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

Strategic Behavior in Sport Contests: Application to Middle-Distance Races From the 2010-2019 Decade

1.1 Problem Definition and Objectives

Lots of economic and social interactions consists in a competition where the players expend their effort in order to increase their probability to win a prize. These situation can be research and development (R&D) rivalry between firms or countries to get a lucrative or strategic innovation, bribery to assure a profitable license, patent or contract from the government, the war for a new global market that has been created by a new innovative product, a political election where candidates fight during long campaigns in order to get elected, or candidates who compete for a job, or to win a promotion.

Example of contests can also be found in sport competition. These competitions can take shape in three different forms. Championships, where each player plays against each other and the ranking of the competition is determined by the results of the players against all the players of the championship. Famous examples of championships are the English Premier League or the Bundesliga in football, the Six Nations Tournament in Rugby. Another type of competition is the tournament. Tournaments is a form of competition which takes the shape of a direct elimination competition. The competitors compete pair by pair and the winner can play against the winner of another pair until only one winner remains and win the prize. Famous examples of tournaments are Grand Slam Tournaments in Tennis, or the play-offs of the National Basketball Association (NBA). Eventually, another type of sport competition takes the form of a race. A race is a competition where the prize is given to the first competitor to cross the finish line. This type of competition is different from a tournament or a championship since in a race, the competitors are competing against one another at the same time. They are not challenging each other by pairs like in a game of football or tennis. Therefore, they are the most interesting competition in order to analyze the behavior of agents in a situation where they are faced to many competitors. Races are therefore in regard of their nature more interesting to analyze agents' behaviors in order to find beginning of answers on the behavior of agents in a R&D rivalry between firms or country because a lot of countries or firms are involved at the same time to develop the same technology and the first one to be able to create it will gain an economic or strategic advantage upon the others that can be seen as the prize.

Some models of the tournament theory take an interest in the strategic behaviors of the players in a case where the intrinsic capacities of the agents are heterogenous, which means in a case where there is a favorite and an underdog. Dixit (1987), shows with a model of game theory that if he plays first, the favorite has always an interest to engage a high level of effort, while the underdog has the opposite incentive. Baik and Shogren (1992) extended Dixit's model considering an endogenous choice for the order of intervention of the agents. They show that the underdog has always an interest to play first while the favorite's best interest is to wait and play in second.

The theoretical models on the strategic behaviors of the favorite and the underdog haven't been much studied empirically. The reason why is quite obvious, it is very difficult to find economic situations where the status of favorite and underdog is clearly established and defined and the strategies of the favorites and underdogs are directly observable, especially the order in which they are engaging their effort. This is particularly true in the economic context used by Dixit (1987) and Baik and Shogren (1992) which is the race to innovation.

Boyd and Boyd (1995) avoid this difficulty by analyzing the strategic behaviors of the athletes during athletics competitions (from 800 meters (m) to 10 000 m) at the Olympic Games of 1992. As part of an athletics race, the model of Baik and Shogren (1992) predicts clearly the following course: underdogs tend to start the fastest, then are caught back before being usually passed by the favorite.

Boyd and Boyd used data coming from the races of the 1992 Olympic Games in Barcelona to test this theory. They only looked at the races for which distance were superior or equal to 800 m, which are the distances for which the tactics have a real role to play and where runners run inside a peloton (and not in lanes). For each distance, they visualized the video recordings of the semi-finals and the final. For the short distances (such as 800 m or 1500 m, they recorded the positions of the runners every 200 m while for the longer distances (3000 m women, 5000 m men and 10 000 m), they recorded the position of the runner every 400 m. At the end, their database contains more than 2300 observations spread over 14 races. Concerning the key variable of the study, which is the measure of the runner status before the race, favorite or underdog, Boyd and Boyd used as a proxy the ranking of the runner in the previous race, which means in the semi-finals if the race studied is a final, or the heats if the race studied is a semi-final.

From a general perspective, the results of Boyd and Boyd are clear and coherent with the theory of Baik and Shogren: the course of races see the relative position of the underdog decline during the race. On the contrary, the favorites improve progressively and end up winning most of the time. Boyd and Boyd also notice that the results are clearer for the men than for the women since for men, all races (semi-finals and finals) confirm the theory, except the 800m final.

However, when I read the article of Boyd and Boyd, I noticed some details that was posing me some problems regarding the model's veracity.

First, the proxy that Boyd and Boyd used to determine who was favorite before the race and who was not, which is the ranking of the runner in the previous round, do not seem to me the best way to measure a runner's chances to win a race. I am myself a French athlete running 800 m at the national level since 10 years, I participated to 12 French National Championships, and I never looked at the ranking of a rival in the previous round in order to determine if his chances where greater than mine to win. Indeed, heats or semi-finals are an unreliable information since they runners who are in my race do not come from the same race. I usually make a complex calculation based on his personal best (PB), his ability to finish quickly his races, his recent shape, the races he has won before, his weather preferences, etc. This way I am able to assess what are the odds for me to beat him and what is the best strategy to apply, or at least try to apply, in order to beat him. One of the goals of this thesis is therefore to create a calculated index based on the different data that can be gathered today for a runner. This way the proxy will be calculated the same way for all runners of all races and will give homogeneity to the analyzed races.

My second observation is about the use of the position of the runner as the parameter which determines the level of effort of the runner over the lap. I do not think it is the most accurate measure we can have today of the level of effort for a runner. Indeed, with the race reports that have been given for the last four World Championships races, we are able to determine who has run the fastest and the slowest on each lap because we have the split times for each runner from every kilometers to every hundred meters. This way if a runner did his first interval much faster than others, then his second interval slower but he had taken such an advantage over the first interval that he has not be caught up by others, he will not be any more considered as the one who has given the greatest effort on the interval. This change is, in my opinion very important if the purpose of the study is to

measure correctly when runners put their highest level of effort in regard of the pre-race status.

The third observation is that they only try to verify if there is a difference in the strategic behavior of men and women. Not only do I believe there should not be a big difference in the strategic behavior of men and women, but I also believe that there are other parameters about the runners or the race that could explain a different behavior, such as the culture of the runner, or the density of the race. These parameters would be, in my opinion, very interesting to test in order to see if one of them changes the strategies of the runners.

The objective of this thesis is to replicate the experiment of Boyd and Boyd with the data of the last world championships and to apply all the modifications that I mentioned above. Moreover, I will be studying the other potential factors of strategic behavioral change in order to see if Baik and Shogren results are still coherent with this new methodology. Therefore, we will be trying to give an answer to the question: how effort is expended over time by a runner in athletics events depending on the pre-race status of the same runner and what other parameters may affect the runner behavior? This question is almost the same that the one Boyd and Boyd asked themselves and the purpose of this study will be to see if we can find similar results when applying a more efficient methodology.

1.2 Course of the Investigation

The thesis design will take the shape of an empirical study. It will therefore analyze through collected data the phenomenon observed by Boyd and Boyd over the 1992 Olympic Games in Barcelona.

It will start with an overview of the relevant literature review over the field of sports as an application to economic theory and over the field of strategic behavior in contests. This theoretical background will help me to justify why sport and athletics are a relevant application in order to prove Baik and Shogren results in R&D rivalry and to have a better understanding of Boyd and Boyd model in order to be able to modify it correctly and improve it. After that, I will define a methodology that would try to improve Boyd and Boyd model. Most of the articles come from EBSco Host database and some others I took on Google scholar, especially for the scientific articles that are not economic science or game theory related.

