Safety Critical Thresholds and Stability Analysis of Collective Cooperation with Heterogeneous Contributions

Abstract

This paper addresses the problem of individual defection (voluntary exit) in collective cooperation. Based on an upper-bounded S-shaped contribution-payoff function, we construct a Homogeneous Equal-Contribution Model and a One-Strong-n-Weak Heterogeneous Model with a high-contribution agent. We strictly define and derive three core safety critical thresholds for collective cooperation: the failure threshold of equal allocation, the failure threshold of proportional allocation, and the full stability threshold. Through rigorous algebraic derivation and functional property analysis, we provide explicit closed-form solutions and solving methods for all thresholds, prove the self-consistency of the model, and reveal the mechanisms by which individual heterogeneity, group size, contribution intensity, and task attributes affect the stability of collective cooperation. This paper provides a complete and unified mathematical theoretical framework for the stability analysis of collective cooperation.