Comparing reactive and memory-one strategies of direct reciprocity

TL;DR

This paper compares the effectiveness of reactive and memory-one strategies in promoting cooperation through direct reciprocity, considering the impact of stochasticity and strategic complexity in finite populations.

Contribution

It systematically analyzes how stochastic and deterministic strategies influence cooperation, highlighting the conditions under which stochasticity enhances cooperative behavior.

Findings

Stochastic reactive strategies promote cooperation with small costs.

Deterministic memory-one strategies like win-stay, lose-shift are most effective with small costs.

Stochasticity can augment cooperation in memory-one strategies with large costs.

Abstract

Direct reciprocity is a mechanism for the evolution of cooperation based on repeated interactions. When individuals meet repeatedly, they can use conditional strategies to enforce cooperative outcomes that would not be feasible in one-shot social dilemmas. Direct reciprocity requires that individuals keep track of their past interactions and find the right response. However, there are natural bounds on strategic complexity: Humans find it difficult to remember past interactions accurately, especially over long timespans. Given these limitations, it is natural to ask how complex strategies need to be for cooperation to evolve. Here, we study stochastic evolutionary game dynamics in finite populations to systematically compare the evolutionary performance of reactive strategies, which only respond to the co-player's previous move, and memory-one strategies, which take into account the own and the co-player's previous move. In both cases, we compare deterministic strategy and stochastic strategy spaces. For reactive strategies and small costs, we find that stochasticity benefits cooperation, because it allows for generous-tit-for-tat. For memory one strategies and small costs, we find that stochasticity does not increase the propensity for cooperation, because the deterministic rule of win-stay, lose-shift works best. For memory one strategies and large costs, however, stochasticity can augment cooperation.