In this paper, we present a game-theoretic model describing 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 non-vulnerable agents with a small cost if they get infected and the vulnerable agents with a higher cost. For the modeling of the epidemic outbreak, we consider a variant of the SIR (Susceptible-Infected-Removed) model involving populations of susceptible, infected, and removed persons of vulnerable and non-vulnerable types. The Nash equilibria of this social distancing game are studied. The main contribution of this work is the analysis of 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 and characterize a notion of Generalized Nash Equilibrium (GNE) for games with a continuum of players and provide characterizations for this type of GNE. It turns out that often there is a continuum of GNE. However, among the GNE, for a given value of the variance bound, there is one that Pareto dominates the others. We also provide conditions under which a more restrictive variance constraint benefits all the game participants. Furthermore, we describe a bargaining-based algorithm for choosing the variance constraint. Through numerical studies, we show that inequality constraints result in a slower spread of the epidemic and an improved cost for the vulnerable players. Furthermore, we present some examples where inequality constraints are also beneficial for non-vulnerable players.

在本文中，我们提出了一个博弈论模型，描述了流行病传播期间的自愿社交距离。代理人的回报取决于他们所实行的社交距离和被感染的可能性。我们考虑两种类型的代理：非易受攻击的代理如果被感染则成本较低，而易受攻击的代理则成本较高。对于流行病爆发的建模，我们考虑了 SIR（易感-感染-移除）模型的变体，涉及易感人群、感染人群和易感人群和非易感人群。研究了这种社交距离博弈的纳什均衡。这项工作的主要贡献是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并描述了广义纳什均衡 (GNE) 的概念，并为这种类型的 GNE 提供了描述。事实证明，通常有一个 GNE 的连续体。然而，在 GNE 中，对于给定的方差界限值，存在一个帕累托支配其他值。我们还提供了条件，在这些条件下，更严格的方差约束使所有游戏参与者受益。此外，我们描述了一种基于讨价还价的算法来选择方差约束。通过数值研究，我们表明不平等约束会导致流行病传播速度减慢，并提高弱势参与者的成本。此外，我们提出了一些例子，其中不平等约束也有利于非弱势玩家。