|
Article Excerpt
Summary. We develop an index theory for the Stationary Subgame Perfect (SSP) equilibrium set in a class of n-player (n [greater than or equal to] 2) sequential bargaining games with probabilistic recognition rules. For games with oligarchic voting rules (a class that includes unanimity rule), we establish conditions on individual utilities that ensure that for almost all discount factors, the number of SSP equilibria is odd and the equilibrium correspondence lower-hemicontinuous. For games with general, monotonic voting rules, we show generic (in discount factors) determinacy of SSP equilibria under the restriction that the agreement space is of dimension one. For non-oligarchic voting rules and agreement spaces of higher finite dimension, we establish generic determinacy for the subset of SSP equilibria in pure strategies. The analysis also extends to the case of fixed delay costs. Lastly, we provide a sufficient condition for uniqueness of SSP equilibrium in oligarchic games.
Keywords and Phrases: Local uniqueness of equilibrium, Regularity, Sequential bargaining.
JEL Classification Numbers: C62, C72, C78.
1 Introduction
Sequential bargaining models of complete information starting with Rubinstein, [39], have provided a fruitful environment for the study of the resolution of disagreements among agents. Unlike cooperative formulations which are silent about the underlying actions of bargaining parties, these games juxtapose equilibrium conditions based on a scrutiny of the optimality of individual choices via which proposals emerge and agreements are crafted. In typical situations, (refined) equilibria of these games exist allowing applications to numerous areas of social interaction. In legislative or other political environments where the non-existence of equilibrium of the cooperative genre is pervasive, such models have been welcomed in celebratory spirits.
Our goal in this paper is to study the structure of the set of equilibria for an important class of these bargaining games and analyze their stability to perturbations of the model. By stability we mean the property that the number of equilibria of these games is finite and each equilibrium is locally expressible as a continuous function of model parameters. Then, a slight change in the bargaining environment results in small changes in equilibrium behavior. Importantly, if we calculate equilibria using parameter values that do not exactly coincide with their true values, the equilibria we obtain are still close to the true equilibria.
Besides its obvious epistemological significance, such a property seems essential in order to build richer models of political interaction. In parliamentary systems, for example, government formation following elections requires us to append a bargaining model at the end of a game preceded by an electoral stage. In these situations, it is important to ensure conditions such that changes in the bargaining environment induced at the electoral stage, result in continuous changes in the distribution of subsequent agreements and policy outcomes.
Bargaining takes place as a sequence of proposal and voting stages. We assume players are recognized to make a proposal with some probability fixed across periods. If the proposal is approved the game ends, else new proposal and voting stages follow. We allow general voting rules, although our results are stronger for a subclass of these. In the main paper we focus on the case of bargaining with discounting, although in the Appendix we show our analysis also applies in the case of fixed delay costs. With varying degrees of generality, related models have been analyzed in, for example, [8, 9, 5, 17, 4, 31, 33, 2, 20, 14, 15, 10, 21, 23], etc. The model also bears resemblance to game forms used in the literature on noncooperative implementation of the core [e.g. 41, 43].
We focus our investigation on Stationary Subgame Perfect equilibria in pure strategies (PSSP). Except in special cases, the behavior of SSP equilibria for these games is not fully understood. Banks and Duggan [2] studied discounted games such as those we consider and showed upper-hemicontinuity of SSP equilibria with respect to parameters. But they also provided an example of a majority rule game that has a continuum of PSSP equilibria, and an equilibrium correspondence that fails lower-hemicontinuity with respect to parameters.
Stronger results arise under special assumptions on the voting rule and/or the agreement space. For n-player unanimity games with discounting, Merlo and Wilson [31] have shown that SSP equilibrium is unique when bargaining emulates the division of a (possibly stochastic) cake and a contraction condition is met. Similarly, Cho and Duggan [10] show uniqueness of SSP equilibrium in discounted bargaining games with one-dimensional agreement space, proper and strong voting rules, and quadratic utilities. Thus, in the games considered in [10] and a subset of the games analyzed in [31] for which the upper-hemicontinuity result in [2] holds, the equilibrium correspondence simply becomes a continuous function of the parameters.
