Collective reinforcement learning and human–machine cooperation in multi-state stochastic games

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With the growing deployment of artificial intelligence agents in collaborative systems, understanding how they jointly shape cooperative behavior with humans under dynamic environmental conditions becomes essential. Stochastic games provide a natural framework for capturing how environmental feedback shapes behavioral choices and supports the emergence of cooperation. Building on this framework, we incorporate collective reinforcement learning dynamics (CRLD) into a multi-state stochastic game and develop a mixed human–machine population model to examine the emergence and stability of cooperation. Unlike existing models that mainly consider homogeneous learning populations or externally determined environmental states, this framework explicitly captures the feedback interaction among human behavioral responses, environmental state transitions, and machine learning dynamics. Our results show that environmental vulnerability, public-goods incentives, human cooperative inclination and its population share, as well as machine discounting, jointly determine the evolutionary path of cooperation. Cooperative humans increase the likelihood of favorable environmental states, lowering the threshold for machines to learn cooperative strategies. In contrast, defect-prone humans can disrupt numerically stable cooperative states and induce cooperation collapse. Increasing the proportion of cooperative humans also reduces the critical discount factor required for machines to sustain cooperation. These findings demonstrate that human behavior reshapes environmental state transitions and alters machine value evaluation, exerting essential influence on learning and strategy evolution. Overall, this study reveals the mechanisms that sustain cooperation in human–machine systems with variable environmental states and provides theoretical insights relevant to coordination in mixed-agent systems.