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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 pari-mutuel 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.

**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 pari-mutuel 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 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 dead-heats, 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 M 3. 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]).

3 The data do not contain information on the amount bet (a) or on the wealth of bettors (M ). 4 The assumption that bettors know the probabilities of winning of the horses running in the race was tested by reestimating the model on a dataset limited to races in which horses are aged 4 or more and have previously run. In these races, bettors are likely to have a very good knowledge of probabilities of winning of horses. Limiting the dataset to this subsample did not modify the 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 ϕ+ the probability weighting function of gains and ϕ− that of losses. Both probability weighting functions are strictly continuous and increasing from the unit interval into itself and satisfy ϕ+(0) = ϕ− (0) = 0 and ϕ+(1) = ϕ− (1) = 1. In general terms, the overall value of a bet on horse i can be written : W (pi, a, M, Oi ) = ϕ+(pi ) ∗ u(M + aOi) + ϕ− (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, ϕ+(pi ) = ϕ− (pi) = pi. In rank-dependent theory, ϕ− (1 − pi) = 1 − ϕ+(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, θ) = 1−e−θx .

The CARA form allows the estimation of the level of absolute risk aversion θ under the assumption that it is constant. Bettors are risk-loving if θ < 0 and risk-averse if θ > 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.

**Functional forms of the probability weighting functions**

The common functional forms presented in Table 1.2 are estimated. More informa-tion on these functions is available in Appendix A3.

Let Ω be the vector of parameters of the probability weighting functions corresponding to each model. Ω, which depends on the shape of the probability function to estimate and on whether the model uses one single probability weighting function (rank-dependent utility) or two probability weighting functions (one for gains and one for losses as in cumulative prospect theory), is defined in Table 1.3.

**Estimation**

Let θ be the parameter of the utility function and Ω the vector of parameters of the probability weighting functions defined in Table 1.3 for each functional form of the probability weighting function and each model to estimate.

pr1 is the probability of winning of the horse which actually won race r, with r = 1, …, J . For each r, the likelihood to observe horse 1 winning the race is l(θ, Ω; O1r ) = p1(O1r ; θ, Ω).

Assuming that outcomes of races are independent and that the expressions of p1 derived from the model are correct, the probability of observing the sample under study is the product of the J individual densities, which corresponds to the following likelihood function: p1(O1r ; θ, Ω) = L(θ, Ω|O1r ) r=1

Hence the log-likelihood function: LL = ln L(θ, Ω|O1r ) = ln p1(O1r ; θ, Ω) r=1

The maximum likelihood estimator has the usual asymptotic properties. It is con-sistent, asymptotically normal, asymptotically eﬃcient and invariant.

ˆ ∼ N (θ0, [I(θ0)] −1 ), I(θ0) = −E0(∂ 2 � (∂ ln L/∂ θ0 � θ ln L/∂ θ0∂ θ0) = −E0 ∗ ∂ ln L/∂ θ0)

Standard errors are computed using both a bootstrap procedure and the previous formula.

**Results**

**The favorite-longshot bias**

The favorite-longshot bias is the finding that betting on favorites (horses with small odds) yields a higher expected return than betting on longshots (horses with rel-atively high odds). It has been shown in a large number of papers, starting with Griﬃth [1949]. It has been observed across various types of races and at diﬀerent times in North America (McGlothin [1956], Weitzman [1965], Ali [1977], Snyder [1978], Asch et al. [1982], Snowberg and Wolfers [2010]) where the pari-mutuel sys-tem prevails, in the UK in both the pari-mutuel and bookmaker systems (Williams and Paton [1997], Jullien and Salanié [2000]), in Australia in both the pari-mutuel (Coleman [2002]) and bookmaker systems (Bird et al. [1987]) and in New Zealand in the pari-mutuel system (Coleman [2002], Gandar et al. [2001]).

The first result of the chapter is that the favorite-longshot bias also exists in France. The expected return for a 1-unit bet on horse i is Ri = πi ∗ Oi + (1 − πi) ∗ (−1), where πi is the probability of winning of horse i and Oi corresponds to its final odds. The probability of winning of a horse, which is the proportion of times the horse would win the same race repeated an infinitely large number of times, is unknown so I compute expected returns using the approach commonly adopted in the literature (see Coleman [2004]), which consists in grouping all horses of the dataset by either intervals of odds or favorite order (the favorite is in the first group, the second favorite in the second group, etc.) and computing the percentage of winners and the average odds of each group.

Expected returns are graphed in Figure 1.2, horses were grouped by odds percentiles and data is presented on a log-odds scale.

Figure 1.2 shows that returns are not equalized across betting odds: betting on favorites yields a higher rate of return than betting on outsiders. The expected return of betting horses with odds of 127 to 1 is −0.6, whereas it is −0.07 for horses with odds 1.43. Hence payoﬀs of favorites are not low enough to compensate for their high probabilities of winning, or equivalently favorites are underbet compared to their probabilities of winning. On the contrary, payoﬀs of outsiders are not high enough to compensate for their low probabilities of winning, or equivalently, they are overbet.

Hence, in a simple model with linear utility and probability weighting functions, the data shows that, in equilibrium, rates of returns are not equalized across horses in a race. Many suggestions have been made to explain this bias, one of which being that the simple model does not properly account for the tastes and beliefs of bettors.

#### Tests of models of decision-making under risk

**Expected utility model**

The results of the parameters obtained in the expected utility framework, as well as the maximum value of the log-likelihood function are presented in Table 1.4.

aθ is significantly negative and has a small absolute value, meaning that at least statistically and in the income range in which I test them, bettors exhibit a small and significant taste for risk. The parameter is smaller to the one obtained by Jullien and Salanié [2000] (−0.055) but similar to that of Snowberg and Wolfers [2010] which fits their data with a CARA utility function of parameter −0.017.

A bet of E20 on a horse with odds 10 and probability of winning of 25% is equiv-alent to the lottery winning E200 with a probability of 25% and loosing 20 with a probability of 75%. The estimated risk-attitude parameter makes a bettor indiﬀer-ent between this lottery and the sure amount of E38. A risk neutral bettor would be indiﬀerent between the same lottery and the sure amount of E35 so that the behavior of bettors exhibits some risk-love.

**Rank-dependent utility model**

The results for all functional forms of the probability weighting functions tested are presented in Table 1.5 and graphed in Figure 1.3.

The risk-attitude parameter of the utility function is negative, statistically significant and small in every specification of the rank-dependent utility model.

**Table of contents :**

**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**