A Stochastic Model of Sequential Bargaining with

A Stochastic Model of Sequential Bargaining with Complete Information Antonio Merlo; Charles Wilson Econometrica, Vol. 63, No. 2. (Mar., 1995), pp. 371-399. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28199503%2963%3A2%3C371%3AASMOSB%3E2.0.CO%3B2-C Econometrica is currently published by The Econometric Society.

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Econornetrica, Vol. 63, No. 2 (March, 1995), 371-399

A STOCHASTIC MODEL O F SEQUENTIAL BARGAINING

WITH COMPLETE INFORMATION

We consider a k-player sequential bargaining model in which the size of the cake and the order in which players move follow a general Markov process. For games in which one agent makes an offer in each period and agreement must be unanimous, we characterize the sets of subgame perfect and stationary subgame perfect payoffs. With these characterizations, we investigate the uniqueness and efficiency of the equilibrium outcomes, the conditions under which agreement is delayed, and the advantage to proposing. Our analysis generalizes many existing results for games of sequential bargaining which build on the work of Stahl (1972), Rubinstein (1982), and Binmore (1987).

KEYWORDS:Noncooperative bargaining, dynamic games, stochastic games

1. INTRODUCTION

NONCOOPERATIVE SEQUENTIAL BARGAINING MODELS have received considerable attention since Rubinstein's (1982) analysis of the basic two-player game with alternating offers. Even within the context of complete information, the model has since been generalized in many respects, allowing for more players and more flexible extensive forms. In this paper, we extend the basic model to a k-player stochastic environment with complete information to provide a unified framework for a large class of bargaining games. We formulate a general characterization of the sets of subgame perfect and stationary subgame perfect payoffs in terms of the fixed points of operators on the set of payoff functions. Besides implicitly providing an algorithm for computing the equilibria of this class of games, these characterizations provide the means for a systematic treatment of such issues as the uniqueness and efficiency of the equilibria. Our analysis also demonstrates how many existing results and techniques may be generalized and exhibits more clearly some of the limitations of their scope. Our main innovation is to allow the surplus to be allocated and the identity of the proposer to follow a general stochastic process. In each period, a state is realized which determines the cake (i.e. the set of possible utility vectors to be agreed upon in that period) and the order in which the players move. The first player to move in any period may either propose an allocation or pass. If he proposes an allocation, each of the remaining players in turn accepts or rejects the proposal. If any player rejects the proposal, a new state is realized and the process is repeated until some proposed allocation is unanimously accepted. This framework incorporates essentially any k-player sequential bargaining game with complete information which satisfies the following two features. First, at each stage, only one agent may make an offer, and, to reach agreement, this ' T h e authors gratefully acknowledge the support of the C. V. Starr Center for Applied Economics. We also wish to thank Roberto Chang, Prajit Dutta, Douglas Gale, two anonymous referees, and the participants of the microeconomics workshop at New York University for helpful comments.

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A. MERLO AND C . WILSON

offer must be unanimously accepted (before any new information is revealed). Second, in the event that no agreement is reached, the continuation game does not depend on the actions of the players in the current period. The model admits the various forms of nonstationarity and randomly chosen proposers studied by Binmore (1987) in the context of a two-player game with a deterministic cake. It allows for an exogenous risk of breakdown in the negotiations as in Binmore, Rubinstein, and Wolinsky (1986), and with some slight modification, it includes games in which the players may take up outside options after rejecting a proposal as in Shaked and Sutton (1984b) and Binmore, Shaked, and Sutton (1989). Our framework also includes the k-player generalization of the alternating offers game studied by Herrero (1985) and Moulin (1986). Although our analytic approach might be applied to a larger class of games, the characterizations of the subgame perfect and stationary subgame perfect payoffs generally do not extend to bargaining games which do not satisfy the two properties listed above. For instance, these characterizations do not apply to games in which players make simultaneous offers (Chatterjee and Samuelson (1990) and Stahl (1990)) or to games with lags before players may recognize and/or respond to offers (Perry and Reny (1993) and Sakovics (1993)). They do not apply to games in which agreement need not require unanimous consent (Baron and Ferejohn (1989)) or to games in which the multilateral bargaining procedure can be reduced to a series of bilateral negotiations (Chae and Yang (1988) and Krishna and Serrano (1991)). Games in which players may impose additional costs when rejecting an offer (Haller and Holden (1990) and Fernandez and Glazer (1991)) or games in which rejected proposals limit the set of feasible future proposals (Fershtman and Seidmann (1993)) also lie outside the scope of these characterizations. Introducing a stochastic and/or nonstationary cake process naturally leads to the possibility that agreement is delayed whenever some player perceives that a better agreement may be achieved by waiting. The most common explanation for delaying agreement is that players are unsure about the true preferences of their opponents. This approach leads to a game of incomplete information. (See, e.g., Sobel and Takahashi (1983), Admati and Perry (1987), or Osborne and Rubinstein (1990) for a survey of this literature.) In the context of complete information, sequential bargaining models generally admit delay only if there are multiple equilibria by which credible punishments may be generated. (The model of Haller and Holden (1990) exploits this idea to explain strikes.) Moreover, since the strategies must depend on the history of offers and counter offers, these equilibria are not stationary. In a stochastic framework, agreement may be delayed even in a unique stationary subgame perfect equilibrium. As in most models of bargaining, we treat the cake and proposer processes as primitives. In a more comprehensive framework, however, the model would be derived from an underlying environment defining property rights, consumption possibilities, and the set of feasible agreements in each state. The cake in any state is then simply the set of expected discounted utility vectors obtainable from currently feasible agreements. It reflects not only physical constraints but

SEQUENTIAL BARGAINING

373

also limitations on the permissible state contingent agreements over allocations in future states. In some bargaining problems, there may be practical limits on the complexity of feasible state contingent proposals. If the anticipated arrival of new information or ideas resolves key elements of uncertainty, the players may delay agreement to achieve a better allocation. Merlo and Wilson (1994) argue that these considerations may motivate the timing of agreements in such diverse contexts as the LDC debt problem, contract litigation, and government formation in parliamentary systems (see also Merlo (1992)). In the next section, we develop the basic framework and define the concepts of subgame perfect (SP) and stationary subgame perfect (SSP) equilibria. In Section 3, we characterize the SSP payoffs as the fixed points of an operator on the set of k-player payoff functions. This characterization is used to establish a contraction condition on the set of feasible outcomes which is sufficient to guarantee a unique SSP payoff. Section 4 contains some results on the efficiency of SSP equilibria, the conditions under which agreement is reached, and the advantage to being the proposer. Given the generality of the underlying stochastic process, a number of perverse results are possible. As in general equilibrium models with incomplete markets, SSP payoffs need not be (even second best) efficient if not all contingent contracts are feasible. We present an example in which two players agree immediately in the unique SSP outcome, even though it is Pareto dominated by an alternative outcome in which agreement is delayed. This source of inefficiency is not possible when the game is nonstochastic. We also provide an example of a three-player game in which an increase in one player's bargaining power in later stages of the game reduces his unique SSP expected payoff in the initial period. Neither of these results are possible if all players have the same discount factor and the cake is a simplex (of random size) in each period. In Section 5, we relax the assumption that strategies be stationary and characterize the set of SP payoffs. Following the basic logic of the argument for the SSP equilibria, we exploit our characterization of the set of SP payoffs to establish sufficient conditions for the SP payoff to be unique. For two-player games, the SP payoff is unique if and only if the SSP payoff is unique. Without additional restrictions, this result does not extend to k-player games as first demonstrated by Avner Shaked. However, under the same conditions which imply a unique SSP payoff, we show that the SP payoff is unique whenever the discount factor is less than l / ( k - 1). 2. T H E GAME

For any positive integer n , let R" denote n-dimensional Euclidean space. For x, y E R n , let x G y denote x, G y,, i = 1,. . . , n, and let x < y denote x G y with xi y. For x E R n , let x-, = (x,, . . . , x,-,, x,,,, . . ., x,), and for z E R', let (z, x-,) = (x,,. .. , x,-,, Z, x,+,, .. . , xn), i = 1,. . ., n. The n-vector (0,. . . , O ) will be denoted by 0 unless the dimension is not clear from the context.

