signature redacted l 'u
TRANSCRIPT
THE ROLE OF TIME IN NEGOTIATIONS
by
ZVI AVIAD LIVNE
B.Sc., Hebrew University, Jerusalem, Israel (1975)
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
(Septenlber 1979)
© Zvi Aviad Livne
Signature of Author ••
J
Signature redacted · · · · · 1 · ~e;a;t~e~t.of Ma~a;e~e~t: ju~e.26, ·1979
Signature redacted Certified by~A· >' • • ~- (
Accepted by • . .. --.
·--~ ••• ----~·: ••••••• Th~sis·c~-su;e~is~~
Signature redacted · · · "'\ · · I · · · · · -· ·~ ~si~ · c~-Su;e~is~~
. Signciture redacted , •••••• , ••• ~· •••. • ••••••••• P'. : •• • II' f "-- l . 'u -_'-1.,,..,,
. .:Chairman, lnterde · . ental· Committee
MCHIVES MAS~~Hli."il:m 1Nsr,rui€
CF TECHNtfUHIV
SEP 2 ~ 1979 LIBRAAIU
-2-
THE ROLE OF TIME IN NEGOTIATIONS
BY
ZVI AVIAD LIVNE
Submitted to the Department of Managementon June 26, 1979 in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
ABSTRACT
The effect of time on negotiating parties and on the environ-ment in which they operate is an important factor, de-emphasized or
even ignored in formal models of negotiations. This thesis formulatesand studies the effects of time by incorporating time-related costsinto existing models and by developing new ones.
The Nash Bargaining Problem, viewed as an arbitration problem,is reformulated by the introduction of discounting and bargainingcosts into the utility functions of the parties. Two solutionfunctions which satisfy a set of arbitration principles and fairnesscriteria are proposed. One solution is shown to pick the NashCooperative Solution of the traditional bargaining problem for thespecial case of equal discounting rates and no bargaining costs.
The economic search problem is reformulated and the possibilityof negotiating during the search is introduced. Optimal search-negotiation strategies with "reservation price" properties arefound. The concept of "characteristic concession patterns", i.e.concession strategies with simple functional representations isdeveloped. Equilibrium strategies for a negotiation game in whichthe participants adhere to their patterns are computed.
Modelling negotiations as problems of adaptive control isexamined. It is shown that introducing time effects may facilitateformulation and may make the model more realistic. Bayesian updatingof probabilities is introduced, and is closely associated with theconcepts of "negotiator type" and "reservation price".
-3-
A variety of other problems are discussed throughout the thesis.Among them: The role of incentives in arbitration; a variant of thePick-the-Best problem with unknown number of observations; and amodification of an existing game theoretical model of reputationin bargaining. A case study involving the coin market in Israelsupplies motivating examples. The results of an experimentalnegotiation game (The Streaker) are examined.
Some applications and possible empirical and theoretical studiesare proposed. It is believed that the results can be helpful toarbitrators and negotiators who are willing to consider explicitlythe effects of time on negotiations.
Thesis Co-Supervisors: Howard Raiffa, Professor of Managerial Economics,School of Business Administration, Harvard.
Gordon M. Kaufman, Professor of Operations Researchand Management, Sloan School of Management, MIT.
-4-
To my mother and father,
to whom I owe so much.
- 5 -
ACKNOWLEDGEMENTS
I was fortunate to have on my thesis committee three warm, generousand intellectually stimulating members, Howard Raiffa, Gordon Kaufman andJoseph Ferreira. Professor Raiffa, my thesis co-supervisor, introducedme to the area which can be called "The Analysis of Negotiations andBargaining". Through his innovative course in the Harvard Business Schooland later through his criticisms and suggestions, I developed a betterunderstanding of the problems and was able to formulate new problems whichare, hopefully, relevant and practical, not purely theoretical. I am alsograteful for the time he spent with me on evenings and vacations trying toimprove the presentation and the style of the manuscript.
Professor Kaufman deserves special thanks both for his perceptiveco-supervision of this thesis and for his important influence on my graduateeducation at M.I.T. He introduced me to the areas of Decision Analysis andBayesian Statistics and helped me financially with Research and TeachingAssistantships. Assisting him in his work in the energy field was a pleasantand rewarding experience.
Professor Ferreira read this thesis carefully and suggested many impor-tant corrections. He was very helpful in proposing new perspectives andinteresting examples to which some of the results in this thesis might beapplied.
I owe special thanks to Abraham Nachmany who spent long hours with medescribing the coin market in Israel, answering questions and thinking abouthis preference structure as a coin dealer. He also let me observe many ofhis business negotiations and by doing so helped me develop some understand-ing of the negotiation activity in his particular market.
I benefitted from discussions with E. Kohlberg, E. Maskin, M. Shakun,M. Hessle, K. Chatterjee, R. Zeckhauser, J. Hammond and the participantsin a seminar conducted at the Harvard Business School. I also benefittedfrom the "field" experience of V. Niederhofer of Niederhofer Zeckhauser andGross (New York), M. Halperin of the Avnet Corporation and my friendY. Shekel, owner of Autoline (New York).
My friends B. Nitay and P. Lubin were very helpful in improving thelanguage in parts of this thesis. D. Brenner and L. Bernozzi typed themanuscript with skill and patience.
My wife, Ruth, so wonderfully supportive and patient, helped me morethan morally. She criticized some of my impractical ideas and made mere-think the real decision problems that negotiators face.
- 6 -TABLE OF CONTENTS
ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
ACKNOWLEDGEMENTS . . . . . . . .a. .0.a.b. .a. .a.. . .. . . .. . ... 5
LIST OF TABLES. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .10
LIST OF FIGURES . . . . . . . . . . . .................. 11
CHAPTER 1: PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . .13
1.1 The Effects of Time on the Processes and Outcomesof Negotiations. . . . . . . . . . . . . . . . . . . . . .13
1.1.1 Introductionf Time. . . . . . . . . . . . . . . . . . . .13
1.1.2 The Effects of Time. . . . . . . . . . . . . . . . . . . . .14
1.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . .18
1.2 Scope of Thesis . . . . . . . . . . . . . . . . . . . . . . .24
CHAPTER 2: A SURVEY AND EXTENSION OF EXISTING MODELS. . . . . . . . . .26
2.1 Approaches to the Modelling of Negotiations. . . . . . . . .26
2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .26
2.1.2 A Taxonomy of Models . . . . . . . . . . . . . . . . . . .27
2.1.3 A Classification of Model Assumptions. . . . . . . . . . .29
2.1.3 A Formal Framework . . . . . . . . . . . . . . . . . . . .29
2.2 Existing Models . . . . . . . . . . . . . . . . . . . . . .33
2.2.1 Zeuthen-Harsanyi . . . . . . .. . . . . . . . . . . . . .33
2.2.2 Hicks 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 9. a. 0. 0. 0. a. a. 0. 0. 0. 9.35
-7-2.2.3 Bishop.......... . . .
2.2.4 Foldes. ..e..... .........
2.2.5 Ashenfelter and Johnson . .
2.2.6 Cross and Coddington. .....
2.2.7 Games of Timing . . . . . . ..
2.3 Extensions . . . . . . ......... . . . . . . . .50
CHAPTER 3: EFFECTS OF TIME IN THE TWO-PERSON BARGAINING PROBLEM......54
3.1 The Problem. . . . . . . . . . . . . . . . . . . . . . . . . .54
3.1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . .54
3.1.2 The Traditional Bargaining Problem and its Solutions. . . .55
3.1.3 The New Bargaining Problem. . . . .
3.2 The Solutions and Their Properties . . . . .
3.2.1 The Extended Raiffa Solution. . . . . . .
3.2.2 The Timing Equilibrium Solution. . . . . . . -a
3.2.3 The Fairness Requirements . . . . . . . .a.e..e
3.2.4 Reexamination of the Nash Cooperative Solution.
3.2.5 Applications. . . . . . . . . . . . . . .
3.3 Mathematical Developments . . . . ...a
3.3.1 The Traditional Bargaining Problem.
3.3.2 The New Bargaining Problem. . . . .
3.3.3 Properties of the Solutions. . . .
Appendix: Incentives and Arbitration Schemes. . . . . . 10
. .a .0 .0 .a .0 .a .a .0 .a .0 .0
. . .& . 0. a. 0. 0. 0. 9. 0. .61
. . a a. a .e64
. a . . - .64
- a -. .o72
. .. ...76
S 0 . a. 0. .80
.. . . 0. .83
. . . . . . . . . . ..85
0 a 0. 0- a- 0. 0- 0 - . .o85
0 & - 0 . - - - - - 0*93
. 0 0 - a - - a - - - 104
. .38
. .40
. .42
..44
..48
a - a - - 109
- 9-
5.2.2 Comments. m m
5.2.3 Two Modifications of the Model.............
5.2.4 Conclusions . . ..........-- - - - - -*-0-
5.3 An Asymmetric Adaptive Control of Negotiations.
5.3.1 Introduction...--e-e-e-a-e-a-a-e-a-o-a-. -
5.3.2 Formulation . - . . - - - . - - - - - - --
5.3.3 An Experiment in Negotiations - The Streaker.
Appendix 1 Formulation of the Model of Rao and Shakun
Appendix 2 The Streaker - Instruction Sheets. - -0 - -
Appendix 3 The Streaker - Data..-.-.-.-.-.-.-.-.-.-.-.
CHAPTER 6:
.193
.196
198
- -200
a - 200
.
202
- - - - - -213
- a - - - w224
.
--.- --226
- - - - - 9229
CONCLUSIONS AND FUTURE RESEARCH - -* . - -.-.-.-.-.-.-.-.-233
.REFERENCES . . . . . . . . . . . . . - . - . . . . . . - . . ' . . - - 0238
BIOGRAPHICALNNOTE. a . O-E.-- . a-0---.. -...--...--...... 243
- 10 -
LIST OF TABLES
Table 4-1......... ..--....---- -... --...... 161
Table 4-2....................... -.. -.. -. -- -0. -. . . . . . 176
Table 4-3.0 .a .0 -0 -a -0 -0 -0 -a - -0-a-0-&-a-a-0-a-0 - -0-0 -0 -0 - 0- 0- 0- 0- 0- 179
Table 5-1- - - - .-.-.-.-.-.-.- -.-.-.-.-.-.-.-.-.-.-.-.-. - - - -216-218
Table 5-2......... .......-.-.-..-.-.-.-- - ' - - . - - - -220
Table 5-3- - - -* -.-.-.- .-.-.- .-.- - - - . ' . - . . . - . - * 222
- 11 -
LIST OF FIGURES
Figure 2-1:
Figure 3-1:
Figure 3-2:
Figure 3-3:
Figure 3-4:
Figure 3-5:
Figure 3-6:
Figure 3-7:
Figure 3-8:
Figure 3-9:
Figure 3-10:
Figure 3-11:
Figure 3-12:
Figure 3-13:
Figure 3-14:
Figure 3-15:
Figure 3-16:
Figure 3-17:
Figure 3-18:
Figure 3-19:
Figure 3-20:
Figure 3-21:
Figure 4-1:
Figure 4-2:
Figure 4-3:
Resistance and Concession Curves . . . . . . . . . . . ..
Negotiation Set; Conflict-Payoffs Point . . . . . . . . .
A Regular Bargaining Domain . . . . . . . . . . . . . . .
The Nash Bargaining Problem - Solutions . . . . . . . . .
Solving a Non-Regular Bargaining Problem . . . . . . . ..
A Normal Bargaining Game . . . . . . . . . . . . . . . .
A Simple, Discretized Bargaining Game (Discounting) . . .
A Simple, Discretized Bargaining Game (Bargaining Costs).
The ERS as a Function of the Deadline T . . . . . . . . .
The Timing Equilibrium Solution . . . . . . . . .. .
Contraction Translation With Respect to 1. . . . . . .
1-Truncation of B . . . . . . . . . . . . . . . . . . . .
A Bargaining Domain With a Focal Point. . . . . . .
The Focal Point and the ERS . . . . . . . . . . . .
The Function U1 is Non-Decreasing for O<t<t0 . . . . . . .
Strong Individual Monotonicity. . . . . . . . . . . . . .
Weak Individual Monotonicity.. . ..... . . . . . . .
Properties of gn' . . . . . . . . . . . . . . . . . .a 0aa
Proof of Lemma 3.2. . . . . . . . . . . . . . . . . .
Proof of Lemma 3.3 (5-a). . . . . . . . . . . . . . .
Proof of Lemma 3.3 (5-b). . . . . . . . . . . . . . .
Proof of Lemma 3.3 (8). . . . . . . . . . . . . . . .0..
Iso-Preference Curves for X1and X3 . . . . . . . . .0..
Iso-Preference Curves forX 2and X . . . . . . . . .
The Three Stage Game . . . . . . ..
* 36
. 56
. 57
. 60
. 60
. 63
. 66
. 67
70
. 73
. 78
. 79
. 81
. 82
. 83
. 87
. 88
90
92
96
97
98
124
124
. . . . . . . . . . . 152
- 12 -
Figure 4-4:
Figure 5-1:
Figure 5-2:
The Acceptance Zones...........a . ... . . . .a..a......166
A Negotiation as a Closed Loop .........0... ....... 186
Sample Means of Normalized Concessions . . . . .a a . . a 221
- 13 -
CHAPTER 1
PREFACE
1.1 The Effect of Time on the Processes and Outcomes of Negotiations.
1.1.1 Introduction
Rigorous modelling of specific negotiations generally leads to intractible
analytical problems. Even the simplest real life negotiations have an intri-
cate gaming structure involving uncertainties and learning, which necessitate
the introduction of simplifying assumptions to obtain useful results.
This situation is common in various branches of the social sciences. Many
models in operations research and in economics are simplifications of reality.
Consequently, the actual problems solved are not the real problems, but
significant insight may still be gained from such abstractions. One of the
dangers is oversimplification to the point where the essence of the real
problem disappears. Models which oversimplify will not survive empirical
testing, but when testing is difficult or impossible the inadequacy of the
models will not necessarily be recognized.
This thesis grew out of my belief that in the modelling of negotiations,
a crucial element has been unduly de-emphasized or even ignored, resulting in
oversimplifying and potentially misleading results. This element is
time, or, more precisely, the effects of time on the negotiating parties and
on the environment in which they operate. The incorporation of the time
element into existing models, and the formulation of new problems in which
time plays a role may lead to models which are better both descriptively and
prescriptively. It may take some time before we can fully assess the use-
fulness of introducing time effects, but as matters stand, current models
could benefit from new approaches. After all, formal models of negotiations
- 14 -
have rarely been found useful by negotiators, and their descriptive quality
is questionable at best.
The role of time in negotiations, especially the effects of the negoti-
ation's duration, have been recognized by writers like Bishop [1964], Foldes
[1964], Cross [1965], and Coddington [1968]. Some games of timing (Karlin
[195911], Dresher [1961]) could, with some effort, be considered as time
models, too. Cross was probably the first to strongly emphasize the importance
of the time factor and to recognize some of its effects. Regrettably, Cross
did not have many followers among economists and apparently. none among game
theorists (although related issues were studied and modeled as stochastic
games). This thesis attempts to re-examine the subject and to study
some bargaining situations which are definitely affected by time. It's
scope is described in section 1. 2. Summaries of the contributions Of
this work are given in the introduction to each chapter and in the concluding
-chapter.
1.1.2 The Effects of Time
-Time affects negotiations In various ways. We can -distinguish -between the
following five categories of time ef fects (each category will be discussed later):
--- The Duration of the negotiation, defined as the absolute time
between the moment the negotiation begins and the moment it/ ends.
(Duration is measured in time units: hours, days, months.)
---The Pace of some process within the negotiation, defined as the
rate by which this process develops. (One example for such a
process is the sequence of concessions of one party. After
smoothing the sequence, one can regard the derivative of the con-
cession's trajectory as the pace of concessions.)
---The Calendar Date, being the date assigned to each activity
within the negotiation. (This date may be as detailed as necessary,
- 15 -
even to hours and minutes.)
---The Order, defined on the set of events that take place during
the negotiation. (An example is a concession offered after a
threat, or before another concession.)
---The Timing, defined for the negotiation as a whole or for parti-
cular events in it in the following way: the event (or the whole
negotiation) is happening before or after one of the parties or
an outsider carries out an action which is not an event of the
negotiation, but which may influence the agreement being negotiated.
The various effects of time to be considered in this thesis can be
associated with each of these concepts. We list here the effects that may
be significant in particular negotiations. Some of them are obviously more
important than others.
Effects of the Duration
---Discounting. Benefits received immediately are preferred to the
same benefits to be received in the future.
-- Bargaining Cost. Bargaining may incur large monetary and non-monetary
costs, depending on time. A labor strike during collective bar-
gaining is a familiar example of damaging costs to both sides.
---Sudden Termination. External forces developed independently of the
negotiation may interfere with or terminate a negotiation before an
agreement is achieved. Many political and business negotiations may
experience this phenomenon.
- 16 -
---Satisfaction. A negotiation terminated successfully but too
quickly may leave one side unsatisfied. If his first proposal
was accepted immediately, the negotiator may even feel cheated.
A merchant in a fish or a fruit market knows very well that when the
customer is satisfied with his performance after hard bargaining,
he may buy more than he had intended.
---Political Effects. The duration of the negotiation may have
political effects on at least one party. These effects may influ-
ence the negotiation strategy. An example is a developing country
negotiating with a multinational corporation. The government may
be accused of "appeasement" or "selling the country" if negotiation
is too fast, or be called "indecisive" or "weak" if it is too long.
---Discomfort. Even professional negotiators tend to feel some degree
of discomfort when involved in hard bargaining. This discomfort
grows as duration increases. The negotiator may be ready to concede
more in order to avoid this psychological inconvenience.
---Internal Effects. Lengthy negotiations tend to reveal the differences
between the members of a bargaining party. These differences (in
objectives, in preferences, and in bargaining costs) may severely
affect the achievement of this party. Furthermore, the possible
coalescence of internal factions who oppose the negotiation may
even cause a break-off. The internal effects may therefore lead
to a Sudden Termination.
Effects Related to the Pace
---Learning, or SignalEffect. Rate of concessions, delays, change of
pace -- all tend to modify the perceptions and beliefs of the
participants. There is a learning (and teaching) process, stimulated
by signals from both sides.
- 17 -
---Pressure. Under rapid succession of events negotiators are less
capable of making the right decisions. They may even make horrendous
mistakes. A skillful negotiator may use time pressure and deadlines
to affect the intellectual capabilities of the other side.
---Irrationality. Certain rates of activities may drive one side
into "wild" behavior, e.g, drastic changes in methods, inconsistent
demands, disregarding available information, etc. This is what can
happen to a kidnapper holding hostages while negotiating with the
police. (Of course, other facts may simultaneously contribute to the
same effect.)
Effects of the Calendar Date
--- PUtilityChange. For a baseball fan, the utility of a $10 bill on
the day of a ballgame may be very high. For this same reason,
agreements are made before Christmas, business deals are approved
before the last date for filing tax reports, and bills are signed
before election day.
Effects of Order
---Ajnda. The actual order of discussions influences the results, an
effect of time which is not duly appreciated. Poor ordering may
prevent efficient agreements or may even lead to a break-off; a
skillful setting of the agenda may be used to gain strategic
advantage.
Effects of Timing
---Threat Power. Timing the negotiation at the height of one's
advantage may increase the threat power. For example, when.
the cellar is flooded a plumber may receive a greater compensation
if he negotiates before rendering his services. After service, his
- 18 -
threat power decreases.
Perception of Time.
Negotiators differ in their perception of time. This difference is re-
lated to professional or even cultural attitudes towards time. Karras [1974]
tells us that the North Vietnamese leased a villa in Paris for two years
when the talks on ending the Vietnam War began, while the Americans expected
much faster negotiations. Differences in attitudes toward time can cause
friction and must be addressed, especially if the negotiating parties consist
of people of different backgrounds (e.g., lawyers versus businessmen,
Americans versus Japanese, experienced negotiators versus inexperienced ones,
etc.).
1.1.3 Examples
In this section we informally describe some examples of negotiations in
which an understanding of the role of time was of major importance. The ex-
amples show the wide range of applications that models incorporating time can
have, although in this thesis we restricted the analysis to business negotia-
tions, mainly over tages and prices. We start with an example from
diplomacy -- the 1971 negotiations between Malta and Great Britain over the
use of military facilities in Malta. The second example is from collective
bargaining -- the 1962-1963 newspaper strike in New York. The last example
is a case (n which the parties must remain anonymous) of an out-of-court
settlement involving a plaintiff and an insurance company. During the prepa-
ration of this thesis we encountered numerous other cases in which the
participating parties were unaware of the effects of time (mostly discounting,
bargaining costs and sudden terminatior) on their negotiations. Prominent
1This example is borrowed from Karras [1974]. His book is full of enter-
- 19 -
among them are cases of lengthy merger negotiations which failed because of
sudden changes in market environment or because of new governmental regula-
tions.1
(a) Malta-Great Britain (1971)2
Malta, a tiny island in the Mediterranean with a population of 350,000
gained independence in 1964, but maintained its close economic and political
ties with Great Britain. Its direct and indirect revenues from the British
facilities on the island amounted to one-fifth of its annual GNP of L100
milliri in 1971, and to one half of Malta's deficit in balance of trade
According to a financial agreement signed in 1964 the British were paying
Malta L5 million annually, while Malta agreed to retain sterling as its
currency and to keep its reserves in pounds sterling. An agreement was also
signed that would slow the firing of Maltese workers which was expected as
the British planned to reduce their use of the port facilities. The agree-
ments were to end in 1974, but due to the new demands of Malta's prime
1One of the businessmen interviewed is the founder and owner of a smallautomotive wire manufacturing company on Long Island, N.Y., who in September1972 received a bid price of $1,500,000 in cash from a large conglomerate.This price was above his "reservation price" of $1,400,000. However, hedecided to try for a bit more, and asked for $1,700,000. The negotiationsmoved very slowly. Nine months after it had started a government rule was
enacted that in effect forbade the conglomerate from expanding into the auto-motive industry. The deal was called off and the owner could not find anotherbidder before the recession of 1973-1974 which almost brought him to bank-ruptcy. In April 1978, after four years of hard work, he valued his companyagain at $1,400,000.2An interesting description is Wriggins [1976], which lists many newspaper
articles describing various parts of the negotiations. Information regardingmoves and countermoves of negotiating parties is difficult to obtain,because interested parties tend to release selective and misleading informa-tion. We feel, however, that the important facts in our examples areaccurate.
- 20 -
minister, Don Mintoff (elected in July 1971), and after nine months of
extensive negotiations, Malta obtained a new agreement granting her
L14 million annually from Great Britain and NATO for the use of port fac-
ilities. Additionally, Malta received L4 million in cash grants from Libya
and Italy andalso an additional L7 million in development loans. They also received
some L3 million from a retroactive rent payment. The strategy employed by
Mintoff was spectacularly successful, at least in the short run, and in-
volved brinksmanship of the highest order. He took full advantage of the
differences between Great Britain, Italy and the U.S. While the British,
who knew him well from previous negotiations and had decreasing interest
in Malta, were ready to play it tough, evacuating their military and start-
ing to dismantle their facilities, the Italians and Americans felt more
threatened by developments and were ready to pay (and to ask Great Britain
to pay) in order to guarantee a peaceful though expensive maintenance of
the status quo. However, in spite of Mintoff's effort to prove the opposite,
Malta was much weaker with respect to the element of time than was Great
Britain. But the British did not succeed in taking advantage of their
superiority, although they initially took some steps in that direction.
Immediately after the abrogation of the treaty in July 1971, the British
stopped their rent payments, slowed and later stopped completely the flow
of their troops to and from Malta, stopped all investments and contracts and
did not replenish stocks. Maltese workers lost their jobs in the port and
a sudden decrease in tourists from England hurt the tourist industry. All
that had important economic effects on Malta; especially the expected unem-
ployment that could destroy Mintoff's political base. Although a L1.5 million
gift from Libya helped to pay salaries for unemployed workers, it was no more
- 21 -
than temporary relief. At this point, in the middle of September 1971, after
a meeting between Mintoff and Prime Minister Heath, Great Britain failed to
recognize Malta's vulnerable situation. Heath was also lead to believe that
an agreement of L9.5 million was at hand and made a L4.75 million payment to
Malta, a move which eased Mintoff's problem and hardened his bargaining
stance. Realizing their mistake, the British turned to an extreme and ex-
pensive operation -- evacuating all their forces from the island. Later on,
when an agreement was reached, the British assessed their bargaining costs
at L6 million. Their move led to the entrance of the U.S. and NATO into the
game. The final agreement apparently gave Malta (and this is conjecture)
at least some L5 million per year above its "reservation price," per-
haps much more. They may still lose in the long run, since the increased
degree of political uncertainty discourages investments, but they succeeded
in reversing the balance of power in the negotiations. They inflicted rela-
tively large bargaining costs on Great Britain while their own costs were
covered.1
(b) The Newspaper Strike (1962-1963)2
The 16-week New York newspaper strike (December 8, 1961-March30, 1962)
'Some reporters observed that Mintoff used time pressures, delaying tacticsand sudden changes in pace in order to overwhelm the British negotiators,who indeed raised their offer from L5 million to L9.5 million in less thanthree months.
2Our references for this example are Raskin [1963] and several articles inthe Editor & Publisher (Nov. 17, 1962; Feb. 23, 1963; Mar. 9, 1963; Mar. 23,1963) and in the Advertisement Age (Mar. 4, 1963). A sizeable article listcan be found in the Business Periodicals Index, July 1962-June 1963, pages455-457.
- 22 -
was lengthy although as early as January 24, 1962, the demands of the
parties were very close. This strike is sometimes presented as an example
of severe personal conflicts and difficult inter-party bargaining. It is
also a good example of integrative, multi-variable bargaining in which an
agreement can be achieved only when the parties recognize the tradeoffs
between the various issues being negotiated. Somewhat less emphasized, but
in our opinion a very important factor, is the effect of the difference in
bargaining costs between the parties. The Publisher's Association, con-
sisting of 9 major daily papers, received a total of $2,750,000 in strike
insurance, but this was enough to cover only a small fraction of its losses
during the strike. Several papers faced a real possibility of permanent
closing due to their already-weak financial situations. The workers, members
of 9 craft unions received high strike benefits from their funds, supple-
mented by state unemployment insurance payments. Their intake amounted to
their regular salaries before the strike. As a result labor negotiators
enjoyed the cooperation of their rank and file. The slow pace of wage
proposals shows that labor was well aware of its advantage with respect to
time. On January 24, the Union demand amounted to a weekly wage increase
of $13.75 while the Publisher's Association proposal was $11.04. After
additional 11 weeks of strike, the final raise was $12.63. For the workers
the difference between $13.75 and $12.63 over a year amounted to only a two-
days salary! Any model trying to explain (or predict) the length of labor
disputes should certainly take bargaining costs of the parties into account.
(c) Out-of-court Negotiations
The plaintiff, Mrs. A., was involved in a car accident and suffered
khis example is based on a Harvard Business School case study (no. 9-175-258(Rev. 12/75)) titled "The Sorensen Chevrolet File." The case was prepared
by Dr. John Hanond. Additional information regarding the termination of thenegotiation was received from Professor Howard Raiffa.
- 23 -
severe injuries. The mechanic was insured by company V for $500,000. His
lawyer, fearing an adverse jury decision, pressed V to settle on a sum below
$500,000, as proposed by A. The company V was clearly in no hurry. The
sequence of pre-trial settlement proposals was the following:
March 15, 1972: A asked for $500,000.
February 1, 1973: V proposed $25,000.
September 8, 1973: A asked for $400,000.
October 18, 1973: V proposed $50,000.
February 1, 1974: V proposed $200,000 (after an appellate court
decides the case should be tried before a jury).
March 15, 1974: V proposed $250,000.
May 14, 1974: A asked for $350,000.
April or May 1974:1 A jury verdict in the plaintiff's county
granted a plantiff $535,000 under similar conditions. The feeling of both
parties is that the amounts awarded by juries in the county are increasing.
After six months of additional negotiations the parties settled out of
court on $325,000. During these negotiations Mrs. A was divorced and re-
married, a move that would have harmed her chances in court, but of which V
was unaware.
This case exemplifies dramatically several effects of time. The in-
surance company, by delaying the settlement, paid $325,000 in 1975 which is
equivalent to $230,000 in 1970.2 The discount effect was therefore substan-
tial. During this long delay, unexpected events did arise (the jury verdict)
which worsened the position of the company. The exact bargaining costs of
the company are not known, but were probably in the neighborhood of $40,000
1Date not specified.
2The discount factor is the compounded prime interest rate.
- 24 -
1975-dollars, a substantial sum. As to Mrs. A, the effect of time on her
family life was devastating. As in many such cases, looking backwards, both
parties would have preferred an earlier settlement.
1.2 Scope of Thesis
This thesis starts with an attempt to formalize the negotiation activity
and establish a uniform approach to its modelling (Section 2.1). This leads
us to describe and compare the existing models (Section 2.2), criticizing
some and extending others (Section 2.3).
Turning next to study the effect of time on arbitration schemes, we
reformulate the Nash Bargaining Problem and propose two new arbitration pro-
cedures that take into account the strength of the parties with respect to
time (Chapter 3). The new solutions exhibit some interesting properties which
shed new light on the traditional Nash Cooperative Solution (Sections 3.2.4
and 3.3.3). We propose some applications for these solutions (Section 3.2.5)
and briefly discuss one related problem (Appendix 1 in Chpater 3).
Next we turn to study some specific problems in negotiations. The mo-
tivating case study is a small exchange market in which two types of agents
negotiate frequently with each other; The particular market is described in
4.2. The decision to postpone negotiation is examined (Section 4.3). It is
viewed as a part of a general search-negotiation model. The role of the
Sudden Termination possibility is studied and the concept of characteristic
concession pattern is developed (Section 4.4). Two related problems are
relegated to the Appendices of Chapter 4: A new version of the Pick-The-Best
problem (Appendix 4.1) and the effect of reputation on bargaining (Appendix 4.2).
Asy etric models of negotiations are the subject of Chapter 5.
- 25 -
We review some related work (Sections 5.1 and 5.2) and then formulate an
asymmetric model of negotiations in an adaptive control framework. In this
model the opponent is characterized by two parameters: his "reservation
price" and his "type" as a negotiator. The accruing information is used
to update the prior assessments of these parameters. In section 5.4 we
describe an experiment which, although not specifically designed for that
purpose, shows that this approach may be useful in the analysis of nego-
tiations.
We conclude withra brief s ummary and directions for further
research (Chapter 6).
In this thesis the mathematical and non-mathematical discussions are
separated into different sections. We hope that this will enable the
reader to get a good qualitative understanding of the problems and their
solutions without being bogged down in mathematical detail. The precise
conditions under which results are obtained can be found in the mathematical
sections.
In a number of places we digress and discuss problems (not necessarily
encountered by negotiators) which are close to those treated in this thesis.
Th-se digressions (mostly in Chapter 4) describe interesting related problems
which can se-rve as points of departure for future research.
- 26 -
CHAPTER 2
A SURVEY AND EXTENSION OF EXISTING MODELS
2.1 Approaches to the Modelling of Negotiations
2.1.1 Introduction
The negotiation process and the factors affecting its outcome have been
studied by researchers in such diverse fields as political science, psycho-
logy, sociology, economics and mathematics (game theory). This is not
surprising in view of the prevalence of negotiation and bargaining activities
in human life. Formal models, however, have been developed only recently.
A few rudimentary models formulated by economists like Cournot, Edgeworth,
Stackelberg, Zeuthen, and Hicks existed before, but most of the developments
occurred after 1950. The publication of the celebrated von -Neumann-Morgen-
stern's "Theory of Games and Economic Behavior" in the '40's contributed
significantly to the emergence of these models. Another source of inspiration
was Schelling's "Strategy of Conflict" published . 1960. Still, in spite of
the profusion of literature nothing that can be called "a theory of
bargaining" has emerged. Instead, several specific problems that arise in
negotiations have been treated. These problems .have mostly been stated and
solved in a way that subjected the results to various contradictory interpre-
tations, sometimes to the point of bitter controversy, Without exception,
misunderstandings have been caused by attempts to generalize results that
were really obtained under severe assumptions. In addition, a myriad of concepts
have been formulated suggesting the absence of an agreed-upon conceptual
framework. A first step in introducing some order into formal models of
negotiations was taken by Stahl [1972], and here we .fellow some of
his proposals.
- 27 -We start by introducing a taxonomy of negotiation models (section 2.1.2).
We then classify the various assumptions that are made in the analysis of
negotiations (section 2.1.3). A formal framework is then presented (section
2.1.4).
The role of time has been rarely emphasized in negotiation models.
Using our conceptual framework we describe here some of the models which do
incorporate time (sections 2.2.1-2.2.4). Some extensions of these and other
models are given in 2.3.1
2.1.2 A Taxonomy of Models
Models of negotiations can be divided into six categories:
--Descriptive Models - Models which explain the steps of the various
parties and the end results as derivatives of some assumptions. These models
serve as explanatory and predictive tools. Examples include the works of
Hicks [1932], Bush and Mosteller [1955], Richardson [1960], Bartos [1974],
Ashenfelter and Johnson [1969] and perhaps Foldes [1964]. Some of the work
of Schelling [1960] fall into this category.
--Normative Models - Models which recommend certain strategies claimed
to be equilibrium, or "almost best" strategies under certain assumptions,
notably the assumption of full reflexivity of thinking. They may be viewed
1For a more comprehensive review of the literature some other references shouldbe consulted, like Luce and Raiffa [1957], Stahl [1972], Bartos [1974], andYoung [1975]. A thorough overview of the works of behavioral psychologistsand political scientists is found in Rubin and Brown [1975]. The December1977 edition of the Journal of Conflict Resolution includes survey articlesand a comprehensive reference list. Game theorists have recently renewedtheir interest in problems related to negotiations, notably the Bargainingproblem. The works of Rosenthal [19761, Kalai [1977], Roth [1976],[1977] andMyerson [19771,[1979] exemplify this development. None of them has, however,dealt with any aspect of time in negotiations.
- 28 -
as a limiting case of descriptive models, and their solutions may be con-
sidered mixtures of prediction and recommendation. Examples include the
works of Zeuthen [1930], Harsanyi [1956], Nash [19501,[1953] (according to one
interpretation), Bishop [1964] and Stahl [1972].
--Asymmetric Models - Prescriptive models, directed for the use of one
side under certain assumptions, especially partial reflexivity of thinking
and limited information. Examples include Hicks [1932] (according to one
interpretation), Cross [1965],[1969], Coddington [1968] and Shakun and Rao
[19741.
--Mediation Models - Prescriptive models for the use of a mediator whose
objective is the facilitation of the negotation process. We do not know of
any formal model in this category.
--Arbitration Schemes - Prescriptive models for the use of an arbitrator.
Unlike mediation models, the arbitration schemes result in a unique agreement.
Examples include Nash [1950],[1953], Raiffa [1953], Kalai-Smorodinsky [1975],
Kalai [1977], Yu [1973], Freimer and Yu [1976], Roth [1977], Rosenthal [1976]
and Myerson [19771[1979].
-Models of Outside Manipulation - Prescriptive models for a third party
who wishes to influence the negotiation by certain acts such as the release of
information or the making of binding commitments, in order to improve its own
position. (When coalition formation is allowed then the theory of n-person
games can be considered as an example of such models.)
Note. The much-studied decision problems related to bidding, auctions,
incentive-schemes, truth-eliciting schemes and so on, are not considered here
as problems of negotiations.
- 29 -
2.1.3 A Classification of Model Assumptions
The assumptions of the models described in section 2.1.2 belong to one
of three categories:1
--Behavioristic assumptions - Assumptions related to the properties of
the parties -- their decision processes, their typical patterns of behavior,
the information available to them and so on.
--Institutional assumptions - Assumptions related to the properties of
the bargaining situation; e.g., the rules of exchanging bids, termination
rules, the possibility of making binding commitments, the possibility of side
payments, the availability of mediation, etc.
--Payoff assumptions - Assumptions related to the payoff received by the
parties under various conditions. Examples are the parties' utility functions,
their conflict payoffs, the cost of arbitration, etc.
The specification of all the assumptions of a particular model is
generally a tedious task but a very important one, since slight changes in
these assumptions may sometimes lead to significantly different problems and
solutions.
2.1.4 A Formal Framework
Two parties are involved in some negotiation. Typically there exists
a set A of possible agreements, one of them must be chosen by both parties
if the negotiation is to end successfully. The benefit derived by the par-
ties depend on the final agreement acA as well as on the duration of nego-
tiation t. We assume the. existence of two von Neumann-Morgenstern utility
functions:
1Adopted from Stahl [1972], pp. 300-302.
- 30 -
U. :A x [0,w)-R , i=1,2 , (2-1)
and of two conflict payoff functions w
wi: [0,0)+lt , i=1,2 . (2-2)
When an agreement a is achieved in time t, party i's utility is VI(at)
(i=1,2). In case that a break-off occurs at time t, party i's utility is
wi (t) (i=1,2).
The utility functions U (at) (the subscript i will be deleted when there
is no danger of confusion) that we choose to use have one of the following
functional forms:
(1) Additive form:
U(a,t) = UA(a) + UT(t) acA, te[O,,) , (2-4)
where UA and UT are unidimensional utility functions. This form is adequate
if and only if the attributes A and T are additive independent, i.e., if and
only if the lotteries LI((a 1,t),(a2 9t2 );.5,.5), L2((a1,t2 ), (a2 ,t1); .5,.5)
are equally preferable by the negotiator for all a1,a2eA, t1,t2 eE. 1 This
condition holds when the only effect of time is a bargaining cost which is
independent of the final outcome.
(2) Multiplicative form:
U(a,t) = UA(a)-UT(t) acA, te[0,e) (2-5)
where as before UA and UT are unidimensional utility functions. This func-
tion is strategically equivalent to a bilinear function U(a,t) = kAUA(a) +
kTUT(t) + kATUA(a)UT(t) where kA , kT, and k are scaling factors satisfyingkA +k kATA .Th r AAAT
k +k + k -=1. This form is adequate if and only if the attributes A andA k.+AT
1This is proved in Fishburn [1965]. For this and the other properties of two-dimensional utility functions mentioned in this section, see Keeney and Raiffa[19761, Chapter 5.
- 31 -
T are mutually utility independent, i.e., if conditional preferences for
lotteries on one attribute i given the other attribute j do not depend on the
level of j , for i, je{A,T}, i j. This condition holds in some cases when
the only effects of time are bargaining costs and discounting. In some other
cases, for example, when bargaining costs are compounded instantaneously, the
utility functions are not of this form.
(3) Additive form with independent product:
U(a,t) = UA(a) + UT(t) + fA(a)fT(t) , aEA, tET , (2-6)
where fA and fT are unidimensional utility functions. The necessary and
sufficient conditions for this form are found in Fishburn [1974]. This form
corresponds to all cases of discounting and bargaining costs.
The advantage of these forms is the small numbers of conditional utili-
ties and constants that have to be assessed (by the negotiators or by a third
party). To find the additive utility one has to assess 2 conditional func-
tions and 1 constant. To find the multiplicative utility the numbers are
2 and 2 respectively. To find the additive with independent product (also
called Bilateral Independence) the number of assessments are 4 and 2 respec-
tively. We will not, however, expound on the assessments of the utility
functions. They will generally be assumed to be known.
We turn now to the formulation of the Sudden Termination possibility.
In every time t during the negotiation the negotiator could assess a distri-
bution Gt(s) on se[t,m), corresponding to the time of . sudden termination,
i.e. the time a such that if no agreement is achieved prior to it then fur-
ther negotiations are not possible, an agreement cannot be achieved, and the
parties receive their conflict payoffs by applying their optimal threats. If
at some tie t, there exists an s* such that Gtt (s*) = 1 and Gt (s)< for all
s<s* then s* is called a deadline. The subjective estimates of the sudden
- 32 -
termination possibility may be updated as time passes, but we want it to be
independent of the events in the negotiation itself. We therefore generally
assume that both parties make the same assessments Gt, for all t, and that
both know that each of them uses the same assessment. One party therefore
cannot gain information about the sudden termination by observing his counter-
part's behavior and trying to assess the counterpart's assessment.
