Multilateral Negotiations and Formation ofCoalitions

Armando Gomes�

Washington University in St. Louis

August 2012

Abstract

This paper analyses properties of games modelling multilateral negotiationsleading to the formation of coalitions in an environment with widespread ex-ternalities. The payo¤ generated by each coalition is determined by a set ofexogenous parameters. We show that in almost all games, except in a set ofmeasure zero of the parameter space, the Markov perfect equilibrium is locallyunique and stable, and comparative statics analysis are well-de�ned and can beperformed using standard calculus tools. Global uniqueness does not hold ingeneral, but the number of equilibria is �nite and odd. In addition, a su¢ cientcondition for global uniqueness is derived, and using this su¢ cient condition weshow that there is a globally unique equilibrium in three-player superadditivegames.jel: C71, C72, C78, D62keywords: Coalitional bargaining, externalities, multilateral negotiations.

�I would like to thank seminar participants at Princeton University, University of Pennsylvania,and the World Congress of the Econometric Society. Address: Washington University in St. Louis,One Brookings Drive, Campus Box 1133, MO 63130-4899. E-mail: gomes@wustl.edu. Tel: (314)935-4569.

1 Introduction

We study multilateral negotiations where contracts forming coalitions can be writtenand renegotiated, and where the formation of coalitions may impose externalities onother players. What is the path of coalition formation? What is the value of players?This paper develops a non-cooperative model of coalitional bargaining that providesanswers to these two questions.The externalities present in the environment are described by a set of exogenous

parameters, conveniently expressed using a partition function form. The partitionfunction form assigns a worth to each coalition depending on the coalition structure(or collection of coalitions) formed by the remaining players. This general formulationallows for the analyses of problems in which the formation of coalitions may imposepositive or negative externalities (see also Ray and Vohra (1999), Bloch (1996), Jehieland Moldovanu (1995), and Gomes (2005)).Our multilateral negotiation procedure follows the traditional approach of using

a dynamic game with complete information where at each stage a player becomesthe proposer (such as Chatterjee et al. (1993), Hart and Mas-Colell (1996), Rayand Vohra (1999), and Okada (1996) among many others). Proposers make o¤ers toform coalitions, followed by players who have received o¤ers making their responsewhether or not to accept the o¤er. Similarly to Gul (1989) and Seidmann and Winter(1998), coalitions after forming do not leave the game and may continue negotiatingthe formation of further coalitions. A variety of situations can be addressed withour game such as the formation of custom unions, merger or carter formation among�rms in the same industry, legislative bargaining, and the signing of environmentalagreements across regions (see, for example, Ray (2007)).The equilibrium concept used is Markov perfect equilibrium (MPE), where the set

of states are all possible coalition structures. TheMPE solutions characterize, jointly,both the expected equilibrium value of coalitions, and the Markov state transitionprobability that describes the path of coalition formation. Thus, the equilibria provideanswers to both questions we posed initially, and our goal is to develop a thoroughanalysis of the equilibrium properties.We establish the existence of equilibria for all games, and show that, for almost all

1

games, except in a closed set of measure zero of the parameter space, the equilibriumis locally unique and locally stable. These properties imply that the predictions of themodel about both the expected payo¤s of players and the path of coalition formationare sharp, in the sense that, at least locally, they are unique and robust to smallperturbations of the exogenous parameters of the game.Therefore, we extend to multilateral bargaining models similar results that hold

for other well-known economic models such as Walrasian equilibrium of competitiveeconomies (Debreu (1970)), Nash equilibrium of n-person strategic form games (Wil-son (1971) and Harsanyi (1973)), and Markov perfect equilibrium of stochastic games(Holler and Laguno¤ (2000)).The problem of �nding equilibria is equivalent to �nding solutions of a mixed

nonlinear complementarity problem (MNCP). Such problems have been extensivelystudied in the mathematical programing literature (see Harker and Pang (1990) andCottle, Pang, and Stone (1992)), and several numerical algorithms have been devel-oped. Hence, the computation of equilibria is a task that can be undertaken usingseveral proven numerical algorithms.The number of equilibrium solutions is �nite and odd for almost all games. We

provide an example of a game with multiple (seven) equilibrium solutions, so theequilibrium is not globally unique. Nonetheless, we derive a su¢ cient condition forglobal uniqueness, and argue that this su¢ cient condition is weak and is likely to besatis�ed by a large class of games. In particular, we prove that the su¢ cient conditionholds for superadditive three-player coalitional bargaining games.How do the equilibrium value of players and the path of coalition formation change

as a result of changes in exogenous parameters such as the partition function formand the probability of being the proposer? Knowing how to address these questionsis of considerable practical interest to negotiators, as they, for example, may be ableto invest in changing the likelihood of being proposers in negotiations. We showhow to answer these questions using standard calculus results (the implicit functiontheorem), which provides a powerful tool for quickly answering comparative staticsquestions by simply evaluating Jacobian matrices at the solution.We demonstrate the applicability of the results using two classic games�apex and

quota games (see Shapley (1953), Davis and Maschler (1965), and Maschler (1992)).

2

Surprisingly, in both games, a player sometimes may not bene�t by investing in ob-taining more initiative to propose in negotiations. Other players may adjust theirstrategies in such a way that lead the proposer to be worse o¤. The analysis alsosuggests several interesting regularities: when the exogenous value of a coalition in-creases, both the equilibrium value of the coalitional members and the likelihood thatthe coalition forms increase as well.The remainder of the paper is organized as follows: Section 2 presents the coali-

tional bargaining game; Section 3 addresses the existence and characterization of theequilibria; Section 4 shows how to compute the equilibria; Section 5 and 6 developthe local uniqueness, stability, and genericity properties of the equilibria; Section 7addresses the number of equilibrium solutions; and Section 8 concludes.

2 The Model

Coalition formation in this paper is modeled as an in�nite horizon complete infor-mation game. In a nutshell, the coalition formation process is such that during anyperiod of the game a player is chosen at random to propose to form a coalition anda payment to all coalition members. Subsequently, all coalition members respond tothe o¤er, and the coalition is formed only if all its members agree.Formally, let N = f1; 2; � � � ; ng be a set of n agents. A coalition is a subset

of agents and a coalition structure (c.s.) � = fC1; :::; CKg; where Ck � N; is apartition of the set of agents into disjoint coalitions (denote by� the set of all coalitionstructures). The game starts at the state where all agents are in solo coalitions (theinitial c.s. is the calligraphic N = ff1g; :::; fngg).The coalition bargaining game is the game with the following extensive form.

Consider that at the beginning of a certain period of the game the c.s. is � (typicalcoalitions in � are often represented by labels i; j). One of the coalitions i 2 � israndomly chosen with probability pi(�) > 0 to be the proposer (let p = (pi (�))�2�

i2�).

Coalition i then makes an o¤er (S; t) where S � � is a set of coalitions in � andt is a vector of transfers satisfying

Pj2S tj = 0: All coalitions in S respond in a

�xed sequential order whether they accept or not the o¤er (it turns out the the orderof response is not relevant). If all coalitions in S accept the o¤er a new coalition

3

S = [j2Sfjg � N; calligraphic S; is formed under the control of the proposingcoalition i. The coalitions j 2 Sni ceding control receive the lump-sum paymenttj and exit the game. The coalition structure evolves from � to �S = S [ (�nS).1

Otherwise, if any one of the coalitions receiving the o¤er rejects it, no new coalitionis formed and the coalition structure remains equal to �. After a lapse of one periodof time, the game is repeated starting with the prevailing c.s. with a new proposerbeing randomly chosen as just described.All agents have the same expected intertemporal utility function, are risk-neutral

and have common discount factor � 2 (0; 1).2 When coalitions form they may imposeexternalities on other coalitions. This possibility is captured by a partition functionform v = (vi (�))�2�

i2�; where coalition i�s payo¤ �ow (during a period of time), when

the coalition structure is �; is equal to (1� �) vi(�) (so if the game stays at c.s. �forever, the value of coalition i is vi(�)). The payo¤s are distributed at the end of eachperiod, after the coalition formation stage, with coalitions ceding control receivinga �nal lump-sum transfer payo¤ and the coalition acquiring control receiving, inaddition to the lump-sum transfer, the payo¤ given by the partition function form(i.e., (1� �) vi(�S) + ti when the c.s. �S forms).We restrict our attention to Markovian strategies. Hence, the proposer�s strategy

only depends on the current state �, and the respondents� strategy only dependson the current state �, the current o¤er she receives and the responses of precedingplayers: A Markov perfect equilibrium (MPE) is a Markovian strategy pro�le whereevery player plays a Nash equilibrium at every stage.

3 Characterization of Equilibrium

Let us be given a MPE strategy �. We represent by �i(�j�) the equilibrium continu-ation value of coalition i which is obtained from the stochastic process induced by �when the c.s. is at � (the value � is computed at the beginning of a period before aproposer is chosen).

1So, for example, if � = ff1; 2g; f3; 4g; f5gg and coalition i = f1; 2g proposes to form coalitionS = ff1; 2g; f3; 4gg; then �S = ff1; 2; 3; 4g; f5gg) and S = f1; 2; 3; 4g: Note that the coalitionstructure becomes coarser as time elapses.

2Thus their utility over a stream of random payo¤s (xt)1t=0 is

P1t=0 �

tE (xt) :

4

The equilibrium continuation value at the end of a period in which � is formed isequal to (gross of lump-sum transfers)

xi (�j�) = ��i(�j�) + (1� �) vi (�) ; (1)

because coalition i receives payo¤ �ow (1� �) vi (�) during the current period and,after a delay of one period, at the beginning of the next period, coalition i �s value is��i(�j�): Let x = (xi (�))�2�

i2�and � = (�i (�))�2�

i2�be the continuation values (where

we sometimes omit the dependency on the strategy pro�le �).An equilibrium � is characterized by several properties which we now summarize.

The minimum o¤er that coalition j receiving o¤er (S; t) is willing to accept is onewhere tj � xj (�j�) (because upon rejection no transfers are made and the stateremains at �). In turn, coalition i proposes o¤ers (S; t) that maximizes the valuexS(�Sj�) �

Pj2Sni tj subject to the constraint that tj � xj (�j�) ; (because a new

coalition S is formed whose value is xS(�Sj�) if the o¤er is accepted by all players).Thus, when an o¤er (S; t) is made the transfers t are uniquely determined by tj =xj (�j�) for j 2 Sni and ti =

Pj2Sni tj.

