Nash social distancing games with equity constraints: How inequality aversion affects the spread of epidemics

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.