Optimal Determination of Bookmakers' Betting Odds: Theory and Tests

JEL Classification D82, G13

John Fingleton and Patrick Waldron

Department of Economics, Trinity College, Dublin 2, Ireland


Abstract

This paper develops a theoretical model of how bookmakers' odds are determined, given varying levels of inside information on the part of punters. Bookmakers' attitudes towards risk and the degree of competition between them will influence bookmaker behaviour. Using a data set of 1,696 races in Ireland in 1993, we find that bookmakers are extremely risk-averse, and estimate that operating costs and monopoly rents combined account for up to 4 per cent of turnover and that between 3.1 and 3.7 per cent of betting is by punters with inside information.


Non-technical summary

The odds or prices set by bookmakers on horse races exhibit four interesting properties which are acknowledged by practitioners and/or observed in empirical studies, the causes of which had been poorly understood by economic theoreticians until recently.

First, empirical odds invariably exhibit a favourite-longshot bias, whereby the prices of the favourites are relatively better value than those of the longshots. This bias is also observed in pool betting, with the interesting exception of data from Hong Kong. Second, the margin implicit in bookmakers' odds increases with the number of runners in the race. Third, this theoretical margin, calculated by summing the probabilities quoted by the bookmakers, overstates their realised operating profits, suggesting that punters can identify horses underpriced by bookmakers and exploit this `inside information'. Fourth, margins vary greatly from country to country, even when market structure does not vary. In particular, bookmakers' prices are significantly higher (i.e. odds lower) in the Irish market than in the British market, although these markets differ only in size.

Shin (1991-3) provided a theoretical explanation for the first two phenomena, assuming inside information on the part of the punters. Since £1 bet on a horse at 3:1 exposes the bookmaker to a smaller potential loss than £1 bet on a horse at 30:1, the bookmaker will require a greater risk premium to insure himself against the possibility of inside information on a longshot. This is achieved by reducing the odds in respect of the longshot. The more horses in the race, the longer the odds on each, and thus the bigger the bookmaker's overall margin. Shin's empirical analysis estimates the extent of inside information without using the outcome of races to confirm the accuracy of that information.

Motivated by the differences between market outcomes in Ireland and Britain, this paper develops a more general model of determination of bookmaker betting odds, based on Shin's model, incorporating (a) infinite risk-aversion on the part of the bookmakers, (b) the possibility of anti-competitive behaviour among the bookmakers and (c) (as Shin does) different degrees of inside information on the part of the punters.

Shin's theoretical analysis is based on perfectly competitive, risk-neutral bookmakers. It ignores operating costs and assumes that any profits are competed away. In practice, bookmakers are often seen balancing their books so as to have identical liabilities across all horses. This would represent infinitely risk-averse behaviour and would not be profit maximising: a bookmaker can increase profits by setting slightly longer odds. While the level of risk-aversion among bookmakers alters the optimal prices, it does not affect the existence of either the favourite-longshot bias or the relationship between margins and the number of runners. Because optimal prices differ, we are able to test whether bookmakers are risk-neutral or infinitely risk-averse. We are interested in whether the higher bookmaker margins in the Irish market can be explained by greater risk aversion or greater levels of anti-competitive behaviour by bookmakers or by greater levels of inside information on the part of punters.

We use the results of 1,696 races in Ireland in 1993 to estimate jointly the extent of inside information, the operating profits earned and the degree of risk-aversion exhibited by bookmakers. Our methodology, using the closed form solution of Jullien and Salaniť (1994) rather than the approximations of Shin (1993) and Paton and Williams (1997), permits the estimation of ''true'' probabilities, which we find, for suitable parameter values, accurately reflect the subsequent outcomes.

Our conclusion suggests that bookmakers in Ireland are infinitely risk-averse and balance their books. We cannot distinguish between inside information and operating costs, merely concluding that combined they account for up to 3.7 per cent of turnover.


Acknowledgements

We wish to thank Hyun Shin and Liam Mason (Irish Racing Services) for data and also participants at various seminars for useful suggestions. We remain responsible for any errors.


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