De�ne the excess

� (�) (S) (x) = xS(�Sj�)�Xj2S

xj (�j�) , (2)

a function of S � �; and x equilibrium continuation value. Proposer i randomizesacross coalitions S that maximizes the excess max

S3i;S��f� (�) (S) (x)g :

Let �i (�) (S) 2 [0; 1] represent the probability that coalition S is chosen by playeri. The (behavioral) strategy �i (�) of proposer i is a probability distribution over�i (�) = fS � � : i 2 Sg, i.e.

PS2�i(�) �i (�) (S) = 1 and �i (�) (T ) = 0 for all

T =2 �i (�) : Also, we de�ne �(�) as the set of strategy pro�les when the c.s. is �;and let � be the set of strategy pro�les (i.e., � 2 �):We use the following standard notation: � is the Cartesian product, jAj is the

cardinality of set A; IA is the indicator function that is equal to one or zero, respec-tively, if statement A is true or false, and the support of �i (�) is supp�i (�) ; the setof all S such that �i (�) (S) > 0:The necessary part of the following lemma, proved in the Appendix, follows di-

rectly from the above discussion and the de�nition of MPE.

5

Lemma 1 A payo¤ structure �i(�) and a strategy pro�le �i (�) is an MPE of thecoalitional bargaining game if and only if the following system of equations is satis�ed,where xi (�) = ��i(�) + (1� �) vi (�):1) the support of the strategy �i (�) is

supp�i (�) � argmaxS3i;S��

f� (�) (S) (x)g ; (3)

2) the expected equilibrium outcome of player i conditional on player j being chosento be the proposer �ji (�) is equal to

�ji (�) =

(max

S3i;S��f� (�) (S) (x)g+ xi(�) j = iP

S�� �j (�) (S)�I[i2S]xi(�) + I[i=2S]xi(�S)

�j 6= i

; (4)

3) the following system of equations holds

�i (�) =

Xj2�

pj (�)�ji (�)

!; (5)

for all � 2 �; and i; j 2 �:

There is a one-to-one relation between �i(�) and xi (�) given by equation (1). Forconvenience we will be solving for the vectors xi (�) instead of �i(�) from now on.The vector x; as well as the partition function form v; belongs to the Euclidean spaceRd; where the dimension d =

P�2� j�j:

Our next result shows that there always exist MPE solutions for all coalitionalbargaining games.

Proposition 1 There exist Markov perfect equilibrium for all coalitional bargaininggames.

We de�ne in the Appendix a correspondence F : X ! Rd such that the �xedpoints of F are the MPE payo¤s of the game. The proposition follows from theKakutani �xed point theorem because we show that F (X) � X; F (x) is convex andnon-empty, and F is an upper hemicontinuous correspondence.

6

4 Computing the Equilibria

The problem of �nding equilibria can be restated as the solution of a certain mixednonlinear complementarity problem (see Harker and Pang (1990) for a survey aboutcomplementarity problems). Mixed nonlinear complementarity problems (MNCP)are the subject of numerous studies in the mathematical programming literature (seeHarker and Pang (1990)), and a large number of algorithms have been proposed forsolving MNCP problems.We now introduce the MNCP problem associated with coalitional bargaining

games. For all x = (xi (�))�2�i2�, e = (ei (�))�2�

i2�; and � 2 �; consider the mapping

f(x; �; e); where coordinate fi(�) is given by

fi(�)(x; �; e) = xi (�)� �pi(�)ei(�)� (1� �)vi (�) (6)

� � XS��

Xj2�

pj(�)�j (�) (S)�I[i2S]xi(�) + I[i=2S]xi(�S)

�!!;

for all i 2 � and � 2 �: Let the maps h (�) and g (e; x) be de�ned by

hi(�) (�) =X

S��:i2S�i (�) (S)� 1; (7)

gi(�) (S) (e; x) = ei(�)� xS(�S)�

Xj2S

xj (�)

!;

for all i; �; S satisfying � 2 �; i 2 �; and i 2 S � �:The mixed nonlinear complementarity problem is the problem of �nding triples

(x; �; e) that satisfy all conditions

f(x; �; e) = 0;

h (�) = 0; (MNCP)

g (e; x) � 0; (8)

� � 0 and �Tg (e; x) = 0;

where �Tg (e; x) = 0 is equivalent to �i (�) (S) :gi(�) (S) (e; x) = 0 for all i; �; Ssatisfying � 2 �; i 2 �; and i 2 S � �:

7

Proposition 2 If (x; �) is an MPE then (x; �; e) is a solution of the problem MNCP,where ei(�) = max

S3i;S��f� (�) (S) (x)g : Reciprocally, if (x; �; e) is a solution of the

problem MNCP then (x; �) is an MPE.

Proof: Consider the necessary part of the proposition, and say that (x; �) isan MPE. Then all the conditions in items 1, 2, and 3 of Lemma 1 hold. Replacingexpression (4) of �ji (�) into equation (5), and considering that, by de�nition, xi (�) =��i(�) + (1� �) vi (�), we obtain the system of equations

xi (�) = �pi(�) maxS3i;S��

f� (�) (S) (x)g+ (1� �)vi (�) +

+�

XS��

Xj2�

pj(�)�j (�) (S)�I[i2S]xi(�) + I[i=2S]xi(�S)

�!!:

Let ei(�) = maxS3i;S��

f� (�) (S) (x)g : We then have that (x; �; e) satis�es the equa-tionf(x; �; e) = 0: Since � is a probability distribution then h (�) = 0 and � � 0 areautomatically satis�ed. Also, by de�nition of ei(�) above; ei(�)�� (�) (S) (x) � 0; sothat g (e; x) � 0: Finally, �i (�) (S) gi(�) (S) (e; x) = 0 follows from de�nition of e andthe support restriction of � in (3). The reciprocal follows using the same arguments.Q.E.D.

Proposition 2 is useful because there are several numerical algorithms for solvingMNCP problems. These algorithms are typically based on both Newton�s methodfor solving a system of equations and on methods for solving linear complementarityproblems (LCP), such as Lemke-Howson�s algorithm (see Cottle et al. (1992) for acomprehensive treatment of the LCP). When solving a system of nonlinear equations,starting from a given initial condition, Newton�s algorithm solves the linear approx-imation of the system of equations to obtain the updating direction. In the case ofMNCP problems, instead of just solving for a linear system of equations, one solvesa linear complementarity problem to obtain the updating direction, so as to satisfythe inequalities in the system.The results of this section thus imply that the computation of equilibrium points

can be accomplished using well-known numerical methods.

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5 Coalition Dynamic Structures

Instead of focusing on the strategy pro�le � 2 �, it is more convenient to focus onthe associated Markov transition probability � = � (�) ; which is de�ned as

� (�) (�) (S) =Xj2�

pj (�)�j (�) (S) ; (9)

where � (�) (�) (S) represents the probability of moving from state � to state �S:The following result shows that uniqueness of strategy pro�les do not hold in

general and thus the best we can hope is to have uniqueness with respect to expectedpayo¤s and the Markov transition probabilities.

Lemma 2 If (x; �) is an MPE then (x; �0) is an MPE for any �0 2 �; with � (�) =� (�0) ; and supp�0i (�) � argmax

S3i;S��f� (�) (S) (x)g.

Proof: By Proposition 2, (x; �; e) solves the MNCP problem, and thus f (x; �; e) =0. Inverting the order of summation in the expression of f (x; �; e) in equation 6), weget,

fi(�)(x; �; e) = xi (�)� �pi(�)ei(�)� (1� �)vi (�)

�� XS��

� (�) (�) (S)�I[i2S]xi(�) + I[i=2S]xi(�S)

�!

where � (�) is as in equation (9). Therefore, for any �0 as above, f (x; �0; e) = 0; andby the reciprocal of Proposition 2 this implies that (x; �0) is an MPE : Q.E.D.

An important di¢ culty in the analysis is that the equilibrium is not locally uniquein the equilibrium strategies �: There exist a continuum of equilibrium strategieswhenever there is one equilibrium in which coalition S is the best response choice ofmore than one player, say i and j: a continuum of strategies can be constructed byhaving player i reducing the weight it chooses S; with player j increasing the weighton S; so that the overall transition probability � is unchanged.Our e¤orts from now on are concentrated on proving uniqueness in terms of the

pair (x; �) : The strategy pro�le � belongs to a space with higher dimensionality

9

than the transition probabilities �; and when passing from � to � some importantinformation is lost. For example, suppose that � (�) (S) > 0 for some S � �:Who arethe players that choose coalition S with positive probability? It is certainly possiblethat the best response strategy for player j 2 S is to choose coalition S; but thatanother player i 2 S is strictly better o¤ choosing a di¤erent coalition.We now introduce the concept of coalitional dynamic structure (CDS) which allow

us to recover all the essential information about the strategy pro�le �; that is notrecorded in the transition probabilities �:Start by de�ning an equivalence relation on the set � induced by the strategy

pro�le �: Given any two players i; j 2 �; say that i ! j if and only if there existsa coalition S � � with i; j 2 S such that �i (�) (S) > 0: Also, say that there is apath from i to j if there exists a sequence of players i1; :::; ik belonging to � such thati ! i1 ! ::: ! ik ! j: Finally, we say that i and j is strongly connected, i ! j; ifthere is a path from i to j and a path from j to i: It is straightforward to verify thatstrong connection is an equivalence relation (transitivity, symmetry, and re�exivityhold). Let Pr (�) be the equivalence classes of this relation (the maximal stronglyconnected components). Also, let Cr (�) = [i2Pr(�)supp�i (�) be the union of theo¤ers in the support of the strategy pro�le of players in Pr (�) :3

The coalitional dynamic structure associated with � is a partition of the set ofplayers and a partition of supp� into the equivalence classes induced by the strongconnection relation.

De�nition 1 The coalitional dynamic structure (CDS) associated with � isC (�) = (C (�) ; P (�))�2� where, for each c.s. �:i) P (�) = (P1 (�) ; :::; Pq(�) (�)) is a partition of � and Pr (�) are the equivalenceclasses of the strong connection relation;ii) C (�) = (C1 (�) ; :::; Cq(�) (�)) is a partition of supp� (�) = [i2�supp�i (�) andCr (�) = [i2Pr(�)supp�i (�) :The set of coalitional dynamic structures is CDS = fC (�) : � 2 �g :

We are interested in analyzing the problem of �nding an equilibrium point (x; �)with a given CDS C. By de�nition of CDS, this implies that the excesses of all

3Note that Cr (�) \ Cr0 (�) = ; if r 6= r0: Otherwise, there exist i 2 Pr (�), j 2 Pr0 (�) ; and S 2Cr (�)\Cr0 (�) such that �i (�) (S) > 0 and �j (�) (S) > 0. But this implies i ! j (contradiction).