374

A. MERLO AND C. WILSON

Let a = (a,, a l , a,, . . . ) denote a temporally homogeneous Markov process realizing values in a Bore1 subset of a complete separable metric space, S. Let K = (1,. . . , k} denote a set of players, where k 2. We refer to an element s E S as a state, and an element i E K as a player. For t = 0,1,2,. .. , let a' = (a,, . . . , a , ) denote the t-period state-history with typical realization (so,. . . , s'). A stochastic sequential bargaining game for K may be indexed by (C, p, P), where, for each state s E S, C(S) c R k is a cake representing the set of feasible utility vectors that may be agreed upon in that state, p(s) is a permutation on K with p,(s) denoting the identity of the player who makes the ith move in that state, and p < 1 is the common discount factor for the players. An allocation is an element x E C(S), where x, is the utility to player i. Let ~ ( s=) p1(s) denote the proposer in state s. We assume p is measurable. The necessary regularity conditions on the cake process will be imposed indirectly below. The game is played as follows. Upon the realization of a state s, ~ ( s chooses ) to either pass or propose an allocation in C(s). If he proposes an allocation, player p2(s) responds by either accepting or rejecting the proposal. Each player responds in turn (in the order prescribed by p(s)) until either some player has rejected the offer or all players have accepted it. If no proposal is offered and accepted by all players, a new state s t is realized in the next period according to the Markov process a . The procedure is then repeated except that the order of moves is determined by p(st) and the proposal must lie in the set C(sl). This process continues until an allocation is proposed and accepted. An outcome ( 7 , ~ may ) be defined as a stopping time T , and ~ a k-random variable, 7 , measurable with respect to a', which satisfies 7 E C(a,) if T < a, and 7 = 0, otherwise. Given a realization of a , r denotes the period in which a proposal is accepted, and 7 denotes the proposed allocation which is accepted in state a,. Define pa = 0. Then, for the game starting in state s, an outcome ( 7 , ~ implies ) a von Neumann-Morgenstern payoff to player i, E[P77,la, = s]. An outcome ( 7 , T) is stationary if there is a measurable subset, SF c S, and a measurable function, p : SF + R k , such that (i) a, E SF, t = 0,. . . , r - 1, (ii) a, E SF, and (iii) 7 = p(u,). That is, no allocation is implemented until some state s E SF is realized, in which case proposal p(s) E C(s) is accepted. Using condition (iii), we may define vF(s) = E[PTp(a,)Ja0= s ] to denote the von Neumann-Morgenstern payoff vector in state s. Since conditions (i) and (ii) imply r = 0 when a, E SF, it follows that vF(s) = p(s) for s E SF, and, from condition (ii), that uF(s) = E[P7qF(a,)lao = s ] for all s E S. We denote a typical stationary outcome by (SF, p ) or, if we need to make the stopping time explicit, by (SF, p , T). Since the remaining concepts are standard, we present them informally. A history is a specification of a finite sequence of realized states and the actions taken at each state in the sequence up to that point. A strategy for a player * A random variable

(0,1,2,. . . , m), and

T(U)

T is a stopping time on u if depends only on ur.

T

is a function of u taking values in


375

specifies a feasible action at every history at which he must act. A strategy profile is a measurable k-tuple of strategies, one for each player. At any history, a strategy profile induces an outcome and hence a payoff for each player. A strategy profile is subgame perfect (SP) if, at every history, it is a best response to itself. We refer to the outcome and payoff induced by a subgame perfect strategy profile as an SP outcome and SP payoff respectively. A strategy profile is stationary if the actions prescribed at any history depend only on the current state and current offer. A stationary subgame perfect (SSP) outcome and payoff are the outcome and payoff generated by a subgame perfect strategy profile which is stationary. Our analysis requires some regularity conditions on the cake process. Rather than explicitly impose these conditions on C, however, it is more convenient to work directly with the following "best response" function. For s E S, i E k , and d E R ~let, 5,(d, s ) = sup({x, E R': x E C(s), x-, d-,I u {O)) denote the payoff from the best allocation in C(s) for player i which yields at least d-, to the other players. If there is no such allocation, then e,(d, s) is 0. We require 5, to satisfy the following restrictions. ASSUMPTION (All: Fix i E K. (a) 5, is bounded and measurable on S x R ~ . (b) For s E S, 5,(., s) is continuous. (c) For s E S and d E R ~(i), (0,d - , ) E C(S) implies (S,(d, s), d-,I E C(S), and (ii) x > ((,(d, s), d-,) implies x @ C(s). Conditions (i) and (ii) of part (c) imply that, subject to the reservation level d - , for the other players, a best allocation for player i always exists whenever > d-,} is not empty. Furthermore, this allocation always extracts {x E C(S): x - ~ all of the surplus from the other players. The conditions of Assumption (Al) are satisfied if C(s) is compact, the set of weakly Pareto optimal and strongly Pareto optimal points coincide, and the Pareto frontier contains no holes. If, in addition, the correspondence C is continuous in the Hausdorff topology and the transition probability of the a process is weakly continuous in s, then 5 is bounded and jointly continuous. Before turning to the analysis of the game, a few remarks are in order. Although the a process is stationary, the game itself need not be stationary in any sense. Since the state space need not be finite, each state may effectively represent the entire history of the process. Consequently, by assuming an absorbing state s* with C(s") = {O), we may incorporate games with a finite horizon, or more generally, game's with an exogenous risk of breakdown in the negotiations as in Binmore, Rubinstein, and Wolinsky (1986).3 There is no real restriction implied by the assumption that players discount utility at a common constant rate. So long as the discounted size of the cake converges uniformly to 0, any variation in preferences generated by state or If the game is slightly generalized to allow the reservation utility of players to vary with the state, we may also incorporate games in which the players may take up outside options after rejecting a proposal as in Shaked and Sutton (1984b) and Binmore, Shaked, and Sutton (1989).

376


player dependent discount factors can always be represented by a different cake process with a common fured discount f a ~ t o r . ~ The information of the players is symmetric at each stage. Responders receive no new information about the future cake process after the proposal is made. We require this restriction to obtain the simple characterization of the SSP payoffs formulated in Theorem 1 below. Whether or not this is an important practical consideration depends on the application. Our own view is that the proposer process is simply a device to determine the bargaining power of the players. Any new information would also be accompanied with the option of making a new offer. We consider only pure strategies and hence only pure outcomes. Permitting mixed strategies may expand the set of feasible payoffs, and with (publicly randomized) mixed strategies, the set of SP payoffs must include at least the convex hull of the SP payoffs attainable with pure strategies. However, for reasons we will explain at the end of Section 3.1, it does not change the set of SSP payoffs. In any case, it is not clear that we should allow any public randomization over and above what is explicitly incorporated in the a process. If the complexity of agreements is an important constraint, the costs of implementing an agreement to randomize allocations may render such allocations infeasible. Where randomization is feasible, that possibility can be explicitly incorporated into the game by splitting states into multiple states, each with the same cake, proposer, and transition function.

3. STATIONARY SUBGAME PERFECT PAYOFFS

An essential feature of an SSP equilibrium is that there is no role for threats to prevent the proposer from fully exploiting his monopoly power. Consequently, in any state in which agreement is reached, the proposer may extract from the other players any surplus in excess of their expected payoffs from delaying agreement until the next period. This property permits a sharp characterization of the SSP payoffs as the fured points of an operator on the set of payoff functions. The methods used by Shaked and Sutton (1984a) and Binmore (1987) to compute the SP payoffs for nonstochastic bargaining games implicitly exploit an analogous characterization. There are two reasons why we study the SSP payoffs before focusing on the entire set of SP payoffs. First, an SSP equilibrium may serve as a natural focal point. In games with more than two players, it may be unique even when there is a serious indeterminacy in the set of SP payoffs. Also, the best response to any stationary strategy lies in the class of stationary strategies. Consequently, 4~ variable discount factor may permit the same process to be represented with a smaller number of cake shapes and hence a smaller number of states, perhaps affecting the set of SSP payoffs. From a more practical perspective, it may also allow a more parsimonious statement of various restrictions on the cake process. We treat the discount factor as a constant to simplify the notation in the statement and proofs of the major results of the paper.

