Logistic analyses for complicated bets

Abstract

The problem discussed is estimating the probabilities of finishing order in a horse race based on simple winning probabilities only. Some models have been proposed based on different assumptions of running time distributions of horses for this problem. However, no detailed data analyses for comparing these models can be found. In this paper, we apply logit models and utilize several data sets and bet types to study the goodness of these models in detail. These complicated bet types include exacta, trifecta and quinella bets. Formal tests for non-nested models are applied whenever possible. Our empirical results suggest that the model based on independent normal running times is better than the others. To predict the winning probabilities of horses, many previous studies suggested that the win bet fractions are reasonable estimates. We utilize this information of winning probabilities to predict the ordering probabilities. Harville (1973) predict the ordering probabilities. Harville (1973) proposed a simple and convenient model that bettors can easily use in practice. In fact, the betting system proposed by Hausch, Ziemba & Rubinstein (1981) used the Harville model in determining the optimal bet amounts to place and show. The Harville model is equivalent to assuming that the running times are exponentially distributed. Henery (1981) and Stern (1990) assumed normal and gamma distributions respectively for the running times. Based on a likelihood approach, this paper considers the comparison among these models and the particular bet fractions. One conclusion is that in exacta and trifecta bets, no method based on the win bet fractions can outperform the exacta and trifecta bet fractions in predicting the relevant ordering probabilities.