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coalitions belonging to the same equivalence class Cr (�) are equal, that is,

er (�) = xS(�S)�Xi2Sxi (�) ; (10)

for all S 2 Cr (�) and r = 1; :::; q (�). In addition, the associated Markov transitionprobability � = � (�) satis�esX

S2Cr(�)

� (�) (S) =X

j2Pr(�)

pj (�) ;

because supp�j (�) � Cr (�) for all j 2 Pr (�) :Therefore, if (x; �) is an equilibrium then (x; �; e) solves the following systems of

equations, or problem F (C) (or FC)

FC(x; �; e) =

0@ fC(x; �; e)EC (x; e)MC (�)

1A = 0 (11)

where the maps fC(x; �; e); EC (x; e) ; and MC (�) associated with CDS C are de�nedby

(fC)i (�) (x; �; e) = xi (�)� �pi (�) er (�)� (1� �) vi (�)

� � X

S

� (�) (S)�I[i2S]xi (�) + I[i=2S]xi(�S)

�!;

EC (�) (S) (x; e) =Xi2Sxi (�) + er (�)� xS(�S); (12)

MC (�) (r) (�) =X

j2Pr(�)

pj (�)�X

S2Cr(�)

� (�) (S) ;

for all r; i; and S satisfying r = 1; :::; q (�), i 2 Pr (�) ; S 2 Cr (�) ; and all � 2 �:The reciprocal result also holds if we impose some additional restrictions on the

solutions of F (C): Any set of payo¤s x that are candidates for equilibrium with anassociated CDS C must satisfy

� (�) (S) (x) � � (�) (T ) (x) for all S 2 Cr (�) and T =2 Cr (�) with T \ Pr (�) 6= ;;(13)

11

because of equalities (10) and inequalities (3). Thus, the set of payo¤s EC consistentwith C is

EC =�x 2 Rd : such that all inequalities (13) hold

:

Moreover, any transition probability � that is consistent with a CDS C satis�es � =� (�) where � is a strategy pro�le with a CDS C (i.e., C (�) = C):4 Thus, the set oftransition probabilitiesMC consistent with C is

MC = f� = � (�) : where � 2 � and C (�) = Cg : (14)

The following lemma, proved in the Appendix, provides yet another useful char-acterization of MPE.

Lemma 3 If (x; �) is an MPE of the bargaining game then (x; �; e) is a solution ofproblem F (C(�)); where � = � (�) 2 MC; x 2 EC; and er (�) = � (�) (S) (x) for anyS 2 Cr (�) : Reciprocally, if (x; �; e) is a solution of problem F (C) satisfying � 2MC

and x 2 EC then there exists an MPE (x; �) of the bargaining game with � = � (�)and C = C(�):

This result allows us to transform the problem of �nding equilibria into a lower-dimensional equivalent problem of �nding solutions of the system of equations F (C).

6 Generic Local Uniqueness and Stability

In this section we show that almost all games, except in a set of measure zero ofthe parameter space (v; p), have equilibria that are locally unique and stable. Theseresults imply that almost all games have only a �nite number of equilibria, and providetools for performing comparative statics analysis in coalitional bargaining games.

6.1 Regular and Nondegenerate Games

We seek to determine in this section conditions under which the equilibrium outcomeis locally unique and stable. Note that global uniqueness does not hold in general,

4Observe that if the strategy pro�le � is such that for all r and i 2 Pr (�), supp (�i (�)) =Cr (�) \ fS � � : i 2 Sg then C (�) = C:

12

as Example 3 illustrates. The next best property is local uniqueness. An equilibriumoutcome is locally unique if we cannot obtain another equilibrium outcome arbitrarilyclose to it. Stability is a property that ensures that comparative statics exercises arewell de�ned. Roughly speaking an equilibrium point is stable if it changes smoothlyfor any small changes of the parameters of the game.According to the previous section the equilibrium outcome are solutions of FC(z) =

0; where z = (x; �; e): In order to obtain (local) uniqueness it is necessary thatthe problem has the same number of equations and of unknowns. Indeed, thereare d =

P�2� j�j equations fC, m =

P�2�

Pq(�)r=1 mr (�) ; where mr (�) = jCr (�) j;

equations EC; and q =P

�2� q(�) equations MC (a total of d + m + q equations).Moreover, the unknowns are the d dimensional variable x; the m dimensional variable�; and the q dimensional variable e (a total of d+m+ q unknowns). So the numberof equations and unknowns coincide.We now introduce the concepts of regularity and nondegeneracy. Games that

satisfy both of these technical conditions are shown to have equilibrium outcome thatare locally unique and stable.

De�nition 2 (Regular game) A solution z of problem FC(z) = 0 is regular if theJacobian dzFC is nonsingular. A CDS C is regular if all the solutions of problem FC

are regular.5 Finally, a game is regular if all CDSs are regular.

The Jacobian dzFC is a matrix of order d+m+ q; and is nonsingular if and onlyif it has full rank (equal to d+m+ q): It has the following special structure

dzFC =

24 d(x;e)fC d�fCd(x;e)EC 00 d�MC

35 ; (15)

where the matrix d(x;e)EC has m rows and d+ q columns, and matrix d�MC (�) has qrows and m columns. This special structure of the Jacobian matrix will be exploredlater on to show that it is nonsingular almost everywhere.6

5By de�nition, if FC has no solutions then the support C is regular.6Note that if Jacobian matrix is nonsingular then the m � (d + q)-matrix dEC must have rank

m: So, for example, support structures with more than d+ q coalitions in the support (m > d+ q)are not candidates for a regular equilibrium point. Reciprocally, we show in the next section that ifmatrix dEC has rank m then for almost all partition functions the Jacobian matrix is nonsingular.

13

Remark: The solution at a given c.s. � only depends on the variables evaluatedat coalition structures that are coarser than �: Thus the Jacobian matrix dzFC canbe partitioned into an upper block triangular structure with diagonal blocks equal todz(�)FC (�) where z (�) = (x (�) ; � (�) ; e (�)) for all � 2 �; where all entries to the leftof the diagonal blocks are zero. Therefore, the Jacobian matrix dzFC is nonsingularif and only if all the diagonal blocks dz(�)FC (�) are nonsingular.Consider now the nondegeneracy technical condition, which is a property of the

support that roughly means that all choices outside the support are not best responsestrategies (Harsanyi (1973) refer to a similar property in the context of n-person non-cooperative games as quasi-strong property). This condition is used in the nextproposition to show that nearby games have equilibrium with the same support.

De�nition 3 (Nondegenerate game) A CDS C is nondegenerate if all solutions of FCsatisfy

� (�) (S) (x) 6= � (�) (T ) (x) for all S 2 Cr (�) and T =2 Cr (�) with T \ Pr (�) 6= ;:(16)

A game is nondegenerate if all CDSs are nondegenerate.

We now show that the implicit function theorem implies that regular and nonde-generate equilibrium points are locally unique and stable. Formally, we show that forany game (v�; p�) and regular and nondegenerate equilibrium point (x�; ��) associatedwith an equilibrium with support C; there exists an open neighborhood B � Rd�Rd

of (v�; p�), an open neighborhood W � Rd � Rm of (x�; ��) and a local mapping(x(v; p); �(v; p)) 2 Rd � Rm such that (x(v; p); �(v; p)) is the only equilibrium pointin the neighborhood W for all games (v; p) 2 B.

Proposition 3 (Local uniqueness and stability) Regular and nondegenerate coali-tional bargaining games have equilibrium points that are locally unique and stable.

Proof: The implicit function theorem immediately implies that, for any game(v�; p�) and regular solution z� = (x�; ��; e�) there exists an open neighborhood B �Rd � Rd of (v�; p�), an open neighborhood fW � Rd � Rm � Rq of (x�; ��; e�) ; and amapping z(v; p) = (x(v; p); �(v; p); e (v; p)) 2 Rn � Rm � Rq such that z(v; p) is the

14

only solution of problem FC in fW for all games (v; p) 2 B: Note that since e (v; p)can be expressed as a function of x(v; p) (see equation (10)) then z(v; p) is the onlysolution in a cylinder W �Rq for W an open neighborhood of (x�; ��) :It remains to show that z(v; p) is indeed an equilibrium point. By Lemma 3,

z(v; p) is an equilibrium point if (i) x(v; p) 2 EC and (ii) �(v; p) 2 MC: We showbelow that (i) and (ii) hold:(i) holds: Because C has full support (nondegeneracy condition), all the inequali-

ties in (13) are strict for x� and thus, by continuity, the inequalities also hold for allx(v; p) in an open neighborhood Q of x�, which is equivalent to x(v; p) 2 EC:(ii) holds: The result follows directly from the following lemma, proved in the

Appendix.

Lemma 4 For any CDS C and �� 2MC there exists an open neighborhood U around�� such that for all � in U that satis�es all equationsP

S2Cr(�) � (�) (S) =P

j2Pr(�) pj (�) there exists � with CDS C such that � (�) = �.

Therefore, since (i) and (ii) hold in the neighborhood W = Q � U; the pair(x(v; p); �(v; p)) is an equilibrium point, which completes the proof. Q.E.D.

6.2 Genericity of Equilibria

Regularity and nondegeneracy are here shown to be generic properties. The parameterspace used to establish the result is the set of all games in (v; p) 2 Rd � �d, where�d = fp 2 Rd : pi(�) � 0 and

Pi2� pi(�) = 1 for all � 2 �g: Formally, a generic

property is one that holds for all games, except possibly those in a subset of Lesbeguemeasure zero on Rd��d (i.e., the property holds for almost all games).7 Combiningwith the results of the previous section, we prove that local uniqueness and stabilityhold for almost all games.This genericity result is established using the well-known transversality theorem

from di¤erential calculus (see Guillemin and Pollack (1974) and Hirsch (1976) andthe Appendix for a restatement of the theorem). The key result of this section is thefollowing.

7We have just argued that the set of regular and nondegenerate games is an open set. Thereforethe set of games that are not regular nor degenerate is a closed set.

15

Proposition 4 (Genericity) Almost all coalitional bargaining games (v; p) in Rd��d

are regular and nondegenerate. Therefore, almost all games are locally unique andstable.