I will try first to improve it by changing the proxy variable, which is the key variable of the study since the purpose of the study is to measure in what way the behavior of a runner is affected by his pre-race status. I will change it by implementing a calculated methodology for the parameter which consider several data such as the PB, the season best (SB), the honors. All the data concerning the different runners background, honors and level, which will determine the index that will inputs the proxy of who is favorite of the race and who is an underdog will be collected from the database of World Athletics, (ex-International Association of Athletics Federations) which is the international federation of athletics.

Then I will analyze all the race reports of the World Championships finals from 2013 to 2019, which makes 4 world championships for the distances of 800 m to 5 000 m (data was not available for 3000m steeplechase and too complicated to collect for 10 000 m) in order to collect data which will enable me to determine the level of effort for each athlete for each interval studied. Only middle-distance races will be analyzed since they are the only track and field event on which a strategic component is involved.

After the compilation of all this data, I will analyze the way the level of effort for a runner was influenced by the pre-race status of the runner and by the place of the interval in the race, or in other words, is it the beginning or the end of the race? However, I will also aggregate the data coming from all the races much more than Boyd and Boyd did in order to be able to find out if my observations are true in particular for specific cases of race taken individually only or a more general phenomenon. I will also study the model separating the population studied under different parameters such as gender of the runners, cultures of the runners and density of the race.

Once the results of these regression will have been obtained, I will be able to verify if my method observe the same phenomenon that Boyd and Boyd did and find similar results to them. I will be also able to discuss the results found and explain why some differences between my method and Boyd and Boyd's method could be found. I will also try to find answers to the possible differences I will find inside the population studied when I will have looked at the variations inside the population looking at the different parameters that I mentioned above.

Once all this will have been done, I hope I will be able to give an answer to the research question I asked myself and be able to assess Baik and Shogren results and Boyd and Boyd results or not. I hope also to find other interesting results that will create something

Boyd and Boyd did not notice when they did their experiment and that these findings will bring new insights in the field of strategic behavior.

2.1 Why sport is a good field to study economics?

First, I would like to present examples where the study of sport cases helped the economists to better understand economic theories. My goal is here to prove that the study of sport is a relevant case in order to understand economic, strategic, or behavioral phenomenon.

Sport economics strongly developed recently (Andreff & Szymanski, 2006). People generally think of it as the use of economic analysis tools to study the sports industry. However, this thinking forgets a second face of sport economics: when sport helps better understand economics. (Eber, 2008) Eber wrote in 2008 an article whose topic was this second face of sport economics. He explains that the observation of high-level athletes can give interesting leads to economists since sport competition delivers a frame particularly fit to the verification of some economic theories as it is a highly competitive context, with actors very motivated and gifted (i.e. rational), clear and stables rules for the game, and perfectly objective results or performances measured without any ambiguity by a score, a distance or a time.

The next part is going to be a bunch of examples where sport have helped verifying economic theories that are very difficult to observe in the economic world due to the complexity to get some data on the level of effort of the agents, their level of skills, their final performance, etc.

Economic theory took a lot of interest starting from the eighties and the articles of Lazear and Rosen (1981) and Rosen (1986), at the type of competition which are tournaments, i.e. highly competitive situation characterized by the fact that the prize of an individual depends only of his ranking compared to the others. Tournaments models describe very well the reward systems used for the corporation management, for sales team members

or in academical environment (bonuses, promotion, etc.). Under hypotheses, the tournament system present interesting normative properties since it is based on an efficient incentives structure. (Lazear and Rosen, 1981).

Few empirical studies have been led on the incentive effects of tournaments. Econo-metrical studies on «real» data turn to be delicate since it is very difficult to identify precisely, within an organization, what are the tournaments set in place, what are these tournaments rules, what are these tournaments prizes, etc. Moreover, it is very difficult to measure the level of effort of the agents. From an experimental point of view, Bull, Schot-ter and Weigelt (1987) found relatively inconclusive results.

In front of this lack of empirical elements, Ehrenberg and Bognanno (1990a) used the results of great tournaments of American golf (tournaments from the Professional Golfers Association (PGA) Tour) that took place in 1984.

It is here important to point out that tournament theory is in opposition to the fair-wage theory based on equity. Indeed, when tournament theory insists on the dispersion of the revenues, the hypothesis of group cohesion proposed (Levine, 1991) insists on the perverse effects (jealousy, mistrust, disincentive) of a very unequal salary structure, especially in the case of teamwork. Therefore, tournament theory applies better to individual sport than collective sport. That's why empirical test on tournament theory are mostly based on individual sports (golf, tennis, running) while those of fair-wage theory are based on team sports (baseball, basketball, hockey, football).

To test the hypothesis that states that tournaments have positive incentive effects Ehrenberg and Bognanno estimate a simple econometrical model with the following structure:

where _sly is the final score of the golfer ] in the tournament i, TPRIZEL the prize money of the tournament i, xi a vector of variables on the difficulty of the tournament, the weather conditions, etc. y, is a vector of proxies for the quality of the player] (average score over the season, etc.), zi is a vector of control variable on the quality of the competition in the tournament i, and eu the error term.

Tournament theory predicts that al < 0. All other things being equal, higher prize money should give an incentive for a bigger effort from participants, and this bigger effort from participants should lead to better performances, which means lower and lower scores. (In golf, the lower is your score, the best you are). The estimations of Ehrenberg

and Bognanno (1990a) lead as predicted to a coefficient ??? significantly negative. More precisely, they find that a raise by a 100,000 $ of the global prize money leads to an average diminution of the number of strokes. In average, every player plays 1,1 strike less during the entire tournament than before the raise of the global prize money which proves that players have an incentive to play better with a raise of the global prize money. Ehrenberg and Bognanno (1990b) replicated their test on the results of the great Europeans tournaments of 1987 and found similar results. However different results were found in a study on the PGA data for the year 1992 (Orszag, 1994).

Other empirical studies based on data coming from sport seem to confirm this result since the positive link between incentives and performances stated in the tournament theory has also been observed in other sport competition such as running in road races (Frick, 1998; Maloney and McCormick, 2000), and tennis (Barget, Llorca, & Teste, 2011) either that you take a population of men (Sunde, 2003) or women (Lallemand, Plasman, & Rycx, 2008). However, it is important to note that concerning the road races in running, more qualified results have been found (Frick & Prinz, 2007) and Lynch and Zax (2000) couldn't find a strong relation between performance and revenues, once the quality of the participants at the beginning of the race was controlled.

A large number of experimental studies have been made in laboratories in order to test the empirical validity of the equilibrium in mixed strategies. The result of these studies is on average not really positive. Usually, except for perfectly symmetric games, observations are very far from the theoretical predictions (Camerer, 2003, Chapter 3). Therefore, the conclusion of experimental works is usually to invalidate the mixed strategies theory. The only cases in which the empirical observations match the equilibrium are the studies of cases lead in the field of professional sport, tennis and football.

The first study mobilizing observations coming from sport in order to evaluate the accurateness of the concept of mixed strategies equilibrium has been done by Walker and Wooders (2001). In this study, Walker and Wooders interest themselves to the case of tennis, and more precisely, to the strategic game which consists in service. Indeed, during a service, two players, the server and the returner, are engaged in a game in the sense of a strategic interaction. The gain issued from the game for each player is the probability to win the point. The server has to choose between two strategies: to serve on the right (R) of the returner or to serve on the left (L) of the returner. As the speed of the serving ball

is very high, often above 200 km/h, the returner has to anticipate the choice of the server. It brings him to play either «Right» if he anticipates that the server is going to serve on his right, or «Left», if he anticipates that the server is going to serve on his left. That way, we can consider that both players play simultaneously and not one after the other.

