The mathematical characterization of social-distancing games in classical epidemic theory remains an important question, for their applications to both infectious-disease theory and memetic theory. We consider a special case of the dynamic finite-duration SI social-distancing game where payoffs are accounted using Markov decision theory with zero-discounting, while distancing is constrained by threshold-linear running-costs, and the running-cost of perfect-distancing is finite. In this special case, we are able construct strategic equilibria satisfying the Nash best-response condition explicitly by integration. Our constructions are obtained using a new change of variables which simplifies the geometry and analysis.As it turns out, there are no singular solutions, and a time-dependent bang-bang strategy consisting of a wait-and-see phase followed by a lock-down phase is always the unique strategic equilibrium. We also show that in a restricted strategy space the bang-bang Nash equilibrium is an ESS, and that the optimal public policy exactly corresponds with the equilibrium strategy.

翻译：经典流行病理论中社会距离博弈的数学刻画仍然是一个重要问题，因其在传染病理论和模因理论中均具有应用价值。我们研究动态有限持续时间SI社会距离博弈的一个特例：其中收益通过零贴现的马尔可夫决策理论进行核算，距离约束采用阈值线性运行成本，且完全距离的运行成本为有限值。在此特例中，我们能够通过积分显式构造满足纳什最优响应条件的策略均衡。我们的构造采用了一种新的变量替换方法，从而简化了几何结构与分析过程。结果表明，该模型不存在奇异解，且由观望阶段与封锁阶段组成的时变bang-bang策略始终是唯一的策略均衡。我们进一步证明，在受限策略空间中该bang-bang纳什均衡是演化稳定策略，且最优公共政策恰好与该均衡策略相一致。