In an epidemic, when should a rational individual start social distancing? Not when told to, but when the personal cost of infection exceeds the personal cost of isolation. Lazebnik & Bhatt (arXiv:2603.12107) solve this game-theoretically for an SI model with threshold-linear distancing costs and find a sharp answer: the unique Nash equilibrium is a bang-bang strategy. Do nothing, then lock down completely. No gradual increase. No partial measures. Wait, then switch.
The wait-and-see phase has a precise duration determined by the infection probability, the cost of distancing, and the remaining time horizon. Below the threshold, the expected cost of catching the disease is less than the cost of avoiding it. Above the threshold — which arrives as cases compound exponentially — the calculus flips. The optimal individual response is binary.
More surprising: the socially optimal public policy exactly coincides with the equilibrium strategy. This is unusual. In most epidemiological games, individuals free-ride on others' distancing, and the Nash equilibrium undersupplies protection relative to the social optimum. Here, under the specific cost structure, individual rationality and collective welfare align.
The alignment depends on the cost being threshold-linear — you pay nothing for small distancing and linearly for serious distancing. Under convex or concave costs, the alignment breaks. The result is a knife-edge: the one cost structure where selfish and cooperative behavior produce the same policy is also the structure where the equilibrium is bang-bang.
The mathematical elegance (explicit integration via a change of variables, no singular arcs) reflects the structural simplicity. When the cost of partial measures is either zero or proportional, there's no point in being half-committed. The rational response to an exponential threat under linear costs is a step function.