Multilateral Bargaining Problems

GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.

19, 151]179 Ž1997.

GA970547

Multilateral Bargaining Problems Elaine Bennett U† Virginia Polytechnic Institute and State Uni¨ ersity, Blacksburg, Virginia 24061; and Uni¨ ersity of California, Los Angeles, Los Angeles, California 90024 Received February 15, 1994

In many situations in economics and political science there are gains from forming coalitions but conflict over which coalition to form and how to distribute the gains. This paper presents an approach to such multilateral bargaining problems. A solution to a multilateral bargaining problem specifies an agreement for each coalition that is consistent with the bargaining process in every coalition. We establish the existence of such solutions, show that they are determined by reservation prices, and characterize these reservation prices as the payoffs of subgame perfect equilibrium outcomes of a non-cooperative bargaining model. Journal of Economic Literature Classification Numbers: C71, C72, C78. Q 1997 Academic Press

1. INTRODUCTION In many situations in economics and political science, there are gains from forming various coalitions but conflicts about which coalition Žor coalitions. should be formed and how the gains should be shared. Examples abound: the formation of a government by political parties in a parliamentary system, trade in an exchange economy, the formation of jurisdictions, and the production of public goods in a local public goods economy, etc. In such situations, the distribution of gains within each coalition involves bargaining within the coalition. If each agent can actually participate in at most one coalition, participation in a particular coalition entails an opportunity cost: the foregone rewards the agent could U

I thank Jan Aaftink, Kenneth Binmore, Randy Calvert, Chew Soo Hong, John Ferejohn, Joseph Greenberg, Bruce Hamilton, John Harsanyi, Martin Hellwig, Michael Maschler, Alvin Roth, Dov Samet, Lloyd Shapley, Richard Small, William Thomson, Myrna Wooders, and especially William Zame for their comments on various drafts. Financial support from the National Science Foundation is gratefully acknowledged. † Elaine Bennett died May 26, 1995, while this paper was under review; final revisions were completed by William Zame. 151 0899-8256r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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have received from participating in another coalition. The bargaining problem of each coalition is therefore interrelated. We call such an interrelated set of bargaining problems a multilateral bargaining problem. The simplest bargaining problem involves only a single coalition. Nash Ž1951. formalized this situation by specifying a set of players, a set of attainable utilities, and a disagreement point Žthe vector of utilities of the outcome that will result if the agents cannot come to agreement.. In the situation of interest to us, however, if agents in a given coalition cannot come to agreement in that coalition, they will form other coalitions. The disagreement vector therefore must be determined endogenously, in terms of the opportunity costs of not entering into other coalitions. Our focus here is on the interrelationships between bargaining problems of the several coalitions, rather than on bargaining within individual coalitions.1 We shall therefore take as the ingredients of a multilateral bargaining problem a set of agents Žplayers. and, for each coalition of players, a set of attainable utility vectors Žrepresenting a description of the potential gains that can be realized by the coalition., and a bargaining function Ža mapping from opportunity costs to agreement vectors for the coalition, representing a summary of the bargaining process within the coalition.. We view the attainable sets and the bargaining functions as parts of the description of a multilateral bargaining problem, in much the same way that endowments and utility functions are part of the description of an exchange economy. To see the way in which the endogenous determination of outside options affect the result, consider four professionals}a biologist, economist, lawyer, and physicist}who each have full-time jobs and also have opportunities for outside consulting. The economist and the lawyer together can represent the electric company at rate hearings for $40,000 per year. The economist and physicist together can consult for the water district for $34,000 a year. The lawyer and the biologist together can consult on toxic cleanup suits for $20,000 per year. Finally, the biologist and physicist together can write environmental impact statements for $34,000 per year. If these are the only available consulting opportunities, which partner should the economist choose and what should the terms be? 2 How should we analyze problems such as these? One way is to treat the bargaining problem for each pair in isolation, and then integrate the results. Assume for definitiveness that, gi¨ en an outside option vector, each pair solves its bargaining problem by equal division from the outside 1 Determining which bargaining function is appropriate is the domain of traditional bargaining theory; see for example, Roth Ž1979., Kalai Ž1985., and Sutton Ž1985.. 2 For simplicity we assume utility is linear in money and that only money matters.

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option.3 Considering the economistrlawyer partnership in isolation suggests the disagreement point Ž0, 0. and hence an equal division of Ž$20,000, $20,000. }conditional on the economist and lawyer forming a partnership. Similarly, the other coalitions would also divide their income equally. In anticipation of these equal divisions, the economist would wish to form a partnership with the lawyer, each of them obtaining $20,000; this would leave the biologist and the engineer to form a partnership, each obtaining $10,000. However, these partnerships are unstable: the economist and the physicist could form a partnership which would yield the economist $22,000 and the physicist $12,000. The source of this instability is plain to see: the economist and the lawyer each have profitable outside opportunities Žthe economist could form a partnership with the physicist, etc.., and the disagreement point in their partnership should reflect these opportunities. It is this endogenous determination of outside options as opportunity costs that separates a multilateral bargaining problem such as this from a disconnected set of simple bargaining problems.4 In this paper, we take player i’s outside option in coalition S to be the maximum utility he could obtain by forming an alternative coalition. By constructing outside options in the way we do, we leave open the possibility that the outside options of members of a given coalition may not be jointly compatible. For instance, for a coalition consisting of the players 1, 2, the outside options of each player may involve partnership with player 3. Since both players cannot simultaneously take up their outside options, our outside option vector does not have the interpretation of the result of disagreement in the coalition w1, 2x. ŽAnd it is for this reason that we prefer the term ‘‘outside option’’ rather than ‘‘disagreement point.’’. Section 2 contains a detailed discussion of this point, including other possible choice of outside options. It is important to keep in mind that the solution to the bargaining problem facing coalition S depends on the outside options of each of the members of S, which depend in turn on the agreements in other coalitions, which depend in turn on the outside options in these other coalitions, which depend in turn on the agreement in the coalition S. Thus, this endogenous determination of outside options links the bargaining problems for all coalitions. A solution to a multilateral bargaining problem specifies an agreement utility for e¨ ery potential coalition Žin a way that is consistent with the 3 This is consistent with the Nash bargaining solution and most other proposed solutions to the simple bargaining problem. 4 Nash uses the term ‘‘disagreement point’’ rather than ‘‘outside option;’’ we prefer the latter term since our outside options should not be viewed as the outcome that will result should the bargaining break down in a particular coalition. See the discussion in Section 2.

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bargaining within each coalition and across coalitions, via outside options..5 In general, some of these agreement utilities will be feasible for the specified coalition and some may not be. We interpret a feasible agreement as a prediction of the outcome of bargaining within that coalition, conditional on the assumption that the coalition forms, and an infeasible agreement as a prediction that the coalition will not form. We do not require that the agreement for the grand coalition be feasible, nor that there be a partition of the player set into coalitions whose agreements are feasible. An outcome consists of a partition P of the player set and a specification of a feasible agreement z S for each coalition S g P; of course, outcomes should be the end objects of any bargaining theory. We say that the outcome Ž P,  z S : S g P4. is consistent with the multilateral solution  x S 4 if z S s x S for every coalition S g P which is not a singleton, and z S s  04 for singleton coalitions in P. It may be helpful to view a solution to a multilateral bargaining problem as a set of beliefs Žconjectures. players hold about the eventual outcomes of bargaining within and across coalitions. A set of beliefs constitutes a solution if these beliefs are consistent Ži.e., all players hold the same beliefs. and stable Ži.e., given the nature of bargaining in each coalition, no player believes he can improve his payoff in any coalition by renegotiating an agreement.. Thus, a solution is a fixed point of the process of forming outside options Žbased on conjectured agreements. and agreements Žbased on outside options.. Of course, only certain outcomes are consistent with a given set of beliefs. For the four professionals, there is a unique multilateral solution: x BL s Ž12, 22., x EL s Ž22, 22., x BP s Ž12, 12., x EP s Ž22, 12.. Note that the agreements for the partnerships of the economistrlawyer and physicistr biologist are not feasible; we interpret this as a prediction that those partnerships will not form. On the other hand, the agreements for the economistrphysicist and lawyerrbiologist are feasible; conditional on formation of those partnerships we predict the first partner will obtain $22,000 and the second will obtain $12,000. Thus, antagonizing the outside opportunity vectors has a big effect on the predicted outcomes. In Section 2 we show that Žunder very standard and quite weak assumptions about the sets of attainable utility vectors and the bargaining functions. multilateral bargaining problems always have solutions, and every solution corresponds to Žat least. one outcome. Moreover, the competition between coalitions enforces a ‘‘law of one price’’: at any solution, the

5

We find it convenient to normalize so that singleton coalitions can achieve only the 0 vector; we therefore suppress singleton coalitions in the solution.

