A Note on the Pure Nash Equilibria for Evolutionary Games on Networks

TL;DR

This paper investigates the conditions under which pure Nash equilibria occur in evolutionary games on networks, linking network topology with equilibrium stability in coordination and anti-coordination settings.

Contribution

It provides necessary and sufficient conditions for pure steady-states to be strict Nash equilibria in network-based evolutionary games.

Findings

Pure steady-states in coordination games can be strict Nash equilibria under certain network conditions.

Pure steady-states in anti-coordination games can be strict Nash equilibria with specific network structures.

The stability of mixed equilibria relates to the eigenvalues of the network's adjacency matrix.

Abstract

Recently, a new model extending the standard replicator equation to a finite set of players connected on an arbitrary graph was developed in evolutionary game dynamics. The players are interpreted as subpopulations of multipopulations dynamical game and represented as vertices of the graph, and an edge constitutes the relation among the subpopulations. At each instant, members of connected vertices of the graph play a 2-player game and collect a payoff that determines if the chosen strategies will vanish or flourish. The model describes the game dynamics of a finite set of players connected by a graph emulating the replicator dynamics. It was proved a relation between the stability of the mixed equilibrium with the topology of the network. More specifically, the eigenvalues of the Jacobian matrix of the system evaluated at the mixed steady state are the eigenvalues of the graph's adjacency matrix multiplied by a scalar. This paper studies the pure (strict) Nash equilibria of these games and how it connects to the network. We present necessary and sufficient conditions for a pure steady-state in coordination or anti-coordination game to be a (strict) Nash Equilibrium.