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

Fluctuations of Martingales and Winning Probabilities of Game Contestants