On Phase Transitions to Cooperation in the Prisoner's Dilemma

TL;DR

This paper investigates how phase transitions can enable cooperation in the Prisoner's Dilemma through mechanisms like adaptive group pressure, which alter payoffs and create bistability, even without changing stationary states.

Contribution

It classifies types of phase transitions to cooperation and introduces adaptive group pressure as a novel mechanism to induce bistability in the Prisoner's Dilemma.

Findings

Phase transitions to cooperation can be of first or second order.

Adaptive group pressure can invert the expected defection outcome.

Bistability arises without changing stationary states or eigenvalues.

Abstract

Game theory formalizes certain interactions between physical particles or between living beings in biology, sociology, and economics, and quantifies the outcomes by payoffs. The prisoner's dilemma (PD) describes situations in which it is profitable if everybody cooperates rather than defects (free-rides or cheats), but as cooperation is risky and defection is tempting, the expected outcome is defection. Nevertheless, some biological and social mechanisms can support cooperation by effectively transforming the payoffs. Here, we study the related phase transitions, which can be of first order (discontinous) or of second order (continuous), implying a variety of different routes to cooperation. After classifying the transitions into cases of equilibrium displacement, equilibrium selection, and equilibrium creation, we show that a transition to cooperation may take place even if the stationary states and the eigenvalues of the replicator equation for the PD stay unchanged. Our example is based on adaptive group pressure, which makes the payoffs dependent on the endogeneous dynamics in the population. The resulting bistability can invert the expected outcome in favor of cooperation.