- 33 -
2.2 Existing Models
2.2.1 Zeuthen-Harsanyi1
Zeuthen [1930] proposed a model of labor-management wage negotiations.
The model was later modified and translated into modern game terms by Harsanyi
[19561 (latest formulation [1977]). In this model two parties, Labor (de-
noted by 1 or L) and Management (denoted by 2 or M) are negotiating over a
finite set of possible agreements A = {ala2 ,...,an 2
The behavioristic assumptions include:
- complete information;
- unlimited computation capacity for both sides;
- each side is a maximizer of the expected utility derived
from the final agreement;
- both sides are Bayesian, ready to assign subjective prob-
abilities to various moves of their counterparts and to
act according to it.
The pay-off assumptions include:
- time plays no role in the utility functions; i.e.,
U (a,t) = Ui(a,O) for all OSt<no and for all aEA;
- the game has a fixed threat w = (w1,wQ2)
The institutional assumptions include:
1This model does not incorporate any effect of time. It is however mentioned
here because other models are influenced by it. Time might be easily addedto the model, as we show in 2.2.3.
2The set of agreements could be infinite, but then an additional institutional
assumption of a minimal concession should be added in order to guarantee asolution.
- 34 -
- time is discretized in sessions. In each session each player
(simultaneously) submits a proposal;
- the only possible moves are the proposing of agreements (no threats
or commitments are allowed);
- no retreats are allowed, i.e., M cannot propose an agreement a
such that UL (a)<UL(b) where b is another agreement already proposed
by M (the same holds for L);
- termination occurs if L proposes aL and M proposes aM such that
UL(aM)<UL(aL) and UM(aL)>UM(aM). In this case the final agreement
is chosen by some lottery between aL and aM or is some "midpoint"
(the exact procedure has no bearing on the results of the model);
- termination occurs also if both L and M do not concede, i.e., they
do not propose an agreement which yields higher utility for their
counterparts. In this case negotiations break off and the parties
get the conflict payoffs.
Zeuthen and Harsanyi proposed the following concession dynamics.
k kSuppose that the (k+l) session just started and let aM, aL be the
agreements recently proposed by M and L respectively. The parties compute
k ktheir risk limits rL and rM defined by
k -Lk
k UL(aL) - ULaMr L k
UL(aL)(2-7)
and
k kk UM(aM) - UHM(Lr- H* a)k (2-8)
UM k
and then the party whose risk limit is higher, say M, maintains his proposal
- 35 -
k+l k k+l k+l k+l(i.e., aM = aM) while L proposes anaL such that rL >rM . (For the
completion derivation of this rule, see Harsanyi [1977], pages 149-152.)
Following Harsanyi, we call this concession rule "Zeuthen's Principle."
Harsanyi proved that if both parties behave according to "Zeuthen's Principle"
then the point of agreement will be the Nash Cooperative Solution. (For a
definition of this solution see 3.1.2.)
How should we classify this model? Harsanyi proposed it as a normative
one, suggesting that following Zeuthen's Principle is the only concession rule
consistent with several "Strong Rationality Postulates." These posulates im-
ply total reflexivity of thinking. It ishowever, difficult to see why the
game described above is an adequate model of labor-management negotiations.
It includes restricting institutional assumptions (like the simultaneous sub-
mittance of bids and the termination rule) and ignores the bargaining costs
and perhaps some other time effects. The model can be viewed only as a first
step in negotiation modelling.
2.2.2 Hicks
Hicks [1932]1 proposed a much-debated labor-management model. The one
institutional assumption is: L may make only increasing proposals while M is
only responding by "disagree" or "agree," making no proposals of his own.
The behavioral assumptions in the model are:2
- if xM is the wage rate proposed by management and xL is the
1 Second edition appeared In 1963. In this edition (pages 350-354) Hicks addssome explanatory remarks. The model is still rather vague and ours is pro-bably only one possible interpretation.
2 The notations are a modern interpretation of Rick's arguments.
- 36 -
wage demanded by labor, then management can find the time t satisfying
UM xM,t) = UMOOL,) for all XL>XM (The utility function itself may not
be known. All that is needed is the set of prices (t, xL).) The graphof
xL vs. t is the "employer's concession curve";
- the union may draw another curve, the "union's resistance curve,"
consisting of pairs (t, x), where x = maxjxLjif xL is the union demand
and if the employer did not give in until time t then the union's
decision is to continue its strikel;
x
Resistance Curve
x2
Concession Curve
t
Fig. 2-1: Resistance and Concession Curves
- M knows L's resistance curve, and L knows that M knows that;
- L does not know with certainty the concession curve of M, and
M knows that, too;
- M will not accept an a such that a>a* (see Fig. 2-1).
- 37 -
Hicks' conclusion is that L cannot expect any agreement better than the
true a* which is not known to him in advance. While this result looks trivial,
helping L very little, it alludes perhaps to the only way L may improve his
conditions; i.e., by trying to change the unknown M's concession curve by
activities outside the negotiation.
The interpretation of Shackle [1957] and Bishop [1964] of the model fits
well our definition of asymmetric models. The model intends to help one party
(Labor in our description and in Hicks' original work) in his negotiations
with another party (Management) which is assumed to be better informed and to
employ a known strategy. Bishop criticizes the restrictiveness of the insti-
tutional assumptions, the vagueness of the construction of the curves, the
implicit assumption that they do not change during the negotiation and the
assumption that the curves intersect; the graphs need not behave like those
in Fig. 2-1.
Stahl [1972] ignores the assumptions of asymmetry and suggests the inter-
pretation of a normative model, in which the sides behave as in the Cross
model (see 2.2.6) and the negotiation takes place during a prolonged strike.
The curves are built as negotiation develops, so that a party may decide to
accept or reject an agreement after updating and evaluating its estimate of
the intersection point.
Our view is that most probably Hicks meant the model to be a descriptive
one. In the negotiation game (during a strike) the management follows the
strategy: choose an opening wage rate and hold to it until the union
surrenders.' The wage xM is chosen so that EUM(xM,t(xM)) is maximized, where
1This strategy is known as "Boulwarism." See Stevens [(1963) pages 34-37]for details about this approach to collective bargaining, advocated and em-ployed by L. Boulware of General Electric Company.
- 38 -
t(xM) is the duration of a strike given that the wage proposal is xM (t(XM)
is a random variable that must be assessed by management). Labor's strategy
is to lower its asking wage rate according to some pattern, and to give in at
time t(xM). The reason that labor does not give in immediately (thus saving
the bargaining costs) is that it has some other objectives in mind (e.g.
political objectives of union leaders) or it hopes for a change in manage-
ment's utility as a result of outside events or the calendar date which may
result in a new price proposal xM. Of course, as a game this model is defi-
cient, because it restricts the strategy space and does not give a clue as to
how updating of assessments should be performed.
2.2.3 Bishop
Bishop [1964] proposed a normative model which is a combination of the
models of Hicks and Zeuthen.
The model assumptions are identical to those of Zeuthen's model (see
2.2.1), with the exception of the introduction of the discounting effect of
time into the utility functions. Bishop considers two types of multiplica-
tive functions. The first is
Ui(at) = (H-t) f(a), aEA aO<t<H , i=1,2 , (2-9)
and w(t) = 0
where H is the expected duration of the negotiated agreement. H also serves
as the maximum duration of the negotiation.
The second function considered by Bishop is
1 f -Rt, acA ,O<t<c , i=1,2 , (2-10)
andw(t) - 0
- 39 -
(instantaneous compounded discounting with infinite horizon, T =
The concession mechanism proposed by Bishop is this. Suppose the
kproposals in session k are a and the time elasped since the beginning of the
knegotiation is tk. Party i is indifferent between his proposal a accepted
ko kat time tk + D and the other proposal a. accepted immediately. Di (called
i 3
the "maximum duration of strike") is therefore the solution of
U (aktk + D.) = Ui(atk) , i # j (i=1,2).(2-11)
The D 's are the basis for the concession rules. The parties compute their
D i's and if D >D , i makes a concession in the k+lst session, while j does
not concede.
If the utilities are of the form (2-9), the solution of (2-11) is
fi(ak) - f (a )D,= (H - t.k) - k1 (i=1,2) (2-12)
f (a )
Bishop proved that the negotiation's outcome in this case is the Nash
Cooperative Solution.
If the utilities are of the form (2-10) we get
D=1 k (213D,= (ln fi(ai) - In fi(a)(2-13)
The result now is an agreement a* satisfying
R 1/R 2 R1 /R 2f1 (a*) * f2(a*) = max{f 1 (a)*f2 (a) } . (2-14)
acA
This agreement is generally not the Nash Cooperative Solution.
Like the model of Zeuthen, this model suffers from several weaknesses
discussed by a number of authors.1I
1Stahl [(1971), page 261] criticizes especially the behavioristic assumptions.Young [(1975), pages 141-144] criticizes the institutional assumptions. Seealso 2.3(2).
- 40 -
2.2.4 Foldes
Foldes [1964] proposed a model which has been considered by most authors
to be equivalent to Bishop's.1 He indeed considers the "delay times", which
are identical to Bishop's "maximum duration of strike", as an adequate measure
of bargaining strength, but unlike Bishop he also makes several assumptions
in order to justify this measure. He also finds necessary and sufficient con-
ditions for the existence of a solution point.
The behavioral assumptions include:
- U are value functions; i.e., only ordinal preference is needed;
- the parties do not change their value functions during the
negotiations;
- complete information;
- the objective of both sides is the maximization of the value of
the final agreement;
- both accept the premise that if one's delay time is shorter than
the other's it means he is weaker and has to make the next
concession;
- full reflexivity of thinking.
The institutional assumptions include:2
- the parties submit proposals whenever they wish; no requirement
of simultaneity;
- the parties could make threats, where a threat b of party i is
a delay time after which i accepts the current proposal of j if
j has not accepted the current proposal of i;
1See Coddington [19681, Stahl [1972], Young [1975], and McCormick [1977].
2For another variant of this set of assumptions, see Foldes [(1964), pages
123-124] and Stahl [1972].
- 41 -
- there is a formal procedure under which threats are allowed
(Foldes [1964], page 123).
Foldes starts with several definitions. Let a1,a2eA be the agreements
proposed by sides 1 and 2 in some time t, and assume that equations (2-11)
hold: if the delay time D,D, satisfy D>Dj( # ij) then a is said to be
"enforceable" against a in time t. If D = D they are "undecidable." It
is easy to see that if a is enforceable against a in time t it is not neces-
sarily so in time s, s 4 t, Foldes is really interested in a stationary en-
forceability and he defines an "equilibrium" to be an agreement a* which is
enforceable against all other agreements for all 0<t<T. It is clear that if
such equilibrium exists, the assumptions of the model imply its immediate
acceptance by both sides.
From now on we assume that A(0) = [0,1] but all the results are easily
extended to [0,1]". Ui are now value functions of x and t, with marginal
rates of substitution denoted by A(xt).
Theorem (Foldes [1964]
(1) A necessary condition for equilibrium at x*, 0<x*<l, is
N1 (x*,0) = -A2 (x*,0) (2-15)
A necessary condition for equilibrium at x* = 1 is
|1(x*,0)I>1A2(x*0)l- (2-16)
(2) Sufficient conditions for the existence of an equilibrium are
(i) Stationary value functions:
X (xt) - f1(x) (i=1,2). (2-17)
(ii) Strictly concave value functions.
Remarks. (1) A(xt) = f (x) if and only if Ui(xt) = f(t + Vi(x))
where f is any strictly monotonic real value function, V (x) a value function
- 42 -
on [0,1].1
(2) By applying the necessary conditions on Bishop's functions (2-9),
(2-10), we get the same results as we got in 2.2.3. The sufficient condi-
tions of the Theorem do not hold for (2-9) and they hold for (2-10) only if-rt
In f (x) are strictly concave. For the special form U(x,t) = xe 1 ,
i -rt
U 2 (xt) = (1 - x)e ,the solution of Bishop and Foldes is
r2
r + r2
Foldes is aware of the weaknesses of his model. In his words:2
"The limitations of our model as a theory of bargaining should be
clear... . They spring directly from our commitment...to a model
which abstracts from uncertainty and from the dynamics of bargain-
ing which predicts efficient trade, and which allows the same bar-
gaining opportunities for both parties. The main positive features
of the model are these. It demonstrates the possibility of determi-
nate static theory of bilateral monopoly in conditions of certainty.
It analyzes market mechanism other than competition which can, in
suitable circumstances, lead without friction or delay to efficient
trade at a theoretically determinable point. Finally, it attempts
to link the traditional economic analysis of bilateral monopolywith the recently-developed theory of threats." 3
2.2.5 Ashenfelter and Johnson
Ashenfelter and Johnson [1969], following Ross [1948], view the collec-
tive bargaining as a three-party negotiation. The parties are the management,
the union leadership, and the union rank and file. Union leadership has to
satisfy the expectations of the rank and file, but has also to reduce these
expectations, if they are too high. The strike is an expedient for that
purpose.
1For this and other properties of value functions see Keeney-Raiffa [(1976),
Chapter 3].
2Foldes [(1964), page 124].
The last sentence refers to Schelling [1960].
- 43 -
The institutional assumptions of the model include:
- negotiation is over a percentage raise in the wage rate (denoted
by x);
- negotiation starts with a simultaneous submitting of bids. If
the bids are not compatible, a strike starts;
- during the strike, management (U) and labor (L) may submit
proposals;
- the negotiation happens only once, i.e., future negotiations
are ignored.
The behavioral assumptions are:
- for all t, OSt<c there is a percentage raise x(t) that is
acceptable to the rank and file. In every time t, x(t) is the
only agreement that can be reached; (Another way is to say that
L has the following decision rule: accept x in time t if xtx(t),
otherwise decline.)
- the function x(t) is common knowledge;
- M is a maximizer of present value.
The payoff assumptions are:
- bargaining costs of management include the loss of profits during
bargaining. It is assumed that the duration of the strike does
not affect future profitability.
- conflict payoffs are not defined -- the strike will eventually
terminate and an agreement achieved.
After assuming a particular shape to x(t), the model is straightforward.
The strike duration (if a strike takes place) is determined by the maximizing
- 44 -
decision of M. A formula for the length of strike is easily obtained.
This "Hicksian" model has clear limitations as a negotiating model.
However, Ashenfelter and Johnson were not interested in developing a refined
negotiation model. They wanted to look at specific industries and to
test a relationship between the average strike length and some variables
which may be derived from the simple negotiation model. Another similar
model, based on the same principle, is presented in Farber [1978].
2.2.6 Cross and Coddington
Cross [1965], [1969] was the first to examine in detail the role of
time in negotiations. He writes [(1965), page 72]:
"As any economist knows, time has a cost, both in money andin utility terms; it is our position that it is preciselythis cost which motivates the bargaining process. If it didnot matter when people agreed, it would not matter whetheror not they agreed at all."
Cross mentions four of the time effects listed in Chapter 1: Discounting,
Bargaining Cost, Learning, and Utility Change. The first two enter his pay-
off assumptions, while the third is the basis for the learning process that
takes place during the negotiations. This learning process is indeed the
main contribution of Cross, and we will describe it in some detail.
Two parties are engaged in distributive bargaining over [0,1]. Their
utility functions are of additive form with independent product:
-.Rit Ci -RitU (xt) = f1 (x)e +jC(e -1) , xe [0,1] 0 t<-,
and Ci -Rit (2-18)
i Ri
where R are the discount rates, C are the bargaining costs per one time-
unit (1 - 1,2).
- 45 -
The institutional assumptions are very simple. They include:
- first proposal marks the beginning of negotiations;
- only proposals are allowed;
- no other requirement as to the order or size of proposals
(retreats are allowed).
The behavioral assumptions include:
- both sides assume that the other side has some concession rate
which is not affected by their concessions;
- both players identify the current rate of concession of the
other side, denoted by ri, and assumes it is constant to the et
of the negotiation;
- whenever he decides to submit a proposal, side i chooses his
proposal xi, given the current proposal xj and his current est:
mate of the expected rate of concessions r of the other party
- so as to maximize (2-18). He assume Shat from now on his
opponent will make all the concessions;
- the estimates of the concession rates are changing continuously
as a result of learning processes discussed later.
Party i identifies the current x and estimates the expected rj, i # i.
Assuming that rj is constant he tries to find a xi that maximizes (2-18).
The first and second order conditions for a maximum give
and
f (x ) f (x - - = 0,
Rf"(xi) - x
(2-19)
(2-20)
nd
i-
7
- 46 -
The optimal value xi, which for a vast set of utility functions is uniquely
determined by (2-19) and (2-20), is now proposed by party 1,
Cross proposes that the expected rate of concession of party j, as
estimated by party i, is modified by learning. The r j rj(t) is a function
satisfying the following requirements:
dr dx >0 if r , (2-21)
Ljand
dr
dtdx > 0. (2-22)
d(-r - );- dt
The learning rates and their first derivatives determine the concession pro-
cesses. From (2-20) we get:
dxi 1 dr=
i ---f II.(2- 23)
dt q (x) dt
fi(x) rj -Ri
Cross examines now several cases, trying to find agreements which are
the unique results of specific pay-off assumptions. For players with
identical learning abilities whose pay-offs are not affected by time, he
gets the Nash Cooperative Solution. Simple formulae are achieved for learning
functions of the form
dr dxa (---- r) (i = 1,2) - (2-24)dt I dt I
Simple formulae are also obtained for the case of linear f . These formulae
yield an expression for the duration and the results of the negotiation as
functions of the various parameters of the model.
- 47 -
Cross views his model as a normative one. As such, it was criticized
for unreasonable behavioristic assumptions. Cross himself observes the
"instability" that is expected if the parties are not identical in their
learning abilities.2
Cross is really trying to introduce overly naive learning and to ignore
the gaming aspects of negotiation. However, he was the first to acknowledge
explicitly the effects of time and to propose a normative theory of learning
in negotiations.
Coddington [(1968), Chapter 3] considers Cross' model to be a closed
loop control system, involving two parties. This view emphasizes both the
predictive and the prescriptive aspects of the model. According to it
Cross attempts to describe how closed-loop systems, consisting of learning,
expectations and acts, stabilize (i.e., reach a solution); he also indicates
how one party has to act in order to arrive at an optimal stable position.
(For detailed treatment of Coddington's view, see 5.1.)
Coddington [(1966), (1968, Chapter 6)] accepts the general framework
of the model of Cross, but introduces several corrcctions. First he replaces
the assumption of constant ri's by an assumption of the existence of t , a
subjectively estimated function describing the expected time to agreement.
The maximization in every stage is constrained by this function. A second
change is a more careful examination of the process of adjustment of expec-
tations. The function t must be of special form if a "consistent"
adjustment is assumed.3
1See Coddington [1966],[1968], Stahl [19711 and Young [1975],
2Cross [(1965), page 771.
2 *Coddington defines a consistent adjustment as a process which leaves t un-changed if former assessments of the opponents' behavior were proved correct.
- 48 -
2.2.7 Games of Timing1
Games of timing are a class of infinite games defined on the unit square
with a payoff kernel K(s,t) satisfying:
L(s,t) s<t
K(s,t) = *(s) s=t
M(s,t) s>t
where L(s,t) and M(s,t) are continuous in s and in t, increasing in s for each
t and decreasing in t for each s. A strategy for player i is a probability
distribution Fi(-) defined over [0,1]. In this game each player is taking an
action in time Ti. where Ti is a random variable distributed according to Fi.
Although never proposed as bargaining models, such games may be presented
as normative models of negotiations with special structure. One example is
this: consider two negotiating parties whose demands are agreements d and
d2 . Suppose that there are no other possible agreements, so that one of the
parties must give in, or else there is no agreement. Let the utility func-
tions of the parties be such that UI(d1) = U 2 (d2) = 1 and U1 (d2) = U2(d1 ) =-l.
There are no time related costs and therefore no conflict-payoffs. Negotia-
tion may theoretically go on forever and indeed there seems to be no reason
why any party should give in. Suppose now that there exists a powerful
third party who is adversely affected by the very fact that an agreement has
not be achieved.2 Each party may turn to this third party (call it 3) for
support, and we assume that the party who is granted support can force his
1References for these james are Karlin [195911]and Dresher [1961].2If the parties are two employees trying to settle an argument of "who isdoing what," the third party is their employer, If the parties are twogovernmental or corporate departments, the third party is top management,If the parties are two countries clashing over disputed territory, the thirdparty is an interested great power.
- 49 -
counterpart to concede immediately. The third party, approached by party i
in time t, grants his support with probability P(t), where Pi(t) is growing
in t, P1(1) = 1 for i = 1,2. The reason for this behavior may be 3's in-
creasing interest in endiag the stalemate. The probabilities P1 (t) and P2(t)
are not identical because the third party may have some prior preferences
regarding whom he should support. If each party may approach 3 only once,
the problem is to find the right timing of asking for support. Let us
assume first that approaching 3 cannot be concealed. The kernel payoff of
this game is:
P1(s) - (1 - P1(s)) s<t
K(s,t) = P1(s) - P2(s) s=t (2-26)
-P 2(t)+ (1 - P2 (t)) s>t
which is exactly the "noisy duel" game with one bullet for each player. If
the parties may approach the third party secretly, the payoff kernel is:1
P (s)- (1 - P1 (s))P2 (t) s<t
K(s,t) = PI(s) - P2(s) (2-27)
-P2 (t) + (1 - P2(t))P1(s) s=t ,
which is exactly the "silent duel" game with one bullet for each party. A
"silent-noisy" game can also be formulated.
Optimal strategies for (2-27) are the pure strategies F1 (s) = 2()
*- 1, where a satisfies
P(s ) + P2(s) 1. ( 28)
we assumed that if no party receives support, 3 imposes one of the agreements
in time t - 1. He decides on the agreement to be imposed by flipping a coin.
- 50 -
Optimal strategies for (2-28) when P1 (s) = P2(s) = s are probability dis--
tributions F. with densities f satisfying1
0 05sssl/3f1(s) = f2(s) = j14s3 1/3s41 . (2-29)
2.3 Extensions
(1) We start by showing that the dubious "concession 3 and "resistance" curves
of Hicks may be meaningful and may even be computed in some special cases.
Consider two negotiating parties whose price proposals are x1 (t), x2(t)
(Ot<w) where x1(t), x2(t)e[O,1], x1(t) is decreasing in t and x2(t) is in-
creasing in t. The curves x1 (t), x2(t) are the resistance and concession
curves. Suppose that a sudden termination possibility exists, and that its cumulative
distribution and density functions are denoted by -G(t) and g(t) respec-
tively. Suppose also that there exist two random variables S, defined on
[O,w), called concession-times, such that if S. = s, then party i makes a
yielding concession in time s. The random variable s arise as the result
of the internal political effects of the negotiation's duration, and is
beyond the control of party i. It is, however, not entirely beyond the con-
trol of party j, j # i. We assume that
FS (t) = Pi(xj(t), t) , i t J, 1Ot<cw , (2-30)
where P1 (x,t) is increasing in x, P2(xt) is decreasing in x, and both are
increasing in t. As party i concedes more, the probability that party j,
j t i , gives in is increasing. This probability is also increasing with
1The general silent duel is solved in Karlin [(195911) Chapter 5 and solution
to problem 5.101.
- 51 -
time, as the other party is eager to end negotiations because of time
effects.
It is easy to see that, as formulated here, the negotiation process
amounts to a game in which each party continuously weighs three factors:
(1) A low rate of concessions increases the negotiation's duration, there-
fore increases the probability of a sudden termination. (2) Generous
concessions may quickly induce a yielding concession from the counterpart,
but the utility received may be low. (3) The possibility that he himself
will be the yielding party has to be considered. Three types of concession-
resistance curves can be drawn:
(a) When possibility (3) is ignored by party 1 (the case of party 2 is
analogous), the problem amounts to an optimization problem:
max f [x1(t)p2(xl(t),t)(1 - G(t))+ w1 (t)g(t)(l - P2(x(t),t))Jdt
x1(t) 0
SAt dx I )< 0 (2-31)
dt
x (t) 0 for all t.
x (0) = 1
(b) When party 1 ignores (3), as in (a), party 2 may compute x(t) and
then solve the problem:
max f [x2 (t)P1 (x2 (tbt)(l - P2(x1(t), t))(1 - G(t)) +
X2(t) 0
+ w2(t)g(t)(1 - P1 (x2 *(t)t))(l - P2(xl(t),t)) + (2-32)
+ x1 (t)P2 (x(Q),t)()(l P1 (x2 (t),t))(l - G(t))]dt
- 52 -
s.t. dx2 (t) > dtx2(t) <1 for all t.
x2(O)=1
(c) When (3) is not ignored, the curve x.(t) are the equilibrium stra-
tegies of the parties, if they exist.
If Hicks' model is interpreted as an asymmetric one then the cases (a)
and (b) are direct extensions of the behavior of labor and management respec-
tively. Note that sudden termination rarely occurs in collective bargaining.
It occurs more frequently in political or business (e.g., merger) negotiations.
Note also that the functions G(-), P1 (-,-), and P2(6,-) are not easy to assess
and are generally not common knowledge. The "naive" approaches of (a) and (b)
are therefore useful as approximations for optimal strategies.
(2) We now turn to examine the Bishop-Foldes model. For the simple case of
linear bargaining costs:
UI(a,t) = a-cI -t , U2(a,t) = 1-a-c2 -.t
the sufficient conditions for the existence of an equilibrium are not satis-
kfied. If we try to apply Bishop's scheme we discover that if ak are the
k kcurrent offers of the parties, then Di=(a1-a2)/c. Therefore cgc> im-
plies D2<D1 for all the stages of the negotiation, and the agreement obtained
is a=1. The party who is weaker on time makes all the concessions.
Let us now introduce the possibility of a sudden termination. Assume
that the duration of the negotiation is distributed exponentially with mean
of . The utility functions are:
xtt .c c x
U(x t) 0 e t(X-C t -C Is~e Ads = eAt(1-& -
- 53 -
and
Uxt (1-s-c - -As -At -x )-U (~) - )-240 c 2 f s e ds e (i-s +-~-
0
From here we get for the marginal rates of substitution A1 (x,t) = -Ax-c1 , and
A2(x,t) = -Ax+A+c 2. Foldes' sufficient condition 1A(xO)I = A2(x*O)I yields
* 1 c2-cy2 2" 111 21
2+ 2A
If c2>c partyI receives more than . The difference between the bargaining2 c1 pry1reevsmoeta
costs, c2-c1 , must be greater or equal to A in order for party 1 to get x*=1.
Some of the ideas developed in the models reviewed in this chapter will
be modified and extended in this thesis. The hypothetical resistance and
concession curves of Hicks give rise to our "characteristic concession pat-
terns" developed in 4.4. The learning model of Cross is close to our control
model, developed in 5. The games of timing motivate one of our proposed
solutions to the new bargaining problem (chapter 3). We do think, however,
that some of the most important decision problems which negotiators face
when time effect are significant have not been addressed. Some of these de-
cisions (e.g. the decision to break the negotiations off, to postpone nego-
tiations, to search for other prices while negotiating, to call an arbitrator
etc.) will be examined here.
- 54 -
CHAPTER 3
EFFECTS OF TIME IN THE TWO-PERSON BARGAINING PROBLEM
3.1 The Bargaining Problem
3.1.1 Introduction
The traditional bargaining problem involves two agents negotiating the
distribution of some good, with the understanding that if they fail to reach
an agreement they receive certain predetermined conflict-payoffs. The
utility which both parties derive from the final outcome, be it an agreement
or a conflict-payoff, is traditionally assumed to be unaffected by the
duration of the negotiation. Most studies to date have been concerned with
specifying, or at least characterizing, the bargaining outcome under various
behavioristic, institutional, and payoff assumptions.
The main contribution of this chapter is to introduce the effect of the
negotiation's duration; this requires a new formulation of the bargaining
problem. Two new solution functions are introduced but no claim is made as
to their descriptive or normative content. They are viewed merely as
"arbitration schemes."1 The solutions satisfy some fairness requirements --
rules used by an impartial arbitrator. Casual examination of some experi-
mental work leads us to believe that the solutions may have some descriptive
quality.2
Our study sheds a new light on the well-known Nash Cooperative
1For a discussion of arbitration schemes vs. normative solutions, see Luceand Raiffa [1957), pp. 119-124.
2See Contini [1967a],[1967b],l[968,Nydegger and Owen [1975].
- 55 -
solution which has been widely accepted as the solution of the bargaining prob-
lem. We show that this solution is indeed a "good" one only if the effect of
time on both parties is identical; i.e., if both discount their utility func-
tions in the same way. When, however, one of the parties is less affected by
time than his counterpart, the Nash Cooperative Solution cannot be recommended
as a "fair"solution, nor can it be proposed as a "rational" outcome of the
bargaining.
Organization of Chapter 3. For the sake of clarity we relegate all the
mathematics to section 3.3. Most of the formal definitions, the theorems and
their proofs can be found there. The problem itself is informally described and
discussed in section 3.1. The solutions, their properties and some applications
can be found in section 3.2. A brief discussion of the role of incentives
supplied by a third party to the negotiating parties is given in the Appendix.
3.1.2 The Traditional Bargaining Problem and Its Solutions
Consider two parties denoted by 1 and 2, and let their von Neumann-Morranstern
utility functions be denoted by U1 (x), U2(x) where x is an element of the set A
of possible agreements. By viewing all the randomizations of possible agreements
as agreements we can assume that the presentation of A on the utility plane is a
compact and convex set B, as shown in Fig. 3-1. The objective has been to find
a point (u1 , u 2 ) in B such that U1(x*) U19, 2 2,(X where x*eA is the
final agreement.
The negotiation set P is defined as the set of all Pareto-Optimal points
of A which dominate the conflict-payoffs point w. This last point corresponds
to the utility levels achieved by the parties if the negotiation fails.
Note that the parties may receive w as a result of the execution of their
optimal threats or as a result of the application of the institutional rules
of the game. The precise reason for getting w is not important so long as
- 56 -
U2
U'
Fig. 3-1: Negotiation Set; Conflict-Payoffs Point
it is a fixed point which cannot be altered by the players during negotia-
tions.
We define the bargaining domain B as the set of all agreements x such
that U1(x2) ? and U 2(x) w2. The tradition has been to require that the
solution be a point in the bargaining set B, preferably in the negotiation
set P.
The common procedure now is to apply a linear transformation which moves
w to the origin and to assume that free disposition of utility is allowed.
The result is a bargaining domain (later called regular) which looks like
the one in Figure 3-2.2
1This condition fits well buyer-seller negotiations where the point w means
"no deal." When the negotiation concerns a change of an existing sharing
rule, as in labor-management bargaining, some caution should be taken in deter-mining the conflict-payoffs point w. A labor strike is viewed here not as athreat but as a negotiation tactic which incurs costs by extending the nego-tiation's duration. The threat point is the ultimate and rarely executedthreat of management-closing the work place.
2 For a definition of a regular bargaining domain, see 3.3.1.
- 57 -
U2
B
w = (0,0) U
Fig. 3-2: A Regular Bargaining Domain
The traditional bargaining problem can now be stated as finding the
point u in the bargaining domain B which corresponds to the final agreement
of the negotiation.
This vague statement of the problem immediately raises several questions.
The most important one relates to the interpretation of the solution concept.
Is it a solution of a descriptive model, of a normative one, or of an arbi-
tration scheme? The last interpretation is obviously the most innocuous
one. The mathematical requirements which should be satisfied by the solution
are then viewed as equity criteria used by an impartial arbitrator. If one
adopts this interpretation one must decide on some way to compare utility
scales. This is exactly what game theorists try to avoid. The advantage
of a solution which does not involve comparison of utility scales is that
it may be robust enough to serve as a determinate solution for many conflict
situations where a unique outcome to the bargaining problem is desired. The
Rash Cooperative Solution, believed to be such a robust solution, has
- 58 -
therefore been widely accepted as a good descriptor and predictor,1
Our view is that the normative and descriptive interpretations of the
Nash Cooperative Solution, and other solutions which do not require compari-
sons of utility scales, are hardly defensible and are probably not very go
approximations. Casual observations of bargaining situations and recent
empirical studies (Nydeggerand Owen [19751) show that inter-comparison of
utilities is common in actual negotiations even without the presence of an
arbitrator. The bargaining problem is therefore basically a problem of
fair division, and the requirements satisfied by the solution are fairness
criteria.
A solution function g is defined on the set Q of all regular bargaining
domains B into R2 such that g(B)eB for all BeQ. The point u = g(B) is the
solution of the bargaining problem B. We now present some of the solution
functions g that have been proposed in the literature.
(i) The Nash Cooperative Solution, denoted by gN(B), defined as the
unique point u=(u1 ,u2) for which u1*u2 is maximized i.e.,
u = gN(B) iff u1 -u2 - max(vl-v21(v1,v2 )CB} . (3-1)
(ii) The Generalized Nash Cooperative Solution2, denoted by ga(B),
defined for all O<a< as the unique point satisfying
u gN(B) iff 0 -u2] .%Max V -2 av 1 2)CB . (3-2)
1nax2{ 1 "v2 lIvD
'Many studies have used the Nash Cooperative Solution (NCP) as a determinatesolution. One example is de Menil's [1971] study of wage equations in whichthe NCP is the outcome of the collective bargaining. A second example isthe prediction model of Hnyilicza and Pindyck [1976] in which future OPECoil prices are viewed as the outcome of internal bargaining between the saverand the spender countries in OPEC. Another example is Pindyck's [1976]calculation of the increased costs to the monetary and fiscal authoritiesin the USA resulting from their conflicting objectives.
2For a development of this solution see Friedman [1977].
- 59 -
(Note that the Nash Cooperative Solution coincides with g(B).)
(iii) A solution due to Raiffa, to be called here the Raiffa Solution,1
denoted by gR(B), defined as the unique point u = (u1,u2) satisfying
u1/u2 ul/U2 (3-3)
and
(upu2 )EP , (3-4)
where u1 = max{v11(v1 ,v2 ) EB} and u2 = max{vI(v ,v2 ) eBl
and P is the Pareto frontier of B. (Refer to Fig. 3-3.) Note
that this solution is the point of intersection of the Pareto frontier of B
and the straight line emanating from the origin with a slope of u/u 2 if
u = v2 this line is the 45 degree line. The arbitrator who chooses to use
this solution makes an ad-hoc comparison of utilities by equating i1 and u29
and then gives the parties equal utilities.
(iv) The Proportional Solution (Kalai [1977]), denoted by g(B), defined
as the point u = (u1 ,u2) satisfying
ul/u2 = 2-l"P2 (3-5)
and
(uu 2 )EP , (3-6)
where p1 >0, p2>0 are constants and P is the Pareto frontier. This solutionisa gene-
_piralization of the Raiffa SolutiongR. Here utilities are compared by equatinguyand 2 1
p2Remark. When the conflict payoffs point is some point w which is not
the origin the bargaining domain is not regular. We solve such a bargaining
IThe solution presented here is only one of those introduced in Raiffa [1953].Other authors called this solution the Raiffa Solution and we follow thisconvention.
- 60 -
U 2 '
Pi
2 p 2
2 9p (B)e0-1 O
g (B)Ne/
/ I
Uli U
Fig. 3-3: The Bargaining Problem-Solutions
U2
(D')
(D) oe
D
U
Solving a Non-Regular Bargaining ProblemFigs 3-4:
- 61 -
problem (call it D) by translating D into a regular bargaining domain
D' = {ulu = v-w, veB}. We define g(D) a g(D') + w. (See Fig. 3-4 in page 60.)
3.1.3 The New Bargaining Problem
Let us assume now that the utilities derived by the bargainers are
functions of the final agreement x and of the duration of negotiation t.
This assumption, discussed in the previous chapters, introduces a major
difficulty in the analysis of the two-person bargaining problem. One of
the parties may claim that he is much "stronger" with respect to time than
his adversary; for example, his discount factor may be smaller, or he may
have no bargaining costs. This strength is not reflected in the solution
of the traditional bargaining problem at all, and therefore the solution
is not "fair". Such a claim may be justified. If negotiations did take
place, the strong party could have taken advantage of its strength by, for
example, threatening a delay. We believe that the effect of time on the
utility functions of the parties should be introduced to the bargaining
problem, and this is what we attempt to do in the new formulation and
solution of the problem.1
1It is worth mentioning here the problem of dividing $100 between a richand a poor man who found it together. The standard answer to "what do youexpect the division to be?" is that the rich gets most of the money, mainlybecause of his superior capability atmaking a -termination threat. However, thepoor man may threaten to delay negotiations, taking advantage of hissuperiority with repect to time -- his bargaining cost and discountingfactor are probably lower than the rich man's. This factor must be intro-duced into-the formulation.
- 62 -
Let us start with some definitions. For each time t, O:St< w , the
bargaining domain in t, denoted by B(t), is the set (in the utility plane)
of all agreements that can be achieved in time t. In each set B(t) there
is a unique element w(t), to be called the conflict-payoffs point in t
which is the utility levels achieved by the players if the negotiation
breaks off at time t. The new bargaining game B is defined as the set
{B(t)10St<w} of all bargaining domains. Two additional sets which will
be useful later are the s-tail of B, denoted by Bs, defined for each s 0
as the set {B(t)IsSt<cc}, and the s-truncation of B, denoted by Ba, defined
for each s> 0 as the set {B(t)O5t:s}. The s-tail of B is the bargaining
game which the arbitrator faces if he is called at time t - s instead of
t = 0, and if both parties held to their most extreme demands during the
initial negotiation period of duration s. The s-truncation of B could be inter-
preted as the effective bargaining game when it is clear to the arbitrator
that under no circumstances can the negotiation continue beyond time s. Such
limitation, if it exists, will be called a deadline. When a deadline exists
at T, only the T-truncation of i is relevant for the solution.1
We are now ina position to state the problem. Given the utility
functions of the parties and the deadline T (if it exists) find an agreement
B(O) which is "fair" according to some equity criteria.
The time effects in most of the bargaining problem situations are dis-
counting, bargaining costs and sudden termination. As a result we are
If T+0 the problem reduces to the traditional bargaining problem B(0) andtime has no effects. If T-+= the party who is "stronger" on time shouldbenefit. It is therefore clear that it is to the advantage of the weakerparty to try to impose a deadline.
- 63 -
dealing with bargaining games B which have some nice properties. The game is
called normal if the utility of all agreements (to both parties) decreases with
time, or formally if the following conditions are satisfied:
(1) Let the Pareto frontier at time t be denoted by P(t). Then P(t) is
dominated by P(s) for all t, s, t>s>O (i.e., for all uEP(t) there is a
veP(s) such that v_>u1 , v>u2).
(2) The conflict-payoffs w(t) are non-increasing in time.
A typical normal bargaining game B is shown in Fig. 3-5, where a deadline
in time T is assumed. The changes in the utility levels of a particular effi-
cient agreement xO are shown by the dotted arrows. The points K(O), K(t1 )
and K(T) are the representation on the utility plane of the agreement x0 at
time t=O, t=t1 and t=T respectively.
U2
K(O)
K(t )
IK(T
B(O)
B(t
B(T)
w(O)
w(t)t=t,L.m a m-aa - a-
w(T) tinT
K(t)
K(t 1)
K(T)=
(U (X0,S) ,U2(x0 ))
=(U1(x0 ,t1 ),U 2(x0 t1)
-(U (x0, T),U2(x0 T))
twno U1
Fig. 3-5: A Normal Bargaining Game
- 64 -
An arbitration solution to the normal bargaining game B is a point on
p(O). A solution function h is defined on the set of all normal bargaining
2games into R2, such that h(B)eP(O). In the next section we describe two such
solution functions, the Extended Raiffa Solution (ERS) and the Timing Equili-
brium Solution (TES).
3.2 The Solutions and Their Properties
3.2.1 The Extended Raiffa Solution
We start by suggesting two arbitration principles that we consider to be
reasonable and appropriate
Arbitration Principle 1. The solution is independent of past negotia-
tions. The demands made by the parties when the arbitrator is called are con-
sidered to be their opening bids. -
This principle absolves the arbitrator from the difficult task of com-
paring and assessing the behavior of the negotiating parties prior to arbitra-
tion.
