforms effort x into effective effort. I make the follow-
ing assumptions concerning this function:
Assumption 2.2.1. 1. f (x) is at least twice continuously differentiable.
2. f (0) > 0.
3. f ′(x) > 0.
4. f ′′(x) < 0.
5. limx→0 f ′(x) = ∞.
Part 2 means that even if a candidate does not spend anything in the cam-
paign his perceived competence remains positive. A direct implication is that
perceived competence is continuous in candidates’ efforts. Part 3 reflects that
spending effort is beneficial to a candidate. Part 4 implies marginal products of
effort are decreasing. Alternatively, it might as well be interpreted as increas-
ing marginal costs of effort, for example, because marginal costs of funding
increase. Part 5 is not a necessary assumption but is sufficient to guarantee the
existence of an interior equilibrium. Assumption 2.2.1 is not necessary but suffi-
cient. Below I will solve a numerical example when f (.) is a linear function. An
example functional fulfilling Assumption 2.2.1 is f (x) = c + xa for some c > 0
and 1 > a > 0.
Apart from increasing the perceived competence of a candidate in an issue,
effort has another effect that is of relevance: it increases the salience of the issue.
If salience does not have an influence on voters decisions this would not make a
difference. However, I assume that increasing the salience of an issue increases
the relative importance of the issue as perceived by voters. In particular, letting
θw and 1 − θw denote the prior importance weights of issues H and S, weights
after the campaign are
w =θwg(κ + xH
1 + xH2 )
θwg(κ + xH1 + xH
2 ) + (1 − θw)g(κ + xS1 + xS
2). (2.2)
for H and 1 − w for S. κ ≥ 0 is a parameter that determines the effectiveness
of creating salience to make the issue more important and is exogenous. It
70 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
may represent institutional features such as the degree to which the media
follow the candidates in creating salience. I make the following assumptions
concerning g(.):
Assumption 2.2.2. 1. g(x) is at least twice continuously differentiable.
2. g(0) > 0.
3. g′(x) > 0 and finite for finite x and limx→∞ g′(x) = 0.
4. g′′(x) < 0 for finite x and limx→∞ g′′(x) = 0.
5. g′′′(x) > 0.
Part 1 reflects that an issue’s importance cannot drop to zero just because
nobody talks about it. Part 2 reflects creating salience affects the relative impor-
tance of the two issues. Part 3 means there are decreasing returns of salience,
and part 4 and 5 imply that returns do not decrease very quickly. The latter two
assumptions are motivated by mathematical necessity rather than following a
real world intuition. The assumptions are also not necessary for the analysis,
but – as we will see below – sufficient to guarantee a nicely behaved prob-
lem for the candidates.3 An example function fulfilling Assumption 2.2.2 is
g(x) = (γ + x)α for some γ > 0 and 1 > α > 0.
Now it is possible to determine voters’ choice. I adopt the assumption of a
probabilistic voting decision as in Snyder (1989) or Klumpp and Polborn (2006),
but I add a particular structure.
Assumption 2.2.3. The probability that candidate 1 wins the election is
Pr1[win] = cHw + cS(1 − w),
and candidate 2 wins with probability
Pr2[win] = 1 − Pr1[win] = (1 − cH)w + (1 − cS)(1 − w).
Hereby I generalize the models of Snyder (1989) and Klumpp and Polborn
(2006) to two issues and possibly endogenous issue weights. The probability to
win is a weighted average of the relative competence measures. Ceteris paribus,
the more competent a candidate is perceived in an issue, the more likely it
is that he wins the election. Similarly, when the strong issue of a candidate
becomes more important, his winning probability also increases. Hence, in
3An alternative assumption guaranteeing a nicely behaved problem, that would allow us to dispensewith Assumptions 2.3-2.5, is g′(x) < α1 and |g′′(x)|< α2 for appropriately chosen α1 and α2.
2.3. WHEN SALIENCE HAS NO EFFECT: A BENCHMARK 71
my model being competent in one or the other issue are imperfect substitutes.
There is no lexicographic ordering of issues.
Finally, before I start with the analysis, I need to make clear what I mean
when I say candidates do or do not specialize. There are two candidates and
two issues. I shall call an equilibrium in which one candidate spends more
on one issue, while his opponent spends more on the other, an equilibrium
with specialization. This definition is intuitive and refers to specialization in
absolute terms, i.e. when a candidate spends absolutely more on one issue and
absolutely less on the other. An alternative and weaker form of specialization,
in which a candidate specializes in an issue if he spends relative to his opponent
more on this issue than on the other, is not considered in this paper.
Definition 2.2.1. If
Sign[xH1 − xH
2 ] = −Sign[xS1 − xS
2 ]
in equilibrium, I shall call this a specialization equilibrium.
2.3 When Salience has no Effect: A Benchmark
To see why specialization as discussed above cannot be reconciled with the stan-
dard model with exogenous issue weights, I now shortly discuss the standard
model and its predictions.
Assume a candidate’s competence is determined as in Assumption 2.2.1.
However, salience does not have an influence on perceived importance of an
issue, and hence wH = w = θw and wS = 1 − θw. The whole problem faced by
a candidate is then separable in two sub-contests for competence only. In each
sub-contest candidate 1 wants to maximize
θ j f (xj1)
θ j f (xj1) + (1 − θ j) f (x
j2)
wj − m1xj1,
while candidate 2 maximizes
(1 − θ j) f (xj2)
θ j f (xj1) + (1 − θ j) f (x
j2)
wj − m2xj2.
This game has been analyzed extensively, see for example Rosen (1986), Dixit
(1987), Snyder (1989), or Yildirim (2005):
Proposition 2.3.1. The game has a unique equilibrium in pure strategies. Candidate 1
spends on both issues more (less) than candidate 2 if m1 < m2 (m1 > m2). Both spend
identical effort on a given issue whenever m1 = m2.
72 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
Proof. See appendix.
The proposition shows that in this model there is no specialization as de-
fined above. Differences in marginal costs may explain why a candidate spends
in total more than his opponent, but do not explain specialization. Differences
in the electorate’s prior belief do not have any influence on relative spending
(but on aggregate spending). A candidate may Hence spend strictly more than
his adversary, but it may not happen that he spends more on one issue and less
on another, as it was for example the case in the Obama vs. McCain example
above.
2.4 When Salience Matters
Now consider the game when salience influences issue weights. The aim of a
candidate is again to maximize his winning probability subject to costs of effort.
Hence, following the assumptions made in Section 2.2, each candidate faces the
following problem:
max(xH
1 ,xS1 )∈R2
+
u1 = cHw + cS(1 − w)− m1(xH1 + xS
1 ) (2.3)
max(xH
2 ,xS2 )∈R2
+
u2 = (1 − cH)w + (1 − cS)(1 − w)− m2(xH2 + xS
2 ) (2.4)
Unlike before, w is now endogenously determined as defined in Assumption
2.2.2. Before going into the details of the equilibrium, I need to establish that
an equilibrium actually exists. This is done in the following proposition:
Proposition 2.4.1. For κ sufficiently large but finite, the game has an interior pure
strategy Nash equilibrium. There cannot exist a corner equilibrium, in which a candi-
date spends no effort on a given issue.
Proof. See appendix.
The intuition for the existence result is straightforward. The effect of the
relative importance on the second order conditions must not be too large, be-
cause then the problem is not concave anymore. By restricting the curvature
of g(.) and letting κ become relatively large, we can avoid this problem and
apply standard equilibrium existence proofs.4 That there cannot exist a corner
equilibrium follows from part 4 of Assumption 2.2.1.
4Note, however, that for this purpose Assumption 2.2.2 is sufficient, but not necessary. There arealternative assumptions guaranteeing existence. What needs to hold is that the first and second deriva-tive of g are sufficiently small.
2.4. WHEN SALIENCE MATTERS 73
Having established that an interior equilibrium exists and that there cannot
be corner equilibria, it follows that I can use first order conditions. From (2.3)
and (2.4) the following system of first order conditions follows:
∂u1
∂xH1
=∂cH
∂xH1
w + (cH − cS)∂w
∂xH1
− m1 = 0 (2.5)
∂u1
∂xS1
=∂cS
∂xS1
(1 − w) + (cH − cS)∂w
∂xS1
− m1 = 0 (2.6)
∂u2
∂xH2
= − ∂cH
∂xH2
w − (cH − cS)∂w
∂xH2
− m2 = 0 (2.7)
∂u2
∂xS2
= − ∂cS
∂xS2
(1 − w)− (cH − cS)∂w
∂xS2
− m2 = 0 (2.8)
Increasing effort influences the probability to win through two different
channels. First, the perceived competence of the candidate increases in the is-
sue he spends the effort on. Second, the relative weight voters assign to this
issue increases. While the former effect is always beneficial, the latter might be
detrimental. Assume candidate 1 is considered to be much more competent in
health care issues than candidate 2, while their relative competence in security
questions is more equal. If candidate 2 now spends effort in a health care cam-
paign, he might play his adversary in the hand; because while he decreases the
gap in perceived competence, he emphasizes his weak spot. Hence, spending
effort on one’s weak issue involves additional costs for a candidate.
From the first order conditions we can derive the following condition, which
has to hold in any interior equilibrium:
w
(∂cH
∂xH1
+ ∂cH
∂xH2
)+ ∆
∂w∂xH
1
=
(1 − w)
(∂cS
∂xS1
+ ∂cS
∂xS2
)+ ∆
∂w∂xS
1
where ∆ = m2 − m1. (2.9)
This equation is fundamental for the analysis. Since the derivatives in the
denominator have opposite signs it follows that the enumerators also must
have opposite signs or be zero. Formally:
Lemma 2.4.1. In any equilibrium it must hold that
Sign
[w
(∂cH
∂xH1
+∂cH
∂xH2
)+ ∆
]= −Sign
[(1 − w)
(∂cS
∂xS1
+∂cS
∂xS2
)+ ∆
].
74 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
Proof. See appendix.
What does this now mean for the equilibrium of the game? I start with the
simplest case, that is when ∆ = 0 and hence candidates face identical marginal
costs of effort. Since 1 > w > 0, the condition in Lemma 2.4.1 then boils down
to
Sign
[∂cH
∂xH1
+∂cH
∂xH2
]= −Sign
[∂cS
∂xS1
+∂cS
∂xS2
].
This tells us that either candidate 1 has a larger marginal product of effort in
equilibrium in the one issue and a lower in the other, or both candidates have
identical marginal products in both issues. I define a candidate’s comparative
advantage in the campaign in the following way:
Definition 2.4.1. A candidate has a comparative advantage in a given issue, if the
voters ex ante belief he is in this issue more competent than in the other issue. If
θH> θS, candidate 1 has a comparative advantage in issue H and candidate 2 in issue
S, and vice versa if θH< θS. If θH = θS, no candidate has a comparative advantage.
Comparative advantages, like in trade theory and other fields of economics,
are among the driving forces for specialization. To see how comparative ad-
vantages affect specialization I assume ∆ = 0 first, such that candidates have
identical marginal costs. In this case, whenever candidates have comparative
advantages, there is specialization in equilibrium:
Proposition 2.4.2. Let ∆ = 0.
i. A no specialization equilibrium exists if and only if θH = θS, that is, when candi-
dates do not have a comparative advantage.
ii. If in addition θw = 12 , a perfectly symmetric equilibrium exists in which each
candidates spends identical effort on each issue.
iii. If θH 6= θS, and hence candidates have a comparative advantage, each candidate
specializes in a given issue.
Proof. See appendix.
A no-specialization equilibrium may only exist if candidates have no com-
parative advantages. Then each candidates spends identical effort on issue i. If
in addition issues have identical ex ante weights, there exists a perfectly sym-
metric equilibrium. However, as soon as there are comparative advantages,
2.4. WHEN SALIENCE MATTERS 75
symmetric equilibria cannot exist anymore. To see why this must be the case
consider to the contrary, that a symmetric equilibrium exists in which candi-
dates spend identical effort on issue i. Then, by definition of comparative ad-
vantages and symmetric equilibria, cH − cS = θH − θS 6= 0. Hence, changing the
relative weights of the issues has an effect on the winning probabilities at the
margin, and this effect goes in opposite directions. Hence, since the marginal
effect of effort on perceived competence in an issue is identical in a symmetric
equilibrium, this cannot be an equilibrium, because there are profitable devia-
tions. Hence, there must be specialization in equilibrium.
I cannot exclude that there may be specialization even when there is no
comparative advantage. Depending on the particular contest technologies f (.)
and g(.) this might be the case. This means I cannot proof uniqueness of Nash
equilibria, either. Actually, in principle it might even happen that candidates in
equilibrium specialize against their ex ante comparative advantage. However,
if this is the case it must hold that the issue a candidate specializes in is the
issue he has an ex post comparative advantage. Hence, when a candidate has a
significant advantage in a given issue, he would have to spend a lot to turn his
comparative advantage. Hence, depending on the marginal costs, at a given
point this cannot be worthwhile and thus when marginal costs are high and
there are ex ante comparative advantages, candidates specialize accordingly.
Now let ∆ 6= 0, so that one candidate has a marginal cost advantage. As-
sume without loss of generality that ∆ > 0, such that candidate 1 has the advan-
tage because his marginal costs are lower. Intuitively, in this situation candidate
1 will increase efforts relative to his opponent. But how is specialization af-
fected? Clearly, if any candidate’s marginal costs approach zero, this candidate
will spend more on both issue than his opponent. But if differences in marginal
costs are not too pronounced, there is still specialization in equilibrium:
Proposition 2.4.3. If |∆| > 0 but sufficiently small, in equilibrium candidates spe-
cialize when there are comparative advantages. However, if |∆| becomes too large, the
candidate with lower marginal costs spends more on both issues.
Proof. See appendix.
How large |∆| may become in order to sustain specialization depends on
the parameters and functional forms of the game. When there is a high degree
of specialization when ∆ = 0, the differences in marginal costs of funding may
be relatively large. If specialization is only marginal, even slight differences in
marginal costs may prevent specialization. Note that ∆ is the absolute differ-
ence in marginal costs. When effort is very cheap at the margin (relative to
76 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
0 5 10 15 20 25
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
κ
xi1/xi
2
Figure 2.1: xH1 /xH
2 as well as xS1 /xS
2 for θw = 4/9, θH = 5/9, θS = 1/3, m1 = m2 = 1,and variable values of κ. As κ → ∞ both ratios converge to 1.
the prize of winning the election, which is assumed to be 1), relative marginal
costs may be very different and specialization may still occur. If marginal costs
are large, relative marginal costs must be close to one for specialization equilib-
rium.
Note that specialization does not mean that a candidate spends more on
issue H than on S or the other way around. A candidate might specialize in
issue H and still spend more on issue S. This is so because I defined specializa-
tion relative to the opponent’s spending. For example, in the 2008 presidential
campaign McCain clearly specialized on security, but nevertheless spend more
on his health campaign. This is likely to happen when one issue is much more
important than the other. For example, if θw is relatively large, the candidate
with a comparative advantage in issue S is likely to spend absolutely more on
issue H, while he still spends more on S than his opponent.
Proposition 2.4.4. A candidate may spend more on the issue he has the comparative
disadvantage in, than on the issue in which in which he has the comparative advantage,
if the issue of his disadvantage is relatively important.
Proof. See appendix.
For example, in Table 2.1 below in rows (vi.) and (vii.) this is the case.
Candidate 1 has a comparative advantage in S and candidate 2 in H. However,
because issue 2 is perceived to be much more important, both spend more on
S than on H.
Now look at the parameter κ and assume ∆ = 0. It determines the marginal
influence of salience on perceived importance. An interpretation of κ is a pa-
rameter determining the responsiveness of the media to the candidates’ polit-
ical advertising. If the media more than proportionally picks up issues made
2.4. WHEN SALIENCE MATTERS 77
salient by the candidates, then the media intensifies salience. This would be
the case when κ is relatively small and so small differences in total spending
for the different issues has a large effect. That makes it similar to the dis-
criminatory power in standard lottery contests, see for example Tullock (1980)
or Gradstein and Konrad (1999). Similarly, Amegashie (2006) interprets κ as
a noise parameter. In standard contest models increasing the discriminatory
power of the contest – i.e. in my framework lowering κ – usually has the effect
of increasing effort in the contest. This is so because a higher discriminatory
power implies increasing returns to effort. In the present framework there are
three different intertwined contests. By decreasing κ, the marginal returns of
effort in the salience contest increase relative to the returns in the competence
contests. Therefore, specialization becomes more and more attractive and thus
candidates tend to specialize more when κ becomes smaller. However, since I
cannot exclude multiple equilibria, a comparative statics analysis is problem-
atic. To avoid this in the following I will assume that the equilibrium is unique.
Define the ratio ρi := xi1/xi
2 as a measure of the degree of specialization in issue
i. If ρi> 1 candidate 1 specializes in issue i and, by Proposition 2.4.2, candi-
date 2 specializes in the other issue. Also assume ∆ = 0. Figure 2.1 shows
the ratios ρH and ρS for increasing values of κ for a parameterized version of
the model. In this version f (x) = x and g(κ + x1 + x2) = κ + x1 + x2. This
contradicts Assumptions 2.2.1 and 2.2.2, but note that these assumptions only
served the purpose to prove existence of an equilibrium in the general model.
They do not affect the comparative statics qualitatively. I chose this functional
form for computational convenience. The figure shows clearly that the degree
of specialization decreases in κ for the chosen parameter values. This was true
for all numerical solutions of the model I computed and seems to be generally
true for the functional forms I used. Note that when ∆ 6= 0 the effort ratios do
not converge to 1, as it is the case in the figure, but to some other constant.
Before I conclude, to get a better understanding of the comparative statics
of the model, I calculated equilibria for varying parameter values in Table 2.1.
There are three different regions. In the first region, rows (i.) to (iv.), no can-
didate has a comparative advantage and marginal costs of effort are identical.
Hence, there is no specialization in equilibrium. In the middle region, rows (v.)
to (viii.), candidates have comparative advantages but marginal costs are still
identical. Hence, there is always specialization. Finally, in the lower region,
rows (ix.) to (xii.), there are still comparative advantages but also differences in
marginal costs. In rows (ix.) and (x.) there is nevertheless specialization, but in
rows (xi.) and (xii.) the difference is too large and candidate 2 spends more on
78 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
θH θS θw m1 m2 κ xH1 xS
1 xH2 xS
2xH
1
xH2
xS1
xS2
(i.) 1/2 1/2 1/2 1/4 1/4 2 1/2 1/2 1/2 1/2 1 1(ii.) 1/2 1/2 1/3 1/4 1/4 2 .268 .732 .268 .732 1 1(iii.) 1/3 1/3 1/2 1/4 1/4 2 .444 .444 .444 .444 1 1(iv.) 1/3 1/3 1/3 1/4 1/4 2 .244 .645 .244 .645 1 1
(v.) 1/3 2/3 1/2 1/4 1/4 2 .330 .466 .466 .330 .708 1.412(vi.) 1/3 2/3 1/3 1/4 1/4 2 .192 .664 .256 .531 .750 1.250(vii.) 1/3 2/3 1/4 1/4 1/4 2 .137 .741 .170 .639 .806 1.160(vii.) 1/4 2/3 1/2 1/4 1/4 2 .226 .489 .355 .323 .637 1.514
(ix.) 1/3 2/3 1/2 1/4 1/5 2 .274 .521 .514 .452 .533 1.153(x.) 1/3 2/3 2/3 1/4 1/5 2 .462 .284 .752 .257 .614 1.105(xi.) 1/3 2/3 1/3 1/4 1/5 2 .159 .730 .274 .733 .580 .996(xii.) 1/2 2/3 1/2 1/4 1/5 2 .455 .478 .688 .490 .661 .976
Table 2.1: Equilibrium values for a parameterized version of the campaign contest.Pr0
1 is candidate 1’s ex ante winning probability, Pr11 his ex post probability
to win, that is after taking into account the campaign.
both issues.
2.5 Conclusion
In this paper I analyzed a multi issue campaign with endogenous issue weights.
When candidates have comparative advantages and marginal costs are similar,
there is specialization in the sense that each candidate spends more than his
opponent on one issue and less on the other. When marginal costs become
too different in absolute terms, the candidate with lower costs spends more on
both issues. The model is helpful in explaining real world campaigns and an
example can be found in the 2008 presidential campaign in the U.S. The driv-
ing force for my results is that efforts cannot be targeted precisely to promote
competence only, but there are spillovers influencing the perceived relative im-
portance of an issue. This is so because promoting a policy platform on a given
issue necessarily makes the issue more salient.
Apart from political campaigns, the model may be applied to persuasive
advertising in goods markets when there are multiple dimensions. While there
exists a large literature analyzing persuasive advertising, see for example the
recent survey by Bagwell (2007), the issue of multiple product features has not
received much attention so far. Goods with multiple features include comput-
ers, cars, or smart phones to name just a few. As an example one can think
of Apple computers vs. Microsoft Windows based computers. While in most
2.5. CONCLUSION 79
Apple commercials style and graphical features are made salient, Microsoft
Windows based computer companies often highlight the computer’s aptitude
for business purposes.
There are some immediate directions for future research that follow from
the current paper. First, it is interesting to study how the results change when
there are more issues and competing candidates. In particular, what happens
when there are more candidates than (important) issues is an interesting ques-
tion. A second immediate route for future research is to give candidates a
fixed budget instead of assuming fixed marginal costs of effort. Finally, it is in-
teresting to explicitly model voters’ preferences and to incorporate a platform
choice decision prior to the campaign. This latter generalization is a project I
am currently working on.
80 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
Appendix
Proof of Proposition 2.3.1
Proof. Existence of an interior equilibrium follows directly from strict concavity
of the candidates’ optimization problem and has been established for example
by Yildirim (2005) and Snyder (1989). Given w is a constant, each candidate
separately optimizes utility in any issue. This yields the following first order
condition in issue i for 1:
θi(1 − θi) f ′(xi1) f (xi
2)
(θi f (xi1) + (1 − θi) f (xi
2))2
wi − m1!= 0.
The first order condition of his adversary in the same issue is
θi(1 − θi) f (xi1) f ′(xi
2)
(θi f (xi1) + (1 − θi) f (xi
2))2
wi − m2!= 0.
Now note that 1’s marginal return of effort is higher / lower than 2’s if and
only if he spends less / more on this issue. If they spend identical effort, the
marginal returns are identical, too. Since in equilibrium marginal costs must
equal marginal returns of effort, the candidate with lower marginal costs must
have lower marginal returns, too, and hence spend more effort. If candidates
have identical marginal costs, m1 = m2, they spend identical effort in equilib-
rium. (See also Comments 3.1 and 3.2 in Snyder (1989)).
Proof of Proposition 2.4.1
Proof. To prove existence of a pure strategy Nash equilibrium it suffices to show
that strategy spaces are convex and compact and utility functions are concave
and continuous. Since any effort larger than 1 is strictly dominated the relevant
strategy space is [0,1]2, which is both convex and compact. Moreover, since
f (0) > 0 as well as g(0) > 0, utility function are also continuous. Hence, it
remains to show that they are also concave. For concavity it has to hold that
∂2u1
∂(xH1 )2
< 0 and∂2u1
∂(xH1 )2
∂2u1
∂(xS1)
2−(
∂2u1
∂xH1 ∂xS
1
)2
> 0
for all relevant effort combinations. I show that for κ sufficiently large this holds
generally, which is then stronger than needed but analytically more convenient.
Because the aim of this paper is not to show conditions that are necessary
APPENDIX 81
for an equilibrium to exist, but to analyze candidate behavior in equilibrium,
providing a sufficient condition for an equilibrium to exist is enough for my
purpose.
I will now show when the payoff functions are concave and without loss of
generality, I focus on candidate 1’s payoff. For candidate 2 the exact same steps
then would show the same.
