A Theory of Strategic Reciprocal

Altruism (the good, the bad and the

unfair)

Nikolaos Georgantzís

LEE/LINEEX , Universitat Jaume I, Castellón, Spain

and University of Cyprus, Economics Dept.

e-mail: georgant@eco.uji.es; phone: 00357 22892397

Miguel Ginés

LEE and Economics Dept., Universitat Jaume I, Castellón, Spain

e-mail: mgines@eco.uji.es; phone: 0034-964-728593

Abstract

We present a model of strategic reciprocal altruism. Contrary to

most models in the literature, undertaking costly actions which only

bene�t others is shown to be an equilibrium in the absence of other-

regarding preferences or reputation considerations. In the general case,

a good and a bad equilibrium may exist. In the former, altruistic in-

vestments in another agent�s capacity to bene�t from synergies arising

from both agents�e¤orts leads to a socially and individually superior

outcome, while in the latter no altruism at all leads to the worst equi-

librium. Under more speci�c conditions, intermediate equilibria are

shown to exist. In the asymmetric case, such equilibria are unfair in

the sense that e¢ cient agents o¤er more to others and enjoy less utility

than ine¢ cient ones. Among other extensions discussed, coordination

against the Pareto inferior equilibria can be implemented by sequen-

tial play which o¤ers a rational explanation why reciprocity has been

1

traditionally seen as sequences of alternating altruistic actions.

Keywords: Reciprocity, Altruism, Fairness, Subgame Perfection.

1 Preliminary thoughts

It is increasingly remembered nowadays1 that Adam Smith in his Theory of

Moral Sentiments (1759) had explicitly referred to the pleasure of contem-

plating others� happiness. This fact alone can explain altruistic behavior.

The formalization of this type of altruism has lead to approaches which,

the more they comply with the axioms of neoclassical economics, the more

they tend to reduce altruism down to the standard story of sel�sh (own)

utility maximization. This is due to the fact that the existing neoclassical

approaches to altruism need some (positive) other-regarding component in

an individual�s utility function to explain why my own actions may bene�t

the society surrounding me.2

1A recent example is Bruni (2006).2An exhaustive review of the vast literature goes beyond the scope of this article, but

the works by Bolton, Brandts and co-authors (for example, Bolton and Ockenfels (2000),

Bolton et al. (1998), Brandts and Charness (2000) and Brandts and Sola (2001)) and

Fehr and co-authors (for example, Fehr et al. (1997), Fehr and Schmidt (1999), Fehr

and Fischbacher (2002), Fehr and Gächter (1998a, 1998b, 2000, 2002)) on the issues of

reciprocity and social preferences, respectively, merit special attention. For example, in

a famous article, Fehr and Schmidt (1999) model inequity aversion, assuming that an

agent�s utility depends on own (i) and n � 1 other agents� (j) payo¤s (�) as captured

by the function: Ui(�) = �i �h�in�1

Pj 6=imax(�j � �i; 0) +

�in�1

Pj 6=imax(�i � �j ; 0)

i.

The �rst term in the square bracket accounts for i�s utility loss due to envy felt if own

pro�ts are lower than the the average among the remaining agents, whereas the second

term is utility loss due to compassion if the contrary holds. From the function presented

here it is straightforward to see that a series of formal problems arise due to recursiveness

when the other agent�s utility rather than payo¤s are introduced in an agent�s own utility

function. Thus, economic agents�other-regarding behavior can only concern other agent�s

payo¤s not happiness (utility) in order to de�ne an economically meaningful approach to

2

A more sel�sh individual is depicted by modern economists who �nd in

altruism a signal triggering reciprocal behavior by those bene�tting from the

�rst mover in this chain of favors. This explanation of altruistic action as-

sumes purely sel�sh individuals engaged in strategic interaction with others.

Formally establishing the conditions under which this second type of altru-

istic behavior emerges usually requires the repetition of human interactions

providing a ground on which reputation of friendly or pro-social behavior

may lead to more cooperative outcomes.3 In terms of observable actions, the

two aforementioned sources of behavior with a positive externality on others

are rather di¢ cult to distinguish from each other, but they would both fail

to explain altruistic behavior by purely sel�sh individuals interacting with

each other only once.