Arbitration Principle 2. The solution depends on the arbitrator's esti-
mate of what happens if he does not intervene, or if his solution is rejected.
If he can assume that the parties would negotiate for some time and ultimately
agree on some point v, then his arbitration problem amounts to solving the
"time-less" traditional bargaining problem induced by v and the Pareto-frontier
at t-o, P(O).
According to the second principle the solution depends on the arbitrator's
assessment of the point v (to be called also a "quasi-conflict-payoffs" point).
This assessment should depend on the utility functions only and not on factors
like the bargaining skills of the parties. Such factors do indeed effect
negotiations but the arbitrator may have little information about them.
- 65 -
One approach the arbitrator can take is to assume that if he did not
intervene, he would be called later, at some time t. His own solution to the
bargainer's problem when called at time t can therefore serve as the quasi-
conflict-payoffs point v. Note that if the arbitrator assumes that during the
time between 0 and t the parties do not concede, then the point v is the arbi-
tration solution of B . Of course, this solution depends on another quasi-
conflict-payoffs point, which again can be chosen as the arbitration solution
in yet a later period of time. The arbitration solution is therefore the last
in a sequence of points, each of which is an arbitration solution of a tail of
the original bargaining problem. Under suitable conditions the existence of
a solution can be demonstrated. This "folding back" approach is reminiscent
of the dynamic programming algorithm or of the decision tree algorithm. In
continuous time it gives rise to a differential equation of the first order.
These two principles lead us to a solution which is reminiscent of the
original Raiffa [1953] solution and is related to the Raiffa Solution gR t
is therefore called here the Extended Raiffa Solution (ERS). Under certain
conditions (discussed in 3.3) this solution exists and is the only one com-
patible with the two principles and the Raiffa Solution of the "time-less"
problem. Suppose, for example, that the bargaining is limited to the period
[0,T] and that an agreement can be achieved at times t=O or t=T only. Assume
also that w(0)sw(T) and that the only duration effect is discounting (See Fig.
3-6). According to Arbitration Principle 2 we have to choose a quasi-conflict-
payoffs point for B. It is natural' to choose a point on P(T) since if there
is no agreement in T-O the parties face the traditional bargaining problem B(T).
The point chosen on P(T) could then be any solution of B(T), viewed as the
traditional bargaining problem.
U2- 66 -
A 2
B(O)
B(T) ?(O)
P(T)
w(O)=w(T) U1
Fig. 3-6: A Simple, Discretized Bargaining Game (Discounting)
We choose AI as the quasi-conflict-payoffs point of B. This point is
the Solution of Raiffa of B(T), viewed as a traditional bargaining problem.
We now look at the regular bargaining domain induced by the Pareto frontier at
t=0 and the point A1 . Applying the Raiffa Solution to this domain, we get
the point A2 as the ERS. For the two-period example, it is the only point
compatible with the two arbitration principles and the traditionalRaif fa Solution.
The same solution procedure applied in the case of a simple, discretized
bargaining game with bargaining costs only is shown in Fig. 3-5.
From the bargaining game B of Fig. 3-7 one can derive 4 different tails.
(1) IT = (B(T)), (2) 1t = {B(T), B(t)}, (3) 1t = {B(T), B(t1 ), B(t0)1,
(4) B - B. The quasi-conflicts-payoffs point of B is w(T). The Raiffa So-0 T
lution of iT is A . This solution now serves as the quasi-conflict-payoffs
point of I , and the solution of the induced "time-less" bargaining problemt.1
is its Raiffa solution, A2 .
- 67 -
U2
B( )
AB (T)A
W(O)
w(t )"M o
w(T)
Fig. 3-7: A Simple, Discretized Bargaining Game (Bargaining Costs)
Continuing in the same fashion we get A3 as the quasi-conflict-payoffs point
of B, and A4 as the ERS.I
In the general case agreements can be -reached at any time prior to T. To.
identify the IRS for this case we can look at the solutions of a sequence of
bargaining games which are finer and finer discretizations of our continuous
game. Under suitable continuity assumptions, the solutions converge to a
iote that because of the 'locations of the quasi-conflict payoffs pointswe had to use the horizontal and the vertical extensions of the Paretofrontiers of 1(t) for some t's in order to guarantee a solution gR in eachstep. These extensions are not needed in the continuous case,
- 68 -
unique point which is chosen as the ERS, For the no-deadline case, the ERS
is defined as the limit point (if it exists) of htR(Ej), 1T being the T-
truncation of B, when T-. This limit exists in the cases of constant pure
discounting or linear bargaining costs.1
The procedure described above supplies solutions to all the t-tails.of- B.
-Let (t)= (9(t),1t2 (t)) be the solution of Bt, for al-I t, Oct<T. The ERS
is 2(0). The curve 'i(t) satisfies the equations.
dt 2 dtt1(t) = - f (u,'t)|utl(t) , OStST, (3-7)
and
t2(t) (fcept),t) ,OstST9, (3-8)
where f(u,t) is the function describing the Pareto frontier in time t. The
boundary condition (which is necessary for a unique solution to exist) is
t(T) = gR(B(T)) . (3-9)
Examples. Some examples will shed light on the Extended Raiffa solution.
Consider first a distributive bargaining problem on [0,11. One party (the
seller) wants the agreement to be as close to 1 as possible, the other (the
buyer) wants it close to 0. The parties are both risk neutral, having
1The limit does not always exist. Consider for example the following game:
The bargaining cost of party 1 is one utile per unit of time for 2n~t<2n+l(n - 0,1,2,...) and is zero for 2n+lSt<2n+2 (n = 0,1,2,...). The bargainingcost of party 2 is [1 utile - bargaining cost of party 1]. It is easy tosee that in this case hR(IT) oscillates between two values as T-. Theregion between these two values on the Pareto-frontier can be considered as a"mediation zone" -- i.e., the arbitrator reduces the size of the Pareto-frontier P(O) to a smaller one P'(0), but not to a unique arbitration point.Such a point could still be chosen according to some ad-hoc procedure. Onethat comes to mind iindiately is the following: Apply the ERS to the newbargaining game induced by P'(0) and w(0). If a unique point is achieved, itis the solution. Otherwise a new mediation zone P"(0) P'(0) is obtained.Repeat the algorithm. Some examples we worked out led eventually to uniquepoints.
- 69 -discount rate r. (where at least one is non-zero) and having no bargaining
costs. If money is compounded instantaneously, the utility functions have
the multiplicative forms
u1(x,t) = e
u2(x, t) = e
-r tx , Ost<co
-rt t2 (l-x),0<tcco
and the Pareto frontier in time t is given by
-r2tu 2 = -ue
(r 1-r2(3-11)
We start by assuming a deadline at t = T.
(3-9) we get:1
I (t) = C(T)e
t2(t) = C(fle
r -r22
r - 12 t
wherer +r2
C(T)=e~ 2 T'.Q6
By applying (3-7), (3-8) and
+ r2 -r tr 1I-r
r1 -r- t+ 1+r2
r r +r1 2
From (3-12) the solution of the T-truncation problem, denotedby ERS(T) is
r2 C r1ERS((T) ( 1(0),t2(O)) r + C(T), r+r2 - C(T)). (3-14)
By letting T-+e- we get
rERS = hR(B) r ft
1 2(3-15)r +r
r12).
'From (3-7) and (3-8) we get the differential equation:
dr(t) -rr2 r 2 t -.rt-d-tt) + -2 11(t) + T 10
The boundary condition is t. (T) = 4 -r1T
the solution (3-12), (3-13) Is obtained.t2(T) - -r2T . From here
and (3-10)
and (3-12)
(3-13)
- 70 -
The effect of the deadline T on the solution in (3-14) is shown for the case
rI > r2 in Fig. 3-8.
ERS1
14
r 52
r1+r2
0
a - m - - m n e - - -
T
Fig. 3- 8 The ERS as a Function of the Deadline T
The solution for the no-deadline case is therefore the point x which
divides [0,1] by the ratio of the discounting rates. The party with the
smaller discount rate is stronger; x is indeed closer to his most
preferred agreement than to his counterpart's.
When linear bargaining costs are introduced and the costs
are subtracted continuously, the utility functions in (3-10) can be
shown to be
- 71 -
u1(xt) =e -r 1 t(x + - ,9<t<
and a a (3-16)
u2 -(x,t)=er2 t( 1 - x + 2 - 0<t<2 r 2 r2
We now get the solution :
ri-a a a1Z1(t) = C(T)e 2 + (1+- + i)e-rt _(3-17)
rr 1 r2 r1
wherer a -a r+r22 1
r1+r 2 + r +r eT(3-18)
From here we get
r2a2-aERS1(T) = (0) =C(T)+ ++ - * (3-19)
r1+2 r+r2
By letting T-w we get
a24-a2ERS= 2+ + . (3-20)
r+r2 r 1r2
This solution may fall outside the interval [0,1]. It can be easily shown
that if the expression in (3-20) amounts to more than 1 the solution is
ERS - 1 . If it is smaller than 0, then the solution is ERS = 0.
Let us examine another example. The utility functions are now2
u1(x,t) - fi(x)-(H - t), O tSH , i = 1,2. (3-21)
The ERS of this bargaining game i (as discussed in section 3.2.4 and proved
in 3.3.3) is the Nash Cooperative Solution of the bargaining domain in t = 0,
gN(B(O)), provided that the deadline H is large enough. Otherwise the
1The differential equation now isd ri-r2 + r2 +G2 --rt a1 r1-r2
4--- e + I- rld (t)+ 2 2(t) +7(1 + +r r r 2.
The boundary conditions are:
t,(T)Me-riT + - A, 1-r1,2.
2 See 2.2'.3 for an example in which these utilities are adequate. (H is interrretedthere as the number of months the agreement holds, x the monthly wage.)
- 72 -
solution lies on the Pareto frontier of B(O) between gR(B(O)) and gN(B(O)).
The solutions of these examples are identical to those proposed by Bishop
(see section 2.2.3) in his normative model given that there is no deadline.
The ERS exists for all normal bargaining games. In most cases it converges
to a point on the Pareto frontier of B(O) as the deadline goes to CO. For
some other properties, see 3.2.3 and 3.2.4.
3.2.2 The Timing Equilibrium Solution
Our second solution for the new bargaining problem is the Timing Equilibrium
Solution, abbreviated TES and denoted by hE. In this section it is described
and compared to the ERS.
We have already mentioned that the aim of the arbitrator is to guess what
happens in the negotiation without his intervention. In certain situations the
arbitrator may find it helpful to contemplate what would happen to the negoti-
ation process if the rules of the game were changed in some specific fashion.
It may happen that optimal strategies to the modified gae do exist and that
the game has a value. This may lead us to other solutions of the bargaining
problem, which the arbitrator might wish to consider. The TES is one such
solution. It is the value of the modified negotiation game that we now describe.
Let the parties make their extreme demands at t=O. Denote their demands at
t by d1 (t) and d2(t). Each party is now told the following:"Negotiate as you
will, but if you are the party who makes the yielding concession (i.e., you
accept an offer that has been previously proposed by your counterpart) you will
receive a prize B from the arbitrator. The prize is calculated so that if the
agreement d1 (t)-d2(t)-d was reached at time t, and if you are party 1, then
(L(d+ B, t), u2 (d,- t)) is a point in' the Pareto frontier in t - 0.
For party 2 the rule .s analgous. If you and your
1Te Appendix of this chapter deals. with the general question of changing therules of the gm in order to obtain solutions.
- 73 -
counterpart yield at the same time, one of you gets the prize, according to
the result of a fair lottery." The rules of this game are common knowledge.
In section 3.3.2 we show that this game has a value and the players
have optimal strategies. The strategy of player 1 is to hold to his initial
demand d1 (t) = d1 (0) until the first time T* satisfying
(u1(d1(T*)), u2(d2(T*))SP(O) , (3-22)
and then to make a yielding concession. The strategy of party 2 is analo-
gous. If the parties apply their equilibrium strategies they hold to their
extreme demands until the first T* such that (u(d(O),T*), u2 (d2(O),T*)eP(O),
and then they yield simultaneously.
U 2K =U 1(d 1(0),0)
K1 - U12(d 1(O),O*)
KJ 2U 2 (d(0),T)2
K- U2(d(OO
J2 U2(d2(O),T*)
/,2
" KU10l
Fig. 3-9: The Timing Equilibrium Solution
- 74 -
The arbitrator can make the following argument: The change in the
rules benefits the parties because the yielding party receives additional
funds from an outside source. The change in rules also guarantees Pareto
optimality. The parties should therefore be content with the point TES =
(u 1 (dI(T*)), u2 (d2(T*))). Of course, the TES solution is a feasible solu-
tion to the original bargaining problem and the revised game need not even
take place. In this case, the TES is an alternative to the ERS. When it
exists, the TES satisfies most of the fairness requirements which are satis-
fied by the ERS. This property alone could be enough to justify the TES as
a solution of the new bargaining problem. But it is indeed a totally
different solution concept. While the ERS is the unique solution that is
compatible with our two arbitration principles and with the Raiffa Solution
of the "time-less" bargaining problem, the TES is the min-max solution of
a hypothetical zero-sum game, invented by the arbitrator. Other such games
may yield other solutions. The TES is therefore one example of several pos-
sible solutions of negotiation games introduced by an arbitrator. This
approach to arbitration will be touched upon in the Appendix to this chapter.
We feel however that further study is needed in order to justify such games
and solutions.
- 75 -
Examples. Let us compare the TES and the ERS for the examples mentioned
in 3.2.1. For utility functions as expressed in (3-10) (instantaneous dis-
counting, no bargaining costs) the TES is the point x satisfying
x + xr2/rl = 1 . (3-23)
Note that for r1 = r2 we get TES = .5 = ERS. For r1 /r2 2 we get TES = .382
vs. ERS = .333. For rI/r2 = 3 we get TES = .318 vs. ERS = .25.
When linear bargaining costs are added, as in equation (3-.16) the TES
is the solution x of the equations
x e-r-it(1 - CL1)
1 = e-rt(1 - at) + e-r2t(l.~12t)l (3-24)
and this may indeed be significantly different from the ERS. This is an
example of the most important difference between the two solution functions,
namely the arbitrary deadline which is imposed on the problem by the TES.
Changes in the utility functions after the time T do not change the TES,
while they may affect the ERS.
Finally, consider the bargaining problem with identical discounting, as
expressed by (3-21). It is easy to show that the TES is the Raiffa Solution
of the bargaining domain in t = 0 , B(0). This is identical to the ERS only
when the deadline H is zero or when the Nash Cooperative Solution and the
Raiffa Solution of R(0) coincide.
Note that the TES is not always defined. When agreements can be
achieved only at discrete times, the modified game may have no value. More
- 76 -
importantly, this is also the case when there exists a deadline T which
is too small, relative to the time effects -- i.e., when the point (u(d1,T)),
(u2(d2 ,T)) is outside the bargaining domain B(O).
3.2.3 The Fairness Requirements1
In this section we describe a set of requirements which we believe ought
to be satisfied by any solution function of the new bargaining problem. Some
of the requirements are simple generalizations of fairness rules which have
been recommended in the literature in the context of the traditional bar-
gaining problem. Others are new requirements which try to capture and to
compare the relative strength of the parties with respect to time. One
requirement (H6) guarantees stability -- it gives the parties an incentive
to resort to arbitration and to accept the arbitrator's decision once it is
made, rather than to involve themselves in pre-arbitration negotiations, or
refuse to accept the arbitrator's solution. The names of our requirements
are chosen. so that their relationship to traditional requirements is clear.
We add the prefix T- (for time) to avoid any possible confusion.
Hi. T-Strong Pareto Optimality. The solution of a normal bargaining
game I, h(l), should be a point on the strong Pareto frontierat t = 0, P(O).
This requirement guarantees that inefficient agreements will be avoided.
After all, a principal reason why the parties resort to arbitration is to
avoid time-related costs.
H2. T-Symmetry. If B(0) is symetric and if the effects of time on
both parties are identical, the parties receive equal utility, i.e.,
h1(li) - h2(i). It is difficult to define "identical effects of time," hut
The discussion here is informal. The exact definitions are given in 3,3,2.
- 77 -
in the case of a symmetric B(O) the effects are identical, by definition, if
for all t20, B(t) are symmetric with respect to the line u1 = u2 .a
H3. T-Independence of Utility Scales. If 5 = {D(t)} is a normal bar-
gaining game satisfying D(t) = {(a 1u1 ,a2u2)I(u,u2 )cB(t)) for all tO for
some fixed a,,a2 then hi(D) = aih(B) (i = 1,2). In other words, multiplying
the utility functions by scalars does not change the arbitrated agreement.
H6. T-Monotonicity. Let B5 be an s--tail of i for some sO. We require
that h (Bs) 1h1(9) (i = 1,2). This requirement guarantees that if the
parties postpone the arbitration to time s, but hold to their extreme demands
until that time, the arbitration point is worse for both than the one at
t = 0. This requirement makes the solution function more "stable", and it
corresponds well to Arbitration Principles 1 and 2. Suppose that party 1
tries to postpone the arbitration and meanwhile to get more concessions from
party 2. If he is indeed successful and the arbitrator is then called,
party lmay achieve aaagreement which is better than the one he could have
gotten without the postponement. It is evident that a new game arises: We
call it the "pre-arbitration negotiation game." Requirement H6 supplies the
parties with strategies that may prevent this game. Party 2, for example,
can hold to his extreme demand until the arbitrator is called; this is
enough to guarantee that party 1 does not benefit from a postponement. A
similar situation occurs when the arbitrator's decision is announced. If
one party, say 1, refuses to accept this decision and to postpone the
negotiation until the arbitrator (or another one) is called again, then
party 2 has a strategy (holding to his extreme demand) which makes this
1A discussion of the requirement of Independence of Utility Scales (in the
context of the traditional bargaining problem) can be found in Luce and
Raiffa [1957], pp. 128-134.
- 78 -
threat unjustifiable.
Requirement H6 gives incentive to the parties to resort to arbitration
and to accept the arbitrator's decision, even if arbitration is not compul-
sory. Therefore it is not a "fairness" requirement in the sense that the
preceding requirements were.
H7.1 T-Individual Monotonicity Under Contraction-Translation. Suppose
that we change B by changing some of its elements B(t) to D={D(t)} where
D(t) = {(u1 u2)1u - av + yt, i 2 = v2 (vv 2)eB(t) for some
05atl, Y 5wlt( -t) such that if at = 1, then yt<0 and if yt = 0, then
Ct<1. Such a unilateral transformation will be called here a contraction-
translation with respect to 1 (the definition for i = 2 is similar). If 1
is the modified game, we can interpret it as a game in which discounting and
bargaining costs affect party 1 more than they affect him in B, while the
effects on party 2 are the same in both games. This is a.fairness ,criteria
because it postulates that as time affects party 1 more severely he should
be worse off.
- P'(0) P(0)
(0)
D(t B()
BWt 't
P'(M
Fig. 3-10: Contraction-Translation With Respect to 1
- 79 -
In Fig. 3-10 weshow a typical case of contraction-translation. The Pareto
frontiers P(O) and P(t) are transformed to P'(0) and P'(t)
respectively.
H7.2 T-Individual Monotonicity Under Truncation. Suppose now that B
is transformed to D in the following way. Some of the sets B(t) are trun-
cated as depicted in Fig. 3-11,while the others do not change. We say that
D(t) is an i-truncation at bt of B(t) if D(t) = {(u 1,u2)JusB(t), u .b t
and D(t) t B(t). The game I is called an i-truncation of B.
U2
B(O) B(t)-D(T)
B(T) B(O)-D(O)
b
Fig. 3-11: I-Truncation of B
We require that if D is an i-truncation of B then hi(5)Sh (i). Intuitively,
this requirement says that if the most desirable agreements of party-i are
somehow infeasible, then party-i would receive a less preferred agreement by the
arbitrator. (This may happen, as in Fig. 3-11,when party 1 changes his
demand at t - 0 to b0 .) This requirement does not encourage generosity
before arbitration. It is, however, in line with our Arbitration Principle 1,
- 80 -
according to which past negotiations do not matter.
H8. T-Uniform Continuity. If B is slightly changed by small modificar
tions of some of the B(t)'s, then the solution does not change by much, This
is certainly an important requirement when the utility functions used by the
arbitrator are his own subjective and inaccurate estimates.
We now add a condition about the effect of changes in the tail of a
normal bargaining game:
HJ0. T-Deadline Monotonicity. Suppose that 9 is a normal bargaining
game with a deadline T. Suppose also that for some t, O5t$T, the tetail of
B is changed in a way that can be interpreted as favorable to party 2, We
then require hI(D)Th1 (B) where B is the modified game. The issue is what
changes in a tail justify such interpretation. We use here the idea of
Arbitration Principle 2 (the existence of a quasi-conflict-payoffs point).
If the solution of the t-tail (viewed as a normal bargaining game) is changed
so that party 2's utility is higher and party l's utility is lower, the
change is postulated to be favorable to party 2. The argument for a change
which favors party 1 is analogous.
3.2.4 Reexamination of the Nash Cooperative Solution
In this section we examine the Nash Cooperative Solution, one of the
most accepted solutions of the traditional bargaining problem, in light of
our Extended Raiffa Solution. The results mentioned in this section are
formalized and proven in 3.3.3.
Consider a traditional bargaining problem (no time effects) with a
bargaining domain B as shown in Fig. 3-12. The point v could be called a
focal point (Schelling [1960]). It suggests itself to the negotiators and
to an arbitrator as a natural bargaining solution. The agreement x
- 81 -
U2
2
Iu
B
01 U1
Fig. 3-12: A Bargaining Domain With a Focal Point
represented by v may have the advantage of being prominent even when the
exact shapes of the utility functions are not known.1 The point v is some-
times indeed the Nash Cooperative Solution of the regular bargaining domain
B.2 The Solution of Raiffa gR(B) is the point u. It does not generally
coincide with v. That the Nash Cooperative Solution coincides with a focal
point is one of its most appealing properties.3 It is therefore interesting
to note that the Extended Raiffa Solution also identifies the focal point
as a worthy arbitration point, but only if the parties are identically
affected by time. We will show in L3 that the focal point
1For example, in labor-management negotiations over the rate of increase inwages, the inflation rate might be such a point.
2The conditions for that are vg>s i = 1,2.
3Note however that if v <J for ie(1,21 or if there are several focal pointsthe Nash Cooperative solution loses its appeal.
- 82 -
of B(O) is the Extended Raiffa Solution of B in the case of equal discounting
of the utility functions and in the case of equal bargaining cost. Consider
for example a game with linear , identical bargaining costs. (See
Fig. 3-13.)
U2
3.
U
T2
3Fig. 3-13: The Focal Point and the ERS
The curves t 1 , C2 , t 3 represent the Extended Raiffa Solutions for three
different deadlines T1 , T2 and T3 . It is proved in 3.3.3 that if T is large
enough then the ERS coincides with the. focal point.
These results make the Nash Cooperative Solution all the more appealing
when the effectsof time on the parties are identical. But note that when
these effects are not identical the Nash Cooperative Solution for B(O) does
not necessarily coincide with the ERS; the solutions may be very far apart.
If one cannot assume identical time effects, one -need not accept the Nash
Cooperative Solution as "fair", since it does not reward the party,, y
stronger with respect to time. It is also not a "rational" solution
- 83 -
because it ignores the threat capabilities of this party.
3.2.5 Applications
The solutions discussed in 3.2 wers proposed as arbitration schemes. As
such they can be used by an arbitrator in cases where the primary concern of
the parties is the effect of the negotiation's duration. An estimate of the
utility functions of the parties is of course needed, and its accuracy is
important especially when the Timing Equilibrium Solution is used. The
Extended Raiffa Solution is somewhat more robust. We have tried it on vari-
ous functional forms and discovered that if the deadline T is large enough,
small changes in the utility functions have a small effect on the outcome.
It is conjectured here that both solutions may be found to have some
descriptive quality. The applications described in this section (except for
a) are based on this conjecture.
a. Explanation of the Length of Strikes. Consider a negotiation in
which party 1 has an "abnormal" utility function U1 (x,t) which is increasing
with t for some agreements x, for OtSt 0 . For t>t0 the utility function
decreases with time for all x. A discrete example is shown in Fig. 3-14.
2
P(0)
P(t) >t
1(t0 )
Uj
Fig. 3-14: The Function U1 is Non-Decreasingfor OStSt0
- 84 -
A formal application of our solution functions may lead in some "good"
cases to points on the Pareto frontier P(O) which are not dominated by any
point on P(t), for any t. However, we may find cases in which the solution
does not exist. We can say that there is no solution to this bargaining
problem when arbitration starts at t = 0. It is clear that if arbitration
started at t = to, a solution would exist, since Bto is a normal bargaining
game.
This anomaly may explain the frequent failure of arbitration of labor
disputes. As discussed in 2.2.5, Ashenfelter and Johnson [1969] view the
collective bargaining process as an implicit 3-party negotiation. The
parties are management, union leaders, and union rank and file. Trying to
keep their personal political power, to strengthen the union as a whole and
to lower the expectations of their constituents, the union leaders may pos-
sess a utility function which is similar to the one discussed above. The
average strike length may therefore correspond to the time t0 after which we
are again in a situation of a normal bargaining game, that is when the
political factors begin to be much less important than the economic ones.
Assessment of the utility functions may give us an estimate of this t00
Alternatively, knowing empirically the.alue of t0 may yield better understanding
of the utility functionsof union leaders.
b. Understanding the Role of a Deadline. Trying to impose a deadline is
a common phenomenon in negotiations. Two standard explanations are:
(1) the sudden termination possibility may materialize if the negotiation is
.too long; and (2) the utility functions of the parties may change, thereby
eliminating the zone of agreement, if there existed one. Another explanation
is this. By-imposing deadlines, parties may improve their situation it. an arbi-
trator Is called sinca the deadline has a strong effecton: thearbitrated solution. It
- 85 -
is in the interest of a party who is weaker with respect to time to impose
a short deadline. It is in the interest of his counterpart to avoid any
deadline. The ERS may help in quantifying the effectsof deadlines since it gives
explicitly the benefits derived by each party for all deadlines.
c. Tendency Towards Focal Points. In many practical cases the effect
of time on the two negotiating parties is roughly the same. In light of our
discussion in 3.2.4 the focal point, if one exists, is justified not only
as a device to coordinate expectations but also as a "fair" arbitration
point which would be chosen by an impartial arbitrator.
d. Mediation Schemes. The set on the Pareto frontier in t = 0 which
include all the ERS (for various deadlines) could serve as a mediation set.
The role of the mediator is to reduce the negotiation set and to encourage
the parties to make concessions. Using our solution functions the mediator
can do just that and reason that this proc "ire is "fair". The mediation
set may be small enough to make further mediation efforts short and
successful.
3.3 Mathematical Development
3.3.1 The Traditional BargainingProblem
Definitions. Formally defined, a regular bargaining domain is a set B
in the non-negative quandrant of R2 which is
(1) Compact,
(2) Convex,
(3) Comprehensive with respect to the origin (i.e.,, if
ueB and wSu, wkO, then nEB),
1Our conventions regarding vectorial inequalities are: the expression wvu
- 86 -
(4) Non-trivial with respect to the origin (i.e., there
is a ueB, u>>O).
We are looking for a solution function g defined on the set of all
regular bargaining domains. For a particular domain B, g(B) = (g1(B),g2(B))
is the solution, where g1(B) is the utility received by player i (i = 1,2).
Requirements. We now list the conditions that have been discussed and
proposed in the literature as "reasonable" requirements of the solution.I
Not all were proposed as "fairness" criteria, mainly because the arbitration
scheme interpretation has not been the only one. Requirement G7 (Strong
Individual Monotonicity) is a new version of G7' (Weak Individual Monotonicity).
G1. Strong Pareto Optimality: if uEB, weB and w<u then wtg(B).
Gl'. Weak Pareto Optimality: if ucB, weB and w<<u then wtg(B).
G2. Symmetry: if B is symmetric (i.e., (u1,u2)cBu>(u2 ,u1)EB) then
91(B) =92(B).
G3. Independence of Utility Scales: for every Noa<M (i = 1,2),
if B'-{(auc 2 u2)|(u 1 ,u2)eB} then g(B')-(agg(B),a292(B)).2
G3'. Homogeneity: for every a>O, if B'={(au1 ,au2 IueB} then
g(B')-ag(B).
G4. Independence of Irrelevant Alternatives: if B'SB and g(B)eB'
means widui(i - 1,2); the expression w<u means widui (i - 1,2) and at leastone inequality holds; the expression w<<u means wi<ui (i - 1,2).
The terminology and the results of section 3.3.1 are based on the works ofNash [1950], Raiffa [1953],Oren [1968], Kalai-Smorodinsky [1975], Kalai [1977]and Roth [1977]. A discussion of the original requirements of Nash is foundin Luce and Raiffa [1957].
It is easy to see that B' is a regular bargaining domain if B is, so thatg(B') is well defined. Requirement G3 is one version of the requirement ofIndependence of Linear Transformations. We choose this version because wewant the bargaining domain to be comprehensive and non-trivial with respectto one point only, namely the origin.
- 87 -
then g(B)=g(B').
G5. Strong Individual Rationality: g(B)>>O.
G6. Monotonicity: if B'25B then g(B').g(B). This requirement
is related to the consistency of the solution when negotia-
tions are done by stages. In stage one the set of possible
agreements is B', and its solution is g(B'). In stage two
additional agreements are possible, the new set of agreements
is B and the solution is g(B). The requirement reflects the
idea that under these conditions both sides should benefit by
the addition of new agreements. However, it is a dubious
requirement when B' and B are two regular bargaining domains,
each reflecting a one-stage negotiation.
G7. Strong Individual Monotonicity: let Ki = {ucR2 Iui"O}, (i = 1,2).
If B'GI but B'tB, and if B'fKi = BAKi for ie{1,2} and if
B'OKOBAK for jti, then g1(B')<gi(B). This requirement
reflects a particular situation, depicted in Fig. 3-15. The
U2
DP'
4
U1
Fig. 3-15m.. Strong Individual Monotonicity
- 88 -
domain B' is bounded by the axis and by P'. It could be inter-
preted as derived from the domain B by a horizontal shift in the
utility level of party A (like the A4 +A5 move described in
Fig. 3-15). Under such transformations it is reasonable to
assume that party 1 will lose utility. If g(B) = A then
g(B') is a point on DA3 of P 2 (if Pareto optimality is required).
G7'. Weak Individual Monotonicity: under the notations of G7, if
BCB and B'lKi = BIKi then g (B').gi(B). This requirement is
almost identical to G7 for the case depicted in Fig. 3-151
but it. is. totally different for the case shown in Fig. 3-16.
In this case, the solution of B' is restricted to the arc A2A3
of the Pareto frontier of B'. For this case G7' is identical
to G6 (Munotonicity) while G7 tells nothing about the solution.
U2
A
B' A2
A3
U
Fig. 3-16: Weak Individual Monotonicity
G8. Continuity: if (B } is a sequence of regular bargaining domains,n
B -B (i.e.,v[(B-B )A(B -B)J-'O where p is Borel measure), then
g(Bn)+(B).
Before stating the next requirement we make the following definition.
Let ) demote the peto! all point. (Yy)isthe plane such that y u for
some i., IB. (B is the "extended" bargaining domain.) A bargaining domain B'e
- 89 -
is induced by the Pareto frontier P and the point u, ucBE if
B' ={vIv eB,, v>u}.
G9. Step by Step: let B'EB, and let B" be the set induced by
the Pareto frontier of B and the point g(B'). If B" is not a
one element set, then g(B) = g(B').
To understand this requirement imagine that the bargainers
divide the negotiations into two steps, denoted by I and II, and
that a failure in stage II means that the outcome of stage I
is the final agreement. The outcome of I can therefore
serve as a conflict payoffs point for stage II. It is required
by G9 that the final outcome should not depend on the particular
division into stages.
The results concerning the solution functions that have been proposed
in the literature are:
(a) The requirements G1, G2, G3, G4 (the original Nash requirements)
imply and are satisfied by the Nash Cooperation Solution gN (Nash [19501).
(b) The requirements G2, G3, G4, G5 imply and are satisfied by the
Nash Cooperative Solution gN (Roth [1977]).
(c). The requirements G3, G4, G5 imply and are satisfied by the
a 2Generalized Nash Cooperative Solution g (Friedman [1977]2).
(d) The Solution of Raiffa gR satisfies the requirements Gi, G2,
G3, G5, G7, G7', G8 (Raiffa [1953]3)a
'This term is taken from Kalai [1977] who introduced this requirement. Myer-son [1977] prop6sed the term Composition for a similar requirement.
2We are not sure as to the first reference in which this property is proved.
3In Raiffa [1953] only -requirements G1, G2, G3 are mentioned. It is easy tocheck that the solution satisfies the rest.
- 90 -
(e) The requirements GI, G2, G3, G7' imply and are satisfied by the
Solution of Raiffa gR (Kalai and Smorodinsky [1975]).
(f) The requirement sets {Gl', G3', G5, G61, {Gl', G3, G5, G9} and
{Gl', G3', G5, G4, G7', G81 are all equivalent and imply the Proportional
Solution gp (Ealai [1977].) ..This -solution is .sometimes viewed as a formali-
zation of the Solution of Raiffa.1
Remark. The Nash Cooperative Solution does not satisfy G6, G7, G7',
G9. In Fig. 3-17 we show counter-examples.
U2
U1gN does not satisfy G6
U 1
(c) does not satisfy G7'
Fig. 3-17:
(b:
U2
gN does not
U1satisfy G7
U1
(d) gdoes not satisfy G9
Properties of gN
Leuma 3.1
Let g be a solution function defined on the set of all regular
'This is the view of Rosenthal [1976] and Roth [1977a) Some caution is neededhere. The Solution of Raiffa is clearly a result of the view that the re-quirements relate to a change in utilities and not in the bargaining game,while Kalai-Smouodinsky [1975] and Kalai [1977] view the requirements as re-sulting from change's in the bargaining game. The solution function gR per-forms an ad-hoc comparison of utilities on every bargaining domain, whilethe proportional solution Sj employs a fixed scaling to compare utilities,and .the proportions must be determined.
U2
(a)
U2
I
- 91 -
bargaining domains. If g satisfies G7 and G8 then it satisfies G7'.
Proof: Let B' be a subset of B, satisfying B1 0 B, B'AK = BAK,
where KC (i = 1,2) are as defined in requirement G7. Now B' is the limit
of the regular bargaining domains B' defined by BI = B'Af{(u,u2 )1~? 1
where V = max {v1 j(v1,v2 )EB' for some v2} . It is easy to see that
B'CB, B'KAK = BnK ,B'A K2 2B's K By G7, g1(B')<g1(B). By G8, g1(B')ii- nA1I BK n,2'AK B'A
g1(B); i.e., G7' is satisfied. 0
Lemma 3.2
The requirement set {Gl', G2, G3, G71 implies and is satisfied by the
Solution of Raiffa gRe
Proof: It is easy to see that gR satisfies the requirements (it satis-
fies G1 as well). Assume now that g is a solution function satisfying Gl',
G3, G2 and G7, and suppose g(B) y gR(B) where B is a regular bargaining
domain. Let g(B) = u, gR(B) = v. Since v is in the strong Pareto frontier
of B,. and u is in the Pareto frontier of B, it is impossible that either
u>v or v<<u hold. Therefore we have either v1<u1 and v22u2 or v2Cu2 and
v .>u. We assume that the first possibility holds. (In the other case
the proof is analogous.) Let a1 - max{u1 1(u,u2 )eB for some u2} 2
max{u2 1(u1,u2) eB for some u11 . It is easy to show that v1 >0 (otherwise
choose l>a>O, and consider the bargaining domain whose Pareto frontier is
the straight line passing through (O,a2) and (Q-1,0). Application of G3, G2
and G7 leads to a contradiction).
Define D1 - (Oa2) D2 - v, D3 - (a2. r, 0). Note that D3 is well2
defined because v2>0. Evidently D2 is in B because the intersection H of
Of2 with the line' (l ,u)IOSu<} is above the intersection Q of the diagonal
of the Raiffa's solution with the same line. From the triangle OMN it is
- 92 -
U2
D/
/ ////
/3N
Fig. 3-17: Proof of Lemma 3.2
- 93 -
V 1 a1 a1seen that = <r therefore a y a . The area D D D 0 is a regular
2 2 2 v abargaining domain (denoted by B') because "2'i 0 = .- vpv> (i.e., B' is con-
2V 2 V2 1
vex). Note that by transforming B' into B" = {(b1 u1 ,b2 u2)I(uiu2)EB'} we
get a symmetric bargaining domain. (We need to choose (bb2)= 1-) if
v2 2lV 22or)(b b2 )I=iv rSVor (b),b2 -2i v .) By G2 (Symmetry), g(B') = D
and by C7 (Strong Individual Monotonicity) D2 cannot be g(B), contradiction. 0
3.3.2 Thejew Bargaining Problem
The normal bargaining game B with a deadline T has already been defined
in 3.1.3. The requirements of a solution function h of B were informally
discussed in 3.2.3. We start here by stating these requirements formally.
Recuirements.
Hi. T-Strong Pareto Optimality: h(B)cP(0) where P(0) is the strong
Pareto frontier of B(0).
H2. T-Symmetry: if w1 (t) - w2 (t) and if B(t) is symmetric with
respect to the line u1 - u2 (i.e., if (v1 ,v2)EB(t) then (v2,v)cB(t))
for all t, OIt T, then h1(B) = h2(B')
H3. T-Independence of Utility Scales: if '- c{aB(t)B(t)e5} for some
a, 05a< w, then h(') - ah(I). Note that a is a two-dimensional
vector (a1,a2) and we allow al 0a 2
H6. T-Monotonicity: let Bi be the s-tail of B for some 0s5T. Then
h(i)Ih(i).
H7.1 T-Individual Monotonicity Under Contraction-Translation: let B
and D be normal bargaining domains, and let SC[O,T] be a set of
indices. If there exists an i, idl,2} such that D(t) is a
contraction-translation1 with. respect to i of B(t) for all
1 For a definition see 3.2.3.
- 94 -
tES, then h1(B)h1(B).
H7.2 T-Individual Monotonicity Under Truncation: let B and D be normal
bargaining games and let SG[O,c) be a set of indices. If there
exists an i, ie{l,2} and a set of scalars {b(t)IteS} such that
D(t) is an i-trunzation at b(t) of B(t)I for all teS then
hi(5)Shi(i).
H8. T-Uniform Continuity: let Bn = {Bn(t)I02tET1 n = 1,2,... be a
sequence of normal bargaining games. Let v be Borel measure. If
p(Bn(t)\B(t))-+O uniformly2 then h(Bn)+h(B).
H9. T-Tail Monotonicity: let B and D be normal bargaining games such
that is =b for all OSslt and it D t for some O5tdT (i.e., i and
B differ only on their t-tail). Then, if hi(Dt)5hi(Bt) and
hj(Dt)Thj(Bt) (i,jE{1,2}, i t J), then hi(D) hi(B).
We now turn to the development of the ERS. We start with a two period
discretization of a normal bargaining game (Lemma 3.3), then move to the
finite discretization (Lemma 3.4) and then to the continuous case (Theorem 3.1).
Lemma 3.3
Let B = {B(O), B(T)} be a (discretized) normal bargaining game. Let
B(0)jp be the set induced by B(0) and a point p, for every peB(0). Define
the Extended Raiffa Solution of i as
hR(I) gR(B(0)I(gR(B(T)))) (3-24)
This solution satisfies the requirements H1, H2. H3, H6, 17.1, H7.2, H8 and
H10.Ro.
1For a definition see 3.2.3.
