A New Mathematical Model for Evolutionary Games on Finite Networks of Players

TL;DR

This paper introduces a novel mathematical model extending the classical replicator equation to finite networks of players, integrating graph theory and evolutionary game dynamics to analyze strategic interactions.

Contribution

It develops a new model for evolutionary games on graphs that generalizes the replicator equation to finite populations with arbitrary network topologies.

Findings

Model behaves like classical replicator equation under certain conditions

Simulations reveal complex dynamical behaviors on different network structures

Potential for analyzing strategic evolution in networked populations

Abstract

A new mathematical model for evolutionary games on graphs is proposed to extend the classical replicator equation to finite populations of players organized on a network with generic topology. Classical results from game theory, evolutionary game theory and graph theory are used. More specifically, each player is placed in a vertex of the graph and he is seen as an infinite population of replicators which replicate within the vertex. At each time instant, a game is played by two replicators belonging to different connected vertices, and the outcome of the game influences their ability of producing offspring. Then, the behavior of a vertex player is determined by the distribution of strategies used by the internal replicators. Under suitable hypotheses, the proposed model is equivalent to the classical replicator equation. Extended simulations are performed to show the dynamical behavior of the solutions and the potentialities of the developed model.