Paper List

期刊: ArXiv Preprint
发布日期: 2026-03-12
Mathematical BiologyGame Theory

Social Distancing Equilibria in Games under Conventional SI Dynamics

Department of Mathematics, Pennsylvania State University | Huck Institute of Life Sciences, Pennsylvania State University

Connor D. Olson, Timothy C. Reluga
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30秒速读

IN SHORT: This paper solves the core problem of proving the existence and uniqueness of Nash equilibria in finite-duration SI epidemic games, showing they are always bang-bang strategies.

核心创新

  • Methodology Introduces a novel change of variables that simplifies the geometry and analysis of the SI social-distancing game, enabling explicit integration and closed-form solutions.
  • Theory Proves that for the specified SI game with threshold-linear costs, the unique strategic equilibrium is always a time-dependent bang-bang strategy (wait-then-lockdown), with no singular solutions.
  • Theory Demonstrates that in the restricted strategy space of two-phase (off-on) strategies, the bang-bang Nash equilibrium is also an Evolutionarily Stable Strategy (ESS), and that it coincides with the socially optimal policy, eliminating free-riding.

主要结论

  • For all parameter tuples (m, I0, tf), there exists one and only one equilibrium point x* (Theorem 10), proving uniqueness in the SI game.
  • The equilibrium strategy is explicitly given by x*(m, I0, tf) = m - 1 - W((1/I0 - 1)e^{m-1-tf}) for intermediate parameters, utilizing the Lambert W function (Eq. 13).
  • The optimal public policy (minimizing population disutility ℰ(x̄)) exactly corresponds with the individual Nash equilibrium strategy (Eq. 18), showing no conflict between individual and social optima in this model.
研究空白： A key unresolved question in epidemic game theory has been whether rational behaviors (Nash equilibria) captured by these models are unique and under what conditions uniqueness fails. Prior work, like Reluga (2013), provided incomplete proofs or was not easily generalizable.

摘要: 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.