In this paper, we present a game-theoretic model describing the voluntary social distancing during the spread of an epidemic. The payoffs of the agents depend on the social distancing they practice and on the probability of getting infected. We consider two types of agents, the vulnerable agents who have a small cost if they get infected, and the non-vulnerable agents who have a higher cost. For the modeling of the epidemic outbreak, a variant of the SIR model is considered, involving populations of susceptible, infected, and recovered persons of vulnerable and non-vulnerable types. The Nash equilibria of this social distancing game are studied. We then analyze the case where the players, desiring to achieve a low social inequality, pose a bound on the variance of the payoffs. In this case, we introduce a notion of Generalized Nash Equilibrium (GNE) for games with a continuum of players and characterize the GNE. We then present some numerical results. It turns out that inequality constraints result in a slower spread of the epidemic and an improved cost for the vulnerable players. Furthermore, it is possible that inequality constraints are beneficial for non-vulnerable players as well.

中文翻译：

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具有公平约束的纳什社交距离游戏：不平等厌恶如何影响流行病的传播

在本文中，我们提出了一个博弈论模型，描述了流行病传播过程中的自愿社交距离。代理人的回报取决于他们实践的社交距离和被感染的可能性。我们考虑两种类型的代理，如果被感染成本较低的易受攻击的代理，以及成本较高的非易受攻击的代理。对于流行病爆发的建模，考虑了 SIR 模型的一个变体，涉及易感人群、感染人群和康复人群，包括脆弱和非脆弱类型。研究了这种社会疏远博弈的纳什均衡。然后我们分析了希望实现低社会不平等的参与者对收益方差构成界限的情况。在这种情况下，我们为具有连续玩家的游戏引入了广义纳什均衡（GNE）的概念，并表征了 GNE。然后我们给出一些数值结果。事实证明，不平等的限制会导致流行病的传播速度减慢，并提高了弱势参与者的成本。此外，不平等约束也可能对非弱势参与者有利。