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

Abstract

SUMMARY To predict ordering probabilities of a multiple-entry competition (e.g. a horse-race), two models have been proposed. Harville proposed a simple and convenient model that can easily be used in practice. Henery proposed a more sophisticated model but it has no closed form solution. In this paper, we empirically compare the two models by 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 complicated bet types which involve more than one position, ordering probabilities (e.g. P(horse i wins and horsej finishes 2nd)) are required. The Harville and Henery models assume different running time distributions 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 strongest and weakest horses. We concentrate on the horse-racing case but the methodology can be applied to other multiple-entry competitions.