A third approach has been adopted by oligopoly theorists4, according

to which voluntarily committing to donating a part of a duopolist�s pro�t

to a rival increases the latter�s incentives to behave in a cooperative way

in the market competition stage. Although this approach has not been

applied so far to model human interaction, it is technically similar to our

framework, although the way in which strategic complementarity emerges

in the oligopoly pro�t-sharing approach is very di¤erent to ours.

Certainly closer to the second and especially the third than to the �rst of

the aforementioned approaches, our framework does not include any other-

regarding component in an agent�s utility function, nor does it depend on

the repetition of human interactions. The resulting behavior looks more like

altruism, envy, reciprocity or even inequity aversion.3Apart from the well known in�nite-repetition theoretical approach to cooperation,

there are several experimental papers which establish the relation between the (probability

of a) �nite repetition of a noncooperative situation with the emergence of cooperative

behavior. Among them, we mention Andreoni and Miller (1993), Sabater and Georgantzís

(2002) and Du¤y and Ochs (2003).4See, for example, recent work by Waddle (2005) and the seminal papers by Malhueg

(1992), FitzRoy and Kraft (1986) and Farrel and Shapiro (1990).

3

reciprocity of the type �I bene�t you in order to increase your e¤ort in a

task which is synergic to my actions�, with the novel features that the mag-

nitude of strategic complementarity is endogenously determined by agents�

actions and that altruistic behavior occurs in the absence of repeated inter-

action, reputation and altruistic components in the agents�utility functions.

Apart from showing the possibility of obtaining such reciprocally altruistic

equilibria we o¤er a rationale of why altruistic actions are easier to observe

when agents act sequentially.

The remaining part of the paper presents the theoretical framework and

discusses possible extensions.

2 A model of strategic reciprocal altruism

Two players, i 2 f1; 2g decide on the level of their e¤ort xi, which yields

them a linear bene�t and a quadratic cost. Apart from these e¤ects of

e¤ort on own utility, the two players�e¤orts may interact positively to yield

a further bene�t to each one of them. However, each player�s capacity to

bene�t from the interaction of e¤orts depends on the other�s fully altruistic

decision to undertake a costly investment in the former�s synergy-absorbing

capacity. This investment is altruistic in the sense that it is costly and has

no direct positive e¤ect on the altruist�s utility. This is captured by a utility

function like the one given by:

Ui = xi +�j

1 + �jxixj �

1

2x2i � C(�i); (1)

where �i is �rm i�s investment in j�s capacity to bene�t from synergies

arising due to the two agents� e¤orts and C(�); C 0(�); C 00(�); xi; xj are non

nonnegative real numbers.

It should be noted that the player�s investment in the other�s synergy-

absorbing capacity has solely a negative direct impact on own utility, and a

4

potentially (if e¤orts are both higher than 0) positive impact on the other�s

utility. The speci�cation of e¤ort synergies in (1) is similar to the way in

which spill-overs are usually modeled in IO models of R&D competition.

There, several results show that �rms may wish to share with other their

technological advances5, but none of the models has gone as far as to assume

that a �rm may want to undertake costly actions to increase the other�s ca-

pacity to absorb the synergies arising from their R&D e¤orts. Despite the

obvious analogies, the literature on spill-overs between �rms has not been

su¢ ciently exploited to provide explanations for reciprocal behavior between

individuals or social groups. Strangely, Vives�(1990, 2005, 2006) work on

strategic complementarities has also remained unexploited by theorists of

strategic interaction between humans. In fact, Vives (2006) explicitly refers

to the way in which multistage interaction may a¤ect equilibria in the pres-

ence of strategic complementarities.6

The two players play a two stage game once. In the �rst stage, play-

ers decide simultaneously on their gift-investment in each other�s synergy-

absorbing capacity. In the second stage, they simultaneously decide on their

e¤ort levels.