The candidates’ first order conditions are
∂u1
∂xH1
=∂cH
∂xH1
w + (cH − cS)∂w
∂xH1
− m1 = 0 (2.10)
∂u1
∂xS1
=∂cS
∂xS1
(1 − w) + (cH − cS)∂w
∂xS1
− m1 = 0 (2.11)
∂u2
∂xH2
= − ∂cH
∂xH2
w − (cH − cS)∂w
∂xH2
− m2 = 0 (2.12)
∂u2
∂xS2
= − ∂cS
∂xS2
(1 − w)− (cH − cS)∂w
∂xS2
− m2 = 0 (2.13)
and second derivatives for 1 are
∂2u1
∂(xH1 )2
=∂2cH
∂(xH1 )2
w + 2∂w
∂xH1
∂cH
∂xH1
+∂2w
∂(xH1 )2
(cH − cS) (2.14)
∂2u1
∂(xS1 )
2=
∂2cS
∂(xS1 )
2(1 − w)− 2
∂w
∂xS1
∂cS
∂xS1
+∂2w
∂(xS1 )
2(cH − cS) (2.15)
∂2u1
∂xH1 ∂xS
1
=∂cH
∂xH1
∂w
∂xS1
− ∂cS
∂xS1
∂w
∂xH1
+ (cH − cS)∂2w
∂xH1 ∂xS
1
(2.16)
Using the specific functional forms the relevant terms as follows:
∂2cH
∂(xH1 )2
=θH(1 − θH) f (xH
2 ) f ′′(xH1 )[θH f (xH
1 ) + (1 − θH) f (xH2 )]
(θH f (xH1 ) + (1 − θH) f (xH
2 ))3(2.17)
− 2θH( f ′(xH1 ))2
(θH f (xH1 ) + (1 − θH) f (xH
2 ))3< 0
∂2cS
∂(xS1 )
2=
θS(1 − θS) f (xS2 ) f ′′(xS
1 )[θS f (xS
1 ) + (1 − θS) f (xS2 )]
(θS f (xS1 ) + (2 − θS) f (xS
2 ))3
(2.18)
− 2θS( f ′(xS1 ))
2
(θS f (xS1 ) + (2 − θS) f (xS
2 ))3< 0
82 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
∂2w
∂(xH1 )2
=θw(1 − θw)g(κ + xS
1 + xS2 )g′′(κ + xH
1 + xH2 )
(θwg(κ + xH1 + xH
2 ) + (1 − θw)g(κ + xS1 + xS
2))3
(2.19)
×[θwg(κ + xH
1 + xH2 ) + (1 − θw)g(κ + xS
1 + xS2)]
− 2θw(g′(κ + xH1 + xH
2 ))2
(θwg(κ + xH1 + xH
2 ) + (1 − θw)g(κ + xS1 + xS
2))3
∂2w
∂(xS1 )
2= − θw(1 − θw)g(κ + xH
1 + xH2 )g′′(κ + xH
1 + xH2 )
(θwg(κ + xH1 + xH
2 ) + (1 − θw)g(κ + xS1 + xS
2))3
(2.20)
×[θwg(κ + xH
1 + xH2 ) + (1 − θw)g(κ + xS
1 + xS2)]
− 2θw(g′(κ + xH1 + xH
2 ))2
(θwg(κ + xH1 + xH
2 ) + (1 − θw)g(κ + xS1 + xS
2))3
∂2w
∂xH1 ∂xS
1
=θw(1 − θw)g′(κ + xH
1 + xH2 )g′(κ + xS
1 + xS2)
(θwg(κ + xH1 + xH
2 ) + (1 − θw)g(κ + xS1 + xS
2))3
(2.21)
×[θwg(κ + xH
1 + xH2 )− (1 − θw)g(κ + xS
1 + xS2)]
From Assumption 2.2.2 it follows that (2.19), (2.20), and (2.21) vanish as κ →∞. Because (2.17) and (2.18) are strictly negative it follows that there exists
some finite κ such that for all κ > κ the problem each candidate faces is strictly
concave. Therefore, existence of a pure strategy Nash equilibrium follows from
Theorem 1.2 in Fudenberg and Tirole (1991), which is due to Debreu (1952),
Fan (1952), and Glicksberg (1952). From part 4 of Assumption 2.2.1 it follows
that a corner equilibrium cannot exist, because a deviation would always be
profitable. Hence, an interior pure strategy equilibrium must exist.
Proof of Proposition 2.4.2
Proof. If we equate (2.10) and (2.12) and (2.11) and (2.13) and by noting that∂w
∂xH1
= ∂w∂xH
2
as well as ∂w∂xS
1
= ∂w∂xS
2
we get after simple manipulations that
−2(cH − cS) =
(∂cH
∂xH1
+ ∂cH
∂xH2
)w
∂w∂xH
1
, (2.22)
−2(cH − cS) =
(∂cS
∂xS1
+ ∂cS
∂xS2
)(1 − w)
∂w∂xS
1
. (2.23)
APPENDIX 83
Equating both yields the fundamental condition discussed above for ∆ = 0:
(∂cH
∂xH1
+ ∂cH
∂xH2
)w
∂w∂xH
1
=
(∂cS
∂xS1
+ ∂cS
∂xS2
)(1 − w)
∂w∂xS
1
(2.24)
Since ∂w∂xS
1
< 0 but ∂w∂xH
1
> 0 it is apparent that the signs of the terms in brackets
musts be different, or both bracket terms must be zero. Hence,
Sign
[∂cH
∂xH1
+∂cH
∂xH2
]= −Sign
[∂cS
∂xS1
+∂cS
∂xS2
].
The derivatives of ck with respect to xki are
∂ck
∂xk1
=θk(1 − θk) f ′(xk
1) f (xk2)
(θk f (xk1) + (1 − θk) f (xk
2))2
, (2.25)
∂ck
∂xk2
= − θk(1 − θk) f (xk1) f ′(xk
2)
(θk f (xk1) + (1 − θk) f (xk
2))2
. (2.26)
It follows that
∂cH
∂xH1
+∂cH
∂xH2
=θH(1 − θH)
[f ′(xH
1 ) f (xH2 )− f (xH
1 ) f ′(xH2 )]
(θH f (xH1 ) + (1 − θH) f (xH
2 ))2, (2.27)
∂cS
∂xS1
+∂cS
∂xS2
=θS(1 − θS)
[f ′(xS
1 ) f (xS2 )− f (xS
1 ) f ′(xS2 )]
(θS f (xS1 ) + (1 − θS) f (xS
2 ))2
. (2.28)
Sinced f ′(x)/ f (x)
dx = f ′′(x) f (x)− f ′(x)2
f (x)2 < 0 we find that
Sign
[∂ck
∂xk1
+∂ck
∂xk2
]=
+ if xk1 < xk
2,
0 if xk1 < xk
2,
− if xk1 < xk
2.
Hence, for (2.24) to hold it must be the case that
Sign[xH1 − xH
2 ] = −Sign[xS1 − xS
2 ].
Thus, unless both spend identical effort in issue H and also in issue S, can-
didates must specialize. Identical efforts in an issue can, however, only be an
equilibrium if θH = θS. To see this consider (2.22) and (2.23). If both spend iden-
tical effort on any issue, it follows from (2.27) and (2.28) that the right hand side
is zero. The left hand side, however, is then only zero if θH = θS. Hence, a no
84 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES
specialization equilibrium can only occur if there are no comparative advan-
tages. To see that there can exist a no-specialization equilibrium look at the
first order conditions. If θH = θS and candidates do not specialize, cH − cS = 0.
Thus, what is left is equivalent to what we have seen when salience does not
matter for perceived importance. Existence of a symmetric equilibrium then
follows immediately. If in addition θw = 1/2, the ’prize’ in each competence
contest is identical and thus a perfectly symmetric equilibrium exists.
Proof of Proposition 2.4.3
Proof. Consider the system of first order conditions in (2.10) - (2.13). I already
established that for m1 = m2 there is specialization whenever θH 6= θS. Note
that the comparative statics of the equilibrium can be derived by totally dif-
ferentiating the (2.10) - (2.13) and using Cramer’s rule. Since all derivatives
are continuous, a marginal change in ∆, evaluated at ∆ = 0, also must have a
marginal effect on efforts. Effort differences in the specialization equilibrium
are, however, discrete. Hence, there must exist a neighborhood of ∆ = 0 in
which there is still specialization as it was before.
To show that for ∆ too large one candidate spends more on both issues,
look at the limit when ∆ → ∞. Candidate 2 will then abstain from the contest
or spend only marginally, since the slope of f (x) at zero is also infinite. This is
independent of the effort spent by 1. Hence, candidate 2’s effort converges to
zero as m2 grows larger and larger. What now is candidate 1’s best response?
He clearly spends strictly positive effort on both issues. The issue he has a
comparative advantage in he clearly wants to promote. The other issue, since
initial spending has a very large impact on perceived competence, but his costs
are relatively low compared to candidate 2, he also spends and he spends more
than his opponent. To put it more technical, since the slope of w is finite by
assumption, the indirect cost of advertising the week issue are bounded for
either candidate. The total marginal costs are those indirect costs of making
the weak spot salient and the direct marginal costs of effort. As the direct
marginal costs of effort increase at one point the difference in marginal costs
outweighs the indirect costs of making the comparative disadvantage salient,
such that for the candidate with lower costs of effort in both issues marginal
costs are lower. Hence, this must also be true for the marginal products of
effort, implying this candidate must spend more on both issues. Hence, for ∆
too large there cannot be specialization anymore.
APPENDIX 85
Proof of Proposition 2.4.4
Proof. Consider a symmetric, no-specialization equilibrium when θH = θS, θw>
12 , and ∆1 = 0. Then both candidates spend more effort on issue H. Now con-
sider a marginal change in θS. Since the first order conditions are continuous in
all variables this marginal change must have a marginal impact on efforts, too.
Effort differences between issues were, however, discrete, and hence a marginal
change cannot overcome this. Hence, there must exist a neighborhood around
the original value of θS for which both candidates spend more on issue H. In
this neighborhood one candidate must have a comparative advantage in S.
Chapter 3
“Where Ignorance is Bliss, ’tis Folly tobe Wise”: Transparency in Contests
Philipp Denter, John Morgan, and Dana Sisak†
†“Where ignorance is bliss, ’tis folly to be wise” is taken from from Thomas Gray’s (1768) poem“Ode on a Distant Prospect of Eaton College”. Many thanks to Stefan Bühler, Catherine Roux, ErnestoDal Bó, Qiang Fu, Martin Kolmar, Jochen Mankart, Johannes Münster, Steve Slutsky, Uwe Sunde,Steven Tadelis, Felix Várdy, Iván Werning, the participants of seminars in Berkeley, St. Gallen, andTübingen, of the 2009 SSES Congress, the 2009 Silvaplana Workshop on Political Economy, the 2009Young Researchers Workshop on Contests and Tournaments, the 2010 Public Choice Society meetingas well as the 2010 APET meeting. The kind hospitality of UC Berkeley and UC Irvine is gratefullyacknowledged. Dana Sisak and Philipp Denter both gratefully acknowledge the financial support ofthe Swiss National Science Foundation. All errors remain our own.
88 CHAPTER 3. TRANSPARENCY IN CONTESTS
3.1 Introduction
On March 20, 2009, U.S. president Barack Obama released a presidential memo-
randum on the subject of ensuring responsible spending of Recovery Act funds.
In this he promises to disclose all lobbying contacts on the distribution of Re-
covery Act funds within three business days to the public. In reaction to this,
on April 7, 2009 the Sunlight Foundation, a nonprofit, non-partisan organi-
zation promoting government openness and transparency, presented its own
proposal for real-time lobbying disclosure on their blog.1 After meeting with a
lobbyist, the government agency immediately submits a summary of the meet-
ing details through a standardized platform, and the results are accessible to
the general public on the internet. Instead of learning about them every quarter
year, journalists as well as the public will have an immediate basis to evaluate
the decisions of policymakers and the influence they were facing. But this is
not the only effect of increased transparency. Real-time disclosure also directly
informs the competing lobbyists about their opponent’s interests and doings.
In this paper we show how this can have bad consequences. Lobbying compe-
tition can become more fierce and less efficient.
This paper addresses the following questions. What information policy is
optimal, if a competitor in a contest can decide and commit to acquire relevant
information about his rival or disclose his own private information to the rival?
Do the competitors agree on information transmission? What is the effect of
mandatory disclosure policy on the outcome of competition? Our main results
are:
• Strong transparency policy in a competitive environment can have detri-
mental side effects for society. We identify conditions where it leads to
increased competition and less efficient outcomes.
• Decentralizing information disclosure instead is often beneficial. We iden-
tify conditions where the competing groups will agree to transparency
decisions, benefiting both the competitors and society at large.
• When outcomes are very sensitive to (lobbying) expenditures (e.g. luck
and outside factors become less important), decentralized agreement be-
comes unlikely. In these circumstances, neither mandatory disclosure nor
a laissez-faire transparency rule are optimal.
1http://sunlightfoundation.com/blog/2009/04/07/a-vision-of-real-time-lobbying-disclosure/.
3.1. INTRODUCTION 89
Our main results may be illustrated through the following simple example:
Two competitors are vying for some prize. One of them (the incumbent) has a
known valuation for the prize while the valuation of the other (the newcomer)
is (potentially) unknown, and may be either high or low. The key intuition
underlying all of the results stems from the following observation: Competi-
tion is fiercest when the two rivals have similar valuations and milder when
valuations diverge. Consider first the decision to acquire information. While
better information helps the incumbent to choose an optimal effort level, if the
decision to acquire information is revealed, then the newcomer will also re-
spond. When the incumbent has a relatively high valuation, he is better off not
acquiring information since, if this information reveals that his opponent has
a high valuation, competition is sharpened while if the opponent is revealed
to have a low valuation, then the incumbent can no longer credibly commit to
deter his opponent through overinvestment. Thus, information acquisition is
unambiguously bad. On the other hand, when the incumbent has a relatively
low valuation, acquiring information is beneficial as it reduces the efforts of the
opponent regardless of valuation—in the case of high valuation, it stems from
the revealed divergence of values while in the case of low valuation, it stems
from discouragement.
Now, consider the decision of the newcomer to disclose information. If the
newcomer faces an incumbent with a relatively high valuation, competition will
be fierce if he discloses a high valuation and mild when his value is revealed to
be low. Since not disclosing leads to an intermediate level of competition, low
valuation newcomers prefer to reveal while high valuation ones do not. The
reverse is true when the newcomer faces a relatively weak incumbent: high val-
uation newcomers prefer disclosure while low valued ones prefer opacity. How
does this translate into a newcomer’s ex ante disclosure policy? His expected
payoffs are dominated by how he fares when he has a high valuation since this
raises both the benefits and chances of winning the contest. As a result, the
optimal policy is to disclose when the incumbent has a relatively high value
and to remain opaque when the incumbent has a relatively low value.
This means that the competing parties agree on disclosure when the value
of the incumbent is relatively high, and on non-disclosure otherwise. Thus, a
central insight to emerge from this analysis is that, despite the fact that the
two sides have opposing interests in that both want to win, they agree that less
“effort”, ceteris paribus, is good. Since information sharing affects the degree
of competition, there is scope for agreement. Furthermore information sharing
not only influences the degree of competition but also the efficiency in allocat-
90 CHAPTER 3. TRANSPARENCY IN CONTESTS
ing the prize to the party who values winning most. Agreement on reduced
competition often also leads to greater efficiency in allocating the prize. When
information sharing is optimal, it results in greater separation in the efforts of
the two parties and, as a result, the prize is awarded to the higher valued party
more often. Likewise, when information sharing is not optimal, it again results
in greater separation of efforts. Thus, endogenous information sharing leads
to ex ante Pareto gains. In this circumstance, mandatory disclosure policies can
increase wasteful competition and distort prize allocations.
Consider some other examples of competitive environments in which trans-
parency policy is relevant. In the U.S., transparency in political campaigning
is regulated by the Federal Election Campaign Act (FECA). It requires candi-
dates to disclose sources of campaign contributions and campaign expenditure
quarterly. Not only is the public opinion affected by disclosure of this infor-
mation but also the campaign decisions of competing candidates and hence
competition. Disclosure of campaign contributions and expenditures conveys
information about the depth of financial support of a candidate and this in
turn influences the decisions of the opposing candidates and hence the election
outcome. This paper suggests mandating transparency can make candidates
compete more fiercely and thus competition more wasteful. Or consider com-
petition between firms. In the U.S., the Securities and Exchange Commission
(SEC) as well as the Federal Accounting Standards Board (FASB) regulate firms’
disclosure of financial information. This information is not only accessible by
stakeholders of a firm but also by its competitors, which has implications for
competition between firms if private information is revealed. Our results shed
light on how mandatory disclosure influences competition in winner-take-all
markets, or more generally markets where competition can be represented by
a contest. This is for example the case in advertising intensive markets, like the
market for soft drinks.
The chapter is organized as follows. Next we survey the related literature.
Section 3.2 introduces the model. Section 3.3 studies information acquisition,
Section 3.4 studies disclosure incentives. Section 3.5 puts the two decisions
together. Section 3.6 considers a more general contest success function and
Section 3.7 discusses the effect of mandatory disclosure policy. Section 3.8
studies the robustness of our findings with respect to the discriminatoriness of
the competition. Section 3.9 concludes.
Literature Review The nearest antecedent to our paper is Kovenock et al. (2010),
who study information disclosure between firms when the contest outcome is
3.1. INTRODUCTION 91
very sensitive to contest expenditures. Our concerns are with both information
disclosure and acquisition and how they relate to the sensitivity of the contest
outcome to expenditures. Baik and Shogren (1995) study the effects of spying
and information acquisition in contest games. To gain tractability, they abstract
away from strategic considerations in the expenditures themselves – essentially,
the contest game is decision-theoretic. Our analysis, however, highlights the
importance of the strategic interaction between acquisition/disclosure and con-
test expenditures. Indeed, our main result is driven by the fact that acquisition
changes the behavior not just of the party gaining new information but also the
party whose information was disclosed.
Information acquisition and/or disclosure decisions have been studied in
three different but complementary settings to ours: Cournot and Bertrand com-
petition, auctions and agency theory. Vives (1984), Li (1985), Shapiro (1986) and
Darrough (1993) amongst others study information transmission in the context
of Cournot and Bertrand competition. With contests we add a third possible
form of competition between firms. Other papers, e.g. Persico (2000) or Eso
and Szentes (2007) have analyzed the incentives to acquire or disclose infor-
mation either about one’s private value or about a common value in auction
settings. With our analysis of an all-pay auction we complement this literature,
while adding a different dimension with the analysis of non-fully discriminat-
ing contests. One of our main results is to show that it can be optimal for a
lobbying group or firm to remain ignorant about the valuation its rival places
on “winning” the contest. The strategic value of ignorance has also been shown
in the context of agency theory. A principal may benefit from ignorance as it
alters the agent’s incentives to exert effort. The agent may benefit as well, as
ignorance may make it harder for the principal to extract rents. Papers high-
lighting these effects are for example Dewatripont and Maskin (1995), Barros
(1997) and Kessler (1998). While this literature focusses on vertical relationships
between two distinct parties, in our model the focus is on competing parties in
a horizontal relationship.
Information transmission from lobbies to the policy maker through lobby-
ing has been studied for example by Potters and van Winden (1992), Lagerlöf
(2007) and Grossman and Helpman (2001). The focus of this literature is on
the welfare implications of lobbying when lobbyists have private information
which is relevant to the policy maker and the policy maker attempts to learn
by observing lobbying expenditures. In contrast we focus on information trans-
mission between lobbyists and its implications for welfare and efficiency, and
highlight consequences for disclosure policy.
92 CHAPTER 3. TRANSPARENCY IN CONTESTS
Information disclosure has also been studied in the context of goods mar-
kets, e.g. Jovanovic (1982), Milgrom (2008) and Daughety and Reinganum
(2008), where the focus is on whether markets lead to optimal incentives for
firms to disclose information about the quality of their goods. This literature
revolves around the trade-off that disclosure is beneficial for the consumer but
costly to the seller. In contrast, we show that mandatory disclosure can be
harmful even without direct monetary costs, purely through its strategic effect.
Finally, our paper is of course also related to the literature on asymmet-
ric information in contests (e.g. Hurley and Shogren (1998), Katsenos (2009),
Moldovanu and Sela (2001) or Hernandez-Lagos and Tadelis (2011)), and the
role of commitment in contests (e.g. Dixit (1987), Morgan (2003), Baik and
Shogren (1992), Morgan and Várdy (2007), Yildirim (2005) and Fu (2006)), al-
though the form of commitment typically consists of committing to a sequence
of moves. In contrast we study contests where players are able to commit to
certain informational regimes.
3.2 The Model
While we couch the model in the context of lobbying, it is easily translated
into other competitive situations.2 Consider two lobbying groups i = A, B who
vie for favorable legislation to be passed. Success yields lobby i a value vi
while failure yields zero. To affect the chances of success, each group chooses
lobbying effort xi. The chance that i is successful depends on the contest success
function (CSF):
pi
(xi, xj
)=
xi
xi + xj. (3.1)
If both groups choose zero lobbying effort (xi = 0) a coin toss determines suc-
cess. Lobbyists are risk-neutral with a constant marginal cost of effort normal-
ized to one. While each lobbying group knows its own valuation for success,
information about the other party differs. In particular, the valuation of group
A is commonly known while group B has private information about its value.
One can think of this situation arising when group A is an “incumbent” who
2We can easily reframe our model in terms of another introductory example – political campaigns.Two politicians i = A, B are campaigning for a political office. The political office yields i a value vi
while failure yields a value normalized to zero. To affect the chances of success, each politician choosessome amount of campaign expenditures xi. The chance that i is successful depends on the contestsuccess function (CSF) defined in equation 3.2. The talent of the incumbent politician is more or lesscommon knowledge and hence his value for office vA is known. For the newcomer we assume thevalue is low with probability q and high else.
3.3. INFORMATION ACQUISITION 93
has engaged in many past fights over related issues while group B is a new-
comer or, alternatively, where publicly available information makes it easy to
estimate A’s value while B’s value, perhaps being more subjective, is harder for
outsiders to estimate. For simplicity, we assume that B’s value is binary—it is
either low, vB = vL, with probability q or high, vB = vH , with the complemen-
tary probability. In Appendix 3.9, we show that qualitatively similar results are
obtained when B’s distribution of values occurs on a continuum. The payoff
functions are equal to
πB =xB
xB + xAvB − xB
πA =
(q
xA
xBL + xA+ (1 − q)
xA
xBH + xA
)vA − xA.
We focus on the case where there is uncertainty as to which lobbying group
has the higher valuation, i.e., when vA ∈ [vL,vH]. Furthermore we assume that
the policy is valuable enough for all lobbying groups to choose strictly positive
lobbying effort.
3.3 Information Acquisition
In this section we consider the incentives to acquire information about one’s
opponent before the contest. In terms of our model, suppose that it were cost-
less for group A to acquire a credible report as to B’s valuation before the start
of the contest and this decision is common knowledge. Afterwards the contest
described in Section 3.2 takes place. One might be tempted to draw an analogy
with a bargaining situation. In effect, A and B are negotiating (through their
efforts) on who will receive the valuable legislative prize. The usual advice
in such situations is to “know thy enemy”. That is, group A should gather
as much information as possible about group B, including its valuation. This
information will enable it to make the best possible decision regarding its ne-
gotiation strategy, which can now be type-specific. Since information gathering
is costless, it seems obvious that the optimal strategy is complete information
gathering.
