Convergence of Replicator Dynamics in the Repeated Prisoner's Dilemma with Restarts

2026-06-17

AI Summary

In the repeated Prisoner’s Dilemma with a restart mechanism, self-interested agents struggle to sustain cooperation spontaneously. This study investigates the replicator dynamics of populations employing strategy sequences of length \( m \) under trigger-based restarts. By modeling the interaction as a parameterized normal-form game and combining stability analysis with combinatorial counting, the work identifies structural properties of the “trial period” necessary for cooperative strategies to be evolutionarily stable: suboptimal strategies featuring longer trial periods possess larger basins of attraction and are thus more readily adopted by the population. The paper provides exact convergence guarantees within a finite strategy space, derives explicit parameter conditions under which cooperation is stable, and offers an analytical expression for the number of such stable cooperative strategies.

Abstract

We investigate a population of self-interested agents playing a repeated Prisoner's Dilemma under the trigger-restart mechanism. Under such a mechanism, agents play a sequence of symmetric games with their partner, and restart the interaction if their actions disagree. Our work focuses on the convergence of replicator dynamics in a well-mixed population of agents, where the emergence of cooperation is challenged by the individual incentive for exploitation. Formulating the corresponding parametrised normal-form game, with agents each adopting a length-m strategy sequence, we show that increasing the strategy length enables cooperation to emerge and stabilise. We provide exact convergence guarantees for restricted strategy lengths and, in the general payoff configuration, provide the necessary parametric conditions for the stability of cooperative strategies. By deriving an exact formula for the number of stable sequences, we find structural properties necessary for stability, as agents must learn to initially defect - the so-called "hazing period" - before cooperating indefinitely. Our analysis shows that, while optimal cooperative sequences exist, agents favour less-optimal sequences with a longer hazing period, which possess larger basins of attraction.

Problem

• Repeated Prisoner's Dilemma
• Replicator Dynamics
• Cooperation
• Restart Mechanism
• Strategy Convergence

Innovation

• replicator dynamics
• repeated Prisoner's Dilemma
• trigger-restart mechanism
• strategy length
• hazing period