Following backward induction, we �rst discuss the unique Nash equilib-

rium of the second stage.

Setting the derivative of (1) with respect to the e¤ort level xi equal to

5For example, Gil-Moltó et al. (2005) show that �rms may deliberately choose to

increase their technological similarities in order to increase their mutual bene�ts from

synergies arising from their R&D e¤orts. More recently, Milliou (2006), in an endogenous

spill-over duopoly model, shows that �rms would make no costly ivestment to protect

their innovations, which is a minimum requirement for the main result presented here to

hold.6 In this sense, our model may be seen as a special case of Vives� (2006) framework,

although, from a technical point of view some of our assumptions contradict those in Vives

(2005). Also, we are more interested in equilibria which are di¤erent from the obvious

candidate of non investment in others�complementarity-absorbing capacity.

5

zero, we obtain the �rst order conditions:

@Ui@xi

= 0) xi = 1 +�j

1 + �jxj (2)

It should be observed that the slope of i�s e¤ort-reaction function in (2)

positively depends (approaching unity asymptotically from below) on j�s

altruistic investment in the former�s capacity to absorb the synergies from

the interaction of e¤orts. Thus, player i�s reaction to j�s e¤ort will positively

depend on j�s investment in the former�s synergy-absorbing capacity. Our

main result in this paper crucially depends on this property of interaction

in the e¤ort-choice stage of the game. It is also worth noting that e¤orts are

strategic complements only for strictly positive values of �. In the absence

of e¤ort synergies, strategic interaction in e¤ort levels disappears, yielding

e¤ort levels of 1 and equilibrium utilities of 1=2.

Solving the system of reaction functions in (2) with respect to xi; xj gives

equilibrium e¤ort levels:

x�i (�i; �j) =(1 + 2�j)(1 + �i)

1 + �i + �j; (3)

which, when substituted into (1), after some nontrivial but standard re-

arrangements, give:

U�i (�i; �j) =(1 + 2�j)

2(1 + �i)2

2(1 + �i + �j)2

� C(�i); (4)

We use this expression to solve the �rst stage of the game.

Di¤erentiating (4) with respect to �i gives:

@U�i (�i; �j)

@�i=�j(1 + 2�j)

2(1 + �i)

(1 + �i + �j)3

� @C(�i)@�i

(5)

Further di¤erentiating the �rst term of the right hand side expression of

(5), we obtain that the equilibrium utility net of altruistic costs is neither

convex nor concave as implied by:

6

@��j(1+2�j)

2(1+�i)

(1+�i+�j)2

�@�i

=�j(1 + 2�j)

2(�j � 2� 2�i)(1 + �i + �j)

4� 0: (6)

Thus, having assumed that C(�) is convex, we can see that, generally

speaking, the equation:

�j(1 + �j)2(1 + �i)

(1 + �i + �j)3

=@C

@�i(7)

may not lead to an interior equilibrium in the �rst stage of the game. We

illustrate here the existence of such equilibria and explore their properties

adopting the following speci�cation:

C(�i) =1

2ki((1 + �i)

2 � 1): (8)

Then, equation (7) is rewritten as:

�j(1 + �j)2(1 + �i)

(1 + �i + �j)3

= ki(1 + �i) (9)

or, simplifying:

�j(1 + �j)2

(1 + �i + �j)3= ki (10)

Now, it is easy to derive a unique best response function in altruistic

investments on the positive reals, using Descartes�sign rule of a polynomial.

�i(�j) = maxf0;3

s(1 + 2�j)

2�jki

� �j � 1g (11)

This corresponds to a maximum since � < �i(�j) implies that:

@U�i (�; �j)

@�i> 0 (12)

and � > �i(�j) implies@U�i (�; �j)

@�i< 0 (13)

7

Now we study the properties of the best response functions which are

graphically represented together with the resulting equilibrium in Figure 6.

First, it is easy to show that it is an increasing function of �j if ki � 4 and

it is a concave function with independence of the values of ki (after some

computation on the derivatives).

We also derive two interesting values for �j .