Where the analogy breaks down is in the form of the “negotiation” be-
tween the two parties. Here, success will be determined by performance in an
imperfectly discriminating contest; thus, there is an integrative as well as dis-
tributive aspect to the “negotiation.” In particular, both lobbying groups benefit
if lobbying efforts are more muted and, since only relative lobbying efforts de-
94 CHAPTER 3. TRANSPARENCY IN CONTESTS
termine the outcome, equilibrium success probabilities would be unaffected if
both sides could agree to scale down their efforts.
But how can ignorance enable the lobbying groups to scale down effort?
Consider a lobbying group A which has a valuation above the average of lob-
bying group B. If it knew for sure it faces a strong group B, competition be-
tween the similarly strong groups would be very intense. But the chance to
encounter a much weaker group B diminishes A’s investment incentive, and
hence also the strong group B’s reaction because from its view investments are
strategic complements. On the other hand, A overinvests against a weak group
B to increase its chances in case its opponent turns out to be strong. The weak
group B will react to this discouragement by lowering its investment because
its investments are strategic substitutes. By optimally choosing to remain igno-
rant about lobbying group B’s valuation, A can on the one hand discourage a
weaker rival and on the other hand appease a stronger rival, thereby softening
the competition between the two lobbies. Thus, unlike a decision-theoretic or
negotiation context, rent-seeking competition between the two parties creates a
value to ignorance.
A sharp illustration of this intuition may be seen for the case where group
A has diffuse priors (i.e. q = 1/2). Here we show that, when group A is strong
compared to B, it prefers to remain ignorant while when it is weak, it seeks
information to mitigate this disadvantage. Formally,
Proposition 3.3.1. If lobbying group A is relatively strong compared to group B (vA >√vLvH) it strictly prefers not to acquire any information about B’s value while a
relatively weak lobbying group A (vA <√
vLvH) always acquires costless information
about group B.
Proof. See appendix.
Figure 3.1 illustrates the intuition behind the value to ignorance graphically.
It shows the best response functions of both groups when A knows the valu-
ation of group B. Optimal lobbying expenditures under complete information
are given where the best response functions intersect. If group A’s value is
relatively high, its lobbying effort under ignorance (vertical line) is higher than
under complete information in case it faces the low value opponent (left panel),
while the opposite is true against the high value opponent (right panel). We
can directly see that this benefits A by decreasing both its opponents’ lobbying
efforts.3
3Technically speaking, our results are due to the non-monotonicity of reaction functions. This
3.3. INFORMATION ACQUISITION 95
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.5
1.0
1.5
2.0
2.5
3.0
xBxB
xAxA
xA(xB)xA(xB)
xB(xA;vH)
xB(xA;vL)
xAIAxAI
A
Figure 3.1: The left panel shows the full-information best response functions whenlobbying group A faces a weak opponent, the right panel when it faces astrong opponent. xAI
A denotes the lobbying effort of A under ignorance.Under ignorance (dot) both types of B expend less than under full infor-mation (square).
Softening competition through ignorance does not always work. If group
A’s valuation is below the geometric mean of vB, ignorance increases compe-
tition. A weak group A invests little when facing a much stronger group B
while it fights hard against the just slightly weaker group B, where competi-
tion is more equal. By staying ignorant A finds itself overinvesting in case it
faces the stronger group B, which reacts to this threat with an increase in in-
vestment. At the same time it underinvests in case it faces the weak group B,
which also reacts with an increase in investment, sensing a good opportunity.
Hence a weak lobbying group A always acquires costless information.
Note that if group A’s decision to acquire information were not observable
to group B, A would always choose to acquire information about B’s value.
Deviating from ignorance to information acquisition enables A to play a best
response while B does not change its behavior as the deviation is unobservable.
In equilibrium this is anticipated by group B and the contest always takes place
under complete information. In this sense observability is a form of commit-
ment opportunity that enables A to commit to a beneficial action which would
otherwise not be feasible, as it is not in its complete-information best response.
implies that efforts are strategic complements for the favorite while they are strategic substitutes forthe underdog, where in our set-up the favorite is the group with the higher valuation. See Dixit (1987)for a discussion.
96 CHAPTER 3. TRANSPARENCY IN CONTESTS
In fact, commitment to ignorance can have a similar effect as pre-commitment
of effort. If group A had the opportunity to be a Stackelberg leader, meaning
it could pre-commit its contest effort in a way observable to B, it would choose
to overinvest relative to simultaneous moves against a lower-valued rival while
it would choose to underinvest against a higher-valued rival. Both rivals react
to this precommitment with a decrease in investment (Dixit (1987)).
3.4 Information Disclosure
Lobbying group A’s decision to stay ignorant could well be obsolete if group
B can credibly disclose its value to A. In fact, it is not clear what happens if A
and B disagree about whether B’s value should be revealed. In this section we
explore the other side of the information transmission decision and focus on
group B’s incentives to disclose its valuation to A. There are many possibilities
how disclosure could work. As a first step we assume that lobbying group B
has the opportunity to commit ex-ante, before learning its value, to a disclosure
policy. In case it chooses to disclose, it discloses its value truthfully and without
cost to A after learning it and before the start of the contest. In this sense we
give B a commitment opportunity to maximize its ex-ante welfare. At the end
of this section we discuss this assumption and analyze an alternative model
where B can only use a costly signal to signal its value to A.
Even though disclosure enables the opponent to make a more informed
decision, this does not necessarily mean that the disclosing group is hurt by
this. For example if the opponent learns that the group has a much higher
valuation it will optimally react by lowering its expenditures, as its chances of
success are so slim, and this is beneficial for both groups. On the other hand, if
the opponent learns the lobbying group has a very low valuation, it might also
find it beneficial to lower its expenditures, as not much is needed for success.
Disclosing a similar valuation on the other hand makes competition fiercer.
If the disclosure decision is made ex-ante, we find that information is only
disclosed when B faces a relatively weak group A. Formally,
Proposition 3.4.1. Assume lobbying group B does not know its value yet but is given
the opportunity to commit ex-ante to a disclosure policy. If lobbying group B expects to
be relatively weak compared to lobbying group A (√
vLvH < vA) it strictly prefers to
commit to non-disclosure. On the other hand, a lobbying group B with a high expected
valuation (√
vLvH > vA) always commits to disclose its value.
Proof. See appendix.
3.4. INFORMATION DISCLOSURE 97
To make the intuition behind Proposition 3.4.1 clearer let us first look at the
incentives of a high- and a low-value lobbying group B separately. A high-value
lobbying group B will prefer disclosure if it can discourage lobbying group
A from expending lobbying effort. This is the case whenever it is relatively
strong, or vA <√
vLvH . For vA ≥ √vLvH disclosing makes A more aggressive,
as it learns that its opponent is of similar strength. The opposite is true for a
weak lobbying group B. When facing a strong group A it prefers to disclose its
valuation, as A will react with lower lobbying effort. If A is weak on the other
hand, revealing its valuation makes competition stronger, as A learns that it is
facing a similarly strong opponent. The weak and the strong lobbying group
B’s incentives are never aligned. If disclosing is beneficial for one, it is harmful
to the other. From an ex-ante point of view, before learning its valuation, the
strong lobbying groups’ interests always dominate though. The reason is that
an increase in success probability in case the value is high is worth more than
in case the value turns out to be low.
Notice that the conditions for information disclosure/withholding in Propo-
sition 3.4.1 are identical to those in Proposition 3.3.1 when group A is determin-
ing whether to pursue this information. That is, despite competing with one
another, both groups agree on information revelation. We formalize this obser-
vation in Corollary 3.5.1 in Section 3.5.
Ex-ante commitment to a disclosure policy is an interesting benchmark but
might not always be feasible. Also costless and truthful revelation can be an
unrealistic assumption in some settings. To test the robustness of our results,
we consider an alternative model. Let us assume that disclosure of the lobbying
group’s value is costly and not verifiable. Instead, a lobbying group has the
option of sending a costly signal in order to try to inform A of its value. We
assume that costs of the signal si are linear, c(si) = si, i = H, L, and signaling
takes place before the start of the contest. Then we find:
Proposition 3.4.2. After lobbying group B learns its valuation and given the chance
to send a costly signal before the contest to group A, only a high-value lobbying group
credibly reveals its valuation. This is only profitable in a situation where group A is
relatively weak (√
vLvH > vA). Otherwise no information is disclosed.
Proof. See appendix.
The intuition for this proposition carries over from the one for Proposition
3.4.1. A lobbying group with a high valuation stands to gain more from a
decrease in A’s lobbying effort. This means that it is willing to expend more
98 CHAPTER 3. TRANSPARENCY IN CONTESTS
�������� ����
��� � �� ����������
���������� �
��������
��� ��� ��������
�� ������ �� ����
�� ���������� ��
��������� �
�������� ������ ��
������ �� �������
�� ������ �
t
t = 1 t = 2 t = 3
Figure 3.2: Sequence of moves.
signaling effort than a low-value group. If it is in its interest, it will always be
able to imitate a low-value group’s signal so that no information is disclosed.
Hence against a strong group A information will never be disclosed because
it is detrimental to the high-value group, while against a weak group A the
high-value group is willing to credibly disclose its valuation through the costly
signal. Our results are in line with the results in Katsenos (2009) who analyzes
costly signaling in a lottery contest with two-sided asymmetric information
and two possible types of valuations, vH and vL for both parties. He finds
that separating equilibria only exist, when the probability to face a strong op-
ponent is sufficiently low. Our result complements this finding in a one-sided
asymmetric information setting where vA can be different from vH and vL.
3.5 Information Transmission
So far we have analyzed the lobbying groups’ disclosure and acquisition deci-
sions separately. Now we combine these analyses to find out, how lobbying
groups exchange information voluntarily. In Section 3.7 we then compare our
findings to lobbying under mandatory disclosure policy.
The game proceeds as follows: Prior to the start of lobbying, each lobbying
group engages in information disclosure/acquisition decisions; that is, group
A decides whether to pursue credible information about B’s valuation while
group B simultaneously decides on its disclosure policy. Following information
acquisition/disclosure, both lobbying groups simultaneously choose lobbying
efforts and payoffs are resolved. Figure 3.2 illustrates the flow of the game.
We assume that lobbying group B has not learned its valuation when decid-
ing on information disclosure. In Proposition 3.4.2 we showed that our results
extend to an alternative set-up where B has learned its valuation and has the
possibility to send a costly signal to group A. Then if both lobbying groups
agree that information should be exchanged (B prefers disclosure and A acqui-
sition) A will learn the value of group B. If on the other hand both lobbying
3.5. INFORMATION TRANSMISSION 99
groups agree not to disclose (B prefers non-disclosure and A ignorance), no
information is transmitted. What is not so clear is what happens if A and B do
not agree. For example A might want to acquire information about B’s value,
but B might not be willing to disclose it. Or B might want to disclose its value
while A does not want to acquire it. The payoff in these situations which we
denote by πDi , could be equal to πCI
i , or πAIi or anything in between depending
on how exactly information transmission works. For our results in this section
we do not need to make any assumption as to what exactly will happen in
these cases as long as πDi ≤ max
{πCI
i ,πAIi
}.
Consider again the case with group A having diffuse priors (i.e. q = 1/2).
Then the lobbying groups always agree on information transmission between
them. Formally,
Corollary 3.5.1. If lobbying group B expects to be relatively weak compared to lobbying
group A (√
vLvH < vA) both lobbying groups agree not to transfer any information
while if lobbying group B expects to have a high valuation compared to A (√
vLvH >
vA) both agree on disclosure.
Proof. This follows from the proof of Propositions 3.3.1 and 3.4.1. There we
found that πCIi > πAI
i for vA <√
vHvL and πCIi < πAI
i for vA >√
vHvL, i = A, B.
For vA =√
vHvL both groups are indifferent. We have the following payoff
matrix.B discloses B doesn’t disclose
A acquires πCIA ,πCI
B πDA ,πD
B
A doesn’t acquire πDA ,πD
B πAIA ,πAI
B
Depending on πDi , multiple Nash equilibria are possible. For example even
though πCIi > πAI
i , i = A, B in case vA <√
vHvL, staying ignorant and not dis-
closing is a Nash equilibrium when information is only transferred if both
parties agree(πD
i = πAIi
). This equilibrium though is Pareto dominated by the
one where A acquires information and B discloses. In this sense the lobbying
parties, given a chance to coordinate, would always agree on the Pareto supe-
rior equilibrium. This is also the only trembling hand perfect equilibrium. Note
that this is also the unique equilibrium when parties can decide sequentially
on information transmission.
We find the lobbying groups’ incentives to be always aligned.The reason
for this is that there exist gains from coordination in the form of reduced com-
petition. By coordinating, both parties can save on lobbying expenditures.
This finding can also be related to the literature on sequential moves and
pre-commitment of effort in contests. Corollary 3.5.1 is in a sense analogous to
100 CHAPTER 3. TRANSPARENCY IN CONTESTS
the findings in Baik and Shogren (1992) and Leininger (1993), who analyze the
choice of the order of moves in sequential rent-seeking contests. They find that
it is in the interest of both lobbying groups to choose the sequence of moves
where the least efforts are expended. This means that both groups always pre-
fer the weak group to go first and pre-commit contest effort. It chooses a low
lobbying effort and the strong group reacts with lower lobbying effort as well.
Even though the weak group ends up winning less often, it is compensated
by lower lobbying costs. When choosing whether to disclose a similar logic
applies. Staying ignorant can have a similar effect as moving first, if it enables
A to move closer to its Stackelberg point. As we have shown, this is the case
for a relatively strong lobbying group A. By staying ignorant it can credibly re-
duce its investment against the high-valuation lobbying group B who will react
by reducing its expenditures as well. Interestingly in this set-up the strategic
complementarities from facing a high-valued rival always dominate, and hence
agreement is possible, even though efforts are strategic substitutes for the low-
valued lobbying group B.
Our results require very little structure in determining how exactly infor-
mation transmission works. The only essential prerequisite is some form of
commitment opportunity. In reality, this could take many forms. For exam-
ple, one purpose of trade associations is to facilitate information exchange (e.g.
Kirby (1988) or Vives (1990)). Members commit themselves to share their pri-
vate information with the help of the trade association, while for non-members
it will be much harder to reveal and receive credible information. Another
example of institutionalized information exchange are strategic marriages. A
strategic marriage policy was pursued by many houses of European rulers dur-
ing the Renaissance and thereafter. The probably best known example is the
House of Habsburg’s strategic marriage to Spain and Italy. Among other things
these strategic marriages can serve as commitments to disclose credible infor-
mation to and acquire credible information about other empires. Another nice
historical example about the voluntary exchange of credible information can be
found in Schelling (1960), “[t]he ancients exchanged hostages, drank wine from the
same glass to demonstrate the absence of poison, met in public places to inhibit the mas-
sacre of one by the other, and even deliberately exchanged spies to facilitate transmittal
of authentic information”. Our analysis provides a rationale for this: exchanging
authentic information can decrease the fierceness of conflict, something that is
good for both parties.
3.6. MORE GENERAL CONTEST SUCCESS FUNCTION 101
3.6 More General Contest Success Function
So far we have assumed that the lobbying process can be represented by a
simple lottery contest. In order to show the robustness of our results, in this
section we assume the political process can be represented by a more general
CSF of the following form:
pi
(xi, xj
)=
f (xi)
f (xi) + f(
xj
) (3.2)
where f ′ > 0 and f ′′ ≤ 0.4
As we have seen in the previous section, whether ignorance is bliss for the
lobbying groups is determined by whether or not group A’s value is above the
average of group B’s valuations. Proposition 3.3.1 shows though, that it is not
the arithmetic average; rather the decision to acquire or disclose information
turns on the geometric mean of B’s value. Next we show that such a criti-
cal value of lobbying group A’s valuation, let us denote it by vA, exists more
generally.
Lemma 3.6.1. For every q, there exists a value vA ∈ [vL,vH] such that, if vA = vA,
lobbying group A is indifferent between acquiring information or not, and lobbying
group B is indifferent between disclosing information or not.
Proof. See appendix.
To illustrate the intuition for the proof of this lemma, assume A knows
its opponent. When A faces a weak opponent B, a relatively small lobbying
effort will basically guarantee success for A. With an increase in B’s value,
A increases its optimal lobbying effort until both groups have an equal value.
Here competition is at its fiercest. Now an increase in B’s value will start to
discourage A from investing, until at one point B becomes so strong that A
invests barely anything. This logic implies that there will always be two possi-
ble values of group B, one larger than A’s, one smaller, such that A expends
exactly the same lobbying effort. If group B has exactly these values, vL and
vH , A’s behavior will be unchanged whether it knows B’s value or not.
It is tempting to reason from Lemma 3.6.1 that Propositions 3.3.1 and 3.4.1
hold for more general prior probabilities of B’s values vL and vH and more
general lobbying technologies. Indeed, we can generalize Propositions 3.3.1
and 3.4.1 as well as Corollary 3.5.1 locally around the critical value vA.
4This is a standard contest success function, see Skaperdas (1996) for an axiomatization.
102 CHAPTER 3. TRANSPARENCY IN CONTESTS
Proposition 3.6.1. In a neighborhood of vA, if lobbying group B expects to be relatively
weak compared to lobbying group A (vA < vA) both lobbying groups agree not to
transfer any information while if lobbying group B expects to have a high valuation
compared to A (vA > vA) both agree on disclosure.
Proof. See appendix.
Is there a reason why Proposition 3.6.1 might not always hold globally, as
does Corollary 3.5.1? It can be shown that under certain circumstances there
can be disagreement between the lobbying groups. The reason is that the criti-
cal value vA for Lemma 3.6.1 is not always the only critical value for group A.
To illustrate, take a very strong lobbying group A with a value close to vH and
assume that the probability of facing a strong group B is small. Then group
A’s lobbying effort under ignorance is similar to the lobbying effort knowing it
is facing a weak group B. But if B happens to be strong and A were ignorant,
it would underinvest by a large amount. Even though this leads the strong
group to reduce its effort, this is not optimal for group A. In fact, there is
an optimal degree of underinvestment against a stronger opponent. If A had
the opportunity to precommit lobbying effort, this would be the effort level it
would optimally choose, the so-called Stackelberg point. Ignorance can enable
lobbying group A to move closer to this optimal effort in certain situations. In
other situations A will surpass the Stackelberg point under ignorance, as in
the example above. If A surpasses the Stackelberg point by too much, acquir-
ing information is the optimal strategy. Consequently, there exist situations
like the one described above where the two lobbying groups will not agree on
information transmission.
3.7 Mandatory Disclosure Policy
Transparency policy is a topic of high relevance in many political debates
around the world. For example in the U.S., transparency laws have been passed
regulating lobbying, political campaigning or financial accounting of firms. A
large part of the U.S. economy is hence affected through transparency laws.
Thus it is important to understand all possible consequences of mandatory dis-
closure policy. In competitive environments like the ones mentioned above,
transparency policy can affect the nature and outcome of competition. Here
we take a closer look at exactly this effect. In the previous section we saw that
typically the competitors agree on whether to disclose information between
themselves. In many cases they agree not to disclose any information to their
3.7. MANDATORY DISCLOSURE POLICY 103
mutual benefit. Transparency policy, on the other hand, forces the competing
parties to disclose certain information to the public, and hence also to their
competitors.
We focus our analysis on two outcome variables: expected aggregate lob-
bying efforts and expected allocative efficiency. It is typically in the interest of
a society to keep lobbying efforts low, since lobbying activities are not directly
productive but serve only to influence policy. In our model this is captured
by the fact that by scaling down efforts proportionally both groups still win
the contest with identical probability. This decrease in lobbying investment can
be used for directly productive activities. Of course, in a frictionless world
one could argue that markets would always allocate these funds efficiently. In
reality, this is certainly not always the case. Furthermore there is also a misal-
location of non-monetary resources, as for example human capital, and hence
reducing lobbying efforts seems a reasonably aim. It is also in the interest of
a society to have the probability that a law or bill which has a relatively high
social value be passed as large as possible. This social value is represented in
our analysis by the lobbying groups’ valuations. We implicitly assume that all
individuals affected by the policy are part of one of the two lobbying groups,
for example a “pro” and a “contra” group. Inside each group there are no trans-
action costs and no externalities, and thus the groups’ valuations for the policy
perfectly reflect societal preferences. This can be seen as an approximation for a
situation where both groups face similar free-rider problems.5 Consequently it
is in society’s interest that a higher valued lobbying group has the best chances
to succeed. We will refer to this as expected allocative efficiency henceforth.
Mandatory disclosure policy can take many different forms, ranging from
disclosure of information about actions (e.g. expenditures or efforts) to disclo-
sure of characteristics (e.g. valuations, costs or productivity), or any mix thereof.
Depending on its form, mandatory disclosure will impact competition to differ-
ent degrees. In this paper we consider disclosure policy about the competitors’
characteristics. Transparency about actions can reveal something about char-
acteristics, but does not necessarily have to (in technical terms there can be
pooling or separating equilibria). Thus our analysis also applies to disclosure
about actions, whenever information about characteristics is revealed.
Let us resume the example of lobbying introduced in Section 3.2 and first
assume that we are interested in keeping the expected wastefulness of the lob-
bying competition low and it is irrelevant for society which lobbying group is
5Free-rider problems in group contests with public goods prizes are discussed for example in Este-ban and Ray (2001) or Kolmar and Rommeswinkel (2010).
104 CHAPTER 3. TRANSPARENCY IN CONTESTS
successful. This could for example be the case in rent-seeking contests. We
then get the following result.
Proposition 3.7.1. Expected aggregate effort is lower under
• information disclosure if lobbying group A is relatively weak (vA ≤√vHvL),
• asymmetric information if lobbying group A is relatively strong (vA >√
vHvL).
Proof. See appendix.
As foreshadowed in Section 3.5 we find that if the uninformed lobbying
group is relatively strong, mandatory information disclosure makes the lobby-
ing process more wasteful in expectation. In addition, we have shown in Corol-
lary 3.5.1 and Proposition 3.6.1 that in many situations the lobbying groups
voluntarily agree not to transfer any information. In these cases a “laissez-
faire” policy leads to less wasteful competition. If we assume that information
can only be transferred when it is in lobbying group B’s interest to disclose its
information, we can conclude the following.
Corollary 3.7.1. If society is interested in keeping lobbying expenditures low a “laissez-
faire” policy is preferable to a policy of mandatory disclosure.
Next we consider expected allocative efficiency. We define expected al-
locative efficiency as the probability that the lobbying group with the highest
valuation wins the lobbying contest. Then we can show
Proposition 3.7.2. Expected allocative efficiency is greater under
• information disclosure if lobbying group A is relatively weak (vA ≤√vHvL),
• asymmetric information if lobbying group A is relatively strong (vA >√
vHvL).
Proof. See appendix.
This finding also relates to the literature on sequential contests. As we
discussed in Section 3.3, asymmetric information enables the uninformed lob-
bying group to act similar to a Stackelberg leader when it is sufficiently strong
relative to the informed lobbying group. Morgan (2003) finds that sequential
rent-seeking contests dominate simultaneous ones in terms of efficiency. Hence
if asymmetric information enables A to get closer to its Stackelberg point, which
is true for vA >√
vHvL, it also improves efficiency. Together with the results in
Corollary 3.5.1 and Proposition 3.7.1 we find the following.
3.7. MANDATORY DISCLOSURE POLICY 105
Corollary 3.7.2. Assume that society is interested in increasing expected allocative
efficiency and keeping expected wastefulness of the lobbying competition low. Then a
“laissez-faire” policy is always weakly superior, independent of the relative weights the
policy maker places on the two goals.
