The Spread of Cooperative Strategies on Grids with Random Asynchronous Updating

TL;DR

This paper investigates how cooperative and selfish behaviors spread on grid networks under asynchronous updates, showing that the proportion of cooperators can be modeled as a polynomial in initial cooperation probability, with theoretical bounds supported by simulations.

Contribution

It introduces a model for the spread of cooperation on grids with randomized asynchronous updates and derives polynomial expressions for cooperator density.

Findings

Cooperator density can be expressed as a polynomial in initial cooperation probability p.

Theoretical bounds for cooperator density are validated through simulations.

The model provides insights into the dynamics of cooperation in grid-based systems.

Abstract

The Prisoner's Dilemma Process on a graph $G$ is an iterative process where each vertex, with a fixed strategy (cooperate or defect), plays the game with each of its neighbours. At the end of a round each vertex may change its strategy to that of its neighbour with the highest pay-off. Here we study the spread of cooperative and selfish behaviours on a toroidal grid, where each vertex is initially a cooperator with probability $p$. When vertices are permitted to change their strategies via a randomized asynchronous update scheme, we find that for some values of $p$ the limiting density of cooperators may be modelled as a polynomial in $p$. Theoretical bounds for this density are confirmed via simulation.