377


the SSP payoff may be the natural outcome of simple learning rules.' Second, the characterization of the SSP payoffs provides an important tool for studying the entire set of SP payoffs. In fact, our characterization of the SP payoffs exploits the same operator used to characterize the SSP payoffs. Furthermore, as we establish in Section 5, if the SP payoff is unique, it is also the unique SSP payoff. 3.1. A Characterization of SSP Payoffs For any positive integer n, let Fn denote the set of bounded measurable functions on S taking values in Rn. For f l, f E F n , let f 1 G f denote f '(s) < f 2(s) for all s E S. For any norm 1 1 . II on R n and any f E F n , let Il f 11 = sup{llf(s)ll: s E S). An operator Q: Fn-t Fn satisfies the contraction property for 1 1 . II if there is a 6 < 1 such that f l, f E Fn implies IlQ( f '1 - Q( f 2)11< 611f 1 - f 211. For any operator Q: Fn-t Fn which satisfies the contraction property, there is a unique f E Fn such that Q( f ) = f (see, e.g., Blackwell (1965, p. 232)). Lemma 1 establishes a useful characterization of the payoff for any stationary outcome. The proof, which is presented in the Appendix, uses a standard argument from dynamic programming. LEMMA1: If (SF, p ) is a stationary outcome, then u p is the unique element of F~ for which up(s) = p(s) for s E SF, and vp(s) = E[/3vp(al)lao = s ] for s E S - SF. Recall that (,(d, s) is the payoff from the best allocation in C(s) for player i , which yields at least d-i to the other players. For f E F~ and i = ~ ( s ) define A,( f )(s> = max{Si(E[/3f(al)lao = sl, s), E[/3fi(al)lao = sll and A-,(f )(s) = E[/3f-,(al)la0 = s]. Then, Assumption (Al) implies that A maps F~ to F ~ . A( f )(s) denotes the equilibrium payoff of the ultimatum game in state s when player i is the proposer and the payoff to disagreement is E[/3f(al)lao = s]. THEOREM1: Given (Al), f is an SSP payoff if and only if^( f )

=f .

PROOF:(a) Suppose f is an SSP payoff. Fix s E S and let i = ~ ( s )Note . first that if no proposal is accepted in state s, then f(s) = E[/3f ( a l ) lao= s]. Now consider an SSP response to some proposal x E C(s). If xj > E[/3fj(al)lao = s], then responder j accepts the proposal, but, if x, < E[/3fj(al)lao = s ] for some responder j, proposal x is rejected. Given these restrictions on the SSP strategies of the responders, a payoff maximizing proposer must obtain a payoff of [i(E[/3f(~l)lao= s], s) from any SSP proposal which is accepted. However, the proposer can also guarantee himself E[/3fi(al)lao = s ] by passing. Therefore, if Si(E[/3f(al)lao = s], s ) < E[/3fi(al)lao = s], no proposal is accepted, in This argument was brought to our attention by Eric Maskin.

378

A. MERLO

AND C . WILSON

which case, f(s) = E [ pf(al)lao = s]. Alternatively, if [i(EIPf(ul)lao = s], s ) > EIPfi(al)lao = s], then, it follows from Assumption (Al) that player i proposes allocation f(s) = ( [ i ( E [ ~ f ( a l ) l a o= s], s), E[/3f-i(ul>luo = s]), which is accepted. If [i(EIPf(ul)lao = sl, s ) = EIPfi(al>lao= sl, then f(s) = EIPf(al)l a, = s] whether the SSP proposal is accepted or not. We conclude that A( f ) =f . ( = ) Suppose A( f ) = f . We establish first that f > 0. Suppose not. Let rn = inf { fi(s): s E S, i E K), and choose i E K and s E S such that f,(s) < prn. Then E [ pf,(al)luo = s] prn > f,(s). By definition, Si(E[Pf(al)lao = s], s ) 2 0. Consequently, f = A( f ) implies f,(s) = max ~ [ , ( E [ ~ f ( a l ) l a=o sl, s), E[/3fi(al)luo = s]) > fi(s). A contradiction. Next, we construct an SSP outcome (SF, p ) for which f = up. Let S p = {s E S: f(s) E C(S)) and let p(s) =f(s) for s E Sp. Consider any s E S, and let i = ~ ( s ) . If s E Sp, then f(s) = p(s) by the construction of p . Suppose s E S - SF. Then, since f = A ( f ) implies fPi(s) = E[pf-,(al)la0 = s], it follows from the definition of A ( . ) that fi(s) = max {ti(f(s), s), E [ p fi(al) l a. = s]). Therefore, since f(s) G C(s) and EIPf(al)lao = sl 0, (All implies that fi(s) = E[/3fi(al)lao = s]. It then follows from Lemma 1 that f = up. To show that ( S + , p ) is an SSP outcome, consider the following strategy. In any state s, the proposer, ~ ( s ) proposes , p(s) if s E Sp, and passes otherwise. , accepts any proposal which yields him a payoff Each responder, j # ~ ( s ) then no less than u;(s) = E[pujC"(al)lao = s]. Since this strategy implements the outcome (Sp, p), the theorem is proved by noting that, given the future payoff function up, any player j who unilaterally defects from the prescribed strategy Q.E.D. in state s earns a payoff no greater than u,+(s). Given Assumption (Al), allowing mixed strategies does not expand the set of SSP payoffs. Consider any state s with proposer i = ~ ( s )An . SSP outcome may be random at state s only if [i(EIPf(al)lao = s], s ) = E [ ~ f , ( a l > l a=o s]. If [i(EIPf(al)luo = s], s ) < E [ Pfi(al)lao = s], no agreement should be reached, and if [i(EIPf(al)lao = s], s) > EIPfi(al)lao = s], the proposer has no best response unless each responder accepts with probability one. However, if [i(E[pf(al)luo = s], s) = E[pfi(al)luo = s], then the SSP payoff is the same whether the proposer's best acceptable proposal in state s is accepted or noL6 3.2. Existence of SSP Payoffs The existence of time dependent Markov equilibria follows easily by induction on finite horizon games and passing to the limit. However, we are aware of no results which imply the existence of SSP payoffs for all games satisfying Assumption (Al). An earlier version of this paper (Merlo and Wilson (1993)) provides an existence proof for a countable state space, but it is not clear what This argument requires that each responder receive his reservation utility level. If Assumption

(Al) is violated in a way which allows for holes in the Pareto frontier, the proposer's best acceptable

offer may give the responders more utility than their reservation levels. Randomization by the

proposer may then lead to different payoffs to the responders.

379


conditions on the cake process are required for a continuous state space. There are some special cases of interest where the existence of an SSP payoff follows from the techniques used in this paper. In Section 5, we show that the monotonicity of the A ( . )operator guarantees the existence of an SSP payoff for the two-player game. In the next section, we provide a contraction condition on the cake process which guarantees the existence of a unique SSP payoff for k-player games.