With a completely altruistic policy maker, transparency is clearly beneficial
for efficiency. Only if it is known which policy is the best, can it be chosen by
the policy maker. If the policy maker follows his self-interests and bases his
decision on lobbying efforts, transparency will have a differential effect on the
lobbying groups, sometimes favoring the “weaker”, sometimes the “stronger”
one. As we have shown, this can lead to another undesirable side-effect of trans-
parency policy, a decrease in expected allocative efficiency. At the same time,
our result has the potential to explain the emergence of mandatory disclosure
policies, even though shown to be inefficient. A policy maker interested in max-
imizing his rent-seeking revenues always weakly prefers mandatory disclosure
to voluntary disclosure.
Furthermore, note that disclosure policy which does not affect current lob-
bying competition, in other words disclosure with a sufficient time lag or with
“soft” disclosure requirements which do not reveal anything about the com-
petitors’ characteristics, does not have these detrimental effects. At the same
time it can still afford possible benefits through increased accountability and
better informed voters. In this respect our findings help evaluate calls for an
increase in transparency, as for example by the Sunlight Foundation in the U.S..
Coming back to our introductory example, the demand for real-time lobbying
disclosure, our findings imply that even apart from the direct costs of increased
transparency such as bureaucratic expenses, this policy is likely to have indi-
rect costs in terms of an increase in expected wastefulness and a decrease in
expected allocative efficiency of lobbying competition, which have to be traded
off against the additional benefits.6
6There may be another negative effect of transparency, not captured in our model. Higher trans-parency makes direct transfers of funds from lobbying groups to policy makers less likely, because thiswould be considered bribery or corruption, which is typically illegal. Of course, this does not meanthat lobbying groups stop exerting pressure. Rather they (partially) substitute away from transfers tolegal sources of effort, which are usually labor intensive. But this has direct negative consequencesfor efficiency and wastefulness of the competition. While bribing is purely distributive and thereforefunds are not “wasted”, labor intensive lobbying directly wastes resources and hence is an allocativeproblem. Therefore, it can be argued from a wastefulness perspective that bribery has an advantageover lobbying, what is in line with for example Lambsdorff (2002). Consequently, transparency may notonly increase lobbying effort, but is likely to influence the composition of lobbying effort in a sociallyundesirable way.
106 CHAPTER 3. TRANSPARENCY IN CONTESTS
3.8 Noisiness of the Contest and the Scope for Agree-
ment
So far we have implicitly assumed that lobbying expenditures do not perfectly
determine the outcome of the competition. By spending more in the contest a
lobbying group can increase its chances to succeed, but there always remains
some uncertainty. Put differently, the lobbying group with the lower expendi-
tures still has a non-zero chance of success – the lobbying process is at least
somewhat noisy. There are different reasons this might be true. For example,
policy makers may have preferences over political outcomes unknown to the
lobbying groups, or face imperfectly observable constraints. Another reason
for a noisy lobbying process from the lobbying groups’ perspective is that lob-
bying efforts are only imperfectly observable by the policy maker. This could
be due to the complexity of the subject so that it is difficult for lobbyists to com-
municate their concerns properly, or because it is not clear ex-ante what the
best strategy to approach a political decision maker is and which consequences
of the favored bill to highlight.
We have captured this uncertainty by using a non-deterministic CSF of the
ratio form, as defined in equation (3.2). We now consider a CSF which can
be interpreted as the limiting case when noise vanishes completely, the all-pay
auction. It represents a situation where the political process is very sensitive
to lobbying effort and where the lobbying group with the highest expenditure
wins with certainty.7 This higher sensitivity implies higher marginal returns to
lobbying effort and therefore increases the fierceness of the competition. It is
interesting to consider this situation as an extreme case, because it is implicitly
assumed that policy makers do not have any private preferences about the po-
litical outcomes, do not face any constraints and the process of communication
between the lobbying groups and the policy maker is free of misunderstand-
ings and noise. In short, the policy maker bases his decision solely on lobbying
expenditures. The next proposition shows how an absence of noisiness influ-
ences the incentives to coordinate on information transmission.
Proposition 3.8.1. When the political process takes the form of an all-pay auction
1. disclosing information is weakly dominated for lobbying group B,
2. staying ignorant is weakly dominated for lobbying group A,
7The standard references analyzing all-pay auctions are Hillman and Riley (1989), Baye et al. (1993,1996), and Krishna and Morgan (1997).
3.8. NOISINESS OF THE CONTEST AND THE SCOPE FOR AGREEMENT 107
3. the lobbying groups’ incentives are never aligned and therefore they will never
agree on transferring information voluntarily.
Proof. See appendix.8
This result reveals that the contest’s degree of sensitivity to rent-seeking ef-
forts influences when the lobbying groups agree on information transmission.
In contrast to ratio form contests, in a fully discriminating contest the lobbying
groups’ incentives are never aligned. The informed group never discloses its
information while the uninformed group always takes an opportunity to ac-
quire information. Because of the fierceness of competition there is no scope
for agreement.
Consider the lobbying groups’ incentives separately. Why does lobbying
group B never benefit from disclosing its valuation? Under a noisy political
process, by disclosing its value, a strong group B discourages a weak group A
from investing. This does not work when the political process is fully discrim-
inating. By disclosing information, a strong lobbying group will only secure
itself a payoff equal to the difference in valuations between itself and its op-
ponent. All other rents are dissipated through competition. With asymmetric
information competition is less fierce and it can in addition earn informational
rents. In fact, it can secure itself the exact same payoff with one-sided asym-
metric information (by marginally overbidding group A’s valuation) and might
even do better. Technically speaking, in all-pay auctions both reaction functions
are monotonically increasing until the valuation of the weakest lobbying group
so there will be no discouragement effect in the relevant range.
Why is there no value to ignorance? When policy makers are perfectly re-
sponsive to lobbying expenditures, there is no advantage to pre-committing lob-
bying expenditures, as has been shown for example in Konrad and Leininger
(2007). In fact, a low-valuation lobbying group is indifferent with respect to tim-
ing while a high-valuation group prefers to decide after its opponent chooses
its expenditures. Hence the advantage from ignorance highlighted under an
imperfectly discriminating political process does not apply this setting — igno-
rance cannot dampen competition to the benefit of both parties, it only benefits
the opponent. Hence lobbying group A always acquires information.
What are the consequences for disclosure policy? First of all, Proposition
3.8.1 shows that lobbying groups don’t agree on disclosure and hence it is no
8A proof for part 2 of the Proposition has first been given in Kovenock et al. (2010) for two-sidedasymmetric information and a continuous distribution of types.
108 CHAPTER 3. TRANSPARENCY IN CONTESTS
a.) b.)
0.0 0.2 0.4 0.6 0.8 1.0
1.0
1.2
1.4
1.6
1.8
2.0
q
vA
0.0 0.2 0.4 0.6 0.8 1.0
1.0
1.2
1.4
1.6
1.8
2.0
q
vA
Figure 3.3: Comparison of aggregate effort (panel a.)) and efficiency (panel b.)).
longer clear what happens under a laissez-faire transparency rule. Further-
more, a reduction in expected aggregate effort and an increase in expected
allocative efficiency, two possible objectives of society, are no longer necessar-
ily compatible as we show now in an example. We find that expected aggre-
gate effort is typically smaller under complete information when A’s value is
not too close to either vH or vL and under asymmetric information else. Ex-
pected allocative efficiency is typically greater under asymmetric information
except if vA is relatively small and q is relatively large. The reason is the fol-
lowing. Asymmetric information has two effects on allocative efficiency when
the policy maker is perfectly responsive to lobbying expenditures. On the one
hand it stratifies the range of efforts of lobbying group B. A low-valuation
group chooses its investment from an interval of the form [0, x] while the high-
valuation group chooses from [x, x]. In contrast, under complete information
they choose from the interval [0, xi], i = H, L. This is beneficial for efficiency. On
the other hand we showed that lobbying group B benefits from informational
rents. Especially when A is very likely to face a low-valuation opponent and vA
is close to vL, this becomes important for efficiency. B’s informational advan-
tage will lead to a low-valuation type winning too often, decreasing efficiency.
In theses cases the detrimental effect of asymmetric information dominates and
expected allocative efficiency is higher under complete information.
Figure 3.3 illustrates this for vL = 1 and vH = 2. In darkgray regions com-
plete information is optimal while in lightgray regions asymmetric information
is preferred. So decreasing expected aggregate effort often implies decreasing
expected allocative efficiency. We can draw the following conclusions regarding
mandatory and voluntary disclosure policy.
Corollary 3.8.1. Policy makers who are perfectly responsive to the influence of lob-
3.9. CONCLUSION 109
byists make decentralized agreement impossible. In these circumstances, neither a
laissez-faire transparency rule nor mandated disclosure is optimal in our framework.
Furthermore, achieving an increase in expected allocative efficiency and a decrease in
expected aggregate effort through disclosure policy becomes unlikely as these two goals
are often in conflict.
Summarizing our results, we find differential effects of transparency policy
on lobbying competition depending on the noisiness of the political process.
While under a sufficiently noisy political process a laissez-faire policy leads to
the best outcome in terms of expected aggregate effort as well as allocative effi-
ciency, this need not be true under a perfectly discriminating political process.
Here the effect of transparency policy is ambiguous and no general results can
be obtained to guide policy decisions.
3.9 Conclusion
How do we evaluate the recent proposals for more transparency in U.S. lobby-
ing? If transparency were free to implement, would more transparency always
be better for society? Even though we cannot give a conclusive answer to these
questions, our analysis highlights a side-effect of transparency policy which has
been absent from the policy debate so far. We show how an increase in trans-
parency can lead to an increase in the wastefulness of lobbying competition and
at the same time to a decrease in the probability that the lobbying group with
the most pressing interests succeeds. Furthermore we show that in the absence
of mandatory disclosure policy, competitors often agree whether or not to share
information and this decision reduces wastefulness and increases allocative ef-
ficiency. Our results have implications beyond lobbying. These considerations
hold weight for the analysis of transparency policy in other competitive settings
like political campaigning or financial accounting of rival firms.
While we focused in our assessment of the welfare implication of trans-
parency on an environment in which effort is considered wasteful, there are
other environments in which effort is considered (socially) beneficial. An im-
mediate example is student’s effort in school or at university. Higher effort
generates better educated graduates, which is beneficial for society as a whole.
Typically grades are based on relative performance (grading on a curve), so
students’ competition for grades is a contest and we can apply our results. We
know from Section 3.7 that transparency leads in expectation to increased effort.
Consequently, to increase students’ efforts a transparent studying environment
110 CHAPTER 3. TRANSPARENCY IN CONTESTS
is likely to be helpful. This can be achieved by promoting studying in groups
or by testing students frequently over the term and publicizing the test scores.
An interesting extension of our analysis would be to allow for common
values. This can be relevant in many settings. In our lobbying example the
lobbyists might posses relevant information about the value of the policy at
stake, as for example when lobbying for a monopoly position and each firm has
done market research. Lobbying groups learn not only about their opponent’s
interest, but also about their own. Most importantly, to draw more precise
policy conclusions a more general model of all affected parties is needed to
evaluate all the possible effects of transparency policy and their interactions.
For example transparency policy in lobbying will also affect the relationship
between the policy maker and the general public. To combine these factors
into one model is an important avenue for future research and will allow a
more thorough evaluation of transparency policy.
APPENDIX 111
Appendix
Proof of Propositions 3.3.1, 3.4.1 and 3.4.2
Equilibrium under Full- and Asymmetric Information. Equilibrium efforts, prob-
ability of success and utility under complete information are equal to (see Nti
(1999))
xCIi
(vi,vj
)=
v2i vj
(vi + vj
)2(3.3)
pCIi
(vi,vj
)=
vi
vi + vj
πCIi
(vi,vj
)=
v3i(
vi + vj
)2.
It is easily verified that A will invest more against a high-value opponent than
against a low-value one iff vA >√
vHvL. Let ∆v :=√
vH − √vL. Under one-
sided asymmetric information, effort, probability of success, and utility in an
interior solution are:
xAIA (vA,vL,vH) =
vLvHv2A ((1 − q)
√vL + q
√vH)
2
(vHvL + vA ((1 − q)vL + qvH))2
xAIH (vA,vL,vH) =
((1 − q)√
vL + q√
vH)vAvH√
vLvH
(vHvL + vA ((1 − q)vL + qvH))2
× (√
vHvL + qvA∆v)
xAIL (vA,vL,vH) =
((1 − q)√
vL + q√
vH)vAvL√
vLvH
(vHvL + vA ((1 − q)vL + qvH))2
× (√
vLvH − (1 − q)vA∆v)
pAIA (vA,vL,vH) =
vA ((1 − q)√
vL + q√
vH)2
(vHvL + vA ((1 − q)vL + qvH))
pAIH (vA,vL,vH) = 1 − vA ((1 − q)
√vL + q
√vH)
(vHvL + vA ((1 − q)vL + qvH))
√vL
pAIL (vA,vL,vH) = 1 − vA ((1 − q)
√vL + q
√vH)
(vHvL + vA ((1 − q)vL + qvH))
√vH
πAIA (vA,vL,vH) =
v3A (q
√vH + (1 − q)
√vL)
2 (qvH + (1 − q)vL)
(vA (qvH + (1 − q)vL) + vLvH)2
112 CHAPTER 3. TRANSPARENCY IN CONTESTS
πAIH (vA,vL,vH) =
(qvAvH∆v + v3/2
H vL
)2
(qvA(vH − vL) + vL(vA + vH))2
πAIL (vA,vL,vH) =
(v3/2
L (vA(1 − q) + vH)− (1 − q)vA√
vHvL
)2
(qvA(vH − vL) + vL(vA + vH))2.
Acquiring Information. Let us consider lobbying group A’s incentives to ac-
quire information. The difference in expected utility is equal to
∆πA =(1 − q)qv3
A (√
vH −√vL)
2 (vA −√vH
√vL)
(vA + vH)2(vA + vL)2(qvA (vH − vL) + vAvL + vHvL)2
×((vH − vL)q
(v3
A − 3v2A
√vLvH − vAvHvL − v3/2
H v3/2L
)
−2qvA√
vLvH
(v2
H − v2L
)+ v3
AvL − 3v2A
√vHv3/2
L − 4vAv3/2H v3/2
L
−vAv2HvL − 2vA
√vHv5/2
L − 2vAvHv2L − v5/2
H v3/2L
−2v3/2H v5/2
L − 2v2Hv2
L
)
For vA <√
vHvL, A clearly prefers to acquire information, while for vA =√vHvL it is indifferent. For vA slightly larger than
√vHvL it prefers ignorance
while for vA approaching vH it might prefer to acquire information again. This
implies we have to be careful about staying in an interior solution, in other
words we need vL ≥ (1−q)2v2AvH
((1−q)vA+vH)2 or vA ≤ vH√
vL
(1−q)(√
vH−√vL)
.
Let q = 12 . Then the difference in utility for group A between complete-
information and asymmetric information is equal to
∆πA |q= 12
=v3
A (√
vH −√vL)
2 (vA −√vH
√vL)
2(vA + vH)2(vA + vL)2(vAvH + vAvL + 2vHvL)2
×((vH + vL)
(v3
A − 3v2A
√vH
√vL − 3vAvHvL − 3v3/2
H v3/2L
)
−2vA√
vH√
vL
(v2
H + 4vHvL + v2L
)− 4v2
Hv2L
)
We can show that this is unambiguously positive for vA <√
vLvH and negative
for vA >√
vLvH given that we are in an interior solution. For vH > 9vL the
condition for an interior solution is binding. So for vH < 9vL vA can be as
high as vH . Let us plug this into the expression in brackets: v4H − 5v7/2
H
√vL −
14v5/2H v3/2
L − 5v3/2H v5/2
L − 2v3HvL − 7v2
Hv2L. This is clearly strictly negative for all
vH < 9vL. For vH > 9vL we insert the highest possible vA into the expression in
APPENDIX 113
brackets carries the sign of:
−(
4v3/2H − 7vH
√vL + v3/2
L
)
which is always negative for vH > 9vL.
Disclosing Information. To see whether group B prefers to disclose or not it is
sufficient to look at group A’s effort difference between full and asymmetric
information. Since less investment of the opponent is strictly preferred given a
fixed investment, it is even more so, if B can in addition optimally react. If A
invests more under complete information against B, B will clearly prefer asym-
metric information. Define ∆xi := xCIi − xAI
i , i = AH, AL. Then the difference
in A’s effort is equal to
∆xAH =v2
AvH
(vA + vH)2− vLvHv2
A ((1 − q)√
vL + q√
vH)2
(vHvL + vA ((1 − q)vL + qvH))2
=qv2
Av3/2H (
√vH −√
vL) (vA −√vH
√vL)
(vA + vH)2 (vHvL + vA ((1 − q)vL + qvH))2
× (qvA√
vH√
vL + qvAvH + 2(1 − q)vAvL
+ qv3/2H
√vL + (2 − q)vHvL
)
∆xAL =v2
AvL
(vA + vL)2− vLvHv2
A ((1 − q)√
vL + q√
vH)2
(vHvL + vA ((1 − q)vL + qvH))2
= −(1 − q)v2Av3/2
L (√
vH −√vL) (vA −√
vH√
vL)
(vA + vL)2 (vHvL + vA ((1 − q)vL + qvH))2
×((1 − q)
(vA
√vH
√vL + vAvL +
√vHv3/2
L
)
+ 2qvAvH + qvHvL + vHvL) .
At vA =√
vLvH A’s effort is identical, while for vA >√
vLvH A underinvests
against a high-value opponent and overinvests against a low-value one under
asymmetric information. The opposite holds true for vA <√
vLvH . Hence it
follows that for vA >√
vLvH a high-value B prefers not to disclose, while a
low-value one prefers disclosure and vice versa for vA <√
vLvH . Now let us
consider the ex-ante expected utility of group B when it has not yet learned its
value. Define ∆πi := πCIi − πAI
i , i = H, L, B. Then
114 CHAPTER 3. TRANSPARENCY IN CONTESTS
E[∆πB] = q∆πL + (1 − q)∆πH
=−(1 − q)qvA (
√vH −√
vL)2 (vA −√
vH√
vL)
(vA + vH)2(vA + vL)2(qvA (vH − vL) + vAvL + vHvL)2
×[(
v2H − v2
L
)qv2
A
(v2
A + vA√
vHvL + 4vHvL
)
+ qvA
(2v2
AvHvL + 2vAv3/2H v3/2
L + 2v2Hv2
L
)(vH − vL)
+ v4Av2
L + v3A
(2v3/2
H v3/2L + 2v2
HvL +√
vHv5/2L + 4vHv2
L + 2v3L
)
+ vA
(4v3
Hv2L + 6v2
Hv3L + 3v5/2
H v5/2L
)
+ v2A
(2v5/2
H v3/2L + 4v3/2
H v5/2L + 2v3
HvL + 7v2Hv2
L + 6vHv3L
)
+ 2v3Hv3
L + 2qv3A
(v3
H − v3L
)].
Hence, for vA =√
vLvH group B is also indifferent in expectation whether to
disclose or not, while for vA >√
vLvH it prefers not to disclose and for vA <√vLvH disclosure is optimal.
Signaling of Valuation. Now lobbying group B has the possibility to expend
money before the contest in order to signal its valuation. To show whether and
when a separating equilibrium exists, consider the following set up. Each group
L and H can send a costly signal to A before the contest, which we denote
by si. The signal is completely unproductive and only serves the signaling
objective. We assume signaling costs are c(s) = s for both groups. The game
has a separating equilibrium when it is possible for L to send a signal which H
does not want to mimic and vice versa.
We first look at vA >√
vLvH . In this situation we know from the above
discussion that L prefers complete information, while H is better off under
asymmetric information. Hence, L would like to signal its type and H would
like to hinder it by mimicking its behavior by setting sH = sL. A’s beliefs are
the following: that any signal sB ≥ sL indicates B has identity L, otherwise
B has identity H. Because the signal is costly individual rationality implies
sB ∈ {0, sL}. In a separating equilibrium we must have that sL = sL and sH = 0.
That is, each group’s incentive compatibility (IC) constraint has to hold and no
group has an incentive to mimic the behavior of the other. The respective IC
APPENDIX 115
constraints are
ICL :v3
L
(vL + vA)2− sL ≥ vL −
2vA√
vLvH
vA + vH+
v2AvH
(vA + vH)2,
ICH :v3
H
(vA + vH)2≥ vH − 2vA
√vHvL
vA + vL+
v2AvL
(vA + vL)2− sL.
It is easily shown that it is not possible to find sL > 0 fulfilling both inequalities
simultaneously. Hence, there does not exist a separating equilibrium when
vA >√
vLvH , and as a result no information is transferred and both groups
engage in an incomplete information contest.
Now turn to vA ≤ √vLvH . In this case, it is H who wants so signal its
identity to overcome incomplete information, while L wants to hinder it. A
believes it is facing H in the contest whenever the signal is sB ≥ sH . Otherwise
it believes it is facing L. Individual rationality implies now sB ∈ {0, sH}. In a
separating equilibrium we must have sL = 0 and sH = sH . The respective IC
constraints are now
ICL :v3
L
(vL + vA)2≥ vL −
2vA√
vLvH
vA + vH+
v2AvH
(vA + vH)2− sH , (3.4)
ICH :v3
H
(vA + vH)2− sH ≥ vH − 2vA
√vHvL
vA + vL+
v2AvL
(vA + vL)2. (3.5)
It is now easily verified that there exists a range of signals sH for which both
inequalities hold simultaneously. From the intuitive criterion (Cho and Kreps,
1987) it follows that the equilibrium value of sH makes L exactly indifferent
between mimicking H or not, so that (3.4) holds with equality. Then we have
s+H = vL +v2
AvH
(vA + vH)2− v3
L
(vL + vA)2− 2vA
√vLvH
vA + vH> 0.
Also, if sH = s+H the beliefs of group A are correct and therefore there exists a
separating equilibrium. Note, however, that all values sH > s+H also support a
separating equilibrium as long as (3.5) still holds. Therefore, we proved that
a separating equilibrium with endogenous information transmission exists if
and only if vA ≤√vLvH , which proves the proposition.
Proof of Lemma 3.6.1
To see this, first note that (i) reaction functions are hump-shaped and (ii) reach a
maximum where xA = xB, i.e. where the reaction function crosses the 45 degree
116 CHAPTER 3. TRANSPARENCY IN CONTESTS
line (for a proof see Yildirim (2005)). Moreover, we find an equilibrium on this
line exactly when vA = vB, i.e. when the game is symmetric. Let us denote
complete-information symmetric efforts for vA = vL by xL and for vA = vH by
xH . Keeping the valuation of the opponent fixed, a group’s effort is strictly
increasing in its own valuation. So let vA increase from vL to vH . Then the
effort of the L-value type is strictly decreasing (strategic substitute) and the
effort of the H-value type is strictly increasing (strategic complement). If the
opponent is of the L-value type, xA increases from xL to some xHL > xL. To
the contrary, if the opponent is of the H type xA increases from some xLH < xL
to xH . Note that xH > xHL > xL > xLH , i.e. if the opponent is of the H-value
type A’s effort is at the beginning lower and at the end higher compared to the
L-value type. Accordingly, by continuity there has to be some vA ∈ (vL,vH) for
which efforts against both types of the other group are identical and equal to
xA.
If vA = vA group A will spend the same lobbying effort in the complete
information games and in the asymmetric information game in equilibrium.
Accordingly, both types of group B will choose the same effort independent of
the informational environment, implying A’s costs and winning probabilities
are identical and thus A is indifferent between both information regimes. �
Proof of Proposition 3.6.1
Proof. We showed in Lemma 3.6.1 that at vA = vA both groups are indifferent
between complete information and asymmetric information. We now prove
also Proposition 3.6.1. To do this we need to analyze the derivative of both
groups’ difference in utilities between complete and asymmetric information.
We derive some preliminary results concerning effort comparative statics at
vA = vA under both informational arrangements with respect to changes in vA.
We then use these results to prove first the information acquisition part of the
proposition and then also information disclosure.
