| Summary: | Considering the uncertainty of game duration and periodic seasonal fluctuation, an <i>n</i>-player switched pollution-control differential game is modeled to investigate a sustainable and adaptive strategy for players. Based on the randomness of game duration, two scenarios are considered in this study. In the first case, the game duration is a random variable, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>f</mi></msub></semantics></math></inline-formula>, described by the shifted exponential distribution. In the second case, we assumed that players’ equipment is heterogeneous, and the <i>i</i>-th player’s equipment failure time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mrow><mi>f</mi><mi>i</mi></mrow></msub></semantics></math></inline-formula>, is described according to the shifted exponential distribution. The game continues until a player’s equipment breaks down. Thus, the game duration is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>f</mi></msub><mo>=</mo><mi>m</mi><mi>i</mi><mi>n</mi><mrow><mo>{</mo><msub><mi>T</mi><mrow><mi>f</mi><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>T</mi><mrow><mi>f</mi><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. To achieve the goal of sustainable development, an environmentally sustainable strategy and its corresponding condition are defined. By using Pontryagin’s maximum principle, a unique control solution is obtained in the form of a hybrid limit cycle, the state variable converges to a stable hybrid limit cycle, and the total payoff of all players increases and then converges. The results indicate that the environmentally sustainable strategy in the <i>n</i>-player pollution-control cooperative differential game with switches and random duration is a unique strategy that not only ensures profit growth but also considers environmental protection.
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