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agreement utilities for each player are the same in every Žpotential. coalition of which he is a member. As a consequence, each multilateral bargaining solution can be characterized by a vector of reservation prices Žone for each player.. As the discussion above suggests, our mechanism for determining outside options and our interpretation of solutions correspond to a particular notion of the way in which bargaining proceeds. We make this correspondence explicit in Section 4 by analyzing a noncooperative model of bargaining in a specific class of characteristic function games, those having the property that at most one coalition can profitably form. Our noncooperative model is an adaptation of a proposal-making model due to Selten Ž1981..6 We show that the stationary subgame perfect equilibria of the proposal-making model correspond precisely to the multilateral solutions Žfor some specification of bargaining functions.. We also use the proposalmaking model to offer some support and further explication of our construction of outside options.7 The idea of antagonizing outside options appears in Rochford’s Ž1984. work on matching markets. Rochford’s purpose is to identify a selection from the core; to do so, she considers only candidate agreements that are in the core. Bennett’s Ž1988. work on matching models is much more in the spirit of the present work, and in fact was an outgrowth of an earlier version of the present work. Binmore Ž1985. analyzes a particular class of three player bargaining problems Žwhich he called ‘‘three-playerrthreecake problems’’.. Binmore presents both cooperative and noncooperative solutions for three-playerrthree-cake problems; his cooperative solution is Žessentially . a multilateral solution in the sense discussed here. A different analysis of three-playerrthree-cake problems is given by Bennett and Houba Ž1992.; their cooperative solution is again a multilateral solution in precisely the sense described here. Less closely related analyses of multiplayer bargaining problems have been presented by Kalai and Samet Ž1985., Chatterjee et al. Ž1987., and Bennett and van Damme Ž1991.. Following this Introduction, Section 2 presents the formalism of multilateral bargaining problems, solutions, and consistent outcomes and establishes the existence of solutions and their basic properties. Section 3 presents a number of illustrative examples. Section 4 presents the noncooperative model and its connections to multilateral solutions.

6

Selten models the bargaining process as a recursive game; we use the more familiar structure of a game in extensive form. 7 Of course the idea of finding mutually reinforcing cooperative and non-cooperative models is due to Nash Ž1951, 1953..

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2. MULTILATERAL BARGAINING PROBLEMS We begin by giving a formal description of multilateral bargaining problems and their solutions, and then establish some of their properties. Following the formal description we provide some additional discussion of the crucial point, the construction of outside options. Illustrative examples will be presented in Section 3. The data of a multilateral bargaining problem consists of a triple ² N, V, f : where: v

N s  1, 2, . . . , n4 is a finite set of players

S for each coalition S Ža non-empty subset of N ., the set V Ž S . ; Rq S is a compact, strongly comprehensive subset of Rq that contains the origin.8, 9 v

f s  f S 4 is a collection of functions, indexed by the coalitions S S ª Rq is the bargaining function of the S ; N; for each S ; N, f S : Rq coalition S. v

The interpretation we intend is that V Ž S . is the set of attainable utility vectors for the coalition S, and that the bargaining function f S summarizes the bargaining process within the coalition S, given outside options.10 Given the outside option vector d S , we interpret f S Ž d S . as the division of the proceeds, conditional on the formation of the coalition S. Throughout, we assume that each bargaining function f S satisfies the following properties: 1. Individual rationality: If d S g V Ž S . then f S Ž d S . G d S 2. Pareto optimality: If d S g V Ž S . then f S Ž d S . is on the Pareto frontier of V Ž S . f S is a continuous function of the outside option vector d S 4. Agreeing to disagree: If d S f V Ž S . then f S Ž d S . s d S 3. Continuity:

8 S By Rq we mean the nonnegative orthant of R S . V Ž S . is strongly comprehensive if S whenever x g V Ž S ., y g Rq with y F x and y / x then y is in the interior of V Ž S ., relative S to Rq . Equivalently, the weak and strong Pareto boundaries coincide. 9 Note that the pair ² N, V : is a game in characteristic function form without sidepayments. 10 The assumption that bargaining within each coalition can be summarized by a function is simply the assumption that the outcome of bargaining is determinate: the same people facing the identical bargaining problems reach identical agreements. A coalition’s bargaining function may reflect the coalition’s standards of fair division, the institutional rules governing bargaining in the coalition, or the relative bargaining skills of its members.

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If the outside option vector d S belongs to V Ž S . then there are attainable utility vectors for the coalition which allocate to each member of the coalition at least the utility of his outside opportunities Ži.e., there are gains from making an agreement.. In this case, the first two assumptions require that the agreement f S Ž d S . be efficient and allocate to each member of the coalition at least as much as he could obtain by not participating. The third assumption requires that small changes in players’ outside options lead to small changes in the agreement. These three properties are familiar from classical bargaining theory and are enjoyed by virtually every solution concept proposed for the simple bargaining problem. The case d S g V Ž S . is the only one considered in traditional bargaining theory. As discussed earlier, however, in our setting we must allow for the possibility that the outside option vector d S does not belong to V Ž S ., because the outside option vector d S represents the opportunities of members of S in other coalitions, and it is certainly possible that no alternative for S is as good for all of its players as their outside options. In this case, the last assumption says that the members of S ‘‘agree to disagree.’’ When extended to allow for infeasible outside option vectors, all of the usual solutions to the simple bargaining problem Žincluding the Nash bargaining solution, the egalitarian bargaining solution, and the Kalai]Smorodinsky solution. enjoy all four properties. To avoid degeneracy, we require that V Ž S . /  04 for at least one coalition S. It is convenient Žand involves no loss of generality. to ‘‘zeronormalize’’ so that V Ž i4. s  04 for each player i; in what follows it is convenient to suppress the trivial singleton coalitions. We write C for the set of nonsingleton coalitions S ; N, and CU for the subset of C consisting of coalitions for which V Ž S . /  04 . For notational convenience, we let Q be an index set that contains a distinct index for each occurrence of a player in one of his coalitions, i.e., for each player position. Since each player position is distinct, < Q < s n2 ny 1. We identify an element x g R Q with a set of vectors, one for each coalition:  x S < S g C 4 , where x S g R S . The set of bargaining functions Q Q f s  f S < S g C 4 defines a mapping f : Rq ª Rq , which is given by f Ž x . s S S  f Ž x .4 . We now turn to the construction of outside options. For each coalition S and each player i g S, we want to use as i’s component of the outside option vector d S the utility he would obtain if player i broke off negotiations in S and took the initiative to form hisrher best alternative coalition. ŽWe return to this point shortly.. Of course, i’s alternatives depend on the agreements that will be reached in other coalitions. We assume that the players in S make accurate Žand therefore identical. conjectures about

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these agreements. To see what this implies, fix conjectured agreement x T for each coalition T / S; given these conjectures, what are the utilities of player i’s alternatives? If i g T and the agreement x T is feasible for the coalition T Ži.e., x g V ŽT .., then player i will obtain x Ti if the coalition T forms. v

T

If i g T and the agreement x T is not feasible for the coalition T Ži.e., x T f V ŽT .., player i cannot obtain x Ti . In view of our previous discussion about the utility of infeasible agreements, the most that player i can obtain in T is the largest utility which allows all of the other members of T to obtain their agreement utilities. That is, the utility to player i of the unattainable agreement x T in the coalition T is max t i < x T rt i g V ŽT .4 . v

We use x Trt i to denote the vector obtained from x T by replacing the ith component by t i . If there is no value of t i for which x T rt i g V ŽT ., then the infeasible agreement x T has no utility for player i; by convention, we take 0 to be the maximum in this case.11 Formally, given agreements  x T < T / S 4 in all other coalitions, we define the outside option vector d S Ž x T < T / S4. for the coalition S in the following way. For each i g S and each coalition T / S with i g T, set: uTi Ž x T . s

½

x Ti

if x T g V Ž T .

max  0, t i : x T rt i g V Ž T . 4

otherwise

and d iS Ž  x T < T / S 4 . s max  uTi Ž x T . i g T and

T / S4 .

