Memoire Online - Strategic behavior in sport contests : application to middle-distance races from the 2010-2019 decade

Contests are a fact of life that can be witnessed everywhere, even in the non-human spheres. Knight (1935, p. 301) saw contests as games as an essential part of economic life:

«The activity which we call economic, whether of production or of consumption or of the two together, is also, if we look below the surface, to be interpreted largely by the motives of the competitive contest or game, rather than those of mechanical utility functions to be maximized.»

They are defined as: «a situation in which players compete with one another by expending effort to win a prize» (Baik, 1994). This part reviews the literature on theory of contests. The literature on theory of contest is very large and diverse and there is not the place in this Master Thesis to cover it entirely, however I will try to present the most basic models alongside models that have been useful in the study of sports. This literature has been remarkably reviewed by Corchón (2007), Konrad (2009), and Fu and Wu (2019). Besides, and inside the scope of my particular topic, Szymanski (2003) made a survey that study the application of contests in sporting events. The following part 2.2 is largely inspired by the recent review of Fu and Wu (2019) that gives a large and efficient vision of the current scientific knowledge. They begin with a summary of important contest modelling frameworks which differ in the mechanisms of selection of the winner, these frameworks will be presented in the 2.2.1 part. They continue by presenting a large range of various contest models in different contexts. Some of these models, the most relevant for this master thesis, will be presented in the part 2.2.2

Fu and Wu (2019) start their review by considering three types of models: perfectly discriminatory contest models, where one secures a win when he outperforms, and which are named as all-pay auctions, rank-order Tournaments (Lazear & Rosen, 1981; Green &

Stokey,1983) and contests with ratio-form contest success functions (Tullock, 1967; Tullock 1980)

Fu and Wu start by present ting all-pay auctions, which are models applied to a large number of contexts like R&D races (Dasgupta, 1986), litigation (Baye, Kovenock, & de Vries, 2005), lobbying (Hillman & Samet, 1987; Hillman & Riley, 1989), etc. In the case of a classical winner-take-all all-pay auction (Hillman & Riley, 1989; Baye, Kovenock, & de Vries, 1996). Then, for an effort profile x = (x?, ... , x_?), a player ?? wins a contest with a probability equal to:

This type of contest has no pure strategy equilibrium and gives a mixed strategy equilibrium. Baye et al. (1996) gave a general determination of the equilibria in a ??-person all-pay auction where the information is complete. For the case of an incomplete information auction, then the bids of the contestants are distributed above zero, therefore a tie is an event with a zero probability.

A framework of large contests was developed that enables for many heterogeneous players and prizes (Olszewski & Siegel, 2016) with complete or incomplete information. Olszewski and Siegel (2020) also show by continuing the research designed by Moldo-vanu and Sela (2001) that many heterogeneous prizes are possibly optimal in a large contest framework where contestants are risk averse with convex costs. Olszewski and Siegel (2019) applied the framework to model college admissions.

Fu and Wu (2019) explain that the result of an all-pay auction is determined easily, as a small difference secures the win of the auction for a contestant. However, the result of a `real life' contest often depends not only on the effort of the contestants but also on perturbations that happen in a random manner. There exists two ways to model contest with a mechanism of selection of the winner with noise.

Lazear and Rosen (1981) proposed a rank-order tournament. If each contestant ?? exert an effort xi which produces an output ??i with:

where the output function ???(???) is usually an increasing function of effort ???.It is also called by convention the impact function.

The function ??? (???) is a settled output and increases with the contestant's effort, ??? is a random effect, which is usually identically and independently distributed among the contestants. Contestants are ordered by their respective output ???, and the contestant with the highest ??? wins. If two players are involved in the contest, the condition for the player 1 to win can be written as:

???(???)	+ ???	> ???(???)	+ ???
? ??? -	??? <	???(???) -	???(???)

