Evolutionary game theory studies the behavior of large populations of agents who repeatedly engage in anonymous strategic interactions that is, interactions in which each agent’s outcome depends not only on his own choice, but also on the distribution of others’ choices. Applications range from natural selection in animal populations, to driver behavior in highway networks, to consumer choice between different technological standards, to the design of decentralized controlled systems.
In an evolutionary game model, changes in agents’ behavior may be driven either by natural selection via differences in birth and death rates in biological contexts, or by the application of myopic decision rules by individual agents in economic contexts. The resulting dynamic models can be studied using tools from the theory of dynamical systems and from the theory of stochastic processes, as well as those from stochastic approximation theory, which provides important links between the two more basic fields.
In these notes, we present a general model of stochastic evolution in large-population games, and offer a glimpse into the relevant literature by presenting a selection of basic results. In Section 2, we describe population games themselves, and offer a few simple applications. In Sections 3 and 4, we introduce our stochastic evolutionary process. To define this process, we suppose that agents receive opportunities to revise their strategies by way of independent Poisson processes.
A revision protocol describes how the probabilities with which an agent chooses each of his strategies depend on his current payoff opportunities and the current behavior of the population. Together, a population game, a revision protocol, and a population size implicitly define the stochastic evolutionary process, a Markov process on the set of population states. In Section 4, we show that over finite time horizons, the population’s behavior is well-approximated by a mean dynamic, an ordinary differential equation defined by the expected motion of the stochastic evolutionary process.
Contents
1 Introduction
2 Population Games
3 Revision Protocols and the Stochastic Evolutionary Process
- 3.1 Definitions
3.2 Examples
4 Finite Horizon Deterministic Approximation
- 4.1 Mean Dynamics
4.2 Examples
4.3 Deterministic Approximation Theorem
4.4 Analysis of Deterministic Dynamics
5 Stationary Distributions
- 5.1 Full Support Revision Protocols
5.2 Review: Irreducible Markov Processes
5.3 Stationary Distributions for Two-Strategy Games
5.4 Examples
6 Asymptotics of the Stationary Distribution and Stochastic Stability
7 Noisy Best Response Protocols in Two-Strategy Games
- 7.1 Noisy Best Response Protocols and their Cost Functions
7.2 The(Double)LimitTheorem
7.3 Stochastic Stability: Examples and Analysis
7.4 Risk Dominance, Stochastic Dominance, and Stochastic Stability
8 Further Developments