Note that, although we take into account coalitions T in which the conjectured agreement x T is infeasible, we only ascribe to player i the utility t i he would obtain if T actually came to a feasible agreement x Trt i . S By definition, d S is a function from collections  x T < T / S4 to Rq , but it S is convenient to view d as a function defined on collections of agreement vectors for all coalitions Žalthough d S will not depend on x S ., so that d S Q S becomes a function from Rq to Rq . We refer to d S as the outside option function for the coalition S. We sometimes write d s  d S < S g C 4 , and Q Q view d as a function from Rq to Rq . A multilateral bargaining problem ² N, V, f : specifies a set N of players, and a set V Ž S . of attainable utilities and a bargaining function f S for each coalition S ; N. A solution to such a multilateral bargaining problem 11

This is harmless, since 0 g V w i x for each i.

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specifies an agreement payoff vector for each coalition in such a way that the agreements are consistent with the bargaining in every coalition and the determination of outside options described above. Formally, x s  x S < S g C 4 is a solution to the multilateral bargaining problem ² N, V, f : if x S s f S Ž d S Ž x .. for every coalition S g C. Our interpretation of solution agreements is that, if x S is feasible for the coalition S Ži.e., x S g V Ž S .., then x S is the agreement which will be implemented in S, provided the coalition S actually forms; if x S is not feasible for the coalition S Ži.e., x S f V Ž S .., then S will not form. By an outcome Ž P, z . we mean a partition P of N and a payoff vector z g R N such that for each S g P, the restriction of z to S, z S, is a feasible utility vector for S, i.e., z S g V Ž S .. We say that the outcome Ž P, z . is consistent with the multilateral solution  x S < S g C 4 if z S s x S for all coalitions S g P l C and z i s  04 for every singleton  i4 g P. That is, for coalitions in the partition, the agreement is feasible and is that specified by the multilateral solution  x S < S g C 4 , and remaining players do not enter into coalitions and obtain a payoff of 0. Note that a solution is not an outcome. A solution is a set of conditional agreements for all coalitions and some of these agreements may be infeasible. An outcome specifies a set of agreements and coalitions so that each player belongs to exactly one coalition of this set and every agreement is feasible for its coalition. Thus it is important to distinguish between a solution and its outcomes. In particular, note that a single solution may give rise to more than one outcome. The following theorem shows the existence of solutions to multilateral bargaining problems and summarizes their basic properties: Ži. solutions are nonnegative and not identically 0; Žii. solution agreements obey the law of one price}all agreement payoffs for a given player are identical; Žiii. if player i’s agreement payoff is positive in some coalition then it is feasible in some Žperhaps different coalition.; Živ. for every player, there is some outcome consistent with the solution at which that player achieves his agreement payoff}in particular, there are outcomes consistent with each multilateral solution. In what follows, recall that C is the set of nonsingleton coalitions. THEOREM 1. If ² N, V, f : is a multilateral bargaining problem, then a solution x s  x S < S g C 4 exists. Moreo¨ er, if x is any solution then: Ži. x G 0 and x / 0 Žii. If i g S and i g T, then x iS s x Ti . Žiii. If x iS ) 0, then there is a coalition T g C with i g T for which T x g V ŽT .. In particular, at least one of the solution agreements  x S : S g C 4 is feasible.

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Živ. For e¨ ery i g N there is an outcome Ž P, z . for which z i s x iS Ž for each S g C .. Proof. In order to establish the existence of solutions, we show first that the outside option function d S of each coalition S is continuous. In X view of the definition, it suffices to show that u Si is continuous for each SX with i g SX . To end this, fix T, i with i g T and define h Ž x T . s max  0, t i < x T rt i g V Ž T . 4 . We assert that hT is a continuous function. To see this, let  x n4 be a Q sequence in Rq converging to x; we show that hT Ž x n . ª hT Ž x .. Note first that, since the range of hT is bounded Žhence compact., some subsequence of  hT Ž x n .4 converges. Passing to a subsequence if necessary we may assume that hT Ž x n . ª w for some w; we wish to show that w s hT Ž x .. For notational convenience, renumber the players in T with player i s 0 and the remaining members of T as 1, 2, . . . , k. Set z n s Ž x n .T and z s x T . Clearly, z n ª z. If hT Ž x . / w, there are two cases to consider. Case 1. hT Ž x . - w s lim hT Ž x n .. By definition, hT Ž y . G 0 for every y, so lim hT Ž x n . ) hT Ž x . G 0 and hence hT Ž x n . ) 0 for n sufficiently large. Hence, Ž hT Ž x n ., z n . g V ŽT . for n sufficiently large. Since V ŽT . is closed, we conclude that Ž w, z . g V ŽT .. But since hT Ž x . is the maximum feasible value, this implies that hT Ž x . G w, a contradiction. Case 2. hT Ž x . ) w. By definition, hT Ž x n . G 0 for each n. Hence hT Ž x . ) 0 so Ž hT Ž x ., z . g V ŽT .. Since hT Ž x . ) w ) 0, strong comprehensiveness guarantees that Ž w, z . is in the interior of V ŽT .. This means that there is a «-ball around Ž w, z . contained in V ŽT . for some « ) 0. In particular, Ž w q «r2, z n . g V ŽT . for n sufficiently large. This implies that hT Ž x n . G w q «r2 for n sufficiently large, again a contradiction. We conclude that hT Ž x . s w, so that hT is a continuous function for each T. By definition, uTi Ž x T . s x Ti

of x T g V Ž T .

uT0 Ž x T . s hT Ž x T .

otherwise

Since the functions x T ª x Ti and hT are both continuous and h i Ž x T . s x Ti whenever x T belongs to the boundary of V ŽT ., the function uTi is continuous. It follows that the outside option functions d S are continuous, as desired. Let m be a number sufficiently large that each V Ž S . fits inside a cube with sides of length m. Let Y be the Q-fold Cartesian product of w0, m x.

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The functions f and d both map Y into itself. By Brouwer’s fixed point theorem, the mapping f ( d has a fixed point x s  x S < S g C 4 . For each S g C, x S s f S Ž d S Ž x .., so x is a solution to the multilateral bargaining problem ² N, yV, f :, as desired. This completes the existence proof. Ži. To see that each x S G 0 note that by definition d S G 0 and that bargaining functions are required to be individually rational. To see that x / 0, recall that to avoid degeneracy we assumed that there is a coalition S with V Ž S . /  04 . Strong comprehensiveness requires that 0 belongs to the interior of V Ž S ., so an agreement x S s 0 would violate Pareto optimality of bargaining functions. Žii. Fix a player i and a coalition S containing i which maximizes x iS , and let U be any coalition containing i from which d iS is calculated. Then d iS F x Ui F x iS. There are two cases to consider. If d iS - x Ui then d iS - x iS and d S / x S, so Property 3 of bargaining functions implies that x S is feasible. For every coalition T containing i and differing from S, the definition of outside options implies that d iT G x iS, and individual rationality implies that x Ti G d iT and maximality of x iS yields x iS s x Ti . On the other hand, if d iS s x Ui s x iS then the definition of outside options implies that x U is feasible, whence the same argument as above Žbut with U playing the role of S . yields that x Ti s x Ui for every coalition T containing i. Setting pi to be the common value of x Ti establishes Žii.. Žiii. Let pi be the common value of x Ti . Since pi ) 0, fix a coalition S containing i, and suppose that, for every coalition T containing i and differing from S, the agreement x T is not feasible for the coalition T. The definition of outside options implies d iS - x Ti s pi for each T, whence d S / x S. But then Property 3 of bargaining functions implies that x S is feasible for S. This establishes Žiii.. Živ. Fix a player i. Let D be the collection of all coalitions S ; N for which x S g V Ž S .. We distinguish two cases. Case 1. pi ) 0. Part Žiii. guarantees that there is a coalition T g D with i g T. Let E be any subset of D which contains the coalition T, consists of disjoint coalitions, and its maximal with respect to these properties. Let EU be the union of the coalitions in E. Define the partition P of N to consist of E and the singleton coalitions in N _ EU . Define the vector z g R N by z S x S for X g EU and z i s 0 for j g N _ EU . It is easily checked that Ž P, z . is an outcome consistent with the solution x and z i s x Ti ; that z j s x iS for each S g C follows from the law of one price. Case 2. pi s 0. Let F be any subset of D which consists of disjoint coalitions, and is maximal with respect to this property. Let F U be the