3.3. The Uniqueness of SSP Payoffs With no additional restrictions on the shape of the cake, the SSP payoff need not be unique, even for a two-player game with a fixed cake size and deterministic order of moves. This possibility was noted by Rubinstein (1982) in an example where two players have an identical constant cost of delaying agreement in each periode7 In this section we formulate a contraction condition on the cake process which is sufficient to guarantee the existence of a unique SSP payoff. For games within our framework, we are aware of no proof of the uniqueness of SSP (or SP) payoff for which this condition is not at least implicitly e ~ p l o i t e d . ~ To illustrate why the SSP payoff may not be unique, we consider the following example due to Binmore (1987). There are two alternating states, s, and s2, with player i proposing in state si and C(s,) = {x E R ~ xi : + 2x, < 3, j # i), i = 1,2. Then, for P > 1/2, Theorem 1 implies the existence of three SSP payoffs, illustrated in Figure 1 for P = 3/4. One of the payoffs, uO,corresponds to a symmetric outcome in which agreement occurs in each state. It is defined by the relations u;(s,) = 3 - 2pup(sj), i # j, i = 1,2, which yield payoffs vO(sl)= (3/(1 + 2P),3P/(1 + 2P)) and u0(s2)= (3P/(1 2p),3/(1 2P)). The other two payoffs, u1 and u2, correspond to outcomes in which agreement occurs only in one of the states and the proposer in that state obtains all of the surplus, u:(s,) = u;(s,) = 3. These are SSP payoffs because a player who expects to receive a payoff of 3 when she is the proposer is unwilling to accept any proposal which yields the other player a nonnegative payoff when he is the proposer. Consequently, the other player can do no better than to accept an offer of 0 when he is the responder. To understand why there are multiple SSP equilibria in this example, notice that a slight increase in the resewation payoff to player 2 in state s2 results in an even greater decrease in the payoff to player 1 in state s,, and vice-versa.

+

+

'That example can be cast into our framework by taking an exponential transformation of the utility function. However, since Rubinstein's axioms only restrict preferences over certain outcomes, that transformation will not preserve von Neumann-Morgenstern preferences. Other approaches are possible. Wilson (1993) shows that the SSP payoff of games with a finite number of states can be represented as the solution to a complementarity problem and exploits index theow to formulate sufficient conditions for uniqueness.


v$s2) v?(s,) v:(s,) FIGURE 1.-Multiple SSP payoffs.

Xf

Consequently, under any norm on F ~ the , A ( . ) operator fails to satisfy the contraction property. One approach to establishing sufficient conditions for the uniqueness of equilibrium, therefore, is to find conditions under which A ( . ) does satisfy the contraction property under an appropriate norm. To formulate these conditions, we require that C ( s ) be defined by an additive function of individual payoffs. (A2): There is a bounded nonnegative function c: S + R' and, for ASSUMPTION each i E K , a strictly increasing function ui: R 1 + R' with u,(O) = 0 such that ) c ( s ) ) for all s E S . C ( s )= { x E R k : C i E K u , : ' ( x l < We may interpret Assumption ( A 2 )as supposing that each player i has a von Neumann-Morgenstern utility ui defined over his consumption of a physical cakee9In state s the players are then bargaining over how to allocate a physical cake of size 4 s ) . To make the payoffs comparable across players, we will measure the payoffs to each player in terms of the certainty equivalent physical cake. For any feasible payoff function g, we call u;'(g,(s)) the physical payoff to player i in state s. Suppose fi(sl) is the certainty equivalent physical payoff to player i in the next period when state s' is realized. Then, by delaying agreement in the current state s, player i obtains an expected payoff of E[pu,(fi(ul>>lu, = sl and = s]).The hence a certainty equivalent physical payoff of u;'(E[pu,( fl(ul))luo contraction condition may then be stated as follows. We have defined u , over negative numbers for notational convenience


CONDITION (C): There is a 6 < 1 such that for any f , g E F k ,

for all i E K and all s E S. Consider two physical payoff functions. Given any player and any state, Condition (C) requires that the difference between the certainty equivalent physical payoffs from delaying agreement is less than 6 times the expectation of their differences in the following period. If ui is linear, Condition (C) reduces to the following: CONDITION (Cr): There is a 6 < 1 such that for any f , g E F k ,

for all i E K and all s E S. Condition (C') is satisfied for any 6 > P . If u is deterministic, we may assume that u = (so, s l , . . . ) In this case, Condition (C) may be expressed as follows: CONDITION (Cfr): There is a 6 < 1 such that for any f , g E F ~ ,

for all i E K and t

=

0,1,2,. . . .

When u, is concave for all i E K, Condition (C") is also satisfied for any

s > p."'

In general, it may be difficult to verify Condition (C), and as a practical matter we may be able to guarantee uniqueness only for special cases such as those which satisfy Conditions (C') or (C"). Nevertheless, since existing proofs of uniqueness almost always exploit a contraction property, we present the argument in the general framework defined by Condition (C). As the model is formulated, two issues must be resolved before we may apply the standard contraction argument. The first problem is that the utility functions for the different players need not be comparable. Therefore, we first transform the problem so that all payoffs are defined in terms of the common physical units. Let u = (u,, . . . , u k ) define the function u: R k + R k and u p ' its inverse. For any pair of functions f : R k + R k and g : R k + Rk, let f 0 g denote the " W e use here the fact that if g : R' -.R' is a monotonic convex function then h < 1 implies Ig(hx) -g(hy)l

E[PTqlu, = S] for all s E S, with ~ [ ~ " q ' l u=,s] > E[P7q la, = S] for some s E S. Since players always allocate the cake efficiently and the reservation values of the players are determined by their future payoffs, they never fail to agree in an SSP equilibrium when agreement in the current state is Pareto optimal. If, however, the expected payoff from delaying agreement can be improved by different allocations in some future states, the SSP outcome may be inefficient. The players may reach an immediate agreement even though, by delaying agreement and implementing an alternative outcome, their expected payoffs could be increased. Such an inefficient agreement is illustrated in the following example. There are three states and two players. An initial state so, where player 1 is the proposer, is followed with equal probability by one of two absorbing states, s, and s,, where player i proposes in state si, i = 1,2. In each absorbing state, si, the players face the same cake, C(s,) = {x E R2: 2x1 + x 2 < 12, x l + 2x2 < 121, i = 1,2. The cake in state s, is defined by C(s,) = {x E R ~ x1 : + x 2 G 61. Cake C(so) and the discounted cakes, PC(si), i = 1,2, are illustrated in Figure 2 for p = .9. Upon reaching either state s, or s2, it follows easily from Theorem 1 that the proposer obtains all of the surplus in any SSP outcome. Consequently, v(s1) = (6,O) and v(s2) = (0,6). Since states s, and s2 are equally likely to follow state so, the reservation payoff for each player i in state s, is EIPvi(ul)IuO= s o ] = 3P, i = 1,2. Consequently, Theorem 1 implies v(so) = (6 - 3P, 3P). For P = .9, u(s,) E int(pC(s,)), i = 1,2. Therefore, any alternative outcome in which the players delay agreement in state s, and which allocates x > u(so) from cake C(si) in state si, i = 1,2, Pareto dominates the SSP outcome. The Pareto improving allocations are indicated by the shaded area in Figure 2. In this example, there is no uncertainty about the cake size. Only the proposer is selected at random. Similar examples can be constructed in which the proposers alternate in a deterministic fashion, but the cake is random. However, inefficient SSP outcomes are not possible if neither the cake nor the


FIGURE 2.-A Pareto improving delay.

proposer processes are random (assuming we consider only nonrandom outcomes). Suppose a is nonstochastic and consider the payoff to any outcome (7, 7). Then, given the initial state, s, 7 and r are nonrandom, and we may index the states so that state s, is realized in period t, t = 0,1,. . . , r. Therefore, 7 E C(s,), and the payoff to ( 7 , r ) is PTq. Now consider some SSP payoff, v, and suppose that v(s) < p T 7 . Since Theorem 1 implies v(s) 2 ~ ' v ( s , )2 . . . 2 pTv(sT), it follows that 7 > v(s,). In this case, however, Theorem 1 and Assumption (Al) imply 7 G C(s,), contradicting the assumption that ( 7 , r ) is a feasible outcome. In interpreting the example above, it is important to remember what the cake represents. In any state, it is the set of expected payoffs from agreements which are feasible in that state. Clearly, if any state contingent outcome can be agreed upon in the current period, there can be no gain from delay. In that case, any outcome which can be achieved by waiting until a later period can also be agreed upon today. In general, it is only in such cases where the current cake contains the conditional discounted expected payoff of any permissible outcome that we can guarantee that an SSP outcome is Pareto optimal.

s

THEOREM 3: Suppose E [ p T 7la, = s] E C(s) for any outcome (7, r ) and all Then, given (All, any SSP outcome is Pareto optimal.

