We show that the standard results for #nitely repeated games do not survive the combination of two simple variations on the usual model. In particular, we add a small cost of changing actions and consider the e#ect of increasing the frequency of repetitions within a #xed period of time. We show that this can yield multiple subgame perfect equilibria in games like the Prisoners' Dilemma which normally have a unique equilibrium. Also, it can yield uniqueness in games which normally havemultiple equilibria. For example, in a twobytwo coordination game, if the Pareto dominant and risk dominant outcomes coincide, the unique subgame perfect equilibrium for small switching costs and frequent repetition is to repeat this outcome every period. Also, in a generic Battle of the Sexes game, there is a unique subgame perfect equilibrium for small switching costs. 1

We analyze a stochastic bargaining game in which a new dollar is divided among committee members in each of an infinity of periods. In each period, a committee member is recognized and offers a proposal for the division of the dollar. The proposal is implemented if it is approved by a majority. If the proposal is rejected, then last period’s allocation is implemented. We show existence of equilibrium in Markovian strategies. It is such that irrespective of the initial status quo, the discount factor, or the probabilities of recognition, the proposer extracts the entire dollar in all periods but the initial two. We also derive a fully strategic version of McKelvey’s (1976), (1979) dictatorial agenda setting, so that a player with exclusive access to the formulation of proposals can extract the entire dollar in all periods except the first. The equilibrium collapses when within period payoffs are sufficiently concave. Winning coalitions may comprise players with high instead of low recognition probabilities, ceteris paribus.

This paper examines existence of Markov equilibria in a class of infinite horizon games in which political institutions are endogenously determined each period. In these dynamic political games (or DPGs) the rules for political aggregation at date t + 1 are decided by the rules at date t, and the resulting institutional choices do not affect payoffs or technology directly. We show that any dynamic political game can be transformed into a stochastic game in which the political institutions are reinterpreted as “public players ” in the game. These players ’ preferences are possibly dynamically inconsistent due to the fact that naturally occurring changes in the economic state, such as evolution of the wealth distribution, alter the way a political institution aggregates preferences of the citizenry over time. The paper characterizes this transformation, and establishes existence of Markov equilibria in which the Markov strategies are smooth functions of the state. Applicability of the result is demonstrated in an example with endogenous voting rules.

The holdup problem arises when parties negotiate to divide the surplus generated by their relationship specific investments. We study this problem in a dynamic model of bargaining and investment which, unlike the stylized static model, allows the parties to continue to invest until they agree on the terms of trade. The investment dynamics overturns the conventional wisdom dramatically. First, the holdup problem need not entail underinvestment when the parties are sufficiently patient. Second, inefficiencies can arise unambiguously in some cases, but they are not caused by the sharing of surplus per se but rather by a failure of an individual rationality constraint.

A framework for studying the evolution of cooperative behaviour in a random environment, using evolution of finite state strategies, is presented. The interaction between agents is modelled by a repeated game with random observable payo#s. The agents are thus faced with a more complex situation, compared to the Prisoner's Dilemma that has been widely used for investigating the conditions for cooperation in evolving populations (Matsuo 1985; Axelrod 1987; Miller 1989; Lindgren 1992; Ikegami 1994; Lindgren & Nordahl 1994; Lindgren 1997). Still, there are robust cooperating strategies that usually evolve in a population of agents.

We present an algorithm to compute the set of perfect public equilibrium payoffs as the discount factor tends to one for stochastic games with observable states and public (but not necessarily perfect) monitoring when the limiting set of (long-run players’) equilibrium payoffs is independent of the state. This is the case, for instance, if the Markov chain induced by any Markov strategy profile is irreducible. We then provide conditions under which a folk theorem obtains: if in each state the joint distribution over the public signal and next period’s state satisfies some rank condition, every feasible payoff vector above the minmax payoff is sustained by a perfect public equilibrium with low discounting.

Miyagawa, Robert Wilson, Henrich Greve, and seminar participants at FEMES 2001 for helpful discussions. Financial

We provide game-theoretic foundations for the median voter theorem in a one-dimensional bargaining model based on Baron and Ferejohn’s (1989) model of distributive politics. We prove that, as the agents become arbitrarily patient, the set of proposals that can be passed in any subgame perfect equilibrium collapses to the median voter’s ideal point. While we leave the possibility of some delay, we prove that the agents ’ equilibrium continuation payoffs converge to the utility from the median, so that delay, if it occurs, is inconsequential. We do not impose stationarity or any other refinements. Our result counters intuition based on the folk theorem for repeated games, and it contrasts with the known result for the distributive bargaining model that, as agents become patient, any division of the dollar can be supported as a subgame perfect equilibrium outcome.

This paper examines Markov Perfect equilibria of general, finite state stochastic games. Our main result is that the number of such equilibria is finite for a set of stochastic game payoffs with full Lebesgue measure. We further discuss extensions to lower dimensional stochastic games like the alternating move game.

Often problems arise where multiple self-interested agents with individual goals can coordinate their actions to improve their outcomes. We model these problems as general sum stochastic games. We develop a tractable approximation algorithm for computing subgame-perfect correlated equilibria in these games. Our algorithm is an extension of standard dynamic programming methods like value iteration and Q-learning. And, it is conservative: while it is not guaranteed to find all value vectors achievable in correlated equilibrium, any policy which it does find is guaranteed to be an exact equilibrium of the stochastic game (to within limits of accuracy which depend on the number of backups and not on the approximation scheme). Our new algorithm is based on the planning algorithm of [1]. That algorithm computes subgame-perfect Nash equilibria, but assumes that it is given a set of “punishment policies ” as input. Our new algorithm requires only the description of the game, an important improvement since suitable punishment policies may be difficult to come by. Keywords: Multi-agent planning, subgame perfect correlated equilibrium, stochastic games. 1