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Horserace betting in France
The betting market on horseraces in France is exclusively a pari-mutuel system1.
The concept of pari-mutuel consists in pooling together all bets corresponding to a race and a bet type, removing a share to cover the taxes and expenses of the betting operator and redistributing the remainder among winning bettors in proportion to their bets. Final payoffs hence depend exclusively on the total pool, the share kept by the betting operator (the “take”)2 and the stakes attracted by each horse.
The more stakes a horse attracts relative to the total pool, the lower the payoff of a bet on this horse. Payoffs on horses are called odds. For the simplest type of bets (which are the focus of the chapter), which consists in finding the winner of a given race, odds of 1.2 on a given horse and race means that a 1 unit winning bet on that horse returns the bet (1) plus 1.2. Odds hence correspond to net returns of a unit bet. A horse cannot have odds inferior to 0.1.
For a race happening on a particular day, the market opens online at about 5 AM on the day of the race. For a bettor which prefers to go to a specialized store, it starts on the day of the race at the opening of stores. A bettor at the track can only bet about thirty minutes before the beginning of the race. The market closes right before the start of the race. Because of the way odds are computed in the parimutuel system, bettors only have access to temporary odds which are computed with the current state of bets and are updated about every minute online.
Data
Data were collected from pmu.fr between April 2013 and May 2015. PMU (Pari-Mutuel Urbain) is the main operator of bets in France. Online, it gathers 84.8% of the total pool and, in-store, it is a legal monopoly. The dataset records information on bets, races, horses and tracks for races which were the subject of bets offered by the PMU. It contains 33,196 races.
For each race, the dataset encompasses the final payoff of each horse, its rank in the race and many of its characteristics. In addition to the time of day, date and track, races are also characterized by their discipline, types and conditions. The data only contains the payoffs of each horse for winning bets, which are the focus in this chapter.
Since I am interested in modeling the process of decision-making regarding the choice of a specific horse in a race, I drop the 7,919 races in which two or more horses in a given race belong to a team, which happens when horses have the same owner or the same trainer. In these cases, all the horses of the team have the same payoff and if one of them wins the race, a bet on any of the horses in the team also wins.
Hence the payoff of a horse that is part of a team does not reflect its probability of winning, but rather the probability that any horse in the team wins.
I also remove races in which several horses arrive in the first position, called deadheats, because I model a race in which only one horse wins the race. I drop races for which payoffs are incomplete or erroneous. It includes races which are not recorded as being over, in which at least one running horse has a missing payoff and for which the final payoff of the winning horse does not correspond to the dividend. I am left with 23,462 races.
As Table 1.1 shows, the average number of races per day amounts to 32. During some days, 80 races take place, while on other days only 13 do. The average number of running horses in a given race is 12. The minimum is 2 while the maximum is 24. Half of the races includes between 9 and 14 running horses. The distribution of odds covers a wide range. The maximum reaches 998 while the minimum is 0.1.
The median amounts to 15.4 and the mean to 27. 90% of odds range between 0.1 and 68.4.
Figure 1.1 shows that the sample contains large favorites, with odds between 0.1 and 0.5 (0.2% of the sample), and very long outsiders with odds above 50 (19% of the sample).
Using these definitions, 64% of large favorites and 0.6% of very long outsiders won their race. Alternatively, defining large favorites as horses that attract twice more bets than the second-most-bet horse in their race, large favorites win 44% of the time.
Model and estimation procedure
The theoretical model
The model describes the decision of a representative bettor who bets a in a given race and is endowed with an initial wealth M3. The choice of a particular horse in the race depends only on its probability of winning and final odds. In a given race r with N horses, the bettor is hence presented with a menu of probabilities and odds ((O1, p1), (O2, p2), …(ON, pN))r, probabilities being non negative and summing to one.
I assume that the menu is known to the bettor when he makes his choice. In practice, final odds are not known until the beginning of the race and the bettor does not have perfect knowledge of probabilities of winning. However, previous studies of horserace bettors imply that bettors have a good knowledge of chances of winning of horses4 (see Sauer [1998]).
Writing Hi the action of betting on horse i, the overall value of a bet on horse i, W(Hi), is a number such that the bettor prefers horse i to horse j or is indifferent between horses i and j if and only if W(Hi) W(Hj). The representative bettor is rational in that he bets on the horse with the highest overall value.
