Coevolutionary Genetic Algorithms for Establishing Nash Equilibrium in Symmetric Cournot Games

TL;DR

This paper demonstrates that social-learning co-evolutionary genetic algorithms effectively lead to Nash Equilibrium in symmetric Cournot games, unlike individual learning algorithms which fail to converge.

Contribution

It introduces and evaluates social-learning versions of co-evolutionary genetic algorithms that reliably establish Nash Equilibrium in Cournot models.

Findings

Social-learning algorithms frequently reach NE at stationary distribution.

Expected Hamming distance to NE state is smaller with social learning.

Large fraction of games played are at Nash Equilibrium.

Abstract

We use co-evolutionary genetic algorithms to model the players' learning process in several Cournot models, and evaluate them in terms of their convergence to the Nash Equilibrium. The "social-learning" versions of the two co-evolutionary algorithms we introduce, establish Nash Equilibrium in those models, in contrast to the "individual learning" versions which, as we see here, do not imply the convergence of the players' strategies to the Nash outcome. When players use "canonical co-evolutionary genetic algorithms" as learning algorithms, the process of the game is an ergodic Markov Chain, and therefore we analyze simulation results using both the relevant methodology and more general statistical tests, to find that in the "social" case, states leading to NE play are highly frequent at the stationary distribution of the chain, in contrast to the "individual learning" case, when NE is not reached at all in our simulations; to find that the expected Hamming distance of the states at the limiting distribution from the "NE state" is significantly smaller in the "social" than in the "individual learning case"; to estimate the expected time that the "social" algorithms need to get to the "NE state" and verify their robustness and finally to show that a large fraction of the games played are indeed at the Nash Equilibrium.