Preliminaries. Here we derive some comparative statics results we need later
on. Because we do not have closed form solutions for equilibrium efforts we
totally differentiate the systems of first-order conditions and use Cramer’s rule.
Under complete information the system of first-order conditions is:
APPENDIX 117
∂πCIAL
∂xCIAL
|vA=vA=
∂pL(xCIAL, xCI
L )
∂xCIAL
vA − 1!= 0
∂πCIAH
∂xCIAH
|vA=vA=
∂pH(xCIAH , xCI
H )
∂xCIAH
vA − 1!= 0
∂πCIL
∂xCIL
|vA=vA= −∂pL(xCI
AL, xCIL )
∂xCIAL
vL − 1!= 0
∂πCIH
∂xCIH
|vA=vA= −∂pH(xCI
AH , xCIH )
∂xCIAH
vH − 1!= 0
Letting i = L, H, totally differentiating these first order conditions yields the
following matrix system:
∂2 pi
∂(xCIAi )
2 vA∂2 pi
∂xCIAi∂xCI
i
vA
− ∂2 pi
∂xCIi ∂xCI
Ai
vi − ∂2 pi
∂(xCIi )2 vi
︸ ︷︷ ︸=Ai
dxCIAi
dvAdxCI
idvA
=
(− ∂pi
∂xCIAi
0
)
Define
Ai1 =
− ∂pi
∂xCIAi
∂2 pi
∂xCIAi ∂xCI
i
vA
0 − ∂2 pi
∂(xCIi )2 vi
, Ai2 =
∂2 pi
∂(xCIAi)
2 vA − ∂pi
∂xCIAi
− ∂2 pi
∂xCIi ∂xCI
Ai
vi 0
.
From Cramer’s rule it follows that∂xCI
Ai∂vA
= |Ai1||Ai| as well as
∂xCIi
∂vA= |Ai2|
|Ai| in equilib-
rium. Hence,
∂xCIAL
∂vA|vA=vA
=−∂2 pL
∂x2L(
∂2 pL
∂x2A
∂2 pL
∂x2L
−(
∂2 pL∂xA∂xL
)2)
v2A
> 0, (3.6)
∂xCIAH
∂vA|vA=vA
=−∂2 pH
∂x2H(
∂2 pH
∂x2A
∂2 pH
∂x2H
−(
∂2 pH∂xA∂xH
)2)
v2A
> 0, (3.7)
∂xCIL
∂vA|vA=vA
=
∂2 pL∂xA∂xL(
∂2 pL
∂x2A
∂2 pL
∂x2L
−(
∂2 pL∂xA∂xL
)2)
v2A
< 0, (3.8)
118 CHAPTER 3. TRANSPARENCY IN CONTESTS
∂xCIH
∂vA|vA=vA
=
∂2 pH∂xA∂xH(
∂2 pH
∂x2A
∂2 pH
∂x2H
−(
∂2 pH∂xA∂xH
)2)
v2A
> 0. (3.9)
These comparative statics show how equilibrium efforts at vA = vA react to
changes in vA if there is complete information.
Under asymmetric information the system of first-order conditions is:
∂πAIA
∂xAIA
|vA=vA=
(q
∂pL(xAIA , xAI
L )
∂xAIA
+ (1 − q)∂pH(xAI
A , xAIH )
∂xAIA
)vA − 1
!= 0
∂πAIL
∂xAIL
|vA=vA= −∂pL(xAI
A , xAIL )
∂xAIL
vL − 1!= 0
∂πAIH
∂xAIH
|vA=vA= −∂pH(xAI
H , xAIH )
∂xAIH
vH − 1!= 0
Totally differentiating yields the following matrix system:
(q
∂2 pL
∂(xAIA )2 + (1 − q) ∂2 pH
∂(xAIA )2
)vA q
∂2 pL
∂xAIA ∂xAI
L
vA (1 − q) ∂2 pH
∂xAIA ∂xAI
H
vA
− ∂2 pL
∂xAIL ∂xAI
A
vL − ∂2 pL
∂(xAIL )2 vL 0
− ∂2 pH
∂xAIH ∂xAI
A
vH 0 − ∂2 pH
∂(xAIH )2 vH
︸ ︷︷ ︸=B
dxAIA
dvAdxAI
LdvAdxAI
HdvA
=
−q∂pL
∂xAIA
− (1 − q) ∂pH
∂xAIA
0
0
Define
B1 =
−q∂pL
∂xAIA
− (1 − q) ∂pH
∂xAIA
q∂2 pL
∂xAIA ∂xAI
L
vA (1 − q) ∂2 pH
∂xAIA ∂xAI
H
vA
0 − ∂2 pL
∂(xAIL )2 vL 0
0 0 − ∂2 pH
∂(xAIH )2 vH
B2 =
(q
∂2 pL
∂(xAIA )2 + (1 − q) ∂2 pH
∂(xAIA )2
)vA −q
∂pL
∂xAIA
− (1 − q) ∂pH
∂xAIA
(1 − q) ∂2 pH
∂xAIA ∂xAI
H
vA
− ∂2 pL
∂xAIL ∂xAI
A
vL 0 0
− ∂2 pH
∂xAIH ∂xAI
A
vH 0 − ∂2 pH
∂(xAIH )2 vH
APPENDIX 119
B3 =
(q
∂2 pL
∂(xAIA )2 + (1 − q) ∂2 pH
∂(xAIA )2
)vA q
∂2 pL
∂xAIA ∂xAI
L
vA −q∂pL
∂xAIA
− (1 − q) ∂pH
∂xAIA
− ∂2 pL
∂xAIL ∂xAI
A
vL − ∂2 pL
∂(xAIL )2 vL 0
− ∂2 pH
∂xAIH ∂xAI
A
vH 0 0
It follows again from Cramer’s rule that
∂xAIA
∂vA=
|B1||B| ,
∂xAIL
∂vA=
|B2||B| ,
∂xAIH
∂vA=
|B3||B| ,
and hence
∂xAIA
∂vA|vA=vA
= (3.10)
− ∂2 pH
∂x2H
∂2 pL
∂x2L(
∂2 pL
∂x2L(1 − q)
(∂2 pH
∂x2H
∂2 pH
∂x2A
−(
∂2 pH
∂xA∂xH
)2)+ ∂2 pH
∂x2H
q
(∂2 pL
∂x2L
∂2 pL
∂x2A
−(
∂2 pL
∂xA∂xL
)2))
v2A
> 0
∂xAIL
∂vA|vA=vA
= (3.11)
∂2 pL
∂xA∂xL
∂2 pH
∂x2H(
∂2 pL
∂x2L(1 − q)
(∂2 pH
∂x2H
∂2 pH
∂x2A
−(
∂2 pH
∂xA∂xH
)2)+ ∂2 pH
∂x2H
q
(∂2 pL
∂x2L
∂2 pL
∂x2A
−(
∂2 pL
∂xA∂xL
)2))
v2A
< 0
∂xAIH
∂vA|vA=vA
= (3.12)
∂2 pH
∂xA∂xH
∂2 pL
∂x2L(
∂2 pL
∂x2L(1 − q)
(∂2 pH
∂x2H
∂2 pH
∂x2A
−(
∂2 pH
∂xA∂xH
)2)+ ∂2 pH
∂x2H
q
(∂2 pL
∂x2L
∂2 pL
∂x2A
−(
∂2 pL
∂xA∂xL
)2))
v2A
> 0
Those comparative statics are the marginal change of equilibrium efforts under
asymmetric information if vA changes at vA.
Information Acquisition. We showed in Lemma 3.6.1 that if vA = vA group
A is indifferent between ignorance and complete information. To prove the
proposition we show that the derivative of the difference of utilities of A with
respect to vA is non-zero at vA = vA. Using
pi =f (xA)
f (xA) + f (xi)
120 CHAPTER 3. TRANSPARENCY IN CONTESTS
and xi = xiB, i = H, L to shorten the exposition, the derivative of ∆πA at vA is
equal to
∂∆πA
∂vA|vA=vA
=
[(1 − q)
(∂pH
∂xA
(∂xCI
AH
∂vA− ∂xAI
A
∂vA
)+
∂pH
∂xH
(∂xCI
H
∂vA− ∂xAI
H
∂vA
))
+ q
(∂pL
∂xA
(∂xCI
AL
∂vA− ∂xAI
A
∂vA
)+
∂pL
∂xL
(∂xCI
L
∂vA− ∂xAI
L
∂vA
))]vA
−((1 − q)
∂xCIAH
∂vA+ q
∂xCIAL
∂vA
)+
∂xAIA
∂vA.
We know that vA > 0, 0 < q < 1.∂pH∂xA
= ∂pL∂xA
= 1vA
and∂pL∂xL
=− 1vL
<∂pH∂xH
=− 1vH
<
0 follow from the first order conditions of the two groups. The derivative
simplifies to
∂∆πA
∂vA|vA=vA
= −((1 − q)
vH
(∂xCI
H
∂vA− ∂xAI
H
∂vA
)+
q
vL
(∂xCI
L
∂vA− ∂xAI
L
∂vA
))vA.
This derivative will only be zero if a change in vA induces the same effect on B’s
complete-information effort as on its asymmetric information effort, or if they
just offset each other for the two types weighted by the probability q and their
valuation. The relevant comparative statics were derived in equations (3.6),
(3.7), (3.11), and (3.12).∂2 pL
∂x2A
< 0,∂2 pH
∂x2A
< 0,∂2 pH
∂x2H
> 0 and∂2 pL
∂x2L
> 0 follow from the
shape of the CSF.∂2 pL
∂xAxL> 0 and
∂2 pH∂xAxH
< 0 come from the fact that at vA = vA
A is an underdog against an opponent with valuation vH but a favorite against
an opponent with valuation vL. Using this, the derivative of the difference in
utilities equals
∂∆πA
∂vA|vA=vA
= −
(∂2 pL
∂x2L
(∂2 pH
∂x2H
∂2 pH
∂x2A
−(
∂2 pH
∂xA∂xH
)2)+ ∂2 pH
∂x2H
((∂2 pL
∂xA∂xL
)2− ∂2 pL
∂x2L
∂2 pL
∂x2A
))
(∂2 pL
∂x2A
∂2 pL
∂x2L−(
∂2 pL
∂xA∂xL
)2)(
∂2 pH
∂x2A
∂2 pH
∂x2H−(
∂2 pH
∂xA∂xH
)2)
vA vH vL
×
∂2 pL
∂xA∂xLvH
(∂2 pH
∂x2A
∂2 pH
∂x2H−(
∂2 pH
∂xA∂xH
)2)
(∂2 pL
∂x2L(1 − q)
(∂2 pH
∂x2H
∂2 pH
∂x2A
−(
∂2 pH
∂xA∂xH
)2)+ ∂2 pH
∂x2H
q
(∂2 pL
∂x2L
∂2 pL
∂x2A
−(
∂2 pL
∂xA∂xL
)2))
+
∂2 pH
∂xA∂xHvL
((∂2 pL
∂xA∂xL
)2− ∂2 pL
∂x2A
∂2 pL
∂x2L
)q (1 − q)
(∂2 pL
∂x2L(1 − q)
(∂2 pH
∂x2H
∂2 pH
∂x2A−(
∂2 pH
∂xA∂xH
)2)+ ∂2 pH
∂x2H
q
(∂2 pL
∂x2L
∂2 pL
∂x2A−(
∂2 pL
∂xA∂xL
)2))
APPENDIX 121
which has the sign of
−(
∂2 pL
∂x2L
(∂2pH
∂x2H
∂2pH
∂x2A
−(
∂2 pH
∂xA∂xH
)2)− ∂2 pH
∂x2H
(∂2 pL
∂x2L
∂2pL
∂x2A
−(
∂2 pL
∂xA∂xL
)2))
.
This term relates∂xCI
AH∂vA
|vA=vAto
∂xCIAL
∂vA|vA=vA
. For∂xCI
AH∂vA
|vA=vA>
∂xCIAL
∂vA|vA=vA
it will
be negative and for∂xCI
AH∂vA
|vA=vA<
∂xCIAL
∂vA|vA=vA
it will be positive. For our CSF
given in equation (3.2) it will always be negative. This means that starting at
xLA = xH
A a slight increase in vA will lead to a relatively higher increase in effort
on the part of group A against the high-type opponent.9 Hence, we find that
at vA = vA the derivative of ∆πA is strictly negative. Thus, there exist some
valuations vA > vA where ignorance is bliss.
Information Disclosure. At vA = vA group B is exactly indifferent whether it
discloses its information or not, ex-ante as well as ex-interim, as group A always
chooses the same lobbying effort. Let us now vary vA marginally from there.
The derivative of the difference in the expected utility of player B between
complete information and asymmetric information with respect to vA at vA
can be written as
∂∆πB
∂vA|vA=vA
= (1 − q)
[vH
(−∂pH
∂xA
(∂xCI
AH
∂vA− ∂xAI
A
∂vA
)− ∂pH
∂xH
(∂xCI
H
∂vA− ∂xAI
H
∂vA
))
−(
∂xCIH
∂vA− ∂xAI
H
∂vA
)]
+ q
[vL
(− ∂pL
∂xA
(∂xCI
AL
∂vA− ∂xAI
A
∂vA
)− ∂pL
∂xL
(∂xCI
L
∂vA− ∂xAI
L
∂vA
))
−(
∂xCIL
∂vA− ∂xAI
L
∂vA
)]
=
((1 − q)vH
(∂xAI
A
∂vA− ∂xCI
AH
∂vA
)+ q vL
(∂xAI
A
∂vA− ∂xCI
AL
∂vA
))1
vA,
where we use pi =f (xA)
f (xA)+ f (xi)and xi = xi
B, i = H, L to shorten the exposition.
We know that vA > 0, 0 < q < 1.∂pH∂xA
= ∂pL∂xA
= 1vA
and∂pL∂xL
=− 1vL
<∂pH∂xH
=− 1vH
< 0
9Note that for more general CSF the opposite case can arise and A increases its effort more againstthe low-type opponent. Then there will be a value of ignorance for vA < vA.
122 CHAPTER 3. TRANSPARENCY IN CONTESTS
follow from the first-order conditions of the two groups. The relevant equilib-rium comparative statics of efforts were derived in equations (3.6), (3.7), and(3.10). Using these in the derivative yields
(∂2 pH
∂x2H
((∂2 pL
∂xA∂xL
)2− ∂2 pL
∂x2L
∂2 pL
∂x2A
)+ ∂2 pL
∂x2L
(∂2 pH
∂x2H
∂2 pH
∂x2A
−(
∂2 pH
∂xA∂xH
)2))
(∂2 pL
∂x2A
∂2 pL
∂x2L−(
∂2 pL
∂xA∂xL
)2)(
∂2 pH
∂x2A
∂2 pH
∂x2H−(
∂2 pH
∂xA∂xH
)2)
v3A
(3.13)
×
(∂2 pH
∂x2H
vH
((∂2 pL
∂xA∂xL
)2− ∂2 pL
∂x2A
∂2 pL
∂x2L
)+ ∂2 pL
∂x2L
vL
(∂2 pH
∂x2A
∂2 pH
∂x2H−(
∂2 pH
∂xA∂xH
)2))
q (1 − q)(
∂2 pL
∂x2L(1 − q)
(∂2 pH
∂x2H
∂2 pH
∂x2A
−(
∂2 pH
∂xA∂xH
)2)+ ∂2 pH
∂x2H
q
(∂2 pL
∂x2L
∂2 pL
∂x2A
−(
∂2 pL
∂xA∂xL
)2)) < 0,
where we use∂2 pL
∂x2A
< 0,∂2 pH
∂x2A
< 0,∂2 pH
∂x2H
> 0 and∂2 pL
∂x2L
> 0, which follow from the
shape of the CSF.∂2 pL
∂xAxL> 0 and
∂2 pH∂xAxH
< 0 come from the fact that at vA = vA
A is an underdog against an opponent with valuation vH but a favorite against
an opponent with valuation vL and
∂2pH
∂x2H
((∂2 pL
∂xA∂xL
)2
− ∂2pL
∂x2L
∂2 pL
∂x2A
)+
∂2pL
∂x2L
(∂2pH
∂x2H
∂2 pH
∂x2A
−(
∂2pH
∂xA∂xH
)2)> 0.
This term relates∂xCI
AH∂vA
|vA=vAto
∂xCIAL
∂vA|vA=vA
. For∂xCI
AH∂vA
|vA=vA>
∂xCIAL
∂vA|vA=vA
it will
be positive and for∂xCI
AH∂vA
|vA=vA<
∂xCIAL
∂vA|vA=vA
it will be negative. For our CSF
given in equation (3.2) it will always be positive. This means that starting at
xLA = xH
A a slight increase in vA will lead to a relatively higher increase in effort
on the part of group A against the high-type opponent. Hence, we find that at
vA = vA the derivative in (3.13) is strictly negative.
Putting together the information disclosure and information acquisition
part, the proof of the proposition follows from the proof of Corollary 3.5.1.
Proof of Propositions 3.7.1 and 3.7.2
Proof. Expected aggregate effort with contest success function pi =xi
xi+xjunder
complete information is equal to
E
∑
i={A,B}xCI
i
=
vA (((1 − q)vH + qvL)vA + vLvH)
(vA + vH) (vA + vL),
APPENDIX 123
while expected aggregate effort under one-sided asymmetric information is
equal to
E
∑
i={A,B}xAI
i
= ((1 − q)
√vH + q
√vL)
((1 − q) 1√
vH+ q 1√
vL
)
(1
vA+((1−q)
vH+ q
vL
)) .
Their difference is equal to
E[∑∆x
]=
vA (((1 − q)vH + qvL)vA + vLvH)
(vA + vH) (vA + vL)
− ((1 − q)√
vH + q√
vL)
((1 − q) 1√
vH+ q 1√
vL
)
(1
vA+((1−q)
vH+ q
vL
))
=(1 − q) qvA (
√vH −√
vL)2 (vA −√
vHvL) (vA (√
vHvL + vH + vL) + vHvL)
(vA + vH)(vA + vL)(qvA(vH − vL) + vL(vA + vH)).
It is easily observed that this is positive for vA >√
vHvL and negative otherwise,
hence, proving Proposition 3.7.1.
Efficiency implies that the informational regime should be chosen to maxi-
mize q xAxA+xL
+ (1 − q) xHxA+xH
as we assume vL ≤ vA ≤ vH . We get
∆
(q
xA
xA + xL+ (1 − q)
xH
xA + xH
)
= − (1 − q)qvA(vH − vL)(v2
A − vHvL
)
(vA + vH)(vA + vL)(qvA(vH − vL) + vL(vA + vH)),
which is positive for vA <√
vHvL and negative else.
Proof of Proposition 3.8.1
Proof. Full information strategies for a match with valuations vi > vj are given
by the bidding distribution functions
Fj(x;vj,vi) =vi − vj
vi+
x
vi
Fi(x;vi ,vj) =x
vj,
for x ∈ [0,vj], see Hillman and Riley (1989) or Baye et al. (1996). In the fol-
lowing let Fi(x;vj) indicate the bidding distribution of group i facing another
group j and denote the corresponding density function by fi(x;vj). The ex-ante
124 CHAPTER 3. TRANSPARENCY IN CONTESTS
expected complete information payoffs are
πCIH = vH − vA
πCIL = 0
πCIA = q (vA − vL) .
Those results are standard and the proofs can be found for example in Hillman
and Riley (1989) or Baye et al. (1996). Using the equilibrium strategies it is
easily verified that expected aggregate effort is equal to
XCI = q∫ vL
0( fA(x;vL) + fL(x;vA)) x dx
+ (1 − q)∫ vA
0( fA(x;vH) + fH(x;vA)) x dx
= q∫ vL
0
(x
vL+
x
vA
)dx + (1 − q)
∫ vA
0
(x
vA+
x
vH
)dx
=q
2
(v2
L
vA+ vL
)+
(1 − q)
2
(vA +
v2A
vH
)
and that expected allocative efficiency (the ex-ante probability that the player
with higher valuation wins) equals
EFCI = q∫ vL
0FCI
L (x;vA) f CIA (x;vL)dx + (1 − q)
∫ vA
0FCI
A (x;vH) f CIH (x;vA)dx
= q∫ vL
0
(vA − vL
vA+
x
vA
)1
vLdx + (1 − q)
∫ vA
0
(vH − vA
vH+
x
vH
)1
vAdx
= (1 − q)
(1 − vA
2vH
)+ q
(1 − vL
2vA
).
Under one-sided asymmetric information consider first the case where vA is
relatively small, vA ≤ vA ≡ vL
q+vLvH
(1−q). We then find that A’s bidding/effort dis-
tribution function has a mass point at zero. The groups’ equilibrium strategies
are given by the distribution functions
FAIA (x;vL,vH) =
{vH−(1−q)vA
vH− qvA
vL+ x
vLfor x ∈ [0,qvA]
vH−vAvH
+ xvH
for x ∈ [qvA,vA]
FAIL (x;vA) =
x
qvAfor x ∈ [0,qvA]
FAIH (x;vA) =
x − qvA
(1 − q)vAfor x ∈ [qvA,vA].
APPENDIX 125
That those distribution functions indeed characterize an equilibrium is easily
verified and we leave this to the reader (a proof is available upon request).
Equilibrium payoffs in this case are
πAIA = 0 < πCI
A = q (vA − vL)
πAIH = vH − vA = πCI
H
πAIL = vL
vH − (1 − q)vA
vH− qvA > πCI
L = 0.
A prefers complete information while B weakly prefers asymmetric informa-
tion – the L-type is better off while the H-type is indifferent.
Expected aggregate effort is equal to
XAIvA≤vA
= q∫ qvA
0
(f AIA (x;vL,vH) + f AI
L
)x dx
+ (1 − q)∫ vA
qvA
(f AIA (x;vL,vH) + f AI
H
)x dx
=∫ qvA
0
(x
vA+
x
vL
)dx +
∫ vA
qvA
(x
vA+
x
vH
)dx
=vA
(q2vA(vH − vL) + vL(vA + vH)
)
2vHvL,
and expected allocative efficiency equals
EFAIvA≤vA
= q∫ qvA
0FAI
L (x;vA) fA(x;vL)dx + (1 − q)∫ vA
qvA
FAIA (x;vH) fH(x;vA)dx
= q∫ q vA
0
x
q vA
1
vLdx
+ (1 − q)∫ vA
qvA
(vH − (1 − q)vA
vH− qvA
vL+
x
vL
)1
(1 − q)vAdx
=q2vAvH − (q − 1)vL[(q − 1)vA + 2vH]
2vHvL.
Now consider vA > vA = vL
q+vLvH
(1−q). Here only L’s effort distribution has a mass
point, which is at zero.
FAIA (x;vL,vL) =
{x
vLfor x ∈ [0, x]
xvH
+(
1 − (1−q)vAvH
)(1 − vL
vH
)for x ∈ [x, x]
FAIL (x;vA) =
x
qvA+ 1 − vL
qvA+
vL (1 − q)
qvHfor x ∈ [0, x]
FAIH (x;vA) =
x
(1 − q)vA+
vL
vH− vL
(1 − q)vAfor x ∈ [x, x],
126 CHAPTER 3. TRANSPARENCY IN CONTESTS
where x = vL − (1 − q)vAvLvH
and x = vL + (1 − q)vA
(1 − vL
vH
). The correspond-
ing expected equilibrium payoffs are
πAIA = qvA − vL +
(1 − q)vAvL
vH< πCI
A = q (vA − vL)
πAIH = vH − vL − vA (1 − q)
(1 − vL
vH
)> vH − vA = πCI
H
πAIL = 0 = πCI
L .
