Cooperation risk and Nash equilibrium: quantitative description for realistic players

TL;DR

This paper extends potential games by incorporating cooperation risks, modeling the Public Goods game as a Hamiltonian, and analyzing phase transitions between cooperation and competition.

Contribution

It introduces a Hamiltonian framework for the Public Goods game that accounts for cooperation risks and explores phase transitions in player behavior.

Findings

Public Goods game modeled as a Hamiltonian with ground state as Nash equilibrium.

Punishments reduce cooperation risk and induce phase transitions.

Rich phase diagram with segregation of cooperative and competitive regimes.

Abstract

The emergence of cooperation figures among the main goal of game theory in competitive-cooperative environments. Potential games have long been hinted as viable alternatives to study realistic player behavior. Here, we expand the potential games approach by taking into account the inherent risks of cooperation. We show the Public Goods game reduce to a Hamiltonian with one-body operators, with the correct Nash Equilibrium as the ground state. The inclusion of punishments to the Public Goods game reduces the cooperation risk, creating two-body interaction with a rich phase diagram, where phase transitions segregates the cooperative from competitive regimes.