Social Distancing Equilibria in Games under Conventional SI Dynamics

The mathematical characterization of social-distancing games in classical epidemic theory remains an important question, for their applications to both infectious-disease theory and memetic theory. We consider a special case of the dynamic finite-duration SI social-distancing game where payoffs are accounted using Markov decision theory without discounting, while distancing is constrained by threshold-linear running-costs, and the running-cost of perfect-distancing is finite. In this special case, we are able construct strategic equilibria satisfying the Nash best-response condition explicitly by integration. Our constructions are obtained using a new change of variables which simplifies the geometry and analysis. As it turns out, there are no singular solutions, and a time-dependent bang-bang strategy consisting of a wait-and-see phase followed by a lock-down phase is always the unique strategic equilibrium. We also show that in a delay strategy space the bang-bang Nash equilibrium is an ESS, and that the optimal public policy exactly corresponds with the game equilibrium. Social distancing only turns out to be a valuable tool in reducing outbreak burden when the outbreak response delay falls in a Goldilocks zone, determined by the speed of outbreak detection and the relative rolling cost of social distancing.