[Paper Review] Social Distancing Equilibria in Games under Conventional SI Dynamics

Connor D Olson, Timothy C. Reluga|arXiv (Cornell University)|Mar 12, 2026

Game Theory and Applications0 citations

TL;DR

The paper analyzes a two-phase bang-bang social-distancing strategy in an SI epidemic game, proving a unique Nash equilibrium and its equivalence to the socially optimal outcome under a zero-discount, threshold-linear cost framework.

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.

Motivation & Objective

• Motivate and formalize social-distancing decisions within an SI epidemic game framework.
• Derive a tractable special-case (zero discounting, constant infection cost, threshold-linear distancing) enabling explicit Nash equilibria.
• Show that the equilibrium is unique and constitutes an ESS in a restricted strategy space.
• Demonstrate that the Nash equilibrium coincides with the socially optimal outcome under the model.
• Characterize how game duration, initial infection level, and distancing efficiency shape equilibrium distancing.

Proposed method

• Model the population with SI dynamics and a social-distancing cost function.
• Impose a threshold-linear reduction in transmission: sigma(z) = (1 - m z)^+ and set C_i constant, h = 0.
• Derive the dynamic game D(c, c̄) and reformulate using two-phase delay strategies with a bang-bang structure.
• Apply Pontryagin’s Maximum Principle to obtain a Filippov system for best-response strategies.
• Transform the adjoint variable V into a decision-potential Phi = I(V+1) to simplify the phase-plane analysis.
• Prove there is a unique Nash equilibrium c* and that it is subgame-perfect; provide explicit expressions for the equilibrium via a transcendental equation and Lambert W when applicable.

Figure 1 : The restricted disutility surface $\mathcal{D}(x,\overline{x})$ (left), the relative restricted disutility $\hat{\mathcal{D}}(x,\overline{x})$ (center), and the emblematic disutility $\mathcal{E}(\overline{x})$ (right) when $t_{f}=6$ , $m=6$ , and initial condition $I_{0}=0.02$ . The stra

Experimental results

Research questions

• RQ1 Does the SI social-distancing game have a unique Nash equilibrium under zero discounting and threshold-linear costs?
• RQ2 Is the equilibrium strategy bang-bang (wait-and-see followed by lockdown) and time-dependent?
• RQ3 Under restricted strategy spaces, is the Nash equilibrium ESS and does it align with the socially optimal outcome?
• RQ4 How do game duration, initial infection level, and distancing efficiency m influence the equilibrium level of distancing?
• RQ5 Can the equilibrium be characterized by explicit analytic forms or closed communities of solutions (e.g., via Lambert W)?

Key findings

• There exists a single equilibrium point x* in the restricted two-phase delay strategy space for all parameter values (m, I0, tf).
• The Nash equilibrium is a time-dependent bang-bang strategy: wait a period then implement full-duration distancing.
• In the restricted space, this bang-bang Nash equilibrium is an ESS (evolutionarily stable strategy).
• The Nash equilibrium coincides with the socially optimal behavior in this SI model under the stated assumptions.
• A closed-form representation expresses x* via a transcendental equation; in special cases, x* reduces to 0 or tf depending on I0 and tf relative to m.
• As tf grows, x* approaches m−1 with a correction term; asymptotics and special-case formulas are provided (Lambert W in the intermediate regime).

Figure 2 : Colorized contour images of the (left) equilibria duration of social distancing $x^{*}$ of the restricted disutility ( 11 ) as a function of the game-duration ( $t_{f}$ ) and the initial proportion infected ( $I_{0}$ ) depends when the linear distancing efficiency $m=6$ . As games get lon