2Definition: p(Bn(t)%B(*0 uniformly itfor every e>O there is an Ne such thatfor all t, OStST, >Ne implies p((Bn(t) a B(t))(B(t) - Bnft)))
- 95 -
Proof. (1) Since gR satisfies GL (Strong Pareto Optimality) and since
the strong Pareto frontier of B(O)IP is a subset of the strong Pareto frontier
of B(O) (for all pEBe(0), and gR(B(T)) is such a point), hgR() is in the
strong Pareto frontier of B(O), i.e., Hi holds.
(2) Since gR satisfies G2 (Symmetry), we have gRI(B(T)) = gR2 (B(T)) and
therefore B(0)I(gR(B(T))) is symmetric. Again by G2 we get \(B)MI2
i.e., H2 holds.
(3) H3 is proved from G3 in the same manner as HI and H2.
(4) Since gR satisfies G5 (Individual Monotonicity) it is clear that
if 8R(B(T)) is not in the Pareto frontier P(O) then Itr (B)>>h (B) if
gR(BV) is in P(O) then hr (B) - hr(BT). Now BT is the only tail of B
which is not B itself. We proved hR(B)ahR(B(T)), therefore H6 holds.
(5) We now prove that H7.1 is satisfied.
(a) Let 5 - {D(O),D(T)} be a normal bargaining game such that:
(i) D(O) = (ulu =a0v 1 'B'(vO'v2)cB() for some v2} where a0 and yo
satisfy O<a01 and y050. By H8 it is enough to assume that g<l and y0<0.
(ii) D(T) - B(T). Clearly gR(B(T)) = gR(D(T)). We denote this point by v.
(See Fig. 3-17.) We have to prove that gR(D(O)Iv)hgR (B(O)Iv). The point
h(B) is on the straight line passing through v and the point (ab). The
point h(B) is on the straight line passing through v and (a2 ,b2) If b1=b2
or if a1 -a2 the proof is immediate since 9R satisfies G7' (Weak Individual
Monotonicity). We can therefore assume b1 >b2 and a1 >a2 , as shown in
Fig. 3-1,9. It is now enough to prove that
b2 - v2 b1 -V 2 (3-25)a2 -v 1 a1 - v .
By some algebraic manipulation (3-25) can be shown to be equivalent to
(b, - b2 )(aI - v1) (a1 - a2)(61 - V2 ). (3-26)
- 96 -
2
b
h(- 5/h(B)
blh-D)2--l --2
/I II// I I
vi I
a3 a2 a1 1
Fig. 3-19: Proof of Lemma 3. 3 (5-a)
From the convexity of B(O) it is easy to see that
b - b2 g b__-__
a3 - v1 a - vi
and from here that
(b1 - b2 ) (a1 - v1) 5 (a3 - v1) (b1 - v2 ) . (3-27)
By comparing (3-26) and (3-27) it is enough to prove that a - v1 -: 1y a2.
But a2 a.c + Y0and v1 - a3.c+y0. From here we get (a3 - v1 ) - (a1 - a2
- (a3 - a )(1- a0) 5 0 and the proof is completed.
(b) Now let D - {D(O), D(T)} be a normal bargaining game such that:
(I) D(0) - B(O). (ii) D(T) - fu u - TV1 + y7 , (v13v2)cB(T) for some 2l
where aT and TT satisfy O<a <1 and y 50 . By H8 it is enough to assume that
- 97 -
aT<1 and YT<. From requirement G3 it is clear that g;(D(T)) = aTgR(B(T))T 1 1
+ YT. Since the Pareto frontier P(O) is sloping downward the geometrical
positioning of the diagonals used in the creation of the ERS is as shown in
Fig. 3-20. It is immediate to show that hR(D)ShR(B).
1 1S
U2
9R
9R(B(t)
_R It
Fig. 3-20: Proof of Lemma 3.3 (5-b)
(c) When D(0) and D(T) are both contraction-translation with respect to
1, the proof is immediate. First we define a normal bargaining B' game in
which the contraction translation occurs in t - T only (i.e., D' = {B(0),D(T)})
and then we apply the proof in (b) to get hR(D')3hY(J). Then we show, by apply-
ing (a), that hR&)hR (5'). From here we get h (')_hR(R).
(6) The proof of H7.2 is based on geometrical considerations, similar
to the ones used in the proof of H7.1. We omit the details.
(7) To prove H8,, let In %(n Bn t c{0,T}} n - 1,2,... be a sequence
of (discretized) normal bargaining games, converging uniformly to B. The
following inequality holds:
- 98 -
d[gR(Bn(O) IgR (Bn(T)))i gR(B(O)I gR(B(T)))]S
S-d IgR (B,(OgR (B n(T))), gR(B n 1R (B(T)))]+
+ d[gR(Bfn(O) SR(B(T))), gR(B(O) IgR(B(T)))]
where d[u,v] = Iiu - vii. From requirement G8, satisfied by gR, we get that
both terms converge to 0 with n. Since there are only two time periods, the
convergence is uniform.
(8) The proof of H10 is similar to the proof of 17.1(b). When B'(T)
CB(T) and h(B'(T)) satisfies h1(B'(T))Th1 (B(T)) and h2 (B'(T))Th2(B(T)) then
the Solution of Raiffa gR(B'(T)) is in the shadowed area of Fig. 3-21. From
here the proof procedes as in H7.1(b).
'U2
B(T)
Fig. 3-21: Proof of Lemma 3.3 (8)
Lemma 3.4
Let B[nJ - (B(O), B(t),...,B(t0 )1 be a (discretized) normal bargaining
game. Define the Extended Raiffa Solution of B[n] as
hR(B[n]) - g (B(O) ih,(i[n]t )) , (3w28)
99 -1where, as usual E[n]t is the t1-tail of B[n]. This solution satisfies
requirements Hl, H2, H3, H6, H7.1, H7.2, H8 and H10.
Proof. The proof is by induction on n. The case n = 1 was proved in
Lemma 3.3. Suppose Lemma 3.4 is correct for n = K and look at Bik + 11=
(B(0), B(t 1),...,B(tk+1 ). The proofs of Hl, H2, and H3 are identical to the
corresponding proofs in Lemma 3. 3. H6 is satisfied because by the properties
of 9R we get hR(M[n])hR([n tl) and by the induction assumption we get
hR ([n])!hR([nit ) for all j, 2jk+l.H7.1 and H7.2 are easily proved by
checking two cases: (i) The changes (contraction-translation, trunction)
occur in the t1-tail only -- the proof is immediate from the induction assump-
tion and from similar geometrical proofs in Lemma 3.3. (ii) The changes
occur in B(O) only -- the proof is identical to part 5(a) in the proofs of
Lemma 3. 3 When changes occur in the tail and B(O), a sequential use of (i)
and (ii) completes the proof. The proof of H8 is analogous to the corresponding
proof in Lemma 3.3. .H10 is immediate from H6 and the induction example. 0
We now turn to the continuous case. Let (= B(t)IO<tST} be a normal
bargaining game and let f(u,t) be the function describing the Pareto frontiers
in times t, for all t, 0tST (i.e., if v = f(u,t) then (u,v)cP(t).) The
bargaining game is called "continuous" if the function f(u,t) is continuous in
u and t. We will, however, restrict ourselves to functions f which are
differentiable.
Suppose that t(t0) is known for some to, O<t0!T. For all At small
enough we define
1 This is an inductive definition. We assume that hR(B[n]t1 ) and generally
the ERS of a normal bargaining game which is discretized into n-1 periods is
well defined. In Lmena 3.3 we dealt with the bargaining game M[].
- 100 -t(t 0 - At) gR (B(t0 - At)(t0)). (3-29)
We then define the time derivative of Zi(t) at t0 by
ti(t0) = - urn t t0 - At) - ti(t0)) (i = 1,2) , (3-30)At+O
if this limit exists. The ERS is then defined by
ERS = 1(0) = lim (t) . (3-31)t+O
It is easy to verify that ti(t0) exists for all t(0), and that 1(t) satisfies
(3-7). We have to prove:
Theorem 3.1
The ERS satisfies the set of fairness requirements H, H2, H3, H6, H7.1,
H7.2, H8, and H10.
Proof. FrOm' the construction 1(t)eP(t), and by continuity t(0)eP(0),
i.e., Hi is satisfied. Requirement H2 is satisfied since t1(T) = t2 (T)
(because B(T) is symmetric) and t2 (T)/eI(T) = 1, and the curve 11(t) = t2
T2t0O, satisfies (3-7). Requirement H3, H7.1, H7.2 and H10 are all proved
immediately from (3-7) and from the concavity of the Pareto frontiers. To
prove H8 note that BnB+ uniformly means f~f 0(ut)-- f(u,t)Idu-+0 uniformly,V v 0
which implies f f (u,t)du + f f(u,t)du uniformly, for all 0$Al. By dif-n
00ferentiation we get f1(v,t)+f(v,t)unifermlyforallv.' Since f and f are
nn
differentiable f (vt) + f (vt)uniforly fOrall v. From (3-7) we -get3v n av
hR(B )-+hRB).
e now turn to the Timing Equilibrium Solution. We fLrat prove that it
is the value of the game described in 3.2.2 and then show that it satisfies
most of the fairness requirements under certain conditions.
- 101 -
Consider our regular bargaining problem on [0,11. The utility functions
u i(x,t) are given and are common knowledge. As usual, party 1 prefers x = 1,
while party 2 prefers x = 0. Party i chooses and announces a sequence of
i I i Ii ipoints a1, a2 , ... in times t1 , t, ... where O= t1 < t2 <e... (i=1,2). The
current proposal of party i in time t such that t1<tckt is a . The partiesk7 k+l akUThpate
are constrained in their choices of points a by the inequalities aI>a1>a3>...
2 2 2 *and a<a2<3< ... . The game ends in the first time t in which the current
s 1<2*proposals a satisfy a'-a . Clearly the game may end only in t = tk for
some k>0 and ie{l,2}, i.e, when one party makes a yielding concession. If
* 1 2t =t and there is no k such that t= t (i.e., the parties did not make
1 *yielding concessions simultaneously), then the payoffs are v2 (ak, t ) for
party 2 and G 1 (u2(a,t ) for party 1, where v2 = G(v1 ) is the equation
* 1 2describing the Pareto frontier in t = 0. If t t = t t (i.e., the parties
yielded simultaneously) then the payoffs are 4u2(a_1 , t*) + 1 *G(u (ak-1, t
1 * -1 2 *for party 2 and iul(ak-1, t9 ) + 4G~ (u2(a 1 t ) for party 1. Finally, if
S 2* 2 *t = t and there is no k such that t = tk, then the payoffs are G(u1(a, t*))
for party 2 and u(a2, t*) for party 1. We call this game N.
Theorem 3.2
Under suitable conditions (expressed in Leuma 3.5) the game N has a
unique pair of equilibrium strategies. The strategy of party 1 is: start
*with your extreme demand x - 1 in t = 0. Make a yielding concession at t ,
where t is the minimal element in the set of all t satisfying
2G(u1 (1, t)) 2 u2(ak*, t) (3-32)
where k* satisfies *22<t, t*+.t. The strategy of party 2 is: start with
**your extreme demand x - 0 in t - 0. Hake a yielding concession at t , where
- 102 -**
t is the minimal element in the set of all t satisfying
G (u(0,t)) ;$ u1(ak**, t)(3-33)
** 1 1and k satisfies tk**t k**+1 t
The application of these equilibrium strategies leads to the payoffs
0 0u1 , u2 which solve the equations
0 0
U Ot0 u 0 (335)U2(0,t ) = 23-)0 0
u0= G(uQ. (3-36)
Proof. It is easy to see that these are equilibrium strategies. Sup-
pose party 2 switches to another strategy. This means that one or several
of the following events occur: (1) Party 2 starts with a demand a>0.
2 2 2 2(2) Party 2 chooses, at least once, a point ak such that ak>a1 and ak is
not a yielding concession. (3) Party 2 makes a yielding concession before
** **t . (4) Party 2 does not make a yielding concession at t . If party 1
holds to his strategy then due to the monotonicity requirement and to the
effects of timeequation (3-32) guarantees that he is better off under (1),
(2), (3) and (4). Therefore, party 2 is worse-off. A change of strategy is
therefore .not worthwhile. A similar argument holds for a change in strategy
by party 1.
Equations (3-34), (3-35), and (3-36) are a direct result of (3-32) and
(3-33) when both parties use their equilibrium strategies. Since party i can
0generate to himself the utility u (i = 1,2) and since there is no pair
0(v ,v2) such that v zui ,V - G(v), it is clear that the equilibrium is
unique. 0
- 103 -
Lemma 3.5
Necessary and sufficient conditions for the existence of a unique
equilibrium in the game N are:
(1) Equations (3-34), (3-35), and (3-36) have a solution.
(2) U1(x,t) are continuous in t (i = 1,2).
Proof. If (1) and (2) hold then there are minimal elements t which
solves (3-32) and (3-33). To see this note that G and G are continuous and
that for two continuous functions f(x) and g(x) if f(x) increases with x and
g(x) decreases with x then the set of all x, such that f(x) = g(x), is
closed. The strategies described in the theorems are therefore well defined,
and the solution t0 of (3-34), (3-35) and (3-36) is unique. Sufficiency is
thus proved.
If (1) is satisfied but (2) is not, it may be the case that A = {t01 0
satisfiepQ-34), (3-35) and (3-36)} has no minimal element. Let t' be the
infimum of A. If U1 (l,t) and U2(0,t) are strictly increasing in a
neighborhood of t' then clearly no optimal strategy exists. Note however
that e-effective strategies can be constructed, and the game may have a value.
When (1) is not satisfied, even the existence of c-effective strategies is
not guaranteed. Q
Theorem 3.3
Let i be a normal bargaining game, and suppose that the conditions (1)
and (2) of Lemma 3.5 are satisfied. Then the TES, denoted h(B), satisfies
requirements Hl, H2, H3. H6, -7.1, H7.2, and 18. (The last requirement,
H10, is not satisfied.)
Proof. Hi. By (3-36) h (B) is on the weak Pareto-frontier of B(0).
However, it must be on the strong Pareto frontier. Suppose h (B) -e
- 104 -
(u (1,t 0 ), u2(0,t0 )) and that there exists a point vEB(O) such that v>he(B).
If v1 = hel(B) and v2>he2(9) then necessarily v, = u(1,0) and there is a
t<t0 such that u2(0,t) = v2 . The point he (9)cannot be the TES. The same
proof holds for v>h(B) and v2 = he2
H2. Symmetry implies u1(1,t) = u2(O,t) for all t. Therefore equations
(3-34) and (3-35) imply he(l ) = h2(B).
H3. If the game is multiplied by (a1,a2) it is immediate to show that
a U0 (i = 1,2) solve the corresponding equations (3-34), (3-35), (3-36).
H6. Immediate from geometrical considerations.
H7.1. If 5 is a contraction-translation with respect to i of 5 it may
happen that a solution for 5 will not exist. Suppose that a solution does
exist. If i = 1, the change in the game corresponds to a change in the
utility function u 1 to u1 such that u1 (x,t)iu1(x,t) for all x and t. In
equations (3-34), (3-35) and (3-36) the solution t 0 satisfies t0 to and
-0 0 -0 0therefore u2-u2 and u 1 u 1
H7.2. The proof is similar to the proof of H7.1.
H8. Let En+ uniformly, and let (u, u2) and tn be the solutions of the
equations (similar to (3-34), (3-35), (3-36)) that could be written for the
games i. It is easy to show that nuo n n + ,and tn 0, thereforean ' u 2 'ui2, t+ ,thrfe
H8 holds. 0
3.3.3 Properties of the Solutions
(1) The ERS of a bargaining game with identical effects of time.
Let the utility functions of the parties be of the following multipli-
cative form:
u1 (x,t) - c(t)f1(x), 05t<c , x[e[0,1] (3-37)
and u2 (x,t) - a(t)f2 (x), O<t<, -, xc[O,1], (3-38)
- 105 -
where 0<a(t)1 is differentiable and is non increasing, f1(x) are differen-
tiable and f2(f11 (x)) is decreasing and concave. The bargaining game B is a
continuous game and the differential equation (3-7) is satisfied by the ERS
of the t-tails of B, for all t.
Theorem 3.4
There exists an a, 0<a<l, such that for all T for which a(T)<a the
ERS of B (the truncation of B in T) is the Nash Cooperative Solution of B(O).
Proof. Let T be some deadline. From (3-7), (3-8) and (3-9) we get the
following equations for the solutions 1(t) of the game BT:
dt 2(t)d (t) 1,-_ i (t) _1'ti (M
dt dt 2 1 a(t) 1 a(t)) O5tST , (3-39)
12 (t) =czyt))
2 2 1 a(t))) OStST , (3-40)
and
t(T) = gR(B(T)). (3-41)
Assume that there is a t , MOt*ST, such that
e(t ) - gN(B(t*)). (3-42)
The problem can be divided into two. One problem consists of (3-39) and
(3-40) holding for OStSt*, with (3-42) as a boundary condition. The second
consists of (3-39) and (3-40) holding for t*5t5T, with (3-41) and (3-42) as
boundary conditions. We are really only interested in the first problem.
We show that it is solved by
1(t) = gN(B(t)), for all t, 0tst* . (3-43)
For convenience we denote gN(B(t)) by gN(t). The boundary condition (3-42)
is satisfied by (3-43). Let'x satisfy 8N (t*) - ui(x*,t*) (iIl,2). Then
from the definition of the Nash Cooperative Solution 9N (t) " ui(x ,t) for
- 106 -
all 0st.t (i = 1,2). From here gN = M(Of (x and therefore
0i
2(t) f2
N1(t) f,(x*) '
Now, 9N (t)' g1N (t) f 'x
r12-1 1 )1 (I ) =2 1 a(t) 1 a(t) f1(x*)
and the equality 2 2
fIt ) f (x* *
holds because x maximizes the expression f(x)-f2 (x). Equation (3-39) is
therefore satisfied by gN(t).*
We now have to prove that if a(t) is small enough, then there is a t
which satisfies (3-42). For convenience we change notation. The
bargaining game is not changed if we assume u1 (x,t) - x-a(t), u2(xt)
f(x)-a(t) where f(x) W f2f1 (x), because the Pareto frontiers P(t) are un-
changed for all t. Let xt satisfy tIt) = xta(t). 1t 2 (t)=f(xt)-a(t) (Oe-tsT).
Denote the derivative of xt ith respect to time byit. From (3-39) weget:
Q(t)f (x + &(t)f(x)-tf'(x
dcz(t x + c(t)xt
and from here. * [f(xpx'
tttf'(x )d(3-44)t
Since a<O, f'(xt)<O we get
S>0 f t tt
1Thecasea = 0 can be excluded since then the ERS "freezes" until ;<0. Iff'(x) - 0 then the strong Pareto optimality of 1(t) (on P(t)) is violated.
t
- 107 -
and t<0iffx tt
* * * *so if xt<x then xttx for t+0, and if xt>x then xt+x for t+0. We have to
prove that convergence occurs in a finite time. Assume xT = gR(B(0))x
Since xT xt x for all t, 05,t!T, we have
0f'(x )x + f(x) f(x)fXTa t t t a t AT
t 2a f(xd) 2c( t Ct 2a T+ ),
therefore
T f(xT) TxC = XT fkdt?:xJ + ;2f(xl+49{ dt
f(xT)
=XT + 'CX + , )lna(T)
= x - 4Blna(T), where =|xT + ,
* T +f'(, )' 2(xT *)/0It is therefore enough that xT - 481na(T)!x , or a(T)e . 0
Theorem 3.4 dealt with the case of identical discount rates. We show
that the. Nash Cooperative Solution of B(O) is the ERS. However, this is not
so when we introduce identical bargaining costs. Let the utility functions
have the additive forms:
u1(x,t) - fi(x) - Q(t) Oft<** x[0,1], i = 1,2 . (3-45)
It can be shown that x is the point on the Pareto frontier P(0) with deri-
vative of -1, which is generally not the Nash Cooperative solution of B(0).
This result is compatible with the focal point property, discussed in 3.2.4,
provided that the right derivative in the focal point is a -1, and the left
derivative is 3 -1. Note that this result is also identical to the condi-
tions for equilibrium, proposed by Foldes, in his normative model (section
2.2.5).
kue to the concavity of f.
- 108 -
(2) The TES as an outcome of a game of timing,
In 3.3.2 we described a game . which leads to the TES. Another version
of the negotiation game leads to the same equilibrium strategies and
thereby to the same-solution. Suppose that the parties are constrained to
either hold to their extreme demands or to make a yielding concession. If
party 1 yields in t while party 2 yields in s, the payoffs are:
G (u2(Ot)) t<s
k1(t,s) = iG~1(u2 0) + 4u(l,s) t=s (3-46)
and u1 u(l,s) t>S
u,(0,t) t<s
k2 (t,s) = (4G(u(1,s) + 4u2 (0,t) t=s (3-47)
SG(u l(1,s)) t>s
where v2 = G(v1 ) is the equation describing the Pareto frontier in t =0.
Assume first that G(v1) = 1 - V 1 . It is then easy to see that the game
is zero-sum. It has a value v and a pair of optimal strategies: Hold to
your extreme demand and then make a yielding concession in time t*. This
* * *time satisfies u2 (1,t*) = 1 - u1 (O,t*). If t is not unique let c' =
min(tlu2 (1,t) = 1 - u1(0,t)} and let t" - max{tju2 (1,t) = 1 - u1 (0,t)}.
The equilibrium strategy is any distribution Qn [t',t"].
We now remove the assumption that G(v1 ) - 1 - v1 . The game is no more a
zero sum game. We look for equilibrium strategies. A pair of distributions
F (t) and F2(t) are optimal (continuous mixed) strategies if they are
equilibrium points, i.e., if
fK1 (tgs)dF(t) fK1 (ts)dF(t)dF2 (s), (3-48)
and
IK2(t OB)dF 2(a)Z . K2 (t 0s) dF ( t)dF 2()("9
- 109 -
Let E = {tIG'A(u2 (0,t)) + u1 (t) = i}, Since conditions (1) and (2) of Lemma
3.5 hold and since G is continuous and non-increasing, E is necessarily close
and connected, E = [t 1 ,t2]. Let F1 and F2 be any two distribution concentra-
ted on E. It is easy to see that (3-48) and (3-49) are satisfied, hence F1
and F2 are a pair of equilibrium strategies.
Appendix: Incentives and Arbitrations Schemes
Arbitration was viewed in this chapter as merely the choice of a point
in the negotiation set in accordance with some fairness requirements. In
actual negotiations the role of an arbitrator may sometimes be much broader.
He may introduce additional rules that facilitate the negotiation process by
encouraging concessions. Final-offer arbitration, which has been introduced
in collective bargaining, is just one example.I In this Appendix we discuss
another incentive system which is actually used in various bargaining
situations -- probably not by arbitrators but by mediators.2
We say that an arbitrator uses an incentive scheme when he introduces
new rules to the negotiation game, and then lets the negotiation go on with-
out any more intervention. If an agreement is not achieved and the parties
are willing to use his services again, he may use an arbitration scheme and
choose an agreement or he may introduce another incentive scheme and send the
1It was first offered by Stevens [1966]. A recent assessment of this approach
can be found in Lipsky and Barocci[1978].
2 Recallfrom Chapter 1 the distinction between an arbitrator, a mediator, and
and outside third party. The arbitrator is choosing a unique point as his
proposed agreement. He does not participate in the negotiations although hemay change the game by introducing new rules. The mediator participates inthe negotiations and tries to facilitate the achievement of an agreement. Anyagreement. A third party tries to affect the negotiation outcome by actingoutside the negotiation - spreading information, making threats and promises,etc. His objective is to affect the final agreement.
- 110 -
parties to another round of negotiations. According to the scheme proposed
here, the arbitrator promises (and his promises are recognized as binding
commitment by the parties) to compensate the party who makes the yielding con-
cession. The amount of compensation is such that the final agreement is
Pareto optimal.
The Timing Equilibrium Solution, discussed in 3.2.2 was based on this
incentive scheme in the following way. Since the utility functions of the
parties are comon knowledge, the incentive described above yields. a unique
agreement (under suitable conditions). Since this agreement can be suggested
by the arbitator, he may save himself the compensation by choosing immedi-
ately the arbitration point. The :.acentive system is therefore not really
in effect.
The effect of this incentive-arbitration procedure can be seen in the
following example of political negotiations. The parties make the initial
demands, d1 and d2 , but It is common knowledge that there is only one agree-
ment d which is jointly preferred to no agredment. The only move is therefore
to propose this agreement d in some time t. It is assumed that the party
which does not propose d (but rather accepts it after it is proposed) is
perceived as "tough" and enjoys more "honor", which can be translated into
utility. The payoff kernals of this game, when arbitration is not available,
are therefore:
1 - a1(t) 1t<(*
K (t,s) 1 - a (t) + b1 (t) t=a
1 -a 1(a) + b1 (s) t >-.8
and
- 111 -
1 - a2(t) + b2(t) t< s
K(t~s) at)+1K ) 1 - a2(t) + b2(t) t= s
1-a2 (s) t > s ,
where b1 (t), b2 (t) are the values of the "honor" enjoyed by party i if he is
"tough" and if the negotiation terminates at t. The functions ai(t) (i = 1,2)
are the bargaining costs. When the arbitrator introduces his incentive
scheme, the kernels are
1t < s
K!',(t,s) = 1 - (t) + (lbt) t = s
1 - a1(s) + b1(s) t> s
and (**)
l - a2(t) + b2(t) t < s
K(ts) = - 2 (t) + 2 (t) t = s
t>s
Note that judging by the payoff kernels only, both parties prefer (**) to (*)
because KI(ts)5IK"i(ts) for i = 1,2, for all t and s.
Suppose that the functions ai(t) and bict) are continuous, satisfying
-a i(t) + bi(t)>O for 0 t<t and -ai(t) + bi(t)<0 for t < t< (i = 1,2).
In (**) it is clear that party i never holds to his demand after t , i.e., the
game duration is bounded by t** - min{t1 , t2 }. The arbitrator will pay no more
than max{a1 (t ), a2(t)}. In game (*) the duration may be much larger,
depending on the particular functions ai(t) and bi(t) (i ( 1,2).
1/2Example. Assume aict) - aiet and bi(t) b b *t (i - 1,2). In game
b1 2 26 2(**) the maximal duration- t** is min{(1-) 2() . In game (*) it can be
a1 a 2 3/2
shown that' the equilibrium strategies are F 2(t) 1 - e(t,(i,2l ,
- 112 -
itj), whereX = . The expected concession time of party i isj
b.(3)5/2fl/(-)3/2 3 5/2 1/2 t** 2/3
( a. , which is greaterthan ( )/ (t )
If t is small enough the last expression is larger than t
- 113 -
CHAPTER 4
COPING WITH TIME EFFECTS IN BUYER-SELLER NEGOTIATIONS
4.1 Introduction and Summary
The examples given in chapter 1 notwithstanding, negotiators do not
ignore the effects of time. They tend to construct a set of "rules of thumb"
related to the duration, pace and timing of the negotiations in which they
are involved. The particular market described in this chapter - a small
exchange market in which price is volatile and information is expensive to
obtain - illustrates this behavior. Transactions in such markets are usually
preceded by extensive, sometimes lengthy, seller-buyer price negotiations.
The market for numismatical items in Israel -which we.call."the coin
market"is one such case. Dealers and collectors are the principal agents in
this market and we are especially interested in the negotiating activities
of the dealers. The dealer's profitability depends mainly on his negotiation
skills. The set of intuitive decision rules which he uses is tested continu-
ously in the market and is therefore worth observing and studying. Some of
these rules will be analyzed in this chapter.
In section 4.2 we describe the background for the rest of chapter 4. It
includes a description of the market (section 4.2.1), a development of the
dealer's utility function (section 4.2.2), an account of some decision rules
which are used by the dealers (section 4.3.2), and a brief discussion of some
time-related decision problems encountered by buyers and sellers in the market
(section 4.2.4).
One observed phenomenon in the market is the frequent division of nego-
tiations into two or more sessions. After spending some time on negotiations,
the dealers and collectors often propose a postponement. By a postponement
- 114 -
we mean an interruption of the negotiation during which the parties may
search for better prices, thereby incurring search costs. When negotiations
resume, the opening bids of the parties are the last bids which they had
proposed before the postponement took place. The postponement is viewed
here as a part of a broad search-negotiation strategy or as a device helping
to limit bargaining costs in lengthy negotiations. Two approaches to
analysis are taken, the first is described in sections 4.3.2 and 4.3.3,
where several search-negotiations models are proposed and the second is
described briefly at the end of section 4.3.3. We believe that this and
future studies of the postponement phenomenon can shed light on the issue
of price spread in small markets. Knowing the serach-negotiation strategy
of their clients, dealers in the market may be justified in charging different
prices and no unique market price should necessarily exist.
The problem of sudden termination, i.e. the abrupt break-off in nego-
tiations due to events which are beyond the control of the bargaingers, is
studied in 4.4. We develop the concept of characteristic concession patterps.
(section 4.4.1), find equilibrium negotiation strategies for bargainers with
known concession patterns (section 4.4.2) and discuss the decision problems
involved in making the initial bids (section 4.4.3). The work on this sub-
ject is preliminary. It should be supported by further empirical studies.
In the course of studying the effects of time in our numismatical market
we dealt with two related problems which are relegated to the Appendices.
One is a variant of the Pick-The-Best problem, in which the number of obser-
vations is uncertain. The other is an extension of a model proposed by
Rosenthal and Landau in which the effect of "reputation" in bargaining is
studied. This model is also related to results about characteristic conces-
sion patterns obtained in 4.4.
- 115 -
4.2 An Example - The Israeli Coin Market.
4.2.1 The Market-General
The Israelis coin marke t i&a'. a small but - ve ry ac t i ve
one. Its activity depends strongly on general economic developments in Israel.
The high inflation rates prevailing since the early 1970's, the frequent
devaluation. of the Israeli Lira (IL) and an increase in trading volume on the- Tel-
Aviv stock exchange in 1976-1977 had important though mixed impacts on the coin
market. Prices have been notoriously volatile. Speculative activity and the
lack of any effective trade regulation have prevented conservative investors
from entering the market. It is, however, an interesting market in which
effective negotiation is the key to success. As such it may serve as a good
source of examples. Of particular interest are the operations of one group of
agents, namely the major dealers. The reallocation of collectible items is
done primarily through the negotiation activities of the members of this
group.
Numismatical Items
The items traded in the market include:
(1) Commemorative Coins, made of gold, silver or cupro-nickel. These
items have been issued by the Bank of Israel since 1958, and they are legal
tender with market prices far exceeding the nominal values.
(2) Comemorative Medals,made of platinum, gold, silver, nickel-silver
alloy, bronze or copper. These items were first issued in 1958 by the Tenth
Anniversary Committee nominated by the Prime Minister and later by The Israel
Coins and Medals Company Ltd., which was later changed to The Israel Govern-
ment Coins and Medals Corporation.
- 116 -
(3) Commissioned Medals (or Presentation Medals), made of gold, nickel-
silver, silver, bronze or copper. These items have been issued by special
government committees.
(4) Trade Coins and Bank Notes, issued since the establishment of the
State of Israel in 1948.
(5) Trade Coins and Bank Notes of the British Mandate (1918-1948).
(6) Historical Items like ancient coins, money substitutes, tickets,
food stamps and other collectible items.
The most traded items are the Commemorative Coins, the Commemorative
Medals and the Commissioned Medals. The prices and the salability of the
different items vary. Some of them are traded for thousands of U.S. dollars,
while most are sold and bought for prices around $100.
The Agents
The - groups and individuals active in the mark e t'ar e
collectors, investors, speculators, small dealers and major dealers.
(a) ' Collectors are interested in having a full collection or a
specialized one, with no more than one unit of each item. They buy
new issues from the Bank of Israel, from the Israel Government Coins and
Medals Corporation or from the dealers. Their number in .1978 was estima-
ted as 10,000 and it is slightly increasing from year to year.
(b) Investors buy and sell itema ;fr, short-term capital
gains. They generally specialize in certain items like gold medals, silver
coins, etc. Their number in 1978 was 500. This number is highly sensitive
to general economic conditions.
- 117 -
(c) Speculators are aggressive investors who specialize in one or
two items, trying to raise their price after they have accumulated large
stocks. In 1978 there were no more than 50 individuals whose main activity
in the market was speculation.
(d) Small dealers buy and sell itmes in their own stores or in
informal meeting places. They are not as aggressive as the major dealers,
and their stocks are relatively small. Their advantage is their contact with
the general public and their readiness to enter into very small deals. In
1978 there were 25-30 small dealers in the market.
(e) Major dealers, to be called "dealers" in the sequel, buy and
sell, invest and speculate in many or all the items. Their trade volume is
larger and they have more information and experience in the market than the
other agents. There are 10-15 groups of dealers, organized mostly in family
units and located in Tel-Aviv.
Market Activity; Information
The volume of business in April 1978 was approximately $30,000 a day,
down from $50,000 a day in the peak years of 1972-1973. Approximately
30% of the business deals took place between major dealers, 30% between major
dealers and small dealers, 30% between major dealers and the public (investors,
collectors) and 10% between small dealers and the public.
The small dealer is sometimes considered by the major dealer as a regular
investor or collector, and sometimes as a major dealer. We make this point in
order to generalize and say that roughly half of the business deals are of the
dealer-dealer type, while the other half is of the dealer-public type. This
distinction is important because the negotiation tactics employed by the dealer
(as well as the rules of ethics which he obeys) depend on the negotiation type.
- 118 -
Price information is difficult to obtain. There is one daily paper
("Sha'ar") which lists prices of some most traded items, but its
reliability i& questionable. There ate also several catalogues
which publish prices considered "indicative" at most. Actors in the
market are in a perpetual search for information; it is thereforenot sur-
prising that one of the rules of ethics in dealer-dealer negotiations is
accuracy in quoting prices of completed deals. There is no official record-
ing of prices, no price index and no reliable record of past prices.
Additional Remarks Regarding the Negotiation Activity
(a) Negotiations can be over one item or many items. Most of the dealer-
dealer negotiations are over several units of one item. Most of the dealer-
public negotiations are over one unit of one or more items.
(b) The means of payment is an important consideration. Dealer-public
negotiations end almost always with a cash payment and with immediate delivery
of the traded items. In dealer-dealer negotiations one witnesses frequent
deals ending with barter, deferred checks, promises of deferred delivery, etc.
The means of payment and its timing serve as bargaining chips.
(c) The speculative activity- in the 'market is notorious.
A speculator ( possibly a dealer ) chooses an item which has a his-
tory of non-active trading and which is known to be mainly in the hands of
collectors. He accumulates quietly a large stock, up to 50% of the regularly
traded volume (it may amount to 10% of the quantity issued), thus creating
artificial scarcity. Then comes the stage of promotion, signified by the
dealer announcing prices which are slightly higher than the market price.
The price starts to rise moderately, and interest in the item is rising too.
The dealer then announces another asking price, which is significantly higher
- 119 -
than the current price level. Since supply is tight this price is quoted
everywhere although there may be only a few buyers. The dealer then
approaches an investor or even a dealer, proposing to sell him a large
quantity of the item for price which is lower than the asking price, but
higher than the price that prevailed before the promotion. If he is
successful he enjoys a large profit. Price of course may fall immediately
after the deal, and the buyer may experience a large loss. This activity
is considered ethical only if the buyer is known to have experience in the
market, but many instances of trapping naive new investors have occurred
and raised public furor. It ought to be mentioned that unsuccessful
speculation (which significantly outnumber the successful ones) may be
very costly because of the non-optimal composition of tb - tock created
by the speculation.
4.2.2 The Dealer's Utility Function
Consider a transaction in which the dealer was the seller. Let us
also assume that the dufation of the negotiation was d and that it started
at the calendar time t 0 .* Let y be the amount the dealer was paid (in cash).
There exists a constant a, O<a<l such that the dealer's utility function
g for the transaction is:
g(y,d,t0 ) - a g(y) + (1-a)g2(d,t0 ). (4-1)
This additive structure was clearly indicated from our interviews. It shows
that no discount effect exists. Indeed this effect is negligible because
typical negotiations take no more than two days, and generally less than
one hour. The bargaining cost is however very important. It consists of
- 120 -
overhead and of opportunity cost. The overhead cost- may be viewed as a
constant increase in debt of r dollars per unit of time. (We discuss later
the attributes of the dealer's global utility function. Debt is one of
them.) The opportunity cost is the amount of profit (say, in cash) that
could be gained if the negotiation did not take place. This cost can be
computed in several ways.1
In dealer-public negotiations the collector has only a vague idea about
the constant a and the functions gand g2 in (4-1). In dealer-dealer nego-
tiations the dealers know each other's g2 , because the effects of time on
them are roughly the same. They may even know the general shape of g1 . It
is the constant a that is unknown. It depends on the current level of the
attributes in the global multiattribute utility function of the dealer.
This function, which will be described now, was built according to the
answers of one dealer to questions regarding preferences among various
lotteries. The interviewed dealer, who holds a B.Sc degree in Chemistry
and has some experience with quantitative models, understood the concepts
of utility theory very well and was. ready to spend much time discussing
his preference structure.
1The dealer we interviewed was able to assess a probability distribution of
the profit per unit of time, denoted by k(t),for every hour t of the day
(10:30 a.m.-1:30 p.m. and 3:30 p.m.-5:00 p.m. being the peak hours). Ifnegotiations start in t0 and last for d units of time then the opportunity
t 0+d
cost is ( k(u)du, which is a function of t0 and d. For simplicity we can
0 t +dassum that d is short enough so that N kudukt) *d.
0
- 121 -
This function will be discussed now in some detail. The six
attributes include:
X, - Cash at hand;
Cash is needed for the . eveTday operations. In the
beginning of a working day he needs a sum of IL 50,000-100,000. If he does
not have this amount of cash he tends to sell from his liquid stock, gener-
ally to other dealers, possibly for a very small profit, if at all.
X2 - Debt (short term);
The dealer receives short term loans from commercial banks. The interest
rate in Israel (1977-78) was almost 1% per week, credit was tight, and large
stocks had to be deposited as collateral. The dealer usually has a range
of debt with which he feels comfortable. This range depends mainly on the
level of X4, the short term component of the stock. The dealer prefers X2
and X to be about equal.
X3 - Liquid Stock;
This is the value of the stock which can be sold within a few hours for
the existing market price. It is slightly preferred to cash when cash is
abundant (because of the appreciation potential) but less preferred when cash
is scarce. The liquid part constitutes generally of 10-20% of the stock.
X4 - Short-Term Stock;
This is the value of that part of the stock which can be sold within one
month in the market, with little effect on prices. It amounts to 50%-60% of
the stock.
- 122 -
X5 - Long-Term Stock;
This is the value of that part of the stock which has no current market
price, or which would have strong effects on prices if sold. The size of
this component depends on the speculative activity of the dealer, his assets
and his risk aversion. Most dealers try to keep it as a constant proportion
of the stock, typically 30%. Because of the difficulties in assessing X5 the
dealer tends to make a careful revision only every several weeks, and then he
quickly adjusts the proportions.
X6 - The Number of Business Negotiations in the Future;
This variable is difficult to measure yet it seems to be a natural one
for the dealer. It serves as a surrogate for future business volume.
Suppose that future business volume in one unit of time is denoted by the
random variable y. The dealer assumes implicitly that y=f(x6)+y, where f is
some increasing function and y is a random variable, independent of X6 , indi-
cating the general market activity. The dealer can affect j by raising X6 ,
I.e. by negotiating with as many collectors and dealers as possible for a
small or no profit in order to raise his future business volume. For example,
the dealer may spend a long time negotiating with a new client, in the hope
that in the future the client will sell to and buy from him rather than from
other dealers.
From the description of the attributes and of the dealer's operations it.
becomes clear that the dealer can increasdehis utility in two ways. He may gain
a monetary profit by, for example, buying and immediately selling so that X1is
increasing while the other variables do not change. Alternatively he may be
involved in a trade which does not change his net worth but which brings the
variables to a more preferred "mix". This last goal is achieved mainly in
- 123 -
dealer-dealer negotiations. Monetary profits in such negotiations are
on the average lower than the profits in the dealer-public negotiations.
The structure of the utility function of one of the dealers was studied
closely. By his testimony he is more aware of the composition of his cash-
debt-stock than the aVierage dealer. We list the results briefly.