E S.

PROOF:Suppose v is an SSP payoff and let ( 7 , ~ )be some alternative , 1 implies that uPi(s)= E [ ~ u - i ( a l ) l u o= s] outcome. Letting i = ~ ( s ) Theorem = s], s). Therefore, if E [ p T 7lao= S] > U(S)for some and vi(s) 2 ~i(EIPv(ul)IuO Q.E.D. s E S, (Al) implies that E [ p T 7lao= S] P C(S).

386


Since we only compare the expected payoffs of the different outcomes at the initial state, our definition of Pareto optimality is an ex ante concept. We might also formulate a concept of ex post Pareto optimality in which the payoffs of different outcomes are compared for every realization of u." Since the timing and the terms of agreement of an arbitrary outcome may depend on the entire history of states, the example above can easily be extended to illustrate an SSP outcome which also does not satisfy even this weaker efficiency concept. However, it is not possible to find another stationary outcome which Pareto dominates an SSP outcome even if we restrict attention to the initial states. THEOREM4: Suppose that ( S p , ~ )is an SSP outcome and ( s i , ~ ' ) is an alternative stationary outcome. Then, given (Al), vi >, up implies vi = v p. PROOF:Let ( S + , F , ~ )be an SSP outcome and consider any alternative stationary outcome ( ~ dI;, , 7'). Let v = v+ and v' = vi. Suppose that v G v'. We will show that u = v'. Consider first any s E ~d and let i = ~ ( s ) From . the definition of t i , u G u' and vl(s) E C(S) imply vj(s) < (i(u'(s), S) G ti(v(s), s). It then follows from Theorem 1 that vi(s) a ti(EIPv(uI)IuO= s], S) = ti(v(s), S) 2 v;(s). We conclude that v(s) = vl(s). Now consider any s E S. Since the definition of 7' implies a,, E s d , we have just established that E [ ~ ' ~ ( U , ~=) s~] U =~ E[P~;'(U,,)~U~ = s]. Also, since Theorem 1 implies v(s) a E[pu(ul)luo = s ] and u is a temporally homogeneous Markov process, it follows from the optional stopping theorem (see, e.g., Karlin and Taylor (1975, p. 267)) that 4 s ) a ~ [ ~ " v ( u , ~ ) l=u s]. , Combining these two Q.E.D. results then yields 4 s ) a E[P'~'(u,~)~u, = s] = vl(s).

4.2. The Timing of Agreement In general, the states in which agreement is reached in an SSP outcome depend in a complicated way on the entire u process including the size and shape of the cake as well as the identity of the proposer in each state. With no other restrictions on the u process, we can rule out delay only under the conditions which imply that immediate agreement is Pareto optimal. To simplify the statement of our results, in this subsection we adopt the convention that, for any SSP outcome (SG, F), S+ = {s E S: up(s) E C(s)}. That is, the players agree immediately whenever they can attain the SSP payoff for that state. THEOREM5: F k s E S and suppose E[P777lao= s] E C(s) for any outcome (77, 7). Then, given (Al), Sp = S for any SSP outcome (Sp, F). l 1 Formally, an outcome ( 7 , ~is) expost Pareto optimal if there is no alternative outcome (TJ',7') such that ~ " T 2 J 'P'TJ for all realizations of u with strict equality for some realization of u.


387

PROOF:Select any s E S. By assumption, cF(s) = E[P7p(u7)lu0= $1 E C(S). Therefore, s E SF = {s' E S: c ~ ( s ' E ) C(s')}. Q.E.D. If we impose Assumption (A2), we may state the conditions of Theorem 5 in terms of the c(u) process and the utility function, u. Recall that ~ ' c ( u is ~ )a supermartingale if E [ P " ~ C ( U-~,)la'] < Pfc(ul). COROLLARY 2: Gi~len(A2), suppose (i) Ptc(ul) is a supermartingale, and (ii) u is concace. Then, SF = S for any SSP outcome (SF, p). PROOF:Since (A2) implies that c(.) is a bounded function, EIPfc(uf)luO= s] as t -, cc. Then, since (pfc(ut)luo= s ) is a supermartingale, the optional stopping theorem (see, e.g., Karlin and Taylor (1975, p. 267)) implies = s ] G 4 s ) . Given (A2), Ku;l(p,(uT)) G c(u,). Therefore, the E[PTc(u7)~u, concavity of u implies +0

c,,

for all s E S. We conclude that E[p'p(uT)Iu0 = sl E C(s) for all s E S.

Q.E.D.

It is easy to verify that condition (i) of Corollary 2 is implied by the aisumption that E [ p c ( u l ) ~ u= o s ] < c(s) for all s E S. In this case, agreement occurs in every state. The concavity of u may be relaxed if u is nonstochastic. Otherwise, it is not difficult to construct counterexamples to Corollary 2. Even if the undiscounted cake is identical in all states, players facing a nonconcave cake may prefer to delay agreement until the uncertainty about their relative bargaining power is resolved. Consider the two-player example with 4 s ) = 2, s E S = {so,s,, s,}, illustrated in Figure 3. In state so, player 1 is the proposer. If no proposal is accepted in so, then states sl and s, are realized with equal probability in the next period. For i = 1,2, s, is an absorbing state in which player i is the proposer. Suppose 4 x 1 = x , x < 1, and u(x) = 2x - 1, x > 1. Then, for s E S, C(s) = {x 2 0: 2x1 x, < 3) u {x 0: x, + 2x, < 3). If p > 2/3, then c(sl) = (3,0), c(s,) = (0,3), and c(s,) = p(3/2,3/2) @ C(s,). A similar example may be constructed if the order in which players move in each period is deterministic but the size of the cake is uncertain.

+


FIGURE 3.-Delay

to resolve uncertainty.

4.3. Adcantage to Proposing Following Rubinstein (19821, the bargaining power of the agents in sequential bargaining models is generally parameterized by the relative magnitude of the discount factors. However, the right to make a proposal is also an important element of bargaining power. In this section, we investigate how the payoffs are affected by changing the identity of the proposer in one or more states. Theorem 1 implies that the proposer always extracts any surplus in the current state in any SSP equilibrium. Therefore, for fixed continuation payoffs, it must be to a player's benefit to be the proposer. At worst, no proposal is accepted and he earns his continuation payoff. The only question, then, is whether the gain from proposing in some future state may perversely affect the available surplus in earlier states. For two-player games, acquiring the right to propose in any state implies an unambiguous gain (or at least no loss) to the player in ecery state. This result follows directly from the monotonicity of the A(.) operator.12 Consider two games, indexed by K', j = 1,2, which may differ only in the identity of the proposer in some states. ) i implies THEOREM6: Suppose K = {1,2), and, for some i E K, ~ ' ( s = ~ ~ (=si. )Giuen (Al), u1 an SSPpayoff for K' implies an SSPpayoff for K ~c 2, , such that u? > u! and u,? < c;, j # i. "We show in Theorem 8 below that the extreme points of the set of SP payoffs for two-person games are SSP payoffs. It then follows from Theorem 6 that increasing the set of states in which player i moves must improve both his best and worst SP payoffs.