B prefers asymmetric information, since the H-type is better off while the L-
type is indifferent, whereas A prefers full information. Ex-ante expected aggre-
gate effort is equal to
XAIvA>vA
=∫ x
0
(f AIA (x;vL) + f AI
L (x;vA))
x dx
+∫ x
x
(f AIA (x;vL) + f AI
L (x;vA))
x dx
=vL(vA + vL)((q − 1)vA + vH)
2
2v2HvA
+(q − 1)(vA + vH)((q − 1)vA(vH − 2vL)− 2vHvL)
2v2H
and expected allocative efficiency equals
EFAIvA>vA
= q∫ x
0FAI
L (x;vA) fA(x;vL)dx + (1 − q)∫ x
xFAI
A (x;vL) fH(x;vA)dx
=vAvH
((q2 − 1
)vA + 2vH
)− vL((q − 1)vA + vH)
2
2vAvH2
.
To complete the proof note that when A and B disagree on information
transmission, we assumed their payoffs to be smaller than max{
πCIi ,πAI
i
}, the
exact value depending on how exactly information transmission works. There-
fore, the disclosure of information is weakly dominated for B and staying igno-
rant is weakly dominated for A.
Continuous uniform distribution
In this section we discuss the model when the type space is not binary but a
continuous uniform distribution. This gives us an idea as to how general our
results are.
APPENDIX 127
Let us assume that B’s value is distributed uniformly on [v,v], with vA ∈[v,v]. In case both lobbying groups know their respective valuations, equilib-
rium efforts are equal to
xCIi
(vi,vj
)=
v2i vj
(vi + vj
)2.
A’s expected utility under complete information is
πCIA =
∫ v
v
v3A
(vA + vB)2
dF(vB) =1
v − v
(v3
A
vA + v− v3
A
vA + v
),
while B gets in this situation
πCIB =
v3B
(vB + vA)2
,
and in expectation this equals
E[πCIB ] =
v3A
vA+v + 3v2A ln[vA + v]− 2vAv + v2
2 −(
v3A
vA+v + 3v2A ln[vA + v]− 2vAv + v2
2
)
v − v.
The expected utility of lobbying group A if it does not know the value of group
B
πAIA =
1
v − v
(∫ v
v
xA
xA + xB(vB)dvB
)vA − xA.
Taking the derivative and setting it equal to zero
∂πAIA
∂xA=
1
v − v
(∫ v
v
xB(vB)
(xA + xB(vB))2
dvB
)vA − 1
!= 0
we get A’s first order condition. Plugging this into group B’s reaction func-
tion xB(xA) = max{√
xAvB − xA,0}
we can solve for the equilibrium efforts.
Focussing on interior solutions we get the following equilibrium efforts:
xAIA =
2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
2
xAIB =
√√√√vB
2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)−
2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
2
128 CHAPTER 3. TRANSPARENCY IN CONTESTS
A and B’s equilibrium utility under one-sided asymmetric information is equalto
πAIA =
2vA(√
v−√v)
vA(ln[v]−ln[v])+(v−v)
v − v
(∫ v
v
1√vB
dvB
)vA −
2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
2
=
2vA(√
v−√v)
vA(ln[v]−ln[v])+(v−v)
v − v2(√
v −√v)
vA −
2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
2
πAIB =
√vBxA − xA√
vBxAvB −√
vBxA + xA = vB − 2√
xAvB + xA
= vB − 2
√√√√ 2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)vB +
2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
2
and B’s expected utility before it learns its type
E[πAIB ] =
v − v
2− 4
3
(v
32 − v
32
)√√√√ 2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
+
2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
2
.
Now we consider the incentives to disclose or acquire information. The
difference in utilities for A and B is equal to
∆πA =1
v − v
(v3
A
vA + v− v3
A
vA + v
)−
(2vA(√
v−√v))
2
vA(ln[v]−ln[v])+(v−v)
v − v
+
2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
2
,
∆πB =v3
B
(vB + vA)2− vB − 2
√√√√ 2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)vB
+
2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
2
.
Ex-ante, before B knows its valuation the difference in expected utility is equal
to
APPENDIX 129
1 2 3 4 5
1
2
3
4
5
v
vA
1 2 3 4 5
1
2
3
4
5
v
vA
1 2 3 4 5
1
2
3
4
5
v
vA
panel a) panel b) panel c)
Figure 3.4: Difference in expected utility for lobbying group A (panel a)) and B (panelb)) as well as zone of agreement (panel c).
∆E[πB] =
v3A
vA+v + 3v2A ln[vA + v]− 2vAv + v2
2 −(
v3A
vA+v + 3v2A ln[vA + v]− 2vAv + v2
2
)
v − v
− v − v
2+
4
3
(v
32 − v
32
)√√√√ 2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
−
2vA
(√v −√
v)
vA (ln[v]− ln[v]) + (v − v)
2
.
These expressions are quite unwieldy and Hence, we illustrate the equivalents
of Propositions 3.3.1 to 3.7.2 only graphically. Normalizing the lowest valua-
tion to one, v = 1, we plot the differences in utility for A as well as B (from an
ex-ante) between full and asymmetric information in Figure 3.4. v is plotted on
the abscissa while vA is on the ordinate. We plot only valuation pairs for which
an interior solution exists. In the lightgray regions the lobbying groups prefer
ignorance/non-disclosure, while in the darkgray region the lobbying groups
prefer to acquire/disclose information. If A is relatively weak, information dis-
closure is favorable for both players while if A is relatively strong both players
prefer asymmetric information exactly as in our baseline set-up in Section 3.2.
We find that players generally agree whether to disclose B’s valuation. In-
terestingly, only in a small region where A has an about average valuation, in
other words vA is close to E[vB], the players’ preferences diverge. In these cases
B prefers disclosure while A prefers to stay ignorant about B’s value. This can
be seen in panel c) of Figure 3.4.
To illustrate Propositions 3.7.1 and 3.7.2 we plot the difference in expected
130 CHAPTER 3. TRANSPARENCY IN CONTESTS
1 2 3 4 5
1
2
3
4
5
vv
A1 2 3 4 5
1
2
3
4
5
v
vA
panel a) panel b)
Figure 3.5: Difference in aggregate effort (panel a)) and expected allocative efficiency(panel b)).
aggregate effort and expected efficiency under complete and asymmetric in-
formation. Figure 3.5 illustrates these differences. In the darkgray region dis-
closure leads to lower expected aggregate effort or higher expected allocative
efficiency while in the lightgray region non-disclosure is preferable.
Overall we find that our results under a continuous uniform distribution
are remarkably similar to the ones under only two types of player B, vH and
vL.
Chapter 4
Imperfect Property Rights: The Role ofHeterogeneity and Uncertainty
Philipp Denter and Dana Sisak†
†We would like to thank Dirk Burghardt, Lukas Inderbitzin, Martin Kolmar, John Morgan, HendrikRommeswinkel, Philip Schuster, Stergios Skaperdas, Rudi Stracke, Darjusch Tafreschi and seminarparticipants at University of St.Gallen and University of California, Irvine for their insightful commentsand suggestions. The kind hospitality of UC Berkeley and UC Irvine is gratefully acknowledged. Bothauthors gratefully acknowledge financial support of the Swiss National Science Foundation.
132 CHAPTER 4. IMPERFECT PROPERTY RIGHTS
4.1 Introduction
Among all rules and laws those that are designed to assign and enforce prop-
erty rights are probably the most important from an economic point of view.
Property rights, if properly enforced, imply the control of economic goods and
resources and are as such fundamental for all market transactions. Maybe
most importantly, property rights allow the holder to obtain income from a
good or resource. As an example, a person’s incentives to work depend on the
expected share of earned income that is obtained. This is not very surprising
and there is a lot of empirical evidence for this claim, see for example Field
(2007). Similarly, the incentives to invest hinge to a great deal on the expected
share of returns obtained, as has been shown for example in a recent study by
Hornbeck (2010). This has of course also important implications for economic
development. Mehlum et al. (2005) show in a theoretical analysis how the lack
of well enforced property rights dampens incentives to invest and so might
lead to poverty traps. Skaperdas and Syropoulos (2002) and Garfinkel et al.
(2008) show that trade may decreases welfare if property rights are insecure.
Similarly, property rights enforcement is important for the internalization of
externalities, as has been noted already in the classical works of Coase (1960)
and Demsetz (1964, 1967). Overall, property rights are important to shape in-
centives in various settings efficiently. Nevertheless, property rights are usually
only imperfectly enforced.
In this paper we identify uncertainty as sufficient for imperfect enforce-
ment. We interpret the process of enforcement as a contest game between an
appropriator and an individual trying to defend his property, the defender. If
the defender deters the appropriator from appropriation, property rights are
perfectly enforced. Our point of departure is a situation absent incomplete
information, where deterrence may be possible if the defender is able to com-
mit to enforcement effort. As soon as we introduce uncertainty about relevant
characteristics of the appropriator, perfect enforcement breaks down and en-
forcement is imperfect. The actual magnitude of uncertainty does not play a
role, and neither does the defender’s attitude towards risk or his strength in the
contest. Introducing a state as an enforcer does not alter imperfect enforcement,
either.
The creation or enforcement of property and other rights has been the sub-
ject of study for many researchers in the social sciences during the past decades.
The early literature started with the seminal works of Coase (1960), Demsetz
(1964) and Becker (1968). While the first two articles are mainly concerned with
4.1. INTRODUCTION 133
the effect of imperfect property rights enforcement and when individuals may
have an incentive to create property rights, Becker tries to answer from a nor-
mative point of view what is the optimal extent of rule enforcement if there
was a social planner. About a decade later another literature developed with
the aim to explain why and how property rights may actually emerge absent
any enforcing institution as the state. The point of departure is usually a state
of anarchy.1 Because in anarchy there do not exist rules or institutions by def-
inition, it is possible to study the prerequisites for rules and institutions, like
property rights, to emerge. Those results are important for our understanding
of modern societies. After all, enforcement of property rights is, even in the
presence of a state, to a great deal the responsibility of individuals or firms.
To quote Friedman (1994), “legal rules are in large part a superstructure erected
upon an underlying structure of self-enforcing rights”. For example, households
buy lockable doors, bike locks, or erect walls and fences around their houses,
department stores hire security personnel, buy video surveillance systems or
burglar alarms. Empirically, this is reflected by the fact that individual spend-
ing for property rights enforcement exceeds public spending (see for example
Shavell (1991) and the references therein). For a self-interested individual there
is no reason to respect others’ possession unless the costs of disrespecting prop-
erty are too high. If somebody can further his goals by stealing from others he
will probably do exactly that. As a consequence, property rights are inherently
challenged and a contest for property emerges. Early contributions to this lit-
erature are due to Umbeck (1977, 1981), who analyzed how claims to property
evolved during the California gold rush and showed that violence was a ma-
jor impediment for the creation of secure property rights. Other important
contributions include the articles of Grossman and Kim (1995) and Grossman
(2001). There it is shown that in a contest for possession the possessor might
be able to secure his possession perfectly if he is sufficiently strong and able
to commit to some level of defensive effort, for example by moving first and
building a wall or fence. The assumption that the defender has the opportunity
to move first has been adopted frequently and reflects incumbency. Another
interesting paper is due to Muthoo (2004), in which it is shown that secure
property rights may emerge over time. This finding is similar to the results
in Hafer (2006). More recent contributions are Gintis (2007), Kolmar (2008), or
Hoffmann (2010). Hoffmann (2010) studies how property rights may emerge
1To our knowledge, the first paper studying the implications of anarchy is Bush and Mayer (1974).Influential articles analyzing economic behavior under anarchy are for example Skaperdas (1992) orHirshleifer (1995). For an overview consult Skaperdas (2006) or Garfinkel and Skaperdas (2007).
134 CHAPTER 4. IMPERFECT PROPERTY RIGHTS
in a barter economy. Gintis (2007) discusses how the endowment effect can ex-
plain the emergence of perfectly secure property rights. Finally, Kolmar (2008)
shows that even if secure property rights emerge – that is property remains
unchallenged – production incentives are nevertheless distorted. The reason
is that securing property rights implies deterrence, and a rational defender of
property only deter appropriation at the margin.
A common theme in many of these papers is that perfect enforcement is
possible if the defender is sufficiently strong, because then he can deter the
appropriator. As a result, power and incumbency determine whether or not
property rights are secure. However, in all those papers uncertainty does not
play a role. In this paper we stress the importance of uncertainty, for it is likely
to prevent deterrence and hence impedes the creation of secure property.
We analyze a game of conflict in which a defender has the opportunity to
publicly commit resources to fend off an appropriator. Because there are appro-
priators of different types (abilities), appropriators are heterogeneous, and the
defender does not know which appropriator is trying to challenge his property.
This seems quite realistic, since, loosely speaking, people usually do not know
in advance who is going to try to break into their houses, or whether somebody
will attempt to break in at all. We show that this simple structure is sufficient to
destroy the possibility of deterrence of all types and thus property rights can-
not be secure. The reason for this finding is, metaphorically speaking, that all
possible appropriators will stand in front of the same wall. In order to defend
property at lowest possible costs, the defender builds the wall only slightly
higher than the appropriator can climb. When there is uncertainty about the
appropriator’s ability it is not clear what is the optimal height and the defender
builds a wall that is optimal in expectation, and thus deters only some average
appropriator. We show that it will never be in the interest of the defender to
build the wall high enough to deter all types of appropriators. In a first step we
analyze the conflict game using the canonical lottery contest success function
(CSF) to model the conflict. We provide exact conditions for different degrees of
security of property rights to emerge and derive a formula which yields the ex
ante probability of successfully defending the good or resource. We interpret
this probability as the degree of enforcement of property rights. We proceed
and show in a more genera model that the result is robust.
While the primary objective of this paper is to highlight the importance of
uncertainty for property rights enforcement, we also contribute to the literature
on the theory of contests. First, we provide a characterization of the equilibrium
of a ratio-form contest with sequential moves and arbitrary continuous type
4.2. THE MODEL 135
uncertainty. We thereby generalize the analysis of Linster (1993) along many
lines. We show that in our setting of type uncertainty the probability to win
the contest may be negatively correlated with the expected relative strength of
the defender, which seems counterintuitive at first glance. Other papers deal-
ing with informational asymmetries in contests and conflict are for example
Hurley and Shogren (1998), Wärneryd (2003), and Slantchev (2010). Hurley
and Shogren (1998) look at rent-seeking games with one-sided asymmetric in-
formation and show how different beliefs about the strength of the opponent
alter equilibrium behavior. Wärneryd (2003) analyzes contest games between
two players that are ignorant about their own type in the setting of a common-
value conflict (that is, both contestants value the prize equally). Slantchev (2010)
analyzes how choosing a position in a bargaining game prior to a possible con-
flict functions as a signal about a player’s actual strength. He finds that it is
often optimal to appeal to Sun Tzu’s principle of “feigning weakness”. Another
recent related article is due to Chassang and i Miquel (2010). Here the authors
highlight the importance of uncertainty for the stability of peace, respectively
for the outbreak of overt conflict or war. They show that uncertainty generates
incentives to start preemptive conflict, and thereby uncertainty makes deter-
rence less likely. This is in line with our findings in this paper.
The chapter is organized as follows: In the next section we describe the
basic model and define different degrees of security of property. Section 4.3 an-
alyzes a simple contest game of self-enforcement of property rights to develop
an intuition for our main result and discusses comparative static results. Sec-
tion 4.4 studies a generalized model and shows that the developed intuition is
quite general. Section 4.5 shows that also in the presence of a state our results
do not change, even though the state may generate economies of scale in the
process of enforcement. In Section 4.6 we discusses our results in relation to
other literatures in economics.
4.2 The Model
4.2.1 Basic Set-up
Consider two players i ∈ {D, A}, where the letters are mnemonics for defender
and appropriator respectively. D is in possession of a good, which A wants
to appropriate. Enforcement of property rights is in the responsibility of D,
which may for example be due to a weak state. In the context of property
rights this assumption is quite realistic. Clearance rates of property crimes are
136 CHAPTER 4. IMPERFECT PROPERTY RIGHTS
consistently low in most countries. For example, the average clearance rate of
burglaries in the U.S. in 2002 was some 14 percent, if victims reported the theft.
However, it is estimated that only every third victim actually reports burglary
(U.S. Department of Justice, 2002). There is also evidence that burglars do
not think about the consequences they face in the event of being caught. The
deterrence effect of punishment is insignificant (U.S. Department of Justice,
2002). That being said for the U.S., public enforcement of property rights is in
most other countries even worse.2 Hence, the major impediment for burglary
and related crimes is usually private spending for self-enforcement.
D and A may be heterogeneous. For specificity, we assume they are hetero-
geneous in their valuation of the good vi. D’s valuation for the good is common
knowledge and is given by vD, whereas A has private information about his
valuation vA. vA is drawn from G(vA), which has the compact and strictly
positive support [vA,vA]⊂ R+ and is differentiable. All this is common knowl-
edge. Throughout the paper we contrast the incomplete information scenario
to one in which there is complete information. In the latter case deterrence
and therefore secure property rights are the outcome if vD is sufficiently high.
As will become apparent later on this does not hold true under incomplete
information.
Note that although we model heterogeneity of players by emphasizing their
different utilities derived from the good, our results are identical if heterogene-
ity in other dimensions is added. In fact, it is uncertainty about A’s willingness
to put in effective effort that matters, which is also influenced by other factors
like marginal costs and effectiveness of effort.
Each player’s probability pi to succeed in the conflict is a function of both
players’ efforts. Specifically, we consider
pD(xD, xA) =
{1 if xA = 0,
xDxD+xA
else.(4.1)
and pA = 1 − pD. In Section 4.4 we show that our results hold more generally.
pA(xD,0) = 1 reflects that the good is in D’s possession in the first place, and A
needs to spend at least some resources to steal it. Each player i chooses effort xi,
and costs of effort are Ci(xi) = xi. We look at sequential moves where D moves
2The Heritage Foundation and the Wall Street Journal rank countries according to the degree ofpublicly enforced property rights on a 0 to 100 scale. For example, a score of 40 suggests “[t]hecourt system is highly inefficient, and delays are so long that they deter the use of the court system”and “expropriation is possible”. Countries with a score equal to or lower than 40 include Argentina,Paraguay, and Bolivia in South America or Croatia, Serbia, and Russia in Europe. See http://www.
globalpropertyguide.com/.
4.2. THE MODEL 137
first, respectively is able to make a commitment. This is a natural assumption
in our context. Fortification or target hardening is usually a durable good and
we usually see somebody erecting a wall or a fence around his house before
somebody else tries to break in.
Given the above assumptions D’s objective is
maxxD∈R+
UD(xD, xA) = vD
∫ vA
vA
p (xD, xA(vA; xD)) dG(vA)− xD. (4.2)
For A, who holds full information after observing D’s effort, maximizing utility
corresponds to
maxxA∈R+
UT(xD, xA) = vA (1 − p(xD, xA))− xA. (4.3)
Because D does not know the type of A we have an incomplete information
game and the equilibrium concept we employ is Bayesian Nash equilibrium.
4.2.2 Concerning Property Rights
Before going into the details of the model a word on our interpretation of secure
property rights is in order. We define a property right analogous to for example
Grossman (2001), Gintis (2007), and Kolmar (2008) as the right of incumbency
or initial claims.3
The appropriator spends effort xA in the contest. If xA(vA; xD) > 0 type vA
is active and challenges D’s property. If xA(vA; xD) = 0 type vA stays passive.
Define π to be the equilibrium probability of observing xA(vA; xD)> 0. We can
use this convention to define three different regimes of security of property:
Definition 4.2.1. Property rights are
1. secure, if the equilibrium probability of appropriative activity is zero, that is the
good is definitively unchallenged and π = 0 in equilibrium. Property rights are
perfectly enforced.
2. imperfect, if the equilibrium probability of appropriative activity is strictly be-
tween zero and one, and 0 < π < 1 in equilibrium.
3This is for example also in the tradition of Locke, who wrote in §27 of his Second Treatise: “Thoughthe Earth, and all inferior Creatures be common to all Men, yet every Man has a Property in his own Person.This no Body has any Right to but himself. The Labour of his Body, and the Work of his Hands, we may say,are properly his. Whatsoever then he removes out of the State that Nature hath provided, and left it in, he hathmixed his Labour with, and joyned to it something that is his own, and thereby makes it his Property. It beingby him removed from the common state Nature placed it in, hath by this labour something annexed to it, thatexcludes the common right of other Men”. See http://press-pubs.uchicago.edu/founders/documents/
v1ch16s3.html.
138 CHAPTER 4. IMPERFECT PROPERTY RIGHTS
3. insecure, if the equilibrium probability of appropriative activity is equal to one,
π = 1.
In defining secure property rights we exactly follow papers such as Grossman
(2001), Gintis (2007), or Kolmar (2008). Secure property rights in our defini-
tion are equivalent to the de facto recognition of incumbency, as extensively
discussed in Gintis’ paper. Many authors including the just mentioned have
shown that property rights can be secure in the sense just defined if incum-
bency enables to commit resources to defense and if the incumbent or defender
is sufficiently strong. If property rights are not secure, they do not have to be
insecure, because we define property rights security in a probabilistic way. In
this we depart from the aforementioned papers. Insecure property rights in
our definition resemble the Hobessian jungle in which there is no recognition
of property whatsoever and property is always challenged. Imperfect property
rights as defined above is a form of security in between the two polar cases.
4.3 Equilibrium
4.3.1 Complete Information: A Benchmark
To establish a benchmark we let G(vA) converge to one point, such that vA =
vA = vA. In this case there is no uncertainty and hence there is complete infor-
mation. The equilibrium of this game is our benchmark.
Proposition 4.3.1. The defender enforces his property with probability
p∗D(x∗D, x∗A) = min
{vD
2vA,1
}.
Proof. See for example Linster (1993).
From this it follows that A does not spend any effort in equilibrium if and
only if vD ≥ 2vA. Therefore, if D is sufficiently strong there is deterrence and
property rights are secure. Otherwise, property rights are insecure. Imperfect
property rights cannot emerge. We now turn to the incomplete information
game.
4.3.2 Incomplete Information
In this section we look at how uncertainty affects the level of security of prop-
erty in equilibrium. If D moves first or is able to make a commitment he
4.3. EQUILIBRIUM 139
considers A’s optimal reaction while optimizing. A’s reaction function is easily
found:
xA(vA; xD) = max{0,√
xD vA − xD}. (4.4)
It follows immediately that the appropriator tries to appropriate only when
xD < vA. For all higher efforts xD there is deterrence. For property rights to be
secure this has to hold for all types vA. Let θ(v) ≡∫ vA
v v−1/2A dG(vA), where v
is the marginal type of A participating actively in the contest, v = max{vA, xD}.
Note that θ(v) = θ(xD), since for all xD < vA the density of vA is zero. To
determine D’s optimization problem this formulation is useful. Plugging (4.4)
in (4.2) and using θ(xD) it is then easily verified that D’s optimization problem
corresponds to:
maxxD∈R+
vD [G(xD) +√
xD θ(xD)]− xD.
A simply reacts according to his reaction function (4.4). Proposition 4.3.2 shows
the equilibrium.
Proposition 4.3.2. D’s unique equilibrium effort in the sequential game is (implicitly)
determined by
x+D =
(vD θ (v)
2
)2
> 0, (4.5)
where v = max{vD, x+D}. Player A’s equilibrium effort as the follower (F) follows
immediately and is
x+A = max
{√vA x+D − x+D,0
}.
Proof. See appendix.
D’s effort in equilibrium is monotonically increasing in vD and always
strictly positive. However, this is not true for A, who might stay passive. If
all types of A stay passive we would have secure property in equilibrium. If
to the contrary all types of A are active property is insecure. We next look for
a condition that tells us when this is the case. It is obvious from (4.4) that the
appropriator with the lowest valuation stays passive in the conflict whenever
x+D ≥ vA. It is this type who is deterred first and therefore this type’s equilib-
rium effort determines whether property is insecure or not.
