Subgame Perfect Correlated Equilibria in Repeated Games

Subgame Perfect Correlated Equilibria in Repeated Games by Pavlo Prokopovych and Lones Smith ABSTRACT This paper investigates discounted infinitely repeated games with observable actions extended with an extensive form correlation device. Such games capture situations of repeated interaction of many players who choose their individual actions conditional on both public and private information. At the beginning of each stage, the players observe correlated private messages sent by an extensive form correlation device. To secure a recursive structure, we assume that players condition their play on the prior history of action profiles and the latest private message they have received from the device. Given a public history, the probability distribution on the product of the players' message sets, according to which the device randomly selects private messages to the players, is common knowledge. This leads to the existence of proper subgames and the opportunity to utilize the techniques developed by Abreu, Pearce, Stacchetti (1990) for studying infinitely repeated games with imperfect monitoring. The extensive form correlation devices we consider send players messages confidentially and separately and are not necessarily direct devices. Proposition 1 asserts that, in infinitely repeated games, subgame perfect correlated equilibria have a simple intertemporal structure, where play at each stage constitutes a correlated equilibrium of the corresponding one-shot game. An important corollary is that the revelation principle holds for such games --- any subgame perfect correlated equilibrium payoff can be achieved as a subgame perfect direct correlated equilibrium payoff. We can therefore focus on the recursive structure of infinitely repeated games extended with an extensive form direct correlation device and characterize the set of subgame perfect direct correlated equilibrium payoffs. In the spirit of dynamic programming, we decompose an equilibrium into an admissible pair that consists of a probability distribution on the product of the players' action sets and a continuation value function. This generalization has allowed us to obtain a number of characterizations of the set of subgame perfect equilibrium payoffs. To illustrate a number of important properties of this set, we study two infinitely repeated prisoner's dilemma games. In the first game, the set of subgame perfect correlated equilibrium payoffs strictly includes not only the set of subgame perfect equilibrium payoffs but also the set of subgame perfect public randomization equilibrium payoffs. In the second game, the set of subgame perfect direct correlated equilibrium payoffs is not convex, strictly includes the set of subgame perfect equilibrium payoffs, and is strictly contained in the set of subgame perfect public randomization equilibrium payoffs. The latter is possible since, in the presence of a public randomization device, the history of public messages observed in previous stages is also common knowledge at the beginning of each stage, which is not the case when messages are private.repeated games with observable actions, correlated equilibrium, private information

Repeated Games with Present-Biased Preferences

We study infinitely repeated games with observable actions, where players have present-biased (so-called beta-delta) preferences. We give a two-step procedure to characterize Strotz-Pollak equilibrium payoffs: compute the continuation payoff set using recursive techniques, and then use this set to characterize the equilibrium payoff set U(beta,delta). While Strotz-Pollak equilibrium and subgame perfection differ here, the generated paths and payoffs nonetheless coincide. We then explore the cost of the present-time bias. Fixing the total present value of 1 util flow, lower beta or higher delta shrinks the payoff set. Surprisingly, unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoff set U(beta,delta) is not separately monotonic in beta or delta. While U(beta,delta) is contained in payoff set of a standard repeated game with smaller discount factor, the present-time bias precludes any lower bound on U(beta,delta) that would easily generalize the beta=1 folk-theorem.beta-delta preferences, repeated games, dynamic programming, Strotz-Pollak equilibrium

Using Turn Taking to Mitigate Conflict and Coordination Problems in the Repeated Battle of the Sexes Game

The Battle of the Sexes game, which captures both conflict and coordination problems, has been applied to a wide range of situations. We show that, by reducing conflict of interests and enhancing coordination, (eventual) turn taking supported by a ???turn taking with independent randomizations' strategy allows the players to engage in intertemporal sharing of the gain from cooperation. Using this insight, we decompose the benefit from turn taking into conflict-mitigating and coordination-enhancing components. Our analysis suggests that an equilibrium measure of the ???intertemporal degree of conflict??? provides an intuitive way to understand the sources of welfare gain from turn taking in the repeated Battle of the Sexes game. We find that when this equilibrium measure is higher, players behave more aggressively and the welfare gain from turn taking is smaller.Battle of the Sexes game, turn taking, conflict-mitigating, coordinationenhancing

Strategic Commitment Versus Flexibility in a Duopoloy with Entry and Exit

We introduce a class of strategies which generalizes examples constructed in two-player games under imperfect private monitoring. A sequential equilibrium is belief-free if, after every private history, each player.s continuation strategy is optimal independently of his belief about his opponents. private histories. We provide a simple and sharp characterization of equililibrium payos using those strategies. While such strategies have desirable robustness properties, they are not rich enough to generate a folk theorem in most games besides the prisoner's dilemma, even when noise vanishes.

On the Limit Equilibrium Payoff Set in Repeated and Stochastic Games

This paper provides a dual characterization of the limit set of perfect public equilibrium payoffs in stochastic games (in particular, repeated games) as the discount factor tends to one. As a first corollary, the folk theorems of Fudenberg, Levine and Maskin (1994), Kandori and Matsushima (1998) and Hörner, Sugaya, Takahashi and Vieille (2011) obtain. As a second corollary, in the context of repeated games, it follows that this limit set of payoffs is a polytope (a bounded polyhedron) when attention is restricted to equilibria in pure strategies. We provide a two-player game in which this limit set is not a polytope when mixed strategies are considered.Stochastic games, Repeated games, Folk theorem

Repeated Games with Voluntary Information Purchase

We consider discounted repeated games in which players can voluntarily purchase information about the opponents’ actions at past stages. Information about a stage can be bought at a fixed but arbitrary cost. Opponents cannot observe the information purchase by a player. For our main result, we make the usual assumption that the dimension of the set FIR of feasible and individually rational payoff vectors is equal to the number of players. We show that, if there are at least three players and each player has at least four actions, then every payoff vector in the interior of the set FIR can be achieved by a Nash equilibrium of the discounted repeated game if the discount factor is sufficiently close to 1. Therefore, nearly efficient payoffs can be achieved even if the cost of monitoring is high. We show that the same result holds if there are at least four players and at least three actions for each player. Finally, we indicate how the construction can be extended to sequential equilibrium.mathematical economics;

Valuating Payoff Streams under Unequal Discount Factors

We study repeated prize allocation problem when the discount factors f the agents are not equal. It is shown that the feasible set of payoffs is not well behaved. In particular, it is not convex as it contains holes and caves. The Pareto frontier is everywhere discontinuous and there is an open subset of discount factors such that the feasible set is totally disconnected.payoffs, differentiated discount factor, repeated games

Achieving Intertemporal Efficiency and Symmetry through Intratemporal Asymmetry: (Eventual) Turn Taking in a Class of Repeated Mixed-Interest Games

Turn taking is observed in many field and laboratory settings. We study when and how turn taking can be supported as an equilibrium outcome in a class of repeated games, where the stage game is a symmetric two-player mixed-interest game with asymmetric joint-payoff-maximizing outcomes that may or may not be Nash equilibria. We consider the "turn taking with independent randomizations" (TTIR) strategy that achieves the following three objectives: (a) helping the players get onto a joint-payoff-maximizing turn-taking path, (b) resolving the question of who gets to start with the good turn first, and (c) deterring defection. The TTIR strategy is simpler than those time-varying strategies considered in the Folk Theorem for repeated games. We determine conditions under which a symmetric TTIR subgame-perfect equilibrium exists and is unique. We also derive comparative static results, and study the welfare properties of the TTIR equilibrium.Conflict, Coordination, Randomization, Turn Taking, Repeated Games