Imitation Dynamics with Payoff Shocks

We investigate the impact of payoff shocks on the evolution of large populations of myopic players that employ simple strategy revision protocols such as the "imitation of success". In the noiseless case, this process is governed by the standard (deterministic) replicator dynamics; in the presence of noise however, the induced stochastic dynamics are different from previous versions of the stochastic replicator dynamics (such as the aggregate-shocks model of Fudenberg and Harris, 1992). In this context, we show that strict equilibria are always stochastically asymptotically stable, irrespective of the magnitude of the shocks; on the other hand, in the high-noise regime, non-equilibrium states may also become stochastically asymptotically stable and dominated strategies may survive in perpetuity (they become extinct if the noise is low). Such behavior is eliminated if players are less myopic and revise their strategies based on their cumulative payoffs. In this case, we obtain a second order stochastic dynamical system whose attracting states coincide with the game's strict equilibria and where dominated strategies become extinct (a.s.), no matter the noise level.Comment: 25 page

Endogenous Correlation

We model endogenous correlation in asset returns via the role of heterogeneous expectations in investor types, and the dynamic impact of imitative learning by investors. Learning is driven by relative performance. In addition, we allow a cautious slow learning pace to reflect institutional conditions. Imitative learning shapes the market ecology that influences price formation. Using the model of non-imitative agents as a benchmark, our results show that the dynamics of imitative learning endogenously induce a significant degree of asset dependency and patterns of non-constant correlation. The asymmetric learning effect on correlation, however, implies a self-reinforcing process, where a bearish condition amplifies the effect that further exacerbates asset dependency. We conclude that imitative learning, even when rational, can to a certain extent account for the phenomena of market crashes. Our results have implications for transparency in regulation issues

Convergence in Models with Bounded Expected Relative Hazard Rates

We provide a general framework to study stochastic sequences related to individual learning in economics, learning automata in computer sciences, social learning in marketing, and other applications. More precisely, we study the asymptotic properties of a class of stochastic sequences that take values in $[0,1]$ and satisfy a property called "bounded expected relative hazard rates." Sequences that satisfy this property and feature "small step-size" or "shrinking step-size" converge to 1 with high probability or almost surely, respectively. These convergence results yield conditions for the learning models in B\"orgers, Morales, and Sarin (2004), Erev and Roth (1998), and Schlag (1998) to choose expected payoff maximizing actions with probability one in the long run.Comment: After revision. Accepted for publication by Journal of Economic Theor

Strategy abundance in 2x2 games for arbitrary mutation rates

We study evolutionary game dynamics in a well-mixed populations of finite size, N. A well-mixed population means that any two individuals are equally likely to interact. In particular we consider the average abundances of two strategies, A and B, under mutation and selection. The game dynamical interaction between the two strategies is given by the 2x2 payoff matrix [(a,b), (c,d)]. It has previously been shown that A is more abundant than B, if (N-2)a+Nb>Nc+(N-2)d. This result has been derived for particular stochastic processes that operate either in the limit of asymptotically small mutation rates or in the limit of weak selection. Here we show that this result holds in fact for a wide class of stochastic birth-death processes for arbitrary mutation rate and for any intensity of selection.Comment: version 2 is the final published version that contains minor changes in response to referee comment

The collapse of cooperation in evolving games

Game theory provides a quantitative framework for analyzing the behavior of rational agents. The Iterated Prisoner's Dilemma in particular has become a standard model for studying cooperation and cheating, with cooperation often emerging as a robust outcome in evolving populations. Here we extend evolutionary game theory by allowing players' strategies as well as their payoffs to evolve in response to selection on heritable mutations. In nature, many organisms engage in mutually beneficial interactions, and individuals may seek to change the ratio of risk to reward for cooperation by altering the resources they commit to cooperative interactions. To study this, we construct a general framework for the co-evolution of strategies and payoffs in arbitrary iterated games. We show that, as payoffs evolve, a trade-off between the benefits and costs of cooperation precipitates a dramatic loss of cooperation under the Iterated Prisoner's Dilemma; and eventually to evolution away from the Prisoner's Dilemma altogether. The collapse of cooperation is so extreme that the average payoff in a population may decline, even as the potential payoff for mutual cooperation increases. Our work offers a new perspective on the Prisoner's Dilemma and its predictions for cooperation in natural populations; and it provides a general framework to understand the co-evolution of strategies and payoffs in iterated interactions.Comment: 33 pages, 13 figure

