A dual theory approach to estimating risk preferences in the parimutuel betting market

Niko Suhonen; Jani Saastamoinen; Mika Linden

doi:10.1007/s00181-017-1258-x

Empirical Economics

May 2018, Volume 54, Issue 3, pp 1335–1351 | Cite as

A dual theory approach to estimating risk preferences in the parimutuel betting market

Authors
Authors and affiliations

Niko Suhonen
Jani SaastamoinenEmail author
Mika Linden

Article

First Online: 28 June 2017

Received: 05 August 2015

Accepted: 16 March 2017

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Abstract

This paper introduces an alternative empirical approach to estimating risk preferences in the parimutuel betting market using a dual theory model which is amended to include bettors’ misperceptions of probabilities. We replicate previous empirical results and test our alternative empirical approach using parimutuel horse race betting data. Our results suggest that while bettors are risk-averse, they are also prone to misperceiving probabilities by overweighting low probabilities and underweighting high probabilities. As an application, these results replicate the choice patterns consistent with the Allais paradox.

Keywords

Allais paradox Dual theory Probability weighting function Rank-dependent utility

JEL Classification

D81 L83

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Notes

Acknowledgements

The authors wish to thank Thomas Epper, Helga Fehr-Duda, David Forrest, Timo Kuosmanen, Timo Tammi, Hannu Vartiainen, Horst Zank and the two anonymous reviewers for their helpful comments and suggestions. This research was supported by Emil Aaltonen Foundation, the Finnish Foundation for Alcohol Studies and the Finnish Foundation for Gaming Research.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Appendix 1

Estimation procedure

Each race c has $n_c $ horses. Since the winning horse is known, our data are constructed so that the dependent variable $y_j $ takes the value 1 if the horse j wins and is 0 otherwise. Different winning horses correspond to the probability that the horse j wins, which is drawn from the multinomial distribution by

$$\begin{aligned} p_j =\Pr [y_j =1]=\frac{H(w_j )}{\sum _{l=1}^n {H(w_l )} }, \qquad j=1,\ldots ,n. \end{aligned}$$

(A1)

The multinomial density for a race can be conveniently written as

$$\begin{aligned} f(y)=p_1^{y_1 } \times \cdots \times p_n^{y_n } =\prod \nolimits _{j=1}^n p_j^{y_j } . \end{aligned}$$

(A2)

The model for the probability that in each race c the horse j wins is

$$\begin{aligned} p_{cj} =\Pr [y_{cj} =1]=F_j (\mathbf{x}_c ,\beta ), \qquad j=1,\ldots ,n, \quad c=1,\ldots ,C, \end{aligned}$$

(A3)

where $\beta $ is a parameter and $\mathbf{x}_i $ are the regressors of the race-specific variables (e.g. turnover, last race). See the additional analysis in Appendix 2.

Since we know the horse that won the race and given that the races are independent of each other, the log-likelihood function for the sample is

$$\begin{aligned} \ln L(\beta )=\sum _{c=1}^C {\ln p_{cj} } , \end{aligned}$$

(A4)

in which $p_{cj} =F_j (\mathbf{x}_c ,\beta )$ is defined in (A3) (for more details, see Jullien and Salanié (2000). Maximizing the likelihood function with respect to $\beta $ gives us an estimator $\hat{{\beta }}$ that is consistent and asymptotically normal and efficient. Furthermore, we allow the parameter $\beta $ to depend on the race-specific variables (Andersen et al. 2008; Harrison and Rutström 2008).

Case 1 Dual theory and power function specification

In the case of the dual theory, we observe that $U(p)=p^{\beta }=w$, and we have

$$\begin{aligned} H(w)=w^{1/\beta }=p. \end{aligned}$$

(A5)

Using the fact that $\sum _{j=1}^n {p_j } =1$, the probability that the horse j wins the race can be written

$$\begin{aligned} p_j =\Pr [y_j =1]=\frac{w_j^{1/\beta } }{\sum _{l=1}^n {w_j^{1/\upbeta } } }. \end{aligned}$$

(A6)

Case 2 Dual theory with misperception of probabilities, i.e. risk-probability function

