Grouping promotes both partnership and rivalry with long memory in direct reciprocity

TL;DR

This study explores how longer-memory strategies, especially friendly rivals, influence cooperation and rivalry in iterated Prisoner's Dilemma, revealing that group structure and memory length significantly promote cooperation.

Contribution

It demonstrates that in group-structured populations, longer-memory strategies enable friendly rivals to dominate, enhancing cooperation levels beyond what is possible with shorter memory strategies.

Findings

Memory length has little effect in well-mixed populations.

Friendly rivals become dominant in group-structured populations with longer memory.

Cooperation levels nearly reach maximum with memory-three strategies in groups.

Abstract

Biological and social scientists have long been interested in understanding how to reconcile individual and collective interests in iterated Prisoner's Dilemma. Many effective strategies have been proposed, and they are often categorized into one of two classes, `partners' and `rivals.' More recently, another class, `friendly rivals,' has been identified in longer-memory strategy spaces. Friendly rivals qualify as both partners and rivals: They fully cooperate with themselves, like partners, but never allow their co-players to earn higher payoffs, like rivals. Although they have appealing theoretical properties, it is unclear whether they would emerge in evolving population because most previous works focus on memory-one strategy space, where no friendly rival strategy exists. To investigate this issue, we have conducted large-scale evolutionary simulations in well-mixed and group-structured populations and compared the evolutionary dynamics between memory-one and memory-three strategy spaces. In a well-mixed population, the memory length does not make a major difference, and the key factors are the population size and the benefit of cooperation. Friendly rivals play a minor role because being a partner or a rival is often good enough in a given environment. It is in a group-structured population that memory length makes a stark difference: When memory-three strategies are available, friendly rivals become dominant, and the cooperation level nearly reaches a maximum, even when the benefit of cooperation is so low that cooperation would not be achieved in a well-mixed population. This result highlights the important interaction between group structure and memory lengths that drive the evolution of cooperation.