Parimutuel contests with strategic risk-sensitive bettors

ABSTRACT

Existing models in the parimutuel betting literature typically explain betting data by either assuming a single, representative bettor with certain risk preferences or by assuming that a number of risk neutral bettors compete strategically within a game theoretic framework. We construct a theoretical framework of parimutuel markets in which we model both strategic interaction and individual bettor risk preferences, distinguishing between sophisticated insiders and recreational outsiders. We solve this model analytically for the optimal insider betting amount in a static symmetric Nash equilibrium. A new data set of 126 million individual horse race bets in New Zealand from 2006 to 2014 allows us to calibrate the model. We find that insiders (those betting $100 or more) outperform outsiders by 7.5% in terms of realized returns. The best fit of the model to the data is obtained when insiders are assumed to be risk neutral and to have an information advantage of 0.08 in probability terms. This finding provides empirical support for the common assumption of risk neutrality in strategic interaction models of parimutuel betting.

KEYWORDS:

• Parimutuel
• betting
• risk aversion
• game theory

JEL CLASSIFICATION:

• C70
• D81
• L83

Acknowledgement

We are grateful for the feedback from two anonymous referees.

ORCID

Paul Geertsema http://orcid.org/0000-0002-1461-3807

Notes

¹ The only other work combining risk aversion and strategic interaction in a parimutuel setting (as far as we are aware) is that of Qiu (2012), in which the author demonstrates the existence of equilibrium in a sequential move framework where agents have utility in accordance with prospect theory. In our model, by contrast, we consider a simultaneous move game between a number of insider bettors with common risk preferences within a two-moment decision model. We also explicitly solve for a closed-form equilibrium solution, which Qiu (2012) does not attempt.

² Betting normally opens a day or two before the start of the race. Bets can be placed at any time until closing time, and the total betting volume on each horse is publicly reported every 5 minutes or so throughout this time. Our decision to model insiders in a simultaneous game is motivated by a previous finding in the literature that around 40% of bets are placed in the last minute prior to closing and that last minute betting is a bettor predictor of final outcomes than earlier bets (Gramm and McKinney (2009)).

³ In an earlier article, Gandar, Zuber, and Johnson (2001) examines a sample of 10 332 horse races in New Zealand between 1994 and 1997 for evidence of the favourite-longshot bias (which they fail to find).

⁴ As of 27 August 2015, 1 NZD = 0.65 USD. In robustness tests, we also consider alternative cut-off points ranging from NZD 50 to NZD 500 (in the Empirical Analysis section).

⁵ At the cost of a simpler model in other respects. We assume a parimutuel contest with only two outcomes (i.e. horses), and we do not model information asymmetries between insiders.

⁶ Recall total insider betting equals 1; we distinguish between an arbitrary but fixed insider bettor that bets $b_1$ and the remaining insider bettors that collectively bet $(N-1)b_n$. In equilibrium $b_1=b_n$ (by symmetry).

⁷ Since the time between placing a bet and collecting the payout is a matter of days, at most we ignore the time value of money.

⁸ Meyer (1987) shows that there exists an equivalence between expected utility maximization and two-moment decision models subject to restrictions on the agent’s preferences or choice sets.

⁹ Subject to second-order conditions and model constraints.

¹⁰ By way of example, let $N=5,q=0.3$, $\alpha =0.1$ and $\tau =0.15$ with $b_0(1)=0.7$ and $q=0.3$. Since $b_0(1)=0.7</q=0.3$, we cannot directly use the equilibrium solution for the optimal amount $b^{\ast }$ for insiders to bet on horse 1, which requires $b_0(1)<q$. However, we can use the formula to calculate the amount that would be bet by insiders on horse 2. Note that $b_0(2)=1-b_0(1)=1-0.7=0.3$ and insiders’ belief that horse 2 will win is given by $1-q=1-0.3=\mathrm{0.7.}$ Now we have $b_0(2)=0.3<(1-q)=0.7$ as is required for the equilibrium insider betting amount, but with the understanding that this insider betting amount now relates to horse 2 rather than horse 1. We then calculate the equilibrium amount insiders will bet individually on horse 2 as $b^{\ast }=f(N,(1-q),\tau ,\alpha ,b_0(2))=f(5,0.7,0.15,0.1,0.3)\approx 0.1480$. (Recall that all amounts are expressed as a fraction of total outsider betting so that $b_0(1)+b_0(2)=1$. So that means each of the 5 insiders will bet 14.80% of total outsider betting, or 73.98% of total outsider betting for all 5 insiders together.)

¹¹ Racing bets include thoroughbred racing, greyhound racing and harness racing

¹² In some races bets from the TAB and other similar overseas, betting agencies are combined in so-called commingled pools. Since we rely on individual bets to calculate pools and win payouts, we exclude events with commingled pools from our analysis.

¹³ For instance, a bet may be recorded as a ‘Place’ bet on horses 2, 5 and 6 with an amount bet of $30. In this case, the bet really consists of 3 atomic $10 bets on each of the three horses.

¹⁴ Gramm and McKinney (2009) finds that late bets better predict actual outcomes than total betting, suggesting that the timing of bets may an alternative method for identifying insiders.

¹⁵ A referee pointed out that it may be possible to construct a more precise proxy for insiders by also considering the past experience of individuals. This could be an interesting direction for further research. For instance, Feess, Müller, and Schumacher (2014) finds that experienced bettors on average bet on lower odds outcomes and earn higher returns than inexperienced bettors.

¹⁶ This follows from the fact that for positive numbers, squaring is an order preserving operation.

¹⁷ Intriguingly, the level of inside information of 0.08 is quite close to the difference in returns between insiders and outsiders in Table 2 of 0.0748. It is also very similar to the coefficient of 0.0749 for the insider dummy in regression 2 of Table 4. While differences in returns are not directly comparable to differences in beliefs, it does suggest that an estimate of inside information of 0.08 is in the plausible range.