389


PROOF:Without loss of generality, we may suppose that ~ ' ( s = ) 1 implies ~ ~ (=s1,)and let A' denote the operator associated with the game determined by KI, j = l , 2 . Suppose A1(ul) = cl. Let f O = cl, and define the sequence (f '1 recursively by f t =A2(f '-'), t = 1,2, . . . . Note first that f: =A:( f O) > A:( f O ) = f:, and fi =A;( f O) (mr,n') = ( ~ ( m ' ~nt-I), mO,and n1 = ~ ( m ' ,no) < no, it follows by mathematical induction, using Lemma lA, that m' m'-l and n' G n'-', t = 1,2,. . . . To establish that these sequences are bounded, we note first that no = t? > 0, since a player can guarantee a payoff of 0 by never agreeing to a proposal. It then follows by induction on the definition of M x N that n' & 0, t = 1,2,. . . . To bound the (m') sequence, we note that (Al) implies a b E Fk such that x < b(s) for x E C(s) and s E S, from which we may conclude that m0 = A G b. Furthermore, since [,(d, s) < b(s) for d E R ~i ,E K and s E S, it follows by induction on the definition of M x N that m t < b , t = 1 , 2 ,... . Since the (m') and (n') are bounded monotonic sequences, there is an ( E , E) E Fk x F~ such that (m', n') + ( E , E) pointwise and E 3 A > t? & E. Since M x N(mt, n') = (m'+ ', nri1) + ( E , E), Lemma 2A implies that M X N ( E , E) = ( E , E). It then follows from Lemma 2 that m* 3 E > A & t? E 3 n*. (e) For i E K, define a stationary outcome (SF', pi) as follows. Let s'*' = {S E S: (mT(s), n*,(s)) E C(s)} and let pi(s) = (mT(s), nTi(s)) for s E SF'. We show first that (m:, n*,) satisfies the conditions of Lemma 1. By definition, (mT,nT,)(s) = pL(s),for s E SF'. Therefore, it is sufficient to show that mT(s) = E[/?m'(al)luo = s] and nTi(s) = E[pnT,f(ul)luo = s ] for s P SF'. If s @ SF', the definition of SF' and (Al) imply that either mT(s) = 0 = [,(n*(s), s) or mT(s) > [,(n*(s), s). In either case, since m* 3 0, it follows from the definition of M(.) that mT(sj = Mi(m*, n*)(s) = E[/3mT(ul)Jao= s]. Similarly, for s P SF' and j # i, (Al) implies that either nT(s) = 0 = [,(E[pm*(ul)luo = s], s) or nT(s) > [,(E[pm*(al>luo = s], s). In either case, n* > 0 and the definition of N ( . ) imply that nT(s) = N,(m*, n*)(s) = E[/3nT(u1)Iu0 = s]. It then follows from Lemma 1 that (mT, n*,) = c ~ ' . The theorem is established by demonstrating that any outcome ( q , r ) for which n* < E [ p T qluo = S] G m*, may be supported as an SP outcome by selecting the appropriate (SF',pi) as a "threat" outcome against defections. We begin by defining a strategy in which the action prescribed at any history for player j depends on only the "state" of the history, 8 E (0,. . ., k } . Consider any realization of the u process, (so,s l , . . . ), and let ( 2 , t^) be the corresponding realization


393

of ( 7 , ~ )Now . consider any period t and suppose that, at the realized state-history ( s o , . . , s,), 8 = 0. If it is the proposer's turn to move and t" = t , he offers 2. Otherwise, he passes. If it is a responder's turn to move, then the current proposal x is accepted if and only if t" = t and x = i .If 8 # 0, then the proposer offers p'(s,), and a responder j accepts a proposal x if and only if (i)x,a nT(s), j # 8, or (ii) xj a mT(s),j = 8. Upon the action taken by the player who last moved, the transition from state 8 to state 8' is determined as follows. The history begins in "state" 0. If the player who moves follows the prescribed action at that history for state 8, then the resulting history remains in state 8. If he deviates from the prescribed strategy, then the resulting history moves to state 8', determined as follows. If the deviation is to pass or to reject an offer, then 8' is the index of some player other than the player who defected. If the deviation is to propose an allocation x, then 8' is the index of some player j other than the proposer for whom X , G mT(s), or, if xj > mT(s) for all other players, then 8' may be chosen arbitrarily. It is straightforward to verify that, if n* G E[PT7luo= S ] G m*, the prescribed strategy profile supports ( 7 ,T ) as an SP outcome. Furthermore, at any "state" 8 # 0, the equilibrium outcome after a realization of ( s o , . . , s t ) is pi($,). Q.E.D. Notice that Theorem 7 provides a general proof of the existence of an SP payoff which does not appeal to the Brouwer fixed point theorem. For two-player games, combining Theorems 1 and 7 implies that the extremal SP payoffs are also SSP payoffs. THEOREM8: Gicen (A21 and k

= 2,

(mT,n;) and (nT,m;) are SSP payoffs.

PROOF:We will prove that M x N(m, n ) = ( m ,n ) implies A(nl,m,) = ( n l , m , ) . It then follows from Theorem 1 that ( n l , m , ) is an SSP payoff. By definition, A l ( n l ,m,) = N,(m, n ) = n,. To demonstrate that A,(nl, m,) = m,, we need to consider two cases. Consider first a state s in which player 2 is the proposer. Then nl(s)= A l ( n l ,m,)(s) = EIPnl(ul)Iuo= s ] . From this it follows that

Az(n1, m,)(s) = max {52(E[Pn(ul>luo = s ] , s ) , EIPm,(ul)luo= s ] ) =

max { t 2 ( n ( s > s ),, EIPm~(ul>IuO =s ] )

=

M,(n, m ) ( s ) .

Next, consider a state s in which player 1 proposes. There are two cases to consider. If

then ( A l ) implies that 5,(n(s),s) = E [ ~ m , ( u l ) l u=os]. Alternatively, if

nl(s) =Al(nl,m,)(s) = EIPnl(ul>Iuo = s ] > S1(EIPm(ul)luo = s ] ,s ) ,

394


then (Al) implies that E[pm2(al)luo= s ] > t2(n(s>,s). In either case, therefore, t2(n(s>,s) ,< E[pm2(u1)Iu0=$], and hence, M z ( m , n ) =max{t2(n(~),$),E[~mz(fll)Iuo= s ] } =~

[ p m ~ ( u ~=)S l] u= A ~ 2 ( n 1 ,m 2 ) .

An identical argument establishes that (m,, n2) is also an SSP payoff. Q.E.D. Theorem 8 does not extend to games with more than two players. However, a comparison of Theorems 1 and 7 imply that a unique SP payoff must be SSP. COROLLARY 3: Given (Al), (i) if k = 2, then the SSP payoff is unique only if the SPpayoff is unique, and (ii) a unique S P payoff is SSP.

if and

PROOF:Part (i) follows from Theorem 8. Part (ii) is established upon noting Q.E.D. that, by definition, N( f , f ) = f implies A( f ) = f. 5.2. Uniqueness of SP Payoffs for k & 2 Part (i) of Corollary 3 does not extend to games with three or more players. For those games, the looser restrictions implied by Theorem 7 for SP payoffs imply that stronger conditions are required for a unique SP payoff. For any stationary nonstochastic game with transferable utility, Herrero (1985) demonstrates that the SP outcome is unique when P < l / ( k - 1). By exploiting the approach used to establish conditions for the uniqueness of SSP payoffs (Theorem 2), we may extend Herrero's result to stochastic bargaining games satisfying Condition (C). The proof of Theorem 9 is in the Appendix. THEOREM9: Giuen (A2) and k > 2, suppose Condition (C) is satisfied for 6 < l / ( k - 1). Then there is a unique S P payoff. Without further restrictions, the upper bound on 6 is exact. Sutton (1986) reports an observation by Avner Shaked that for a three-player nonstochastic game in which the identity of the proposer cycles, any feasible allocation in the first period may be sustained as an equilibrium whenever P & 1/2. The idea is an application of the argument in the proof of Theorem 7. We may suppose that the SP outcome is implemented by a strategy in which the proposer passes if no agreement is to be reached in the current period. Then, if the proposer deviates by proposing an offer other than that prescribed by his SP strategy, at least one responder can benefit from a rejection if the continuation SP outcome awards her the entire cake in the following period. Consequently, no agreement will be reached in the current period, and, since some other player receives the entire cake in the following period, the payoff to the deviant is zero. Likewise, if a proposer deviates by passing or a responder deviates by rejecting an offer, her payoff is zero if some other player is awarded the entire cake in the next period.

395


Consequently, no player has an incentive to deviate and the prescribed outcome is subgame perfect. The argument easily generalizes to the k-player game whenever p > l / ( k - 1). Notice, however, that this restriction on p is vacuous for the two-player game.

