COALITION BARGAINING IN REPEATED GAMES

• Date: 01 November 2018
• Author: Fernando Vega‐Redondo,Arnold Polanski
• Published date: 01 November 2018
• DOI: http://doi.org/10.1111/iere.12325

INTERNATIONAL ECONOMIC REVIEW

Vol. 59, No. 4, November 2018 DOI: 10.1111/iere.12325

COALITION BARGAINING IN REPEATED GAMES∗

BYARNOLD POLANSKI AND FERNANDO VEGA-REDONDO

University of East Anglia, U.K.; Bocconi University, Italy, and IGIER, Italy

We consider an intertemporal framework where different coalitions interact repeatedly over time. Both the

terms of trade and the endogenous cooperation structure are characterized, in a protocol-free manner, when:

(C1) A coalition is formed with positive probability if, and only if, the shares obtained by its members weakly

exceed their respective share expectations.

(C2) Each matched coalition distributes the entire surplus among its members.

(C3) Members of any coalition are treated symmetrically with respect to their share expectations.

We show, in particular, that the cooperation structure and the shares are unique when the game ends each

date with vanishing probability.

1. INTRODUCTION

The lion’s share of economic activity takes place in groups and organizations that cooperate

repeatedly. For example, in production economies, employees meet regularly in their respective

firms and each meeting results in the production of some output. Similarly, goods and services

are repeatedly exchanged among a given set of traders. Our objective in this article is to

provide a general approach to these intertemporal situations that, abstracting from the varying

features and contrasting implications of specific matching and bargaining protocols, provides a

unified and robust understanding of the main issues involved.

Formally, our model considers an environment where multiple productive coalitions of agents

can meet over time, each of them limited to participating in one coalition at a time.1Then, each

agent demands a share of the created (and transferable) surplus taking into account her op-

portunities in other coalitions. Such demanded shares, in turn, determine the probability with

which coalitions may form over time. As stressed, our approach to the problem abstracts from

the particular mechanism that may be at work in the process of formation of the different coali-

tions and their internal bargaining. Instead, our aim is to characterize the outcome (i.e., payoff

shares and coalitions formed) under the assumption that, independently of the aforementioned

procedural details, the following conditions hold at any point in the process:

(C1) Coalitions formed with positive probability are those, and only those, in which each

member obtains at least her expected share.

(C2) Each matched coalition distributes the entire surplus (as shares) among its members.

(C3) Members of any coalition are treated symmetrically with respect to their expected shares

when the surplus of this coalition is distributed.

The first condition determines the cooperation structure given some expected payoff profile

over the whole game. It asserts that a coalition will meet whenever it generates enough surplus

to satisfy the expectations of all its members. The second condition requires that all surplus be

distributed among any set of agents who decide to form a coalition. Importantly, this condition

∗Manuscript received October 2016; revised October 2017.

Please address correspondence to: Arnold Polanski, School of Economics, University of East Anglia, Norwich

Research Park, Norwich, NR4 7TJ, U.K. E-mail: a.polanski@uea.ac.uk.

1Therefore, nonintersecting coalitions can meet at each date and produce a separate surplus.

1949

C

(2018) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1950 POLANSKI AND VEGA-REDONDO

rules out transfers to nonmembers. The third condition embodies fairness in the sense of

egalitarian treatment of (rational) expectations.

As implicitly suggested in our motivation of the model, we can think about the payoff shares

that satisfy the former three conditions as the outcome of dynamic negotiations undertaken by

several parties prior to the actual repeated cooperation. Relatedly, they can be thought of as

embodying some appealing normative criteria that agents will abide by, and insist upon, when

choosing the way in which to allocate their resources. Both views are largely reflected in the

following quote from Shapley and Shubik (1972, p. 116):

A prudent ‘economic’ man . . . would be loath to enter a partnership for a stated share of the proceeds

until he had satisfied himself that more favorable terms could not be obtained elsewhere. We can

imagine that each player would set a price on his participation, and that no contracts would be signed

until the prices . . . are in harmony.

There is a vast literature on surplus sharing and coalition formation. One branch of this

literature is based on noncooperative models, as surveyed by Ray (2007). Our approach can

be seen as complementary to that pursued by this branch as it abstracts from the details of

the matching and bargaining procedures and turns instead its attention to the properties that

any outcome should satisfy independently of the specific protocol. This is relevant because it

is well known that, in general, small changes to the postulated rules can have a major impact

on equilibrium outcomes. There is therefore the concern that such a theoretical approach

may not be robust to minor modeling details. By way of illustration, the models considered in

Baron and Ferejohn (1989), Chatterjee et al. (1993), Hart and Mas-Colell (1996), Krishna and

Serrano (1996), or Okada (2011) lead to quite different outcomes (e.g., the Shapley value, the

nucleolus, or a point belonging to the core) depending on the specific features of the matching

and bargaining environment contemplated in each case.

Protocol dependence is not an issue for the strand of the literature that relies on cooperative

game theory and, as in our present approach, is axiomatic and outcome based. The stable set

(von Neumann–Morgenstern, 1944), core (Gillies, 1953; Shapley, 1953), Shapley value (Shapley,

1953), and the bargaining set (Aumann and Maschler, 1964) are prominent examples in the long

tradition concerned with such an axiomatic characterization of solutions to cooperative games.2

These concepts, however, are not designed to predict coalition formation. They answer the

fundamentally different question of how the coalitional gains should be distributed, provided

that the grand coalition (or some other given coalition structure) has formed. Moreover, these

concepts offer solutions for one-shot games instead of games that are played repeatedly. There

is, however, some literature that adds an intertemporal dimension to cooperative games (e.g.,

Lehrer, 2003; Predtetchinski et al., 2006; Lehrer and Scarsini, 2013). This literature is concerned

with a context very different from ours in that the payoffs accrue to coalitions over time and

any coalition that forms is required to be robust to “internal” deviations at intermediate stages

of the payoff-accumulation process. Its main focus is also on extensions of the static solution

concepts of the classical cooperative literature to such a context.

Although our framework shares with the classical cooperative game theory its axiomatic

approach, it differs from its standard solutions in two important respects. First, it is explicitly

designed to predict both coalition formation and surplus sharing. Second and more importantly,

it is driven by the dynamic structure of our game and is not restricted by an (arbitrary) time limit.

It is precisely this structure that allows for the joint computation of the matching probabilities

and shares, relying alone on the conditions 1–3 that embody their mutual consistency.

In particular, our conditions 1 and 3 formalize the idea that expectations should play a

prominent role in the computation of players’ shares. “Rational” (in the sense of consistent)

expectations have, of course, long been an integral part of equilibrium analysis in noncooperative

2Some cooperative solutions allow players/coalitions to contemplate alternative paths of the game as in Chwe (1994)

or in Ray and Vohra (2015), who modified the concept of the stable set to incorporate farsighted behavior. The

sequentiality of the contemplated moves is, however, hypothetical and not factual, as in our game.