This makes clearly a constant-sum game since the probability to win the game for a player is the probability to lose it for the other. This game, in which it is essential for the players to be unpredictable, admits a unique Nash equilibrium in which the server and the returner decide both to take a mixed strategy.

What predictions can we make starting from the theoretical analysis of the game? By the definition of an equilibrium in mixed strategies, both pure strategies (L and R) have to, at the equilibrium, bring the exact same Esperance of win to the server. This way, for the server, the percentage of success (i.e. the rate of won points) has to be the same when he serves on the right of the returner that when he serves on his left. Walker and Wooders studied ten finals of Grand Slam Tournaments or Masters. The Table 1 gives the aggregated results for all ten games.

This table is really impressive. On the all set of ten games studied, which represent in total 3026 services, the rate of points won are almost identical for both strategies (64% for L, 65% for R). This agrees to the theoretical prediction of mixed strategies equilibrium.

Another characteristic of the Nash equilibrium in mixed strategies is that players have to choose randomly. This way, there should not be a correlation between the present choices and the past choices. This characteristic is nevertheless not verified by the study

of Walker and Wooders. Indeed, players modify their choice too frequently in comparison to a random decision, which means, like in the other experiments made in laboratory, that there is no independence between the present actions and the past actions. This result come to balance a little bit the conclusion of Walker and Wooders in favor of the mixed strategies theory.

A replication of Walker and Wooders' study was done on another set of games (Hsu, Huang, & Tang, 2007). They find even clearer results in favor of mixed strategies theory since both properties of the mixed strategies equilibrium, i.e. an identical rate of success on both pure strategies and a random choice between strategies, are verified on this replication study.

The same type of study has been made on penalties in football (Chiappori, Levitt, & Groseclose, 2002, Palacios-Huerta, 2003). Like for the service in tennis, the penalty in football is a simple situation of strategic interaction between two players, the striker and the goalkeeper with the characteristics of a constant-sum game. They consider that both players have two pure strategies. The striker can choose to shoot on his right (R) or the shoot on his left (L). To simplify, they suppose that the striker does not have the possibility to shoot in the middle, however the analysis would be exactly the same if we included this third pure strategy. Like the ball only takes 0,3 seconds to reach the goal line because of the strength of the strike, the goalkeeper has to anticipate a strike on his right (R) or on his left (L) before the striker has even touched the ball. Here again, they do not consider the possibility for the goalkeeper not to dive, which means to stay put in the middle of the goal. To support this choice, observations have been made that a psychological bias exists in favor of action that pushes goalkeepers to dive too systematically compared to strikes distribution. (Bar-Eli, Azar, Ritov, Keidar-Levin, & Schein, 2007). In other words, this situation is a simultaneous game, both players ignoring the choice of the other at the moment they make their decision.

In this constant-sum game, the gain of the striker is the probability for him to score a goal while the gain for the goalkeeper is the probability for the striker to fail. On the basis of the 1417 penalty strikes analyzed by Palacios-Huerta (2003), the values of gains are presented in the following table:

In every cell of the table, the first number is the gain of the striker, i.e. the probability, in percentage that he scores, and in second the gain of the goalkeeper, which is the complementary probability.

They name ??? the probability at the equilibrium that the striker shoots on the left, ??? the probability that, at the equilibrium, the striker shoots on the right, ??? the probability at the equilibrium for the goalkeeper to dive on the left and ??? the probability that, at the equilibrium, the goalkeeper dive on the right. It is very easy to verify that the unique equilibrium in mixed strategies is characterized by:

The data of Palacios-Huerta concerning 1417 penalties shot in games of the English, Spanish or Italian league. The following table compares, at the aggregated level, the observed strategies with the theoretical predictions.

The observations are very close to the theoretical predictions. This way, the mixed strategy equilibrium turns out to be a good predictive model of the strategies actually adopted by the strikers and the goalkeepers.

For a given player, the concept of mixed strategies implies that he has to have the same success rate on both pure strategies. Let's be French a little and take the case of Zinédine Zidane, one of the best football players ever and one of the players studied by Palacios-Huerta. On 40 observed penalties, he shot 19 times on the left (48%) and 21 times on the right (52%) with success rates almost identical of 74% and 76% respectively. This way Zinédine Zidane, aside of being a great football player, is a pretty good game theorist too!

However, the strategic choices of penalty strikers or goalkeepers can be largely unconscious, but their huge expertise of the game drive them to take naturally the optimal strategies (Palacios-Huerta, 2003, p.406).

Palacios-Huerta also shows that the decisions of the players are random, as a consequence, the present choices available are independent from the past choices. This way, the second implication of Nash equilibrium in mixed strategies, which is that the decisions are taken randomly, is also verified by this experiment on the way strikers and goalkeepers act strategically in the penalty game.

From a methodological point of view, it is important to note that a major difficulty comes from the heterogeneity of the strikers and the goalkeepers. Indeed, the basic characteristic, the equality of probabilities to score for each of both pure strategies) is not preserved by the aggregation of heterogenous players. Thanks to about forty observations for each player, Palacios-Huerta is able to realize individual tests (i.e. by players). He proposes also aggregated tests that make it possible to evaluate if, on the global study, the probabilities to score are the same on the right and on the left for each player of the sample, even if they can be potentially different between the players.

Very similar results to those of Palacios-Huerta were found (Chiappori et al., 2002; Coloma, 2007) and we can therefore affirm that players choose optimal mixed strategies and their decisions are random. Another study, which is this time studying another typical phase of the game, when the striker is at the beginning of the penalty area (Moschini, 2004). This study also finds results that match the theoretical predictions by the Nash equilibrium in mixed strategies.

What conclusion can be taken from all these studies based on data coming from sport competition? Classical experiment done in laboratory show that beginners in a game that just started to play this game do not adopt the optimal strategies in games with a mixed strategies equilibrium. However, the studies considering the case of tennis or football show that professional athletes find the equilibrium strategy. On the spectrum of expertise, we find at one extremity the beginners without a lot of experience, for whom theory does not work very well, and at the other extremity, we have professional athletes, for whom theory seems to apply correctly. Of course, the majority of «real» constant-sum game happen in an intermediary context between these two extreme cases, however, this intermediary level has not been much studied and not a lot of things is known about it in the scientific world.

In the contract theory, the employer (the principal) has to conceive an incentive contract in order to assure that the employee (the agent) will give the desired level of effort. This theory stipulates that the assurance to keep his job thanks to a long-term contract can create a classical problem of moral hazard giving an incentive for the employee to reduce his efforts, i.e. shirking. The evaluation of this theory turns out to be complex, because it is difficult to estimate in a reliable way the individual performances of the employees. Once again, the interest of data coming from sport comes from the fact that the individual performances, so the contribution of the employee (the player) to the company (the team) can be measured without any ambiguity. Some studies have mobilized the data on professional American team sports to test the hypothesis of a disincentive effect of long-term contracts. The purpose of the study is simply to see if, as the hypothesis of a disengagement of the effort after the signing of a long-term contract states, the performances of the players diminish just after the signing of such a contract.

