Deterministic Approximation of Stochastic Evolution in Games

Abstract

This paper provides deterministic approximation results for stochastic processes that arise when finite populations recurrently play finite games. The processes are Markov chains, and the approximation is defined in continuous time as a system of ordinary differential equations of the type studied in evolutionary game theory. We establish precise connections between the long-run behavior of the discrete stochastic process, for large populations, and its deterministic flow approximation. In particular, we provide probabilistic bounds on exit times from and visitation rates to neighborhoods of attractors to the deterministic flow. We sharpen these results in the special case of ergodic processes.

The Evolutionary Foundations of Preferences

“…Under this approach, evolution would equip us with a utility function that would provide the goal for our behavior, along with a learning process, perhaps ranging from trial-and-error to information collection and Bayesian updating, that would help us pursue that goal. 11 11 There are, of course, other aspects of our preferences that evolution may prefer to place outside our learning. Many people have a deep-seated fear of snakes (cf.…”

Green consumers and public policy: On socially contingent moral motivation

“…As a practical rule of thumb, if the payoff they experience falls short of an exogenous satisficing requirement 13 , they revise their strategy choice and simply mimic an arbitrary other person (they try the strategy of their neighbor, regardless of what this is). Benaïm and Weibull (2003) show that his type of adaptation rule gives rise to a specific dynamic originating from evolutionary biology, the replicator dynamics (Taylor and Jonker 1978), which has been studied extensively in evolutionary game theory. The replicator dynamics says that the growth rate of a strategy is proportional to the success of that strategy, where "success" is measured by payoff minus average payoff:…”

Probabilistic Interpretations of Integrability for Game Dynamics

“…3 For background on population games and evolutionary dynamics, see Sandholm (2010). 4 For formal analyses linking the stochastic decisions of individual agents to a deterministic description of aggregate behavior via a differential equation, see Benaïm and Weibull (2003) and Roth and Sandholm (2013). defined in terms of a Lipschitz continuous function ρ : R n → R n + , called a revision protocol, that maps vectors of excess payoffs π to vectors of switching rates ρ(π). Specifically, ρ i (π) is the rate at which a revising agent switches to action i, and this rate does not depend on the action the agent was playing when the revision opportunity arose.…”