1) If ki � 4 then there is a �j such that �i(�j) = 0 and � > �j implies

�i(�) > 0.

2) If ki � 1=2 then �i(�) is a function that does not cross the diagonal,

otherwise ki < 1=2 implies that b� = ki1�2ki is the point where the function

�i(�) crosses the diagonal.

Finally if there is a bound � > b� = ki1�2ki on the possible � and ki < 1=2

there is �j � � such that �i(�j) = �.

Similarly, the best response of agent j satis�es the same properties since

the agents are equal except for the gift-production technology expressed in

j�s gift-cost parameter kj .

The �rst result we could derive is that the bad equilibrium always exists

with independence of the values of k.

Result 1: Existence of a Bad equilibrium (��i ; ��j ) = (0; 0).

Proof: If �rm j sets �j = 0, the right hand side of (7) is always larger

than the left hand side. Thus, �rm i�s best response will also be �i = 0. By

symmetry, this implies that this is always an equilibrium.�

Now, depending on the values of ki we could also �nd an interior equilib-

rium and other extreme equilibria. Before exploring the properties of these

equilibria, we summarize the conditions under which they may emerge.

CASE 1: If ki � 1=2 for all i = 1; 2 then no other equilibrium than the

bad one described above exists since none of the best responses cross the

diagonal.

This case is depicted in Figures 1 and 2, which show that such a situation

8

may emerge either in a symmetric or an asymmetric setting. A straightfor-

ward proof is also available from the authors of the fact that, when unique,

this equilibrium is also globally stable.

CASE 2: If there is a ki > 1=2 and kj < 1=2 then we could �nd asym-

metric situations in which either interior and extreme equilibria co-exist, or

only the aforementioned bad equilibrium exists. The former of these situ-

ations is depicted in Figure 3, while an example of the latter has already

been mentioned in Figure 2.

CASE 3: If ki � 1=2 for all i = 1; 2 then we surely �nd three equilibria,

the bad one, an interior one and the good one, which implies full reciprocity.

Figures 4 and 5 present examples of a symmetric and an asymmetric situa-

tion in which such equilibria emerge. A proof is available form the authors

that, in the case in which good and bad equilibria co-exist, they are both

locally stable. On the contrary, interior equilibria are, unstable.

Now we characterize the interior and good equilibria of the game.

Result 2: Existence of an Interior equilibrium (��i ; ��j ).

Proof: Given ki � 1=2 for all i = 1; 2, we know that �(�) > � and the

interior equilibrium is the solution to the following system of equations:

�1(�2) = maxf0;3

s(1 + 2�2)

2�2k1

� �2 � 1g (14)

�2(�1) = maxf0;3

s(1 + 2�1)

2�1k2

� �1 � 1g (15)

De�ne the following function: S(�) = � � 3

q(1+2�0)2�0

k1+ �0 + 1 with

�0 = 3

q(1+2�)2�

k2� � � 1. S(0) = 3

q1k1and S(�) < 0 since �0 is increasing in

� and �0(�) > �. Then, by continuity there exists a �� such that S(��) = 0

and ��1 = �� and ��2 = �2(�

�1) �

Result 3: Existence of a Good equilibrium (��i ; ��j ) = (�; �).

9

Proof: If there is a bound on the maximal investment � for each agent,

the good equilibrium is characterized by agents investing their full capacity.�

The three aforementioned equilibria can be easily ranked using the Pareto

criterion.

It is also interesting to show that if agents di¤er in their e¢ ciencies to

increase each other�s synergy-absorbing capacities (ki 6= kj), asymmetric

equilibria arise. For example, if k1 = 0:16 and k2 = 0:13, the resulting

interior equilibrium is given by: (��1; ��2) = (1; 1:1). In such an equilibrium,

the e¢ cient agent invests more in the ine¢ cient one�s absorptive capacity

than does the latter in favor of the former. This leads the e¢ cient agent

to enjoy lower levels of utility than does the ine¢ cient one (1:8858; 1:83).

Therefore, the interior equilibria obtained here are unfair, in the sense that

e¢ cient and, thus, generous agents enjoy less utility than their ine¢ cient

counterparts.

