2012/11/09 by David Aldous, Aldous, David, Mykhaylo Shkolnikov +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #60G44 (Primary) 91A60 (Secondary) #Bayesian Methods and Mixture Models #CONTEST #Combinatorics #Complex Systems and Time Series Analysis #FOS: Mathematics #Martingale (probability theory) #Mathematical Dynamics and Fractals #Mathematical economics #Mathematics #Population #Probability (math.PR) #Statistics #math.PR #msc:60G44 #msc:91A60
paper · pdf · doi:10.48550/arxiv.1211.2045
published in arXiv (Cornell University) (Cornell University) · 18 pages
arxiv created 2012/11/09 · openalex publication_date 2012/11/09 · arxiv updated 2012/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Within a contest there is some probability Mi(t) that contestant i will be\nthe winner, given information available at time t, and Mi(t) must be a\nmartingale in t. Assume continuous paths, to capture the idea that relevant\ninformation is acquired slowly. Provided each contestant's initial winning\nprobability is at most b, one can easily calculate, without needing further\nmodel specification, the expectations of the random variables Nb = number of\ncontestants whose winning probability ever exceeds b, and Dab = total number\nof downcrossings of the martingales over an interval [a,b]. The distributions\nof Nb and Dab do depend on further model details, and we study how\nconcentrated or spread out the distributions can be. The extremal models for\nNb correspond to two contrasting intuitively natural methods for determining a\nwinner: progressively shorten a list of remaining candidates, or sequentially\nexamine candidates to be declared winner or eliminated. We give less precise\nbounds on the variability of Dab. We formalize the setting of infinitely\nmany contestants each with infinitesimally small chance of winning, in which\nthe explicit results are more elegant. A canonical process in this setting is\nthe Wright-Fisher diffusion associated with an infinite population of initially\ndistinct alleles; we show how this process fits our setting and raise the\nproblem of finding the distributions of Nb and Dab for this process.\n