On the maxmin value of stochastic games with imperfect monitoring

We study zero-sum stochastic games in which players do not observe the actions of the opponent. Rather, they observe a stochastic signal that may depend on the state, and on the pair of actions chosen by the players. We assume each player observes the state and his own action. We propose a candidate for the max-min value, which does not depend on the information structure of player 2. We prove that player 2 can defend the proposed max-min value, and that in absorbing games player 1 can guarantee it. Analogous results hold for the min-max value. This paper thereby unites several results due to Coulomb.Stochastic games; partial monitoring; value

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

A minority game with bounded recall.

This paper studies a repeated minority game with public signals, symmetric bounded recall, and pure strategies. We investigate both public and private equilibria of the game with fixed recall size. We first show how public equilibria in such a repeated game can be represented as colored subgraphs of a de Bruijn graph. Then we prove that the set of public equilibrium payoffs with bounded recall converges to the set of uniform equilibrium payoffs as the size of the recall increases. We also show that private equilibria behave badly: A private equilibrium payoff with bounded recall need not be a uniform equilibrium payoff.folk theorem; de Bruijn sequence; imperfect monitoring; uniform equilibrium; public equilibrium; private equilibrium;

Uniform Folk Theorems in Repeated Anonymous Random Matching Games

We study infinitely repeated anonymous random matching games played by communities of players, who only observe the outcomes of their own matches. It is well known that cooperation can be sustained in equilibrium for the prisoner’s dilemma (PD) through grim trigger strategies. Little is known about games beyond the PD. We study a new equilibrium concept, strongly uniform equilibrium (SUE, which refines the notion of uniform equilibrium (UE) and has additional properties such as a strong version of (approximate) sequential rationality. We establish folk theorems for general games and arbitrary number of communities. Interestingly, the equilibrium strategies we construct are easy to play. We extend the results to a setting with imperfect private monitoring, for the case of two communities. We also show that it is possible for some players to get equilibrium payoffs that are outside the set of individually rational and feasible payoffs of the stage game. In particular, for the PD we derive a bound on the number of “free-riders” that can be sustained in society. A by-product of our analysis is an important result relating uniform equilibrium and strongly uniform equilibria: we show that, in general repeated games with finite players, actions, and signals, the set of UE and SUE payoffs coincide