The Application of Ranking Probability Models to Racetrack Betting
结合Lo和Bacon-Shone的排序概率近似模型与HZR投注系统，使用美国和香港的最终投注数据，在假设零计算成本下，相比原系统获得了更低风险下的更高利润；日本数据则显示利润差异不大。
Hausch et al. (HZR) (Hausch, D. B., W. T. Ziemba, M. Rubinstein. 1981. Efficiency of the market for racetrack betting. Management Sci. 27 1435–1452.) developed a betting system that demonstrated positive profits at two racetracks. The system assumes running times are distributed exponentially, but other distributions for running times (Henery [Henery, R. J. 1981. Permutation probabilities as models for horse races. J. Roy. Statist. Soc. B 43(1) 86–91.] and Stern [Stern, H. 1990. Models for distributions on permutations. J. Amer. Statist. Assoc. 85(410) 558–564.]) have been shown to produce a better fit in Bacon-Shone et al. (1992a), Lo (Lo, V. S. Y. 1994. Application of running time distribution models in Japan. D. B. Hausch, V. S. Y. Lo, W. T. Ziemba, eds. Efficiency of Racetrack Betting Markets. Academic Press, 237–247.), and Lo and Bacon-Shone (Lo, V. S. Y., J. Bacon-Shone. 1994. A comparison between two models for predicting ordering probabilities in multi-entry competitions. The Statistician 43(2) 317–327.) using data from Hong Kong, the Meadowlands, and Japan. The better fit is at the cost of severely increased complexity in computing ranking probabilities, though. In response, Lo and Bacon-Shone (Lo, V. S. Y., J. Bacon-Shone. 1992. An Approximation to Ordering Probabilities of Multi-entry Competitions. Research Report 16, Department of Statistics, University of Hong Kong.) proposed a simple model of computing ranking probabilities which closely approximates those based on the Henery and the Stern models and fits the data as well. This paper couples the Lo and Bacon-Shone model and the HZR system. For data sets from the United States and Hong Kong, we show improved profit over the HZR system at lower levels of risk using final betting data assuming zero computational costs. With data from Japan, our model shows little difference in profits from the HZR system.