A Gibbsian approach to potential game theory

TL;DR

This paper explores how potential games with gradient-based strategy adjustments converge to Gibbs measures and investigates phase transitions in oligopoly models, revealing complex states influenced by local competition and goods distribution.

Contribution

It introduces a Gibbsian framework for potential game dynamics and analyzes phase transitions in oligopoly models with varying competition and distribution patterns.

Findings

Potential game dynamics equilibrate to Gibbs measures.

Standard Cournot model shows no phase transition.

Increased local competition leads to rich phase diagrams with multiple states.

Abstract

In games for which there exists a potential, the deviation-from-rationality dynamical model for which each agent's strategy adjustment follows the gradient of the potential along with a normally distributed random perturbation, is shown to equilibrate to a Gibbs measure. The standard Cournot model of an oligopoly is shown not to have a phase transition, as it is equivalent to a continuum version of the Curie-Weiss model. However, when there is increased local competition among agents, a phase transition will likely occur. If the oligopolistic competition has power-law falloff and there is increased local competition among agents, then the model has a rich phase diagram with an antiferromagnetic checkerboard state, striped states and maze-like states with varying widths, and finally a paramagnetic state. Such phases have economic implications as to how agents compete given various restrictions on how goods are distributed. The standard Cournot model corresponds to a uniform distribution of goods, whereas the power-law variations correspond to goods for which the distribution is more localized.