Key for the proof is to show that, for all CDS C; the Jacobian dzFC; at any solutionsof problem FC (z) = 0; is nonsingular, for almost every parameter (v; p) in Rd ��d.That is all CDS are regular almost everywhere.In order to illustrate the arguments involved to prove regularity, consider �rst

that C is a CDS where matrix dEC has rank m. The solutions of problem FC can berepresented as the zeros of the augmented problem FC(z; v) = 0; where we take intoaccount the dependency with respect to the game. The Jacobian of this mapping is

d(z;v)FC =

24 � � �(1� �)Id(x;e)EC 0 00 d�MC 0

35 ;where dzFC =

24 � �d(x;e)EC 00 d�MC

35 and dvFC =24 �(1� �)I0

0

35, and � denotes arbi-trary coe¢ cients. The augmented Jacobian is a surjective matrix (with rank equalto the number of rows) because all blocks dEC; dMC, and �(1� �)I have rank equalto the number of rows, and because of the disposition of zeros. Thus FC is transver-sal to zero (i.e., FCt0) or 0 is a regular value of the augmented problem. By thetransversality theorem, for almost every v, FC(v) is also transversal to zero, FC(v)t0.Thus the square Jacobian matrix dzFC(v) is surjective at all solutions of FC, and thusnonsingular at all solutions for almost all v 2 Rd and all p 2 �d.Note that when the CDS C is such that matrix dEC has rank smaller than m the

Jacobian dzFC is singular for all parameters. However, in the Appendix, we showthat the further augmented problem FC(z; v; p) = 0 is such that d(z;v;p)FC(z; v; p) = 0in surjective, and thus, by the transversality theorem, dzFC is nonsingular almosteverywhere in the parameter space Rd ��d:

The argument to prove that almost all games are nondegenerate is as follows.Given any support C consider an hyperplane H in the space Rd+m+q de�ned byequality (16), � (�) (S) (x) = � (�) (T ) (x) for some pair S; T (so C is nondegenerateif there is no solution of FC(z) = 0 such that z 2 H): Applying the transversality

16

theorem again (see Guillemin and Pollack (1974)) it follows that, for almost no pa-rameters, there are solutions FC(z) = 0 such that z 2 H : because the codimensionof H in the space Rd+m+q is 1, the transversality theorem applied to the surjectiveproblem FC(z; v; p) = 0 restricted to the domain H �Rd��d implies that for almostno parameters (v; p) there are no solutions of FC(z) = 0 such that z 2 H. Using thefact that a �nite union (there are only a �nite number of pairs S; T ) of sets of measurezero is a set of measure zero, we conclude that there exists a set of parameters, withcomplement of measure zero, where FC is regular and nondegenerate.

6.3 Comparative Statics Analysis

Understanding how the value of coalitions and the path of coalition formation changesin response to changes in the exogenous parameters of the game v and p is a relevantcomparative statics exercise. Regular and nondegenerate games are very convenientbecause they allow us to perform comparative statics analysis using standard calculustools.The following corollary is an immediate application of the implicit function the-

orem and Proposition 3. The sensitivity matrix SC allow us to evaluate how theequilibrium point changes �z = SC (�v;�p) in response to local changes of thegame.

Corollary 1 (Comparative Statics) Let (v; p) be a regular and nondegenerate gameand z = (x; �; e) be an equilibrium with CDS C: The �rst-order e¤ects of a changein the exogenous parameters (v; p) on the solution z is given by the sensitivity matrixSC = � [dzFC]�1 d(v;p)FC (i.e., �z = SC (�v;�p)). In particular, the e¤ect of a localchange �v of coalitional values are given by �z =

�[dzFC]

�1��x (1� �)�v; where�

[dzFC]�1�

�x denotes the submatrix with the �rst d columns of the inverse Jacobian.

The �rst-order e¤ects with respect to changes in value �v are given by the sensi-tivity matrix � [dzFC]�1 dvFC: But since

dvFC =

0@ dvfCdvECdvMC

1A = �

0@ (1� �) I00

1A ; (17)

17

the sensitivity matrix � [dzFC]�1 dvFC simpli�es to�[dzFC]

�1��x (1� �) : So evaluating

the inverse of the Jacobian matrix at the solution yields the �rst-order e¤ects ofchanges in value.We illustrate with the next examples the comparative statics properties of quota

and apex games.

6.4 Examples

Comparison of the equilibrium payo¤s predicted by our model with established so-lution concepts from cooperative game theory, such as the nucleolus, bargaining set,kernel, core, and Shapley value, shows that our predictions are di¤erent than all othercooperative solution concepts.Example 1: Quota GamesQuota games have been studied by Shapley (1953) and Maschler (1992). Consider

a four-player quota game, where each pairwise coalition gets vfi;jg = !i + !j for alldistinct pairs i; j 2 N; where the quotas of the four players are (!1; !2; !3; !4) =(10; 20; 30; 40); and all remaining coalitions get vS = 0 for all S � N; S 6= fi; jg (tosimplify notation we omit the c.s. � in vi (�)): Players are very patient (i.e., we areinterested in the limit when � converges to 1); and they all have an equal chance tobe proposers.The equilibrium point and the transition probabilities are depicted in Figure 1

(in order to simplify the notation, in the �gure the c.s. ffi; jg; fk; lgg is denoted by(ij)(kl); the numbers below the c.s. in parenthesis are the corresponding equilibriumvalues � of each coalition; and the percentages above the arrows are the transitionprobabilities � (S)). TheCDS at the initial state is C = (ff2; 3g; f2; 4g; f3; 4gg; ff1gg) ;and the excesses are e(S) = 4:938 for S = f2; 3g; f2; 4g; and f3; 4g; and it can alsobe easily veri�ed that this solution is a strong regular solution.

The solution � = (17:41; 17:53; 27:53; 37:53) is di¤erent from the nucleolus (Schmei-dler (1969)) and the core (both of which coincide with the quota (10; 20; 30; 40)); thekernel, the bargaining set (Maschler (1992)), and the Shapley value (which is equal to

18

Figure 1: MPE solution of the quota game (10; 20; 30; 40):

(17:5; 20; 28: 33; 34: 16)).8 In our solution, player 1 gets 7:41 more than his quota andplayers 2, 3 and 4 get each 2.47 less than their quota values. This example illustratesthat the solution proposed in the paper is di¤erent from all the other major existingsolution concepts.Interestingly, the equilibrium strategy of player 1 is to wait for a pairwise coalition

to form, an strategy that allows player 1 to get signi�cantly more than his quota.The solution thus makes predictions that are consistent with experimental resultsreported in Maschler (1992), where player 1 realized that he was weak and that hiscondition would improve if he waited until a pairwise coalition formed, and capturesan important strategic element of the game. Indeed, player 1 is better o¤ if thecoalition f2; 3g forms, rather than f2; 4g or f3; 4g; because in the ensuing pairwisebargaining with 4, player 1 can get a payo¤ equal to 25.9

8Since the bargaining set contains the kernel, and the kernel contains the nucleolus, this impliesthat our solution is di¤erent from the kernel and bargaining set (see Maschler 1992).

9However, strategies considered in this paper rule out the possibility that player 1 makes sidepayments to players 2 and/or 3 in order to encourage them to form coalition f2; 3g:

19

How do players�value change with changes in quotas and proposers�probabilities?Evaluating the value-sensitivity matrix with respect to changes in quotas, as we haveseen in Section 6.3, yields2664

��1��2��3��4

3775 =26640:366 0:549 0:062 0:0220:211 0:816 �0:020 �0:0070:211 �0:183 0:979 �0:0070:211 �0:183 �0:020 0:992

37752664�!1�!2�!3�!4

3775 ;and the coalition formation sensitivity matrix satis�es @�(fi;jg)

@!i> 0 and @�(fj;kg)

@!i< 0

for all distinct i; j; and k in {2,3,4} (for the sake of space we report only the signs ofthe transition probabilities).The information contained in the value sensitivity matrix yields the following

results: the value of all players increases when their quotas increase, but increases inthe quota of player 1 are shared by all players, while increases in the quotas of eitherplayer 2, 3 or 4 are almost completely appropriated by them (in fact, the other twoplayers distinct from player 1 su¤er a loss). Moreover, when a player�s quota goesup, all coalitions including this player become more likely to form (and coalitions notincluding this player are less likely to form).The comparative statics with respect to changes in proposers�probability is de-

scribed by the value-sensitivity matrix2664��1��2��3��4

3775 =26640 �5:42 1:96 3:450 1:80 �0:65 �1:150 1:80 �0:65 �1:150 1:80 �0:65 �1:15

37752664�p01�p02�p03�p04

3775 ;where, in order to preserve the sum of probabilities equal to one, we consider pi =p0i=(

P4j=1 p

0j); and the coalition formation sensitivity matrix satis�es

@�(fi;jg)@p0i

< 0 and@�(fj;kg)@p0i

> 0 for all distinct i; j; and k in f2; 3; 4g.This comparative statics analysis reveals a surprising result: When player 2 has

more initiative to propose, he bene�ts and player 1 loses from it. Interestingly, though,the opposite happens when players 3 and 4 have more initiative. Their equilibriumpayo¤s decrease when they have more initiative to propose!10 �10This result can be rationalized as follows: when p2 increases, coalitions {2,3} and {2,4} are less

20

Figure 2: MPE solution of the apex game.

Example 2: Apex GamesApex games, introduced by Davis and Maschler (1965), are another interesting

class of n-person games that have received considerable attention. In this game, onlytwo types of coalitions create non-zero value: any coalition with the Apex player(player 1), or the coalition with all the n � 1 remaining players (the Base players).For concreteness, consider the 5-player game N = f1; 2; 3; 4; 5g; where vf1;jg = 100

for j = 2; :::; 5, vf2;3;4;5g = 100; and vS = 0 otherwise. Players are very patient (� isin�nitesimally close to 1), and all players have equal chance to be proposers.The solution is depicted in Figure 2 (we use the same notation as in Figure 1).

The CDS at the initial state is C = (ff1; 2g; f1; 3g; f1; 4g; f1; 5g; f2; 3; 4; 5gg) ; andthe excesses are e(S) = 300

7for S 2 C:

likely to form and coalition {3,4} more likely; since player 1�s gains are lowest when coalition {3,4}forms he indirectly su¤ers when p2 increases. By similar reasoning, when p4 increases, coalitions{2,4} and {3,4} are less likely to form and coalition {2,3} more likely, which bene�ts player 1 andhurts the other players.

21

The solution for the game is � = (42:9; 14:3; 14:3; 14:3; 14:3). This solution coin-cides with the kernel of the game and the nucleolus. However, it is di¤erent from thebargaining set, the core (which is empty), and the Shapley value (which is equal to(60; 10; 10; 10; 10)):

Moreover, the model also predicts that any of the four apex coalitions f1; jg;j = 2; :::; 5; form with 20% probability, and the base coalition f2; 3; 4; 5g forms with20% probability.Comparative statics results for the apex game can also be easily obtained. The

sensitivity matrix describing the changes in value is266664��1��2��3��4��5

377775 =2666640:23 �0:48 0 00:74 0:17 0 0�0:26 0:17 0 0�0:26 0:17 0 0�0:26 0:17 0 0

3777752664

�vf1;2g�vf2;3;4;5g�p01�p02

3775 ;

where pi = p0i=(P5

j=1 p0j); and the sensitivity matrix describing the changes in coalition

formation is266664��(f1; 2g��(f1; 3g��(f1; 4g��(f1; 5g

��(f2; 3; 4; 5g

377775 =2666640:0309 �0:0009 �0:2 �2:2�0:01 �0:0009 �0:2 0:8�0:01 �0:0009 �0:2 0:8�0:01 �0:0009 �0:2 0:8�0:0009 0:0036 0:8 �0:2

3777752664

�vf1;2g�vf2;3;4;5g�p01�p02

3775 :

Surprisingly, these results indicate that changes in proposer probabilities haveno e¤ect on the players�values. Also, as was the case with the previous example,whenever the value of a coalition increases then both the equilibrium value of allcoalitional members and the probability that this coalition forms increase as well.