(1) {cash(X1 ), liquid stock (X3)1 and {X29X4,X5 ,X6 } are utility independent
as long as a 21 4(x52 3+x4 +x5 2)cx1 +x 2(x2+X4+x5) and n1 cx6 <- n2 for some
constants a1 , a2 , n1 and n2 which can be determined by the dealer.
These conditions are satisfied for most negotiations.
(2) {debt (K2), short term(K ) and {XKx3,X5X6I are utility independent as
long as a!(x+x3+x 5) Sx4 s a(x 1+x 3+x 5) and a"x4 S x2 <ax 4 and
n < x c n' for some constants a', a a", a n' and n' which can be1 -6-2 l2'l'21 2
determined by the dealer.
(3) (long term (X5)} is utility independent of its complement as long as
(x+s3+x2-x) < c (x+x+x4-x 2), for some constants 81, 82'
(4) {number of negotiations (X6)) is utility independent of its complement
as long as n"< x6 < nand < x+x4 2 < for some constants
nt, n, 8{ and 8.
Further questioning showed that the iso-preferences of the dealer in the
(X, 3 ) and in the (X2 ,X6 ) planes are asdepicteduislug.4-1 and in Fig. 4-2.
(Qhe numbers in these figures are in IL'OOO)
- 124 -
Cash - xl,150
125
100
75
50
35q
25'
21 29 38 52 75 98 120 140 X3 - LiquidStock
Fig. 4-1: Iso-Preference Curves for K and X
x2f0J
200d
400
600
8004
1000
0
K
- I -
200 400 600 800 1000 x4 - ShortTermStocks
Fig. 4-2: Iso-Preference Curves for X2 and X
Short -TermDebt
ft A
- 125 -
These independence properties lead to a utility function u(x) which is (in
a large region of parameters, where most transactions take place) of the
multiplicative form with two additive forms. This means that there exist
constants k, kl3, k24, k5, Al A29 A3 , A4 such that:
ku(x)+l = (kk13 yj 1',x3 )+l)-(kk2 4u24 (x2 x4)+1Xkk5u5(x5 )+.Xkk6 u6(x 6)+,
(4-3)
where
u1 3(X19X3)1= u1(X1 ) + A 3u3(x3) (4-4)
and
u24(x2'x4) = 2 u2 (x 2 ) + A4 u4 (x. (4-5)
From this global utility function the function g1 (y) in (4-1) can be computed.
The constant a depends on the level of attributes which did not change during
the transaction.
The utility function in (4-3) can be further simplified, to an additive
form, if the attributes I 1, .. ,,X6 were additive independent, i.e. if preferences
over lotteries on X1,...I6 depended only on their marginal probability dis-
tribution. After extended questioning we concluded that the additive form
could serve as rough approximation. The dealer, when confronted with some
of the behavioral consequences of an additive form agreed that they roughly
corresponded to his behavior. An additive form
6
u(x) Zk %a1 (x1 ), (4-6)isl
is very useful. AUl that w have to assess are the constants ki and the one
disasional utility functions a x). This is not difficult to do. For example,
the fimctions a (x ) were alt very close to -a , that is they exhibited
constant risk aversion (the canstants c were not identical).
- 126 -
Suppose -for -example that the dealer sold a coin from his liquid stock
and received y dollars (in cash). His utility function g(y) for this
transaction is b1(1-e-1 y + b2 , for some constants b and b2 , and-so is
strategically equivalent to -e c y
4.2.3 Some Observed Decision Rules
In this section we list some decision rules that are used by dealers
and by collectors in the market. In later sections we explain these rules
as the negotiator's response to the effects of time.
(a) Characteristic Concession Patterns. It was observed that every dealer
in the market conceded in a typical pattern. Furthermore these patterns seem
to be common knowledge. Some of the dealers are known to hold to their initial
price proposal for a long time and then to make a sudden large concession.
Others keep a constant rate of concession (i..e. the graph of price proposals
as a function of time is a straight line). Others yield generously in the
beginning and then yield less and less. In constrast, some dealers increase
their rate of concessions. These patterns obviously depend on the responses
of the other party in the negotiation, but the general functional form of
the concession pattern remains the same.
Randomization over concession patterns does take place, as could be
expected, but the dealer' adheres to his pattern, most often.
(b) "Irrationality". The interviewed dealer asserted that almost every
dealer behaves "irrationally" from time to time. He may retreat, stop the
negotiation with no apparent reason, demand unreasonable prices, etc. This
is generally done deliberately. The dealer, by exhibiting irrationality,
may lose a deal (generally a small one), but he tends to believe that this
- 127 -
behavior enhances his future prospects. Looking at the grand game in which
"behave rationaly" and "behave irrationally" are the only two strategies, the
dealer is probably right In thinking that a randomized strategy is better than a pure one.
(c) Negotiations' Duration; Postponements. Dealer-dealer negotiations are
typically shorter than dealer-public negotiations and terminate in one
session. Many dealer-dealer negotiations break off immediately as the initial
proposals are made. The successful ones may involve from three to seven
price offers by each party. The dealer-public negotiations are generally
longer and are frequently divided into several sessions. The dealer, reali-
zing that the price gap between the parties is large, proposes a postpone-
ment. During this postponement the dealer negotiates other transactions, while
the collector is searching for a better price. If the collector returns in
a relatively short time (within a day or two) negotiations resume and the
last price proposals serve as the opening bids.
4.2.4 Time-Related Decision Problems
Time-related decision problems encountered by a dealer in the coin
market which we study here are:
(a) In dealer-public negotiations, when should the dealer propose a postpone-
ment?
(b) In dealer-public negotiaions, assuming the collector is planning a
search for the best price, can the dealer find an "optimal" rate of concessions?
An "optimal" opening proposal?
(c) In dealer-dealer negotiations, under what conditions should the negotia-
tions be terminated imediately after the opening bids?
- 128 -
(d) What are the implications of the existence of typical concession patterns?
How should the dealer negotiate given the concession pattern of his counter-
part?
(e) What is the role of irrational behavior (as described in 4.2.3)?
(f) If the sudden termination possibility is introduced, how should it af-
fect the concession rate?
The decision problem of the collector, which we study here, is:
(g) What is the "optimal" search-negotiation procedure?
We write "optimal" because optimality has various meanings in various
contexts. A decision is sometimes optimal because it is a part of an equili-
brium strategy in a well defined game, and sometimes because it maximizes some
objective function without taking into consideration the responses of the
other party. Note that we make a clear distinction between the negotiation
behavior of,and the problems encountered by dealers and collectors. We do so
because iris our belief - that the dealer's negotiation skill is superior to
the collector's. Consequently the latter should be interested more in find-
ing a good search-negotiation procedure, i.e. a rule telling him in what
order to conduct the search and how much time to spend with each dealer, ra-
ther than in developing a sophisticated negotiation strategy. In contrast,
the dealer should pay more attention to his negotiation strategy, and cer-
tainly to the gaming nature of the dealer-dealer negotiations.
- 129 -
4.3 The Postponement of Negotiations
4.3.1 Introduction
In this section we consider two parties, B (for Buyer) and S (for Seller)
involved in distributive bargaining. Bids are alternated until one party,
say B, proposes a postponement, i.e. a time period in which no exchange of
bids, nor any other negotiation activity takes place. When the postponement
period ends (its length could be specified in advance) the parties continue
the negotiations. It should be emphasized that when negotiations resume it
may happen that both parties concede, although in some cases it may be clear
that an optimal strategy for one party is to make a termination threat right
away.
It is not difficult to give examples in which both B and S act optimally
when they postpone their negotiation, resume the negotiations and concede
further.1 The postponement of negotiations is therefore a phenomenon that
should be studied. It is observed in many contexts and is very common in the
coin market d6scribed in 4.2.
In sections 4.3.2 and 4.3.3, we deal with the search-negotiation process
experienced by buyers, sellers or, in other contexts, job-seekers. The example
to which we refer again and again is a collector travelling between .. deal-
ers, trying to get the best price for some coins he owns.
'We give here one example. Let the seller be a collector who has no time re-lated costs nor search costs. Let R be the reservation prices of dealer i(i-l,...,n) and assume R t R- for all ij, i # J. It is clear that if all thedealers are careful enoukh aAd if they can verify the price quotes that thecollector claims to have received, then the collector will eventually receiveslightly more then the second largest reservation price. The dealers may notknow if their reservation prices are the highest. Their optimal strategies,under suitable assumptions regarding bargaining costs, are to propose priceswhich are slightly higher than the last proposal the collector has receivedand to postpone negotiations iumsdiately thereafter. When they propose theirreservation prices they make termination threats and leave the game.
- 130 -
For this collector, the postponement of negotiations is simply a part of an
overall search-negotiation strategy. This strategy, under various assump-
tions, is described in section 4.3.2. The mathematical proofs are given in
section 4.3.3.
The computation of optimal search-negotiation strategies can serve
several purposes. (1) It may be used as a decision aid by a dealer whose
objective is to prevent lengthy, futile negotiation. By understanding the
collector's strategy he either can offer a relatively high initial bid and
then propose a postponement immediately, or he can try to affect the collector's
reservation price by extending the negotiation. (2) It could be used in
descriptive economic models, for example in models of market share. The
introduction of negotiations into search tends to lower the expected number
of observations. Thereby leading to more centralization. (3) Above all
it can he used by a collector in the coin market, by a firm choosing
among several projects (which have to be tested) or in any other search
context.
- 131 -
4.3.2 Search-NegotiationjModels
Various economic search models have been proposed in the economic and
statistical literature, mainly in the context of job-seeking. These models
generally ignore negotiation activities that take place during search.
The searcher is assumed to receive, at some cost, non-negotiable price quotes.
His decision problems are the order in which he should carry the search and
the rules of optimal stopping of the search. He cannot affect his search costs,
nor can he receive more than one price quote from each source. Such models
seem to be oversimplifying in markets where bargaining over price (or wage) is
common. The searcher may prefer to spend more time on negotiations than on
sea rch. In this section we propose search models which give the searcher
such flexibility. The searcher (let us assume he is a seller) may negotiate
with each buyer for some time, and then may continue his search. At any point
in time he may contact a buyer with whom he had previously negotiated and ac-
cept this buyer's last offer. The case in which previous offers are always
available will be called here the perfect recall case. The case in which
the probability that previous offers are available depends on the time since
these offers were made: will be called here the decreasing recall case. When
past offers are never available we are in the no-recall case.
In the context of the .coin market, the searcher is "generally a
collector moving between the dealers, trying to sell one or more items. In
this market both the perfect recall and the decreasing recall assumptions may
be reasonable, depending on the particular items for sale and on the stock-
cash-debt mix of the buyers (as explained in 4.1).
ltecent references are Landsberger and Peled [1977], Lippman and McCall [19761,Rothschild[1974J and Weitzmmn [1978]. An extensive reference list can befound in Lippuan and McCall [19761 and in de Groot [1970].
- 132 -
Other assumptions made in our models are concerned with the number of
buyers ( f inite and -small number in the . coin market
but very large in other: markets), the -learning process
taking place during the search,and the bargaining and travelling cost functions.
In the first three models, we assume that each dealer concedes according to
some known characteristic concession pattern which depends on the initial bids.
Under this assumption the final negotiation agreement and the duration of the
negotiation are determined by the opening bid. Later we assume that the final
result is not known with certainty, but that the searcher is capable of as-
sessing the dealer's final price and to update his assessment during the ne-
gotiation.
According to the classification of negotiation models proposed in.Chapter
1, our models are asymmetric ones - They prescribe optimal behavior
forone party (the searcher) assuming certain behavior on the part of the deal-
ers (whose strategy space is indeed very restricted).
Consider now a seller S travelling between - buyers and spending time
on negotiation with some or all of them. The travelling cost between any two
buyers is a constant d, and the bargaining cost is a linear function of dur-
ation; the seller's utility function is thus
Ua(x,t,k) = x - cot - d-k - ra , (4-5)
where x is the final selling price, t is the total time spent on negotiation
during the search, k is the number of trips between buyers and r0 is the
seller's reservation price, i.e. a price at which the seller may find a buyer
without incurring any cost.
In our models the buyer is assumed to have some fixed negotiation stra-
tegy which is represented by a certain characteristic concession pattern. We
make this restriction in order to prevent the enormous complication which
- 133 -
arises if the buyers are allowed to excercise full strategic behavior. We
assume here that buyer i proposes an opening bid xi(0) which is the realiza-
tion of a random variable with a probability distribution Fi(e). He then
concedes according to some concession curve x (t) (O<t<Co) which depends deter-
ministically on the opening bid x1(0) and on one parameter %g.1 If negotia-
tion is postponed at time t (measured from the time it started) then the last
price offer of buyer i is x (t). When buyer i is contacted again, after a
postponement of length s, his offer is still available with a probability p(s).
Our objective is to study the optimal search -negotiation procedure and
to point to some implications of this procedure. The results obviously depend
on the assumptions regarding the distributions Fi(s), the concession curves x Ct),
the process of learning during the search and the recall function p(Q). We
examine several sets of assumptions, and characterize the results for the no-
recall, perfect recall and decreasing recall cases. The results in Lemmas 4.1,
4.2 and 4.3 are directly obtained from known results.
Model 1 No learning, identical buyers, infinite number of buyers.
We assume that Fi(0) - F(*) and A - A for all i (identical buyers). We
also assume that the initial bids x1(0) are independent and that A is known
with certainty (no learning). The number of buyers is infinite.
Since the buyers are a-priori identical they are visited in a random
order. The problem is to establish an optimal stopping rule. Define functions
y (i - 1,2,...) by
y(x) - oimax {x(t) - cetixi(O) - x } , (4-6)O<t<*
iSome characteristic concession patterns which depend on one parameter areproposed and discussed in section 4.4.
- 134 -
and for each i let t (x) be the first time in which this maximum is achieved.'
Since the buyers are identical, yi(x) = Yj(x) and t (x) = t (x) for all i, j
and x. We therefore omit the subscript i and denote these functions by y(x)
and t(x). The following result will be proved in 4.3.3.
Lemma 4.1
For the no-recall, the perfect recall and the decreasing recall cases,
the optimal search-negotiation process under the conditions of Model 1 involves
searching for the first buyer i whose initial bid xi(0) satisfies x1 (0) > x
negotiating with this buyer for t(xi(0)) units of time; and accepting his
proposal in that time. The switchpoint x satisfies
d = I y(x)dF(x) - y(x*)(1-F(x*)). (4-7)x*
Model 2 No learning; identical buyers; finite, known number of buyers (denoted
by n)
The assumptions of this model are identical to those of Model 1, but the
number of buyers is a known finite number n. The following result will be
proved in 4.3.3.
Lemma 4.2
For the no-recall case, the optimal search-negotiation process, under the
assumptions of Model 2, involves searching for the first buyer (denoted by
n-i if he is the i-th buyer to be visited) whose opening bid xn-i(0) satisfies
X (0) > x*i; negotiating with him for t(xni(0)) units of time; and accept-
ing his proposal in that time. The points x* satisfy
eO
y(x*j41 ) + d -"/ y(x)dF(x) + y(x* )F(x*M) (j - ,...,n-2), (4-8)
1In order for y1 (x) and ti(x) to be well defined it is enough to assume that
for all L, xc(t) are continuous and bounded.
- 135 -
where x0* = -co and y(x0*)F(x0* 0.
Lemma 4.3
For the perfect recall case, the optimal search-negotiation process,
under the assumptions of Model 2, is identical to the optimal search-nego-
tiation process with an infinite number of buyers, with the exception that if
there is no i such that xi(0) > x* (x* is defined in Lemma 4.1) then when the
search ends the searcher negotiates with the buyer i whose xi(0) is the largest
among xI(0),...,xn(0).
Before commenting on the decreasing recall case we have to distinguish
between two sub-cases: (1) The travel time between buyers is zero, and (2) the
travel time between buyers is a constant, denoted by t0. In the first case
the search cost d is the result of some fee, an application fee, for example,
which is imposed on the searcher before he can observe the buyer's opening bid.
In the second case the cost is the searcher's opportunity cost of t0 units of
time. It is easy to see that when the travel time is zero (case 1) then the
decreasing recall problem is very similar to the perfect recall problem. Only
when negotiations take place does the probability that a previous bid is
still available start to decrease. Since negotiations take place only with
the actual buyer, the decreasing recall does not affect the results expressed
in Lemma 4.3. When travel time is not zero (case 2) the situation is differ-
ent. It is then possible that previous buyers will be visited in order to
verify that their initial bids are still available. Again, it is easy to see
that negotiations will take place only in the last stage of the search-nego-
tiation process. A problem similar to ours was studied recently by Karni and
Schwartz [19771, [1978]. In their search model the number of buyers is infinite
- 136 -
and the search cost is decreasing as the search proceeds. They prove the
existence of an optimal strategy and show that for a certain type of recall
function p(-) the search terminates whenever an offer made in the past is
solicited and found available. Our problem is finite hence an optimal search
policy certainly exists. The problem can be formulated as a finite optimal
control problem and can be solved:by a dynamic programming algorithm (although
dimensionality- is a problem). -Of special interest are cases in which the
"reservation utility" property, as in Lemma 4.2, holds. One sufficient con-
dition for this property to hold is given in the following Lemma.
Lemma 4.4
A sufficient condition for accepting a past observation whenever it is
solicited and found available is:
y(x*) [p)- p(t+s)J _d (4-9)
for all t, s > 0, where x* is given in (4-7).
Model 3 No learning; finite number of non-identical buyers
In this model the order of search must be determined. Once this order
is determined, the search proceeds as in Model 2, although the formulae for
the reservation prices are slightly different. The results are summarized in
the following Lemmas. The proof of the first one is trivial. The second is an
mediate extension of the result in Weitzman [1978].
- 137 -
Lemma 4.5
With no-recall and n buyers, the order of search is determined by the
expected values E(y (x )). If E(y(x ))> E(yk(xk)) then buyer j is visited
before buyer k.
Lemma 4.6
With perfect recall, the order of search is determined by the numberseO
z. satisfying d = f [y.(x) - y (z )]dF(x). If z > z then buyer j is visited21 Z.
before buyer k. J
The results of Models 1-3 are essentially identical to the results of
models of search without negotiation, as reviewed for example in Gilbert and
Mosteller [1966] and in Lippman and McCall [1976]. The searcher negotiates
only at the end of the search, with the actual buyer. This is of course a
direct result of the assumption that there exist known characteristic conces-
sion patterns xi(t) with known parameters A . Once the initial bid is made,
all uncertainty is removed and the final agreement can be determined. This
situation is changed if Aiare unknown in advance, but are assessed subjectively
and updated during the negotiations. The A 's are assumed to be independent
of x (0) and of the updating process, as will be explained later. We consider
here two sets of assumptions. In Model 4 we assume that -the A 's are independ-
ently drawn from a known distribution. In Model 5 we assume that they are in-
dependently drawn from an unknown distribution which is reassessed during the
search. A special case is when all the A 's are identical. In Model 5 we
also introduce the concept of learning during the search.
Other sets of assumption regarding A 's and the characteristic concession
patterns could be made, but the analytic treatment becomes very difficult. The
- 138 -
problem can always be formulated as a control problem to which a dynamic
programming algorithm is applicable, but the formulation becomes extremely
messy and no clear "reservation price" type properties exist.
Model 4
We now assume, for simplicity, that F (-) = F(-) for all i, 1 < i < n.
The concession patterns x (t) depend on x (0) and on one parameter A . The
A 's are independently drawn from a known distribution G(-) with density
function g(-). Once the second price proposal of buyer i is made, A becomes
known with certainty. The time t between the first and the second price pro-
posals is distributed according to a known distribution H(-) with density
function h(e). This time is independent of A and of x1(0). For simplicity
we assume that y1(x (0), Ai), which is the utility obtained if the searcher
chooses to sell to buyer i, is increasing with A and with x1(0). Without
these assumptions the formulae in the following Lemmas would have a more com-
plicated form.
Since the buyers are a priori identical, the order in which the search-
negotiation is conducted is random. We prove in section 4.3.3 the following
two Lemmas.
Lemma 4.7
Under the conditions of Model 4, assuming no recall, the optimal search-
negotiation process is the following:
(1) Observe the intitial price x (0) of the i-th buyer.
(2) If x (0) < x , go to the next buyer. Otherwise start negotiations.
(3) If the second price offer of buyer i is not given within T, units
of time, go to the next buyer.
Let V(
when t
A 4*(
- 139 -
(4) If the second price offer of buyer i is made at s < T .,n-i n-1
and if it impliesA < A*- (x-i(0), s go to the next buyer.
Otherwise continue negotiations and accept buyer i's proposal at
t .(x (0))o1n-i n-i
n-i) denote the expected value of the optimal search-negotiation strategy
here are n-i+1 buyers. The parameters x* and t ,he functions
-,-) and the values V(n-i) are given by the following equations :
Boundary conditions:
X*= -o, t 0 *0 = V() = Ex(o), 0y(x (4-10)
The functions A*1 (.,.) are given implicitly by:
y(x-(0), X*n-i(Xn(0), si)) + cosn-i= -d + V(n-i-l), (4-11)
for i = n-1, n-2,...,l.
The parameters T -satisfy:n-i
E [Y(xni(O) -A*ini(0),s)) +c.s] = -d + V(n-i-1), (4-12)
for i - n-1, n-2...,l.
Denote A*n- i(x,t) by pi(X). Th e parameters x* can be computed from:
Tn-i
f h(t) g[ (y.(x*-(0),3 ) dX+ G(x*n- (-co t- - d + V(n-i-l)Jdt.
(4-13)Pi(x*n-i(0)
+ (1 - H(Ti-)) ( --- d + V(n-i-l)) - -d + V(n-i-1), for tin n-*l,n-2,.'..,l.+'(1 B(T-i))(;Cn-i3
The value V(n-i) is given by:
V(n-i) - F(x* -)[V(nil)dJ + (1F(x* -i))e(1-H(Ti-)[V(n-i-1)-d-c.T,
'Rcall that t>) is defined as the first time in which the expression in(4-6) is maximized.
- 140 -
+ f(x).[T 7i(t) G(X* n i(x,t))(V(n-i-l)-d-c-t)dt]dx
x* (0)n-i
T -+ f(x).[ I h(t) [ f g(A)y(x,A)d]dt] dx (4-14)
f I- 0 for i~n-1...l.x* (0) y (x)
n-i
The optimal search-negotiation process, as described in Lemma 4.7 can be
considered as a generalization of the "reservation utility property". After
passing the first test, that of the size of the initial bid, the buyer is
tested as to his second bid. These two bids are enough to determine th' char--
acteristic concession pattern. The buyer cannot postpone his second bid too
much - the searcher will leave and go elsewhere. It is however in his best
interest to announce his second price as late as possible, because then as
seen from equation (4-11) he can offer a lower price. The qualitative nature
of the optimal search-negotiation process does not change if more than two
offers are needed in order to determine the characteristic concession pattern.
We now examine the optimal behavior under the assumption of perfect
recall. By a repeated application of the result in Lemma 4.6, the following
can be proved imnediately.
Lemma 4.8
Under the conditions of Model 4, assuming perfect recall, the optimal
search-negotiation process starts as the one in the no-recall case with the
exception that for all i , x*n-i x*, Tni - T and e* - X*. If an agree-
ment is not achieved before the n-th buyer is visited, a new stage of search-
negotiation starts. In this stage each buyer i is assigned a number z, the
buyer with largest z being visited first. The number z may increase or
decrease during the negotiations with buyer 1. When z becomes smaller than
z for son J #I i, the searcher stops negotiations with buyer i and travels
- 141 -
to negotiate with buyer j.
The decreasing recall case is an in-between case. A direct computation
of the switchpoints x* , t* and A* is difficult, but their values are
always between the values of the two extreme cases. The solution of the de-
generate case (when y(x1 (0)) are determined by xi(0)) is found in Landsberger
and Peled [1977].
The optimal search-negotiation strategies, presented in Lemma 4.7 and
in Lemma 4.8 have some interesting implications. It can be shown that the
number of buyers visited is larger and the amount of time spent on negotiations
is smaller when perfect recall is assumed than when no recall is assumed. The
results are sensitive to the travel cost d. It should be noted that in our
numismatical market d is small relative to c, and consequently the search is
more emphasized than the negotiations in the first stage of the search-nego-
tiation process. However, high price volatility, which corresponds to de-
creasing recall or even to no recall, prevents the collector from doing too
much search. Our rough estimate is that the average collector visits 2-5
dealers and is involved in negotiations with only one or two bof these
dealers.
Model 5
We now assis that the A are independently drawn from an unknown distribu-
tion G(e) which can be reassessed during the search. For example G() may
be the uniform distribution on [O,a] where a is unknown. A special case is
the degenerate case A - A with probability 1, where A is unknown. For this
i 0 0
'A similar analysis of the traditional search is found in Landsberger and Peled[1977]. Some of their "comparative statics" can be applied to our searchnegotiation model.
- 142 -
special case we can prove:
Lemma 4.9
(a) The no-recall case. Under the assumptions of Model 5, and assuming
A = A for some unknown AO, the optimal search-negotiation process startsi 0
as does the process described in Lemma 4.7, but with different switchpoints
x n , ,T niand An- (.,) When X0 becomes known, the process continues
as a regular search process, as described in Lemma 4.2.
(b) The perfect recall case. The optimal search-negotiation process starts
as does the process described in Lemma 4.8, but with different switchpoints
x, T and K(., -). When A becomes known, the process continues as a regular
search process, as described in Lemma 4.3.
The general search-negotiation process in. Model 5 involves learning about
the underlying distribution G(-) of A . We assume that the searcher is a
Bayesian who can assess a prior density function g'(") on Aand to update it
after every observation. The optimal search-negotiation procedure depends of
course on the value of the information gained while observing the A 's. It
is obvious that as the value of one observation increases, the searcher nego-
tiates with one of the buyers in an earlier stage of the search. When the
value of this information is small (relative to bargaining costs) the nego-
tiations may take place only in the last stages of the search.
To make the formulation easier we will now assume that the time between
the first and the second offers of all dealers is a known constant denoted by
a Consequently the searcher either waits for the second offer or leaves
immediately after observing the initial bid.
1 This Lema is a direct resu.t of LT a 4.10. When A becomes known, Model 5becomes identical to Models 2 and 3.
- 143 -
Lemma 4.10
(a) The no-recall case. Under the assumptions of Model 5, and assuming the
time between the first and the second price offers is a known constant sop
the optimal search negotiation strategy is the following:
(1) Let the current assessment of A be denoted by g'(.). Observe the
initial bid of buyer i, x (0).If xn-i(0) > x* (g') wait for
the second offer. Otherwise go to the next buyer.
(2) If the decision is to continue negotiations, observe An- and
update the assessment of A. Let the new assessment be denoted by
g"(-). If Ani > A* (x (O),g") then continue the negotiations.
Otherwise go to the next buyer.
Let V (g) denote the expected value of an optimal search-negotiationn-i
process given that g is the probability assessment of A and given that there
are n-i+l buyers. The functions x*ni() and A*ni(-,-), and the values
V ni() can be computed from the following equations:
Boundary conditions:
and x*0 (g') = -cc for all g'; A*0(x,g') - -c for all x and g';
(4-15)V0 (g') - ffy(x,A)g'(A)dF(x)dA.
00
The functions A* (,.-) are given by:
y(x-(O)9 A* (Xi(O), g")) + cs0 - -d + V 1 1 (g"), in-,...l.(4-16)
Denote A* i(x,g") by 6(x). The parameters x* (e) can be computed from:
f-c-s 0-d + vni.(g"))s'(A)dX + Y(x_ i(0), X)g'(X) dA = -d + Vn-i- 1(')
or 6)if(4-17)x 6 (X (0)) x > (X '(0)) &r --. ,.
- 144 -
The value Vn-i(-) is given by:
Vn-i(g') = F(x*n-i(g'))[V 1 1 (g')-d] + f(x) - 7 c-s0 -d+Vn-i-1 (g"))g'(X)dX +
x*_i (g') Ai<6(x)
+f(x,x)g'(x)dx] dx, for i=n-i,...l. (4-18)
A> cdi(x)
(b) The perfect recall case. Under the assumptions of Model 5 and of part (a)
of this Lemma the optimal search-negotiation strategy is similar to the one in
(a), with the exception that AXn-ig) = n-j(x,g) and x (g) = x (g) for
all i and j. (Formulae (4-15) to (4-17) do not hold). If the last buyer is
visited and an agreement has not yet been achieved a new stage starts, iden-
tical to the first but with all xn-i(0) known and some of the An-i known.
(This part of the optimal strategy is similar to the one described in Lemma 4.8.)
Examples
We conclude this section by listing a few examples of price distributions
and of characteristic concession patterns for which the optimal search-nego-
tiation strategy could be computed. Inson -cases computation is dif ficult.
The characteristic concession patterns are discussed further in 4.4.
(a) Known final price, random opening bid. In this case the A indicate the
rate of concessions. For example, xi(t) = 1-(l-xi(0))e it, where the final
price is always below 1. The initial bids are distributed on [0,a), O<a<l.
(b) Unknown upper bound on price, random opening bid, fixed rate of conces-
sion. In this case the A indicate the final price itself. For example,
-01__ wxhe(0)x (t) x (0) + a-t for t < where x (0) is distributed on [0,1]
and A is uniformly distributed on [1,A]. By assuming that the prior dis-
tribution of A is Pareto we can use a convenient prior-posterior scheme.
- 145 -
(c) Final price depends on opening bid. Here A is the duration of nego-
tiations. For example, xi(t) = x1(0) when 0 < t< A1 , and xi(t) =
x (0) -(l+a) when t > A . The x (0)"s are distributed on [0,1],a is a known
constant, and A is distributed on some positive interval.
- 146 -
Remark
Based on his understanding of the collector's optimal search-negotiation
process, the dealer can reduce his expected bargaining costs by properly choos-
ing his initial bid x(O), his parameter A and the time s in which the second
bid is made. His opening bid should certainly be above the switch points com-
puted in Lemmas 4.7 (if no-recall is assumed) and 4.8 (if perfect recall is
assumed). The highest past bid, quoted by the searcher, may serve as a lower
bound for x(O). If the searcher is ready to continue negotiations and his ask-
ing price is not too high, the optimal decision of the deale? is probably to
make the second bid right away. But if the gap between the initial bids is
still large the dealer has to weigh the advantage of waiting (the critical
value A(x(O),s) decreases as s increases) and the disadvantage (additional
bargaining costs). Intuitively, it is clear that as the dealer's bargaining
cost increases, his optimal waiting time decreases. The very-busy dealer will
announce his second price much faster than other dealers. As a result his
price should be higher. This means that the searcher can expect to know the
final price quicker when the bargaining cost of the dealer is higher.
One important advantage of the dealer is the information he has regard-
ing the market prices. This advantage enables him to terminate negotiations
immediately when he observes that the collector overestimates the market price
and to continue negotiations and achieve an agreement when the searcher under-
estimates the price. His advantage increases with the travel cost d.
- 147 -
4.3.3 Proofs of Results
Proof of Lemma 4.1
Suppose that there is an optimal search-negotiation policy 6, i.e. a rule
specifying the conditions under which the searcher negotiates with buyer i,
the length of each negotfation, and the conditions for termination. Suppose
now that for some i, the rule 6 implies that there is one session of negotia-
tion, with some buyer i, which lasts for t, t t ti(xi(0)) units of time. It
is easy to see that this rule is dominated by a rule 6' which is identical to
6 except that under the same conditions the length of negotiation with buyer
i is t (X 1(0)).
It is also clear that for the cases of no recall and perfect recall, 6
cannot be optimal if it involves negotiations with buyer i for ti(xi(0)) units
of time followed by further search. This is so also for the case of decreas-
ing recall because of the way the recall function p(s) is defined. The proba-
bility that yj(x(0)) is available given that s units of time have passed from
the time it was proposed is equal to the probability that it is available given
that s units of time have passed from the time that x (0) was proposed, and
that negotiation with buyer i did not take place. As a result it is never
optimal to negotiate and then to continue the search.
Since uegotiation takes place only at the end,. the search-negotiation
problem is really identical to the traditional search problem with an infinite
number of buyers. The result regarding the switchpoint x* is the known "res-
ervation price property" of search and it holds for the cases of perfect recall,
no-recall and decreasing recall as proved, for. example, in de Groot [(1970),
Chap. 13]. 0
- 148 -
Proof of Lemma 4.2
The general form of the search-negotiation process, i.e. searching and
negotiating only with the actual buyer, is proved in the same way that Lemma
4.1 was proved. As to the sequence of switchpoints x* they are computedi
in the following wayI:Let Vn-i be the expected value of the search-negotiation
under the optimal strategy whenn-i buyers have yet to be observed, and let
xn-i+(O) be the current observation. The searcher should negotiate with the
current buyer if y6 n-i+1())>Vn-id and continue the search otherwise. The
recursive relation 'is therefore:
Vnil (V -d) -Prob [(x i+PO)) < Vn g-d]+ Eyx +(klyx ()> -n-i+l n-i- n- n n-n-i n-
Let x*J(0) satisfy y(x*.(0)) = V.-d. We get
Y(X*j (0)) + d = y(x*.(O)) F(x*.) + f y(x)dF(x),
which is exactly equation (4-8).
Proof of Lemma 4.3
This Lemma is a direct result of the proofs obtained independently by
Lippman and McCall [(1976), page 169] and Landsberger and Peled [(1977),
page 23]. o
Proof of Lemma 4.4
Let xi(0) be an opening bid, which was proposed t-t ttits of time before
the current observation is made. Compare two strategies, 4$1 and 6 2. In 6
the searcher checks the availability of xi(0) and continues with the optimal
This computation appears in many references for example see Gilbert andMosteller [1966].
- 149 -
strategy thereafter. In 62 the searcher continues as in 61 without checking
the availability of x (0). Suppose that s units of time after it was checked,
x (0) is solicited, as a part of the optimal strategy. In 61 it is still
available with probability p(t) j while in 6 it is available with proba-P~t) 2
bility p(t + s). In expected value terms the difference between 6 and 62 is
smaller or equal to y(x (0))* p(t) - p(t+s)]. Since y(x1 (0))<
y(x*), equation (4-7)is obtained.
Proof of Lemma 4.7
The proof is straightforward, based on the comparison of expected
utilities at every stage of the search-negotiation process. It should be
noted that an implicit assumption of this model is that the time in which the
second price offer is made, x 1 , is smaller than t -(x(n-i0)s X ) the
optimal negotiation's duration, for all xn-i(0) and An-i. Otherwise the
model is slightly more complicated. 0
- 150 -
4.4 Concession Patterns and Sudden Termination
4.4.1 Introduction
In this section we examine and model one of the observed phenomena
in dealer-dealer negotiations, namely the adherence of the dealers to their
characteristic concession patterns. By constraining themselves to known
patterns the negotiators knowingly limit their strategy space, thereby reduc-
ing their probability of achieving the most favorable agreement; however, they
shorten the negotiations' duration thereby limiting the effects of time. Since
bargaining costs are large, the dealer prefers saving on duration rather than
attempting to get the best possible agreement.
A characteristic concession pattern I is a set of functions y which can
be considered "similar" to each other. Each function yCI, corresponds to a
particular sequence of price proposals. For example, the set I= {yy(t)=bt,
O<t<co, for some b>O} includes all the increasing functions with a constant
concession rate. The set J = {yly(t)=e-at, O<t< 00 , for some a>0} includes
all decreasing functions for which the ratio /y is constant. These sets
may be justified as characteristic concession patterns on the ground that the
concession patterns of each dealer are observed to be almost always elements
of one set. There may be various reasons for the particular concession pat-
tern chosen by the dealer, but this is not our concern here. We will simply
assume that each dealer made this choice once in the past, and that the char-
acteristic concession patterns are common knowledge.
Game .The dealer-dealer negotiation game is viewed here as a three-stage: game.
Stage 1 consists of the (simultaneous or consecutive) submittance of: initial
- 151 -
bids. When this stage ends, each party may continue the gaze or terminate it,
with no cost incurred. If the game is continued, Stage 2 starts. In this
stage the parties must concede according to their characteristic concession
patterns. Throughout stage 2 the parties are allowed to terminate the nego-
tiation.
During stage 2, in time t 0 which is pre-specified, the parties play a 2x2
game in which their strategies are Accept and Do Not Accept (A and X.) The
motivation for this game will be given later. Stage 3 starts at time t0 if
an agreement has not been achieved before t0 and if both parties play their
A strategy at to. Once this stage starts, the parties behave as if it is stage
2 that just started, with the last bids serving as the initial bids. Intui-
tively, by starting stage 3, the parties are -telling each other: "Forget every-
thing that happened in the negotiation. Let us start negotiating all over
again with the last two price bids as starting points".
The motivation for the 2 x 2 game is this. When stage 2 starts, the parties
must choose to concede according to their characteristic concession pattern.
Since further cooperation is not allowed, they can be expected to choose the
Nash Equilibrium solutions, which depend on the patterns and on the initial
bid. When conceding along their characteristic pattern the parties have no way
to shorten the negotiation's duration, even though they can both compute the
final agreement long before they agree on it. This inefficiency can be cor-
rected if both parties are ready to start "all over again", provided that the
new equilibrium strategies lead to the same solution-but shorten' the
duration.
This game therefore corresponds to the situation in which the parties,
after negotiating for some time, are convinced that they both follow their
optimal Nash strategies and are willing to cooperate to reduce time effects.
Stage number 2 of the negotiation game is discussed in 4.4.2.
- 152 -
In that section we examine several characteristic concession patterns and
compute the equilibrium strategies. The expected duration and the expected
utility of each party are also computed and compared. Our assumption in 4.4.2
is that the initial bids of the parties are 1 and 0, and that the interval
[0,1] is a subset of the zone of agreement. Under these assumptions the
reservation prices play no role. They are important in stage 1, when the
initial bids are proposed. This stage is dicussed in section 4.4.3.
The game we study is a descriptive model of the dealer-dealer negotiation.
It yields some qualitativeresults, which can be studied empirically, although
it was not attempted here. Some of these results are:
(1) The characteristic concession patterns divide the dealers into equiva-
lence classes. Negotiating parties tend to have identical concession patterns.
(2) There are concession patterns which encourage the parties to revise
their strategies and to start stage 3, thus saving on time. Other patterns
discourage such cooperation.
(3) The gap between the opening bids depends on the concession patterns of
the parties. It is wider when the patterns encourage cooperation.
xInitial
Price Seller's Concession Curve
Revised Curves
Buyer' a Concession Curve
o Stage 2 t0 Stage 3 t
Fig. 4-3: The Thiee Stage Game
- 153 -
4.4.2 Game Models
Consider distributive bargaininggame over [0,1]. The utility functions of
the parties are assumed to be of the additive form:
U (x,t) = x - Cl*t, x EO111, O<t<c, (4-21)
and
U2(xt) = 1 - x - C2 *.t, x e[O,19, O<t< o. (4-22)
Suppose that the Sudden Termination is a Poisson process with parameter A.
The distribution of the time of the first arrival is
P(d<t) = 1 - e-Xt , 0<t<O. (4-23)
The proposal of party i in time t will be denoted by yi(t) (i = 1,2, O<t<Coo).
The parties want to achieve an agreement as fast as possible, because of
time effects, and they do that by committing themselves to y (t) of a certain
functional form. The particular concession patterns that we examine here de-
pend, for simplicityon a unique parameter. They are:
(a) Exponential with decreasing rate of concessions:
IE=,{y 1 )|y( = e-at for some a>0, 0<t< cc}, (4-24)
and
I - (y2 y2(t) =1--ebtfor some 'b>O, 0C<t<0. (4-25)
S B(Note that the superscript S in I denotes Seller, and the superscript B in I
denotes Buyer.)
