Evolutionary Games on Networks and Payoff Invariance Under Replicator Dynamics

TL;DR

This paper introduces a modified payoff scheme for evolutionary games on networks that ensures invariance under payoff shifts and demonstrates that cooperation can emerge in structured populations under this scheme, unlike in traditional models.

Contribution

It proposes a new payoff scheme that maintains invariance under affine transformations and shows its effectiveness in promoting cooperation in network-structured populations.

Findings

Modified payoff scheme ensures invariance under payoff shifts.

Cooperation emerges in structured populations for Prisoner's Dilemma, Hawks-Doves, and Stag-Hunt.

Standard replicator dynamics predict no cooperation in Prisoner's Dilemma.

Abstract

The commonly used accumulated payoff scheme is not invariant with respect to shifts of payoff values when applied locally in degree-inhomogeneous population structures. We propose a suitably modified payoff scheme and we show both formally and by numerical simulation, that it leaves the replicator dynamics invariant with respect to affine transformations of the game payoff matrix. We then show empirically that, using the modified payoff scheme, an interesting amount of cooperation can be reached in three paradigmatic non-cooperative two-person games in populations that are structured according to graphs that have a marked degree inhomogeneity, similar to actual graphs found in society. The three games are the Prisoner's Dilemma, the Hawks-Doves and the Stag-Hunt. This confirms previous important observations that, under certain conditions, cooperation may emerge in such network-structured populations, even though standard replicator dynamics for mixing populations prescribes equilibria in which cooperation is totally absent in the Prisoner's Dilemma, and it is less widespread in the other two games.