The results of the studies are rather inconclusive. Some authors observe a modification of performances in the expected way, with a diminution of the performances of baseball players (Woolway, 1997; Marburger, 2003) and NBA basketball players (Stiroh, 2007). But other researchers do not find a significative difference in the performances before and after the signing of the new long-term contract for baseball players (Krautmann, 1990; Maxcy, Fort, & Krautmann, 2002). In fact, in the case of baseball (Scoggins, 1993) and NBA basketball (Berri & Krautmann, 2006), it is shown that the conclusion depends in a crucial manner in the way the performance measure is done. Another observation can also be made by saying that in team sports, the performance and influence of a player is not always perfectly measurable. Indeed, some players have a style of play based on moves without the ball that are going to draw the opponent's defense to him, giving space for his teammates to go to the goal or basket and score more easily. His presence is therefore absolutely essential to the team, but it will not be measured in statistics.

Behavioral economics consist in finding stronger psychological foundations to the standard economic theory. This leads to take into consideration the judgements bias of individuals, the social norms or the social pressure. A few studies have worked from the sport field to measure empirically the impact of social pressure on favoritism (2.1.4.1), or more generally, the role of emotions on the behaviors and the performances (2.1.4.2).

Social pressure plays an important role in a large number of economic contexts. In the wave of the new behavioral economics, it is today clear that social pressure impacts the individual behaviors and can lead to the shape of implicit corruption that is favoritism. (Prendergast & Topel, 1996) While many experimental results observe a decisive impact of social pressure on the individual behaviors, econometrical verifications on real case are very rare, because it is very difficult to find reliable data. Sport competition have made possible to test in a real-life environment the effect of social pressure.

Three studies considering the case of professional football refereeing analyze the behavior of referees in order to verify the role of social pressure on favoritism (Garicano, Palacios-Huerta, & Prendergast, 2005; Sutter & Kocher, 2004; Dohmen, 2008a). The idea to study if referees, surely pushed by the pressure of the stadium crowd, really have a tendency to favorize the home team (home bias). The problem is of course to be able to find an objective measure of the referees' behavior. However, there exists one decision that fits quite well to such a measure. It is the extra-time that the referees grant at the end of the game in order to compensate the stops that happened during the regular time for injuries, substitutions, unsporting, etc.). Indeed, if the law is fixed concerning the general principles guiding the management of this extra time, the field referee remains free to decide when to blow the end-of-the-game whistle.

A study (Garicano et al., 2005) use a database on refereeing int the Spanish league in order to see if the referee are not more in a hurry to blow the end-of-the-game whistle if the home team is leading with a weak margin, which means with a one goal advantage, and less in a hurry to blow the whistle when the home team is lead by one goal. The results are very clear. For example, in the case where the home team is lead by one goal, the extra-time is around 35% longer than on average, and when the home team leads by one goal, it is 29% shorter than on average. Similar studies have been made (Sutter & Kocher, 2004; Dohmen, 2008) on the German Bundesliga. Their results confirm largely those of Garicano et al. in favor of the hypothesis of favoritism for the team playing at home.

One of the great contributions of behavioral economics is to show the fundamental role of emotions in the behaviors of individuals. Here again, the observation of the world of sport can turn out to be very informative.

A study on missed penalties on German Bundesliga (Dohmen, 2008b) confirms that emotion and social pressure have a major role in individual performances, even for high level athletes. In particular, individuals can «crack under the pressure», an effect much studied in social psychology, but neglected by economists. However, like Dohmen notes: «There are plenty of situations in which pressure arises in the workplace. Knowing how individuals perform under pressure conditions is crucial because it has implications for the design of the workplace and the design of incentive schemes» (Dohmen, 2008b, p. 636). It is clear that it is very difficult to obtain in the world of enterprise the kind of data necessary to the evaluation of an eventual perverse effect of the pressure on the performances. Dohmen (2008b) goes around this difficulty by studying the missed penalties of German footballer since the foundation in 1963 of the German professional football league, the Bundesliga. He chooses to give a strict definition of what is «cracking under the pressure» in the case of a penalty since he considers that it returns to an off-target strike (or on the posts), i.e. a situation of complete failure without any interference of the goalkeeper. He observes then that the proportion of missed penalties «under the pressure» is higher for team that are playing at home (7.54% against 5.57% for teams playing away from home), but that it does not depend significantly of what is in stake with the success of this penalty's importance (score at the penalty's time, decisive game at the end of the season). This way, the players seem more sensitive and fragilized by the pressure of the public than by the context of the game. This could have very interesting development in the enterprise field as pointed out Dohmen: «The empirical result of this paper implies, for example, that workers who might feel they are being observed, especially by well disposed co-workers or spectators, perform worse than they otherwise would» (Dohmen, 2008b, p.652)

Besides, it has been noticed on the basis of a thorough analysis of professional tennis games (Paserman, 2007) that men and women do not behave in the same way under the pressure. More precisely, his data on the speed of the service, the percentage of first service or the length of the exchanges for the points suggests that men maintain the same strategy and the same level of performance in key moments of the game, while women turn to a less aggressive strategy (slower services, longer exchanges) for important points. This result that there is a difference between men and women in the manner to manage the pressure in a highly competitive context confirms a whole set of experimental results. (Gneezy, Niederle, & Rustichini, 2003; Gneezy & Rustichini, 2004)

Another study based on data coming from sport have brought forward the role of emotions in the performances. (Palomino, Rigotti, & Rustichini, 1998) They estimate, on the basis of 2885 professional football games, the probability to score a goal at the different moment of the game. They study how this probability is linked to three founding determinants of the performance of a football team:

- the abilities, measured by indicators such as the number of scored goals or conceded goals over the whole season.

- the strategy, defined by the choice to attack or to defend in reaction to the score of the game depending on the time remaining of the game (beginning, last fifteen minutes). This is measured by the way the probability to score a goal depends of the score and the time remaining for a team.

- the emotions, that regroup an entire set of emotional and psychological factors about the game, with the advantage of playing at home.

The results show clearly that the three factors intervene simultaneously and interact in the determination of the performance, i.e. the probability to score. In detail the results show that:

- A better ability than the opposite team multiplies the probability to score by a factor between 2.1 and 2.4

- Different strategic situations make the probability to score vary following a factor between 1.4 and 2.2

- To play at home (emotion factor) multiplies the probability to score by a factor between 1 and 2

In total, these results show that the three forces have equivalent importance in the determination of the performance, which is the probability to score.

Palomino et al. (1998) think their results could have important implications in economics since they clearly show that psychology and rationality occur simultaneously in the behavior of actors and the result of the game. This type of phenomenon is possibly also existing in the economic world. Very general factors occurring in the behavior and performances of a football team is most probably also occurring in other type of organizations that are living in a highly competitive environment, such as enterprises. Palomino et al. (1998) even note that:

«Soccer teams are examples of economic organizations who face each other in a very standardized, repeated, situation (a soccer match), which is therefore easy to study. Their behavior can provide insights on the way an economic organization

works and in particular on the way strategic and emotional factors interact in its life» (Palomino et al., 1998, p.30)

This way, they hope that their results to stimulate a new axis in economic theoretical research with the building of new models able to take explicitly into account the interactions between reason and emotions.

2.2 Theory of Contests

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.

2.3 Strategic Behavior in Contests

The purpose of this part is to remember the key theoretical background and the surrounding empirical studies about the experiment of Boyd and Boyd (1995), which is the key experiment of this study, and around which this whole thesis is built. I will therefore in a first part remember the findings of Dixit (1987) and Baik and Shogren (1992a) on which the experiment of Boyd and Boyd (1995) is built. Then I will present other empirical studies about the strategic behavior in contests. After that, I will present thoroughly Boyd and Boyd's experiment that will be replicated in this thesis.