Furthermore, I assume that the race is only won by one horse. In this perspective, the few races won by several horses were removed from the sample.
Given the stated assumptions, bettors continue to bet in a given race until odds make them indifferent between betting on any horse in the race and not betting5.
So in equilibrium: i 1, …,N, W(Hi) = w, w constant (1.1)
Let a be the amount bet by the representative bettor, M her initial wealth, Oi the odds associated to horse i and pi the probability of winning of horse i. I also define u, a continuous and strictly increasing utility function with u(0) = 0 and write + the probability weighting function of gains and − that of losses. Both probability weighting functions are strictly continuous and increasing from the unit interval into itself and satisfy +(0) = −(0) = 0 and +(1) = −(1) = 1. In general terms, the overall value of a bet on horse i can be written : W(pi, a,M,Oi) = +(pi) u(M + aOi) + −(1 − pi) u(M − a) (1.2)
The specific expression of the overall value of a bet on horse i depends on the model of decision-making studied. In expected utility, +(pi) = −(pi) = pi. In rank-dependent theory, −(1 − pi) = 1 − +(pi) and in cumulative prospect theory M = 0, so that I assume that the reference point is not betting6. More details on each model are given in Appendix A1.
The model is solved using the procedure of Jullien and Salanié [2000] which consists in computing w and then the probability of winning of the horse which actually won the race (p1), which is in turn used to compute the likelihood function. The procedure used to obtain p1 is explained for each model in Appendix A2.
Functional form of the utility function
Following Jullien and Salanié [2000], I assume that the utility function has the following CARA form throughout the chapter: u(x, ) = 1−e−x
The CARA form allows the estimation of the level of absolute risk aversion under the assumption that it is constant. Bettors are risk-loving if < 0 and risk-averse if > 0. This expression of u is convenient since M, which is not observed, cancels out in the expression of the probability used in the likelihood function.
Table of contents :
Introduction (French)
Introduction (English)
1 Testing Models of Decision under Risk: The Case of Horserace Bettors in France
1.1 Introduction
1.2 Horserace betting in France
1.3 Data
1.4 Model and estimation procedure
1.4.1 The theoretical model
1.4.2 Functional form of the utility function
1.4.3 Functional forms of the probability weighting functions
1.4.4 Estimation
1.5 Results
1.5.1 The favorite-longshot bias
1.5.2 Tests of models of decision making under risk
1.6 Conclusion
Appendices
A1 Overall value of a bet in each model
A1.1 Expected utility model
A1.2 Rank-dependent utility model
A1.3 Cumulative prospect theory model
A2 Obtaining p1 to estimate the parameters of the models
A2.1 Expected utility model
A2.2 Rank-dependent utility model
A2.3 Cumulative prospect theory
A3 Details on probability weighting functions by model
A3.1 Rank-dependent utility model
A3.2 Cumulative prospect theory
2 Merger Efficiency gains
2.1 Introduction
2.2 The French urban public transport industry
2.2.1 Organizational background
2.2.2 Transport groups and competition in the industry
2.2.3 Financial situation
2.3 The merger
2.3.1 The story
2.3.2 Competition concerns of the French Competition Authority
2.3.3 Potential efficiency gains
2.4 Merger evaluation
2.4.1 Empirical strategy
2.4.2 Data and variables
2.4.3 Main analysis
2.4.4 Robustness checks – balanced panel
2.4.5 Robustness checks – unbalanced panel
2.5 Discussion
2.6 Conclusion
3 Buyer’s discretionary power and the selection of efficient firms
3.1 Introduction
3.2 Literature
3.3 The institutional context
3.4 Data
3.4.1 Datasets
3.4.2 Variables
3.4.3 Descriptive statistics on the estimation sample
3.5 Empirical strategy
3.5.1 Identification
3.5.2 Two-step estimation
3.6 Results
3.6.1 Determinants of the choice of a procedure
3.6.2 Impact of the award mechanism on supplier selection (TFP of selected suppliers)
3.6.3 Robustness checks
3.7 Discussion
3.7.1 Adapted procedure and number of bidders
3.7.2 Adapted procedure and characteristics of winners
3.8 Conclusion
Appendices
A1 Descriptive Statistics on the Amadeus sample
A2 Measures of total factor productivity
Bibliography
