# Thermodynamics of Evolutionary Games

## TL;DR

This paper models the evolution of cooperation using Hamiltonian dynamics similar to Ising models, revealing phase transitions and the role of punishment as a magnetic field influencing cooperative behavior.

## Contribution

It introduces a novel Hamiltonian formalism for evolutionary games, linking cooperation dynamics to phase transitions in spin systems, and explores punishment effects within this framework.

## Key findings

- Cooperation fraction corresponds to a magnetization-like observable.
- A phase transition between cooperation and defection is identified.
- Punishment acts as a magnetic field promoting cooperation.

## Abstract

How cooperation can evolve between players is an unsolved problem of biology. Here we use Hamiltonian dynamics of models of the Ising type to describe populations of cooperating and defecting players to show that the equilibrium fraction of cooperators is given by the expectation value of a thermal observable akin to a magnetization. We apply the formalism to the Public Goods game with three players, and show that a phase transition between cooperation and defection occurs that is equivalent to a transition in one-dimensional Ising crystals with long-range interactions. We then investigate the effect of punishment on cooperation and find that punishment plays the role of a magnetic field that leads to an "alignment" between players, thus encouraging cooperation. We suggest that a thermal Hamiltonian picture of the evolution of cooperation can generate other insights about the dynamics of evolving groups by mining the rich literature of critical dynamics in low-dimensional spin systems.