The first to consider the case where one of the players is given a chance to pre-commit strategically his effort is Dixit (1987). He is the first to impose an order of moves based on a Stackelberg leadership which brings sequential moves in the contest. In his paper, he studies two models. A model with two players, and a model with several contestants. For the model with two players, he finds that the player which is the favorite is the player who has an incentive to overexert. He explains it by the Figure 1 coming from Dixit (1987) paper. In the panel (a) of Figure 1, The curve TT shows the total growth return for player 2, [1 - p(x1, x2)]K, where p(x1, x2) is the probability for the player 1 to win and K is the prize to win, as a function of his effort x₂ for a fixed level of x₁. In the part (b) of the graph, the curve MM is the corresponding marginal. Dixit (1987) explains that a slight increase in x1 shifts these curves to T'T'and M'M' respectively and the marginals cross at the point P. If there is an increase to its left in x_lreduces the marginal return in

x₂, which leads the player 2 to reduce x₂if x_?is pre-committed. If the player 1 is the favorite, then (1 - p) is low and x₂lies to the left of P. Therefore, it is in player 1 strategic interest to commit to overexertion. Dixit (1987) notes however that the opposite holds if the player 1 is the underdog.

Dixit also develops the corresponding intuition with the Figure 2 using reaction or best response functions. In the case where player 1 is the favorite, in the neighborhood of the Nash equilibrium N, the favorite's best response function (R?) is sloping upward and the one of the underdog (R2) is sloping downward. In this case, the player 1 has an incentive to make a strategic pre-commitment to ma higher x?, which moves the outcome from N to S?. Player 2 wins he can commit to a lower x2, which will move the outcome from N to S2. When the pre-commitment is made by the means of a separate variable Y?(respec-tively Y2), then the effect is to push the best response functions R? to the left to R? ^?and R2 downward to R2 ^?. If you compare the position of these curves compared to the p =

? line which represents the line where both players have the same probability of win-2 ning, we can see that at the Nash equilibrium, the outcome of the favorite (player 1) gets

higher if he increases his level of effort x? while the underdog (player 2) sees his outcome increase if he lowers his level of effort x2.

This result support the idea that in a situation where there is a favorite and an underdog, the favorite has to overexert his effort compared to the underdog, Dixit (1987) explains it with the phenomenon that can be seen in sport where the manager of the team declares that «since he is expected to win, he mut try all the harder» while the underdog says to be «under no pressure, and is just going to enjoy the occasion».

However, Baik and Shogren (1992a) demonstrates in a comment of Dixit's model that the favorite will never overcommit effort if you extend Dixit's model to allow for endogenous order of moves. They take an endogenous order of moves, in this case, the favorite will tend to find it advantageous to wait until the underdog moves while the underdog's best strategy is to make the first move and not to wait.

For this, they consider a situation in which two risk neutral players (1 and 2) compete against each other to a K prize. If xland x2 are the players' effort levels and p(xl, x2) the probability that player 1 wins. For this situation the players' expected payoffs are:

Following Dixit, they assume that p(x_l, x₂) takes either the logit form p(x_l, x₂) = fl(xl)/[fl(xl) + f2(x2)] or the probit form p(xl,x2) = G[fl(xl) - f2(x2)]. They also assume that f?(0) > 0 for i = 1, 2 and that f_l, f₂, and G [f_l(x_l) - f₂(x₂)] are increasing functions.

They then draw the Figure 3 by defining QQ as the locus of points that satisfy f₁(x₁) = f₂(x₂). Since f₁(x₁) and f₂(x₂) are increasing functions, the curve QQ is sloping upwards. Moreover, f1(x1) < f2(x2) above the curve, and f1(x1) > f2(x2) below the curve. Given the assumptions, they find that the reaction function of player 1 r1(x2) is increasing in x2, reaches the maximum on the curve QQ and then decreases, while the reaction function of player 2 r2(x1) is increasing in x1, reaches the maximum on the curve QQ and then decreases. They assume finally that f1(x1) > f2(x2) at the Nash equilibrium, which is equivalent to saying that the player 1 is the favorite and the player 2 is the underdog.

Figure 3. Reaction functions when player 1 is the favorite (Baik & Shogren, 1992, p. 360).

This graph allows them to say that Player 1's expected payoff is decreasing when

one moves up along his own reaction function while the Player 2's expected payoff is decreasing when one moves right along his own reaction function.

Then Baik and Shogren (1992a) consider the following game: The players decide in a first time and announce simultaneously publicly the periods in which they will choose their level of effort. The players then choose simultaneously their effort level knowing when the opponent will choose his level of effort. In the case where both players announce the same period, the subgame would be a simultaneous-move game.

If the favorite announces the first period and the underdog announces the second, then the favorite is the Stackelberg leader and the underdog the Stackelberg follower. Baik and Shogren call this subgame the favorite-leader subgame. In this subgame, the underdog

chooses his own level of effort after having observed the favorite's level of effort. The opposite game is the underdog-leader subgame. Baik and Shogren note that the simultaneous move subgame payoffs are (????, ??? ?), those of the favorite-leader subgame are

(?????, ??? ??), and those of the underdog-leader subgame are ???? ??, ??? ???. From the analysis

of the graph, they obtain that ????? < ??? ? < ??? ?? for the underdog. Therefore, they demonstrate that in the announcement stage of the game, the underdog will choose the first period without caring at what strategy is the favorite taking. They explain that the underdog has no incentive to wait for the favorite's choice as this strategy of selecting effort in the second period is a dominated strategy. And since the underdog chooses the first period,

the favorite chooses the second since ??? ? < ?????. The unique subgame-perfect equilibrium shows that the underdog will choose to move first in the announcement stage and the favorite will choose to move second. This demonstration enables them to make the following proposition:

«In the equilibrium, the favorite moves after the underdog. Endogenous order of moves results in both players expending less effort compared with the Nash equilibrium. Consequently, strategic behavior leads to under-commitment of effort associated with a contest over a fixed reward.» (Baik and Shogren, 1992a, p. 360)

This result is very interesting in the theory of contests because this suggests that in a situation with endogenous order of plays will lead to see the underdog acting as a leader in terms of strategic behavior. They explain it by taking an R&D analogy where two firms are trying to develop a product based on their relative power derived from the research stage. The firm that has an advantage in the research is considered the favorite while the other is considered the underdog. Then if their model prediction is true, the underdog will lead research and the favorite will follow. They note however that they do no eliminate the potential for industrial sabotage (Hirschleifer, 1988) with the example of a firm that decides to aim for a «victory at all costs» and which will be putting all its effort to disabling its rival.

If strategic behavior in contests is used a lot in R&D, innovation, academic publishing and sports (Vojnovic, 2016), they are however much better understood in principle that in practice (Carpenter, Matthews, & Schirm, 2010). Besides, strategic behaviors of contestants that are unintended like self-promotion or sabotage are not often studied. Indeed, it is difficult to study such behaviors since contestants usually try to hide these behaviors

because they are usually associated with the idea of being illegal or immoral (Charness & Levine, 2004). Because of this, the studies investigating strategic behavior are usually restricted to laboratory experiments or formal models of behavior. (Harbring & Irlen-busch, 2011; Lazear, 1989; Münster, 2007).

However, a few empirical studies exist that try to investigate the «real» strategic behaviors of individuals in «real situations».

