The survival of the weakest in a biased donation game

TL;DR

This study explores how biased Tit-for-Tat strategies influence cooperation dynamics in social dilemmas, revealing a counterintuitive 'survival of the weakest' phenomenon in structured populations.

Contribution

It introduces a biased Tit-for-Tat model with independent bias parameters, demonstrating complex phase behavior and the emergence of a 'hidden T phase' where weaker strategies dominate.

Findings

A 'hidden T phase' emerges with small T-bias and large C-bias.

Weaker T strategies can dominate due to non-transitive ecological mechanisms.

Structured populations are crucial for the observed phenomena.

Abstract

Cooperating first then mimicking the partner's act has been proven to be effective in utilizing reciprocity in social dilemmas. However, the extent to which this, called Tit-for-Tat strategy, should be regarded as equivalent to unconditional cooperators remains controversial. Here, we introduce a biased Tit-for-Tat (T) strategy that cooperates differently toward unconditional cooperators (C) and fellow T players through independent bias parameters. The results show that, even under strong dilemmas in the donation game framework, this three-strategy system can exhibit diverse phase diagrams on the parameter plane. In particular, when T-bias is small and C-bias is large, a ``hidden T phase'' emerges, in which the weakest T strategy dominates. The dominance of the weakened T strategy originates from a counterintuitive mechanism characterizing non-transitive ecological systems: T suppresses its relative fitness to C, rapidly eliminates the cyclic dominance clusters, and subsequently expands slowly to take over the entire population. Analysis in well-mixed populations confirms that this phenomenon arises from structured populations. Our study thus reveals the subtle role of bias regulation in cooperative modes by emphasizing the ``survival of the weakest'' effect in a broader context.