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union of the coalitions in E. Define the partition P of N to consist of F and the singleton coalitions in N _ F U . Define the vector z g R N by z S s x S for S g F U and z j s 0 for j g N _ F U . It is again easily checked that Ž P, z . is an outcome consistent with the solution x; the construction guarantees that z i s 0 s x iS for each S g C. This completes the proof. B In view of part Žii. of Theorem 1, corresponding to every solution Q Ž x g Rq i.e., x s  x S < S g C 4. we may associate the vector p g R N defined by pi s x iS for any S with i g S. Note that pi is the utility player i would obtain in any feasible agreement to which he is a party. Moreover, parts Žiii. and Živ. guarantee that pi actually is feasible for player i Žif pi ) 0 then there is a coalition S containing i such that x S is feasible and if pi s 0 then player i can obtain pi on his own.. It is natural to interpret pi as a reservation price for player i; we refer to p as the reservation price vector of the solution x s  x S < S g C 4 . Discussion: The Meaning of Outside Options Outside options play a crucial role in our theory, so it seems appropriate to conclude this section with a more elaborate discussion of the meaning of outside options and the motivation for our choice. If the players in the coalition S fail to reach an agreement one or more players may enter into alternate coalitions; the opportunity cost for each member of S is thus the utility of foregone agreements in alternative coalitions. We should therefore take as i’s outside option, d iS, some utility level that represents a summary of these foregone agreements. Many different summary methods seem possible, each corresponding to a particular view of the way in which bargaining proceeds. We have taken d iS to be the utility i would obtain from agreement in his best alternative coalition.12 Of course, this represents an optimistic assessment. Below we discuss a particular alternative Ž‘‘reductionist’’. scheme and argue that it corresponds to quite a different view of multilateral bargaining. In Section 4 we present a formal noncooperative bargaining model which formalizes our particular view of the way in which bargaining proceeds and leads in effect to our particular method of assigning outside options. Our construction of outside options leaves open the possibility that the outside options of members of a coalition S may not be jointly compatible. For instance, players 1, 2 g S may each rely on a pairing with player 3, but 12

We rely on the self-interest of i’s partners in the alternative coalition to guarantee that their components of the agreement are at least as large as they can obtain in their alternative coalitions. One property of the solution we present is that it is compatible with this type of self-interest.

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these pairings cannot both be realized. In particular, that means we cannot interpret our outside option vectors as the result of disagreement in the coalition S. ŽAnd it’s for this reason that we prefer the term ‘‘outside option’’ rather than ‘‘disagreement point.’’. To motivate our interpretation of outside options, it is useful to first recall Binmore’s Ž1985. analysis of the role of outside options in two-player simple bargaining problems. Binmore analyzes a sequential model of bargaining between two players who must choose a utility vector Ž x 1 , x 2 . in the Žconvex. set V, with a given Žfeasible . outside option vector d s Ž d1 , d 2 .. Binmore’s analysis is based on Rubinstein’s Ž1982. alternating offer model: player 1 makes an offer, player 2 may accept Žin which case play ends. or decline; if player 2 declines, he may terminate bargaining or make an offer in turn, etc. Players discount future payoffs according to the common discount factor r - 1, and infinite plays result in a payoff of 0 to both players. Binmore analyzes two different scenarios. In the first, the result of termination of bargaining by both players is that they obtain their component of the vector d. In the second scenario, the result of termination of bargaining by player i is that player 1 obtains his component of d while the other player obtains nothing. In the first of these scenarios, d plays the role of the ‘‘disagreement point’’ or ‘‘status quo point’’ in the sense that Nash Žand others. discussed. In the second scenario, however, d plays the role of defining strategic choices for each player. Despite the difference in interpretations, Binmore shows that two scenarios lead to precisely the same Žsubgame perfect equilibrium. outcomes. As the discount factor r tends to 1 Žso that players become more patient., this common outcome is the Nash bargaining solution Žsee Section 3 for a definition.. The conclusion in each of these scenarios is driven by the same intuition that drives the conclusion of the basic Rubinstein alternating offer model. When player 1 makes an offer, he is committed until player 2 responds; while 1’s offer is on the table, player 2 can take up his outside option, but player 1 cannot. And player 2’s decision to terminate bargaining and take up his outside option is affected only by his own component of d, and not by player 1’s component. Our interpretation of outside options corresponds to a similar view of the way in which bargaining proceeds: Players bargain sequentially, and an offer binds those who make it Žand those who have accepted it., but not those who have yet to respond to it. From the time that player 1 makes an offer to player 2 Žfor participation in some particular coalition S . until the time that the offer expires, player 1 is committed to the offer. But, until player 2 responds, player 2 is not committed, so player 2 can take her outside option but player 1 cannot take up his. In our multiplayer setting, 2’s outside option represents the utility she could obtain if she broke off

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negotiations with 1 in the coalition S and took the initiative to form some other coalition. In this circumstance, of course, it would be player 2 that would have the initiative, and not player 1. It therefore seems reasonable to take as 2’s outside option the utility of her best alternative agreement. What utility should player i anticipate if he broke off negotiations in the coalition S in order to enter into an agreement in the coalition T ? If it were common knowledge that members of T would come to a feasible agreement z T, the answer would be z iT. However, if bargaining proceeds simultaneously in all coalitions, the agreement in T cannot be common knowledge Žin advance.. On the other hand, player i and every other player will form conjectures about the potential agreement in the coalition T. At a Žperfect foresight. equilibrium, these conjectures will be correct Žand therefore equal., so we should impute to player i the utility resulting from this common conjecture. Why should player i believe that if he breaks off negotiations in S he will be able to form his most preferred alternative coalition, say T ? After all, members of T will try to obtain their most preferred alternatives}which might not include forming the coalition T. If this were the case, player i might find that when he breaks off negotiations in S and initiates negotiations in T, he cannot obtain the cooperation of other members of T. It would seem, therefore, that we should take for player i’s outside option the maximum utility he could obtain in any alternative coalition T, subject to the additional requirement that all other members of T find T to be their most preferred alternative. However, as we show in Theorem 1, at a multilateral solution the difference between these choices disappears: at a multilateral solution it will always be the case that all the members of i’s most preferred alternative coalition T also find T to be at least as good as any other alternative. If the outside option vector d S for the coalition S is not feasible, members of S will not agree to any feasible division in S Žbecause for any feasible division, at least one of them could do better by making some other agreement., so the coalition S will not form. In this case, we adopt the convention that the agreement coincides with the outside option vector, keeping in mind that we interpret such an infeasible agreement as an agreement not to form the coalition S. The possibility of infeasible agreements raises one final question. We use as i’s outside option Žin the coalition S . the maximum utility he or she could obtain by entering into an agreement in some alternative coalition T, but we allow for the possibility that the coalition T will not come to a feasible agreement; what utility should we impute to player i in this case? Depending on the point of view taken, this question could have several different answers. The most obvious is that we should impute zero utility to player i when the Žconjectured. agreement in T is infeasible. An

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alternative view, and the one we shall adopt, is that we should impute to player i the maximum utility he or she could obtain in any feasible agreement which yields all other members of T at least their utilities at the Žconjectured. infeasible agreement. In Fig. 1 below, tUi is the utility we impute to player i for the infeasible agreement x T in the coalition T s w i x. In the first graph x Trt Uj is the feasible agreement which maximizes player i’s utility while yielding j her agreement utility x Tj , and in the second graph tUi s 0 because no feasible agreement for T yields j at least her agreement utility x Tj . To see why this evaluation of infeasible agreements is quite natural in this context, recall that if x T is infeasible then the agreement in T coincides with the outside option vector, d T. In that case, each member j of T could obtain at most her outside option d Tj by coming to agreement in her most preferred alternative coalition Uj . Presumably, player j would accept any offer from player i that yielded j a payoff of d Tj q « , for any « ) 0. Hence player i can expect to obtain as much as the residual value of the coalition T after all of his partners have been paid their outside option values, d Tj . Note that part Žii. of Theorem 1 justifies our earlier discussion about outside options and most-preferred alternatives. Player i’s outside option d iS in the coalition S is the maximum he could obtain in any alternative coalition; why should he expect to be able to form his best alternative coalition? If d iS s 0, nothing more needs to be said, since i can obtain 0 by himself. If d iS ) 0, the definition of outside options requires that there be some coalition T for which the vector y T s x Trd iS is feasible. Were player i to offer members of T their components of y T q « for any « ) 0, each of them would be willing to accept, since Žii. guarantees that, for j g T and j / i, y jT s x Tj is the maximum utility player j could obtain in an agreement in any coalition. Finally, our construction of outside options may be illuminated by discussing an alternative route we did not take.13 To calculate the outside option values of the players in S, determine the solution for the reduced bargaining problem in which the coalition S has no value. Formally, given a multilateral bargaining problem ² N, V, f : and a coalition S, consider the reduced bargaining problem ² N, V X , f : such that V X Ž S . s  04 and V X ŽT . s V ŽT . for all T / S. Given the solution agreements for the reduced bargaining problem ² N, V X , f :, we may compute outside options for the coalition S on the basis of this outcome. Given the outside option for the coalition S, we may then compute the agreement in S using the bargaining function f S. Of course, we want to compute the predicted outcome for the reduced bargaining problem ² N, V X , f : by the obvious inductive proce13

We thank Martin Hellwig for stimulating discussion on this point.

