Imitation Processes with Small Mutations

Drew Fudenberg,Lorens A Imhof

doi:10.2139/ssrn.619203

Abstract

This note characterizes the impact of adding rare stochastic mutations to an imitation dynamic, meaning a process with the properties that any state where agents use the same strategy is absorbing, and other states are transient. The work of Freidlin and Wentzell [10] and its extensions implies that the resulting system will spend almost of its time at the absorbing states of the no-mutation process, and provides a general algorithm for calculating the limit distribution, but this algorithm can be complicated to apply. This note provides a simpler and more intuitive algorithm. Loosely speaking, in a process with K strategies, it is sufficient to find the invariant distribution of a K x K Markov matrix on the K homogeneous states, where the probability of a transit from all play i to all play j is the probability of a transition from the state all agents but 1 play i, 1 plays j to the state all play j.

Highlights

Many papers in economics and evolutionary game theory study various sorts of “imitation dynamics,” according to which agents are more likely to adopt strategies that are popular and/or successful.1 Under the starkest version of these dynamics, it is impossible for agents to adopt a strategy that is not currently in use, so that any “homogeneous”state where all agents use the same strategy is absorbing
One way to formalize the idea that some of the homogeneous states are “more persistent” than others it to assume that a small mutation term makes the system ergodic, and analyze the limit, as the mutation probability goes to zero, of the invariant distributions
The work of Freidlin and Wentzell [10] and its subsequent applications and extensions2 shows that the resulting system will spend almost all of its time at the absorbing states of the underlying no-mutation process, and provides a general algorithm for calculating the limit distribution, but this algorithm can be complicated to apply, especially in games with more than two strategies

Summary

Introduction

Many papers in economics and evolutionary game theory study various sorts of “imitation dynamics,” according to which agents are more likely to adopt strategies that are popular and/or successful. Under the starkest version of these dynamics, it is impossible for agents to adopt a strategy that is not currently in use, so that any “homogeneous”state where all agents use the same strategy is absorbing. It is typically the case that all of the interior states are transient, so that the dynamics converges to one of the homogeneous states This does not mean that all of these states are plausible, as. The work of Freidlin and Wentzell [10] and its subsequent applications and extensions shows that the resulting system will spend almost all of its time at the absorbing states of the underlying no-mutation process, and provides a general algorithm for calculating the limit distribution, but this algorithm can be complicated to apply, especially in games with more than two strategies. When mutations are of order ", the process will spend 1 O(") of the time at the homogeneous states, which are the vertices of the state space, O(") time on the "edges" where only two strategies have positive probability, and o(") time at interior points. A second implication is that the ratio of the mutation probabilities will matter, even when this ratio is bounded away from zero and in...nity; this is related to the fact that a single mutation is enough to cause a transition from one homogeneous state to another

Anonymous Populations

Examples

General Imitation Processes