7 Uniqueness and the Global Number of Equilib-ria

We show in this section that even though there can be multiple equilibrium points,as illustrated by the next example, almost all games have a �nite and odd number

22

of MPE equilibria. Moreover, we derive a su¢ cient condition for the global unique-ness of equilibria. The result states that if the index of each equilibrium solution isnon-negative, where the index is equal to the sign of the determinant of the Jacobianmatrix dzFC; then there is a globally unique equilibrium. We prove that the su¢ -cient condition holds for three-player superadditive games, and thus there is only oneequilibrium in this class of games.Let us start by showing that coalitional bargaining games may have multiple equi-

librium points.

Example 3: War of Attrition (Multiple Equilibria)The following three-player symmetric example have seven MPE solutions. The

partition function that describes this game is vi(ff1g; f2g; f3gg = 0; vfi;jg(ffi; jg; fkgg) =1; and vfkg(ffi; jg; fkggg) = 3: The three-player coalition are not allowed (or has avery low value) and we assume that proposers are chosen with equal probabilities and� 2 (0:5; 1) :We describe below all the equilibria (results are derived solving equations(3), but we omit the details).There is an equilibrium in which the expected equilibrium value is x = (0:5; 0:5; 0:5) ;

the transition probabilities are � (fi; jg) = 1��5�; for all pairs fi; jg; and � (;) = 8��3

5�;

where ; represents no proposal (or remaining at the initial state): In this equilibrium,each of the three players refrains from proposing with high probability, and only pro-poses with a small probability to the other two players. They all reject any proposalsbelow 0:5; and thus players are indi¤erent between proposing or not.There are three other equilibria (they are all symmetric so we just focus on one

of them), in which the expected equilibrium values are x = (0; 1; 1); the transitionprobabilities are � (f1; 2g) = � (f1; 3g) = 1��

2�; and � (;) = 2��1

�: In this equilibrium,

players 2 and 3 reject any proposals lower than 1, make no proposal with high proba-bility, and, when proposing, choose to form a coalition with player 1. Player 1 cannota¤ord to pay more than 1 to form a coalition and thus it makes no proposals withprobability one.Finally, there are three additional equilibria (they are also symmetric), in which

the expected equilibrium values are x =�6�3�� ;

�3�� ;

�3���; which converges to x =

(3; 0:5; 0:5) when � ! 1; the transition probabilities are � (;) = 13; � (f2; 3g) = 2

3:

23

In this equilibrium, player 1�s strategy is to refrain from proposing and reject anyproposal worth less than 3, and player 2�s and player 3�s strategies are to always�give in�and propose to form the coalition f2; 3g. �

The main result of this section establishes a formula for counting the numberof equilibrium points; the formula is based on the Index theorem (see, for example,Mas-Colell et al. (1995)). The index is a number that is assigned to each equilibriumpoint of a regular game. Say that z is an equilibrium point with support C; the indexof z is de�ned as the sign of the determinant of the Jacobian dzFC evaluated at z(and is either +1 or �1). We denote index z = sign det (dzFC) :The result is useful because it implies that there is an odd number of equilibria.

In particular, the number of equilibrium points is not zero, so there exists at least oneequilibrium. We will see next that the result can be used to obtain global uniquenessfor speci�c classes of games.While the Index theorem has been applied to establish similar results for competi-

tive economies (Debreu (1970)) and normal form games (Wilson (1971) and Harsanyi(1973)), the application to coalitional bargaining games is a bit more involved dueto the fact that the equilibrium is not locally unique in terms of the equilibriumstrategies, but only in terms of the equilibrium points (see Section 5). To prove theresult we use a stronger version of the Index theorem for correspondences developedin McLennan (1989)-see the Appendix for the restatement of the Lefschetz �xed pointtheorem (LFPT) which is used in the proof. On the other hand, for both competitiveeconomies and normal form games a standard version of the Index theorem devel-oped in di¤erential calculus textbooks su¢ ces to develop the formula for the numberof equilibria.

Proposition 5 Almost all games (all regular and nondegenerate games) have a �niteand odd number of equilibrium points. Moreover,X

C

Xz:2MPEC

sign det (dzFC) = +1;

where the summation is over all CDS C and MPEs with equilibrium points z and CDSC:

24

Proof: Consider the correspondence F : X ! X where F (x) � Rd is de�nedby8>><>>:

where yi (�) = �pi(�)maxS3if� (�) (S) (x)g+ (1� �)vi (�)

y 2 Rd : +��P

S��P

j2� pj(�)�j (�) (S)�I[i2S]xi(�) + I[i=2S]xi(�S)

��;

and supp (�i (�)) � argmaxS3i

f� (�) (S) (x)g

9>>=>>; : (18)The set X is the convex and compact set X � Rd de�ned by X = �

�2�X(�), where

X(�) = fx (�) 2 Rj�j such thatXi2�xi (�) � v and xi (�) � vig;

and vi = min�3ifvi(�)g and v = max

�2�

�Pi2� vi (�)

: By the de�nition of X and F it is

easy to verify that indeed F (X) � X:The set of �xed points of F , F� = fx 2 X : x 2 F(x)g ; corresponds to the equi-

librium points of the game (see Section 3). The set F� is �nite for all regular andnondegenerate games v: all the equilibrium points are, by Lemma 3, solutions ofFC(x; �; e) = 0 for some some support C: But since the game is regular the solutionsare locally isolated (Proposition 3), and since the solution belongs to the compact Xthen there is only a �nite number of solutions.Moreover, it can be easily shown that F : X ! X is an upper hemicontinuous

convex-valued correspondence (thus F(x) is contractible for all x 2 X). The setX � Rd, Cartesian product of simplexes, is a simplicial complex and thus F satis�esthe conditions of the LFPT.Let Ux� be an open neighborhood around each x� 2 F�; so that x� is the only

�xed point in Ux� : The Additivity Axiom of the Lefschetz index implies

�(F ; X) =Xx�2F�

�(F ; Ux�): (19)

In addition, the Lefschetz index is

�(F ; X) = 1; (20)

25

because F can be approximated by a continuous map f 0 : X ! X such that�(F ; X) = �(f 0; X) (Continuity Axiom), and X is a contractible set, and thus thereis an homotopy ' : X � [0; 1] ! X where '1 = IX and '0 = z0 2 X: Therefore,any continuous map f 0 : X ! X is homotopic to the constant map so, by the WeakNormalization and Homotopy Axioms, �(F ; X) = �(f 0; X) = 1: Equations (19) and(20) imply,

�(F ; X) =Xx�2F�

�(F ; Ux�) = 1:

The next lemma, proved in the Appendix, establishes a formula for computing theLefschetz index �(F ; Ux�); which shows that it is equal to sign det(dz�FC) wherez� = (x�; ��; e�) is the equilibrium point.

Lemma 5 The Lefschetz index of a regular and nondegenerate equilibrium point z� =(x�; ��; e�) is equal to �(F ; Ux�) = sign det(dz�FC); and is equal to either +1 or �1.

Q.E.D.Proposition 5 implies a su¢ cient condition for global uniqueness of equilibria.

Corollary 2 All regular and nondegenerate coalitional bargaining games have a glob-ally unique equilibrium if det(dzF ) � 0 where the Jacobian is evaluated at any solutionof problem FC for all CDS C.

Because of the special structure of the Jacobian dzFC; we conjecture that a su¢ -cient condition for det(dFC) � 0 at all solutions of problem FC is that the inequalitiesxi (�)� xi(�S) � 0; for all S 2 Cr (�) and i =2 S; hold. Direct computation of deter-minants reveals, for all games with three players and all CDSs C; that the inequalitiesimply that det(dFC) � 0 (see Proposition 6).The inequality xi (�) � xi(�S) � 0, where i =2 S; is a weak condition that has a

natural economic interpretation: player i is excluded from the o¤er S if and only ifmoving from c.s. � to �S imposes a negative externality on player i (i.e., xi(�S) �xi (�)):We show in the next proposition that these inequalities hold for all three-player

games where the grand coalition is e¢ cient. Therefore, the equilibrium point com-puted explicitly in Gomes (2004) is the unique equilibrium point for almost all games.

26

Proposition 6 Almost all three-player games with externalities where the grand coali-tion is e¢ cient, i.e. v(fNg) �

PS2� vS (�) for all �, in particular superadditive

games, have a globally unique equilibrium.

We remark that in the war of attrition Example 3, a strongly regular game withseven equilibria, the grand coalition was not e¢ cient, and therefore it is not in con-tradiction with Proposition 6.

8 Conclusion

This paper studied the equilibrium properties of n-player coalitional bargaining gamesin an environment with widespread externalities (where the exogenous parametersare expressed in a partition function form). The coalitional bargaining problem ismodeled as a dynamic non-cooperative game in which contracts forming coalitionsmay be renegotiated. The equilibrium concept used is Markov perfect equilibrium,where the set of states is all possible coalition structures.A comprehensive analysis of the equilibrium properties is developed. We show

that for almost all games (except in a closed set of measure zero) the equilibrium islocally unique and stable to small perturbations of the exogenous parameters, and thenumber of equilibria is �nite and odd. Global uniqueness does not hold in general,but a su¢ cient condition for global uniqueness is derived, and this su¢ cient conditionis shown to prevail in three-player superadditive games.Comparative statics analysis can be easily performed using standard calculus tools,

allowing us to understand how the value of players and the path of coalition formationchanges in response to changes in the exogenous parameters. Being able to answercomparative statics questions is valuable to negotiators, because they may be able,for example, to invest in changing the likelihood of being proposers. Applications ofthe technique are illustrated using the apex and quota games, and some interestinginsights emerge: surprisingly, a player may not bene�t from having more initiativeto propose (other players may adjust their strategies in such a way that lead theproposer to be worse o¤). The analysis also suggests several interesting regularities:when the exogenous value of a coalition increases, both the equilibrium value of thecoalitional members and the likelihood that the coalition forms increase.