If the noise term (??? - ???) is distributed with a cumulative distribution function ??(·). The probability that contestant 1 wins is then given by:

which is the probit winning probability specification that we are going to see later. (Dixit, 1987)

This framework has been applied in many studies in order to explore the optimal prize structure in small tournaments (Krishna and Morgan, 1998) and demonstrate that optimal tournaments do not necessarily induce first-best outcome, contestant's incentive for risk taking in tournaments (Hvide, 2002), dynamic versions of the model to investigate optimal interim feedback policies in this framework (Aoyagi, 2010). There is also a study (Balafoutas, Dutcher, Linder, & Ryvkin, 2017) that investigated the optimal prize allocation in tournaments of heterogenous contestants. They observe that a loser-prize tournament that gives as a reward a low prize to bottom performances can be optimal.

However, Fu and Wu (2019) note that the most adopted modeling approach that permits a noisy mechanism for the selection of the winner is the one that takes a ratio-form contest success function. The Tullock (1980) contest model is the most popular case of this modeling form. For the case of a winner-take-all contest with a ratio-form contest success function, the probability that a contestant ?? wins, ???, is given by the ration of the output of his effort to the total output supplied by the entire group of contestants such as:

Under the assumption that all the contestants follow a linear effort cost function ??(x?) = x?. Then this framework gives a measurement for the winning probability as a function of effort in a winner-take-all imperfectly discriminatory contest. Tullock (1980) adopts ???(x?) = x?^?with r> 0. He then applies this model to the rent-seeking contest. These types of model are called Tullock contests. If r is large, the marginal pay back for effort will be higher and other factors that have an influence on the winner's selection will count less. As a consequence, if for the case of a symmetric contest, contestants tend to increase their bids when r increases. In a traditional symmetric Tullock contest game, there is a unique Nash Equilibrium in pure strategy only if the contest is not very discrim-

In this part are presented a few of the many models on which Fu and Wu (2019) reviewed the existing literature. The choice of these models is set on the applicability of these models to the case of this thesis: running races.

The model studied in this Thesis is a model which works in sequential moves. Therefore, it is important to see where it is coming from. Fu and Wu (2019) point out that the first to bring the structure of Stackelberg is Dixit (1987). He identifies the conditions under which a contestant benefits from being the first mover, as he can strategically impact the late mover behaviour by his action. It has been shown many things about these types of contests. First, that only the contestant with the lowest costs has a positive payoff in return (Konrad & Leininger, 2007), that a head start given to the first mover can have an impact of the contest's performance (Segev and Sela, 2014), or that sequential moves lead to a higher rent dissipation if more than two players are involved in the auction. (Klunover, 2018)

Fu and Wu (2019) also note that there has been a continuous research trying to en-dogenize the timing of moves of contestants with for example a three-period model. (Baik and Shogren, 1992a; Leininger, 1993). In this model, the contestants can furnish their efforts in either period 1 or 2, and a winner is selected after both contestants have completed their entire effort. They decide the timing to commit their effort simultaneously before the game. It is shown that in the case where contestants are asymmetric in regard of their capacities, the underdog always choose to commit effort earlier while the favorite

chooses to commit effort later. In a model with settings of an all-pay action (Konrad & Leininger, 2007), it is also shown that the strongest contestant always chooses to move later. This theory is very important for the following master thesis because it is the one we are trying to verify.

Contests with budget constraints are highly relevant in regard of a running race. Indeed, every athlete knows that his capacities are not unlimited and that if he goes too fast for too long, then he is going to be biologically forced to slow down by the rise of the lactic acid in his muscles which is going to tetanize him.

