Social Distancing Equilibria in Games under Conventional SI Dynamics

TL;DR

This paper analyzes social-distancing game equilibria under SI epidemic dynamics, showing a unique bang-bang strategy as the Nash equilibrium and its alignment with optimal public policy, using explicit construction and new variable transformations.

Contribution

It provides an explicit construction of Nash equilibria in social-distancing games with SI dynamics, introducing a new change of variables for simplified analysis and proving the equilibrium's uniqueness and stability.

Findings

Unique bang-bang equilibrium strategy identified

Equilibrium strategy is evolutionarily stable (ESS)

Optimal public policy matches the equilibrium strategy

Abstract

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 with zero-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 restricted strategy space the bang-bang Nash equilibrium is an ESS, and that the optimal public policy exactly corresponds with the equilibrium strategy.