27

Appendix

Proof of Lemma 1: The necessary part follows directly from the discussion beforethe statement of the result and the de�nition of MPE solution. Let us prove thesu¢ cient part of the proposition. Suppose that we are given payo¤s and strategypro�les

��; �j; �

�satisfying all the conditions of the lemma. We use the one-stage

deviation principle for in�nite-horizon games. This result states that in any in�nite-horizon game with observed actions that is continuous at in�nity, a strategy pro�le� is subgame perfect if and only if there is no player i and strategy �0i that agreeswith �i except at a single stage t of the game and history ht; such that �0i is a betterresponse to ��i than �i conditional on history ht being reached (see Fudenberg andTirole (1991)).Note �rst that the coalitional bargaining game is continuous at in�nity: for each

player i his utility function is such that, for any two histories h and h0 such thatthe restrictions of the histories to the �rst t periods coincides, then the payo¤ ofplayer i; jui (h)� ui (h0)j ; converges to zero as t converge to in�nity: It is immediatelyclear that the negotiation game is continuous at in�nity because jui (h)� ui (h0)j �M��t+1 + �t+2 + � � �

�= M

1���t+1; for M large enough.

But the strategy pro�le �i is such that, by construction, no single deviation �0i atboth the proposal and response stage can lead to a better response than �i. Therefore,by the one-stage deviation principle, the stationary strategy pro�le � is a subgameperfect Nash equilibrium. Q.E.D.

Proof of Proposition 1: Any payo¤ x candidate for equilibria must satisfysome obvious restrictions. For example, the payo¤ x satis�es xi (�) � vi; where thelower bound is vi = min

�3ifvi(�)g ; because player i can get at least vi by refusing to

participate in any coalitions. It must also be the case, due to a feasibility constraint,that

Pi2� xi (�) � v; where the upper bound is v = max

�2�

�Pi2� vi (�)

: Therefore

any MPE payo¤ x must belong to the convex and compact set X � Rd de�ned byX = �

�2�X(�), where

X(�) = fx (�) 2 Rj�j such thatXi2�xi (�) � v and xi (�) � vig:

28

Consider the mapping f(x; �; e) into Rd; where for all i 2 � and � 2 �; thecoordinate fi(�) is given by

fi(�)(x; �; e) = xi (�)� �pi(�)ei(�)� (1� �)vi (�) (21)

� � XS��

Xj2�

pj(�)�j (�) (S)

!�I[i2S]xi(�) + I[i=2S]xi(�S)

�!;

and also the correspondence F : X ! Rd where y 2 F (x) if and only if y =x � f (x; e; �) and � and e are such that supp (�i (�)) � argmax

S3if� (�) (S) (x)g and

ei(�) = maxS3if� (�) (S) (x)g :

We show that the correspondence F satis�es all the conditions of the Kakutani�xed point theorem.We �nd convenient to use the map f ji (�)(x; �j) where

f ji (�)(x; �j) =

(maxS3i

nxS(�S)�

Pj2Sni xj(�)

oj = iP

S�� �j (�) (S) (I(i 2 S)xi(�) + I(i =2 S)xi(�S)) j 6= i:

Note that,

fi(�)(x; �) = �

Xj2�

pj (�) fji (�)(x; �(j))

!+ (1� �)vi (�) :

First, by de�nition, X is non-empty subset that is compact and convex. In addition,(1) F (X) � X : Take any x 2 X and y = f (x; �) with � 2 �(x) := f� 2 � :

x = x(�)g: We �rst show thatP

i2� yi (�) � v and then yi (�) � vi which impliesthat y 2 X: First,X

i2�yi (�) =

Xi2�fi(�)(x; �) =

= �

Xj2�

pj (�)Xi2�f ji (�)(x; �j)

!+ (1� �)

Xi2�vi (�)

where the order of the �rst summation has been inverted. ButXi2�f ji (�)(x; �) =

XS��

�j (�) (S)

xS(�S) +

Xi=2S

xi(�S)

!� v

29

becauseP

S �j (�) (S) = 1, �j (�) (S) � 0; and xS(�S)+P

i=2S xi(�S) � v; for x 2 X:Therefore X

i2�yi (�) � �

Xj2�

pj (�) v (�) + (1� �)v (�) = v:

Also, we have that yi (�) � vi: First note that fji (�)(x; �) � vi for all j, because

f ji (�)(x; �) � xi (�) � vi as player i can always choose not to make any o¤er (S = fig)and x 2 X. Also, f ji (�)(x; �) =

PS�� �j (�) (S) (I(i 2 S)xi(�) + I(i =2 S)xi(�S)) �

vi because x 2 X: Therefore,

yi(�)(x; �) = �

Xj2�

pj (�) fji (�)(x; �j)

!+ (1� �)vi (�) � vi:

(2) F (x) is a convex (and non-empty) set for all x 2 X: Say that y; y0 2 F (x)with y = f(x; �) and y0 = f(x; �0) where �; �0 2 �(x): Then, for any � 2 [0; 1] ;�y+ (1� �) y0 = f(x; ��+ (1� �)�0) 2 F (x) because ��+ (1� �)�0 2 �(x) (�(x)is convex).(3) F is u.h.c., that is, for any sequence (xn; f(xn; �n))! (x; y) with �n 2 �(xn)

then y 2 F(x) (i.e., there exists an � 2 �(x) such that f(x; �) = y): The sequence(�n) belongs to � a compact subset of a �nite-dimension Euclidean space. Therefore,there exists a subsequence of (�nk) that converges to � 2 �: Rename this subsequenceas (�n) for notational simplicity. We have that �ni (�) (S) ! �i (�) (S) for all S � �and i 2 � 2 �, and that f(xn; �n) ! f(x; �); due to the continuity of f; and thusy = f(x; �):

It is su¢ cient to show that � 2 �(x). By the de�nition of �(x), � 2 �(x)if and only if � 2 � and �i (�) (S) = 0 for all S � � and i 2 � 2 � such thatxS(�S)�

Pj2S xj(�) < maxS3i

�xS(�S)�

Pj2S xj(�)

�: Consider any S � � for which

the inequality above holds. By continuity, we have that there exists a large enoughn0 such that for all n � n0, xnS(�S) �

Pj2S x

nj (�) < max

S2�i

�xnS(�S)�

Pj2S x

nj (�)

�:

But since �n 2 �(xn); this implies that �ni (S) = 0, and �i (S) = 0. Q.E.D.

30

Proof of lemma 3: For all � satisfying C = C(�) then � = � (�) satisfyXS2Cr(�)

� (�) (S) =X

S2Cr(�)

Xj2�

pj (�)�j (�) (S) =

=Xj2�

pj (�)X

S2Cr(�)

�j (�) (S) =X

j2Pr(�)

pj (�) for all r;

because if j 2 Pr (�) then supp (�j (�)) � Cr (�) ; which corresponds to the last setof equations in F (C).Now, if i! j then there exist a coalition S such that i; j 2 S and �i (�) (S) > 0:

But because supp (�i (�)) � argmaxfS��:i2Sg

fe (�) (S)xg then

ei := maxfS��:i2Sg

(xS(�S)�

Xj2S

xj(�)

)� ej:

Repeating the same argument, if there is a path from i to j then ei � ej; and if i isstrongly connected to j then both have the same excess ei = ej: Thus, er (�) = ei =xS(�S)�

Pj2S xj; for all S 2 Cr and all i 2 Pr (�) : Substituting the expressions for

the excesses into equation (7) �nishes the if part of the proof. The reciprocal followsdirectly from the construction of the polyhedral setsMC and EC. Q.E.D.

Proof of lemma 4: The same steps of the proof applies to each c.s. � separately,so to simplify notation we eliminate explicit references to � below. The following claimimplies the lemma, as shown below.Claim: Let C = (C;P ) be a CDS, �� a strategy pro�le with C = C (��), and let

�i = supp��i : Given any � = (� (S))S2Cr close to zero satisfyingP

S2Cr � (S) = 0

there exists � = (�i (S))S2�ii2Pr

close to zero satisfyingP

S2�i �i (S) = 0 for all i 2 Prsuch that � (�) = �:Suppose that the claim holds. Let �C := f� 2 � : C = C (�)g: There exists

�� 2 �C such that � (��) = �� (as �� 2 MC). Let �i = supp��i : Given any � closeto �� de�ne �� = �� �� (which is close to zero). Consider a �� given by the claim(related to ��) and let � = �� + ��: Such � satis�es �i (S) > 0 for all S 2 �i(because ��i (S) > 0 for all S 2 �i and �� (S) are close to zero) and

PS2�i � (S) =

31

PS2�i �

� (S) +P

S2�i ��i (S) = 1: So � 2 �C and, by linearity of � (�) ; � (�) = �.Therefore, it is su¢ cient to prove the claim.Proof of claim: It is enough to analyze each component r = 1; :::; q separately

so we drop the subscript r (so P = Pr and C = Cr; and say that #P = p). Byconstruction (de�nition of P and C), there exists an ordering (i1; :::; ik; :::; ip) of Pwhere P k = fi1; :::; ikg and Ck = [i2Pk�i satisfy Ck \�ik+1 6= ? for all k = 1; :::; p�1 (starting from any element in P; each new element in the order is chosen so thatit has a support connected to some of the previous elements chosen).The proof now proceeds by induction. The induction hypothesis is: Suppose that

the following statement holds for P k and Ck: Given any �k =��k (S)

�S2Ck close to

zero satisfyingP

S2Ck �k (S) = 0 there exists �k =

��ki (S)

�S2�ii2Pk

close to zero satisfy-

ingP

S2�i �ki (S) = 0 for all i 2 P k such that �

��k�= �k: The statement also holds

for k+1: Consider any � = (� (S))S2Ck+1 close to zero satisfyingP

S2Ck+1 � (S) = 0:

Let S 2 Ck\�ik+1. De�ne �ik+1 (S) = � (S) for all S 2 �ik+1nfSg and let �ik+1�S�=

�P

S2�ik+1nfSg� (S) : Also de�ne �k (S) = � (S) for all S 2 Ckn�ik+1 ; �k (S) = 0 for

all S 2 Ck \ �ik+1nfSg; and let �k�S�= �

PS2CknfSg �

k (S) : Thus, using the in-duction hypothesis, there exists � = (�i (S)) S2�i

i2Pk+1which is close to zero, satis�esP

S2�i �i (S) = 0 for all i 2 Pk+1 and is such that � (�) = �: Since the statement is

true for k = 1 (just let � = �) and, by induction, the statement is also true for k = p;which is exactly the claim (as P k = P and Ck = C), this completes the proof.Q.E.D.