Fu and Wu (2019) enumerate a large number of models that take the assumption that «contenders have budget constraints and have to allocate their resources among parallel battles to maximize the sum of the expected rent they can receive from the whole set of battlefields» (Fu & Wu, 2019) These types of model are called a Colonel Blotto game of duopoly conflicts in multiple battlefields. This game was proposed by Borel (1921) and analyzed in the case of three markets (Borel & Ville, 1938). The analysis was generalized to a certain number of markets for symmetric players, with one asymmetric case being solved by Gross and Wagner (1950). Kovenock and Roberson (2012) investigated a Colonel Blotto game in which two players make an alliance and compete against a common rival, where allied players can transfer resources to each other. This situation is interesting for sport because it can model the situation of a race where two runners are going to unite their forces to beat the favorite. This happens a lot in cycling for example where leaders from different teams can command their whole teams to work together in order to beat the leader of the general ranking or the favorite.

These studies take as an assumption that the contestants allocate their budgets simultaneously. A study about a two-stage tournament with each player effort being constraint by a limit show that the underdog tends to behave more aggressively in the first period (Harbaugh and Klumpp, 2005) which agrees with Baik and Shogren (1992a). Therefore, it is really interesting in order to affirm that theoretically, underdogs over commit their effort in races because it brings the idea of the limitation of the resources of the runner.

Fu and Wu (2019) remark in their review that most modeling frameworks of contests have been assumed with risk-neutral contestants. However, the result of a contest is most of the time. Even if Real Madrid will in 99% of the cases crush a fifth division team, there

may be a chance they might get beaten. This can also be observed in cycling races where there are some cyclists who are very cautious in their strategy, waiting for the last hundred meters to attack, and cyclists who launch attacks 200km from the finish line even though they know their chances of success is very poor because this strategy is very much riskier.

A way to bring risk in a model is to take the assumption that contestants are risk averse. Hillman and Katz (1984) investigated contenders' incentive with the presence of risk aversion. The contestants' utility function u(·) is assumed to be strictly increasing and concave, which is different from models with risk-neutral players who have a linear utility. For a given award w, with a personal valuation of the prize by the contestant v? the expected payoff of contestant j in the contest game is calculated by:

In a two-player contest, this model admits a pure strategy Nash equilibrium if both contestants are constantly absolute averse. (Skaperdas & Gan, 1995) Moreover, Cornes and Hartley (2012) generalized this result in order to prove the existence of a unique equilibrium in the case of an asymmetric contest.

Fu and Wu (2019) also explain that the impact of risk aversion on the effort of contestants is ambiguous. Indeed, Skaperdas and Gan (1995) pointed out that a more risk-averse contestant has an incentive to exert less effort in the contest because he reduces his safe payment by doing so but a contestant becomes also at the same time more risk averse, which gives him an incentive to increase his level of effort because he reduces the probability of losing this game that way. This is called the self-protection effect.

Fu and Wu (2019) explain that the competitive balance between contenders is key in the performance of a contest. If there is a too large difference between contestants, then the underdog is discouraged while the favorite will be allowed to slack off. Baye, Kovenock, and de Vries (1993) illustrate this logic with a multiplayer all-pay auction model with complete information. In the case where the favorite possesses an excessive advantage, the context is paradoxically generating a higher revenue by excluding him. While keeping only the underdogs in the contest. This can be assimilated once again to cycling races, where if a rider is considered too strong by others, nobody will want to take him a relay, then the favorite will be forced to slow down because he would be beaten if he was pushing forward. This way the average speed of the group is reduced. An experiment using

data from professional golf tournaments demonstrates empirically that the presence of a superstar in a competition tends to lead to a lower general performance.

Fu and Wu (2019) note this has inspired research efforts to study the incentive effects of identity-dependent rules and research for the model that exploits the heterogeneity of the ability of contestants optimally and manipulate the balance of the playing field to make desirable equilibrium behaviors. An all-pay auction model considering two players and complete information showed that a contestant can be favored in two ways: by having his bid being scaled up by a fixed percentage to simulate a handicap or by the addition of a fixed constant from his bid. (Konrad, 2002). Other studies (Siegel, 2009; 2014) describe more general settings enabling for discriminatory rules for the contests.

Strategic behavior in sport contests : application to middle-distance races from the 2010-2019 decade

2.2 Theory of Contests