(b) Linear (constant rate of concession):
\ -{yI(t)|y 1 (t) = 1 - at for some &>0, Oq<t<O}, (4-26)
and
- 154 -
= {y2(t)Iy2 (t) = bt for some b>0 , 0<t<O}
(c) Quadratic with increasing rate of concessions:
= {y1(t)1y1(t) = 1 - at2 for some
B = 2I ={y2(t)1y2(t) =bt for some b>
(d) Constant with a sudden
I S= yl(t)|y 1(t) -
aO, 0<t<o}
>0 , O<t< 0}.
concession:
1 0<t<a
for some
y2(a)
0
I y2(t)Y2 (t) =
a<t
a>0,. o<t<oJ, (4-29)
O<t<b
for some b>0 , 0<t<ox}. (4-30)
b<t
We divide the possible pairs of strategies (y1(t), y2(t))into three cate-
gories. A pair of strategies is said to encourage cooperation if for all t0,
0<t<w, the negotiators, by starting stage 3 at t - t, shorten the negotia-
tions' duration. A pair of strategies is said to discourage cooperation if for all
such t , the .ctual duration is lengthened. A pair of strategies is
neutral if for all such t. thenegotiationsP duration is not changed.0
Lema 4.11
SThe equilibrium strategies of the linear vs. linear case (i.e.yy(O L
y2 (t)eI ) are neutral. They are:
(4-27)
and
(4-28)
and
- 155 -
y1(t) = C2(i+c+c )t1(2
y2(t) = C(1 + cC1 C2
(4-31)
(4-32)
Proof:
Let yi(t) be continuous concession patterns satisfying y(t)<O, j2(t)>0,
*
y 1(O)=1, y2 (0)= 0. Assume the curves intersect and let t be the first t sat-
isfying y1 (t) = y2 (t). From (4-21), (4-22), (4-23), (4-26) and (4-27) the kernal payof f s
are: ** 2
are: (a,b= e (y(t*) - Cl-to) - fC-A td ,
0
and
K 2(a,b) = et
** t
(1-y2 2(t) -C2.t ) - fc2't-Ajtdt
0
By computing3k1 (a,b)
aa and3k2 (a, b)
3b and by equating them to zero we get:
6,y (t *)Ay1(t ) - ayt*
1 3a
e
(t ay2 (t )
2 2(
atI- + C =0
*- at
I- c 2 -A-O.
(4-33)
(4-34)
For the linear vs. linear case, these two equations are:
A(-at )- b + C 0
and
A(bt)+a-C2 -A0.
S2From hereweet a =C 2 (+ c C)2 b-=C(l + c 1 C ).Suppose now that
and
- 156 -
*the parties follow their equilibrium strategies from t=O to t=t0 , O<t <t
At time t=t0 they comp te new equilibrium strategies with parameters
*a1 , b1 . The new termination time t (t ) (measured from t=t ) satisfies
t *(t 0 1 - at - bt o . The new functions yiare y(t)=1-at- aIt,aI + b0
1
y2(t) bt0 + b1twhere a and b are the revised parameters. From (4-33)
and (4-34) we get:
X(l-at -aIt (t0))- bI + C = 0 ,
and
A(bt + b1 t (t0))+ a - C2 - A=0 ,
which are solved by aya, b =b.
Corollaries
S B * 1(a) In the Lvs. B case, an agreement is achieved at time t = C1+2+ A
with probability eI ( 1 2
(b) The expected utilities of the parties are
Ix -CCC
Ei = e 1 2 (C CC1 2
It is easy to show that E1>E2 if and only if C1>C2 . Note that the party whose
bargaining cost is higher can expect higher utility.
Lemma 4.12
S BIn the constant vs. constant case (i.e y(t) e I , y2 (t) E I ) the
equilibrium strategies are neutral. They are the following probability dis-
tributions:
F(a) = 1 -ec2a O<a<cec , (4-35)
- 157 -
and
F2 (b) = 1 - e~ Ob <b<w . (4-36)
Proof:
Suppose that at some t %O both parties still hold to their initial bids.0-
The kernal payoff of party 1 is:
Ie-a -1 - C t a<b
K1(a,b) =
- +- Cito
We try to find an "equalizer" distribution f2(b) such that fK1 (a,b)f2(b)db
is constant. The result can be shown to be f2(b) = C1e 1b. Similarly,
-C af(a) = C2e 2. Since these results are independent of t0, these are neutral
equilibrium strategies. 0
Corollaries
S B(a) In the IC vs. IC case the expected time of agreement (if one is achieved)
1CCIC2is CIC2 and an agreement is achieved with probability 2
(b) The expected utilities of the parties are both zero. Note that the party
whose bargaining cost is higher is expected to yield after his counterpart.
Lena 4.13
In the exponential vs. exponential case (i.e y(t)E I , y2 (t)e I) the
equilibrium strategies encourage cooperation. They are:
y1(t) - e-at , O<t<eO,
and
y2(t) - 1 -eCbt , o<t<eo,
- 15 -
where a,b are the unique solutions of:
b - C + a - C2b+A + a+X 1 * (4-37)
and
b-C b a-C2 a
b+ 2a+ 2 ) (4-38)
Proof:
Suppose y )(t) e ,yt) = 1 - e-bt for 0<t<t0, and suppose that a
revision of parameters is performed at t=t . Denote the new parameters by
a1,b1 . In the time of agreement t*(t ), measured from the revision, we have
y (t') = (k2-k1)e-aIt*+ ki, y2(t*) = (k1-k2)e-bIt*+ k, whereki = 1 - e-bt0,
kc2 = e -ato. (We write t* instead of t*(t0 ).) From (4-33) and (4-34), we get:
n(k2- k)e-alt* + b(k1- k2)e-bt* + C,=0 ,
and
X(k- k2)e1-bIt* + a1(k2- k1)e-a1t* - C2 =0.'
We also have y1(t*) = y2(t*), or e-alt* + e-b1 t*
From these equations we get:
-at* - - + by)+b) , (4-39)e 1kI- k 2 11
and
C-b t* c2e ( + a)/ (+a(4-40)
If t0-o we get k2=19k 1 0, and (4-37), (4-38) are obtained immediately.
1 C1 . 2Now letC k - k , c2 k2 - k . Equations (4-39) and (4-40) are
2 1 2 1
- 159 -
identical to (4-37) and (4-38), with C1 and C2 in (4-37), (4-38) replaced by
C1 , C2 . Evidently C1 > C1 , and C2 > C2, and from here it can be shown
immediately that a1I> a, b 1 > b and that t*(t ) + t0< t*(o) where t*(o) is the
duration computed for t =0.0
Corollaries
(a) For C= C2=C we get a-b=2C + A. If C > C2 it is easy to see that b> a,
i.e. the party who is stronger on time concedes more.
(b) For C = C-2=C the time to agreement is t* = Aln2 . An agreement is
achieved with probability e-Aln2/(2C +)-At* Ci Ci(c) The expected utilities are Ei - e ( + - . Note,however, that
by making frequent revisions of parameters the parties may increase their
utilities.
Lemma 4.14
In the quadratic vs. quadratic case (i.e y1 (t) E I, y2 (t) E I ) the equi-
librium strategies discourage cooperation. They are
yl(t) = 1 - at2 O<t< w,
and
y2 (t) bt2 O<t< w,
C2(C1+ C2 + A)2
with a= , -2(4-41)
C 1+ C 2
and 2
b - 1 2 (4-42)C1+ C2
Proof:
- 160 -
We start with to = 0, equation (4-33), (4-34) are now:
(1-at*2) - 2bt* + C1 = 0,
and
bt*2 + 2at* - C 2 -A=0.
The termination time t* is (a + b)~. Equations (4-41) and (4-42) are ob-
tained immediately.
Suppose now that a revision of parameters is performed at t=t . The
new strategies are y1 (t) =1-at02- a1t2,2and y 2(t)=bt0 2+ b1 t. The
solution for a is
(C2 + Aat 2)(C 2+ c2+X)2= 2 2 2 2 22> a.
(1 - at - bt 2)(C1+ C2+ A) - A(1 - at - bt )
Similarly b1 > b. From here we get immediately t*(o) < to + t*(t ).
Corollary
(a) Heret*(o)2+ and the probability of achieving an agreement
is e 2A/ (C1+ C2+A) The expected utilities are
E -C Fe2X /(Cl+C2+(X)( 1 L)+1) 1-(i=l2)
Examle et asue 1 2 C +C - X12
Example Let us assne CC1 C2 = 1, A - 1, and compare the results of the
last four corollaries. In column 1 of Table 4-1 we denote the characteristic
pattern which both negotiators follow. In other columns we compare the
expected utilities, the expected duration and the probability of achieving
an agreement.
0
- 161 -
Pattern Expected Expected Probability
Utility Duration of Agreement
Linear (IL) .07 .33 .72
Constant (IC 0.0 .50 .67
Exponential (IE) .19 (at least' .23 (at most, .79 (at least)
Quadratic (IQ) -.23 .67 .51
Table 4-1
This example shows clearly how severe the effects of time might be. Note that
an arbitrator, called in to propose an agreement at t=0, would assign utility
of .5 to each - far above what they can expect to achieve if they start their
negotiation game. Note also the clear superiority of the exponential conces-
sion pattern.
Our next step is to examine negotiations between two bargainers who have
different characteristic concession patterns. Equilibrium strategies do not
necessarily exist. When they do exist we can talk about the expected utilities
(under equilibrium strategies) of the bargainers. A bargainer with a char-
acteristic concession pattern1I may then have a preference order on the con-
cession patterns I , ... , In. He prefers to negotiate with I rather than
with Ik (k#j) when his expected utility in an I vs. I negotiation is higher
than is an Ik vs. I negotiation. The preference orders of bargainers with one
of the four characteristic concession patterns discussed in this section are
given in the next Lemma.
- 162 -
Lemma 4.15
The preference order of all the negotiators over the concession patterns
of their counterparts is I > I > I > I .C E L Q
Proof:
B B BIt is easy to show that fork IL''E, and I ,an optimal strategy
against an IC negotiator is to concede as little as possible (i.e. to chooseCS
b +0). The optimal strategy of the IC negotiator is to concede immediately.
The expected utility is therefore 1 for the buyer and 0 for the seller.
The equilibrium strategies of any two negotiators can be computed by
using (4-33) and (4-34). The Lemma can, however, be proved without carrying
Bthe full computation through. As an example we show that an IB negotiator
LS S S S
prefers an IE opponent to an IL one,, and an opponent to an IQ one.
The equations of equilibrium for the IL vs. IL case are:
(l - aLtL*) - bL + Cl = 0
LtL* + aLC 2 - 0 (4-43)
1 - aLt* = bLt*
where the subscript L corresponds to the concession pattern of the seller,
S
B SThe equations of equilibrium for the I vs. IE are:
-a t *
eE E -bE + C -0
-a 1t*
XbEtE* + ae EEC -(444)
.-'Et F.*m E%tE -bt
- 163 -
where the subscript E corresponds to the concession pattern of the seller,
SE*
The equations of equilibrium for the B vs. I case are:
X(l-a tQ*2 )-b +C, =0
Xb t* + 2at*-C - X=0 (4-45)Q Q Q Q 2
1 - a t *2 = b t*Q Q Q Q
SAgain, the subscript Q corresponds to the seller's concession pattern, I .
We want to prove
t*C 2 -At* C2 -At*+ C2e E (1 - bEtE* + r)>e L(1-b tL* + ) >e Q (1- b t*+ )
(4-46)
It is-enough to prove that tE* < tL * t * and b tE* < b t* < b t *.E -L - Q E E -LL -Q Q
From the first and third equations in (4-43) and (4-44) we get (bLtL*
bE E bE) Wealso have tL* = -id tE* = - SupposeL E
b Lt * < b E*, then bL < bE and tL* < tE*. From the second equations in
(4-43) and (4-44) we get aEbEtE* < aL . From the first and the second equa-
tions of (4-44) we have aEb E* < a . SincebEt*>bL eta b tC2+EE aEEe E L L
Now aE + XM bEtE* M(C2 + )e E > (C2+ X)(1 + aEtE) >( 2+X)(1 + aEtL*),
and from here aE >bt * a contradiction. Therefore bEtE* < bLtL*, bE <bLCtC
and the first inequality in (4-46) is proved. The proof of the other inequal-
ity is similar. 0
- 164 -
From Lemma 4.15 an interesting result emerges. Suppose that in some
market the search for a counterpart to the negotiation is costless. Then
most negotiations take place between parties with identical concession pat-
terns. Consider for example a set of bargainers whose concession patterns
are elements of the set {IL , IEvQ}S The IE bargainers prefer IEopponents,
therefore they negotiate with each other. The IL bargainers have no free
IE opponents, therefore they negotiate with each other. So do the IQ bar-
gainers. The concession patterns thus induce a division of negotiators
into equivalence classes.
Since the exponential concession patterns are "superior" in some sense
to others (they encourage cooperation and reduce the time related costs) it
can be expected that negotiators will tend to choose this concession pattern.
This is a hypothesis that could be tested empirically. The market of numis-
matical items that we observed, while too small to yield results which are
statistically significant, does confirm this hypothesis. Among 15 dealers
there are between 5 to 7 dealers whose concession patterns are approximately
exponential. 1
A game theoretical model which attempts in specifying the distribution
of negotiating types (who are having each a known, Markovian reputation) can
be found in Rosenthal and Landau [1979]. It is described briefly in Appendix
2. In their model, Rosenthal and Landau assume that ill-reputed negotiators
have the same number of opportunities to negotiate as do reputable ones.
Based on the hypothesis made in this section, we modify the model (in Appendix
2) by assuming that reputed players have more opportunities to negotiate than
others. The results, although depending on the particular probabilities
1Our rough estimate is that 3-4 dealers adhere to linear concession patternsand one or two to quadratic patterns. These estimates are based on impressionsof the interviewed dealer and on some observations of actual negotiations.
- 165 -
assumed in the model, are compatible with the results of this section.
We conclude this section by proposing a qualitative explanation of the phe-
nomenon of irrationality in dealer-dealer negotiations, described in 4.2.3(b).
By occasional deviation from his characteristic concession pattern, the dealer
creates uncertainty as to this pattern. This uncertainty may yield a higher
expected return in every negotiation in which he participates, but the number
of the expected negotiations decreases. On an expected value basis it may be
that an optimal long-term negotiation strategy is to exhibit some irrationality.
A more elaborate game model is needed in order to treat this problem rigorously.
4.4.3 The Opening Bids
In the negotiation game described in section 4.4.1, there is only one un-
certainty - the actual reservation prices of the bargainers. In section 4.4.2,
we assumed that the first bids are already within the zone of agreement and that
both parties know that. In this section we examine the decision problem of
making initial bids which can fall outside the zone of agreement. Both parties
continue negotiations after the initial proposals are announced (i.e. after
stage 1) only if the final agreement, which can be computed as was shown in
4.4.2, is acceptable to them.
As shown in Fig. 4- 3-, for every pair of initial bids, x, x2 the final
agreement x(xl, x2)and the duration t(x , X2) can be computed. Party 1 continues
negotiations if x(x, x2) < Rb(t(xls x2)), where Rb(t) is a curve which we
compute later. Similarly party 2 continues negotiations if x(x , x 22)
R (t(xl, x2)). The two curves R(t) and R(t) bound the triangular-like arc
ABC which we call the acceptance zone.
I
- 166 -
x
A rRb(t)
R (t)s
x B
X2
t (x x2 ) t
Fig. 4-4 : The Acceptance Zone
The negotiations continue past stage 1 only if the computed outcome falls within
the acceptance zone.
To compute the curves Rs(t), Rb(t) we have to assume the existence of
reservation prices r, trb, i.e. to assume that the seller can always find a
buyer for a price of rs, and that the buyer can always find a seller for a price
of rb. If the negotiation is broken off because of a sudden termination, the
assumption is that the seller and the buyer can still settle for r and rb
respeLtively.1
The expected utility of the seller, given that the equilibrium strategies
-At t -Asof stap 2 yield an agreement x and a duration t, is e (x-c1 t-r) - c ifAse ds-
0
'Tis means that the sudden termination is caused not by a sudden change inmarket prices but by a sudden change in the utility functions. In this ac4'eleach party assumes that after termination he can get his reservation prictr(i - sor b).
- 167 -
= j. (x-r +. The function Rs(t) is therefore- the solution ofsA A s
or
R(t)= -e +r - (4-46)s A s A
Similarly, Rb(t) is given by
C2C 2
Rb A + rb +A (4-47)
The decision problem of party 1 is therefore:
ax e-2At K 2 (x(xx2) - 1 t(xx2 ) - r) -1 P(xi, X2),
where (4-48)
P (xl,x2 )uProb [ R5 (t(xlx2)) :Sx(xl3x2) i Rb(t(xl' 2))I rs
The decision problem of party 2 is:
C2max ( e- L2t(xiX2 (rb x(xlpx2 ) - C2 t 'x2)) - p2C2 pb(Xl.x2 )
where 2 (449)
Pb(Xlx 2)= Prob [ R s(t(x ,x2) x(xIx2) Rb(t(xx2 ))1 rbl
Note that the resolution of these problems depend on the way the initial bids
are announced (simultaneously or consecutively), on the concession patterns of
both parties and on the reservation prices r and rb (which are not common
knowledge).
The decision problems as formulated here may be very difficult to solve
analytically. Some restrictive assumptions should be made to facilitate the
analysis. One simple version of this game (with A 0, C1 - C2-+0) was studied
by Chatterjee [1978] who derived (for the case of simultaneous bidding) equa-
tions for the optimal xx 2 as functions of r5 and rb. The results depend on
- 168 -
the probability distributions underlying rs and rb, which are assumed to be
conmon knowledge.
Some qualitative results can be obtained immediately. Consider three
S B S B S Bnegotiation games: I vs. IL ,
1 E vs. IE and IQ vs. IQ In all three games
the buyers have a reservation price rb and the sellers have a reservation
price r s. Let xE (xlx2). xL(xl'x2) and xq(xx 2) be the outcomes in the three
games when the initial (simultaneous) bids are x1and x2 . Let tE(xlx2)9
tL (xlx2 ) and tQ(x1 ,x2) be the corresponding durations. Denote the optimal
E E L L Q Qopening bids in the three games by'(x 1 , x2 ) (x1 , x22) and (xQ , x2 ), and
let Z L, Z E and ZQ(i = 1,2) be the expected utilities in the three games. They
satisfy ZE = ZL = ZQ (i 1,2). This is so because the equilibrium is
characterized by the derivatives of the concession curves at their point of
E E L Lintersection. If x1 and x2 are known, then x1 , x2 are the unique solutions
of the equations
E E) L LtE(Xl 'E2 tL(x 1 L t2 )
(4-50)and E 2E L 2L
The bids x 1Q and x2Q could be computed analogously.
The last result shows that "linear" and "quadratic" players can achieve
the same utilities as "exponential" players, but in order to do -so, their in-
itial bids should be closer to each other. In other words, they should behave
more "cooperatively" than exponential players in the first stage of the game.
Another result that can be proved immediately is that the opening bid of
the seller in the simultaneous bidding case is smaller than his opening bid in
the consecutive bidding case when he is the first to bid. In both cases the
initial demands are more extreme when A is larger. The sudden termination
possibility has therefore the effect of making the initial bids closer to each
- 169 -
other.
The three stage game described in 4.4.1 - 4.4.3 is, we believe,'a
bounded rationality model the results of which lead to conjectures.that can
be tested empirically. Some of these conjectures, stated in a non-formal
language, are:
(1) The gap between the initial bids decreases as the bargaining costs in-
crease and as price volatility increases. (Price volatility is related to
the sudden termination possibility.)
(2) The gap between the initial bids depends on the reputation of the parties.
(Reputation is related to chracteristic concession patterns.)
(3) The duration of negotiation is not affected by the reputation of the
parties. It is affected by the bargaining costs, by price volatility and by
the initial bids.
- 170 -
Appendix 1
On a Variant of the Pick-the-Best Problem
In this Appendix we discuss a variant of the Pick-the-Best problem, a sequen-
tial decision problem in which the objective is to maximize the probability
of choosing the best in a group of candidates.I The motivation for this
variant is the problem of search for other offers conducted during a postpone-
ment in a negotiation. The price quotes offered during this search are
assumed to be non-negotiable. The number of quotes is not known in advance,
but is rather a random variable. If no offer is accepted before the post-
ponement ends then negotiation resumes. This problem arises in several other
contexts. One example is the search for another job, conducted by an em-
ployee before he negotiates the terms of a new contract with his current em-
ployer. Another example is a search conducted by a firm for alternative in-
vestment opportunities, conducted prior to the date in which resources must
2be committed to an on-going project.
Suppose that the elements x1,x2 ... p , N'N+1 of a set I are observed
sequentially. One of these elements is the maximum of I, according to some
preference order defined by the searcher. The latter "wins" if he ends up
with the maximum in his possession. The objective is to find a strategy such
that the probability of winning is maximized.
In the traditional "secretary problem" the rule is that only one element
can be chosen, and that back solicitation is not allowed. The assumption is
1For an overview of the traditional problem and for some extensions see
Gilbert and Msteller [1966] and de Groot [1970]. Some of the terms we use(like "candidate", "win") are taken from the first reference. The problem isalso known by the names: Secretary Problem, Dowry Problem, Beauty Contest, etc.
rer+ irn t i ^f this Acnenne"Aw s eps wi th seciuertial dect son problems
- 171 -
also that the order of observations is random, i.e. all (N+l)! permutations
are equally likely. Yang [1974] allows backward solicitation and arrives at
a system of equations which can be solved recursively to yield the optimal
strategy.
In this model we allow backward solicitation and we assume that the num-
ber N of elements in the set I is unknown. We also assume that only the ele-
ments x, ... , sxN appear at random, while x.+, is a "special" element, appear-
ing only after the xI's have all been observed. The value of the element
xN+l is itself a random variable with some known distribution. In our nego-
tiation context 'N+l is the result of the negotiations if resumed after the
postponement.
The x 's appear sequentially, and for simplicity we denote them by
xl, ... , xN according to the order of appearance. An element xk is a candi-
date if it is the maximum in the subset {xI , ... , xk of I. When some xm ap-
pears the searcher could either pick the last candidate xk (lck<m) or wait
for the next observation. If xk is chosen, the searcher may find that it is
not available any more. We assume that there is a known probability p(m,k)
such that at stage m,xk is available with probability p(m,k). We generally
assume p(m,m)=1 for all m. The number N is unknown, but when the special
element xN+l appears it becomes known that Nn0 , xn being the last x to0
appear before xN+l. The element xN+l is not immediately observed, but its
rank is decided by a lottery under which there is a probability r that XN+1
is the maximum in I, and a probability 1-r that it is not. It is easy to
see that if r the optimal decision is to wait for the special element, i.e.
no search is carried. We therefore assume r<-.2i
Suppose that xm is observed. Let Q(mk) be the probability of winning
when the decision is to pick the last candidate xk (p-for jick). Let Q(m,k).
- 172 -
be the probability of winning when the decision is to wait and to continue
with the optimal strategy (Qw for wait). Define
Q(m,k) = max{Q (mk), Qw(mk)) 1<k<m , (4-51)
and
A(m,k) = Qw(mk) - Q (m,k) l<k<m . (4-52)
The strategy is to pick the last candidate if A(m,k)<O and, to wait otherwise.
We want to get a set of equations for the Qp's and the Qw's. Let pn(n=1,2,
... ) be the prior probability on N, and let qn(n=1,2,...) be the probability
that N=n given that n observations have already been made. Let tn(n=1,2...)
be the probability that max{x1 9,.., X}= max{x1 ,... xn) given that N>n. For
all m>2 define Q(mo) to be the probability of winning given that the last
observation is x and that the last candidate is known to be not available.M
With these notations the optimality equations are:
Qw(mk) = qm[r+(1-2r)p (m+,k)] + (1-qm) [gQ(m+lm+l) + Q(m+1,k)]
1<k<m<- , (4-53)
and
Q (m,k) = p(mk)(1-r)tm + (l-p(mk))Q(mo) 1<k<<c , m>2 , (4-54)
and
Q(mo) - qc.r + (1-qn)(i;r Q(m+lm+l) + mjQ(m+1,o)), m>2 . (4-55)
Yang [1974] examined the case where r-O; N is known with certainty;
p(m,k)-f(m-k) where f is a decreasing function, i.e. the case of backward
- 173 -
solicitation with partial recall.'
In our problem, if there exist an n0 such that pn=0 for all n>n0, then
we have a finite set of equations which can be solved recursively. If this
is not so, a direct solution may be difficult to find. In the rest of this
Appendix we study some general properties of the solution and give some
examples.
Lemma 4.16 Equation (4-53), (4-54) and (4-55) imply
r<Q(m,O)<Qw(mk) 1<k<<c , m>1 . (4-56)
Proof:
From (4-53) and (4-55) we get:
Qw(mk) - Q(mO) = q(l-2r)p(mr-lk) + (1-q)Ry (Q(m+l,k) - Q(m+1,0) >
> (--qm mQw(m++,k)-Q,(m),0)) >
>(1-qmrni m+2 w(m+2,k)-Q(m+2,0)) >1 m+ m2*n
> 1(1-q nl w(m+n+1,k) - Q(m+n+1,0))Um
'If f(j)-O for j>O, f(j)-l for j=O, we get from (4-53), (4-54) and (4-55) that
Q (mm)>QW(m'm)<-- >> I m- + 1 -m+2-( - I-Qp m m N m1 N m+i N wi+2
(l+ M +M +..1+ 1M + + 1+
i.e. Qp(m-m) > Q-(m-m)<=>l> + + .. which is exactly the classica a--io mN-i
solution of the secretary problem.
- 174 -
By sending n- we get Qw(mk) - Q(m,0)>0 for all k,m,l<k<m<e, m>1. From
(4-53), (4-54) and (4-55) we get:
Q(mtk)-r = qm(1-2r)p(ml,k) + (1-qm) [1 Q(m+lm+l) + 4y Q(m+l,k) - r] >
> q(1-2r)p(m+1,k) + (1-qm) 1Q(ml,m+l) + Q(m+1,k)-
LyQm1,Q m+) + lj Q m Qlk)- 0
-yQ(m+m+l) -yQ(r+1,O)] =
= Q (mk) - Q(m,0),
therefore Q(mO)>r.
Lemma 4.17 If tm+r and q,-r when m+-, and if p(m,k) is strictly decreasing
in m for all k, then if N is large enough the searcher tries to pick a candi-
date before the end of the search.
Proof:
From (4-53), (4-54) and (4-55) we have (for all k, l<k<m)
Qw(mk) - (r+(l-2r)p(m+l,k))0 as mre
and
Q (m,k) - p(m,k)(1-r) - (1-p(m,k))r 0 as m+
Therefore
A(m,k) - (1-2r)(p(m+l,k) - p(m,k))+O as m- , i.e.
for a large enough A(m,k) is negative and the searcher tries to pick a candi-
date. 0
Lemma 4.18 A sufficient condition for waiting when xm is observed is t 4 .
Proof:
We can express A(m,k) as:
- 175 -
A(m,k) = p(mk)[Q(mO)-(l-r)tm+m] + (1-qm)$ [Q(m+lk) - Q(m+1,0)] +
+ qm(1-2r)p(m+l,k)
From (4-56) the second term is non-negative. So is the third term. There-
fore,
Q(m,0) - (1-r)t .0 implies A(m,k)>O. But from (4-46) r<Q(m,O)m
therefore
r- (1-r)t >0(or t < ) implies A(m,k)>O.m m -l-r0
Examples:
(1) Suppose N is known with certainty, pn = r. Then
tm = (4-57)
0 otherwise
and m< ar implies A(m,k)>0, for all k, lk<m. For r=-4 the optimal stra-l-r
2tegy involves waiting until at least xm appears, m = jn0 . For r=.3 we get
3
(2) Suppose that the duration d of the postponement is distributed exponen-
tially, and that the observations appear in a constant rate. We then have
for all n>O:
P- Prob(d<n+l) - Prob(d<a) (1-e n+l) - (- f),
-A -
- e. (1-.),
where is the mean of the duration. We also have -q.1-e-x for all m, and
Am A 1 -n-tg-e m.(1-e ) Ej -e It can be shown that E e- < E1(Am) whereV nmm n u-m n1
e-t -Am -A(X) - dt -is a tabulated function. Therefore t a-E (Xm)e (1-e
tKa
- 176 -
We can establish a rough lower bound on the number m* such that for all m<m*,
when x is observed the optimal decision is to wait. This m* is the largest nm
satisfying m-E1(xm)em (1-e ) < For example, when X=.l (i.e. the expec-
ted number of observations is- = 10), the bounds m* as function of r are asXshown in the following table:
r m* Prob (N>m*)
.25 2 .82
.30 4 .67
.35 8 .45
.40 17 .18
.45 53 .005
.50 O 0
Table 4-2
Note that the third column is a rough upper bound on the probability that the
researcher attempts to pick any observed x. For r=.4 we have m*=17 (while
the expected number of observations is only 10) and the probability that the
searcher attempts to pick an observed x is less than .18. These results are
independent of the recall probabilities p(m,k).
Lemma 4.19 (The no-recall case)
If p(m,k)w0 for all 1<k<m<e, and if t3 is non-decreasing in m, then a
necessary and a sufficient condition for waiting when xM+1 is observed is
tm < (i - 1,2,...). (4-58)m--r
- 177 -
Proof:
From equations (4-53), (4-54) and (4-55) we get:
Qw(m,k) = qm.r + (1-q) [y Q(m+1,m+1) + gy Qw(m+1,k)]
1<k<m<cm, (4-59)
and
Q (m,m) = (l-r)t m1<m<cm. (4-60)
Denote f(m) = qm(r -(l-r)tm). We first claim that A(m+1,m+1) and A(m,m)-f(m)
have the same sign for all m.
If A(m+l,m+l)>O then Q(m+l,m+l) = Qw(m+l,m+l)
> QP (m+l,m+l)
= (1-r)tM+1
therefore
A(mm) = QW(m,m) - Qp (mm)
= qm.r + (l-qm)O %(m+l,m+l) - (1-r)tm
> q M. r + (1-qm) (1-r) tm+l - (1-r) tm
> qm.r + (1-qm)(1-r)t m - (1-r)tm
= f(m).
If A(m+lm+l)<_0 than Q(mIl,m+l)=Q (m+l,m+1) = (1-r)tm+1 , thereforep
A(mm) = Qw(m,m) - Q (m,m)
c q.r + (1-qm) Qp (m+1,m+l) - Q (m+lmm+l)
= f(m).
- 178 -
The claim is thus proved. Now
A(m,m) = Qw(mm) - Q (m,m)
> qm.r + (1-qm)r - (1-r)tm
= r + (1-r)t
= f(m)/qM '
therefore, A(m,m) - f(m) > f(m) (- - 1), and thus the signs ofqm
A(m+l,m+l) and f(m) are the same. (Note that the cases q M=1 and qV=0 needn't
be treated because the decision then is evidently to pick the last candidate
in the first case and to wait in the second.) Since f(m)<O if and only if
t < , the Lemma is proved. 0
Remark
The Pick-the-Best problem has received much attention in the statistical
literature, but it rarely reflects real search situations. It involves no
utility functions, no prior information, no learning and no search costs.
Its use is therefore limited. In our extension of this problem we allow
some prior information regarding the number of observations and some learning,
but the applications are still limited.
- 179 -
Appendix 2
Reputation in Bargaining
The reputation of a bargainer is often a very important piece of infor-
mation in negotiation. When search is involved, as in the coin market,
the searcher prefers to start the search with a -reputable-
dealer. The dealers themselves, negotiating with each other, plan their
negotiation strategy according to the reputation of their counterpart. A
game-theoretic model, introducing the effect of reputation in bargaining, was
proposed recently by Rosenthal and Landau[1979]. In this. Appendix we propose
a small change in their model which makes it somewhat more realistic and
comment on other possible extensions.
Rosenthal and Landau consider a set of a large number of negotiators
who are playing repeatedly the same 2 x 2 bargaining game shown in Table 4-3.
Y Y
Y 2,2 15,3
Y 3,1 0,0
Table 4-3: The Rosenthal-Landau Game
The strategies Y and Y correspond to [yield] and [do not yield] respectively.
The negotiators do not choose each other, rather they are paired randomly at
every stage. Each negotiator has a reputation index, which is a number
between 1 and n. This index changes according to the negotiator's behavior
in the games which he plays. When he plays Y, his reputation decreases by
one unit, unless his index is 1 (in which case his reputation does not change.)
When he plays Y, his reputation increases by one unit, unless his index is n
(in which case his reputation does not change)1. The objective of each player
This is Variant 2 in Rosenthal-Landau. Their Variant 1 will not be consideredhere.
- 180 -
is to maximize his utility from the infinite stream of payoffs u1 , u2,9...,
where u is the payoff received after the i - th game was played. The pay-
offs are discounted with a factor a, 0<a<1. The objective is therefore
to maximize 00 kk=1
Two -possible decision rules are considered. Each one, if held by all
negotiators, is called a custom. The rules are:
C1 : players yield to opponents of higher reputation, do not yieldto opponents of lower reputation, and play some mixed strategyagainst opponents of the same reputation (the mixed strategypossibly depending on the reputation level);
and
C2 : players yield to opponents of lower reputation, do not yield toopponents of higher reputation, and play some mixed strategyagainst opponents of equal reputation (the mixed strategypossibly depending on the reputation level).
Rosenthal and Landau proved that if a is sufficiently small both C and C2
generate equilibrium i.e. a fixed, mixed strategy employed by any negotiator
whose counterpart's reputation is identical to his. They also proved that
when a is close to zero, C is socially preferable to C2. (The.social loss1 2
is defined as the steady-state probability that two randomly paired players
play their - Y strategies.)
There is one important factor that is not captured by the model. In
real markets players with bad reputations do not find partners to negotiation
as frequently as do reputable players. Negotiators are usually searching for
easy-yielding counterparts, and only when search costs are relatively high do
they choose a "tough" opponent to negotiate with. This factor can be incor-
porated in the model in the following way. AM each stage the probability
Rosenthal and Landau [(1979), pages 2-3].
- 181 -
that a player with reputation i is paired with another player with a
random reputation is tie The probability that he is not paired, therefore
not playing in the current stage, is 1-t.. When a player does not play,
his reward is0 and his reputation does not change. By choosing t1 > t2 >
> tn we make it less probable for a negotiatorwithabad reputation to find
a counterpart.
With this modification in the game, players have an additional tradeoff
to consider. By not yielding when his reputation index is i a player may
gain a reward of 3, but his probability of being paired with a counterpart
at the next stage decreases from t to ti+1. Future rewards are therefore
decreased.
We first show that the rule C is a societal equilibrium if a l+4t
and that rule C2 is a societal equilibrium if a < 1+3t For t = 1 these
are Theorems 4 and 5 of Rosenthal and Landau. We then propose another rule,
denoted by D, and show that it is a societal equilibrium if a <1 .The1 +2t . Th
general form of the steady state probabilities W% of custom D is presented.
In the new formulation we follow the wtations and definitions of Rosen-
thal and Landau (see especially pages 5-7, 15-18). For the justification of
the various formulae the reader should consult their paper.
(a) Let Q bean n xn transition matrix, where q i is the probability of
transition from the i - th reputation state to the j - th.
(b) Let (i,..., w n) be the steady state probabilities that thereputation
of a randomly chosen negotiator is (1,..., n) respectively.
(c) Let r be the expected return from a single trial of the game for a
player in state i before he is informed that he plays in the current stage
and before he studies his current opponent's state.
(d) Let v be the expected (discounted) value of the infinite stream for a
- 182 -
player who begins the game in state i.
(e) Let p be the probability that a negotiator with reputation i plays
his yielding strategy Y.
(f) Let t be the probability that a negotiator with reputation i does
have a counterpart in any single stage (ti > 0).
The equilibrium equations are:
r= 1Q , (4-61)
and
v = r + aQv, (4-62)
where nSir = 1 ,(4-63)
and
0 < n < 1 =1,.,n.
Additionally, we require1
3p + avi+11 + p + av i =" 1,... n, (4-6'4)
where v0 = V1 3 vn+1 =vn
tIiLet 1 = n be the probability that a player who is currently
Etkkk=l
playing is paired with a player with a reputation index j. The transition
matrix Q under C is given by:
ql -1 - t1(1-p) 1 q192 -t(1-P
q,- t pijTi + tI S iqit1i+ -ti qii1
= ti(1-Pi)rI +
t iEJ (i = 2,..., n-l)j
lSee Rosenthal-Landau [(1979), page 16].
nn- t n n n
- 183 -
,n tnpn7rn
q1. = 0, otherwise.
The proof of the condition on a follows the proof of Theorem 4 in
Rosenthal and Landau [1979]. For v' = r + aQv we have to show IV, -
iS 1/a for all i, 1 < i < n.
I - =11
(We denote v' =v', vI+=V'0 1 n+l n
+0 Qi 1 - ri+1 - +
= t 1 ( S Wijj1i-l
2p+ j + ici-(4pl
+ a(tp 1 rr 1 vi-2 + t i-
+ t _(1-p )w v + t
-ti+i +.3S +J; . ~j J<i+l
S v - +j>i-l1 -
(1-t _ )v1
S Trv)j<i-l
+ r +1 (p +12
a~ti+ +7i+ vi + t i+ Swjv + (+-i+l)vi+1j>i+l
+ ti+ 1 (1-Pi+ 1 )wi+lvi+ 2 + ti+lJ<i+1iVi+2
j <l(t + t 1 v1 -2'~ 1+1 i+l v )wr +j <i+1
+ t + at _1v - 3t i+ -atvi+12)ij<i-+
- 184 -
+ Ti- 1 (t 1 (4p _ - 2p ) V + t p v1 + at 1(1-P iy)v
i+1 -1ati+1vi+2)
+ (t + at v - -2 3 t -at )i+1+2)
1( i- i-i 1 v-2 1+1 1+1 -1+
+ T ti1+ ti.1v 1 2 t (4 - 2i~l 1-2 1+1 i+l l
. at -a~ vi Yt IPil)v1+2)- a1+11+1 - 1+1 '91+1 ) v )
+ a(1-t_11)v 1 - a (1-ti+ v 1+1
< t _.1 i-i 1+1 - - Tr11) + Trii(5ti1) + i (4t 1)
+ i+ (3t ) + 1-t
< 1 + 4t -1
< 1 + 4t
1 1if a <1+4t
The proof for C2 is similar.
Our results' are not significantly different from those of Rosenthal and
Landau. After trying various other extensions and failing to find other inter-
esting customs, we tend to agree with their conclusion that the model is too
structured and too simple to yield more insight. We do believe that the game
can be extended to more elaborate models of reputation with larger strategy spaces.
- 185 -
CHAPTER 5
MODELLING THE NEGOTIATION AS A PROBLEM OF OPTIMAL CONTROL
5.1 Introduction
The negotiation process has been generally viewed in the game-theory
literature as pre-play communications of a cooperative game. Alternatively,
it is sometimes viewed as a game which in principle can be solved in the sense
of equilibrium strategies, but only if it is severely restricted by insti-
tutional rules. Both approaches are of little help to the negotiator at work
and to the observer who tries to predict, or at least to interpret, the final
outcome of a particular negotiation. The first approach is not applicable in
what is probably the most common type of negotiation: - the one-shot, fixed
threat distributive bargaining. The second approach, even when applicable,
generally ignores the sequential aspect of negotiations, thereby leaving no
role for learning, persuasion, bluffs, and other activities which make ne-
gotiations interesting and complicate social interactions. This is not to
say that game theory does not help our understanding of the negotiation process.
Its effect has been significant, but not always fully comprehended. In many
cases it may be difficult to see clearly the "correct" insight to be gained
from the theory. The frequent misuse of game-theory concepts by students of
conflict situations points to the difficulty in properly interpreting negotia-
tion as a classic game.
An alternative theory of negotiations, emphasizing the sequential nature
and the adjustment of expectations, while playing down the importance of the
strategic moves of the players, is a theory of expectations which began with
Cross's learning model [1965], [1968]1, and was later corrected, extended,
'See Section 2.2.6 for a discussion of the Cross model.
- 186 -
and formalized by Coddington [1968]. It was applied, as a control model, by
Rao and Shakun [1974]. In addition to his important corrections to Cross's
work, Coddington proposed to view the negotiation as a closed feedback loop
with two steering systems. A block diagram of this loop is shown in Fig. 5-1.