Concerning the behavior in an asymmetric contest, Baik and Shogren (1992b) study in a laboratory experimental design the model of strategic behavior proposed by Dixit (1987). For this, they designed an experiment as follows: the subjects enter a room and are divided in two groups, favorite and underdogs. Each subject read the experimental instructions concerning their group without being able to communicate in any possible manner. The instructions told the subjects that they would be in competition against an opponent to try winning 360 tokens, with each token being worth $0.0025. The probability that a contestant will win depends on the number, which represents the expenditure, he selected and the number the opponent selected in an expected payoff matrix. Therefore, the higher number a contestant selects, the more chances he gets to win, but his cost would increase accordingly. If a player loses, his number is subtracted from his opportunity cost of participating in the experiment ($5/hr.) To represent the asymmetry, favorites and underdogs are given distinct payoff matrices but both players have both payoff matrices at their disposal. Then the two players chose a number from their payoff sheets simultaneously in order to test the Nash benchmark game. To test the favorite first-mover game, the favorites were told to select their number first while the underdog had to wait the favorite's choice before selecting his level of effort. 20 games were done in the experiment. Their results are that the support to Dixit's model where the favorite will expend more effort in the Stackelberg case is only partial. They found indeed in this experiments that in some cases, the underdog tries harder because probably he has nothing to lose.

Concerning the unintended behaviors such as sabotage, Riedl, Grad, and Lettl (2019) made an experiment made on the digital environment, which is really interesting nowadays since many contests take place in these environments. The particularity of this environment is that the barriers of entry are very low (Boudreau, 2018) which means that even low skilled contestant can enter the competition, which increases the heterogeneity between contestants. This phenomenon is called the «rise of the amateur» (Howe, 2008). Moreover, this environment enables large number of contestants in the contests thanks to

Figure 4. Hypothesized relative race positions over time (Boyd & Boyd, 1995, p. 1039).

the internet compared to prior research. Riedl et al. (2019) major findings are that self-promotion decreases as the contest size increases and that the contestants that have the higher ability are less likely to self-promote compared to contestants who have a lower ability. Moreover, they found that the contestants with the higher ability are the most likely to choose the strategy of sabotage.

Boyd and Boyd (1995) observe the opposite theoretical findings of Dixit (1987) and Baik and Shogren (1992a) and try to find an empirical verification to these theory by finding a real situation in which to observe the behavior of agents. But this cannot be easily done in situations involving strategy, where the economic actors are loath to reveal any information concerning their strategic intention because of competitive reason. They decide therefore to apply these theoretical results to the field events of the Olympic Games of 1992 in Barcelona. They choose this example because in athletics, athletes have the freedom to make decisions on how to strategically expend or conserve effort during the event, which makes it possible to apply Baik and Shogren (1992a) findings because this incorporates the endogenous order of moves. Therefore, it will be the findings of Baik and Shogren that they will be trying to prove, which are, when applied to running events, the pre-race favorite being conservative at the beginning of the race, then increasing his effort as the race develops while the underdog will start quickly before being caught up by the favorites at the end of the race.

The figure 4 present these predictions graphically showing the evolution of the position of the favorite and the underdog as the race progress towards the finish line. Their analysis carries of the running distance from 800m to 10 000m and the distance field events which are the long jump, triple jump, shot put, discus throw, javelin throw, and hammer throw.

Boyd and Boyd (1995) empirical model is built the following way: they consider a race with T successive intervals, or laps. They define _Pit as the relative position of the runner i at the end of lap t, with t = 1, 2, ... , T. Their interest is to analyze the manner in which Pit evolves over the course of the race. This is modelized with the relationship:

If the theoretical results of Baik and Shogren are correct, underdogs have an incentive to overexert effort early in a race whereas favorites have an incentive to let them do, saving energy for the end of the race. In this case, 13o is negative, 131 is positive, and both are statistically significant.

Boyd and Boyd (1995) also note that the way a runner choose to exert effort rationally and strategically depends in part on his physical attributes. For example, some runners choose strategically to use their superior foot speed by waiting the final hundreds meters to unleash their final kick to the finish, but tall runners may choose to go to the front of the peloton to minimize the risk of being jostled by the other runners with their long legs. These unobservable individual characteristics may be a form of misspecification of the model and is potentially a bias for the estimates ??? and ???. Boyd and Boyd use therefore fixed effects (Mundlak, 1978; Greene, 1993) to separate individual characteristics that affect the relative position over time (???) from lap-to-lap strategic behavior stemming from the individual status as a pre-race favorite or underdog (???). Each ??? is unobservable, but constant over time, therefore its effect can be removed by taking deviations from the mean. Furthermore, they observe that by specifying ??? as a function of pre-race status as given in equation 2, what seemed to be individual-specific slope characteristics in equation 1 are estimated by ??? and ???. This way, Boyd and Boyd (1995) are able to obtain unbiased estimators for equation (3) using fixed effects model:

(5) ???? - ??? = ???(??- ??) + ???(??- ??)????? + ????
where ??? = 1 /?? ? ????

event, ?? is the mean lap of the event and the error terms ???_?are again independent and identically distributed. They eventually pool data for each athlete within an event which enables them to estimate equation 5 using ordinary least squares, which provide them consistent estimates of ??? and ??? for each race.

For the running events, they analyzed the races by observing the videotapes in order to record the position of each athlete at the end of the lap. Positions were recorded every 200m for the 800 and 1500m while it was recorded every 400m for the longer distances which are 3000m for women, 3000m steeplechase for men, 5000m, and 10 000m. They did this for the semi-finals and the final of the competition which spread out over 14 races. The proxy for the underdog-favorite variable (?????) was the runner's finishing position in the previous race of the competition, i.e. the semi-finals if the analyzed race was the final, or the heats if the analyzed race was a semi-final. The regression for 14 men and women races is shown in the table 3.

Table 4. Regression Results for Men's and Women's Running Event. (Boyd & Boyd, 1995, p. 1041)

They find that the signs of 85% of the estimates are consistent with Baik and Shogren's hypothesis that underdogs tend to overexert effort early in races with 75% of the estimates being statistically significant at the level of at least 10%. This is shown by the fact that 85% of the estimates have a negative ??? and a positive ???. They also note that the magnitude in the ??? coefficients in absolute value is inversely proportional with the distance of the race. Which means that pre-race favorites move to the front of the peloton much faster in short races, which can be explained by the fewer number of laps of short events. In a similar manner, the ??? coefficient varies inversely with the race distance in absolute value. Boyd and Boyd explain it again by the greater number of laps but also by the greater number of participant to the races (8 participants for an 800m against 15 for a 5000m) which increase the likely range of the underdog-favorite proxy.

For the distance field events, Boyd and Boyd bring a slight modification to the equation 5 to estimate the following model:

where ???? is the distance of the ??th athlete's throw or jump on her ??th attempt, ??? =

? ??? is the ??th athlete average distance over her ??? throws or jumps ???? is the ??th

athlete's mean number of non-foul attempts, and the error terms are again independent and identically distributed. For the events they took for the underdog-favorite variable the distance of the athlete best attempt in the qualifying round. This way the hypothesis of Baik and Shogren are the same for ??? and ??? than for the running races. They obtained for the distance field events the following results:

Table 5. Regression Results for Men's and Women's Field Distance Events. (Boyd & Boyd, 1995, p. 1042)

The regression results show that the coefficients are consistent with Baik and Shogren hypothesis for three of the events (men's discus and hammer, women long jump). But from a general perspective, the distance fields events results are more ambiguous than the running events results. They give for an explanation that they do not take in the analysis the weather since in a race, the runners face the same weather since they are running simultaneously. However, in distance field events, competitors are jumping or throwing one after another, so the weather conditions are not always exactly the same for every athletes. They explain also this ambiguity by the fact that in qualifications of a field event there is a distance of direct qualification to the final. If an athlete reaches this distance, he gets automatically directly qualified, therefore the ????? may get biased since the athlete who reaches this distance at his first attempt in qualification will not try to do better afterwards since he will be willing to save energy for the final.

