Driving key nodes to learn cooperation in social dilemma

This paper presents a method that drives key nodes with minimal control cost to influence the agents in social dilemmas to learn to cooperate. Cooperation exists widely in human society and nature. Discovering the mechanisms that promote the evolution of cooperation has always been a concern across disciplines. In this context, continuous action social dilemma games are a fitting model to explore individual interactions and the spread of cooperative behaviors in social networks. This paper proposes a novel framework that applies optimal control theory to steer the evolution of cooperation within these games. Existing research lacks a definitive method for judicious selecting critical nodes capable of guiding the entire system toward a desired state with minimal cost. To bridge this gap, our framework offers a control mechanism that influences cooperative behavior and determines the optimal number and identity of nodes to be controlled, thereby maximizing outcomes. Building upon continuous action social dilemma games in social networks, we formulate a set of coupled Hamilton-Jacobi-Bellman (HJB) equations, employ a value iteration reinforcement learning (RL) algorithm to solve for the HJB function, and demonstrate the convergence of the proposed algorithm. Furthermore, we conduct a comprehensive qualitative and quantitative investigation into the selection of controlled nodes across diverse networks, focusing on optimizing control performance while minimizing associated costs. Our research substantiates the efficacy of the proposed algorithm across a spectrum of social networks, affirming its utility and potential impact in promoting the evolution of cooperative behavior.