Defensive withdrawal or benefit radiation: Two regimes of cooperation under adaptive participation

Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University - Changsha, Hunan 410081, China and Key Laboratory of Applied Statistics and Data Science, School of Mathematics and Statistics, Hunan Normal University - Changsha, Hunan 410081, China

Received: 27 April 2025
Accepted: 24 July 2025

Abstract

Cooperation is ubiquitous in biological and social systems, yet it faces persistent challenges from free-riding behavior. While voluntary participation has been recognized as a key mechanism for sustaining cooperation, existing studies predominantly assume static decision-making rules, namely loner strategy or fixed participation probability, overlooking the dynamic nature of human participation strategies. To address this gap, we employ the Bush-Mosteller reinforcement learning algorithm to model aspiration-driven adaptive participation in public goods games. Our results reveal that cooperation peaks when the aspiration level equals the potential maximum payoff of cooperators, with distinct evolutionary mechanisms emerging on either side of this critical value. Below the threshold, cooperators form self-organizing defensive barriers through strategic withdrawal, effectively mitigating exploitation risks. Above the threshold, enhanced reciprocity within cooperative clusters generates positive network externalities, enabling cooperative expansion through benefit radiation effects. These findings provide novel insights into how adaptive participation strategies shape the evolution of cooperation, highlighting the importance of dynamic decision-making processes in social dilemmas.