PARTY 1 PARTY 2
ACT ACT
STRATEGY MODIFICATION STRATEGY MODIFICATION
EXPECTATIONS ADJUSTMENT EXPECTATIONS ADJUSTMENT
INFORMATION PROCESSING INFORMATION PROCESSING
Fig. 5-1: A Negotiation as a Closed Loop
The process starts when one party, say party 1, executes some act which
is defined as the beginning of the negotiation, perhaps simply making a de-
mand. The second party responds with some act which is permissible by the
institutional rules.2 From now on the negotiation consists of a sequence of
This diagram and the description following it are our interpretation, withsome modifications, of Coddington's model.
2Our opinion here is that negotiation continues even if party 2 does not agreeto negotiate at all. It is enough that one party executes the initial actand perhaps additional permissible acts, and that these acts are communicatedto party 2. Otherwise, imagine a case in which, after many proposals by par-ty 1, party 2 suddenly agrees to negotiate and accepts the last proposal of 1.Is this a one-stage negotiation? Obviously not. We have to accept the firstact of party 1 as sufficient for opening the negotiation, even though we donot know how party 2 will respond, if at all.
- 187 -
acts, carried out by both parties. (It may be difficult to decide what consti-
tutes an act and what is only a persuasion measure, but under specific insti-
tutional assumptions it is possible to define permissible acts, e.g., a price
concession, a termination threat, an explicit commitment, etc.)
Party 1 acts on the basis of some information, with given expectations
of future events and a strategy to meet them. His act is observed by party 2
as new information to be evaluated. After processing this information, party
2 reviews his expectations, modifies his strategy, and then acts. The pro-
cess goes on until some acts which signify the achievement of an agreement
take place. Note that if an agreement is not achieved it may still be diffi-
cult to know when the negotiations really break off.
This scheme emphasizes the feedback between the acts of both parties.
Additional influencing factors, not yet mentioned, may be added to the model;
one such factor is the internal forces, acting within one party. These forces
are created and developed by acts of both parties and influence the main loop
by serving as sources of new information, and by changing the set of feasible
strategies (generally by adding constraints). External forces developing out-
side the main loop are another factor. These and more could be added to the
model as secondary feedback loops, making the model more realistic but also
much more complex.
Time has an important role in the model. According to Coddington, time
is really the driving force of the process. It serves as an input in the
stage of information processing and in the stage of expectations adjustment.
It is also a component in the parties' strategies. This is in direct opposi-
1This act may be "doing nothing", so that two consecutive acts of one party
do not contradict our description. One could also define a time interval Asuch that any two or more acts carried out by one party within A units, areconsidered one act.
- 188 -
tion to the game theoretical models in which time is abstracted out.
Based on the conceptualization of negotiation as a closed-loop process, var-
ious types of models could be developed -- descriptive, normative, asymmetric
and third-party models. A normative model, for example, would attempt to look
for equilibrium strategies which bring the system to a stable condition (i.e.,
to an achievement of an agreement) as fast as possible. An asymmetric, one-
side descriptive model would emphasize the steering actions of one party,
assuming some constraints on the strategies of the parties.
Rao and Shakun [19741 attempted to apply some of these ideas. They con-
sidered their model to be a normative one, but according to our definitions
it is really an asymmetric model. In their model a dynamic programming algor-
ithm is used for the first time to compute optimal negotiation decisions. An-
other novelty is the classification of negotiators into "types". The opti-
mizing decision depends onthe negotiator's type. The effects of time are not
introduced explicitly in the model and this is the main reason for several
restrictive assumptions.
We begin the next section by describing the model of Rao and Shakun,
then comment on their assumptions and results.. Next we propose two variants of
this model which make it work under more realistic conditions. The detailed
mathematical formulations are relegated to Appendix 1 of this
chapter.
The main contribution of this chapter is found in section 5.3. There
we propose a framework for an asymmetric adaptive control model in which du-
ation-related costs are introduced, and the possibility of sudden termination
is included. In the formulation we assume imperfect state information, but
the problem is essentially equivalent to a control problem for which the
'Note that there is no conceptual difficulties that prohibit the introductionof time in game models.
- 189 -
dynamic programming algorithm can be applied. We then examine the crucial
problem of sequential updating of probabilities. We assume that the counterpart's
responses depend on a set of two unobserved states -- namely the reservation
price of the counterpart and his type as a negotiator. The approach we
propose can be easily extended (although the formulation may be cumbersome)
to a larger set of unobserved acts.
5.2 The Model of Rao and Shakun
5.2.1 A Description
Consider two parties negotiating the division of the prospective profits
from a joint project. Their interests are in complete opposition since each
wants a larger share of the expected pie. If they do not reach an agreement,
then the project is cancelled and there is no pie to be divided. The parties
do not know their opponents' utility functions. They are not even sure that
a zone of agreement exists. Suppose also that the parties are ready to abide
by a set of institutional rules (to be discussed later) which restrict their
strategic possibilities as negotiators and turn the negotiation into well
defined game of incomplete information (in extensive form). The problem is
to find optimal negotiation strategies for the game.
Without loss of generality we can assume that the parties negotiate over
the segment [0,1]. An agreement at a (Ot<1) means that one party, denoted by
H, receives 100aZ of the profits while the other, denoted by L, receives the
rest. Party H (high) prefers the agreement to be close to 1. He can also
specify a number . z. -C [0,11 such that he prefers breaking off to any agreement
a, c>ZH Party L (low) preferes the agreement to be close to 0. He can
specify a number mL' 3L (0,1, such that he prefers breaking off to any agree-
- 190 -
ment a, a > zL. The parameters z L and z4 are called "reservation prices".
The zone of agreement is [zr, z] if zL<zH and is the empty set, otherwise.
In each session one party is allowed to propose a value of a. When the two
last offers are identical, an agreement is achieved. The parties are not
allowed to retreat, and their concessions (aconcession is -defined as. the
absolute value of the difference between the last two price offers of one
party) must be a multiple of some minimal concession unit.. If two consecu-
tive concessions are zero, then the game terminates and the parties receive
the conflict payoffs. With all these rules, and with the addition of some
behavioristic assumptions, the game can be solved for optimal strategies.
Let xt(i=L or H) denote the proposal of i when there are still t
sessions remaining to the end of the negotiation. Let c = xt+2 - x
(i-L or H) be the concession of i in this stage. The model assumptions are:
Institutional Assumptions:
Il. There are 2n negotiation sessions.
12. Only submittance of bids is allowed. In the first session of the
2nnegotiation, L submits a bid x . In the second session H submits
2n-1The parties maintain a pattern of alternating bids.
2n+2 2n+113. Prenegotiation demands are xL = 0 and -M 1.
S t> t+2 s s+2I4. No retreats are allowed;i.e., x.H -x and L - x for every s,t
satisfying 1 < s < 2n+2, 1 < t < 2n+1.
I5. The concessions amust satisfy 4 -M26 , where m C{0,1,2...}, where
6 >0 is the minimal unit of concession. It is assumed that 6>1/2n..
16. The negotiation terminates sucessfully if xt -X for some t,
such that x S+1 for all sw>t (i#J.). There is a break-off in the
t+1negotiation after two consecutive nonconcession, i.e., if c71
C, 0 and xt+l t0xfor some t, 2n > t >l1.
- 191 -
Payoff Assumptions:
Pl. The conflict payoffs, denoted WL and wH, are constant throughout
the negotiation.
P2. Both sides have von Neumann-Morgenstern utility functions defined
on [0,11, where @L is decreasing with x and $H is increasing with x.
Behavioristic Assumptions:
Bl. The data on which party i bases his decision at stage t are {xt+2
t+l t+lc , x } . These are the only state variables.
B2. Each party can be one of the three negotiator types discussed in
5.2.2.
B3. The parties can assess the subjective probabilities p 1 for all
m, k, 1 and t where
t PThe opponent The opponent conceded med atPmkl PROB concedes 1.6 stage t+1 and the party con-
at stage t-1I ceded k6 at stage t
tB4. The assessments P of each party are supplied by him at every stage,
but the parties are ready to assume at every stage t that p -
t-2pmkl =
B5. Complete information is not required unless the neogtiatior is of
type-3.
Define now the reservation price zL., zH and the zone of agreement D. Let zL
w max[O, inf (xix e[0,1 L(x)>wL)land z - min[l, inf (K Ixe[0,1], H(x)>wHl
The zone of agreement D is the segment [z . ELI if zL>zf, and the empty set,
otherwise. An agreement, if achieved, must clearly fall in D. These notations
will be used in 5.2.2.
- 192 -
Rao and Shakun define three types of negotiators (for the methematical
formulation see Appendix 1 of this chapter). He is a type-1 negotiator if
his objective at each stage is to choose a policy which maximizes his total
expected utility when the negotiation terminates. He is a type-2 negotiator
if his objective at each stage is to maximize the minimum of the total utility
that he may receive at time of termination. He is a type-3 negotiator if
he computes at each stage two quantities called risk limits, and bases his
decision on a comparison of these quantities. If his risk limit, which he
can compute accurately, is larger than the risk limit of his adversary, which
he can only assess subjectively, then he does not concede. If his risk
limit is smaller, then he makes the minimal concession.
A negotiator may change his type during the negotiation. The behavior
sequence could be described by a 2n-tuple of the form (a1 ,b ,a2 ,b2 ,...) where
a is L's type in the first session, a2 his type in the third session, b1 is
H's type in the second session, etc.
The decision problem of a negotiator i who is a type-1 or a type-2 nego-
tiator can be formulated as a stochastic terminal control problem, and be
solved by dynamic programming. The type-3 negotiator does not need any
algorithm. He simply has to assess the risk limits in each stage. Note that
the negotiator type of the opponent is used by each party in the assessment of
probabilities.
Rao and Shakun work out a small numerical example and check the sensi-
tivity of the results to various assumptions. The final agreements, the
probability of a break-off and the expected number of sessions are all de-
pendent upon the negotiator's types. In this particular example the nego-
- 193 -
tiators are not confronted with the difficult problem of updating the prob-
ability assessments. The probabilities are assumed to. be constant through-
out the process.
5.2.2 Comments
The model is based on institutional, behavioristic, and payoff assump-
tions which are similar to those in the models ofZeuthen, Hicks and Bishop.
These assumptions pose several problems that need to be addressed. The three
main problems are: Termination Rules, Probability Generation, and Model Clas-
sification.
Termination Rules
According to the institutional assumption 16, the negotiation terminates
when there are two consecutive non-concessions (one from each party). This
means that a party may knowingly break-off the negotiation by refusing to con-
cede after a similar move by his counterpart. This restriction does not ne-
cessarily hold true for real life situations; in actual negotiations the num-
ber of consecutive non-concessions may be larger than 2 and even then nego-
tiations may continue. The model's arbitrary termination rules force
some undesirable conclusions. The point which we want to stress here is not
that the assumptions are oversimplifying but that the outcome of the model de-
viates considerably both from real life behavior and from the outcome that is
implied from the assumptions. The first problem is that each party knows
with certainty that if he concedes in stage t he would still have stage t-2
to make another move. This certainty may not exist in real negotiations.
Also, by not conceding a party may bring about the undesired ternimation of
the negotiation without the opportunity to make a last minute concession, an
opportunity that is usually present in real life.
A simple example may illustrate the last point. A potential customer and
- 194 -
a shop-keeper haggle over the price of a product. After several consecutive
non-concessions, the customer walks out of the store thereby signalling to
the shop-keeper: "I am serious, this was my last offer". Under the model the
shop-keeper is chained to his counter, but in real life he may decide to rush
out and make a new concession. We can sum up by saying that even if experience
in real negotiations shows that two consecutive non-concessions often result
in a break-off, it is still true that parties reserve for themselves the op-
tion to make a large concession in the last round.
It is important to understand why a Rao-Shakun negotiator chooses to
break off negotiations. Since the modtl assumes implicitly that both parties
have initially a genuine interest in achieving an agreement, and since there
are no costs or utility changes associated with the duration or the calendar
date, there may be only two reasons for breaking off: (1) one party, say H,
can offer no more agreements which he prefers to a break-off; i.e., his last
proposal 4 satisfies 4 - 6< z ; (2) the development of the negotiation
process itself changed the preference structure, thus making a break-off more
1desirable for one party. We now examine the implications of these two pos-
sibilities.
Suppose that both sides know that reason (1) is the only reason for
termination for both. The whole process necessarily looks like this. L starts
by not conceding. He lets H concede until 4 < zL for the first time. Now
he has to assess z, an assessment that depends on prior information and on
the 'past sequence of H's bids. Once he assesses a probability distribution
1This may happen if the other party behaves in a "non-cooperative" or "stub-
born" way which raises doubts about the relationship between the parties af-ter the negotiation ends. 'It may also happen when one party is making largeconcessions, thus causing the other to re-evaluate his reservation price z.
- 195 -
for zH on [0,x4) he must choose when to concede for the first time. One rule
of thumb which he could use is: Don't concede at a stage t if ic i > 6. It
is clear that once L conceded, H refuses to concede anymore because he is cer-
tain that his last bid is below zL. The final agreement is necessarily the
last bid of H. Both parties are therefore trapped by the institutional rules
of the model, and the result depends almost exclusively on L's prior assess-
ment of zHB Viewed as a game in extensive form, the optimal strategy can be
computed without using the algorithm proposed: by Shakun and Rao.
Suppose now that L suspects that H may terminate the negotiation because
of (1) and (2). It then may be important for him to let H know that he, L
may also terminate because of (1) and (2). This is so because H may try to
trap L after L's first concession if he thought that L had conceded because
of (1); knowing this, L is led to trap H, thereby increasing the probability
of a break-off! It may be the case that both parties communicate to each
other the fact that both (1) and (2) hold. Note that it is then clear that
t+2 t+1 t+1the equations of optimality do not hold if xL , c , x are the only state
variables. The whole sequence of bids up to stage t must serve as the state
of the system in stage t.
Probability Generation
Another problem, commented on by Rao and Shakun, is the process of gen-
erating the probability distributions at every stage. The quality of any
tadaptive optimization process depends largely on the quality of the p, the
assessment of which was left out of the model. The recommendations of the
model cannot be judged without having some information on the process that
generates the probability assessments.
Model Classification
Translated into our conceptual framework the model is evidently an asym-
- 196 -
metric one. A party is advised to behave as a type-i or type-2 negotiator,
or to switch from one type to another during negotiations. He may simulate
his opponent's behavior before starting negotiations, as Rao and Shakun do in
their example. It is hoped, but not verified, that even if the quality of
the probability assessments is not high, the strategy is robust enough to
result in a good performance.
5.2.3 Two Modifications of the Model
In this section we propose two variants of the model of Rao and Shakun.
In Variant 1 we introduce new termination rules. In Variant 2 we introduce
time effects. Our objective in proposing these two variants is to correct
the problems related to the termination rules discussed in 5.2.2 without changing
the model of Rao and Shakun significantly.
Variant 1.
The assumptions of this asymmetric model are slightly different from those
of the model of Shakun and Rao. The number of negotiation sessions is now not
known in advance. The parties are allowed to threaten a break-off and to
terminate the negotiation and not only to submit bids. Termination occurs if
the parties submit identical bids or if one party threatens a break-off and the
other party does not respond with a yielding concession. Two consecutive non-
concessions do not terminate the negotiation. Some of the assumptions are
similar to the ones of Shakun and Rao. They are:
12. Only submittance of bids, threatening a break-off and breaking off
are allowed.
2N+2 2N+lI3. Pre-negotiation demands are xL -0, x -1, where N=miu{NLN H1
(See explanation for NLNiin B6.)
I5. Concessions are in units of 6.
Pl. Constant conflict pay-offs wL and wE.
- 197 -
P2. Both sides have von Neumann-Morgenstern utility functions (x).
B3. The parties are ready to compute for all m, k, 1 and t.
assmest t-2B4. In stage t each party assumepkl = p.kl =
The new assumptions are:
t t-117. Termination occurs if: (1) xH =xL for some t satisfying l<t <
2N+2 or (2) party i threatens a break-off in stage t; party j .responds
with a partial or no concession; party i must then declare a break-
off in t-2.
B6. Each party decides in advance on a number N such that if there is
no agreement after 2N -2 sessions he threatens a break-off.
B7. Each party constructs (before the negotiation starts) a set S of
ordered bids.
S -{(Ni Ni-1 2Ni-2,...,9 x )} i =L or H,
such that if (x2Ni,..., x) eS he threatens a break-off Instage n-1.
B8. The parties do not know each other's Si and N1 , but they know that
both S are not empty, and that both Ni are finite.
This model insures against trapping because two consecutive non-concessions
do not terminate the negotia;ion. Both sides also know that both (1) and (2)
of page 194 are reasons for termination. The impending threat of break-off
limits obstinate non-conceding behaviors. The problem of probability generation
still remains but the opponent may be now classified according to his S, Ni
and his strategy (Rao-Shakun considered a classification by the strategy alone).
A Bayesian scheme on S and N is theoretically possible. However, our remark
about the proper state of the system is still valid. The whole past history
should serve as a system statew This adds considerable complexity to the
formulation.
- 198 -
Variant 2.
In this model we-introduce -the effects of time explicitly. Its assump-
tions include all the assumptions of Variant 1 except the payoff assumptions.
We now assume equal time intervals between the permissible acts. We also
introduce the effects of time into the utility functions and we allow for a
sudden termination possibility. The payoff assumptions are:
Pl. The conflict pay-of fs w depend on time:
wt:i{l..., 2NI+2}+R , i = L, H,
where w (n) is the pay-off of i if a break-off occurs in stage n.
P2. The utility functions @j depend on time:
* : {(x,n)lx E[0,1, 1<n<2NIl}+R, i = L, H,i
where 4i(xn) is the pay-off of i if x is agreed upon at stage n.
We add the following assumptions:
18. Sudden termination possibility. There is a random variable T which
corresponds to the remaining number of sessions (i.e., the duration)
of the negotiation when some external events and the deliberations.
t tIf t =T, and x L then termination occurs at stage t.
B9. Both parties have identical assessments of T. These assessments are
updated during the negotiations.
We can now formulate the sequential decision problem of L, with the whole
past history serving as a system state. This is a control problem with timeL1
lags and with probabilistic responses which are not independent. (The formu-
lation is similar to the one of the model of Shakun and Rao, and is therefore
ommitted.)
5.2.4 Conclusions
The technical difficulties resulting from the termination rules in the
model of Rao and Shakun are solved by the introduction of time effects and
1See Bertsekas [1976] for a detailed treatment of these problems.
- 199 -
more realistic termination procedures. With these modifications, the dy-
namic programming algorithm seems (theoretically at least) a useful tool
in negotiations modelling. Two major problems still remain: the computa-
tional difficulties in the execution of the algorithm, and the generation
of probabilities. The essence of the negotiation process is really the trans-
lation of accruing information into probabilistic assessments of the behavior
of the other party, his type, his expectations, his reservation price, etc.
Without an adequate process for probability generation the applicability
of the model remains questionable.
Our belief is that a control model may make sense only when there is
some ground to assume that the counterpart uses some "rigid" strategy, i.e.
that he adheres to a particular strategy which is not optimal by any defini-
tion but which he prefers to use because of perhaps psychological reasons,
his past experience with this strategy, its simplicity, etc. In this case
the controller can optimize against his opponent's strategy. He can devise
probability updating schemes if the strategy is an unknown element of a set
of strategies, with a generic parameter that has to be assessed, The model
presented in the next section is such amodel. Our argument there is that
when time effects are very severe and when the negotiator is involved in many
negotiations in a short period of time, he may wish to use a simple strategy
which is easy to compute and use. The reader is referred to chapter 4 in
which the arguments for the characteristic concession patterns of the dealer
in the Israeli coin market were made. These dealers, very experienced nego-
tiators, do choose simple strategies against which a control model might be
useful.
- 200 -
5.3 An Asymmetric Adaptive Control Model of Negotiations
5.3.1. Introduction
In this section we follow Coddington's suggestion and formulate the nego-
tiation-as anadaptive control problem. The controller is one of the parties,
to be called our party, and the analysis is done from his point of view. It
is his objective function that is maximized. The other party, to be called
here the counterpart, produces the responses which are the "random" distur-
bances against which our party tries to optimize. These responses are of course
the results of some decision model used by the counterpart. Simplifying
assumptions concerning these responses are needed in order to prevent the model
from turning into an extremely complicated game of strategy. These assumptions
are, we believe, weak enough to make the model a useful decision aid.
As in the model of Rao and Shakun, the dynamic programming algorithm can
be used to solve the control problem, although- the "curse of dimensionality"
may prove to be insurmountable. In order to facilitate the computations, we
can assume that the number of sessions and the number of possible price pro-
posals are small. Alternatively we can apply some sub-optimal method of
control. Three such methods are: (i) Open Loop Policy, according to which our
party does not utilize anyaccumulated information; (ii) Open Loop Feedback
Policy, according to which the controller observes information but while
making his decision he assumes that no future information will be available;
and (iii) M-Measurement Feedback Policy, according to which learning from M
future observations is assumed when the decision is made.1 We will not,
1For a review of methods and applications of adaptive control, see Pekelmanand Rausser [1978].
- 201 -
however, deal here with problems of computation and approximation. Our main
interest is in the formulation of the problem and in its justification. Of
special interest is the problem of updating the probabilities during the
negotiation, a crucial problem on which we have already commented. The main
idea of this section is that the opponent's negotiation strategy can some-
times be assumed to depend on two factors: His reservation price and his
type as a negotiator. The assessment of these two factors can be updated
during the negotiation and be used by the controller.
In section 5.3.3 we present some experimental data which indicates that
this approach may indeed be useful. Still, its limitations should not be
ignored. The model is based on strong behavioral assumptions (summarized
in 5.3.3) and it requires tight prior distributions on at least one parameter.
It is also assumed in the model that the number of price proposals made
during the negotiations is high, otherwise the data is not powerful enough
for purposes of inference and decision. In the - S.A. this assumption is
rarely true in business negotiations. It is, however, true in labor-manage-
ment negotiations. When-only few offers are expected the model loses much
of its appeal.
- 202 -
5.3.2 Formulation
(1) The basic problem
Consider the following class of problems.
in discrete time according to the equation
yk= fk(k, Ck, vk),
1 A dynamic system evolves
k = 01.,-,
where yk denotes the state of the system at stage k, ck is the control
variable at stage k and vk is the disturbance at stage k. We assume
the existence of state spaces k control spaces C and response
spaces Vk such that yk k' ck CCk, ' k CVk for k- 0l1...n-l.
The state yk is a vector
y k (gk' Ok), k =0,1,...n-1,
where gk is itself a vector of observed quantities, k is a vector of
unobserved ones.
The disturbance vk is a vector
vk - (dk' ek), K - ,,.. -,
(5-1)
(5-2)
(5-3)
1In this section we follow the approach and some of the notation ofBertsekas [1976].
- 203 -
where dk and ek are independent of each other for all k. However ek+l
depends on {ek ki , (k - 0,l...,n-l),while dk depends on ykik} and is
independent of {dkl...d}.
The utility function is given by
U(yc,v) = U(k*, yk*, ck*, vk*)for 1< k* < n,(5-4)
where k* is the first k such that at least one of the following conditions
holds1: (1) dk* = ck* and ek1 = 0, (2 ) dk*t# ck* and ek* = 1, (3) ck* = Fs
(4) vk* = F2 . The symbols F and F2 satisfy F1 E Ck, (F2 ek) e Vk for all k.
The problem is to find a set of optimal controls c0 ,...,cck* such that
EU(y,c,v) is maximized.
(2) Interpretation
This formulation is general, but we are really interested in its inter-
pretation as a negotiation model. After describing the meaning of each
element in the basic problem, we will continue to use the negotiation inter-
pretation throughout this section.
1 2 3The observed quantities kM - k , g2 , g ) consist of the last price
demand of our party (g1), his last act which is not a demand (g, 2) and the
last act (or demand) of the counterpart (g3).
The unobserved quanties 0k - (zk k) consist of the reservation price, ZkP
of the counterpart at stage k and his type, s-, as a negotiator at stage k.
The control ck is the act of our party at stage k.
The disturbance vk - (dk'ek) is composed of the act of the counterpart (dk
'These conditions will be explained later.
- 204 -
and of a zero - one variable indicating the feasibility of further negotiations
when stage k + 1 starts (e ).
The number k* indicates the stage at which negotiation is terminated.
This happens .when an agreement is achieved, in which case the last demands of
the parties are identical (case (1) in the definition of k*), when a sudden
termination occurs (case (2)), when the negotiation is broken off by our party
(case (3)) or by the counterpart (case (4)).
As formulated here, our basic problem falls into the class of control
problems with imperfect state information. The dynamic programming algorithm
can be applied to solve our problem provided that we specify the following
conditional probabilities: (1) Pz(zk+1 Iyk, ck, vk, k), (2) Ps(sk+ yk, ck
vk, k), (3) Pd k y k , k), (4) Pe (ek 5k-, k). The first three condi-
tional probabilities are concerned with the effect of our party's actions on
the counterpart's reservation price, type and action. The last is concerned
with the sudden termination possibility.
Note that ours is a finite dimensional problem. If we formalized it as
an infinite dimensional one it would become complicated. To justify the
finite dimensionality we could make the same ad-hoc assumption that was made
in 5.2, namely that there exists an N such that both sides prefer a break-off
in stage N to further negotiating. This is however a restrictive assumption,
justified only when there is a discontinuous change in the utility function
at stage N. We prefer to assume the existence of an N such that eN = 1 with
probability 1, i.e. sudden termination happens with certainty if nego-
tiations are lengthy enough.
(3) The Control Spaces
We now examine in more detail the elements of our basic problem. The
- 205 -
Control Space Ck is the union of a set of permissible demands Dk, a set of
binding termination threats Hk, a set of permissible postponements Qk and
a one element set $FJ called the break-off set. According to this definition
a permissible control in state k is one and only one of the following: (1) a
price demand, (2) a binding termination threat, (3) a postponement, or (4) a
break-off. If we allowed our party to make demands combined with threats or
with postponements we would have to look at the Cartesian product of the sub-
sets of Ck and not at their union. This would complicate notation without
contributing much to the reality of the model.
The sets of demands Dk are finite sets satisfying DkC [0, 1], D Ck+1- Dk
for k = 0,1,...n-1. We assume that our party's most preferred value is 1
and that this is also the maximum he can demand. The set D is therefore the
segment [0, 1]. In later stages our party lowers his demands so that
Dk = [0, Ck -l] where ck -1 is the last control which is a demand, or formally:0 0
k, is the largest i, i < k, such that cie D .Without much loss of0 _i-V-
- 1 2 -l0- 1generality we can assume that D0 n0 n o'' n0 , 1} where
9 0 {0 n90an0 91 h n0sthmi-
mal concession unit, and that Dk are subsets of this finite set.
The set of binding termination threats Hk= {hk+l,...,hn-l},(k=l,...,n-2),
contains N-k-l threats, where the threat h corresponds to the following bind-
ing commtent announced by our party: "If an agreement is not achieved
until stage J, I will terminate the negotiations..at stage j+1 (i.e. cj+ 1= F).1"
Note that the particular threat h.+, when announced at stage k, is to ter-
minate the negotiations if the counterpart does not yield immediately.
Threats are permissible only when the can be proved to be binding. Other-
wise (and this is the common situation) they are not permissible.
1This condition corresponds to our regular institutional assumption of no
retreat.
- 206 -
The sets of postponements Qk {qk+29' .qn-1} (0 < k < n-3), contain
n-k-4 binding commitments. A typical commitment q is of the following form:
"I postpone negotiations until stage j", by which our party means that if the
counterpart does not respond by terminating the negotiations nor by making a
yielding concession, then the negotiations stop until stage j. During a post-
ponement the observed states g (i = j,...,k) do not change but the unobserved
state 03 may be different from 0k. As to the feasibility indicator e, it may
change from 0 to 1 due to the realization of the sudden termination possibility
during the postponement. Note that the postponement proposal is a binding
threat (but not a termination threat). If the counterpart does not respond to
a postponement q with a yielding concession, our party is not available for
negotiations until stage j. At this stage the disturbance v (the counter-
part's response) is observed. This response may be any element of the response
space V1 , including F2 , i.e. termination.
(4) The Response Space
The Response Space Vk is the Cartesian product of EkU{F(F2} and the set
{0, 1}. The set Ek consists of the counterpart's demands, while' {F2} is the
termination set. Since the counterpart prefers the agreement to be close to
0, and this is his maximum demand, the sets Fk are of the form [ vk 0_I1
where k0 is the last i such that ci_ is a demand. (These definitions are
analogous to the ones in (3)).
The feasibility indicator ek satisfies eke {0,1) for all k, and if
e lfor some j then elI for all m > j its value is determiSed by
1 Postponement is given a role in this model, not only because of its im-portance in some markets, as discussed in Chapter 4, but also in order toshow how one can introduce activities other than concessions into the model.It is implicitly assumed here that proposing a postponement may. be a part ofan optimal control.
- 207 -
a random variable T (Sudden Termination) which takes values on [0,...,n]
with a probability distribution which is common knowledge.
(5) The Unobserved States
The vectors 0 k are two dimensional vectors ak C Z x S where Z C [0,1] is
the set of possible reservation prices and where S is a set of negotiator types.
By saying that 0k = (z,s) we mean that the counterpart in stage k is a nego--
tiator of type s whose reservation price is z. The reservation prices and
the negotiator types are key elements of our model and will be discussed here
in more detail.
The reservation price zk in stage k is defined as the largest price which
is acceptable to the counterpart if he is faced with a "take it or leave it"
termination threat. This price may change from stage to stage, mainly as a
function of time. Note that when offered his reservation price the counterpart
may refuse to accept it, since he may think that the final price will be
better than his current reservation price. The reservation price zk affects
the counterpart's response through the probability distribution Pd mentioned
in (2). Its assessment by our party is given by the probability distribu-
tion P also mentioned in (2). These two probabilities will be discussedz
later.
The concept of negotiator type was introduced in chapter 4 of this thesis,
when the characteristic concession patterns were discussed. The classifi-
cation of negotiators into types that we propose in this section seems at
first totally different, but it is in fact the stochastic variant of the
deterministic concession patterns. This point will be explained in the remark
In. the end- of this section, -after we define and discussthe types of negotiators.
- 208 -
In each stage of the negotiation the counterpart faces three alternatives:
to make a yielding concession, to concede nothing or to make some concession.
The probability that he chooses each of these alternatives should be assessed
by our party. But given that he chooses to make some concession, we wish to
assume that he gives in a fixed portion of the maximum amount that he can con-
cede. Formally, let his last price offer in some stage k-1 be dk-1, and sup-
pose that our party's offer in stage k is ck. Suppose the counterpart is a
seller, and denote the maximum of ck and the counterpart's reservation price zk
by mk. The counterpart's offer in stage k, dk, is obviously a point in the
interval [mk-dk l]. Let F(*) be some probability distribution defined on (0,1).
The counterpart is called "an F negotiator", if the distribution of
A d , his "normalized concession" is F(e). To assume that a negotiatork mkk-l
remains of type F means that his normalized concessions A are independent,
identically distributed random variables.
Note that by assuming that the counterpart is an F negotiator for some
probability distribution F and parameters vll and p2 , we implicitly assume that
he employs a strategy which is independent of our party's normalized conces-
sions. This does not mean that his concessions are independent of the respec-
tive concessions of our party, but it means that his strategy is rather rigid.
This assumption will be examined in 5.3.3.
The number of price proposals in typical negotiations is almost never
large enough to make good use of prior-posterior analysis, except when F is a
narrow distribution, with small variance. It is our conjecture that in some
situations U2 is indeed very small. If this is so, we can have a tight assess-
ment of p1 (by doing prior-posterior calculations) after our counterpart makes
only a few concessions. Note that the assessments of
- 209 -
depend on the assessment of the reservation prices zkP and vice versa. The
assumption regarding the reservation prices is again that they change slowly
if at all. It is probably safe to assume that zk is fixed at least during the
initial stage of the negotiations. At this stage a maximum-liklihood estimate
of PI, P2 and z could be computed.
Do negotiatiors really concede in the way described here? In section
5.3.3 we bring some experimental data which indicates that in some circumstances
this assumption may be reasonable. Our conjecture is that it is especially so
when time-related costs are high. The negotiators are then pressed to decide
quickly of their concession, and they may resort to the easily computed con-
cession behavior described in this section.
Remark
The concession mechanism proposed here is a generalization of the char-
acteristic concession patterns discussed in Chapter 4. Suppose that our party,
after making his initial offer c1 , decides to hold to this offer and to concede
nothing during the negotiations. Suppose also that p2 - 0, i.e. the counter-
parts normalized concessions A are all identical. If the time intervals be-
tween the counterpart's proposals are fixed, it can be shown that his conces-
sion pattern is exponential. If P2 0 0 but is fairly small, then the realized
sequence of concessions is close to this exponential curve. If our party de-
cides to concede in- the same way (making constant normalized concessions) then
the functional form of the concession curve turns to be flatter than exponential.
(6) Updating Probabilities
The probability Pd(d |yk, ck, k) of the counterpart's action in stage k
1The exponential concession pattern is defined in 4.4.2.
- 210 -
is updated in part subjectively and in part according to a Bayesian scheme.
The probabilities of a zero or a yielding concession as well as the proba-
bility of termination should be assessed subjectively. It is reasonable to
assume, for example, that the probability of a yielding concession increases
as the gap between the parties decreases and that the probability of no con-
cession decreases as the duration increases. The probability of no concession
depends also on our party's responses. The probability distribution of the
counterpart's concession. (given that it is not a zero concession nor a yield-
ing one) can becomputed from F(-)and depends cnzkPsk and ck. In the finalstages
of the negotiation, when it is clear that zk > ck, the mean of this distri-
bution is 0 *[ccdk-lJ and the variance is V' [ckdkli2
When it is not clear that zk > ck,
the mean of the concessions is distributed like Pl-[Min(ck, Zk)-dk-ll and
the variance like P 2 .[Min(ck zk)-dkl]2
To update the assessments of P6(sk+1 Yk, ck, Vk, k) (the probability
k kof the negotiator type in stage k+1, which gives us P1 and p2) and of
Pz(Zk+1 Ik Ck, ,k) (the probability of the reservation price in stage
k+l) we have to distingquish between two cases. - (1) The reservation price is
a constant, zi M z for all i. (2) The reservation price changes with k or with
Ck or with both.
Let us start with case (1). Judging by the counterpart's responses dk it
is possible, after several concessions, to determine if z is significantly
Our party can be quite certain about which case holds. If the effects oftime on the counterpart are not strong, and if he is well informed about mar-ket prices, then his reservation price is probably a constant. If he haslittle knowledge of market prices, and if time effects are strong - hisreservation price may change significantly during the negotiation.
- 211 -
-1larger or smaller than the current offer of our party. This is not easy to
do if past offers were all close to the reservation price. In this case
what can be done is to compute the most likely value of z from a likelihood
function of which the prior assessment of z is a par, a computation which
is fairly complicated. In case (2), when the reservation price is changing,
little can be done. Unless there is a reason to assume a certain relation-
ship between zk, dk and ck . A changing reservation price will have to be
assessed subjectively (it is of course bounded from below by the counter-
part's responses).
In conclusion, a prior-posterior analysis on P and 12 could be applied
if it is known that our party's proposals are within the zone of agreement.
Otherwise, a maximum likelihood computation can yield an estimate of the
reservation price of the counterpart and of the parameters Py and p2. This
scheme breaks down if the reservation price changes in an unpredictable way.
Note that sometimes a sudden change in the reservation price may be evident
(for example, after our party applied some persuasion measures which are not
activities of the formal process). In this case the new reservation price has
to be updated. Of course, our party then faces the problem of assessing, in
advance, his ability to convince the counterpart to increase his reservation
price. Such assessments start to be -very vague. They -are really outside the
realm of our model.
1 d-dBy computing - di i 1 i - 1,...j for some j and by rejecting the
i c -d
hypothesis that A are the realizations of independent random variables with
Distribution F(e) (with a small variance) we show ,that c1 > rB for i = 1...,j-k, for some small k.
- 212 -
(7) Conclusions
As developed here, the model can be solved by the dynamic programming
algorithm, although not all the necessary probabilities are generated by the
model. Some probabilities still have to be assessed. Still, by assuming
that the counterpart keeps his normalized concessions more or less equal, we
reduce the burden of assessing too many probabilities. Again, this assumption
is reasonable only in certain circumstances. The problem of probability
generation, commented upon in 5.2.2, is therefore only partially solved.
In applications, our party needs to have good reasons to believe that
his counterpart does use a strategy of almost constant normalized concessions.
As mentioned before, this assumption implies independence between the
normalized concessions of the parties. Past data may be needed for the
assessment of tight priors on the parameters V1 and V12. If no such data
exist and if the number of price proposals in the negotiation is small our
party will not be able to use the control framework to improve his assess-
ment of his opponent's reservation price.
- 213 -
5.3.3 An Experiment in Negotiations - The Streaker
"The Streaker" is a negotiation game played by graduate students At the
Harvard Business School.1 In this game a seller, whose reservation price is
$300, tries to sell a used car for as much as he can. The buyer, whose re-
servation price is $550, tries to buy the car for the lowest possible price.2
The game was conducted in the following way. The class of 100 students
was divided randomly into 50 pairs. Each pair consisted of a randomly
chosen buyer and seller. The pairs had several days to negotiate over the
sale of the car, but most of them completed their negotiations in one session,
lasting probably less than two hours.3 The students were instructed to list
the offers and counter-offers made during their bargaining session. Since
the parties reported independently, we have two accounts of every negotiation.
By comparing these accounts ( and paying attention to comments written by the
parties during the negotiations) we could establish the actual sequences of bids
with high confidence.4
The game is one of several negotiation games played in the course CompetitiveDecision Making, taught in the HBS in the Fall terms of 1977 and 1978 by Pro-fessor Howard Raiffa. The instruction sheets given to the participants can befound in Appendix 2 of this chapter. For analysis of the relationships betweenthe opening bids and the final agreements see Kimball [1979]. We are gratefulto Prof. Raiffa for letting us use the data(the data can be found in Appendix 3).2Somebuyers mistakenly assumed that their reservation prices are $600 and not
$550. As a result some cars were bought for more than $550.
3This is our rough estimate. The. students were' not asked to report theduratlon of .their negotiation.
tThe reporting was not always independent. Some parties compared their accountsafter ending the negotiation. Others simply copied the accountsprepared by their
counterparts. We know it from our observations of some negotiations and from
the striking fact 'that the reports of all the buyers were exstY1 identi-.cal to the reports of their respective sellers. Theb dependencies are not veryimportant because the studentshad no reason to cooperate in making false reports.
- 214 -
The game was intended to acquaint students with a concept of reservation
price and to give them some experience in negotiation. It was not designed
to test the hypotheses made in this section. As a result the data poses some
problems. Two important problems are: (1) The role of time, and (2) The
number of price proposals.
Time effects, which were not controlled, may have been very important.
Consider for example two students starting their negotiation at 1:30 PM. One
of them has a class to attend at 2:00 PM, while the other's next class is at
3:00 PM.I If at 1:55 PM, negotiations are still taking place, the first
student finds himself under severe time pressure. Comparing his reward from
selling or buying the car for a good price and his disutility for missing all
or part of his next class, the student may decide to terminate-negotiations.
quickly; This decision must affect his performance, probably hurting it.2
Judging from. the- data alone, it is difficult to decidewhich negotiators
experienced this Problem. It is however reasonable to assume that very
short sequences of bids correspond to time pressure experienced by at least
one of the parties. Since these sequences will be deleted from the data any-
way, we may have reduced the uncontrolled effects of time.
Our data consists of the sequences of price proposals. (It can be found
in its entirety in Appendix 3 of this chapter.) Note that the accounts of
only 41 out of 50 pairs are presented. The other 9 accounts were eliminated
Yost negotiations took place in school, between classes. Few students met onweekends or in the evenings. For these students differences in time-relatedcosts were not so large as our example indicates.
