Markov Chains of Evolutionary Games with a Small Number of Players

2025-06-29

AI Summary

This study addresses the challenge of modeling evolutionary game dynamics in finite, small populations. We propose a unified analytical framework based on Markov chains, explicitly constructing transition probability matrices to model canonical games—including iterated prisoner’s dilemma, stag hunt, and rock–paper–scissors—under three update rules: best response, pairwise comparison, and proportional imitation. Our key contributions are threefold: (i) the first analytically tractable matrix-based modeling of multiple games and update mechanisms within small-population settings; (ii) exact characterization of absorbing state structures, stationary distributions, and convergence rates; and (iii) identification of equilibrium selection’s sensitivity to and structural dependence on the choice of update rule in finite populations. The framework yields a computationally feasible, quantitatively comparable theoretical tool for predicting evolutionary outcomes in small-scale interactive systems.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Machine Learning: Evolutionary Learning

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Model transition matrix for small-player evolutionary games
• Analyze game dynamics in Iterated Prisoner's Dilemma scenarios
• Compare revision protocols like Best Response strategies

Innovation

• Finite player Markov Chains for evolutionary games
• Simplified Sandholm population games framework
• Multiple revision protocols transition matrices