Duality between cooperation and defection in the presence of tit-for-tat in replicator dynamics

TL;DR

This paper explores the replicator dynamics of cooperation, defection, and tit-for-tat strategies in the prisoner's dilemma, revealing a duality that influences the emergence of cooperation and how mutation affects these dynamics.

Contribution

It introduces a self-duality property in the replicator dynamics involving tit-for-tat, unconditional cooperation, and defection, and analyzes how mutation impacts cooperation.

Findings

Dynamics is self-dual under time reversal and strategy exchange.

Tit-for-tat can equalize strategy fractions despite defection dominance.

Mutation favors cooperation when cost-to-benefit ratio is low.

Abstract

The prisoner's dilemma describes a conflict between a pair of players, in which defection is a dominant strategy whereas cooperation is collectively optimal. The iterated version of the dilemma has been extensively studied to understand the emergence of cooperation. In the evolutionary context, the iterated prisoner's dilemma is often combined with population dynamics, in which a more successful strategy replicates itself with a higher growth rate. Here, we investigate the replicator dynamics of three representative strategies, i.e., unconditional cooperation, unconditional defection, and tit-for-tat, which prescribes reciprocal cooperation by mimicking the opponent's previous move. Our finding is that the dynamics is self-dual in the sense that it remains invariant when we apply time reversal and exchange the fractions of unconditional cooperators and defectors in the population. The duality implies that the fractions can be equalized by tit-for-tat players, although unconditional cooperation is still dominated by defection. Furthermore, we find that mutation among the strategies breaks the exact duality in such a way that cooperation is more favored than defection, as long as the cost-to-benefit ratio of cooperation is small.