Using Prelec’s (1998) two-parameter specification $U(p)=\exp \left[ {-\beta (-\ln p)^{\alpha }} \right] $, the inverse function is

$$\begin{aligned} H(w)=\exp \left[ {-\left( {-\frac{\ln w}{\beta }} \right) ^{1/\alpha }} \right] . \end{aligned}$$

(A7)

As was the case with the dual theory power function specification, the probability that the horse j wins a race can be written

$$\begin{aligned} p_j =\Pr [y_j =1]=\frac{\exp \left[ {-\left( {-\frac{\ln w_j }{\upbeta }} \right) ^{1/\alpha }} \right] }{\sum _{l=1}^n {\exp \left[ {-\left( {-\frac{\ln w_l }{\beta }} \right) ^{1/\alpha }} \right] } }. \end{aligned}$$

(A8)

In both cases 1 and 2, given that the races are independent from each other, the log-likelihood function for the sample is

$$\begin{aligned} \ln L(\beta )=\sum _{c=1}^C {\ln p_{cj} } . \end{aligned}$$

(A9)

Appendix 2

Covariates

Next, we investigate whether race-specific covariates—turnover, a weekend race, the last race of the day, purse, the main racetrack in the country (Vermo), a small track, and a time trend—affect risk behaviour. Their impact on the risk parameter $\beta $ is captured by

$$\begin{aligned} \hat{{\beta }}=\hat{{\beta }}_0 +\hat{{\beta }}_1 x_{c1} +\ldots +\hat{{\beta }}_k x_{ck} . \end{aligned}$$

(A10)

Estimates of the power function are reported in Table 4 ¹⁰. Note that we tested several other variables, e.g. the day of the week, the two last races, the number of horses, a racing distance. Some potential sources of heterogeneity are reported in Table 4.

The most important finding is that the effects of race-specific covariates are marginal. First, the last race effect does not exist. This contradicts earlier studies (e.g. Ali 1977) which argue that bettors’ risk taking becomes more aggressive in the last races because they want to recover their earlier losses due to loss aversion suggested by Kahneman and Tversky (1979)¹¹. Second, betting behaviour does not change between weekends and weekdays. So, even though the population of bettors may differ between weekdays (e.g. professionals or gambling addicts) and weekends (e.g. amateurs or casual bettors), their attitude towards risk remains unchanged.

Table 4

Results of estimation with race-specific covariates

Covariate	Estimate	SE	p value	Log-likelihood
Power function
$\beta _0 $	0.871633	0.018398	0.000	−29907.63
Turnover	0.000017	0.000012	0.130
Weekend	0.009113	0.016185	0.573
Last race	0.022889	0.025698	0.373
Purse	-0.000002	0.000001	0.071
Vermo track	-0.046202	0.022561	0.041
Small track	0.039058	0.030931	0.207
Time (trend)	0.000002	0.000025	0.947

Next, we attempt to capture the level of market information by using variables such as turnover, purse and track size. We assume that a high purse in a race indicates that the horses are familiar to the bettors, and the more successful horses are likely to race in large tracks. In addition, the turnover of a race may represent a level of general interest towards the race. However, our data do not support these views. Betting behaviour was more aggressive in Vermo, which is the largest track and the races which have higher purses (marginally statistically significant). Furthermore, the turnover of a race is not statistically significant. Finally, we attempt to approximate the influence of off-track betting. Since we know that it has increased over the sampling period, we use a time trend to capture this effect. However, no effect was found for this.

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Authors and Affiliations

Niko Suhonen
- 1
Jani Saastamoinen
- 1
Email author
Mika Linden
- 2

1.University of Eastern Finland Business SchoolJoensuuFinland
2.Department of Social and Health ManagementUniversity of Eastern FinlandJoensuuFinland

A dual theory approach to estimating risk preferences in the parimutuel betting market

Abstract

Keywords

JEL Classification

Notes

Acknowledgements

Compliance with ethical standards

Conflicts of interest

References

Copyright information

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Covariate	Estimate	SE	p value	Log-likelihood
Power function
\(\beta _0 \)	0.871633	0.018398	0.000	−29907.63
Turnover	0.000017	0.000012	0.130
Weekend	0.009113	0.016185	0.573
Last race	0.022889	0.025698	0.373
Purse	-0.000002	0.000001	0.071
Vermo track	-0.046202	0.022561	0.041
Small track	0.039058	0.030931	0.207
Time (trend)	0.000002	0.000025	0.947