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### Place Pricing in Horse Racing

There have been many previous studies that focus on different approaches to the problem of pricing a horse to place rather than to win. If we assume that the win market is efficient then at the start of a race the last traded win price for any horse represents the true probability of that horse winning.

Harville (1973) assumed that any horse that won is automatically discounted from placing and the win probabilities of the remaining horses recalibrated to sum to 1. This naive Bayesian approach suffers from various problems.

1) It is rare for a race to finish in order of win probabilities. In other words, the favourite does not always win and the second favourite does not always finish second etc.

2) The "Silky Sullivan" effect. Some horses either win or finish well off the pace. One such horse was Silky Sullivan that sat at the back of the race until the final furlong and would make a spirited attempt to win (or fail miserably).

3) Trainer orders. A jockey will be under orders from the trainer, which cannot be modeled mathematically. A jockey may be told to go all out for the best placing possible. Another jockey may be told to strike out for the win but canter in if the attempt fails. Yet another jockey may be told to treat the race as a training run. And so on.

Further studies by Henery (1981) and Stern (1990/2008) have sought to model the fact that favourites tend to place less than their place odds say they should and long-shots tend to place more often than their place odds suggest.

A comparative study of all three approaches has been carried out by Lo (2008) and demonstrated the superiority of the Henery and Stern models over the Harville model. However, these superior models are computationally taxing and Lo demonstrates a simplified approximation that is easily coded.

A Study of Betfair Place Odds - An article demonstrating the futility of using place pricing models to discover pricing errors on the Betfair place market.

References

Harville, D.A. (1973) Assigning probabilities to the outcomes of multi-entry competitions. Journal of the American Statistical Association, 68, pp. 312-316

Henery, R.J. (1981) Permutation probabilities as models for horse race Journal of the Royal Statistical Society, B 43, pp. 86-91

Lo, V.S.Y. & Bacon-Shone, J. (2008) Approximating the Ordering Probabilities of Multi-entry Competitions by a Simple Method. Handbook of Sports and Lottery Markets, pp. 51-65. Elsevier

Stern, H.S. (2008) Estimating the Probabilities of the Outcomes of a Horse Race (Alternatives to the Harville Formulas). Efficiency of Racetrack Betting Markets. Hausch, Lo & Ziemba. pp. 225-235.

Stern, H.S. (1990) Models for Distributions on Permutations. Journal of the American Statistical Association 85, No. 410 (June) : pp. 558-564