Stochastic Imitation in Finite Games

In this paper we model an evolutionary process with perpetual random shocks where individual behavior is determined by imitation. Every period an agent is randomly chosen from each of n finite populations to play a game. Each agent observes a sample of population-specific past strategy and payoff realizations. She thereafter imitates by choosing the strategy with highest average payoff in the sample. Occasionally the agents also experiment or make mistakes and choose a strategy at random. For finite n-player games we prove that in the limit, as the probability of experimentation tends to zero, only strategy-tuples in minimal sets closed under the better-reply graph will be played with positive probability. If the strategy-tuples in one such minimal set have strictly higher payoffs than all outside strategy-tuples, then the strategy-tuples in this set will be played with probability one in the limit, provided the minimal set is a product set. We also show that in 2x2 games the convergence in our model is faster than in other known models.Evolutionary game theory; bounded rationality; imitation; Markov chain; stochastic stability; better replies; Pareto dominance

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.

Location, Information and Coordination

In this paper, we consider K finite populations of boundedly rational agents whose preferences and information differ. Each period agents are randomly paired to play some coordination games. We show that several ``special`` (fixed) agents lead the coordination. In a mistake-free environment, all connected fixed agents have to coordinate on the same strategy. In the long run, as the probability of mistakes goes to zero, all agents coordinate on the same strategy. The long-run outcome is unique, if all fixed agents belong to the same population.

Cooperation through Imitation

This paper characterizes long-run outcomes for broad classes of symmetric games, when players select actions on the basis of average historical performance. Received wisdom is that when agent's interests are partially opposed, behavior is excessively competitive: ``keeping up with the Jones' '' lowers everyones' welfare. Here, we study the long-run consequences of imitative behavior when agents have sufficiently long memories --- and the outcome is dramatically different. Imitation robustly leads to cooperative outcomes (with highest symmetric payoffs) in the long run. This provides a rationale, for example, for collusive cartel-like behavior without collusive intent on the part of the agents.Evolution, Imitation

Ordinal Games

We study strategic games where players' preferences are weak orders which need not admit utility representations. First of all, we ex- tend Voorneveld's concept of best-response potential from cardinal to ordi- nal games and derive the analogue of his characterization result: An ordi- nal game is a best-response potential game if and only if it does not have a best-response cycle. Further, Milgrom and Shannon's concept of quasi- supermodularity is extended from cardinal games to ordinal games. We ¯nd that under certain compactness and semicontinuity assumptions, the ordinal Nash equilibria of a quasi-supermodular game form a nonempty complete lattice. Finally, we extend several set-valued solution concepts from cardinal to ordinal games in our sense.Ordinal Games, Potential Games, Quasi-Supermodularity, Rationalizable Sets, Sets Closed under Behavior Correspondences

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

Games between responsive behavioural rules

I study recurrent strategic interaction between two responsive behavioural rules in generic bi-matrix weakly acyclic games. The two individuals that play the game in a particular period choose their strategy by responding to a sample of strategies used by the co-players in the recent history of play. The response of a player is determined by his behavioural rule. I show that the game reaches a convention whenever the behavioural rule of each player is `weakly responsive' to the manner in which strategies were chosen in the past by the co-players, and stays locked into the convention if the behavioural rules are `mildly responsive'. Furthermore, amongst `mildly responsive' behavioural rules, individuals described by the behavioural rule of `extreme optimism' perform the best in the sense that their most preferred convention is always in the stochastically stable set; under an additional mild restriction that differentiates the behavioural rule of the other player from extreme optimism, the convention referred to above is the unique stochastically stable state

Games between responsive behavioural rules

I study recurrent strategic interaction between two responsive behavioural rules in generic bi-matrix weakly acyclic games. The two individuals that play the game in a particular period choose their strategy by responding to a sample of strategies used by the co-players in the recent history of play. The response of a player is determined by his behavioural rule. I show that the game reaches a convention whenever the behavioural rule of each player is `weakly responsive' to the manner in which strategies were chosen in the past by the co-players, and stays locked into the convention if the behavioural rules are `mildly responsive'. Furthermore, amongst `mildly responsive' behavioural rules, individuals described by the behavioural rule of `extreme optimism' perform the best in the sense that their most preferred convention is always in the stochastically stable set; under an additional mild restriction that differentiates the behavioural rule of the other player from extreme optimism, the convention referred to above is the unique stochastically stable state

Imitating the Most Successful Neighbor in Social Networks

We consider a model of observational learning in social networks. At every period, all agents choose from the same set of actions with uncertain payoffs and observe the actions chosen by their neighbors, as well as the payoffs they received. They update their choice myopically, by imitating the choice of their most successful neighbor. We show that in finite networks, regardless of the structure, the population converges to a monomorphic steady state, i.e. one at which every agent chooses the same action. Moreover, in arbitrarily large networks with bounded neighborhoods, an action is diffused to the whole population if it is the only one chosen initially by a non--negligible share of the population. If there exist more than one such actions, we provide an additional sufficient condition in the payoff structure, which ensures convergence to a monomorphic steady state for all networks. Furthermore, we show that without the assumption of bounded neighborhoods, (i) an action can survive even if it is initially chosen by a single agent, and (ii) a network can be in steady state without this being monomorphic