Optimal Intervention and Strategic Response in the Social Distancing Game

Abstract

Social distancing during the COVID-19 pandemic exposed two intertwined dilemmas. The first—balancing public health against socio-economic cost—is well known. The second—coordinating voluntary individual behavior with utilitarian policy—has received far less attention. Even when individuals are willing to contribute, aligning their actions with societal goals remains a structural challenge. We model social distancing as an investment problem, where reduced transmission comes at the cost of mutual contributions by individuals and public authority. While prior work often neglects individual rationality, we frame the problem as one of optimal incentive design: how should policy be shaped when individuals act voluntarily and respond strategically? We develop a differential game—the social distancing game—in which a central authority provides time-varying investment, and individuals adjust behavior in response to both policy and population-wide effects. Assuming ideal responsiveness and compartmental epidemic dynamics, we derive the optimal intervention analytically using Pontryagin’s maximum principle. We found that even under idealized conditions, public intervention cannot fully resolve inefficiencies from misaligned goals. This reveals structural limits of public investment-based distancing policies and delineates their scope, while also indicating when such measures are essential and when other interventions—such as vaccines or healthcare support—should take precedence.