Corollary 4.3.1. All types of appropriators are active in the equilibrium of the conflict
140 CHAPTER 4. IMPERFECT PROPERTY RIGHTS
game if and only if
vD ≤ vD ≡ 2√
vA
θ(vA).
Accordingly, if this condition is met property rights are insecure and π = 1−G(vA) =
1. If this condition is not met some types of A stay passive and property rights are
imperfect or secure.
Proof. Follows immediately from Proposition 4.3.2 and (4.4).
The corollary tells us that when D gets too strong relative to A he spends
so much effort in equilibrium that some types of A prefer to stay inactive in
equilibrium, that is partial deterrence occurs. In order to have secure property,
however, full deterrence is necessary. To answer whether property can be secure
in an equilibrium we need to look more closely at x+D. In particular, property is
secure if and only if x+D ≥ vA.
Corollary 4.3.2. In equilibrium D always chooses effort
x+D < vA, (4.6)
implying π = 1 − G(x+D) > 0 and hence property rights are never secure.
Proof. If we let x+D → vA on the RHS of (4.5), θ(v) gets zero in the limit, and so
does the RHS. This yields a contradiction since then 0 = x+D = vA > 0.
The corollary shows that uncertainty is sufficient to explain imperfectly en-
forced property rights. Note that the result does not depend on the possible
range of vA, since any interval [vA,vA] with positive measure suffices. Only in
the limit, when vA = vA = vA, secure property can be sustained in an equilib-
rium.
What is the intuition for this result? Assume vD ≥ 2vA. Without uncer-
tainty D would spend exactly xD = vA ∈ [vA,vA], hence deter A and enforce
property rights perfectly. But when there is uncertainty he prefers not to do
that and enforces only imperfectly. To see the intuition behind the result it is
illustrative to look at a simple example. Assume vD = 8 and vA ∈ {2,4} with
equal probabilities. Without uncertainty D would deter both types of A and
secure property rights perfectly. In the presence of asymmetric information D
faces uncertainty in his decision making. By spending xD = 4 he would deter
both types of A, however, he would also spend too much with a probability of
50 percent. Effort higher than 2 has, with a 50 percent probability, no benefit
at all. By increasing effort the expected marginal benefit from effort decreases
4.3. EQUILIBRIUM 141
1 2 3 4 5
-1
1
2
3
1 2 3 4 5
-1
1
2
3
xDxD
∂UD
∂xD
∂UD
∂xD
Figure 4.1: Derivative of D’s utility function with respect to xD against both playerswith vD = 8, vA = 4, and vA = 2. In the right panel we see the derivative ofD’s objective function in case vA = 4 is zero at xD = 4. In the left panel wesee that at this effort the derivative in case of vA = 2 is negative, implyingthe overall derivative has to be negative, too, and therefore the first ordercondition for an optimum cannot hold.
faster in the presence of uncertainty, because at the same time the marginal ben-
efit of effort and the probability that effort has a benefit at all decrease. Because
by increasing effort the fraction of deterred types of A increases, the marginal
benefit of effort decreases not only due to diminishing marginal products but
also as a consequence of a decreasing probability that effort has a positive ben-
efit at all. However, efforts have always to be paid in full, whether or not effort
is effective. In our example the marginal benefit of effort at xD = 4 is with equal
probability zero and one, while marginal costs are equal to one for sure. Hence,
the first order condition for an optimum does not hold but effort has to be lower
in equilibrium. This intuition is also graphically illustrated in Figure 4.1. In the
case of a continuous distribution as described in the model, the probability of
deterrence is continuously increasing in xD in the interval [vA,vA]. Therefore,
as xD → vA the probability of deterrence approaches one, and therefore the
expected marginal benefit of effort approaches zero.
Having shown that property rights are not secure in equilibrium, it is of
interest to look more closely at the ex ante probability that D actually keeps
the good. If the probability to lose possession is economically not significant
the implications of uncertainty for an individual are rather negligible and our
relatively strict definition of security drives the results, rather than economic
significance. There are again two different sources of security of property: first,
some types of A might be deterred, and this increases the probability of keep-
ing the good. Second, those types that are not deterred do not succeed for
sure but get the good only with some probability, which is determined by the
142 CHAPTER 4. IMPERFECT PROPERTY RIGHTS
CSF. In other words, because of deterrence, there is some probability not to
encounter an appropriator at all, and there is also some probability to keep the
good conditional on having a conflict. We need to consider both probabilities,
what simply corresponds to integrating over the CSF, given the equilibrium
strategies:
Corollary 4.3.3. The ex ante probability that D keeps the good is
φ =∫ vA
vA
x+Dx+D + x+A(·, ·)
dG(vA)
= G(x+D) +θ(x+D)
2vD
2,
where x+D and xSA are defined as in Proposition 4.3.2.
Proof. This follows immediately from the definition of the CSF in (4.2) and
Proposition 4.3.2.
To get an idea of the magnitude of the impact of uncertainty it is instructive
to proceed with an example distribution. For specificity, assume the appropri-
ator’s valuation is uniformly distributed on [(1 − σ)v,v], σ ∈ [0,1], and assume
vD = 2v. This ensures that the defender would perfectly secure property rights
against all realizations of the appropriator’s valuation if he had perfect informa-
tion. σ is a spread parameter and may vary between zero and one, the former
being the limit when uncertainty vanishes. When σ increases uncertainty gets
larger. The expected valuation of the appropriator, however, decreases. The fol-
lowing formula establishes the connection between the spread and the ex ante
probability to keep possession in our example:
φu(σ) =σ(σ + 3) + 4
(σ + 2)2. (4.7)
A nice property is that φu is independent of v, what makes it quite easy to
analyze. If there was no uncertainty, σ = 0, we get φu(0) = 1, as we should
have expected. As uncertainty increases φu decreases. In Figure 4.2 we plot φu
against σ. An interesting question is now how uncertainty translates into the
degree of enforcement of property rights. For example, if we are interested in
finding the value of σ necessary to generate an ex ante chance of 1% or 5 % to
lose possession, we find σ1% = 0.04168 or σ5% = 0.25403 respectively. Therefore,
if uncertainty is about 25 % relative to v, which is less than 13 % relative to vD,
the probability to lose is some 5 %. However, φu cannot fall below 8/9 ≈ 88.9%,
reflecting that D is still the stronger player. Actually, as σ increases the expected
4.4. GENERALIZING THE MODEL 143
0.0 0.2 0.4 0.6 0.8 1.0
0.85
0.90
0.95
1.00
0.0 0.2 0.4 0.6 0.8 1.0
2.0
2.5
3.0
3.5
4.0
σσ
φu κ
Figure 4.2: Left panel: The ex ante probability φu to keep property as a functionof σ, if vA ∈ [(1 − σ)v,v] and vD = 2v. Right panel: Expected relativevaluation κ ≡ E[vD]/E[vA ] =
42−σ as a function of σ. While φu is strictly
monotonically decreasing in σ the opposite is true for κ. Therefore, in thepresence of uncertainty κ may be a poor measure of strength in a contest,contrary to complete information contests.
valuation of A decreases, and as a consequence D becomes relatively stronger
in expectation.
Note that in the example the probability to win the contest is actually
negatively correlated to the ratio of expected valuation, κ ≡ E[vD]/E[vA ] =
vD/E[vA], something that never happens absent imperfect information. The in-
tuition is that higher uncertainty benefits D because with a given level of effort
the fraction of deterred appropriators is increasing. This makes it attractive to
lower effort to save costs, but this necessarily brings some formerly deterred
types back into the game. As a consequence, D is better off but loses more
often. A consequence of this finding is that standard measures of strength in
contests, as for example the valuation, marginal costs, or effort productivity,
are not necessarily meaningful in the presence of uncertainty.
4.4 Generalizing the Model
In this section we now show that our main result in Corollary 4.3.2 does not
depend on the particular CSF we employed above to model the conflict, nor
on the assumption of uncertainty neutrality. We proceed by first proving that
deterrence is generally possible, and then show that under uncertainty full
deterrence cannot be an equilibrium outcome.
Each player i ∈ {D, A} values the contested good by vi. Both derive utility
144 CHAPTER 4. IMPERFECT PROPERTY RIGHTS
from wealth, which is given by
wi =
{vi − xi if i wins in the contest,
−xi else.
A player’s utility is defined over wealth and given by ui = ui(wi) where u′i is
strictly positive and finite and u′′i ≤ 0. So we allow for risk aversion. Both
players put effort xi ≥ 0. D’s probability of winning the contest is
pD(xD, xA) =
{f (xD)
f (xD)+ f (xA)if f (xD) + f (xA) > 0,
1 else,(4.8)
where f (0) = 0, f ′ > 0 and f ′′ ≤ 0.4 A wins with a probability pA = 1 − pD.
(4.8) is a standard CSF, see for example Konrad (2009). If we let f (x) = x this is
the CSF we employed before.
Player i’s von Neumann–Morgenstern utility function is
Ui(xi, xj) = pi(xi, xj)ui(vi − xi) + (1 − pi(xi, xj))ui(−xi). (4.9)
As before D is able to publicly commit effort, for example by moving first.
The following lemma is now important to prove our main result later on.
Lemma 4.4.1. Assume each player maximizes a von Neumann–Morgenstern expected
utility function as defined in (4.9) and probabilities are determined by (4.8). Denote
A’s best response by xA(vA; xD). Then there exists µ(vA) such that
1. if xD ≥ µ(vA) the optimal response is xA(vA; xD) = 0,
2. if xD < µ(vA) the optimal response is xA(vA; xD) > 0,
3. and µ(vA) is increasing in vA, µ′(vA) > 0.
Proof. See appendix.
The lemma states that in the above described framework for each type of A
there exists a unique finite level of effort xD that is necessary to deter A and so
to secure property rights. This threshold level is increasing in the valuation of
A. The lemma is important because it states that deterrence is technologically
possible, so we do not assume the result of the following proposition by making
deterrence infeasible:
4In an earlier version of this paper we show that our results also hold for a more general class ofCSFs that allow for deterrence under full information.
4.5. WHEN THERE IS A STATE 145
Proposition 4.4.1. Let the distribution of types of A be G(vA) with density g(vA) and
support [vA,vA] ⊂ R+, where vA < vA and denote D’s optimal effort by x+D. Then
in equilibrium it always holds that x+D < µ(vA) and as a consequence thereof there is
never full deterrence and property rights are never perfectly enforced.
Proof. See appendix.
The result that a very small amount of incomplete information suffices to
guarantee imperfect enforcement is quite general. Apart from property rights
enforcement this is also interesting in other circumstances. For example, in
international relations piling stocks of defensive weapons is often used as a
deterrent for other countries to declare overt conflict or war. However, in the
presence of uncertainty deterrence is possible only probabilistically, as we have
shown. This result is similar to a recent result of Chassang and i Miquel (2010),
who show that a little amount of incomplete information can impede deterrence
and hence lead to overt conflict, because uncertainty may create incentives for
preemptive attacks.
4.5 When There is a State
The above reasoning immediately carries over to the case of state enforced prop-
erty rights, or property rights which are partly state and partly self-enforced, as
for example in Konrad and Skaperdas (2010). Even if the state is able to gener-
ate economies of scale in the process of enforcement, this does not change our
results as long as marginal costs of enforcement are positive. To see this assume
the following simple extension of the model in Section 4.2. As before, D may
spend xD for self-enforcement of property rights. However, now total effective
enforcement effort is e(γ1,γ2, xD) ≡ γ1xD + γ2, where the γ1 and γ2 are en-
forcement instruments under the control of the state. γ2 may be interpreted as
a guaranteed minimum level of enforcement effort, whereas γ1 influences D’s
marginal costs of effective enforcement effort, for example because the state
subsidizes lockable doors and burglar alarms (γ2 > 1). As a consequence, the
probability of enforcement is
∫ vA
vA
e(γ1,γ2, xD)
e(γ1,γ2, xD) + xA(vA; e(γ1,γ2, xD))dG(vA).
Let the state’s cost functions be C1(γ1) and C2(γ2) with strictly positive marginal
costs. The sequence of events is the following:
1. The state determines γ1 and γ2.
146 CHAPTER 4. IMPERFECT PROPERTY RIGHTS
2. D determines xD.
3. A determines xA.
We assume the state moves first because usually states are longer lived than
individuals and an individual takes usually as given the level of state enforce-
ment. That D moves second follows from the same reasoning as above. The
state then maximizes
maxγ1,γ2
∫ vA
vA
e(γ1,γ2, xD)
e(γ1,γ2, xD) + xA(vA; e(γ1,γ2, xD))dG(vA)vS − C1(γ1)− C2(γ2),
taking into account the optimal reactions of both D and A. It is straightforward
to show that for A and D this situation is now strategically equivalent to the
one in Section 4.3, except that D’ marginal costs of effort are now 1/γ1 and
he is constrained to choose effort e ≥ γ2. Hence, he will always choose xD <
(vA − γ2)/γ1 unless γ1 = ∞ or γ2 = vA. In those cases, however, the state
is the de facto single enforcer of property rights. So it suffices to focus on
the state’s behavior without taking into account D’s optimal reaction to study
whether there may be full deterrence. The state is in the identical position as
D in our baseline model above with the only difference being the cost function.
But since marginal costs are strictly positive, it follows immediately from the
earlier discussions that the state does not fully deter A, either. The derivative
of the state’s objective function at γ2 = vA is −C2(vA) < 0, and at γ1 = ∞ is
−C1(∞) < 0. Therefore, the state will not fully deter A, either, even though it
may be able to generate economies of scale in the process of enforcement.
4.6 Discussion
Property rights and other rules and laws will not be perfectly enforced in an
equilibrium. But what is the deeper reason for that? In our framework infra-
marginal losses against all already deterred types drive the result. However, on
a more fundamental level this is due to the inability of the defender to make
type contingent defensive efforts. Uncertainty about the appropriator’s type
forces the defender to built the same defense for all possible contingencies, and
it is then not optimal to deter all appropriative activities. This enables some
types of appropriators to make positive expected rents, which would not nec-
essarily be the case without uncertainty. This parallels many findings from the
economics of information. For example, it is also the impossibility to write type
contingent contracts that drive the results in the seminal work of Akerlof (1970).
4.6. DISCUSSION 147
Because sellers have to charge the market price for all types of cars, leading to
a race to the bottom in quality, only bad quality cars survive. In our example
it is not possible to built up type or state contingent fortification. Note that
this holds true for any number of policy instruments, as long as contingent
instruments are not feasible.
Moreover, the appropriator gains from uncertainty, and thus receives in-
formation rents. That uncertainty benefits some players and allows them to
receive information rents is something we find quite often in the contract the-
ory and principal-agent literature. For example, in a Baron and Myerson (1982)
regulation problem, where the authority does not know the cost function of the
monopolist, we usually find the low cost types to benefit from the possibility of
high cost types. Another example for such information rents is second degree
price discrimination.
Finally, on a more technical note, for deterrence to be technically possible
appropriators’ and violators’ reaction functions need to be downward sloping
and must cross the abscissa at some point. Equivalently, efforts need to be
strategic substitutes over a given range of the reaction function. This is what we
find in standard oligopoly models with quantity competition. In this respect
Cournot oligopoly models and standard (imperfectly discriminating) contest
models are similar, as has been noted by Mehlum and Moene (2002) and others.
However, while in Cournot competition we find strategic substitutes over the
whole range of the strategy space, this is not true in contest models, in which
we usually find hump-shaped best responses. For low values of the opponent’s
effort we find strategic complements and strategic substitutes only later (Dixit,
1987). However, if we want to analyze deterrence this is exactly the region
we are interested in. So, because we find property rights are never secure in
a contest with uncertainty, in Cournot oligopoly models with uncertainty a
Stackelberg leader will never deter all possible types of competing firms, either.
This is in line with Gal-Or (1987), who showed that in an environment with
uncertainty “the preemptive capabilities of a Stackelberg leader are reduced”.
148 CHAPTER 4. IMPERFECT PROPERTY RIGHTS
Appendix
Proof of Proposition 4.3.2
Proof. D maximizes
vD
∫ vA
vA
[xD
xD + xA(vA; xD)
]dG(vA)− xD
s.t. xA = max{√vA xD − xD,0} ,
or equivalently
G(v)vD + vD√
xDθ(v)− xD,
where v = max{x+D ,vA} and θ(v) ≡∫ vA
v v−1/2A dG(vA) as defined above in Sec-
tion 4.3. To prove existence of a maximum it is sufficient to note that this
function is continuous (since it is differentiable) on the compact and convex
set [0,vD], and therefore it follows from the extreme value theorem that a maxi-
mum exists. Uniqueness and the applicability of the first order condition follow
from its strict concavity, the second order condition is
−vDθ(xD)
4x3/2D
− vDG′(xD)
2xD< 0.
The first order condition reads
θ(v)vD
2√
x+D
+ vD G′(v)− vD
√1
xD
√xD G′(v)− 1
!= 0.
Simple manipulations reveal that
x+D =
(θ(v)vD
2
)2
.
Proof of Lemma 4.4.1
Proof. Given the above assumptions A maximizes
EuA = pA(xA, xD)uA(vA − xA) + (1 − pA(xA, xD))uA(−xA) (4.10)
=f (xA)
f (xA) + f (xD)uA(vA − xA) +
f (xD)
f (xA) + f (xD)uA(−xA)
APPENDIX 149
It is easy to see that his utility from staying passive in the contest, that is
from spending effort xA = 0, is u0 ≡ uA(0). Moreover, all efforts xA > vA are
strictly dominated. Denote the utility level from putting xA = vA by uvA≡
pA(xA, xD)uA(0) + (1 − pA(xA, xD))uA(−vA) ≤ u0.
The first derivative of (4.10) is
∆ =∂EuA
∂xA
=f ′(xA) f (xD)
( f (xA) + f (xD))2uA(vA − xA)−
f (xA)
f (xA) + f (xD)u′(vA − xA)
︸ ︷︷ ︸≡I
− f ′(xA) f (xD)
( f (xA) + f (xD))2uA(−xA)−
f (xD)
f (xA) + f (xD)u′(−xA)
︸ ︷︷ ︸≡I I
.
Except for the first term all of those terms are negative in sign. The first two
terms are the impact of increasing effort marginally on winning, while the
third and fourth term describe the effect on losing. Increasing effort has two
opposing effects on the winning utility. First, winning becomes more likely,
what is positive. Second, if he wins he gets less, what is negative. The third
and fourth term are both negative, reflecting that increasing effort decreases
the probability to lose and at the same time decreases the utility in the event
of losing. It is straightforward to verify that I is strictly decreasing in xA. That
directly implies f (xA)/( f (xA) + f (xD))uA(vA − xA) is strictly concave and is
hence either a monotonic function in xA, increasing or decreasing, or its graph
is inverse U-shaped. Because I I is negative EuA must also be either monotonic
in xA, increasing or decreasing, or it is inverse U-shaped. However, we know
that u0 > uvA, implying that it cannot be monotonically increasing and must be
decreasing on some interval. Of course, if EuA is inverse U-shaped ∆ must be
zero at some point, at which utility is strictly higher than uA. Then the best
response must be positive, xA(vA; xD)> 0. If the derivative is strictly decreasing
A’s optimal choice is xA(vA; xD) = 0. To find out which of those two possible
solutions is the correct one we have to evaluate ∆ at xA = 0:
∆|xA=0 =f ′(0)f (xD)
[uA(vA)− uA(0)]− u′A(0).
It is now easily verified by inspection of ∆|xA=0 that this is a monotonic function
150 CHAPTER 4. IMPERFECT PROPERTY RIGHTS
in xD and that there exists a threshold value xD = µA(vA), such that
Sign[∆|xA=0] =
+ if xD < µA(vA),
0 if xD = µA(vA),
− if xD > µA(vA).
If xD ≥ µA(vA) it is true that ∆|xA=0 is either zero or negative. Otherwise ∆|xA=0
is positive. From the above discussion it follows immediately that xA(vA; xD)>
0 if and only if xD < µ(vA). The threshold is given by
µ(vA) = f−1
(f ′(0)
u′A(0)
[uA(vA)− uA(0)]
),
from which it immediately follows that µ(vA) is continuously differentiable
and µ′(vA) > 0.
Proof of Proposition 4.4.1
Proof. In Lemma 4.4.1 we found that there exists a threshold level of effort xD
denoted by µ(v), which is strictly increasing and differentiable. Therefore, its
inverse vA = µ−1(xD) exists and is also continuously differentiable. We then
define
ω(xD) ≡ µ−1(xD).
ω(xD) is the marginal type vA that is deterred given effort xD. If ω(xD) < vA
all thieves are active. ω(xD) ≥ vA implies all thieves are deterred.
We are now able to define D’s expected utility function:
EuD =∫ vA
vA
p(xD, xA(vA; xD))uD(vD − xD)
+ (1 − p(xD, xA(vA; xD))) uD(−xD)dG(vA)
=
(G(ω(xD)) +
∫ vA
ω(xD)p(xD, xA(vA; xD))dG(vA)
)uD(vD − xD)
+
(1 − G(ω(xD))−
∫ vA
ω(xD)p(xD, xA(vA; xD))dG(vA)
)uD(−xD)
The derivative with respect to xD is the following:
APPENDIX 151
Γ = uD(vD − xD)
×[∫ vA
ω(xD)
(∂pD(·, ·)
∂xA
∂xA(·, ·)∂xD
+∂pD(·, ·)
∂xD
)dG(vA)
− ω′(·)g(ω(·))pD(·, ·) + ω′(·)g(ω(·))]− uD(−xD)
×[∫ vA
ω(xD)
(∂pD(·, ·)
∂xA
∂xA(·, ·)∂xD
+∂pD(·, ·)
∂xD
)dG(vA)
− ω′(·)g(ω(·))pD(·, ·) + ω′(·)g(ω(·))]
− u′D(vD − xD)
(G(ω(xD)) +
∫ vA
ω(xD)pD(·, ·)dG(vA)
)
− u′D(−xD)
(1 − G(ω(xD))−
∫ vA
ω(xD)pD(·, ·)dG(vA)
)
Fortunately, we are able to boil down this expression significantly. To show that
xD cannot optimally be equal to or larger than µ(vA), it is sufficient to show that
Γ|xD=µ(vA)is negative. Therefore, using pD(xD, xT) = f (xD)/( f (xD) + f (xT)),
let xD = µ(vA) and note that ω(µ(vA)) = vA and G(vA) = 1 by definition. This
simplifies the derivative already significantly:
Γ|xD=µ(vA)= uD(vD − µ(vA))
×(
g(vA)ω′(µ(vA))−
f (µ(vA))g(vA)ω′(µ(vA))
f (xA(vA,µ(vA))) + f (µ(vA))
)
+ u(−µ(vA))
×(
f (µ(vA))g(vA)ω′(µ(vA))
f (xA(vA,µ(vA))) + f (µ(vA))− g(vA)ω
′(µ(vA))
)
− u′D(vD − µ(vA)).
Now we can use that xA(vA;µ(vA)) = 0 by definition of µ(vA) as well as f (0) =
0 to see that the only remaining term is
Γ|xD=µ(vA)= −u′
D(vD − µ(vA)) < 0.
This is the marginal cost of effort weighted by the probability to encounter a
thief with a valuation lower than vA. The derivative is negative, and therefore
the necessary condition for a maximum is not fulfilled. Hence, it cannot be
optimal to deter type vA. Moreover, continuity of Γ implies this is true not only
for this type, who has measure zero, but for some interval of types [v′A,vA] with
strictly positive measure. Therefore, property rights must be imperfect.
Bibliography
Akerlof, G. A. (1970). The market for ’lemons’: Quality uncertainty and the
market mechanism. The Quarterly Journal of Economics, 84(3):488–500.