This thesis is born by reading the paper of Boyd and Boyd (1995) and observing some methodological details that could be improved in order to prove Baik and Shogren theoretical hypothesis.

The first observation I could make is about the idea to analyze the semi-finals in the paper. In my opinion, semi-finals are not as interesting to study as finals since, for starters, it is not the final race of the competition. Therefore, there is a bias created by the fact that runners that are a little bit above others in terms of level may try to save some energy in order to qualify for the final. In a middle-distance international championship, there are two ways to qualify for the next round. Either you get your «qualification through the

place» by finishing in the first positions of your race, (2 for the 800m, 4 for the 1500m, 5 for the 5000m) or you hope to be in the best times of all the semi-final in order to get your «qualification through the time» in order to get the last places remaining for the final. If we take the example of the 800m there are usually three semi-finals of eight runners at an international level for eight tickets for the final. In order to get to the final, a runner has the choice between finishing in the first two places of his semi final or to finish third or fourth and hope to have a time sufficient to be in the eight best runners. This disposition completely biases the analysis of semi-finals since usually the runners of the third semifinal have a huge advantage compared to others since they know what time they have to do in order to qualify through the time. Therefore, we can observe frequently much faster races in the last semi-final because runners are incented to do so since if they succeed to be faster than other semi-finals runners (which often happen because these runners do not have this incentive) they have more chances to qualify. If by allying themselves, the race is fast, they will have a probability to get to the final of 0.5 (4 places out of 8) instead of 0.25 (2 places out of 8). Their chances to qualify have been doubled! Therefore, only finals will be taken into account in the races that I will analyze in my study.

This phenomenon creates another problem in Boyd and Boyd model, but this time about the UF1 variable. Indeed, the settings of the UF1 variable are such that it is the place of the runner in the previous turn that is taken into account. However, how the races are not the same because a semi-final can be fast and the other slow, therefore, biases are created since for example the best runner can win the slowest race and be nevertheless ranked behind a runner who finished fourth of the fastest race in the settings of the underdog-favorite variable proxy. The best way to determine the favorite of a race prior to the race is to analyze loads of data such as the PBs, the races the runner has run recently, the weather conditions, his results against the opponents, his final kick, the previous races he has won, etc. This is what I'm going to try achieving in this thesis by creating a method to determine the underdog-favorite variable proxy by an aggregation of different and various data.

The third observation is about the fact to choose the place of a runner at the end of an interval or lap is not the perfect way to measure the level of effort of an individual during an interval. Indeed, if we take the example of two competitors A and B on an 800m, the runner A starts quickly and run his first 200m in 24.0 seconds. The competitor B starts slower his race and cover the first 200m in 25.5 seconds. At the 400m, the competitor A passes the line in 50.5 seconds which means he has run the second 200m in 26.5 seconds.

The competitor B is still behind him in 51 seconds but has closed the gap since he has run his second 200m in 25.5 seconds. The following table represents the respective rankings of both runners if we take into account the place at the end of the lap or the ranking of the runner over the lap.

				Position Method		Ranking Method
	0-200m	200-400	400m	200m	400m	0-200m	200-400m
Runner A	24.0	26.5	50.5	1	1	1	2
Runner B	25.5	25.5	51.0	2	2	2	1

We can see from this table that if we follow Boyd and Boyd methodology to determine who has exerted the greatest effort on the second lap, the runner A will be designated as the runner who will have done so. However, we see with this concrete example that he was not the fastest runner over this interval. Therefore, it is more relevant to my opinion to take into account the ranking of the runner over the interval rather than the position of the runner at the end of the lap because we can see with this example that the runner who has run the fastest, i.e. who has committed the greatest effort is the runner B. I will therefore take the ranking of the runner over the interval rather than his position. Of course, at the time Boyd and Boyd did their experiment it was impossible to analyze the times over all the laps of every runner since the data was not available. But now the data is partially available for the last world championships and I think it is a good occasion to test Baik and Shogren's hypothesis using more precise data.

This change is important and has major impacts. First by taking the ranking over the lap, the volatility of the positions of the runners will be increased, especially for the longer distances. Indeed, for long distances, the races start usually very slowly, and the runners run in pack. Therefore, the difference between the fastest runner over the lap and the slowest is usually very slow (sometimes less than a second) and a runner ranking over different laps can be second on the lap 2 but fourteenth on the lap 3 because he has run 0.15 seconds slower than the other runners before getting back to fourth place over lap 4 because he has run 0.22 seconds faster than others. The ranking of this runner is too unstable to be able to find a significant regression. In order to get results, I will have to reduce the number of laps for longer races to gain in stability of the regression. But this decision makes sense, indeed the time over a longer interval will be more representative of the effort of the runner than the time over a shorter interval which will be almost the

same for every runner of the race at the beginning. This decision is also taken from a practical point of view since the data collected if we take the example of the 5000m gives only the times for every 1000m for every runner. There is another advantage of reducing the number of laps which is we will be able to see if the variation in absolute value of the estimates in function of the distance in Boyd and Boyd experiment was real or simply caused by the greater number of laps since all races will have a similar number of laps.

Eventually, my last observation is upon the choice to analyze distance field events in order to verify Baik and Shogren hypothesis. I think this choice to be not a very good one. Indeed, if middle-distance races are a highly relevant example to test strategic behavior in contests. It is because these races are not done with a maximum effort during the whole event. Indeed, over this distances, it is not possible to start in pure sprint and finish the race. Therefore, runners have to strategically decide when to exert their effort and when to save energy. Distance fields events on the contrary are pure explosive efforts where the effort lasts a few seconds or even a fraction of a second for throws. Athletes are therefore giving a hundred percent of their energy on each jump or throw and do not calculate if they have to save some energy for the next ones. Therefore, there is no strategic choice in trying to exert effort at the best timing, they are trying their best every time. They may improve their distance a bit along the event since they can better their technique are adapting their marks to the weather condition. But there is no strategic choice to overexert at the beginning of the event. This is, in my opinion, why Boyd and Boyd have ambiguous result for these types of event. This makes me say that the study of distance fields event is irrelevant in order to determine if an underdog overexerts his effort at the beginning of a contest. Therefore, these events will not be studied in this thesis.

The objective of this thesis is to improve Boyd & Boyd (1995) in order to verify if Baik and Shogren hypothesis can still be verified after these changes. As a reminder, these changes would be: defining the events most relevant in order to prove Baik and Shogren hypothesis the calculation of the underdog-favorite variable proxy via a combination of data, and to build an empirical model based on the ranking of the runner on the interval rather than on the place of the runner at the end of the interval. The methodology of this thesis will consist therefore in delimiting the frame of the study in a first time, defining what data to collect and how. Then in a second time to determine what data to collect and

how to combine it in order to build an underdog-favorite variable proxy only with data available prior to the race that will be a good predictor of the issue of the race. Eventually, it will be to build an empirical model that will make it possible to verify Baik and Shogren hypothesis that the underdog overexerts his level of effort at the beginning of a contest.

The first task to be able to improve Boyd and Boyd empirical model is to determine precisely what data is relevant for the study and what is not. The events in track and field that will be considered will be middle distance races. These races are the most interesting to study in order to analyze strategic behavior of contestants since strategic choices are key in determining who will win in these races. Moreover, these races are not event that are realized in a purely explosive effort, such as 100m, long jump, hammer throw. The athletes do have to choose when to exert their effort in the race and therefore make a strategic choice. Therefore, it is a perfect situation to verify Baik and Shogren hypothesis that underdogs tend to overexert their effort level in the first stages of a sequential contest. The other events are less interesting to study since they are mainly based on the explosive qualities of the contestants and their technical qualities. There is no choice on how much effort you should exert for this try or this sprint in these events since these events are competed with full energy delivered all race or try long. This important delimitation will make me study only the race that will be longer than 800m included. However, I decided not to include the 10 000m since this race is not really considered as a middle-distance race but a long-distance race.

