Dynamic Games of Social Distancing during an Epidemic: Analysis of Asymmetric Solutions

TL;DR

This paper models social distancing during an epidemic as a dynamic game with asymmetric solutions, analyzing how individual perceptions and behaviors influence disease spread and equilibrium outcomes.

Contribution

It introduces a dynamic game framework with asymmetric solutions for social distancing, characterizes Nash equilibria, and reduces computational complexity through monotonicity properties.

Findings

Players exhibit diverse behaviors even with identical parameters.

Numerical results show parameter impacts on policies and costs.

Existence of Nash equilibrium is established and characterized.

Abstract

Individual behaviors play an essential role in the dynamics of transmission of infectious diseases, including COVID--19. This paper studies a dynamic game model that describes the social distancing behaviors during an epidemic, assuming a continuum of players and individual infection dynamics. The evolution of the players' infection states follows a variant of the well-known SIR dynamics. We assume that the players are not sure about their infection state, and thus they choose their actions based on their individually perceived probabilities of being susceptible, infected or removed. The cost of each player depends both on her infection state and on the contact with others. We prove the existence of a Nash equilibrium and characterize Nash equilibria using nonlinear complementarity problems. We then exploit some monotonicity properties of the optimal policies to obtain a reduced-order characterization for Nash equilibrium and reduce its computation to the solution of a low-dimensional optimization problem. It turns out that, even in the symmetric case, where all the players have the same parameters, players may have very different behaviors. We finally present some numerical studies that illustrate this interesting phenomenon and investigate the effects of several parameters, including the players' vulnerability, the time horizon, and the maximum allowed actions, on the optimal policies and the players' costs.