Social Distancing Equilibria in Games under Conventional SI Dynamics

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.

Figures (8)

• 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 strategy at the saddle point ($x^* \approx 2.87$) of the center plot is the Nash equilibrium among all off-on two-phase strategies and is also the minimizer of the emblematic disutility (right), so is also the social optimal.
• Figure 2: Colorized contour images of the (left) equilibria duration of social distancing $x^*$ of the restricted disutility \ref{['eq:DSISDG']} 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 longer, the duration of distancing increases from $0$ to about $5$. The white line divides the surface into three regions depending on whether no distancing is used ($x^*=0$), distancing is used for the full game-duration ($x^*=t_f$), or some intermediate amount is used. The equilibrium per-capita burden (right, Eq. \ref{['eq:burden']}) increases monotonically as both the game duration and the initial case count increase.
• Figure 3: A dimensional example of equilibrium (bottom) and corresponding burden (top, Eq. \ref{['eq:burden']}). For a community of 10,000 people the epidemic is detected when the first $\hat{I}_0 = 20$ are infected. The epidemic is initially doubling in size each 1 week ($=\ln 2 / \hat{\beta} \hat{N}$), infected people expect to lose $1,000 ($=C_i$) from the infection, but are not willing to pay more than $10 ($=1/\hat{m}$) a week in social distancing costs, then social-distancing can only reduce the burden significantly if the vaccine is rolled out in less than 100 weeks.
• Figure 4: Epidemic burden as functions of $t_f$ and $m$ for given values of $I_0$. Dotted black line represents $t_f = m-1$. The case of $I_0=1$ (left) is gameless, as infection-risk is constant and independent of player actions. We see the burden increases monotonically as duration $t_f$ get longer but decreases monotonically as social-distancing is made more efficient. In the case of $I_0=10^{-4}$ (right), burden is small for short games because infection risk never becomes substantial, but for long games, the burden only decreases significantly once $m/t_f \approx 1$.
• Figure 5: Colorized contour image of the reduction in burden by social distancing ($\Delta B^*$, Eq. \ref{['eq:improve']}) (left), and the relative reduction in burden (right) when $m=6$. The plots clearly show that for this epidemic model, the public-health value of social-distancing is concentrated at intermediate durations of time between detection and mass-vaccination. The dashed red line is our approximation of the time $t_f^@(m,I_0)$ (Eq. \ref{['eq:tpeak']}) when the improvement-over-indifference $\Delta B^*$ is maximal.

Theorems & Definitions (18)

• Lemma 1
• proof
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6