A comparison between two models for predicting ordering probabilities in multi-entry competitions

Abstract

To predict ordering probabilities of a multi-entry comlpetition (e.g. horse race), two models have been proposed. Harville (1973) proposed a simple and convenient model that people can easily use in practice. Henery (1981) proposed a more sophisticated model but it has no closed form solution. In this paper, we empirically compare the two models using a series of logit models applied to horse-racing data. In horse-racing, many previous studies claimed that the win bet fraction is a reasonable estimate of the winning probability. To consider more complicated bet types (e.g. exacta, place & show), ordering probabilities (e.g. P(horse i wins and horse j finishes second)) are required. The Harville and Henery model assume different running time distribution and produce different sets of ordering probabilities. This paper illustrates that the" Harville model is not always as good as the Henery model in predicting ordering probabilities. The theoretical result concludes that if the running time of every horse is normally distributed, the probabilities produced by the Harville model have a systematic bias for the extreme cases (the strongest and weakest horses). We concentrate on horse-racing case but the methodology can he applied to other multi-entry competitions.