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dure: For each T, form a reduced bargaining problem ² N, V Y , f :, compute outside options on the basis of the agreements in ² N, V Y , f :, and then compute agreements for ² N, V X , f : using the bargaining functions f T , etc. ŽA similar strategy has been followed by Crawford and Rochford Ž1986... Although this ‘‘reductionist’’ scheme has some appeal, it is not without its drawbacks. We cannot really expect to be able to predict a unique outcome or even a unique solution for each multilateral bargaining problem. But if the reduced bargaining problem ² N, V X , f : admits multiple solutions, the outside option vector for the coalition S cannot be the unique vector of payoffs to members of S that would result if S were not to form. That leaves two possibilities. The first is to settle for multiple outside option vectors for the coalition S; this seems unsatisfactory. The second is to take as the outside option vector for the coalition S some summary of outcomes of the reduced bargaining problem, but it seems unclear what sort of summary to use. More importantly, the ‘‘reductionist’’ scheme fails to capture what seems to us to be an essential feature of multilateral bargaining. Suppose for simplicity that only the coalitions S, T yield nonzero gains; to avoid triviality, suppose also that S l T / B. The bargaining within S is influenced by the bargaining within T because the latter defines the opportunities which members of S must forego if T forms. In our scheme, the bargaining within T is in turn influenced by the bargaining within S because the latter defines the opportunities which members of T must forego if S forms, and so forth. Thus the bargaining problems of both coalitions are inextricably linked. The ‘‘reductionist’’ scheme would view the bargaining within S as influenced by the bargaining within T, but would then image the bargaining within T proceeding in isolation, ignoring the influence of the bargaining within S. To make the same point dynamically, think about the behavior of an individual i g S l T. In our scheme, i might negotiate first with members of S; if that negotiation broke down, i might then negotiate with members of T, and later with members of S again, and so forth. ŽThe noncooperative model we present in Section 4 has exactly this feature.. By contrast, the ‘‘reductionist’’ scheme would allow for i to negotiate first with members of S, and then, if negotiations broke down, to negotiate with members of T but without the possibility of returning to members of S.

3. EXAMPLES In this section we present a series of illustrative examples to give some insight into the nature of multilateral solutions and their outcomes, and to provide comparisons with other solution concepts. Recall that the data of a

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multilateral bargaining problem include the specification of a bargaining function for each coalition. The most familiar bargaining functions are the solutions to the simple bargaining problem, such as the Nash bargaining solution ŽNash, 1950., extended to allow for outside option vectors which are infeasible. Formally, given the coalition S and the convex feasible set S S V Ž S ., we define the Nash bargaining function N S : Rq ª Rq in the S S S following way: For d g V Ž S ., N Ž d . is the vector which maximizes the Nash product Ł i g S Ž x iS y d iS . over all x S g V Ž S .; for d S f V Ž S ., N S Ž d S . s d S.14 If all coalitions use the Nash bargaining function, we refer to the multilateral bargaining problem as a Nash multilateral bargaining problem and to its solution as a Nash multilateral solution.15 EXAMPLE 1 ŽThree-PlayerrThree-Cake Problems.. The simplest interesting multilateral bargaining problems are the three-player situations in which individuals and the coalition consisting of all three players earn nothing, so that only the two-player coalitions are profitable to form. If the coalition w i, j x forms, it has the ‘‘cake’’ V w i, j x to divide, but since there are only three players, at most one of these three cakes will actually be divided. This class of bargaining problems has recently been the subject of papers by Binmore Ž1985. and by Bennett and Houba Ž1992.. Depending on the nature of the core of the underlying game we have three cases to consider. Case 1. The core of ² N, V : is empty. In this case, there is a unique vector q s Ž q1 , q2 , q3 . such that q S is on the Pareto efficient boundary of V Ž S . for S s w1, 2x, w2, 3x, w1, 3x; i.e., there is a unique vector q such that all three coalitions can afford to pay their members exactly their components of q. For e¨ ery choice of bargaining functions, q is the vector of reservation prices of a multilateral solution.16 For the Nash, egalitarian, or Kalai]Smorodinsky bargaining functions, or indeed for any bargaining

14 Recall that the Nash bargaining solution}as well as some others}are defined only for con¨ ex attainable utility sets; in such cases we will also assume convexity. 15 Egalitarian E S and Kalai]Smorodinsky K S bargaining functions and multilateral solutions may be defined by similarly extending the egalitarian and Kalai]Smorodinsky bargaining solutions. See Kalai Ž1985. for a description of these solutions for simple bargaining problems. 16 To see why, note that in each coalition each player’s outside option is his reservation price Žsince it is part of a feasible agreement in another coalition. and since the coalition’s outside option vector lies on the boundary of the attainable set it is the only feasible, individually rational agreement. ŽWhen there is no surplus to divide, the division rule is irrelevant..

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FIG. 1. Nash multilateral solution for a game with an empty core.

functions that are strictly individually rational,17 this vector q is the vector of reservation prices of the unique multilateral solution. See Fig. 1. Note that none of the Nash multilateral agreements in Fig. 1 is the Nash agreement for the simple bargaining problem with Ž0, 0. as the disagreement point, i.e., competition among the coalitions forces the agreements away from the solutions the coalitions would have reached in isolation. For

17 We say that f S is strictly individually rational if f S Ž d S . 4 d S whenever D S g int V Ž S . S . Žrelative to Rq .

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this solution there are three possible outcomes:

w 1 x , w 2, 3 x 4 ; Ž 0, q2 , q3 . .

Ž  w1, 2x , w3x 4 ; Ž q1 , q2 , 0. . Ž  w1, 3x , w2x ; Ž q1 , 0, q3 . .

i.e., any of the three cakes might be divided, but whatever cake is divided will be divided according to the reservation price vector q. Case 2. The core of ² N, V : contains a unique point q s Ž q1 , q2 , q3 .. This core point is supported by a partition of coalitions, say w1x, w2, 3x4 . For this case q1 s 0 and Ž q2 , q3 . g V wy2, 3x. It follows from strong comprehensiveness of V w2, 3x and the fact that q is the unique point in the core that Ž q1 , q2 . g V w1, 2x and Ž q1 , q3 . g V w1, 3x. As in Case 1 above, it follows that, for every choice of bargaining functions, q is the reservation price vector of a multilateral solution. For this solution there are three possible outcomes:

Ž  w1x , w2, 3x 4 ; Ž 0, q2 , q3 . . Ž  w1, 2x , w3x 4 ; Ž 0, q2 , 0. . Ž  w1, 3x , w2x 4 ; Ž 0, 0, q3 . . i.e., the payoff in the coalition w23x is according to q and if player 1 succeeds in forming a coalition Žeither w12x or w13x., he will obtain 0. Case 3. The core of ² N, V : contains more than one point. It is not hard to see that all points in the core are supported by the same partition of coalitions, say w1x, w2, 3x4 . For i s 2, 3 let t i s max t <Ž0, t . g V w1, i x4 . The fact that the core of ² N, V : contains more than one point implies that the point Ž t 2 , t 3 . lies in the interior of V w2, 3x. Let N 23 Ž t 2 , t 3 . be the Nash solution in the coalition w2, 3x, given the outside option vector Ž t 2 , t 3 .. Then Ž0, N 23 Ž t 2 , t 3 .. g R 3 is the reservation price vector of the unique Nash multilateral solution. See Fig. 2. Since the agreements for the coalitions w1, 3x and w2, 3x are infeasible the Nash multilateral solution has a unique outcome

Ž  w1x , w2, 3x 4 ; Ž 0, N 23 Ž t 2 , t 3 . . . , i.e., the cake V w23x will be according to the Nash solution for the outside option vector Ž t 1 , t 2 . and player 1 obtains nothing. Similarly, w1x, w2, 3x4 ; Ž0, E 23 Ž t 2 , t 3 ... is the unique outcome of the Žunique. egalitarian multilateral solution, and Žw1x, w2, 3x4 ; Ž0, K 23 Ž t 2 , t 3 ... is the unique outcome of the Žunique. Kalai]Smorodinsky multilateral solution. Indeed, whenever the bargaining functions f S are strictly individually rational, Ž0, f 23 Ž t 2 , t 3 .. is the reservation price vector of the unique multilateral solution and Žw1x, w2, 3x4 ; Ž0, f 23 Ž t 2 , t 3 ... is its unique outcome.

