Markov Chains of Evolutionary Games with a Small Number of Players

TL;DR

This paper develops a framework for analyzing finite-player evolutionary games using Markov chains, explicitly constructing transition matrices for classic games and revision protocols to study their properties.

Contribution

It introduces a simplified approach to modeling small-player evolutionary games with Markov chains, including explicit transition matrices for various classic games and protocols.

Findings

Explicit transition matrices for Iterated Prisoner's Dilemma, Stag Hunt, Rock-Paper-Scissors

Analysis of properties of these Markov chains

Comparison of different revision protocols

Abstract

We construct and study the transition probability matrix of evolutionary games in which the number of players is finite (and relatively small) of such games. We use a simplified version of the population games studied by Sandholm. After laying out a general framework we concentrate on specific examples, involving the Iterated Prisoner's Dilemma, the Iterated Stag Hunt, and the Rock-Paper-Scissors game. Also we consider several revision protocols: Best Response, Pairwise Comparison, Pairwise Proportional Comparison etc. For each of these we explicitly construct the MC transition probability matrix and study its properties.