Nonlinear Nonlocal Diffusion Equations for the Analysis of Continuous Coordination and Anti-Coordination Type Games

TL;DR

This paper extends coordination and anti-coordination games into a continuous spatial setting, modeling their dynamics with nonlinear nonlocal PDEs, and proves key mathematical properties including existence, uniqueness, and regularity of solutions.

Contribution

It introduces a novel continuous-space framework for coordination games, deriving a nonlinear nonlocal diffusion PDE, and establishes foundational mathematical results for these models.

Findings

Proved existence and uniqueness of solutions for the PDE model.

Established maximum principle and regularity results for true coordination games.

Provided numerical simulations illustrating solution behaviors.

Abstract

Coordination games with explicit spatial or relational structure are of interest to economists, ecologists, sociologists, and others studying emergent global properties in collective behavior. When assemblies of individuals seek to coordinate action with one another through myopic best response or other replicator dynamics, the resulting dynamical system can exhibit many rich behaviors. However, these behaviors have been studied only in the case where the number of players is countable and the relational structure is described discretely. By giving an extension of a general class of coordination-like games, including true coordination games themselves, into a continuous setting, we can begin to study coordination and cooperative behavior with a new host of tools from PDEs and nonlocal equations. In this study, we propose a rigorously supported extension of structured coordination-type games into a setting with continuous space and continuous strategies and show that, under certain hypotheses, the dynamics of these games are described through a nonlinear, nonlocal diffusion equation. We go on to prove existence and uniqueness for the initial value problem in the case where no boundary data are prescribed. For true coordination games, we go further and prove a maximum principle, weak regularity results, as well as some numerical results toward understanding how solutions to the coordination equation behave. We present several modeling results, characterizing stationary solutions both rigorously and through numerical experiments and conclude with a result towards the inhomogeneous problem.