Stochastic Game Theory: Adjustment to Equilibrium Under Noisy Directional Learning

This paper presents a dynamic model in which agents adjust their decisions in the direction of higher payoffs, subject to random error. This process produces a probability distribution of players' decisions whose evolution over time is determined by the Fokker-Planck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This "logit equilibrium" forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data.bounded rationality, noisy directional learning, Fokker- Planck equation, potential games, logit equilibrium, stochastic potential.

Liability Rules and Evolutionay Dynamics

We consider the convergence properties of behavior under a comparative negligence rule (CN) and under a rule of negligence with contributory negligence (NCN), assuming bilateral care with three care levels. Using an evolutionary model, we show that CN reduces the proportion of the population using low care more rapidly than does NCN. However NCN increases the proportion of the population using high (efficient) care more rapidly than does CN. As a result, the mean care level increases more rapidly and the mean social cost falls more rapidly under CN than under NCN.tort law, evolutionary game, liability rules, economic analysis of law

The Role of Opportunistic Punishment in the Evolution of Cooperation: An application of stochastic dynamics to public good game

This paper discusses the role of opportunistic punisher who may act selfishly to free-ride cooperators or not to be exploited by defectors. To consider opportunistic punisher, we make a change to the sequence of one-shot public good game; instead of putting action choice first before punishment, the commitment of punishment is declared first before choosing the action of each participant. In this commitment-first setting, punisher may use information about her team, and may defect to increase her fitness in the team. Reversing sequence of public good game can induce different behavior of punisher, which cannot be considered in standard setting where punisher always chooses cooperation. Based on stochastic dynamics developed by evolutionary economists and biologists, we show that opportunistic punisher can make cooperation evolve where cooperative punisher fails. This alternative route for the evolution of cooperation relies paradoxically on the players' selfishness to profit from others' unconditional cooperation and defection.Comment: 30 page, 9 figure

The Evolutionary Processes for the Populations of Firms and Workers

This paper analyzes the cultural evolution of firms and workers. Following an imitation rule, each firm and worker decides whether to be innovative (or not) and skilled (or unskilled). We apply evolutionary game theory to find the system of replicator dynamics, and characterize the low-level and high-level equilibria as Evolutionarily Stable Strategies (ESS) “against the field.” Hence, we study how a persistent state of underdevelopment can arise in strategic environments in which players are imitative rather than rational maximizers. We show that when the current state of the economy is in the basin of attraction of the poverty trap, players should play against the field if they want to change their status quo. The threshold level to overcome the poverty trap can be lowered if there is an appropriate policy using income taxes, education costs and skill premia. Hence, we study the replicator dynamics with a subsidy and payoff taxation to overcome the poverty trap.Imitative behavior, conformism, poverty traps, skill premium, strategic complementarities

On the robustness of learning in games with stochastically perturbed payoff observations

Motivated by the scarcity of accurate payoff feedback in practical applications of game theory, we examine a class of learning dynamics where players adjust their choices based on past payoff observations that are subject to noise and random disturbances. First, in the single-player case (corresponding to an agent trying to adapt to an arbitrarily changing environment), we show that the stochastic dynamics under study lead to no regret almost surely, irrespective of the noise level in the player's observations. In the multi-player case, we find that dominated strategies become extinct and we show that strict Nash equilibria are stochastically stable and attracting; conversely, if a state is stable or attracting with positive probability, then it is a Nash equilibrium. Finally, we provide an averaging principle for 2-player games, and we show that in zero-sum games with an interior equilibrium, time averages converge to Nash equilibrium for any noise level.Comment: 36 pages, 4 figure