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FIG. 2. Nash multilateral solution for a game with an interval core.

The reader should not be misled. In three-playerrthree-cake problems strict individual rationality of bargaining functions is enough to guarantee uniqueness of the multilateral solution. For more general multilateral bargaining problems, strict individual rationality of bargaining functions substantially restricts the range of possible multilateral solutions, but is not generally enough to guarantee uniqueness. In the Nash bargaining solution, outside options are viewed as the basis of negotiations; an alternative view is to treat outside options as constraints on the solution. Adopting this view leads us to define the constrained Nash bargaining function in the following way. Given a coalition S S and an attainable set V Ž S . ; Rq , the constrained Nash bargaining S S S S solution CN : Rq ª Rq maps an outside option vector d S g Rq to the SŽ S . agreement CN d defined by: If d S f V Ž S . then CN S Ž d S . s d S If d S g V Ž S . then CN S Ž d S . g V Ž S . is the arg max of the Nash product Ł i g S x i over the set  x g V Ž S .: x G d S 4 .

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Note that CN S is individually rational but not strictly individually rational. That is, CN S Ž d S . G d S for all d S , but equality CN S Ž d S . s d iS is possible for some components i g S, even when d Sf int V Ž S .. Loosely speaking, the distinction between individual rationality and strict individual rationality may be seen in the response to an outside offer: individual rationality requires that outside offers be met Žwhen possible., and strict individual rationality requires that outside offers be exceeded Žwhen possible.. For three-playerrthree-cake bargaining problems, Bennett and Houba Ž1992. characterize the constrained multilateral Nash solutions and Binmore Ž1985. characterizes the multilateral solutions of a variant of the constrained Nash bargaining function. EXAMPLE 2 ŽNash and Constrained Nash Multilateral Solutions.. To best illustrate the difference between the Nash and constrained Nash bargaining solutions Žand, more generally, between the multilateral solutions of bargaining functions that are strictly individually rational and those that are only weakly individual rational. consider the threeplayerrthree-cake Žtransferable utility. problem with attainable utility sets given by: 2 < V w 12 x s  Ž y 1 , y 2 . g Rq y 1 q y 2 F 50 4 2 < V w 23 x s  Ž y 2 , y 3 . g Rq y 2 q y 3 F 30 4 2 < V w 13 x s  Ž y 1 , y 3 . g Rq y 1 q y 3 F 10 4

V Ž S . s  0 4 for every other coalition S ; N. This bargaining problem has a unique Nash multilateral solution generated by the reservation price vector Ž15, 35, 0. and a unique outcome Žw12x, w3x4 ; Ž15, 35, 0... If, however, we consider the constrained Nash bargaining functions, the situation is quite different: the reservation price vectors of constrained Nash multilateral solutions form the interval Ž25 y l, 25 q l, 5 y l.<0 F l F 54 . For all l - 5, w1, 2x and w1, 3x are the feasible coalitions. The endpoint l s 0 corresponds to equal division in the coalition w1, 2x and, intuitively speaking, is the solution for situations in which player 1 is unable to ‘‘play off’’ his two potential partners to obtain a higher payoff for himself; while the endpoint l s 5 corresponds to the point where player 1 cannot obtain a higher payoff from player 3 in w1, 3x }because 3 is obtaining nothing already}and hence cannot bargain for more from player 2; this corresponds to situations where player 1 has exhausted all gains from playing off his potential partners. Cooperative solution concepts typically consider candidate payoff vectors with one component for each player; multilateral solutions consider

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candidate payoff vectors with one component for each player in each potential coalition. One may wonder whether ‘‘all this additional complication’’ is really necessary. To see what is gained, first recall that cooperative solution concepts, such as the core, bargaining set, kernel, nucleolus, follow a common approach: consider a particular candidate payoff vector and ask whether it is stable Žthe notion of stability employed distinguishes the various solution concepts.. When testing for stability, the payoff vector is treated as the ‘‘status quo’’ in the negotiations. The multilateral bargaining approach instead uses the Žconjectured. agreements in other coalitions. The following example highlights the difference this can make. EXAMPLE 3 ŽThe Bargaining Set and The Nash Multilateral Solution.. Consider the 5-player sidepayment game in which only the cyclic threeplayer coalitions are profitable: CU s  w 1, 2, 3 x , w 2, 3, 4 x , w 3, 4, 5 x , w 4, 5, 1 x , w 5, 1, 2 x 4 3 < V w i , j, k x s  Ž yi , y j , y k . g Rq yi q y j q y k F 6 4

for w i , j, k x g CU

V Ž S . s  0 4 for all other coalitions S ; N. For the coalition structure w1, 2, 3x, w4x, w5x4 the payoff vector Ž3, 0, 3, 0, 0. is the unique payoff vector in the bargaining set of Aumann and Maschler Ž1964.. Although the game is cyclically symmetric, the payoff within w1, 2, 3x is not. To see why, note that once the coalition w1, 2, 3x has formed and any candidate payoff vector Ž y 1 , y 2 , y 3 , 0, 0. is selected, the symmetry of the situation is broken. Players 1 and 3 have alternate coalitions in which their prospective partners Žplayers 4 and 5. are currently earning nothing}and so presumably would accept nearly nothing. Player 2’s alternative coalitions, however, involve either player 1 or player 3, either of whom must receive at his yi . As a consequence, there are ‘‘justified objections’’ to every payoff vector that allocates player 2 a strictly positive payoff. That there are no ‘‘justified objections’’ to the payoff vector Ž3, 0, 3, 0, 0. seems entirely appropriate gi¨ en that Ž3, 0, 3, 0, 0. is the status quo vector. If, however, no outcome can be viewed as the ‘‘status quo,’’ this reasoning loses much of its force. In that case, it seems appropriate to use as a basis for negotiations within the coalition w1, 2, 3x the agreements that would be reached through bargaining in the alternative coalitions. This of course is the approach of multilateral bargaining solutions. As a consequence the multilateral solution respects the underlying symmetry of the game: the unique Nash multilateral solution is generated by the reservation price vector Ž2, 2, 2, 2, 2.. If the coalition w1, 2, 3x forms, the associated outcome has the payoff vector Ž2, 2, 2, 0, 0..

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Simultaneous ¨ s Sequential Coalition Formation As the next example illustrates, the problem of modeling coalition formations may become more complicated if coalition formation may take place sequentially, rather than simultaneously. EXAMPLE 4. Consider again the bargaining problem Žfrom the Introduction. of the economist Žplayer 1., lawyer Žplayer 2., biologist Žplayer 3., and physicist Žplayer 4.. The attainable utility sets 18 for each profitable coalition are: 2 < V w 1, 2 x s  Ž y 1 , y 2 . g Rq y 1 q y 2 F 40 4 2 < V w 2, 3 x s  Ž y 2 , y 3 . g Rq y 2 q y 3 F 34 4 2 < V w 3, 4 x s  Ž y 3 , y4 . g Rq y 3 q y4 F 20 4 2 < V w 1, 4 x s  Ž y 1 , y4 . g Rq y 1 q y4 F 34 4

V Ž S . s  0 4 for all other coalitions S ; N. This problem has a unique Nash multilateral solution Ždisplayed in Fig. 1. with the reservation price vector p s Ž22, 22, 12, 12.. The only feasible agreements are for the coalitions w1, 4x and w2, 3x. If these coalitions form, the outcome will be Žw1, 4x, w2, 3x4 ; Ž22, 22, 12, 12... This outcome is entirely sensible if we imagine that coalitions form simultaneously. To see what might happen if coalitions form sequentially, let us examine this bargaining problem in a bit more detail. Consider player 4. If 4 agrees to form a coalition with player 1 the division of the payoff in the coalition w1, 4x will be Ž22, 12.. On the other hand, if 4 can wait for players 2 and 3 to form a coalition and lea¨ e, what will remain will be a simple bargaining problem for the coalition w1, 4x in which they have 34 units of utility to divide, and neither of them has a ¨ iable outside option. In such a situation the Nash bargaining solution will yield player 4 a payoff of 17 Žhalf of 34.. Hence, player 4 might prefer to wait rather than agreeing to form a coalition initially Žsimilarly, player 3 might also prefer to wait.. We note that this problem arises only if multiple profitable coalitions can simultaneously form. In Section 4, we rule out this possibility by restricting to environments for which only one coalition can profitably form. ŽFor instance, this would be the case if very pair of coalitions in CU had nonempty intersection. . If coalition formation is sequential, and profitable disjoint coalitions may actually form, a multistage analysis seems necessary; at this point, we have none to offer. 18

Again we assume that utilities are linear in money and that only money matters.