Transversality theorem: Suppose f : U � V ! Rn is continuously di¤er-entiable where U � Rn and V � Rm are open sets. If the n � (n + m) Jacobiand(x;q)f has rank n whenever f(x; q) = 0 (i.e., ft0) then the system of n equation andn unknowns f(�; q�) = 0 is regular for almost every q� 2 V:

Proof of proposition 4: The proof is by induction on the number of playersand the induction hypothesis is: for games with less than n players, almost all para-meters (v; p) in Rd��d are regular and nondegenerate and all such games have localsolution mappings x (v; p) that are surjective.The hypothesis holds for one player games: the only support is C = ff1gg and the

Jacobian matrix of problem FC is obviously nonsingular. Now, let � be a c.s. with n

32

players, and let us represent by a subscript 0 the references to the c.s. � and by thesubscript �0 the references to all its proper subgames. Let V0��0 represent the setof all (vi(�); pi(�)) and V�0 ���0 the set of all (vi(�0); pi(�0))�02�

�0 6=�.

Let R�0 � V�0���0 be the set of games that are regular and nondegenerate andthe local mappings x�0 (v; p) are surjective. According to the induction hypothesisalmost all games of V�0���0 belong to R�0. Consider the solutions of the augmentedproblem FC0(z0; v0; p0; v�0; p�0) = 0 where z0 = (x0; �0; e0) and we consider thatx�0 (v�0;p�0) changes with v�0;p�0 (even though expressions

Pi2S xi (�) + er (�) �

xS(�S) do not depend directly on the parameters v; p, the term xS(�S) is a functionof v�0; p�0). The Jacobian matrix at the solution, d(z0;v0;p0;v�0;p�0)FC0, is24 � � �(1� �)I0 0 �

� 0 0 0 �d(v�0;p�0)g � x�0 (v�0; p�0)0 d�MC0 0 dp0MC0 0

35 ; (22)

where g : V�0 ! Rm is the linear map g (x�0) (S) = xS(�S) for all the sets in thesupport C0 = (�i(�)) ; and � denotes arbitrary coe¢ cients. Note that the linear map gis surjective, and thus the composition g�x�0 (v�0; p�0) is surjective (the compositionof surjective maps is surjective). But then we have that FC0(z0; v0; p0; v�0; p�0)t0because all blocks �(1� �)I0; d�MC0 ; and �d(v�0;p�0)g � x�0 (v�0; p�0) are surjective.Therefore, by the transversality theorem, for almost every (v; p) 2 Rd��d; FC0(z0)t0.Because of the block triangular structure of the Jacobian matrix dzFC(z) (see remarkon Section 6.1) this shows that dzFC(z) is nonsingular (C regular) almost everywhere.The argument to show that C is nondegenerate almost everywhere is the same one

discussed in Section 6.2.To complete the proof, it still remains to show that the local solution mappings

x (v; p) of problem FC are surjective. But

d(v;p)x =

�d(v0;p0)x0 d(v�0;p�0)x0

0 d(v�0;p�0)x�0

�;

because x�0 does not depend on (v�0; p�0) ; and it is thus enough to prove that x0 (v; p)is surjective (by the induction hypothesis d(v�0;p�0)x�0 is surjective). The implicitfunction theorem gives us the expression of the derivative of the local mappings (referto (12)) as, d(v;p)x0 (v; p) = � [dz0FC0 ]

�1n d(v;p)FC0, where [dz0FC0 ]

�1n is the submatrix

33

of [dz0FC0 ]�1 restricted to the �rst n rows, and d(v;p)FC0 is given by (22). But both

[dz0FC0 ]�1n and d(v;p)FC0 are surjective so d(v;p)x0 (v; p) is surjective. Thus we conclude

that x0 (v; p) is surjective. Q.E.D.

Lefschetz Fixed Point Theorem (LFPT) (McLennan 1989): Let T be thecollection of admissible triples (X;F; U) where X � Rm is a �nite simplicial complex,F : X ! X is a upper hemicontinuous contractible valued correspondence (u.h.c.c.v.),U � X is open, and there are no �xed points of F in U � U: Then there is a uniqueLefschetz �xed point index �(X;F; U) that satisfying the following axioms (when Xis implicitly given we just say �(F;U)):(Localization axiom): If F0; F1 : X ! X are u.h.c.c.v. correspondences that agree onU , and (X;F1; U); (X;F0; U) 2 T ; then �(X;F1; U) = �(X;F0; U):(Continuity axiom): If (X;F; U) 2 T , then there is a neighborhoodW of Gr(F ) suchthat �(X;F 0; U) = �(X;F; U) for all u.h.c.c.v. correspondences F 0 : X ! X withGr(F 0) 2 W:(Homotopy axiom): If h : [0; 1]�X ! X is a homotopy with (X; ht; U) 2 T , for allt, then �(X; h0; U) = �(X; h1; U):(Additivity axiom): If (X;F; U) 2 T and U1; :::; Ur is a collection of pairwise disjointopen subsets of U such that there are no �xed points of F in U � ([rk=1Uk) then�(X;F; U) =

Prk=1 �(X;F; Uk):

(Weak Normalization axiom): For y 2 X; let cy be the constant correspondencecy(x) = fyg. If y 2 U then �(X; cy; U) = 1:(Commutativity axiom): If X � Rm and Y � Rn are �nite simplicial complexes,f : X ! Y and g : Y ! X are continuous functions, and �(X; g � f; U) = �(X; f �g; g�1(U)):

Proof of Lemma 5: De�ne the correspondence F(x) = x � F(x), where F(x)is the correspondence de�ned in (18). The Lefschetz index of F and the degree of Fare related by �(F ; U) = deg(F; U; 0) (see McLennan (1989)), and, for convenience,we work in the remainder of the proof with the concept of degree.The mixed nonlinear complementarity problem is the problem of �nding triples

(x; �; e) that are the solution of problem (MNCP) in Section 4.

34

For each point x consider the mixed linear complementarity problem MLCP(0);

h (�) = 0;

g (e; x) � 0; (23)

e free variable, � � 0 and �Tg (e; x) = 0;

Let z (x) = (e (x) ; � (x)) be a solution of the MLCP(0) (there can be multiple solu-tions). Note that F(x) = ff(x; z (x)) : z (x) is a solution of MLCP(0)g :Let (x�; e�; ��) be any regular and nondegenerate MPE with an associated CDS

C = (C;P ); with C = (C1; :::; Cq) and P = (P1; :::; Pq): By Lemma 3, there exists�� 2 �C such that �� = �� (��) ; and (x�; ��) is MPE. Furthermore, because allpoints in Pr are connected, we can choose a strategy pro�le �� satisfying supp (��i ) =Cr \ fS � � : i 2 Sg for all i 2 Pr:Consider now the perturbed mixed linear complementarity problem, or MLCP(")

h(")(e; �) = h (�) + " (e� e�) = 0;g(")(x; �; e) = g(x; e) + " (� � ��) � 0;e free variable, � � 0; �Tg(") = 0;

(24)

where " > 0: The Jacobian matrix M(") of MLCP(") is a P -matrix (i.e., a matrixwith all its principal minors positive). This is so because (see Cottle et al. (1992,pg. 154)), M(") = M + "I, where M is the Jacobian of MLCP(0); is a P0-matrix(i.e., a matrix with all its principal minors nonnegative). Let us prove that M is aP0-matrix: Consider the principal matrix M�� associated with a subset � of lines (orcolumns).11 We now show that either det(M��) is equal to zero or one. Note �rstthat det(M��) =

Qi2�det

�M�i�i

�where � = [i�i and �i are the elements of � with

entry i (either ei or �i (S) for some S 3 i). But det�M�i�i

�= 1 if �i = fei,�i (S)g

and is zero otherwise. Therefore, we conclude that all principal minors of M arenonnegative, and thus M is a P0-matrix.Given thatMLCP(") has a P -matrix then there is a unique solution z" (x) (Cottle

et al. (1992, pg. 150)) for all x: MLCP(") can be transformed into a standard

11We refer to the lines corresponding to @hi and @gi (S) as lines �i and �i (S), and the columnscorresponding to @�

@�iand @�

@�i(S)as columns �i and �i (S) : Also, we use the standard notation that

A��; A��; and A�� represent the submatrix of A with, respectively, rows and columns, columns, androws extracted from the index set �: Also, � denotes the complementary set of �.

35

LCP eliminating the variable e and the equation h(") = 0 (this is possible becauseMee(") = "I is nonsingular), and the transformed LCP also has a P -matrix (theSchur complement ofMee(") inM(")). Note that, in addition, we have that z" (x�) =(e�; ��) ; and that z" (x) converge to a solution of MLCP(0) when " ! 0 (Cottle etal. (1992, pg. 442)), and that z" (x) is piecewise linear in x:We now show that, because x� is a strong solution, there exists an " > 0 such

that for every 0 < " < " there exists an open neighborhood U" of x� such that z" (x)is smooth in U". Moreover, if we let � represent the index set

� = f�i (S) : for all S 2 Cr and i 2 Sg; (25)

then all �i (S)-coordinates of the solution z" (x) that do not belong to � are zero, andz" (x) are explicitly given by (M��("))

�1 q�(x); where M��(") is

M��(") =

�"I�� (deg)�d�h "Iee

�;

and the vector q�(x) has ei-coordinate equal to ("e�i � 1) ; and �i (S)-coordinate in �equal to "�i (S)

� + e(S) (x) ; for all 0 < " < " and x 2 U":In order to prove the above claim consider the function

'(x) = min[qr=1 fe(S) (x)� e(T ) (x) : S 2 Cr; T \ Pr 6= ;; and T =2 Crg :

Naturally, the function ' is continuous in x and, because x� is a strong solution,'(x�) > 0: Therefore, there exists an " > 0 and an open neighborhood U � Ux� of x�;such that all x 2 U satisfy '(x) > 2": Now suppose that the solution z" (x) for x 2 Uis such that a �i (T )-coordinate is non-zero for T =2 Cr and i 2 Pr. Then gi(") (T ) = 0which is equivalent to ei + " (�i (T )� ��i (T ))� e(T ) (x) = 0; and implies e(T ) (x) �ei � ": Also, gi(") (S) � 0 for all S; and thus ei + " (�i (S)� ��i (S)) � e(S) (x) � 0;which implies that e(S) (x) � ei + ": Therefore, e(S) (x) � e(T ) (x) � 2" � 2" forx 2 U; in contradiction with '(x) > 2" for all x 2 U: Now, since z" (x�) = (e�; ��) ;and supp (��i ) = Cr \ fS � � : i 2 Sg, and z" (x) is continuous, then there existsan open neighborhood U" � Ux� of x� where all �i (S)-coordinates of the solutionbelonging to � are non-zero. This implies that gi(") (S) = 0 holds for all �i (S) in �;and thus z" (x) = (M��("))

�1 q�(x):

36

De�ne the mapping F" (x) = f(x; z" (x)) (this mapping is well-de�ned due to theuniqueness of z" (x)); where F" (x�) = 0: Since f is smooth and z" (x) ! z (x) thenF" (x)! F (x) : Therefore, for every � > 0 there exists " such that dist

x2U(F" (x) ;F (x)) <

�; for all 0 < " � ": But since F (x) has no zeros in the boundary of @U then F" (x)also does not have any zeros in @U . By the homotopy and continuity property of thedegree, deg (F; U; 0) = deg (F"; U; 0), for " close to zero.Therefore, it only remains to show that deg (F"; U; 0) = sgn (det(dFC (z�))) for "

close to zero, where z� = (x�; e�; ��). This result follows from sgn (det(dxF" (x�))) =

sgn (det(dFC (z�))) 6= 0; as we will show. Indeed, this implies that F" is nonsingular at

x�; and thus there exists an open neighborhood V � U of x� where x� is the only zeroof F": But since the point x� is the only zero of F(x) in U � Ux�, and F" (x)! F (x)

then there are no zeros of F" in the compact region UnV; for " small enough, andthus x� is the only zero of F" in U . A well-known property of the degree then impliesthat �(F ; U) = deg (F"; U; 0) = sgn (det (dxF" (x�))) = sgn (det(dFC (z�))) :We now show that sgn (det(dxF" (x�))) = sgn (det(dFC (z�))) ; for " small enough.

