Statistical Physics of the Spatial Prisoner's Dilemma with Memory-Aware Agents

TL;DR

This paper presents an analytical model linking statistical physics and evolutionary game theory to explain how memory-aware agents in spatial Prisoner's Dilemma can reach cooperative equilibria, influenced by system temperature and phase transitions.

Contribution

It introduces a novel gas-like model for agents with payoff memory, revealing how temperature affects equilibrium states and enabling analytical understanding of cooperation emergence.

Findings

Higher temperature can promote cooperation in the spatial Prisoner's Dilemma.

The model identifies phase transitions between order and disorder in agent dynamics.

Analytical relations between temperature and equilibrium states are established.

Abstract

We introduce an analytical model to study the evolution towards equilibrium in spatial games, with `memory-aware' agents, i.e., agents that accumulate their payoff over time. In particular, we focus our attention on the spatial Prisoner's Dilemma, as it constitutes an emblematic example of a game whose Nash equilibrium is defection. Previous investigations showed that, under opportune conditions, it is possible to reach, in the evolutionary Prisoner's Dilemma, an equilibrium of cooperation. Notably, it seems that mechanisms like motion may lead a population to become cooperative. In the proposed model, we map agents to particles of a gas so that, on varying the system temperature, they randomly move. In doing so, we are able to identify a relation between the temperature and the final equilibrium of the population, explaining how it is possible to break the classical Nash equilibrium in the spatial Prisoner's Dilemma when considering agents able to increase their payoff over time. Moreover, we introduce a formalism to study order-disorder phase transitions in these dynamics. As result, we highlight that the proposed model allows to explain analytically how a population, whose interactions are based on the Prisoner's Dilemma, can reach an equilibrium far from the expected one; opening also the way to define a direct link between evolutionary game theory and statistical physics.