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4. THE PROPOSAL-MAKING MODEL In this section we present a noncooperative analysis of multilateral bargaining problems that formalizes some of the discussion in Section 2 concerning outside options and our interpretation of solutions. Our noncooperative model is an adaptation of a proposal-making model due to Selten Ž1981.. Selten considers only games with transferable utility and models bargaining as a recursive game; we allow for nontransferable utility and use the more familiar structure of a game in extensive form. We take as given an NTU game ² N, V :, so that N s  1, 2, . . . , n4 is a S finite set of players and for each coalition S ; N, the set V Ž S . ; Rq is a compact set containing the origin. As before, we assume that ² N, V : is nondegenerate Žso that V Ž S . /  04 for some S . and 0-normalized Žso that V w i x. s  04 for each i g N ., and that each V Ž S . is strongly comprehensive Žso that the weak and strong Pareto sets coincide.. Write C for the set of nonsingleton coalitions and CU for the subset of C consisting of coalitions S for which V Ž S . /  04 . As Example 7 demonstrates, the possibility of sequential coalition formation presents special difficulties, which we wish to avoid. To this end, we construct an extensive form game in which bargaining terminates with the formation of any coalition, and we restrict our attention to games ² N, V : having the property that every pair of profitable coalitions has nonempty intersection Žthat is, S l T / B whenever V Ž S . /  04 and V ŽT . /  04.. Our extensive form game can be described in the following way. Players bargain by making, accepting, and rejecting proposals. A proposal Ž S, x . specifies a coalition S containing i and a feasible payoff distribution x g S. The game is played according to the following rules: 0. Start of game: Nature has the first move. Nature selects each player i g N with probability p i ) 0; the selected player has the initiative ŽSee a1.. 1. The initiator role: player i with the initiative can either Ža. make a proposal Ž S, x . for which i g S, and name a player j g S as the responder Žsee a2., or Žb. pass the initiative to any other player j, who becomes initiator Žsee a1.. 2. The responder role: player j responding to a proposal Ž S, x . can either Ža. accept the proposal Ž S, x . and name a player k g S who has not yet accepted to be the next responder Žif j is the last player in S to accept, see a3. or Žb. reject the proposal and become the initiator Žsee a1..

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3. End of game: The game ends when all members of a coalition S have agreed to a proposal Ž S, x .. In this case members of S receive their components of x, and other players receive 0. 4. Perpetual disagreement: Infinite plays result in 0 payoff for all players. A sketch of the game tree is in Bennett Ž1991a, p. 58.. The solution notion we adopt here is a stationary subgame perfect equilibrium in pure strategies. By stationarity, we mean that player i, Ža. when the initiator, always makes the same proposal and chooses the same responder Žb. when the responder, considering a proposal Ž S, x . on which players in SX ; S have already acted, makes the same response and chooses the same next responder Of course, stationarity is a strong requirement, but it does not seem unnatural in this context. Moreover, it appears that without stationarity}or some other requirement beyond subgame perfection}very little can be said: Bennett Ž1991b. shows that every individually rational feasible outcome can be supported as a subgame perfect equilibrium outcome. By contrast, the restriction to pure strategies is made here only for expository convenience; there would be no substantial difference if we allowed for mixed strategies with finite support. ŽSee Bennett Ž1991a. for the proofs.. Given a stationary subgame perfect equilibrium s Ža profile of pure strategies ., we write P Ž s . for the set of proposals Ž S, x . made and accepted with positive probability Ži.e., along the equilibrium path., and Pi Ž s . for the set of proposals Ž S, x . g P Ž s . for which player i belongs to the coalition S.19 Since players not in the coalition which actually forms obtain payoffs of 0, we may in the obvious way identify P Ž s . with the set of outcomes consistent with s Ži.e., the outcomes that occur with positive probability given that players follow s .. Write qi Ž s . for i’s payoff when he is initiator Žstationarity implies that qi Ž s . is well-defined., and set q Ž s . s Ž q1Ž s ., . . . , q N Ž s ..; we refer to q Ž s . as the initiation ¨ ector of the equilibrium s . If q g R N , we say that a proposal Ž S, x . is consistent with q if x i s qi for each i g S. As the following proposition shows, only proposals consistent with q Ž s . are made and accepted, so in particular it follows that each player is indifferent among proposals in Pi Ž s ., i.e., if Ž S, x ., ŽT, y . g Pi Ž s ., then x i s yi . 19

Keep in mind that Nature has the first move; hence P Ž s . will generally contain more than one proposal.

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PROPOSITION. If s is a stationary subgame perfect equilibrium with q Ž s . as its initiation ¨ ector, then e¨ ery proposal in P Ž s . is consistent with q Ž s .. Proof. Consider first the behavior of player i responding to the proposal Ž S, x .. If player i rejects and becomes initiator, he will obtain qi Ž s .. If i is the last member of S to act on Ž S, x ., subgame perfection requires that player i accept if x i G qi Ž s .. Backward induction implies that if i is not the last member to act on Ž S, x . then player i will accept if x i G qi Ž s . and x i G q j Ž s . for every j who has not yet acted. Now consider any proposal Ž S, x . g P Ž s .. The first part of the discussion above shows that x i G qi Ž s . for each i g S. If x i ) qi Ž s . for some i, strong comprehensiveness of V Ž S . implies that there is a vector y g V Ž S . with y j ) q j Ž s . for every j g S. The second part of the discussion above shows that the proposal Ž S, y . will certainly be accepted. But then any player j g S can obtain y j ) q j Ž s . when he is initiator, contradicting the definition of q j Ž s .. We conclude that x i s qi Ž s . for each i g S, so that Ž S, x . is consistent with s , as asserted. B Although only proposals consistent with q Ž s . are made and accepted with positive probability, there may be other proposals consistent with q Ž s . which are never made at all.20 We are now in a position to describe the relevant connections between the proposal-making model and multilateral bargaining problems. As we have noted earlier, if we begin with a game in coalitional form ² N, V :, then each specification of bargaining functions  f S 4 leads to a multilateral bargaining problem ² N, V, f :. Our first goal is to identify the initiation vectors of stationary subgame perfect equilibria of the proposal making model with those vectors R N which are reservation price vectors of multilateral solutions for some choice of bargaining functions. THEOREM 2 X .

For a ¨ ector q g R N, the following are equi¨ alent:

Ža. there is a stationary subgame perfect equilibrium s of the proposalmaking model such that q s q Ž s ., the initiation ¨ ector of s Žb. there are bargaining functions  f S 4 such that q is the reser¨ ation price ¨ ector of a solution of the multilateral bargaining problem ² N, V, f :. N Proof. Following Bennett and Zame Ž1988., say that a vector q g Rq is an aspiration for the game ² N, V : if Ži. for every i g N there is a coalition S ; N with i g S such that the restriction q S of q to the players in S belongs to S Ž q is coalitionally feasibility., and Žii. there does not exist a

20 If we allow for mixed strategies, and, following Selten, require that a player who is indifferent among several proposals select randomly among them, then P Ž s . consists precisely of those proposals which are consistent with q Ž s ..