Consider F (x; �; e) ("),

F (x; �; e) (") =

0@ f(x; �; e)h (�) + " (e� e�)g (e; x) + " (� � ��)

1A :Simple linear algebra shows that the Jacobian dxF" (x�) is the Schur complement ofM�� (") in dF�� (") (dxF" (x�) = dF�� (") =M��); where

dF�� (") =

24 (dxf) d�f def(dxg)� "I�� (deg)�0 d�h "Iee

35 ; (26)

is evaluated at point (x�; e�; ��) : Therefore, det (dxF" (x�)) = det (dF�� (")) = det (M��)

(see Cottle et al. (1992, pg. 75)). But since det (M��) > 0 (M is a P -matrix) thensgn (det(dxF" (x

�))) = sgn (det (dF�� ("))) :

We claim that sgn (det (dF�� ("))) = sgn (det(dFC (z�))) : In order to prove the

claim we use the following formula for the determinant (Cottle et al. (1992), pg.60): for an arbitrary diagonal matrix D; det(A+D) =

P detD detA where the

summation ranges over all subsets of lines. Observe that matrix dF�� (") = A+D;

37

where A = dF�� (0) and D is the diagonal matrix,

D =

24 0 0 00 "I�� 00 0 "Iee

35 :Developing the expression for det (dF�� (")) using the formula above we get a polyno-mial in " (detD is a power of " ). We are only interested in the non-zero coe¢ cientwith lowest order because, when " converges to zero, this is the coe¢ cient that de-termines the sign of det (dF�� (")) :The rows and columns of matrix A = dF�� (0) corresponding to �i (S) and ei are

R (�i (S)) =Xj

I[j2S]e(xj) + e (ei) ;

R (ei) = �XS2Cr

e (�i (S)) ;

C (�i (S)) =Xj

I[j =2S]xj(S)e(xj)� e (ei) ;

C (ei) = �pie(xi) +XS2Cr

e (�i (S)) ;

where vectors e(xi); e (ei) ; and e (�i (S)) are the unit vectors at, respectively, coordi-nates xi; ei; and �i (S) :Consider A�� the submatrix of A corresponding to the rows � of A: Let � be a

maximal subset of � such that rank (A��) is di¤erent from zero (j�j = rank (A��)

and rank (A��) = rank (A��)). Note that A where is the set of lines = � [fei : i 2 �g[ fxi : i 2 �g is equal to A = dF�� (0) ; according to the de�nition (26).Also, detA 0 0 = 0 for set of lines 0 that strictly contains because � is a maximalsubset of � such that rank (A��) 6= 0:We now show that det (dF�� (0)) = det(dFC (z

�)) 6= 0; which proves that thelowest-order non-zero coe¢ cient is equal to a positive integer (the number of max-imal subsets � � �) multiplied by det(dFC (z�)); and thus sgn (det (dF�� ("))) =sgn (det(dFC (z

�))) ; for " small enough.We now propose an algorithm replaces all rows and columns �i (S)�s with the

same S by only one row and column �i (S) for all S 2 Cr; and also replaces all rowsand columns ei for all i 2 Pr by only one row and column er for each r = 1; :::; q:

38

Algorithm: Start with matrix A = dF�� (0) :Step 1 : Choose an element r; that have not yet been chosen, from the set

f1; 2; :::; qg and proceed to the next step, or else, stop if the choice is not possible.Step 2 : Choose two distinct rows �i (S) and �j (S) of A with j 6= i and S 2 Cr

and proceed to the next step, or else return to step 1 if the choice is not possible.Step 3 : Subtract row �i (S) from row �j (S) (i.e., R (�j (S)) = R (�j (S)) �

R (�i (S))); and add column ej to column ei (i.e., C (ei) = C (ei) + C (ej)). Thematrix that is obtained after the two operations have the same determinant as ma-trix A: Let this matrix be the new matrix A. After these two operations, row �j (S)of A has only one non-zero entry at column ej; with a value equal to 1. The de-terminant of A can be computed by a co-factor expansion along row �j (S) ; andjAj = (�1)(#�j(S)+#ej)jA0j; where A0 is the submatrix obtained after deleting row�j (S) and column ej of matrix A:Now, perform the following symmetric transformations on the submatrix A0: Sub-

tract column �i (S) from column �j (S) (i.e., C (�j (S)) = C (�j (S)) � C (�i (S)))and add row ej to row ei (i.e., R (ei) = R (ei) +R (ej)). The matrix that is obtainedafter the two operations have the same determinant as A0: Let this matrix be the newmatrix A0. After these two operations, column �j (S) of A0 has only one non-zeroentry at row ej; with a value equal to �1. The determinant of A0 can be computed bya co-factor expansion along column �j (S) ; and jA0j = (�1)� (�1)(#�j(S)+#ej�1)jA00j;where A00 is the submatrix of A0 obtained after deleting column �j (S) and row ej:observe that the column �j (S) of A0 is in the same location as row �j (S) of A0; butrow ej appears one entry before column ej of A (because the row �j (S) that has beenremoved appears before row ej). Putting together the expressions for the determinantyields jAj = jA00j: Let matrix A00 be the new matrix A, and return to step 2.Because � is a maximal subset of � with rank (A��) 6= 0 and rank (EC) 6= 0, the

algorithm starts with matrix A = dF�� (0) and ends with matrix A = det(dFC (z�))(maintaining the same determinant in all steps).Therefore, det (dF�� (0)) = det(dFC (z�)); as we claimed. Q.E.D.

Proof of Proposition 6: We focus on the c.s. � = ff1g; f2g; f3gg because wealready know that two-player games have a unique equilibrium (Rubinstein (1982)).We �rst show that X(i; S) = xi (�)�xi(�S) � 0; where i =2 S; if there is a positive

39

probability that S is chosen in equilibrium. Say that S = fj; kg (if S = ; (no proposalcase) then xi(�S) = xi and if S = N = f1; 2; 3g then there are no elements i =2 S): Inorder to simplify the notation, let xi = xi (�) ; xi(�S) = xi (jk) ; xS(�S) = xjk (jk) ;and V = vN(fNg). Suppose that S is chosen in equilibrium with positive probability.Then e (S) (x) � e (N) (x) ; which is equivalent to,

xjk (jk)� xj � xk � V � xi � xj � xk; (27)

andxjk (jk) + xi (jk) + xi � xi (jk) � V: (28)

But since there is no delay in the formation of the grand coalition when the game isat the c.s. ffjkg ; figg ; we have that

xjk (jk) + xi (jk) = �V + (1� �) (vjk (jk) + vi (jk)) :

Replacing this expression into (28) yields

X(i; jk) = xi � xi (jk) � (1� �) (V � (vjk (jk) + vi (jk))) � 0:

We now compute det(dFC) for all admissible CDS C = (C;P ); and show thatdet(dFC) � 0. From the de�nition of CDSs it follows that P = (P1; :::; Pq) is apartition of N and C = (C1; :::; Cq) is an ordered disjoint collection of subsets S � Nsatisfying: for all S 2 Cr then S \ Pr 6= ; and S � [rs=1Ps; and also [S2CrS �Pr: Moreover, there is no S = fig that is chosen in equilibrium, and thus Cr �ff1; 2g; f1; 3g; f2; 3g; f1; 2; 3gg: A list of all admissible CDSs (except for permutationsof the players) follows with the corresponding value for det(dFC) (i; j; and k aredistinct elements of N; and di = 1 � �

PS �(S)I[i2S], z(i; jk) = �X(i; jk); and wi =

�pi):

40

CDS C det(dFC)(ff1; 2g; f1; 3g; f2; 3g; f1; 2; 3gg) z(2; 13)z(1; 23)z(3; 12)(ff1; 2g; f1; 3g; f1; 2; 3gg) z(3; 12)z(2; 13)(w1 + d1)(ff1; 2g; f1; 2; 3gg) z(3; 12)(d2w1 + d1d2 + w2d1)(ff1; 2; 3gg) d1w3d2 + d1d3d2 + d1d3w2 + d2d3w1(ff1; 2g; f2; 3g; f1; 3gg)

Pi;j;k (di + 2wi) z(k; ij)z(j; ik)

(ff1; 2g; f1; 3gg)P

i;j 6=1(diw1 + did1 + wid1)z(j; 1i)

(ff1; 2gg; ff1; 2; 3g; f1; 3g; f2; 3gg) z(2; 13)z(1; 23)(w3 + d3)(ff1; 2gg; ff1; 2; 3g; f1; 3gg) z(2; 13)(w1 + d1)(w3 + d3)(ff1; 2gg; ff1; 2; 3gg) (d2w1 + d2d1 + w2d1)(w3 + d3)

(ff1; 2gg; ff1; 3g; f2; 3gg)�P

i;j 6=3(2wi + di)z(j; i3)�(w3 + d3)

(ff1; 2gg; ff1; 3gg) (d2w1 + d2d1 + w2d1)(w3 + d3)

Note that the �rst 6 entries of the table corresponds to CDSs with P = (f1; 2; 3g)and the remaining entries to CDSs with P = (f1; 2g; f3g):The determinant for all CDSs are nonnegative because it is a sum of nonnegative

terms. Corollary 2 implies that there is a unique global MPE solution. Q.E.D.

41

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