  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.