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coalition T ; N and a vector x T g V ŽT . such that x Ti ) qiT for each i g T Ž q is unblocked.. Bennett Ž1991a. shows that q is an aspiration if and only if it is the initiation vector of some subgame perfect equilibrium of the proposal-making model. Hence to prove the desired result it suffices to show that q is an aspiration if and only if there are bargaining functions  f S 4 such that q is the reservation price vector of the multilateral bargaining problem ² N, V, f :. One direction of this equivalence is easy. Let q be the reservation price vector of a solution for the multilateral bargaining problem ² N, V, f :. For each coalition S g N, q S is the agreement in the coalition S. By definition, q S is either infeasible or on the Pareto boundary of V Ž S .; hence q is unblocked. In view of Theorem 1, for each i g N, there is a coalition T g N such that i g T and q T g V ŽT .; hence q is coalitionally feasible. To obtain the reverse direction, fix an aspiration q and let GC Ž q . be the set of coalitions S for which q S g V Ž S .. For each coalition S, define the agreement x S s q S and define the outside option vector d S using the set of agreements  x T 4 . Note that d S F q S for each S and that d S s q S for S f GC Ž q .. ŽIf S f GC Ž q ., then q S f V Ž S .. Since q is coalitionally feasible, for each i g S there is a coalition T with i g T and q T g V ŽT ., whence d iS G qiT .. For each coalition S, we need to construct a bargaining function f S such that f S Ž d S . s q S . It is convenient to distinguish two cases. If S g GC Ž q . then d S , q S g V Ž S .. For y g V Ž S ., consider the ray  y q lŽ q S y d S .<0 F l - `4 . Since V Ž S . is strongly comprehensive, this ray meets the Pareto boundary of V Ž S . in exactly one point, call it f S Ž y .. For y f V Ž S ., set f S Ž y . s y. It is easily checked that the function f S satisfies our criteria to be a bargaining function, and that f S Ž d S . s q S . If S f GC Ž q ., we take f S to be an arbitrary bargaining function; the fact that d S s q S g V Ž S . entails that f S Ž d S . s d S s q S . It is easily checked that  x S 4 s  q S 4 is a solution for the multilateral bargaining problem ² N, V, f :, and that q is the reservation price vector of this solution. B Although we have formulated this equivalence in terms of initiation vectors and reservation price vectors, it is easy to recast it in terms of outcomes. In particular, an outcome is consistent with a multilateral solution if and only if it can be supported by a stationary subgame perfect equilibrium. The proposal-making model also provides another framework in which to think about the interpretation of outside options. Fix a multilateral bargaining problem ² N, V, f : and a solution  x S 4 with reservation price vector q; let  d S 4 be the corresponding set of outside option vectors. We have defined d iS as the highest payoff player i could obtain, given the

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solution agreements  s S 4 , in any coalition other than S; below we show that this definition is entirely consistent with Žstationary subgame perfect. equilibrium behavior in the proposal-making model. According to Theorem 3, we can find a stationary subgame perfect equilibrium s of the proposal-making model whose initiation vector is q. Fix a player i and consider any node of the game tree at which player i has the initiative and is to make a proposal. The choice of this proposal depends of course on player i’s expectations about how other players will respond to every possible proposal Žassuming that other players follow s .. In particular, for S any coalition containing i, the decisions to make some proposal to S and the choice of which proposal to make are influenced by the set of acceptable proposals that could be made to coalitions other than S. Let ŽT, y T .4 be the set of all such acceptable proposals, and let a Si be the supremum of i’s payoffs over all these acceptable proposals; it seems natural to view a Si as player i’s outside option relative to the coalition S and the strategy profile s . We claim that a Si s d iS, player i’s outside option as we have previously computed it Žwith respect to the solution agreements  x S 4.. To establish the claim we need to show two things: Ži. If T / S is any coalition with i g T and ŽT, y . is a proposal for which yi ) d iS , then ŽT, y . will be rejected with probability 1. Žii. for every « ) 0, there is a coalition T / S with i g T and a proposal ŽT, y . which will be accepted with probability 1 such that yi ) d iS s « . To verify Ži., consider any coalition T with i g T and proposal ŽT, y . that will be accepted with positive probability. If y j - q j for any j g U, j / i, then player j will do better by rejecting the proposal ŽT, y . and obtaining the initiative, after which he will be able to obtain q j . Hence y j G q j for each j g U, j / i. By definition, therefore, yi F uTi Ž q T .. Since X S T Ž  . X d i s max u i qT : T X / S, it follows that yi F dX iS , asX desired. To verify Žii., fix « ) 0. Since d iS s max uTi Ž q T .: T X / S4 there is a coalition T such that d iS s uTi Ž q T .; that is, the vector q T rd iS is feasible for the coalition T. Strong comprehensiveness implies that there is a vector y which is feasible for the coalition T and satisfies the inequalities yi ) d iS y « and y j ) q j for each j g T, j / i. Subgame perfection guarantees that the proposal ŽT, y . will be accepted with probability 1, as asserted.

REFERENCES Aumann, R., and Maschler, M. Ž1964.. ‘‘The Bargaining Set for Cooperative Games,’’ in Ad¨ ances in Game Theory ŽM. Dresher, L. S. Shapley, and A. W. Tucker, Eds.., Annals of Math. Studies, Vol. 52, pp. 443]476.

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Bennett, E. Ž1988.. ‘‘Consistent Bargaining Conjectures in Marriage and Matching,’’ J. Econ. Theory 45, 392]407. Bennett, E. Ž1991a.. ‘‘Three Approaches to Bargaining in NTU Games,’’ in Game Equilibrium Models, Vol. III, Strategic Bargaining, ŽR. Selten, Ed.., pp. 48]69. Berlin: Springer-Verlag. Bennett, E. Ž1991b.. ‘‘A Folk Theorem for the Proposal Making Model,’’ in Game Equilibrium Models, Vol. III, Strategic Bargaining, ŽR. Selten, Ed.., pp. 70]79. Berlin: Springer-Verlag. Bennett, E., and Houba, H. Ž1992.. ‘‘Odd Man Out: Bargaining Among Three Players,’’ Working Paper Series, Dept. of Economics. Johns Hopkins University, Baltimore. Bennett, E., and van Damme, E. Ž1991.. ‘‘Demand Commitment Bargaining: The Case of Apex Games,’’ in Game Equilibrium Models, Vol. III, Strategic Bargaining, ŽR. Selten, Ed.., pp. 118]140, Berlin: Springer-Verlag. Bennett, E., and Zame, W. R. Ž1988.. ‘‘Bargaining in Cooperative Games,’’ Int. J. Game Theory 17, 279]300. Binmore, K. G. Ž1985.. ‘‘Bargaining and Coalitions,’’ in Game Theoretic Models of Bargaining, ŽA. E. Roth, Ed.., pp. 269]304. Cambridge: Cambridge University Press. Chatterjee, K., Dutta, B., Ray, D., and Sengupta, K. Ž1987.. ‘‘A Non-Cooperative Theory of Coalitional Bargaining,’’ working paper, Pennsylvania State University. Crawford, V. P., and Rochford, S. C. Ž1986.. ‘‘Bargaining and Competition in Matching Markets,’’ Int. Econ. Re¨ . 27, 329]348. Kalai, E. Ž1985.. ‘‘Solutions to the Bargaining Problem,’’ in Social Goals and Social Organization ŽL. Hurwicz, D. Schmeidler, and H. Sonnenschein, Eds.., pp. 77]105. Cambridge: Cambridge University Press. Kalai, E., and Samet, D. Ž1985.. ‘‘Monotonic Solutions to General Competitive Games,’’ Econometrica 53, 307]327. Nash, J. F. Ž1950.. ‘‘The Bargaining Problem,’’ Econometrica 18, 155]162. Nash, J. F. Ž1951.. ‘‘Non-Cooperative Games,’’ Ann. Math. 54, 286]295. Nash, J. F. Ž1953.. ‘‘Two-Person Cooperative Games,’’ Econometrica 21, 129]140. Rochford, S. Ž1984.. ‘‘Symmetrically Pairwise-bargained Allocations in an Assignment Model,’’ J. Econ. Theory 34, 262]281. Roth, A. E. Ž1979.. Axiomatic Models of Bargaining. New York: Springer-Verlag. Rubinstein, A. Ž1982.. ‘‘Perfect Equilibrium in a Bargaining Model,’’ Econometrica 50, 97]109. Selten, R. Ž1981.. ‘‘A Noncooperative Model of Characteristic Function Bargaining,’’ in Essays in Game Theory and Mathematical Economics in Honor of Oskar Morgenstern ŽV. Boehm and H. Nachtkamp, Eds.., pp. 131]151. Vienna, Zurich; Wissenschaftsverlag Bibliographisches Institut Mannheim. Sutton, J. Ž1985.. ‘‘Noncooperative Bargaining Theory: An Introduction,’’ Re¨ . Econ. Studies 53Ž5., No. 176, 709]725.