Department of Economics, Unicersity of Minnesota, 271 19th Avenue South, Minneapolis, Minnesota 55455, U.S.A. and Department of Economics, New York University, 269 Mercer Street, New York, New York 10003, U.S.A. Manuscript receiued March, 1993; final reuision received May, 1994.

APPENDIX PROOF OF LEMMA1: Given ( S F ,p ) , define V : F k + F k , by V (f X S ) = p ( s ) for s E S p , and V (f )(s) = E I P f ( a l ) l u o = S ] for s E S - S F . The Lemma will be established if we can show that u p is the unique solution to V ( f ) = f. We show first that V ( u p )= u p . Recall that the stopping time for agreement is defined by a, E S p and a, E S p , t = 0,. . . ,T - 1. Define i-! by a T lE S p and a, E S p , t = 1,.. . ,T , - 1 , to be the stopping , s ]= time for agreement starting in perlod 1. Then, for any s E S , c p ( s )= E I P T c p ( a T ) l a= ~ [ p ' 1 - ' u ~ ( a , ~ )=l s]. a ~ Consider any s E S - S F . Then, since a , E S - S p implies T = T I , it follows that

If s E S p , then, V ( v p X s )= u p ( s )follows immediately from the definition of u p . T o complete the proof, we need only to verify that V satisfies the contraction property. Let II.II, denote the sup norm on R~ defined by llxllI = sup{lx, I, i E K } . Consider f f E F k .Then, for any s E S , either I~v(f ' X s ) - V( f 2)(s)ll, = 0 or

',

Ilv(f' ) ( s )- V ( f 2 ) ( ~ ) 1 1 2 =I I E[ p ( f l ( u l )- f

'

2 ( u 1 ) ) l u o = llr~ ]< ~ l l f- f

211m

Q.E.D. The following two lemmata are used in the proofs of Lemma 2 and Theorem 7. LEMMA1A: Gicen ( A l ) , m' M ( m 2 ,n2).

m 2 and n' 6 n 2 imply ~ ( m 'n ,' ) 6 N ( m 2 ,n 2 ) and ~ ( m 'n ,' ) >

PROOF: Suppose m' m 2 and n' < n2. T o establish that ~ ( m 'n ,' ) < N ( m 2 ,n 2 ) , consider any s E S. Then, for i = ~ ( s )[ , ( E [ p m l ( a l ) l a o= sl, s ) < [ , ( E [ p m 2 ( a l ) l a o= sl, s ) and E [ n l ( o l ) l a o= sl 6 ~ [ n ~ ( a , ) I=us]. , The definition of A ( . ) implies N,(m2,n 2 ) ( s )> Ni(ml,n l X s ) . Similarly, for j # K ( s ) ,the definition of A ( . ) implies

~ , ( m 'n, l ) ( s ) = E [ p n ~ ( a , ) l a = os] S]

x , x

E

C(s), s E S. For

t = l , 2 , . . . , define m' and n' recursively by ( m ' , n ' ) = ( ~ ( m ' - 'n'-'), , ~ ( m ' - ' n'-I)). , W e first

establish by induction that ( m ' ) and ( n ' ) are respectively monotonically decreasing and increasing sequences. Since 0 = n n ( s )= n l ( s )< pmO(s)= m l ( s )< mn(s),s E S , it follows that no < n' < m' < mn. Suppose nl-' < n' < m' < m l - ' . Then Lemma 1A implies n' < n" and m'+' < m'. T o establish that nl+' < m l + ' , consider any s E S and let i = K ( s ) . Then,

'

n l , ( s )=~ [ p n ' _ ; ' ( u , ) l= u ~s ] < ~ [ p n L ~ ( u ~ =) ls a] < , ~ [ ~ m L ~ ( u =~ s)]l u , implies [ , ( E [ p m ' ( u l ) l u o= sl, s ) < t i ( n l ( s ) s). , Then, since E[n:(al)Iun= s] < E [ m f ( u l ) l u = o s ] , it follows that n:+' ( s )= N,(ml,n1)(s)< Mi(ml,n'Xs) = m:+'(s). Similarly, for j + ~ ( s ) , n j + ' ( s ) = q ( m ' , n l ) ( s )= E [ p n j ( u l ) l a o= s ]

< E [ p m ~ ( u 1 ) l u=0s] < ~ ; ( r n ' , n ' ) ( s=) m i + ' ( s ) . Since ( m ' ,n ' ) is monotonic and bounded, there is a pair o f functions m*, n* E Fk such that m1(s)Lm * ( s ) and nl(s)T n*(s), s E S. By definition, therefore, ( m l + ' ,n l + ' )= M x N ( m l ,n ' ) + ( m * , n*), pointwise. But Lemma 2A implies that M x N(m1,n'Xs) + M x N(m*, n*)(s). W e conclude that M(m*, n * ) = m* > n* = N(m*, n*). This establishes (i) and (ii). T o establish (iii), suppose ( m ,n ) = ( M ( m ,n ) , N ( m , n)). Then no < n and m < m n and, therefore, Lemma 1A implies that n' = ~ ( m ' n, o )< N ( m , n ) = n and m = M ( m , n ) < ~ ( r n ' n, o )= m'. It follows by induction on Lemma 1A that n' < n and m < m', t = 1,2,3,.. . . Since ( m ' ,n') + ( m * , n*), Q.E.D. we conclude that m* > m and n* < n. The following Lemma is used in the proof o f Theorem 9. LEMMA3A: Gic'en ( A 2 ) , suppose M X N(u m', u n i ) = ( u m', u n'), i = 1,2. Then, Condition ( C ) implies that, for any s E S: l ) m?(ul)l lao = s ] , E[ln:(u,) (i) For i = K ( s ) , Ini(s) - n f ( s ) l < 6 max ( E j , i E [ l m ~ ( a n:(ul)l loo= sl). (ii) For j # K ( s ) , In:(s) - n?(s)I < 6 E [ I n j ( a l )- n?(al)I la, = s]. (iii) For i E K , Im:(s) - m;(s)l < m a x ( E j t rInj(s) - n?(s)l,6~[Irn,'(cr,) - mf(a,)I lao = sl). PROOF:W e will establish part (i). Parts (ii) and (iii) are established similarly. For s E S and i = K ( s ) ,the definition o f N implies

Without loss, we may assume that N,(u m', u n l ) ( s )- N,(u m 2 ,u n 2 X s )3 0. Then, there are two cases to consider. I f

then, since ui(n:(s)) = 4 ( u o r n ' , ~

> u ! ( c ( s )- ~ u ; ' ( E [ ~ ~ ~ ( m ~ ( u ~ ) ) ~ ~ = s J f l


Condition (C) implies

u;'(E[puj(m?(~l))l~o=s])

-

jti

d6

C E[lmj(u,)

- m:(ul)l

lao= s ]

jti

Alternatively, if u,(n;(s)) = 4 ( u m', u nlXs) = E[pui(n:(ul))luo = s], then, since u,(n?(s)) = N.(u m2, u n2)(s)> ~ [ p u , ( n ~ ( u , ) ) l u=, s], Condition (C) implies In:(s) - n?(s)l Q 6E[ln:(u1) Q.E. D. n f ( u , ) ~lao= SI. PROOFOF THEOREM9: For x, y E R ~ define , Ilx, ylla = sup{lx,l + Iyil, i E K ) . Suppose M X N(u mi, u n') = (u m', u ni), i = 1,2. Given Condition (C), we will use Lemma 3A to show that Ilm' - m 2 , n' - n211, s 6(k - l)llml - m2, n' - n211,. The assumption that 6 < l / ( k - 1) then implies that (m', n') = (m2, n2). Then, since u is strictly monotonic, it follows that (u m', u n') is the unique fixed point of M x N. But, by definition, A( f ) = f implies M X N( f , f ) = ( f , f ). Therefore, since Theorems 1 and 2 imply a unique fixed point of A(.), it follows that m' = n1 which, combined with Theorem 7, establishes the result. T o show that lim' - m 2 , n' - n211, d 6(k - l)llrn1 - m2, n1 - n211,, consider any s E S and i E K . Since we assume u m' = M(u mi, u n'), recursive application of Lemma 3A (iii) implies a stopping time w such that (i) Imt(s) - mf(s)l s E[GwE1+ , ln:(u,) - n:(u,)l Iuo = sl, and (ii) for any stopping time T < o, Im,'(s) - m?(s)( d E[6'+' lm:(u7+,) - m:(u,+,)l lao= s]. Similarly, Lemma 3A (i)-(ii) implies a stopping time v such that (i) Int(s) - n:(s)l d E [ S w f'Ej+ i l m ~ ( u u + l-) r n ? ( ~ , + ~ )Iuo l = s ] and (ii) for any stopping time T d v , In$) - n:(s)l < EIGTlnf(uT)- nf(uT)l lao = 31. Let A denote the indicator function on realizations of u denoting w < v. Then, 0