But neither the restriction on the dimensionality of the agreement space in [10], nor unanimity rule in [31] guarantee uniqueness of SSP equilibrium in general. The uniqueness condition in [31] that bargaining emulates the division of a cake, does not cover agreement spaces with public goods or political spaces in which disagreement is of ideological nature. In Section 3 of this paper we specify a four-player discounted unanimity game with a one-dimensional agreement space and strictly concave, continuously differentiable utilities that admits a continuum of PSSP equilibria. (1)
Thus, at most we can hope to show that this type of pathological behavior of the equilibrium set is not generic. Binmore [8] has pursued such arguments for Rubinstein's two player game with delay costs, but we know of no general study of that nature. Our results specialize in three versions of decreasing strength which guarantee that, for almost all discount factors: (a) when the voting rule is oligarchic, a class that includes unanimity rule, the number of SSP equilibria is odd and equilibrium correspondence is lower-hemicontinuous. (b) If the space of agreements is one-dimensional, then all SSP equilibria are locally unique and finite in number for general voting rules. Finally, (c) for non-oligarchic voting rules in multidimensional agreement spaces there is a finite number (possibly zero) of PSSP equilibria.
Our results are weaker in the last case since PSSP equilibria may not exist in these games. Still, our result for the subset of PSSP equilibria has been used in a majority rule application in [21] to show that minority governments in parliamentary government formation bargaining almost always occur with positive probability when utility from cabinet positions is small relative to the ideological disagreement of political parties.
Before we move to the formal analysis, we tie our work to two strands of related literature and outline the arguments that permitted our results. The first related literature sprung from the pioneering study of the local uniqueness of Walrasian equilibria in general equilibrium theory by Debreu [11]. The second literature is game-theoretic and started with, Harsanyi [18] who provides an alternative proof of the fact that almost all finite games in normal form have an odd number of Nash equilibria. Kreps and Wilson [27] show determinacy of equilibrium outcome distributions for finite games in the extensive form. A similar result is obtained for a class of cheap talk games by In-Uck Park [34]. Haller and Lagunoff [16] show genericity of behavior in Markovian equilibria of dynamic games with finite action and state spaces.
When it comes to games with continuous action spaces, Dubey [13] offers a general result for simultaneous move games. The bargaining games we analyze involve both a continuous action space for the proposer as well as multi-period dynamic interaction. Thus, these games are not covered by any existing studies. Yet, due to their particular structure, these games are amenable to similar techniques. A key insight is that a proposer, ostensibly choosing from a continuum of agreements, has a unique optimal agreement for every winning coalition (under typical assumptions), and a finite number of winning coalitions to choose from. Exploiting this fact, our proof strategy proceeds as follows.
First, we introduce the notion of an agenda setting plan corresponding to a player/proposer and a winning coalition: it is a mapping from the possible reservation values of players to optimal proposals by the proposer that are acceptable by members of the coalition. The role of agenda setting plans in our analysis is very akin to that of demand functions in the study of economic equilibrium. Much like demand functions shift the focus from the consumers' optimization problem to reduce economic equilibrium to a set of equations that ensure that markets clear, agenda setting plans sidestep the proposer's optimization problem and reduce equilibrium to a set of equations that ensure that players' coalition choices produce reservation values that are consistent with these choices. (2)
PSSP equilibria emerge when each proposer chooses a unique coalition. Since there is a finite number of possible combinations of coalition choices by the players, we ensure that every PSSP can be expressed as the solution to one among a finite number of systems of equations. Thence, our result only requires that each of these systems of equations has finitely many solutions.
Because they involve solutions to optimization problems (the agenda setting plans) our equilibrium equations are not sufficiently smooth to allow us to apply Sard's theorem (or the transversality theorem). In the theory of general economic equilibrium this problem has been confronted early on by Rader [37] who was able to extend Debreu's [11] results to cases when the demand functions are not differentiable but satisfy certain stability properties. More recently, Shannon [42] developed a degree theory for non-smooth equations. As an application, she strengthened Rader's work to a conclusion similar to Dierker's [12]. Our theorem is derived by...
|
