When should one stop the most exciting game? Sequential Inference for win-martingales
Prediction markets have become a prominent way of aggregating beliefs about binary future events, and their price processes are often interpreted as evolving win probabilities, or ``win-martingales.'' Motivated by this perspective and recent work on Aldous' ``most exciting game,'' we study when a decision maker should stop observing a win-martingale and make a decision about the outcome. In particular, we allow the true outcome to be revealed at a fixed finite horizon, as in a sports game or election. Under a general terminal loss and running cost, we reduce the Bayes risk to an optimal stopping problem for the win probability process. When the win-martingale is a diffusion and its volatility separates into a deterministic time factor and a state-dependent factor, a deterministic time change transforms the problem into one for a time-homogeneous diffusion with a generally time-inhomogeneous running cost. Under explicit structural assumptions, we obtain a complete free-boundary characterization of the solution to the stopping problem in both finite and infinite horizons without discounting. We prove smooth-fit and $C^1$ regularity of the value function, $C^1$ regularity of the optimal stopping boundaries before the horizon, and derive a nonlinear integral equation that characterizes the boundaries uniquely. Taken together, these results yield a common decision-theoretic framework and solution theory for a broad class of posterior dynamics that includes the Aldous, Bass, and binary sequential-inference martingales as special cases. Our analysis requires no temporal monotonicity of the running cost and therefore accommodates highly nonmonotone stopping boundaries. In particular, we exhibit an example in which the optimal boundary has no limit as the calendar-time horizon is approached.