0

0

0

0

0

0

Since w,v > 0 and 6 < 1, the theorem will be established if we can show that (a) E j E K l n l ( s )~ s ) dI S(k - l)llml - m2, n 1 - n211,, and (b) Cj,,lrn,'(s) - m:(s)l < ( k - l ) m l - m< n' n211,, s E S. T o establish (a), define a stopping time T to be the first period in which Ini(,,,(uT) - n:(,7,(uT)l s GE[Ejt K(u,,lmj(u7+,) - m?(u7+,)l lao= s]. Then, recursive application of Lemma 3A 6)(ii) implies that, for all s € S, (i) In:(s) - n:(s)l b E[6'lnj(u7) - n?(uT)I lao= s], j E K , (ii) E[C,+K~,7,6rlnj(u,) - n?(u,)l lug = ~1 d E[Ej+K(,7,6T+11n~(uT+1) - n ? ( ~ ~ + luo ~ ) =l $1, and

398


(iii) E[STln;(, ,(uJ - nZ(,T)(u,)l Iuo = sl s E [ C j + K(,7,6T+'lm:(uT+1)- m:(u,+ ,)I Iuo = sl. Combining these refations, we obtain, for all s E S,

Similarly, to establish (b), we may define a stopping time T to be the first period in which, for some player 1 , Im!(u,) - mf(u,)I s Cj+,lnj(u,) - n:(uT)I. Since i is measurable with respect to r , recursive application of Lemma 3A (iii) implies that, for all s E S, j E K, (i) Imj(s) - m?(s)l s ~ ~ ~ [ l r n ~-( m:(u,)l u,) lao = $1, and (ii) E[lmi(uT) - m:(uT)l lao = s l < ~ [ C , + , ( n : ( u , ) n:(uT)( luo = s]. Therefore,

< ( k - l)(iml - m 2 , n1 - n2ilm.

Q.E.D.

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MERLO,A. (1992): "Bargaining over Governments in a Stochastic Environment," New York University, C. V. Starr Center for Applied Economics, Research Report #92-55. MERLO,A,, AND C. WILSON(1993): "A Stochastic Model of Sequential Bargaining with Complete Information and Nontransferable Utility," New York University, C. V. Starr Center for Applied Economics, Research Report #93-06. -(1994): "Efficient Delays in a Stochastic Model of Bargaining," unpublished manuscript, New York University, C. V. Starr Center for Applied Economics. MOULIN,H. (1986): Game Theory for the Social Sciences. New York: New York University Press. OSBORNE,M. J., AND A. RUBINSTEIN (1990): Bargaining and Markets. San Diego: Academic Press. PERRY,M., AND P. J. RENY(1993): "A Non-Cooperative Bargaining Model with Strategically Timed Offers," Journal of Economic Theory, 59, 50-77. RUBINSTEIN, A. (1982): "Perfect Equilibrium in a Bargaining Model," Econometrica, 50, 97-109. SAKOVICS, J. (1993): "Delay in Bargaining Games with Complete Information," Journal of Economic Theory, 59, 78-95. SHAKED,A,, AND J. SUTTON(1984a): "Involuntary Unemployment as a Perfect Equilibrium in a Bargaining Model," Econometrica, 52, 1351-1364. -(1984b): "The Semi-Walrasian Economy," London School of Economics, ST/ICERD, Discussion Paper #84/98. (1983): "A Multistage Model of Bargaining," Review of Economic SOBEL,J., AND I. TAKAHASHI Studies, 50, 411-426. STAHL,D. 0. I1 (1990): "Bargaining with Durable Offers and Endogenous Timing," Games and Economic Behavior, 2, 173-187. STAHL,I. (1972): Bargaining Theory. Stockholm: Economic Research Institute, Stockholm School of Economics. SUTTON,J. (1986): "Non-Cooperative Bargaining Theory: An Introduction," Review of Economic Studies, 53, 709-724. WILSON,C. (1993): "A Characterization of the Index of Solutions to a Generalized Complementarity Problem," unpublished manuscript, C. V. Starr Center for Applied Economics, New York University.

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References On the Theory of Infinitely Repeated Games with Discounting Dilip Abreu Econometrica, Vol. 56, No. 2. (Mar., 1988), pp. 383-396. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198803%2956%3A2%3C383%3AOTTOIR%3E2.0.CO%3B2-B

Strategic Delay in Bargaining Anat R. Admati; Motty Perry The Review of Economic Studies, Vol. 54, No. 3. (Jul., 1987), pp. 345-364. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28198707%2954%3A3%3C345%3ASDIB%3E2.0.CO%3B2-%23

Bargaining in Legislatures David P. Baron; John A. Ferejohn The American Political Science Review, Vol. 83, No. 4. (Dec., 1989), pp. 1181-1206. Stable URL: http://links.jstor.org/sici?sici=0003-0554%28198912%2983%3A4%3C1181%3ABIL%3E2.0.CO%3B2-N

The Nash Bargaining Solution in Economic Modelling Ken Binmore; Ariel Rubinstein; Asher Wolinsky The RAND Journal of Economics, Vol. 17, No. 2. (Summer, 1986), pp. 176-188. Stable URL: http://links.jstor.org/sici?sici=0741-6261%28198622%2917%3A2%3C176%3ATNBSIE%3E2.0.CO%3B2-E



An Outside Option Experiment Ken Binmore; Avner Shaked; John Sutton The Quarterly Journal of Economics, Vol. 104, No. 4. (Nov., 1989), pp. 753-770. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28198911%29104%3A4%3C753%3AAOOE%3E2.0.CO%3B2-F

Discounted Dynamic Programming David Blackwell The Annals of Mathematical Statistics, Vol. 36, No. 1. (Feb., 1965), pp. 226-235. Stable URL: http://links.jstor.org/sici?sici=0003-4851%28196502%2936%3A1%3C226%3ADDP%3E2.0.CO%3B2-V

Striking for a Bargain Between Two Completely Informed Agents Raquel Fernandez; Jacob Glazer The American Economic Review, Vol. 81, No. 1. (Mar., 1991), pp. 240-252. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28199103%2981%3A1%3C240%3ASFABBT%3E2.0.CO%3B2-1

Perfect Equilibrium in a Bargaining Model Ariel Rubinstein Econometrica, Vol. 50, No. 1. (Jan., 1982), pp. 97-109. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198201%2950%3A1%3C97%3APEIABM%3E2.0.CO%3B2-4

Involuntary Unemployment as a Perfect Equilibrium in a Bargaining Model Avner Shaked; John Sutton Econometrica, Vol. 52, No. 6. (Nov., 1984), pp. 1351-1364. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198411%2952%3A6%3C1351%3AIUAAPE%3E2.0.CO%3B2-4

A Multistage Model of Bargaining Joel Sobel; Ichiro Takahashi The Review of Economic Studies, Vol. 50, No. 3. (Jul., 1983), pp. 411-426. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28198307%2950%3A3%3C411%3AAMMOB%3E2.0.CO%3B2-D



Non-Cooperative Bargaining Theory: An Introduction John Sutton The Review of Economic Studies, Vol. 53, No. 5. (Oct., 1986), pp. 709-724. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28198610%2953%3A5%3C709%3ANBTAI%3E2.0.CO%3B2-M