Fluctuations of Martingales and Winning Probabilities of Game Contestants

Abstract

Within a contest there is some probability M_i(t) that contestant i will be the winner, given information available at time t, and M_i(t) must be a martingale in t. Assume continuous paths, to capture the idea that relevant information is acquired slowly. Provided each contestant's initial winning probability is at most b, one can easily calculate, without needing further model specification, the expectations of the random variables N_b = number of contestants whose winning probability ever exceeds b, and D_{ab} = total number of downcrossings of the martingales over an interval [a,b]. The distributions of N_b and D_{ab} do depend on further model details, and we study how concentrated or spread out the distributions can be. The extremal models for N_b correspond to two contrasting intuitively natural methods for determining a winner: progressively shorten a list of remaining candidates, or sequentially examine candidates to be declared winner or eliminated. We give less precise bounds on the variability of D_{ab}. We formalize the setting of infinitely many contestants each with infinitesimally small chance of winning, in which the explicit results are more elegant. A canonical process in this setting is the Wright-Fisher diffusion associated with an infinite population of initially distinct alleles; we show how this process fits our setting and raise the problem of finding the distributions of N_b and D_{ab} for this process.