Amegashie, J. A. (2006). A contest success function with a tractable noise pa-
rameter. Public Choice, 126(1/2):pp. 135–144.
Amoros, P. and Puy, M. (2011). Issue convergence or issue divergence in a
political campaign? Public Choice, pages 1–17. 10.1007/s11127-011-9865-0.
Aoyagi, M. (2010). Information feedback in a dynamic tournament. Games and
Economic Behavior, 70(2):242 – 260.
Aragonès, E., Castanheira, M., and Giani, M. (2012). Electoral competition
through issue selection. UFAE and IAE Working Papers 903.12, Unitat de
Fonaments de l’Anàlisi Econòmica (UAB) and Institut d’Anàlisi Econòmica
(CSIC).
Athey, S. and Schmutzler, A. (2001). Investment and market dominance. RAND
Journal of Economics, 32(1):1–26.
Bagwell, K. (2007). The economic analysis of advertising. volume 3 of Handbook
of Industrial Organization, pages 1701–1844. Elsevier.
Baik, K. H. and Shogren, J. F. (1992). Strategic behavior in contests: Comment.
American Economic Review, 82(1):359–62.
Baik, K. H. and Shogren, J. F. (1995). Contests with spying. European Journal of
Political Economy, 11(3):441–451.
Baron, D. P. and Myerson, R. B. (1982). Regulating a monopolist with unknown
costs. Econometrica, 50(4):911–30.
Barros, F. (1997). Asymmetric information as a commitment in oligopoly. Euro-
pean Economic Review, 41:207–225.
154 BIBLIOGRAPHY
Baye, M. R., Kovenock, D., and de Vries, C. G. (1993). Rigging the lobbying
process: An application of the all-pay auction. American Economic Review,
83(1):289–94.
Baye, M. R., Kovenock, D., and de Vries, C. G. (1996). The all-pay auction with
complete information. Economic Theory, 8(2):291–305.
Becker, G. S. (1968). Crime and punishment: An economic approach. Journal of
Political Economy, 76:169.
Berg, J. E., Nelson, F. D., and Rietz, T. A. (2008). Prediction market accuracy in
the long run. International Journal of Forecasting, 24(2):285–300.
Bernhardt, D., Duggan, J., and Squintani, F. (2009). Private polling in elections
and voter welfare. Journal of Economic Theory, 144(5):2021–2056.
Bhagwati, J. N. (1982). Directly unproductive, profit-seeking (dup) activities.
Journal of Political Economy, 90(5):988–1002.
Bikhchandani, S., Hirshleifer, D., and Welch, I. (1992). A theory of fads, fashion,
custom, and cultural change as informational cascades. Journal of Political
Economy, 100(5):pp. 992–1026.
Blais, A., Gidengil, E., and Nevitte, N. (2006). Do polls influence the vote?
In Brady, H. E. and Johnston, R., editors, Capturing campaign effects, pages
263–279. Ann Arbor: University of Michigan Press.
Bélanger, E. and Meguid, B. M. (2008). Issue salience, issue ownership, and
issue-based vote choice. Electoral Studies, 27(3):477 – 491.
Brams, S. J. and Davis, M. D. (1973). Resource-allocation models in presidential
campaigning: Implications for democratic representation. Annals of the New
York Academy of Sciences, 219:105–123.
Brams, S. J. and Davis, M. D. (1974). The 3/2’s rule in presidential campaigning.
The American Political Science Review, 68(1):pp. 113–134.
Bulow, J. I., Geanakoplos, J. D., and Klemperer, P. D. (1985). Multimarket
oligopoly: Strategic substitutes and complements. Journal of Political Econ-
omy, 93(3):488–511.
Burke, J. and Taylor, C. (2008). What’s in a poll? incentives for truthful report-
ing in pre-election opinion surveys. Public Choice, 137(1):221–244.
BIBLIOGRAPHY 155
Bush, W. C. and Mayer, L. S. (1974). Some implications of anarchy for the
distribution of property. Journal of Economic Theory, 8(4):401–412.
Cabral, L. M. B. (2002). Increasing dominance with no efficiency effect. Journal
of Economic Theory, 102(2):471 – 479.
Callander, S. (2007). Bandwagons and momentum in sequential voting. Review
of Economic Studies, 74(3):653–684.
Chassang, S. and i Miquel, G. P. (2010). Conflict and deterrence under strategic
risk. The Quarterly Journal of Economics, 125(4):1821–1858.
Cho, I.-K. and Kreps, D. M. (1987). Signaling games and stable equilibria. The
Quarterly Journal of Economics, 102(2):179–221.
Coase, R. (1960). The problem of social cost. Journal of Law and Economics,
3:1–44.
Congelton, R. D., Hillmann, A., and Konrad, K. A. (2008). 40 Years of Research
on Rent Seeking. Springer, Berlin.
Corchón, L. (2007). The theory of contests: A survey. Review of Economic Design,
11(2):69–100.
Dalton, R. J. and Wattenberg, M. P. (2001). Parties Without Partisans: Politi-
cal Change in Advanced Industrial Democracies. New York, Oxford University
Press.
Damore, D. F. (2004). The dynamics of issue ownership in presidential cam-
paigns. Political Research Quarterly, 57(3):pp. 391–397.
Damore, D. F. (2005). Issue convergence in presidential campaigns. Political
Behavior, 27(1):71–97.
Darrough, M. N. (1993). Disclosure policy and competition: Cournot vs.
bertrand. The Accounting Review, 68(3):534–561.
Daughety, A. F. and Reinganum, J. F. (2008). Communicating quality: A unified
model of disclosure and signalling. RAND Journal of Economics, 39(4):973 –
989.
Demsetz, H. (1964). The exchange and enforcement of property rights. Journal
of Law and Economics, 7:11–26.
156 BIBLIOGRAPHY
Demsetz, H. (1967). Toward a theory of property rights. American Economic
Review, 57:347–359.
Dewatripont, M. and Maskin, E. (1995). Contractual contingencies and renego-
tiation. RAND Journal of Economics, 26(4):704–719.
Dixit, A. K. (1987). Strategic behavior in contests. American Economic Review,
77(5):891–98.
Drazen, A. (2001). The political business cycle after 25 years. In NBER Macroe-
conomics Annual 2000, Volume 15, NBER Chapters, pages 75–138. National
Bureau of Economic Research, Inc.
Ederer, F. (2010). Feedback and motivation in dynamic tournaments. Journal of
Economics & Management Strategy, 19(3):733–769.
Egan, P. (2009). Issue ownership and representation: A theory of legislative
responsiveness to constituency opinion. Technical report.
Ellingsen, T. (1991). Strategic buyers and the social cost of monopoly. American
Economic Review, 81(3):648–57.
Epstein, G. S. and Nitzan, S. (2004). Strategic restraint in contests. European
Economic Review, 48(1):201–210.
Erikson, R. S. (1971). The advantage of incumbency in congressional elections.
Polity, 3(3):pp. 395–405.
Erikson, R. S. and Palfrey, T. R. (2000). Equilibria in campaign spending games:
Theory and data. American Political Science Review, 94:595–609.
Eso, P. and Szentes, B. (2007). Optimal information disclosure in auctions and
the handicap auction. Review of Economic Studies, 74(3):705–731.
Esteban, J. and Ray, D. (2001). Collective action and the group size paradox.
American Political Science Review, 95(3):663–672.
Field, E. (2007). Entitled to work: Urban property rights and labor supply in
peru. The Quarterly Journal of Economics, 122(4):1561–1602.
Friedman, D. (1994). A positive account of property rights. Social Philosophy
and Policy, 11(02):1–16.
Friedman, L. (1958). Game-theory models in the allocation of advertising ex-
penditures. Operations Research, 6(5):699–709.
BIBLIOGRAPHY 157
Fu, Q. (2006). Endogenous timing of contest with asymmetric information.
Public Choice, 129(1):1–23.
Fudenberg, D. and Tirole, J. (1991). Game Theory. The MIT Press.
Gal-Or, E. (1987). First mover disadvantages with private information. Review
of Economic Studies, 54(2):279–92.
Garfinkel, M. R. and Skaperdas, S. (2007). Economics of Conflict: An Overview,
volume 2 of Handbook of Defense Economics, chapter 22, pages 649–709. Else-
vier.
Garfinkel, M. R., Skaperdas, S., and Syropoulos, C. (2008). Globalization and
domestic conflict. Journal of International Economics, 76(2):296–308.
Gassebner, M., Jong-A-Pin, R., and Mierau, J. O. (2008). Terrorism and electoral
accountability: One strike, you’re out! Economics Letters, 100(1):126–129.
Gelman, A. and King, G. (1990). Estimating incumbency advantage without
bias. American Journal of Political Science, 34(4):pp. 1142–1164.
Gershkov, A. and Perry, M. (2009). Tournaments with midterm reviews. Games
and Economic Behavior, 66(1):162–190.
Gintis, H. (2007). The evolution of private property. Journal of Economic Behavior
& Organization, 64(1):1–16.
Goeree, J. and Grosser, J. (2007). Welfare reducing polls. Economic Theory,
31(1):51–68.
Gradstein, M. and Konrad, K. A. (1999). Orchestrating rent seeking contests.
Economic Journal, 109(458):536–45.
Green, J. and Hobolt, S. B. (2008). Owning the issue agenda: Party strategies
and vote choices in British elections. Electoral Studies, 27(3):460 – 476.
Grossman, G. M. and Helpman, E. (2001). Special Interest Politics. MIT Press.
Grossman, H. I. (2001). The creation of effective property rights. American
Economic Review, 91(2):347–352.
Grossman, H. I. (2002). “Make us a king”: Anarchy, predation, and the state.
European Journal of Political Economy, 18(1):31–46.
158 BIBLIOGRAPHY
Grossman, H. I. and Kim, M. (1995). Swords or plowshares? A theory of the
security of claims to property. Journal of Political Economy, 103(6):1275–88.
Gürtler, O. and Münster, J. (2010). Sabotage in dynamic tournaments. Journal
of Mathematical Economics, 46(2):179–190.
Hafer, C. (2006). On the origins of property rights: Conflict and production in
the state of nature. Review of Economic Studies, 73(1):119–143.
Harris, C. and Vickers, J. (1987). Racing with uncertainty. Review of Economic
Studies, 54:1–21.
Healy, A. J., Malhotra, N., and Mo, C. H. (2010). Irrelevant events affect voters’
evaluations of government performance. Proceedings of the National Academy
of Sciences, 107(29):12804–12809.
Hernandez-Lagos, P. and Tadelis, S. (2011). Reputation in repeated contests.
Working Paper, Haas School of Business, UC Berkeley.
Hillman, A. and Riley, J. (1989). Politically contestable rents and transfers. Eco-
nomics and Politics, 1:17–40.
Hirshleifer, J. (1991). The paradox of power. Economics and Politics, 3(3):177–200.
Hirshleifer, J. (1995). Anarchy and its breakdown. Journal of Political Economy,
103(1):26–52.
Hoffmann, M. (2010). Enforcement of property rights in a barter economy.
Social Choice and Welfare, 34(2):249–263.
Hong, C. S. and Konrad, K. (1998). Bandwagon effects and two-party majority
voting. Journal of Risk and Uncertainty, 16(2):165–172.
Hornbeck, R. (2010). Barbed wire: Property rights and agricultural develop-
ment. The Quarterly Journal of Economics, 125(2):767–810.
Hurley, T. M. and Shogren, J. F. (1998). Effort levels in a Cournot Nash contest
with asymmetric information. Journal of Public Economics, 69(2):195–210.
Iaryczower, M. and Mattozzi, A. (2009). On the nature of competition in alter-
native electoral systems. Technical report.
Iaryczower, M. and Mattozzi, A. (2011). The pro-competitive effect of cam-
paign limits in non-majoritarian elections. Economic Theory, pages 1–29.
10.1007/s00199-011-0613-y.
BIBLIOGRAPHY 159
Jacobs, L. R. and Shapiro, R. Y. (1994). Issues, candidate image, and priming:
The use of private polls in Kennedy’s 1960 presidential campaign. The Amer-
ican Political Science Review, 88(3):pp. 527–540.
Jovanovic, B. (1982). Truthful disclosure of information. The Bell Journal of
Economics, 13(1):36–44.
Kamstra, M. J., Kramer, L. A., and Levi, M. D. (2003). Winter blues: A sad stock
market cycle. The American Economic Review, 93(1):pp. 324–343.
Katsenos, G. (2009). Long-term conflict: How to signal a winner? Mimeo,
University of Hannover.
Kessler, A. S. (1998). The value of ignorance. RAND Journal of Economics,
29(2):339–354.
Kirby, A. J. (1988). Trade associations as information exchange mechanisms.
RAND Journal of Economics, 19(1):138–146.
Klumpp, T. and Polborn, M. K. (2006). Primaries and the New Hampshire
effect. Journal of Public Economics, 90(6-7):1073–1114.
Kolmar, M. (2008). Perfectly secure property rights and production inefficien-
cies in Tullock contests. Southern Economic Journal, 75(2):441–456.
Kolmar, M. and Rommeswinkel, H. (2010). Contests with group-specific public
goods and complementarities in efforts. Working Paper.
Konrad, K. and Skaperdas, S. (2010). The market for protection and the origin
of the state. Economic Theory, pages 1–27. 10.1007/s00199-010-0570-x.
Konrad, K. A. (2004). Inverse campaigning. Economic Journal, 114(492):69–82.
Konrad, K. A. (2009). Strategy and Dynamics in Contests. Oxford University
Press.
Konrad, K. A. and Kovenock, D. (2005). Equilibrium and efficiency in the tug-
of-war. Technical report.
Konrad, K. A. and Leininger, W. (2007). The generalized Stackelberg equilib-
rium of the all-pay auction with complete information. Review of Economic
Design, 11(2):165–174.
Kovenock, D., Morath, F., and Münster, J. (2010). Information sharing in con-
tests. Mimeo.
160 BIBLIOGRAPHY
Krasa, S. and Polborn, M. (2010). Competition between specialized candidates.
American Political Science Review, 104(04):745–765.
Krishna, V. and Morgan, J. (1997). An analysis of the war of attrition and the
all-pay auction. Journal of Economic Theory, 72(2):343–362.
Krueger, A. O. (1974). The political economy of the rent-seeking society. The
American Economic Review, 64(3):pp. 291–303.
Lagerlöf, J. (2007). A theory of rent seeking with informational foundations.
Economics of Governance, 8(3):197–218.
Lambsdorff, J. G. (2002). Corruption and rent-seeking. Public Choice, 113(1-
2):97–125.
Lazear, E. (1995). Personell Economics. The MIT Press, Cambridge.
Lazear, E. and Rosen, S. (1981). Rank-order tournaments as optimum labor
contracts. Journal of Political Economy, 89(5):841–864.
Leininger, W. (1993). More efficient rent-seeking – a Munchhausen solution.
Public Choice, 75 (1):43–62.
Li, L. (1985). Cournot oligopoly with information sharing. The RAND Journal
of Economics, 16(4):pp. 521–536.
Linster, B. G. (1993). Stackelberg rent-seeking. Public Choice, 77:307–21.
McAllister, I. and Studlar, D. T. (1991). Bandwagon, underdog, or projection?
opinion polls and electoral choice in britain, 1979-1987. The Journal of Politics,
53(3):pp. 720–741.
McBride, M., Milante, G., and Skaperdas, S. (2011). Peace and war with en-
dogenous state capacity. Journal of Conflict Resolution, 55(3):446–468.
Mehlum, H. and Moene, K. (2002). Battlefields and marketplaces. Defence and
Peace Economics, 13(6):485–496.
Mehlum, H., Moene, K., and Torvik, R. (2005). Crime induced poverty traps.
Journal of Development Economics, 77(2):325–340.
Meirowitz, A. (2005). Polling games and information revelation in the downsian
framework. Games and Economic Behavior, 51(2):464–489.
BIBLIOGRAPHY 161
Meirowitz, A. (2008). Electoral contests, incumbency advantages, and cam-
paign finance. The Journal of Politics, 70(03):681–699.
Milgrom, P. (2008). What the seller won’t tell you: Persuasion and disclosure
in markets. Journal of Economic Perspectives, 22(2):115–131.
Moen, E. R. and Riis, C. (2010). Policy reversal. American Economic Review,
100(3):1261–68.
Moldovanu, B. and Sela, A. (2001). The optimal allocation of prizes in contests.
American Economic Review, 91(3):542–558.
Moldovanu, B. and Sela, A. (2006). Contest architecture. Journal of Economic
Theory, 126(1):70–96.
Morgan, J. (2003). Sequential contest. Public Choice, 116:1–18.
Morgan, J. and Stocken, P. C. (2008). Information aggregation in polls. American
Economic Review, 98(3):864–96.
Morgan, J. and Várdy, F. (2007). The value of commitment in contests and tour-
naments when observation is costly. Games and Economic Behavior, 60(2):326–
338.
Moscarini, G. and Smith, L. (2011). Optimal dynamic contests. Technical report.
Working paper.
Mueller, D. C. and Stratmann, T. (1994). Informative and persuasive campaign-
ing. Public Choice, 81(1-2):55–77.
Muthoo, A. (2004). A model of the origins of basic property rights. Games and
Economic Behavior, 49(2):288–312.
Nadeau, R., Cloutier, E., and Guay, J.-H. (1993). New evidence about the exis-
tence of a bandwagon effect in the opinion formation process. International
Political Science Review, 14(2):pp. 203–213.
Nalebuff, B. J. and Stiglitz, J. E. (1983). Prices and incentives: Towards a general
theory of compensation and competition. Bell Journal of Economics, 14(1):21–
43.
Nitzan, S. (1994). Modelling rent-seeking contests. European Journal of Political
Economy, 10(1):41–60.
162 BIBLIOGRAPHY
Nordhaus, W. D. (1975). The political business cycle. Review of Economic Studies,
42(2):169–90.
Nti, K. O. (1999). Rent-seeking with asymmetric valuations. Public Choice, 98(3-
4):415–30.
O’Keeffe, M., Viscusi, W. K., and Zeckhauser, R. J. (1984). Economic contests:
Comparative reward schemes. Journal of Labor Economics, 2(1):27–56.
Persico, N. (2000). Information acquisition in auctions. Econometrica, 68(1):135–
148.
Petrocik, J. R. (1996). Issue ownership in presidential elections, with a 1980 case
study. American Journal of Political Science, 40(3):825–850.
Petrocik, J. R., Benoit, W. L., and Hansen, G. J. (2003). Issue ownership and
presidential campaigning, 1952-2000. Political Science Quarterly, 118(4):pp.
599–626.
Polborn, M. (2006). Investment under uncertainty in dynamic conflicts. Review
of Economic Studies, 73(2):505–529.
Potters, J. and van Winden, F. (1992). Lobbying and asymmetric information.
Public Choice, 74:269–292.
Prendergast, C. (1999). The provision of incentives in firms. Journal of Economic
Literature, 37(1):7–63.
Reynolds, A., Reilly, B., and Ellis, A. (2005). Electoral System Design: The New
International IDEA Handbook. International Institute for Democracy and Elec-
toral Assistance.
Riker, W. (1996). The strategy of rhetoric: Campaigning for the American constitution.
J-B NCR Single Issue National Civic Review Series. Yale University Press.
Rosen, S. (1986). Prizes and incentives in elimination tournaments. American
Economic Review, 76(4):701–15.
Sahuguet, N. and Persico, N. (2006). Campaign spending regulation in a model
of redistributive politics. Economic Theory, 28(1):95–124.
Saunders, Edward M., J. (1993). Stock prices and Wall Street weather. The
American Economic Review, 83(5):pp. 1337–1345.
Schelling, T. (1960). The Strategy of Conflict. Harvard Univ Press.
BIBLIOGRAPHY 163
Shapiro, C. (1986). Exchange of cost information in oligopoly. The Review of
Economic Studies, 53(3):pp. 433–446.
Shavell, S. (1991). Individual precautions to prevent theft: Private versus so-
cially optimal behavior. International Review of Law and Economics, 11(2):123–
132.
Siegel, R. (2009). All-pay contests. Econometrica, 77(1):71U92.
Sigelman, L. and Buell, E. H. (2004). Avoidance or engagement? Issue conver-
gence in U.S. presidential campaigns, 1960-2000. American Journal of Political
Science, 48(4):650–661.
Sisak, D. (2009). Multiple prize contests - the optimal allocation of prizes. Jour-
nal of Economic Surveys, 23(1):82–114.
Skaperdas, S. (1992). Cooperation, conflict, and power in the absence of prop-
erty rights. American Economic Review, 82(4):720–739.
Skaperdas, S. (1996). Contest success functions. Economic Theory, 7(2):283–290.
Skaperdas, S. (2006). Anarchy. In Weingast, B. and Wittman, D., editors, Oxford
Handbook of Political Economy. Oxford University Press.
Skaperdas, S. and Grofman, B. (1995). Modeling negative campaigning. The
American Political Science Review, 89(1):49–61.
Skaperdas, S. and Syropoulos, C. (2002). Insecure property and the efficiency
of exchange. Economic Journal, 112(476):133–146.
Skaperdas, S. and Vaidya, S. (2009). Persuasion as a contest. forthcoming Eco-
nomic Theory.
Slantchev, B. L. (2010). Feigning weakness. International Organization, 64:357–
388.
Snyder, J. M. (1989). Election goals and the allocation of campaign resources.
Econometrica, 57(3):pp. 637–660.
Spangenberg, F. (2003). The Freedom to Publish Opinion Poll Results – Report on a
Worldwide Update. Amsterdam: Foundation for Information.
Straffin Jr., P. D. (1977). The bandwagon curve. American Journal of Political
Science, 21(4):pp. 695–709.
164 BIBLIOGRAPHY
Stromberg, D. (2008). How the electoral college influences campaigns and pol-
icy: The probability of being Florida. American Economic Review, 98(3):769–
807.
Tullock, G. (1967). The welfare costs of tariffs, monopolies, and theft. Economic
Inquiry, 5(3):224–232.
Tullock, G. (1980). Efficient rent seeking, pages 97–112. J. Buchanan, R. Tollison
and G.Tullock: Towards a Theory of the Rent-Seeking Society. Texas A&M
University Press.
Umbeck, J. (1977). The California gold rush: A study of emerging property
rights. Explorations in Economic History, 14(3):197–226.
Umbeck, J. (1981). Might makes rights: A theory of the formation and initial
distribution of property rights. Economic Inquiry, XIX:38–59.
U.S. Department of Justice (2002). Burglary of single-family houses. Problem-
Oriented Guides for Police Series, 18.
Vives, X. (1984). Duopoly information equilibrium: Cournot and Bertrand.
Journal of Economic Theory, 34(1):71 – 94.
Vives, X. (1990). Trade association disclosure rules, incentives to share informa-
tion, and welfare. RAND Journal of Economics, 21(3):409–430.
Vowles, J. (2002). Did the polls influence the vote? A case study of the 1999
new zealand general election. Political Science, 54(1):67–78.
Wärneryd, K. (2003). Information in conflicts. Journal of Economic Theory,
110:121–136.
Yildirim, H. (2005). Contests with multiple rounds. Games and Economic Behav-
ior, 51(1):213–227.
Curriculum Vitae
Education
2007 - 2012 PhD in economics and finance,
University of St. Gallen,
2001 - 2006 Diploma in economics (Diplom-Volkswirt),
Johannes Gutenberg University, Mainz,
Dongbei University of Finance and Economics, Dalian.
Professional Experience
2007 - 2012 Research and teaching assistant,
University of St. Gallen,
2011 Visiting researcher,
University of California, Irvine,
2007 - 2008 Student assistant,
Johannes Gutenberg University, Mainz.
July 2012 Philipp Denter