## Fluctuations of martingales and winning probabilities of game contestants.(English)Zbl 1300.60060

Summary: 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 that 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 $$N_b = \text{number}$$ of contestants whose winning probability ever exceeds $$b$$, and $$D_{ab} = \text{total}$$ number of downcrossings of the martingales over an interval $$[a,b]$$. The distributions of $$N_b$$ and $$D_{ab}$$ 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.

### MSC:

 60G44 Martingales with continuous parameter 91A60 Probabilistic games; gambling
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