2The reward from getting a good price is a slightly higher score index, whichaffects the final grade, but only marginally. There is also the satisfactionof doing well with respect to other students in the class.
- 215 -
for several reasons.1 The data is used to examine the idea developed in
5.3.2(5) namely that negotiators make their concessions by conceding at each
stage a fixed portion of the maximum auvunt they can concede. The problem
is that in some cases only a few price proposals were made. We will therefore
have to eliminate some of the data and to acknowledge the fact that the
testing of hypotheses performed in this section is preliminary in nature.
1 2 nFormally, let 1B9 xB92.Bo9xn be the price bids of the buyer, and let
1 2 m i i+lx, xS,...,xS be the price bids of the seller. These bids satisfy xB < xB
for all i and x > x+ 1 for all J. The numbers n, m satisfy n - ml< 1.
Denote the reservation prices by rB (=550, or 600 in some cases) and r (=300).
Suppose the buyer is the first bidder.2 We define the variablesy and y4., to
be called normalized concessions, by:
i+1 i
yi B9 , i = 1,...,n-l, (5-5)minfrB'
and j- _l
yan Sj = 1,...,m-1. (5-6)
S x - max{r 5,41
The y variables take on values in [0, 1]. When y 0 for some i, we
i+l iknow that xB KB, i.e. the buyer made no concession in stage i. When
y- 1 for some i, we know that i+l -min{x. In this case our datay KB mnrl.
i+1 i ishows that with no exception xB - x, i.e. when yB = 1 the buyer made a
'There were three pairs who could not arrive. att:an agreement,. in spite of- thelarge zone of agreement. Two pairs did not document their negotiations ade-quately. Two other pairs proposed the reservation prices of their counter-parts and then split the differences - a behavior which can result eitherfrom their having full information or from complete cooperation. In any case,real negotiation did not take place there. Two more pairs exhibited strangenegotiation behavior (like retreating) which may result from inexperiencein negotiations.2 The definitions in the other case are analogous.
- 216 -
yielding concession. We eliminate the y's whose values are 0 or 1 from the
data. We are then left with sequences of values from (0,1). Some of these
sequences consist of one or even of no elements. These sequences are elimi-
nated too (they correspond to negotiations in which very few price proposals
were made). The data which we use is shown in Table 5.1. It consists of
sequences of normalized concessions of 52 out of 82 negotiators. (For remark
concerning the * 's and the numbers in square brackets, page 218.)Sample
Pair Negotiator Normalized Concessions (O's & l's deleted) Sample StandardNo. (B or S) Mean Deviation
.67
.64, .50
.36
.50
.67, .50
.20, .13
.04, .049
.08, .19,
.03, .039
.28
.50, .48
.50
.50
.20, .25,
.25, .24,
.33, .62
.36
.50
,.09, .07, .05, .02, .04,
,.09, .50* .20, .13, .14
.12, .14, .14, .08, .05,
,.16, .07, .08
.07,
1
1
2
3
3
4
4
5
6
6
7
8
8
9
9
10
.59
.55
.62
.50
.56
.13[.10]
.09
.37
.49
.50
.50
.29
.37[.32J
.45
.51
.45
12
.08
.36
0.0
.10
.12[.08J
.05
13
.01
0.0
0.0
.12
.14[.081
15
.21
.07
B
S
S
B
S
B
S
B
B
S
B
B
S
B
S
B
.50,
.50,
.87,
.50,
.50,
.29,
.04,
.13,
.03,
.46,
.50,
.50,
.50,
.29,
.29,
.40,
.65,
.40,
.50, .20
.38, .43, .60*
- 217 -(cont.)
Pair Negotiator Normalized Concessions (O's & l's deleted)No.j (B or S) I
-Y U
10
11
11
12
12
13
14
15
15
16
16
17
19
20
20
22
22
23
23
24
24
26
S
B
S
B
S
B
B
B
S
B
S
B
B
B
S
B
S
B
S
B
S
B
SampleMean
SampleStandard
Deviation- i4
.47
.41
.44
.37
.65
.60
.50
.13[.
.11,
.33,
.50,
.33,
.46,
.71,
.50,
.05,
.09,
.15,
.18,
.20,
.20,
.78,
.25,
.19,
.43,
.25,
.33,
.07,
.22,
.23,
.50,
.33,
.83
.40,
.31,
.40
.83
.50,
.50
.10,
.44
.21,
.24,
.12,
.43
.83
.50
.45,
.50,
.17,
.31,
.08,
.07,
.38,
.50
.30,
.50
.45, .50
.60
.11, .10, .08, .10, .12, .07,
.23, .18, .14, .17, .11, .14,
.50*
.50
.33
.50
.25
.79
.08, .24, .09, .05, .28, .40*
.08, .25, .63, .65, .05, .53
.70
.28, .15, .56*
.20[.18]
.27
.32
.81
.38
.32
.48
.22
.44
.16[.13]
.31
.44
.50
.30[.25]
09]
.51
.09
.09
.05
. .26
.11
0.0
.11[.02]
.11[.04]
.20
.16
.04
.18
.13
.04
.05
.24
.13[.091
.26
.24
0.0
.14[.07]
- 218 -
(cont.)
Pair NegotiatorNo.1 (B or S)I
26
27
27
30
30
36
36
38
38
39
39
40
41
41
Normalized Concessions (O's & l's deleted) SampleI Mean.
__ _ _ __ _ _ _i 0I
S
B
S
B
S
B
S
B
S
B
S
B
B
S
.50,
.22,
.13,
.18,
.33,
.22,
.50,
.29,
.22,
.50,
.67,
.27,
.13,
.91,0
.53,
.40,
.29,
.43,
.25
.17,
.30,
.63,
.36,
.33,
.50
.04
.40
.67
.71
.50
.33
.66
.14,
.17,
.50
.17,
.50
.16, .60*
.52, .50
.80*
.58
.37
.25
.42
.29
.26[.17]
.40
.47
.39[.25]
.44
.59
.16
.27
.79
SampleStandard
Deviation
.11
.14
.11
.24
.006
.19[.03]
.16
.17
.29[.10]
.10
.12
.16
.19
.17
Table 5-1
Remarks:
(1) In 8 of the sequences of normalized concessions we computed the sample
means and the sample standard deviations for both the whole sequence and for the
sequence with. one observation deleted. The tesults for- the latter are indicated
by square brackets. The deleted observations sare stared. The deleted con-
cessions are all much larger than any of the other concessions in the same
sequences. In 7 cases they are the last (not necessarily final) concessions.
In the analysis of the results we will work with the deleted sequences only.
- 219 -
(2) Of 52 sequences of concessions, 21 consist of 2 concessions only. Such
short sequences are not as reliable as longer ones. We will state all our
results for both the whole data and for the set of long sequences (i.e. the
31 sequences coaprised of 3 or more concessions).
(3) The sample standard deviation computed in the Table is defined as the
square root of the unbiased sample variance, i.e. [ (y 2-y) 1/2S = n-l
where y1, ... y are the observations and y is the sample mean.
Discussion
(1) A quick look at the data shows that 6 out of 52 negotiators behaved in the
way our controller in section 5.3.2 wanted them to behave: they kept their
normalized concessions constant, at .5. These six (3B, 6B, 6S, 7B, 14B, 24S)
did not do particularly well in comparison with the whole group.1 Additional
21 negotiators had small variations in their normalized concessions, with
standard deviations smaller or equal to .10. As a group, their performance
was average.
(2) It is important to know if the normalized concession of the parties depend
on each other. For each pair let x be the i-th normalized concession of the
negotiator who made the initial bid, and let y be the i-th normalized concession
of his counterpart. We wish to examine the simple regression
Y -81 + B2xi + c, - - 1,2,...,n, (5-7)
where C are normally and independently distributed with mean zero and a common
variance a . By using a diffuse prior on 0 1,0 2 and a, i.e. by assuming
1only one (6S) did better than the average negotiator of his type. The others
did significantly worse than the average.
- 220 -
p2 1 823 a)'Cc< 2p C 2 < , O <a< , (5-8)
the marginal posterior for S and a is the t-distribution and for a it is1 2
the inverted gamma distribution. In the following table we summarize the
results for 2 for the five longest sequences.
Number ofNormalizedConcessions
(n)
15
6
10
8
5
Posterior Mean2
of.fi(E"(0 B
Posterior Standadof 6 Deviaton(v( ))1/2
2- ,
.14
.41
1.53
1. 71
.37
.10
.67
3.88
1.28
.49
As can be seen from the few
in a small neighborhood of zero,
Table 5-2
examples, 82 is not necessarily concentrated
i.e.theremaybe dependencies between the
1The starred observations in Table 5-1 are included. We constructed Table 5-2from 5-1 and from the Appendix. The zero concessions are included too.
Computed from E"(82) -M S(x -)27 *
3Computed from v"( 2 2 2 Y2,where
S n-2 YfE"(81) - E"( 2)1/2, E"(8) - y -
a-
PairNo.
4
8
15
23
36
- 221 -
normalized concessions of the parties. This, we believe is not a refutation
of our conjecture about constant, independent normalized concessions. The
experiment was not planned with the objective of testing this hypothesis.
As a result the time effects were not controlled. We suspect that all the
negotiators in Table 5-2 negotiated for a relatively long time, had low
bargaining costs, and had enough time to prepare their strategies. The con-
ditions under which we believe the conjecture may hold (time pressure, high
bargaining cost, many negotiations in a short time) simply did not hold.
(3) From Table 5-1 we can construct an histogram of the sample means of
the normalized concessions (Fig. 5-3).
Number ofSequences
7
6
5
4
3
2
.05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 SampleMean
Fig. 5-3: Sample Means of Normalized Concessions
The underlying distribution of the sample means is of course unknown.
It seems to be a two modal distribution, with modes around .2 and .5.
- 222 -
(4) How can a negotiator benefit from knowing that his opponent's normalized
concessions are F -d istributed, independent random variables? The
simplest way is to offer an extreme initial bid, well beyond the opponent's
(unknown) reservation price and to concede relatively little in the first
stages of the game, staying outside the zone of agreement, trying to make in-
ferences about the opponent's reservation price. Consider for example the
pair number 8. The sequences of their concessions are reproduced from Appendix 3
in Table 5-3.
Pair No. 8 S is the first B: 200 300 350 350 400 475- 500bidder I S: 2000 1500 1200 1000 750 600 525
Table 5-3
It is advisable for th'e seller to start with an opening bid which is, with
a very high degree of belief, above the reservation price of the buyer, but
not too high so that B will not be antagonized and threaten not to move "until
S is reasonable". By starting at 2000 the seller 8S did just that, although
we do not knowwhat wps his prior assessment of zE. Immediately after
observing B's first bid (200) the seller could construct a prior distribution
on zB. (For simplicity he could assume that z B is an integer multiple of
100 and make his assessment for zB - 300, 400,....) He then combines this'
prior with his prior distribution on the size of the normalized concessions
(which may be A = .1, .2, .3, .4 and .5 with probability .2 each.) When the
next concession (300) is observed, the likelihood function is created from
(5-5). A maximum likelihood estimate for p and more importantly for zB is
then readily available. The process continues as new observations become
known.
From the analysis above it becomes clear that a tight prior on the dis-
tribution F of the normalized concession may compensate for diffuse prior on
- 223 -
zBS After few observations the posterior on zB becomes tight too. If
the prior on F is not so tight, the negotiator needs many observations.
This, however, he is not likely to get. With the exception of labor-manage-
ment negotiations, most business negotiations in the U.S. are very brief.
Only few price proposals are alternated, and consequently one cannot hope
to make strong inferences from the data. A negotiator who wishes to study
more on his opponent, should lead to alternations of many bids. He has to
be confident that his opponent still makes almost constant normalized
concessions.
- 224 -Appendix 1
- Formulation of the Model of Rao and Shakun
Let x be the proposal of party i when there are t sessions lef t,and let c 1
be his concession inthat time. Let Fit be the expected utility of the best overall
policy for party i for the next t periods, i.e.,
F t+2 , t+l t= maxt E($it), (5-11)
C i ,Ci 90
wxt+2P- t+l t+l t t-2 t-l4t-3
where *it =it ix+,ct1 xj ,cici-2,...,cj ,cj-,...) is the utility of
party i for the last t periods, given that the negotiation is in state
t+2 t+l t+l t t-2xi ,Cj ,x. I andiselects ci, c1 ,9...e.
Let Git be the minimum utility of the best overall policy of party i for
the last t periods, i.e.,
t+2 t+l t+l t t+2 t+l t+1 tGit(i ,c ,x ,c1 ) = min it(xi ,c ,xj ,c,
t-l t-3ctl , ... j(5-12)
t-2 t-4 t-l t-3
t-2 t-4where ci-, ci , ... , are the optimal concessions of i.
The negotiator i is a type-l negotiator if his objective is to maximize F
in (5-11). He is a type-2 negotiator if his objective is to maximize G in (5-12)
The maximized values are denoted by f and hi, i.e.,
ft(x t+2,Ct+1 t+i) = max F (xt+2, Ct+l t+1 Ct) (5-13)fit(x1 *C9 xt)it i C .j , 1
and
h (x t+2,Ct+1 t+l) )=max G (xt+2,Ct+l t+, t).(5-14)
c titx J
Now define f by
,-t+2 t+}.t+2 t+{4t j - 6, Oct-2, ...)),(5-15)
- 225 -
and let0 t+2 t+1 t+2 t+1 t+l
= st-2(xi ,6I1 -6) - Git (Xi ,c ,x )rit~ 0 t+2 t+1w
fit-2 i ,6,x - ) -
be the extended risk limit of party i in stage t. The negot:
rit, denoted by rjt, by assessing fjit-2.
The negotiator i is a type-3 negotiator if he concedesc
rit .rt and does not concede when rit> CtFor a type-1 negotiator, say L, who observes Lt+ = M*6
H
be formulated as follows:r K t+2
Max Max S pmk ft-2'+2+k-6,<k<K 1=0
t+2 t+l K t+2St LN mot t-2
2=1
Kt+Max maL'x E fokt t-2X +k[L 0<k<KL=0Att2 X
with the boundary conditions:
ft( t+29 -S +2 (L t+2)*t4+2) L XL
Max OLP
k k+L<(x+_L +2)/mk6 +
E L +2+kLE UL(2CL ke
k+% >( +lt+2tt+2 t++
f2 XL xHMax L * S Pok1k>O k+t< +1 +2
I kt<xH XL)/
(5-16)
iator i can assess
C=6 when
the problem can
t+l
t+1R 6 )+moo ] (m>O)
,e2" 6 t+l-z6) (m=f)
(5-17)
(5-18)
pmkt (m>0)
(5-19)
+
E L +2-k)1 (mO)k+ t+1 t+2 I
k~f(z XL )/6 JFor the problem of the type-2 negotiator, see Rao and Shakun [1974] formula(6).
'For the complete derivation of rit see Rao and Shakun [1974] pages 1366-67.
- 226 -Appendix 2
THE STREAKER
Negotiating Exercise - Seller
Your son, now serving overseas in the military, has asked you to sell his
car, a 1955 Street Streaker. This model was a limited production, high per-formance vehicle manufactured by Marvel Motors before that Company went out of
business in 1956. The car body shows its age, but your son has taken meti -culous care of the engine and drive train, and the car is in excellent opera-
ting condition. Knowing that used car purchasers are greatly influenced byexterior appearance, you wish that your son had spent more time polishing thantinkering, but you realize that making major improvements in the body nowwould-Tcost more than would be recovered by the increased selling price.
You have researched the market, but have found no published market value
for the Street Streaker. Your insurance agent informs you that his companyplaces a collision damages limit of $?)0 on the car, assuming it is in top
condition, and a local used car dealer has offered you $300, explaining thatthe car is unlikely to be resold except for scrap value. Hoping to find an
ad in the local paper. Three people have come by to look at the car, but
you did not receive an offer from any of them. Earlier today, deciding tobring the matter to a close, you called all three prospects and told them you
had an offer for the car, and intended to sell if they were no longer inter-
ested. One of the prospects expressed a continuing interest, and stated that
he would come by to see you.
This one remaining prospect had inspected the car yesterday, and spenta good deal of time examining the engine. He did not seem overly concerned
about the body, which you considered a good sign. He asked for a price, youasked him for an offer, and he responded that he would have to think about it.
You are reasonably sure that you will find no other prospects, and you intend
to sell the car to the dealer for $300 if you cannot strike a bargain with the
prospect who is to arrive.
Copyright 1978 by the President and Fellows of Harvard College
- 227 -
Negotiating Exercise - Buyer
You are an amateur mechanic who enjoys repairing old cars. You havemade your hobby pay for itself by selectively purchasing autos at "junk"prices, and reselling them after reconditioning at substantial mark-ups. Twoweeks ago you paid $225 for a 1953 Road Buffer, a low production model thatwas manufactured by Marvel Motors before it went out of business in 1956. Thebody of the car is in excellent condition, which is what attracted you, butyou find that several engine parts hould be replaced if the car is to be putin optimal operating condition. Since you intend to give the Road Buffer toyour daughter on her seventeenth birthday, you do not want to cut any corners.Marvel Motors parts are no longer available on the market, but you have dis-covered that similar parts can be modified to suit your purposes. To pur-chase these parts and have them machined will cost $600.
Two days ago you found a classified ad offering a. 1955 Street Streakerfor sale on a best offer basis. You know that this model, also a MarvelMotors product, has a very similar engine to the Road Buffer, and has severalimproved parts that are interchangeable with the Road Buffer. You examinedthe car yesterday and found strong signs that the engine has been very wellmaintained. You are certain that the key parts you are interested in wouldbe at least as satisfactory as the machined parts you had intended to buy.Further, in inspecting the Street Streaker, you found other items you could
cannibalize to improve the Road Buffer. On the whole, the purchase of theStreet Streaker seems like a very good idea. However, it will cost you $50in addition to the purchase price to remove the needed parts and make minormodifications to them so that they can be used in your Road Buffer.
The owner of the Street Streaker was very quiet as you inspected thevehicle, pointing out only that while the body showed its age, his son hadkept the car in excellent running condition. He refused to give you a price,asking instead for an offer. You told him that you would have to think aboutit, but would contact him shortly. He agreed to let you know before he soldthe car to anyone else.
This afternoon the owner called you to say that he had received an offerfor the vehicle, and had promised to make a decision on it by tomorrow after-noon. You told him you would stop by later on.
As you prepare to meet him, you are aware of your two alternatives:
1. Purchase Machined Parts 2. Buy Street Streaker; Remove andModify Parts
Cost - $600 Cost - Purchase Price + $50.
You are certain that you will not be able to find another car that willsuit purposes in the time available to you.
Copyright * 1978 by the President and Fellows of Sarvard College
- 228 -THE STREAKER
Summary of What Happened
Name of Buyer:
Name of Seller:
FINAL PRICE
ELYour role (check one): Buyer
To be filled out during negotiations:
'Seller
Sequence of Proposals
Comments
Suggested by Price (including amounts you expect to get)
Continue on back if necessary.
Did you tell your opponent your own reservation price (either truthfully orfalsely!)?
If so, when did you disclose it? At what point in the above sequence?
Did you tell the truth about your reservation price? If no, what price didyou aamoince?
- 229 -
Appendix 3
The Streaker-Data'
Pair The FirstNo. Bidder
1
2
3
4
6
7
8
a aI
Sequence of Offersv
420 425
630 615
515 515
575 570
1The data is ordered according to the final outcome (in decreasing order).Sellers are denoted by S, buyers by B.
B
S
S
B
B
B
B
S
B: 300 450 550 600
S: 1000 650 600
B: 550 550 585
S: 985 605 585
B: 400 500 550 575
S: 1200 800 600 575
B: 200 300 350 375 390 400 410 417
S: 1050 950 900 800 750 700 660 640
B(cont.): 430 435 440 450 500 510 515
S(cont.): 610 605 600 595 580 575 575
B(cont.): 520 550
S(cont.): 550
B: 800 570 550
S: 300 500 550
B: 350 450 500 512
S: 650 550 525 512
B: 400 450 475 500
S: 500 500 500
B: 200 300 350 350 400 475 500
S: 2000 1500 1200 1000 750 600 525 500
- 230 -
(cont.)
Pair The FirstNo. Bidder Sequence of Offers
9 S B: 300 400 450 500
S: 900 575 575 530 500
10 S B: 350 450 475
S: 800 750 500 475
11 B B: 450 450 500 520 520 535 542.5
S: 750 600 600 575 550 542.5
12 B B: 250 350 430 475
S: 1000 700 475
13 B B: 150 400 450 465
S: 500 500 475 465
14 B B: 300 425 437.5
S: 600 450 437.5
15 S B: 250 260 275 287 295 300 305 330 312
S: 465 440 410 385 370 360 350 345 340
B(cont.): 314 321 325
S(cont.): 335 330 325
16 B B: 300 350 375 425
S: 600 550 475 425
17 B B: 200 375 400
S: 425 405 400
18 S B: 350 400
S: 425 400
19 B 1 : 350 375 400
1 S: 450 425 400
- 231 -
(cont.)
Pair The FirstNo. Bidder
20
21
22
23
24
25
26
27
28
29
30
B
S
S.
B
B
B
B
B
B
B
B
Sequence of Offers
B: 150 225 350 375 400
S: 650 500- 425 40
B: 450 400
S: 300 400
B: 150 250 300 350 397
S: 900 700 700 575 397
B: 200 225 250 275 275
S: 750 650 625 600 525
B(cont.): 350 375 385
S(cont.): 440 400 385
B: 225 300 350
S:
B:
S:
B:
S:
B:
S:
B:
S:
B:
S:
B:
S:
600 450 400
175 350 380
425 380
200 300 330
500 400 400
250 250 300
500 475 425
250 350 375
450 400 375
195 300 360
400 360
175 225 300
450 400 375
335 345
450 445
385
385
350 360 366 375 377
400 400 382 377
350 375
400 375
350
350
- 232 -
(cont.)
Pair The FirstNo. Bidder Sequence of Offers
31 B B: 200 400 350
S: 500 350
32 B B: 250 350
S: 575 350
33 B B: 100 350
S: 500 350
34 B B: 225 325 345
S: 350 345
35 B B: 190 300 337.5
S: 400 337.5
36 B B: 100 200 250 275 295 325 335
S: 700 500 425 400 345 335
37 B B: 200 325
S: 450 325
38 B B: 200 200 250 300 312.5315
S: 425 375 330 325 315
39 B B: 100 275 300 312
S: 450 350 325 312
40 B B: 117 180 185 301
S: 350 300 301
41 B B: 175 225 275 300
S: 850 350 300
- 233 -
CHAPTER 6
CONCLUSIONS AND FUTURE RESEARCH
In this condluding chapter we summarize the main results of our study
and propose some directions for future reserach. We do not repeat here the
discussion of the importance of incorporating time effects into formal models
of negotiations. The reader is referred to chapters 1 and 2 for such a
discussion, for some motivating examples and for a literature review.
In chapter 3 we examined the Nash Bargaining Problem interpreted as an
arbitration scheme. We introduced the effects of discounting and bargaining
costs into the utility functions, and thus changed the problem in two ways:
(1) We created a more realistic (and more complicated) version of the tra-
ditional bargaining problem. (2) We identified several types of solutions
for the new problem by extending the concept of arbitration. The arbitra-
tor. was allowed to design incentive games whose outcomes serve as arbitra-
tion solutions. The Extended Raiffa Solution (ERS) corresponds to the first
problem change. This solution is the unique function satisfying some arbi-
tration principles and a set of fariness requirements. The Timing Equili-
brium Solution (TES) corresponds to the second change. It is the value of
an incentive game in which the arbitrator compensates the party who makes the
yielding concession. These two solutions, when applied to the same problem,
yield different results. The way the arbitrator views his role may therefore
have significant effects on his arbitrated solution. In this thesis we
examined closely the ERS and studied its properties. Various aspects of the
TES were left for future study.
The U.S was found to contain the Nash Cooperative Solution of the original
- 234 -
"time-less" problem in the special case of equal discounting rates and no
bargaining costs. This result weakens the significance of the Nash Bar-
gaining Solution because in most cases the discount rates are not identical
and bargaining costs do exist.
In chapter 4 we addressed several problems which had been observed in
the coin market in Israel. These problems exist also in many similar small
markets, where price information is difficult to obtain. The dealer-collector
negotiation in this market was viewed as a part of a search-negotiation
strategy used by the collector. This strategy may involve negotiating with
one dealer for a certain period of time, meanwhile-getting better. information
about the dealer's reservation price, then postponing the negotiation and re-
newing the search, possibly returning to the dealer at a later stage for
further negotiations. Optimal search-negotiation strategies under various
assumptions were computed. These strategies have in some cases "reservation
price" properties which make them simple to understand and easy to compute.
The optimal strategies may also serve as decision aids for the collector and
the dealer and may be used in descriptive economic models. Our model is
somewhat restricted by the assumption that dealers concede according to their
"characteristic concession patterns", so that the collector can gain much
information from observing the first two or three concessions. Some restric-
tions are however inevitable if one wants to introduce learning into nego-
tiations. The collector has to be able to decide to terminate the negotiation
on the basis of his counterpart's behavior. In order to make this decision
he needs to use some behavioristic model of his opponent. The characteristic
concession patterns represent such a model.
Conceding along a characteristic pattern is a frequently observed phe-
noanon. We assumed the existence of such patterns without trying to justify
- 235 -
them, although intuitive explanations can be given. Under the assumption that
negotiators adhere to their patterns, a model involving a three stage nego-
tiation game was descrioed anid solved. The results gave rise to some con-
jectures regarding the relationships between the final outcome and the model's
parameters. These conjectures can be examined empirically and, if found to
be right, explain some of the effects of time-related costs on final outcomes
of negotiations. They can also help in explaining the distribution of
different reputation levels in the market. A game theory model of reputations
was discussed in Appendix 2 of chapter 4. In this model certain negotiat.' on
rules were shown to yield stationary distribution of reputations in the
market.
In chapter 5, after reviewing previous approaches, we examined the pos-
sibility of modelling negotiations as problems of adaptive control. Some
severe problems obviously existed, mainly the problem of generatingand up-
dating of probabilities and the computational difficulties resulting from the
high dimensionality. Control models seem to be helpful when some strong
assumptions on the behavior of the players can be made. For example, that
players belong to certain "negotiator types", that they have fixed reserva-
tion pricesetc. Experiments can be useful in deciding if various such con-
ditions do hold. An experiment in negotiations examined in chapter 5 (The
Streaker), although not designed specifically for our purpose, suggested that
one particular behavioristic model of concessions that we tried is plausible,
or at least worth studying.
Throughout the thesis we mentioned some directions for future
research, involving both theoretical and empirical work. We now samrize
the main suggestions and cosent on the prospects and the expected difficulties.
- 236 -
(1) Empirical studies of characteristic concession patterns, of normalized
concessions and of other concession mechanisms (see chapter 4 and 5). Such
studies may discover some of the practical rules used by negotiators in specific
markets. The difficulty with these studies is the low reliability of the data.
Classroom experiments tend to be too structured and the negotiation environ-
ment is artificial. Observing real negotiation may be a long and tedious
process; additionally, there is no guarantee that all the activities in the
negotiation under study are indeed observed.
(2) Explaining negotiation outcomes by the bargaining costs and the discount-
ing rates of the parties. The data may demonstrate the actual effects of time,
but good data will again be difficult to obtain. Needed are the sequences of'
proposals, the utility functions of the parties and their time-related costs.
Some experiments have already been conducted by Contini [1967a], [1967b),
[19681.
(3) Axiomatizing the incentive-arbitration schemes and designing incentive
games with the objective of reducing the inefficiencies caused by the time
effects. The real problem here is to define what constitutes a fair incentive
scheme. There may be important applications of "fair" schemes which succeed
in shortening the negotiation duration.
(4) Studying the equilibrium properties of classes of characteristic con-
cession patterns. In this thesis we assumed the existence of these patterns,
but a theoretical foundation still ought to be built. The relationship between
these patterns and the notion of perfect equilibrium of repetitive games should
also be explored.
(5) Extending the search-negotiation models by allowing more uncertainty in
the negotiation process. Differences in risk aversion should affect the stra-
tegies. Note that the search-negotiation can easily be extended to a market
- 237 -
game with many searchers and non-searchers. In this case equilibrium proper-
ties may perhaps be found. Linking the order of search to reputation levels
(a concept introduced by Rosenthal-Landau [1979]) may be particularly useful.
(6) In spite of the difficulties, we still believe that specialized bounded-
rationality, asymmetric negotiation models can be built. The adaptive con-
trol framework may be helpful although it involves strong assumptions about what
opponent's behavior.
(7) Examining the effects of time-related costs on other models of conflict
(not necessarily negotiations).
In conclusion, the introduction of time effects into existing models and
the formulation of new models with time may yield better models, descriptively
and normatively. It is our belief that these models will contribute to the
understanding of negotiations' outcomes and will be helpful in the planning
and in the analysis of negotiations.
- 238 -
REFERENCES
Ashenfelter, 0. A. and G. E. Johnson [1969], "Bargaining Theory, Trade Unions,and Industrial Strike Activity", American Economic Review, 59, 35-49.
Bartos, 0. J. [1974], Process and Outcome of Negotiations, New York: ColumbiaUniversity Press.
Bertsekas, D. P. [1976], Dynamic Programming and Stochastic Control, New York:Academic Press.
Bishop, R. L. [1963], "Game Theoretic Analysis of Bargaining", QuarterlyJournal of Economics, 77, 559-602.
[1964], "A Zeuthen-Hicks Theory of Bargaining", Econometrica,32, 410-417.
Britto, D. L., A. M. Buoncristiani and M. D. Intriligator: "A New Approachto the Nash Bargaining Problem", Econometrica, 45, 1163-1172.
Bush, R. R. and F. Mosteller [1955], Stochastic Models for Learning, New York:Wiley.
Chatterjee, K. [19781, "A One Stage Distributive Bargaining Game", Working Paper,Harvard School of Business Administration.
Chow, Y. L. [1969], Statistical Analysis, New York: Holt, Rinehart and Winston.
Coddington, A. [1966], "A Theory of the Bargaining Process: Comment", AmericanEconomic Review, 56, 522-530.
[1968], Theories of the Bargaining Process, London: Allen andUnwin.
Contini, B.[1967a], "The Value of Time in Bargaining Negotiations: Part I, ADynamic Model of Bargaining", University of California, Center forResearch in Management Science, Working Paper No. 207, Berkeley.
[1967b], "Part II: Experimental Evidence", Working Paper No. 208,Berkeley.
[1968], "The Value of Time in Bargaining Negotiations: Some Experi-mental Evidence", American Economic Review, 58, 374-393.
Cross, J. G. [1965J, "A Theory of the Bargaining Process", The American EconomicReview, 55, 66-94.
[1966], "A Theory of the Bargaining Process: Reply", The AmericanEconomic Review, 56, 530-533.
mm___m___[1969], The Economics of Bargaining, New York: Basic Books.
- 239 -
De Groot, M. H. [1970], Optimal Statistical Decisions, New York: McGraw-Hill.
De Menil, G. [1971], Bargaining: Monopoly Power versus Union Power, Cambridge,Massachusetts: MIT Press.
Dresher, M. [1961], Games of Strategy: Theory and Applications, EnglewoodCliffs, N.J.: Prentice-Hall.
Farber, H. S. [1978], "Bargaining Theory, Wage Outcomes and the Occurrenceof Strikes", The American Economic Review, 68, 262-271.
Fishburn, P. C. [1965], "Independence in Utility Theory with Whole ProductSets", Operations Research, 13, 28-45.
[1974], "Von Neumann-Morgenstern Utility Functions on TwoAttributes", Operations Research, 22, 35-45.
Foldes, L. [1964], "A Determinate Model of Bilateral Monopoly", Economica,N.S., 31, 117-131.
Freimer, M. and P. L. Yu [1976], "Some New Results on Compromise Solutionsfor Group Decision Problems", Management Science, 22, 688-693.
Friedman, J. W. [1977], Oligopoly and the Theory of Games, New York: North-Holland.
Gilbert, J. P. and F. Mosteller [1966], "Recognizing the Maximum of a Sequence",Journal of American Statistical AssociationL 61, 35-73.
Harsanyi, J. C. [1956], "Approaches to the Bargaining Problem Before and Afterthe Theory of Games: A Critical Discussion of Zeuthen's, Hicks' andNash's Theories", Econometrica, 24, 144-157.
[1977], Rational Behavior and BargaininZ Equilibrium in Gamesand Social Situations, Cambridge: Cambridge University Press.
Hicks, J. R. [1932], The Theory of Waes, London: McMillan (2nd ed.-1963).
Hnyilicza, E. and R. S. Pindyck [1976], "Pricing Policies for a Two-PartExhaustible Resource Cartel", European Economic Review, 8, 139-154.
Kalai, E. [1977), "Proportional Solutions to Bargaining Situations: InterpersonalUtility Comparisons", Econoutrica, 45, 1623-1630.
Kalai, E. and K. Smorodinsky [1975), "Other Solutions to Nash Bargaining Problem",Econometrica, 43, 510-518.
Karni, E. and A. Schwartz [19771, "Search Theory: The Case of Search withUncertain Recall", Journal of Economic Theory, 6, 38-52.
[1978), "Two Theorem on Optimal Stopping with BackwardSolicitation", Journal of Applied Probability, 4, 869-875.
- 240 -
Karlin, S. [1959 II], "Mathematical Methods and Theory in Games, Programmingand Economics", (Vol. II), Reading, Mass: Addison-Wesley.
Karras, C. L. [1974], Give and Take, New York: Crowell.
Keeney, R. L. and H. Raiffa [1976], Decisions with Multiple Objectives:Preferences and Value Tradeoffs; New York: John Wiley & Sons.
Kimball, C. E. [1979], "Haggling", Unpublished Term Paper (prepared forEconomics Course 2240a, Harvard University).
Landsberger M. and D. Peled [1977], "Duration of Offers, Price Structure,and the Gain From Search", Journal of Economic Theory, 16, 17-37.
Lippman S. A. and J. J. McCall [1976], "The Economics of Job Search: A
Survey", Economic Inquiry, 14, 155-189.
Lipsky, D. D. and T. A. Barocci [1978], "Public. Employees in Massachusettsand Final-Offer Arbitration", Monthly Labor Review, 101(4), 34-37.
Luce, R. and H. Raiffa [1957], Games and Decisions: Introduction and Critical
Survey, New York: John Wiley & Sons.
McCormick, B. [1977], "On Expectations, the Value of Time and Uncertainty, in
the Bargaining Process", Unpublished note (Cambridge University).
Myerson, R. M. [1977], "Two-Person Bargaining Problems and Comparable Utility",Econometrica, 45, 1631-1637.
[1979], "Incentive Compatibility and the Bargaining Problem",
Econometrica, 47, 61-74.
Nash J. F., Jr. [1950], "The Bargaining Problem", Econometrica, .8, 155-162.
Nydegger, R. and G. Owen [1975], "Two Person Bargaining: An ExperimentalTest of the Nash Axioms", International Journal of Game Theory, 3, 239-249.
Owen, G.- [1968], Game Theory, Philadephia: W. B. Saunders Co.
Pekelman D. and G. C. Rausser [1978], "Adaptive Control: Survey of Methods
and Applications", MS Studies in the Management Sciences, 9, 89-120.
Pindyck, R. S. [1976], "The Cost of Conflicting Objectives in Policy Formula-
tions", Annals of Economic and Social Measurement, 5, 239-248.
Raiffa, H. [1953], "Arbitration Schemes for Generalized Two-Person Games",in Annals of Mathematics Studies (Kuhn and Tucker, Eds.), 28, Princeton:Princeton University Press.
Rao, G. A. and M. F. Shakun [1974], "A- Normative Model for Negotiations",Management Science, 20, 1364-1375
- 241 -
Raskin, A. H. [1963], "The New York Newspaper Strike", New York Times, April 1,1963. (Also in The 50% Solution (I.W. Zartmen, ed.), Garden City, NewYork: Anchor Press, 1976).
Richardson, L. F. [1960], Arms and Insecurity, Pittsburg: Boxwood.
Rosenthal, R. W. [1976], "An Arbitration Model for Normal-Form Games",Mathemetics of Operations Research, 1, 82-88.
Rosenthal, R. W. and H. J. Landau [1979], "A Game-Theoretic Analysis ofBargaining with Reputations", Working Paper, Bell Laboratories, Murray Hill, N.J.
Roth, A. E. [1977], "Individual Rationality and Nash's Solution to theBargaining Problem", Mathematics of Operations Research, 2, 64-65.
[1977a], "Independence of Irrelevant Alternatives, and Solutionsto Nash's Bargaining Problem", Journal of Economic Theory, 16, 247-251.
[1978], "The Nash Solution and the Utility of Bargaining",Econometrica, 46, 587-594.
Rothschild, M. [1974], "Searching for the Lowest Price When the Distributionof Prices is Unknown", Journal of Political Economy, 82, 689-711.
Ross, A. M. [1948], Trade Union Wage Policy, Berkeley: Uriversity ofCalifornia Press.
Rubin, J. Z. and B. R. Brown [1975], The Social Psychology of Bargaining andNegotiation, New York: Academic Press.
Schelling, T. C. [1956], "An Essay on Bargaining", The American EconomicReview, 46, 281-306.
[1960], The Strategy of Conflict, Cambridge, MA. : HarvardUniversity Press.
Shackle, G. L. S [19571, "The Nature of the Bargaining Process", in Theory ofWage Determination (J. Dunlop, ed.), New York: St. Martin's Press.
Stahl, I. [1972], Bargaining Theory, Stockholm: EFI.
Stevens, C. M. [1963], Strategv and Collective Bargaining Negotiation- New York:McGraw-Hill.
[1966], "Is Compulsory Arbitration Compatible With Bargaining?",
Industrial Relations, j2f, 38-52.
Von Neumann, J& -and 0. Morgenstern, [1953], Theory of Games and EconomicBehavior, 3rd ed., Princeton, N.J.: Princeton University Press.
Wiggins, W. H. [1976], "Up for Auction: Malta Bargains with Great Britain,
1971", in The 50% Solution (I.W. Zartman, ed.) Garden City, New York:Anchor Press.
- 242 -
Weitzman, M. L. [1978], "Optimal Search for the Best Alternative", MIT EnergyLaboratory Report No. MIT-EL-78-008.
Yang, M. C. K. [1974], "Recognizing the Maximum of a Random Sequence Basedon Relative Rank with Backward Solicitation", Journal of AppliedProbability, 1, 504-512.
Young, 0. R. [1975], Bargaining: Formal Theories of Negotiation, Universityof Illinois Press.
Yu, P. L. [1973], A Class of Solutions for Group Decision Problems,Management Science, 19, 936-946.
Zellner, A. [1971], An Introduction to Bayesian Inference in Econometrics;New York: Wiley.
Zeuthen, F. [1930], Problems of Monopoly and Economic Warfare, London: GeorgeRutledge and Sons.
- 243 -
BIOGRAPHICAL NOTE
Zvi A. Livne earned a B.Sc. in Mathematics and Physics
cum laude from the Hebrew University of Jerusalem in 1975.
His undergraduate studies were interrupted by his military
service as an officer in the Israeli Defence Forces.
During his graduate studies at M.I.T., Zvi A. Livne
participated in a research on the supply of oil from newly
discovered fields and in other energy-related projects. Part
of his work was published in an M.I.T. Energy Laboratory
Working Report (January 1979, with G.M. Kaufman and W.
Runggaldier) titled: "Predicting The Time Rate of Supply
From a Petroleum Play". He was supported by research grants
from the Electric Power Research Institute and by teaching
assistantship in the M.I.T. Sloan School of Management.
Aside from his interest in the analysis of negotiations
and bargaining, Zvi A. Livne is interested in the methodology
and applications of Decision Analysis, in the analysis of Public
Policy and in several topics related to uncertainty in economics
(insurance, search, incentives and more).
Zvi A. Livne is a member of The Institute of Management
Science. He delivered a paper based on Chapter 3 of this
thesis at the New Orleans TIMS/ORSA Conference (May, 1979).