Replicator dynamics with feedback-evolving games in heterogeneous populations

Understanding the dynamics of cooperation in public goods games is critical for enhancing collaborative efforts across various domains, yet existing literature often overlooks the complexities introduced by heterogeneous participants. In order to characterize the heterogeneity of populations, we divide individuals into knowledgeable individuals and ordinary individuals, and the games organized by different focal individuals may have different efficiency. In this study, we introduce feedback evolution games where strategies coevolve with the multiplication factor of the defector. Our results indicate that full cooperation is frequently unfeasible in crowdsourcing contexts, consistent with real-world observations. Moreover, for organizers, the increase in the proportion of knowledgeable individuals carries not only direct costs but also indirect costs that may arise from defectors. Importantly, we highlight that the speed of feedback updates plays a crucial role in fostering group cooperation, with a sufficiently high update speed being essential to overcoming social dilemmas. These insights not only advance our understanding of strategic decision-making in collaborative environments but also provide actionable implications for promoting cooperation in complex social systems.