On differences between fractional and integer order differential equations for dynamical games

TL;DR

This paper demonstrates that fractional order differential equations better model complex adaptive systems in game theory, providing stable solutions where integer order models do not, with implications for dynamic game analysis.

Contribution

It introduces fractional order differential equations into replicator dynamics for non-cooperative games, showing they yield stable equilibria absent in integer order models.

Findings

Fractional order models have stable internal solutions in Rock-Scissors-Paper game.

Integer order models lack stable equilibria in the same game.

Fractional order asymmetric games exhibit local asymptotic stability.

Abstract

We argue that fractional order (FO) differential equations are more suitable to model complex adaptive systems (CAS). Hence they are applied in replicator equations for non-cooperative game. Rock-Scissors-Paper game is discussed. It is known that its integer order model does not have a stable equilibrium. Its fractional order model is shown to have a locally asymptotically stable internal solution. A FO asymmetric game is shown to have a locally asymptotically stable internal solution. This is not the case for its integer order counterpart.