Game Theory of Social Distancing in Response to an Epidemic

During an epidemic, the rational thing to do — the thing that actually maximizes your own wellbeing — is not to follow public health advice. Let that sit for a second. A mathematician named Timothy Reluga proved it with a differential equation, and the gap between what is optimal for you and what is optimal for everyone is the whole story. The problem Reluga is solving is older than any particular disease. When a pathogen spreads through a population, every individual faces a personal calculation: how much of my time, money, convenience, and social life am I willing to give up to reduce my own risk of infection? Social distancing — reducing contacts, working from home, skipping gatherings — is costly. People adopt it only when the personal incentives outweigh those costs. They are not doing a calculation for the community. They are doing a calculation for themselves. To formalize this, Reluga starts with the classical Susceptible-Infected-Recovered model, or SIR. The population is split into three groups: people who can still catch the disease, people who are currently infectious, and people who have recovered or died. The epidemic evolves continuously over time as people move from susceptible to infected to removed. This is standard epidemiology. What Reluga adds is a layer of strategic choice sitting on top of those dynamics — and that addition changes everything. The framework he builds is called a differential game. The word "differential" signals that this isn't a one-shot decision; it's a game played continuously over time, where each person chooses a different level of distancing effort at every moment. Think of it as each individual tracing a curve of behavior through the entire arc of the epidemic. At each instant, you decide how hard to distance. Your choice affects your own risk. But because you're one person in a large population, your choice doesn't meaningfully change the epidemic trajectory — that's determined by everyone else's average behavior. So you take the population strategy as given and optimize for yourself. The individual's payoff has a clean structure. It's the expected present value of two things: the stream of daily costs from investing in distancing and the risk of paying the full cost of infection. The effectiveness of distancing is captured by a relative-risk function — call it s of c, where c is your distancing effort. When effort is zero, your relative risk is roughly one. As effort increases, your risk drops toward zero, but with diminishing returns. Doubling your effort doesn't halve your risk; the curve bends. A key parameter in the whole analysis is what Reluga calls m — the maximum efficiency of social distancing. Think of m as an upper bound on how much protection you can buy per unit of cost. Reluga finds a clean threshold: if m is at or below e, roughly 2.718, social distancing is effectively worthless in equilibrium. No matter what R-naught is — R-naught being the average number of new infections caused by one person in a fully susceptible population — if your distancing tools are weak enough, rational individuals simply don't use them. The equilibrium concept Reluga targets is called a subgame-perfect Nash equilibrium. A Nash equilibrium is a strategy where no single individual can lower their expected cost by changing behavior unilaterally while everyone else holds steady. "Subgame perfect" means this best-response property holds at every possible state the epidemic could pass through — not just at the start, but throughout. To find these equilibria, Reluga solves a boundary-value problem numerically, integrating backward from the end of the epidemic. Now here's the result that lands hardest. In the infinite-horizon case — no vaccine on the horizon, epidemic running its full course — optimal individual distancing never recovers more than 30 percent of the per-capita cost of infection. Thirty percent is the ceiling. And that ceiling is reached only near a specific value of R-naught, around 2. Why is R-naught equal to 2 the sweet spot? The logic is intuitive once you see it. If R-naught is very low, the epidemic burns itself out quickly regardless of what anyone does — there's not much cost to recover. If R-naught is very high, distancing can't put enough of a dent in transmission to justify its costs — the epidemic steamrolls through anyway. It's the middle range, around 2, where individuals have both enough threat to motivate action and enough leverage to make distancing matter. Influenza sits close to this range. So does the original strain of SARS-CoV-2. The picture changes substantially when a vaccine is coming. Reluga also studies the finite-horizon version of the game: a vaccine will arrive at some future time and instantly protect all remaining susceptibles. The question becomes how the total epidemic cost depends on when that vaccine shows up. For a specific parameterization — maximum distancing efficiency m around 20, R-naught around 3, and a tiny initial outbreak — the numbers are striking. If vaccination arrives 8.6 transmission generations after the epidemic begins, individuals save 50 percent of expected per-capita infection cost compared to no behavioral response. If vaccination arrives just 6.5 generations in, individuals save 80 percent. Two fewer generations of waiting, and the benefit nearly doubles. This points to something important for policy. Better distancing tools and better epidemic detection don't just reduce infections in the present — they buy time. They slow the epidemic enough that a vaccine arriving later can still prevent a large fraction of infections, rather than landing after the epidemic has already peaked. Reluga frames this explicitly: improved distancing efficiency and smaller initial outbreaks both lengthen the window of opportunity during which vaccination substantially reduces total cost. There's a race between epidemic spread and vaccine deployment, and behavioral tools are what slow the clock on the epidemic side. Now for the part that game theory was built for. The equilibrium behavior that emerges from rational individuals minimizing their own costs is not the same strategy that would minimize total per-capita costs across the community. This is not a subtle difference. The mechanism is an externality: when you distance, you reduce not only your own risk but also the risk you pose to others. That community benefit doesn't show up in your private payoff calculation. So you systematically under-invest in distancing from the community's perspective. The Nash equilibrium is self-consistent — no individual has an incentive to deviate — but it is collectively inefficient. Reluga shows that the community-optimal strategy always costs less per capita than the equilibrium strategy. The gap between the two is largest for moderate values of R-naught, near 1, and narrows as R-naught grows. Structurally, this is the same architecture as a Prisoner's Dilemma: individually rational choices produce a collectively worse outcome than coordination would. What can't the model see? Reluga is straightforward about this. The SIR framework assumes a homogeneous, well-mixed population — everyone has equal contact rates, there's no network structure, no geography, no age heterogeneity. Real epidemics spread on contact networks where some people have dozens of connections and others have very few, and that structure changes dynamics in ways the model can't capture. Vaccination is treated as an instantaneous event rather than a gradual rollout. Reluga points toward several extensions: structured contact networks, infection-age models with delay equations, economic capital dynamics, and time-varying vaccine rollout. These are real limitations. But the core result doesn't depend on those refinements. What Reluga establishes, with precision, is that even perfectly rational individuals facing a real epidemic will under-invest in social distancing — and we can calculate exactly how large that shortfall is. The 30 percent ceiling without a vaccine. The shift to 50 or 80 percent with one on the way. The efficiency threshold at e. These are not rough intuitions. They are quantitative answers to the question that epidemiologists and policymakers face every time a new pathogen emerges: how much can we expect people to protect each other when they're only trying to protect themselves? The answer, it turns out, is less than we need. And knowing exactly how much less is where the work of closing that gap can begin. This lecture was created by ennepō. Go to https://ennepo.ai to Discover, Create and Follow the latest research in your field. Read when you can. Listen when you want to.