A Short Note on Nonlinear Games on a Grid

TL;DR

This paper explores complex dynamics in nonlinear grid-based games where players update strategies based on payoff rankings, extending traditional linear models to include all possible payoff orderings inspired by Conway's Game of Life.

Contribution

It introduces a framework for analyzing nonlinear payoff rankings on a grid, broadening the scope beyond linear sums and revealing diverse dynamic behaviors.

Findings

Rich array of dynamics observed with nonlinear payoff rankings

Linear sums account for only a small fraction of possible behaviors

Framework inspired by Conway's Game of Life enhances understanding of strategic evolution

Abstract

Players are arranged on a regular lattice and coded with a specific strategy for a pre-defined game. Each player sums their payoffs from playing the game with each of their neighbors, and then adopts the strategy of the most successful player in the neighborhood. Dynamics are thus determined by the relative ranks of all possible payoff sums. Linear sums of payoffs, however, generate only a small proportion of all possible rankings. Allowing for any ranking (motivated by Conway's Game of Life) creates a rich array of dynamics.