Insights into the Structured Coordination Game with Neutral Options through Simulation

TL;DR

This paper explores the equilibria of structured coordination games with neutral options using graph partitioning, analyzing small cases and extending insights through simulations on larger random graphs to understand factors influencing consensus stability.

Contribution

It introduces a novel graph partitioning approach to analyze equilibria in structured coordination games with neutral strategies, expanding beyond symmetric reductions.

Findings

Catalogued all Nash equilibria for graphs with up to seven vertices.

Used simulations on larger Erdős-Rényi graphs to observe relationships among graph properties.

Proposed conjectures relating edge density, cluster number, and stability of consensus.

Abstract

Coordination games have been of interest to game theorists, economists, and ecologists for many years to study such problems as the emergence of local conventions and the evolution of cooperative behavior. Approaches for understanding the coordination game with discrete structure have been limited in scope, often relying on symmetric reduction of the state space, or other constraints which limit the power of the model to give insight into desired applications. In this paper, we introduce a new way of thinking about equilibria of the structured coordination game with neutral strategies by means of graph partitioning. We begin with a few elementary game theoretical results and then catalogue all the Nash equilibria of the coordination game with neutral options for graphs with seven or fewer vertices. We extend our observations through the use of simulation on larger Erd\H{o}s-R\'enyi random graphs to form the basis for proposing some conjectures about the general relationships among edge density, cluster number, and consensus stability.