In a more general model in which fairness considerations are relevant, the

fact that interior equilibria are, generally speaking, unfair makes them more

di¢ cult to emerge and more vulnerable to opportunistic, non-altruistic be-

havior. The di¢ culty of such unfair equilibria to emerge is further enhanced

by the fact that they are unstable. An interesting, straightforward and re-

alistic way of implementing the good equilibrium is by sequential play. We

show the following result:

Result 4: Sequential play implements the Good equilibrium.

Proof:

We will show that, given ki � 1=2 for all i = 1; 2 and the best response

function of agent 2, the best choice for agent 1 is to choose �1 = �. Fur-

thermore the equilibrium attained is the good one.

Let U�i (�i; �j) be the utility attained by agent i with generic �i; �j ,

which is strictly increasing in �j . Then for any �i, U�i (�i; �j) � U�i (�i; �)

and since the best response of agent i, �i(�) = �. Finally we conclude that

10

max�i;�j2[0;�]U�i (�i; �j) = U

�i (�; �).

The equilibrium in the sequential play is given bymax�i2[0;�] U�i (�i; �j(�i)).

But the utility attained in this way is always lower thanmax�i;�j2[0;�] U�i (�i; �j) =

U�i (�; �). And we know that �i = � attains this utility. Then we conclude

that there may be more than one equilibrium in the sequential play but at

least the full reciprocity equilibrium always exists. �

3 An example from peer review processes

Among numerous examples of strategic reciprocal altruism, peer review

processes exhibit interesting analogies with our framework. There are many

reasons why peer reviewing is used by most scienti�c journals and confer-

ences as a way of screening and selecting among the papers submitted to

them. Ideally, reviewers should be interested in letting the best of the con-

tributions go through, in order to maximize the journal�s or the conference�s

overall quality. However, unlike the editors, who are dealing with all the

submissions at a time, referees have local information and cannot compare

across di¤erent submissions. On the other hand, editors have the ability to

compare across di¤erent referee reports but may be misled by the fact that

some referees have spent more time than others in understanding a paper,

in order to reach positive conclusions yielding useful, constructive advice to

the authors. It should be made clear that, here, the terms positive, good

quality or, simply, good referee reports are used to talk about reports which

contain encouraging and insightful comments on the authors�work, or in any

case constructive criticisms which help the authors increase their e¤ort and

improve their future results from the research reported in the papers under

review. Thus, by de�nition, destructive, personal, arrogant and, obviously,

wrong comments on a paper are never part of such a report. To make things

simpler, a good report is assumed to be the result of an altruistic dedication

of time and e¤ort by the referee, trying to understand the authors�work

11

and then advice them in a constructive way on the future of their research.

Of course, it can be argued that some papers are totally unworthy, in which

case the best advice to the authors would be to abandon any e¤ort on this

direction. We rule this possibility out, because we are not aware of any

case in which a destructive, bad or arrogant referee�s report was the best a

paper could receive as a feedback by the scienti�c community. There is a

widespread feeling that a referee�s dedicated and careful involvement in the

work under review is more likely to yield a positive outcome.

In a world like the one described above, explaining why an individually

rational agent would undertake any cost to write a careful, insightful and

helpful referee report is not a trivial task.

Our framework would explain the altruistic dedication of a referee as

an attempt to improve the author�s work on a given topic as a strategic

investment aimed at increasing the author�s optimal e¤ort and eventually

abilities to deal with a research question whose answer is synergic to the

referee�s own research. The peer review example discussed here o¤ers a rich

source of empirical data (although not always observable but abundant in

informal narration among economists, for example) on whether a real-world

referee recognizes the complementarity between his own research and that

of other researchers, or whether, on the contrary, looks at others�e¤ort as

a strategic substitute of his or her own research agenda, in which case a

negative (bad) referee report is the optimal strategy.

