Regret Matching with Finite Memory

We consider the regret matching process with finite memory. For general games in normal form, it is shown that any recurrent class of the dynamics must be such that the action profiles that appear in it constitute a closed set under the “same or better reply” correspondence (CUSOBR set) that does not contain a smaller product set that is closed under “same or better replies,” i.e., a smaller PCUSOBR set. Two characterizations of the recurrent classes are offered. First, for the class of weakly acyclic games under better replies, each recurrent class is monomorphic and corresponds to each pure Nash equilibrium. Second, for a modified process with random sampling, if the sample size is sufficiently small with respect to the memory bound, the recurrent classes consist of action profiles that are minimal PCUSOBR sets. Our results are used in a robust example that shows that the limiting empirical distribution of play can be arbitrarily far from correlated equilibria for any large but finite choice of the memory bound.Regret Matching; Nash Equilibria; Closed Sets under Same or Better Replies; Correlated Equilibria.

Decision Making in Uncertain and Changing Environments

We consider an agent who has to repeatedly make choices in an uncertain and changing environment, who has full information of the past, who discounts future payoffs, but who has no prior. We provide a learning algorithm that performs almost as well as the best of a given finite number of experts or benchmark strategies and does so at any point in time, provided the agent is sufficiently patient. The key is to find the appropriate degree of forgetting distant past. Standard learning algorithms that treat recent and distant past equally do not have the sequential epsilon optimality property.Adaptive learning, experts, distribution-free, epsilon-optimality, Hannan regret

Players with limited memory

This paper studies a model of memory. The model takes into account that memory capacity is limited and imperfect. We study how agents with such memory limitations, who have very little information about their choice environment, play games. We introduce the notion of a Limited Memory Equilibrium (LME) and show that play converges to an LME in every generic normal form game. Our characterization of the set of LME suggests that players with limited memory do (weakly) better in games than in decision problems. We also show that agents can do quite well even with severely limited memory, although severe limitations tend to make them behave cautiously

Decision making in uncertain and changing environments

We consider an agent who has to repeatedly make choices in an uncertain and changing environment, who has full information of the past, who discounts future payoffs, but who has no prior. We provide a learning algorithm that performs almost as well as the best of a given finite number of experts or benchmark strategies and does so at any point in time, provided the agent is sufficiently patient. The key is to find the appropriate degree of forgetting distant past. Standard learning algorithms that treat recent and distant past equally do not have the sequential epsilon optimality property.Adaptive learning, experts, distribution-free, e-optimality, Hannan regret

Finding Any Nontrivial Coarse Correlated Equilibrium Is Hard

One of the most appealing aspects of the (coarse) correlated equilibrium concept is that natural dynamics quickly arrive at approximations of such equilibria, even in games with many players. In addition, there exist polynomial-time algorithms that compute exact (coarse) correlated equilibria. In light of these results, a natural question is how good are the (coarse) correlated equilibria that can arise from any efficient algorithm or dynamics. In this paper we address this question, and establish strong negative results. In particular, we show that in multiplayer games that have a succinct representation, it is NP-hard to compute any coarse correlated equilibrium (or approximate coarse correlated equilibrium) with welfare strictly better than the worst possible. The focus on succinct games ensures that the underlying complexity question is interesting; many multiplayer games of interest are in fact succinct. Our results imply that, while one can efficiently compute a coarse correlated equilibrium, one cannot provide any nontrivial welfare guarantee for the resulting equilibrium, unless P=NP. We show that analogous hardness results hold for correlated equilibria, and persist under the egalitarian objective or Pareto optimality. To complement the hardness results, we develop an algorithmic framework that identifies settings in which we can efficiently compute an approximate correlated equilibrium with near-optimal welfare. We use this framework to develop an efficient algorithm for computing an approximate correlated equilibrium with near-optimal welfare in aggregative games.Comment: 21 page

Players with Limited Memory

This paper studies a model of memory. The model takes into account that memory capacity is limited and imperfect. We study how agents with such memory limitations, who have very little information about their choice environment, play games. In particular, the players do not know if they are playing a game. We show that players do better in games than in decision problems. This is because the players, unknowingly, improve the environment they face in games. We also show that people can do quite well in games even with severely limited memories, although memory restrictions tend to make them behave cautiously. Lastly, we introduce a solution concept approiate for analysis games in which the players may have limited knowledge of their environment and have some memory restictions. We show hos this solution concept is related to other like the iterated removal of strictly dominated strategies.

Stochastic uncoupled dynamics and Nash equilibrium

In this paper we consider dynamic processes, in repeated games, that are subject to the natural informational restriction of uncoupledness. We study the almost sure convergence to Nash equilibria, and present a number of possibility and impossibility results. Basically, we show that if in addition to random moves some recall is introduced, then successful search procedures that are uncoupled can be devised. In particular, to get almost sure convergence to pure Nash equilibria when these exist, it su±ces to recall the last two periods of play.Uncoupled, Nash equilibrium, stochastic dynamics, bounded recall