4 Concluding remarks-Extensions

Reciprocity may arise in the absence of any reputation e¤ects or cooperation

due to repeated play. In this paper, we show how strategically reciprocal be-

havior may emerge from the Homo Economicus�interest to increase others�

capacity to bene�t from his own actions.

We have shown this in a two-person world in which agents�bene�ts from

12

own e¤ort are enhanced by the e¤ort of the other player. In this context, an

agent will altruistically incur a cost, without directly bene�tting from it, in

order to help the other to bene�t from synergies arising due to both agents�

e¤orts. This result is due to the strategic e¤ect arising from such an altruistic

behavior, leading to an increase of the two players�interdependence in the

e¤ort-choice stage of the game.

Apart from the sequential-move extension discussed above, it would be

interesting to investigate whether explicit cooperation in the stage of e¤ort

choice facilitates the emergence of the reciprocal equilibrium and whether

increasing the group size a¤ects the reciprocity equilibrium described here.

Our conjecture is that, if e¤ort synergies enter into an agent�s utility as the

sum of bilateral interaction terms, agents may (under speci�c conditions)

prefer to specialize in two-person reciprocal relations, whereas if multilateral

synergies appear in an agent�s utility as the product of many agents�e¤orts,

global (social) reciprocity equilibria are more likely to emerge.

To summarize, our results predict that if human interaction is the result

of situations like the ones described here, three types of outcomes should

be expected. The good one, in which everyone does everything possible for

others to bene�t from his own e¤ort, extracting everybody�s maximal e¤ort

and yielding maximal utility to all. The bad one, in which nobody invests

in others�synergy-absorbing capacities, extracting minimal, individually op-

timal e¤ort levels and a positive but minimal equilibrium utility resulting

from isolated human action. The unfair ones in which e¢ cient agents con-

tribute more and receive less than ine¢ cient ones, yielding e¤ort levels and

utilities lying between the two aforementioned equilibria.

Surprisingly, in many real world situations people �nd it hard to reach

the good equilibrium. So far, this has been explained as a consequence of

people�s lack of altruism or, alternatively, as dynamic instability of the co-

operative outcome in situations of repeated interaction. In our framework,

13

failure to reach the good equilibrium can only be due to lack of synergies

arising from interacting agents�e¤orts. Certainly, a more plausible expla-

nation could be that agents often fail to recognize the bene�ts from actions

which bene�t others, if such actions do not have a direct positive impact

on their own utility. Then, it would be due to strategic myopia not lack of

altruism that good equilibria may not be reached as often as they should.

14

5 Appendix: Figures

k1=k2>0.5

β1(β2)

β2(β1)

β2

β1

Existence of a unique“bad” equilibrium(symmetric case).

Existence of a unique“bad” equilibrium(symmetric case).

Figure 1: A unique "bad" equilibrium (symmetric case).

15

k1<0.5,k2>0.5

β1(β2)

β2(β1)β1

β2

Existence of a unique“bad” equilibrium(asymmetric case).

Existence of a unique“bad” equilibrium(asymmetric case).

Figure 2: A unique "bad" equilibrium (asymmetric case).

16

k1<0.5, k2>0.5

β1(β2)

β2(β1)

β1

β2

Existence of one “bad”and two “unfair”equilibria.

Existence of one “bad”and two “unfair”equilibria.

Figure 3: A "bad" equilibrium co-existing with two asymmetric ones.

17

k1=k2<0.5

β1(β2)

β2(β1)

β1

β2

Existence of a “bad”, an“interior” and a “good”equilibrium (symmetric case).

Existence of a “bad”, an“interior” and a “good”equilibrium (symmetric case).

Figure 4: A "bad", an interior and a "good" equilibrium (symmetric case).

18

k1,k2<0.5

β1(β2)

β2(β1)

β1

β2

Existence of a “bad”, a“good” and an “unfair”equilibrium.

Existence of a “bad”, a“good” and an “unfair”equilibrium.

Figure 5: The "bad the good and the unfair" case.

19

Equilibrium in the effort choiceEquilibrium in the effort choicestagestage

xi

xj

Figure 6: Best response functions and equilibrium in the e¤ort choice stage.

20

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