The fact that I want to use the ranking of a runner over an interval as the endogenous variable rather than his position at the end of the interval like Boyd and Boyd had done gave me constraints. Indeed, in order to be able to collect this type of data, you need to have electronic cells such as the ones over the finish line at every 100m if you want the splits for every 100m. However, this technology is very expensive therefore it is usually only available for the major events of the athletics season such as World Championships or Diamond League meetings, which is the circuit with all the greatest athletics meetings in the World. However, to study the strategies of runners in meetings is to my opinion not particularly relevant. Indeed, this races do not happen exactly the same way than the championships since a runner, the pacemaker is paid by the organization in order to sacrifice himself and to make the race start quickly by overexerting his level of effort on

purpose at the beginning of the race. This is done in order to give an opportunity to the runners to beat their PBs, to qualify for the great championships, or beat the World Record. Every other runner know that he will be doing so and therefore try to place themselves behind him. Usually, this pacemaker does not finish the race since he has overexerted so much effort at the beginning of the race that he is too tired to finish it with the hope to get a good time. This changes also completely the chances for the favorite and the underdog to win. Indeed, usually the faster is the race, the less chances have the underdogs to beat the favorite. Therefore, as meeting races are usually fast races, underdogs incentive to try to beat the favorites are usually low, since they know they have really small chances to do so. Instead, they will try to remain behind the favorites for as long as they can hoping to beat their PBs with the aspiration, motivation and pace provided by the favorites. With these observations, I believe that only championships races are relevant to study strategic behavior in order to prove Baik and Shogren hypothesis. Therefore, the studied race will be the races from 800m to 5000m from the world championships between 2013 and 2019, which makes 4 events, world championships being biannual events. The world championships of 2011 were not studied since the technology giving the split times of every runner along the event did not exist at the time.

Besides, and as explained in the part 2.3.3.4, I decided that only races that are finals of the championships are going to be studied in reason of the strategic biases that are created by semi-finals because of the qualification procedures to the final.

Moreover, the choice to use ranking over the interval rather than the position at the end of it limits me because in order to be able to assess which runner has run the fastest on an interval, I need to have the split times of every runner for every interval. This means, if I want to replicate the exact methodology of Boyd and Boyd every 200m for the 800m and 1500m and every 400m for the 3000m steeplechase and the 5000m. However, the collection of this data was not always possible homogenously. Indeed, the race analysis reports, coming from the World Athletics (i.e. the international federation of athletics) website that were analyzed in order to collect the data were not splited in a homogenous way. For 2013 and 2019 world championships for example the detail of split times was given for every 100m for every runner while for 2015 and 2017 world championships, they were only given every 400m for the 1500m (splits being 400-800-1200-1500) and every 1000m for the 5000m. Table 7 present the different split levels that were available for each races over the studied period. The table shows that if for the 800m, there are not problems about the availability and the homogeneity of the data for split times of runners,

it is quite different for 1500m and 5000m. Moreover, it shows also that the data was not available for the 3000m steeplechase. Indeed, this event is a 3000m over which runners have to pass 4 barriers of 91.4 cm for men and 76.2 cm for women and a river made by a barrier and after that barrier a pit of 2m deep filled this water. The problem is that this «river» is usually situated inside of the track in the second turn of the lap. Therefore, runners have to «cut» the lap in order to pass over that river. Their lap is therefore shorter than 400m and to compensate it, the starting line of the 3000m steeple chase is 25m farther from the finish line than the 3000m flat starting line. Therefore, every turn, runners catch up 3.57m which makes it impossible to measure the split times every 400m since every lap the line of the 400m is virtually moving. Therefore, the international federation do not release the data concerning the split times of each runner for the steeplechase events and I am forced to rule it out of the study for this reason.

Level of	800	800	800	800	1500	1500	1500	1500	5000	5000	5000	5000
Detail	2013	2015	2017	2019	2013	2015	2017	2019	2013	2015	2017	2019
Split 100	X	X	X	X	X			X	X			X
Split 400						X	X
Split 1000										X	X

We face here a situation where we have to choose what level of number of laps we want to study. In order to get comparable and homogenous data for every race, I decided to study races with the minimum number of laps possible. This choice has many advantages. First, it will give me homogenous data that will make all 800m, 1500m, and 5000m comparable between each other since all races of a same event will be comparable. I will be therefore able to aggregate the data from all 800m together, all 1500m together, and all 5000m together. Second, it will improve the volatility of the data. Indeed, by choosing the ranking over the interval over the place at the end of the lap, the volatility of the place of runners has increased a lot. As explained in the 2.3.3.4, the reduction of the number of laps is going to help to get a statistically significant regression since it is going to reduce the number of laps which are potentially subject to the increase of volatility of the ranking of runners over laps. Eventually, this choice is interesting because races are going to be more comparable in terms of number of intervals. Therefore, we will be able to assess if the variation of the estimates in absolute value in function of the distance is a real thing or just a statistical phenomenon due to the bigger number of laps.

Because of these practical but also methodological reasons, the number of studied intervals for 800m races will be of 4, e.g. each 200m interval, the number of studied intervals for 1500m will also be of 4, following the splits of the race analysis reports (400-8001200-1500), and for 5000m the number of studied intervals will be of 5, e.g. each 1000m.

This frame delimitation will, I hope, enable me to measure as accurately as possible the strategic behavior of contestants in the case of middle-distance running depending on their pre-race status. This pre-race status, which is based on the place of the runners in the previous round in Boyd and Boyd's experiment, will be for this study defined by a combination of data available prior to the race. The methodology used to determine this proxy will be discussed in the following part.

The game of predicting who is going to win a race is very popular among observant of athletics competitions. In order to predict correctly the outcome of a race, a lot of parameters is usually taken into account by the persons who play it. The interesting fact about this game is that it can be assimilated to the pre-race favorite variable since this variable is a perception of who is going to win the race, which is the same process than when someone is predicting it.

To predict who is favorite of a race and who is not, the relevant and easily collectable data are, to my opinion, the following:

- The personal best of the runner, this is the most obvious data to collect. Indeed, the first thing that runners look about their opponents is their respective PBs since it is the best measure they can have on how fast the opponent can run the distance in the case the race is fast, and what margin they have between their PBs pace and the race pace if the race is slow.

- The season best of the runner, which is the best time the runner has run the distance during the year. This data is interesting for the same reason than the PBs. Moreover, it brings information on the current shape of the runner, indeed by taking SB into account in the parameter, we are able to ponder the PB of the runner if his PB has been done 10 years ago for example. For example, in 2019, if a runner A who has a PB of 3'28»53 at 1500m done in 2009 and a SB of 3'34»45, he will be probably ranked behind a runner B who has run 3'30»32 in 2019 since the «real» level of runner A in 2019 is much closer to 3'34»45 than 3'28»53.

There are other data that are highly relevant in my opinion to take into consideration but that are note easily collectable even though they can be estimated by other data easier to collect:

- the final kick of the runner. In middle distance races, and especially for championships races, one key asset to have in order to be able to win the races is to be able to have a high finishing speed. This can be also seen in championships by the fact that, often, runners from the below distances try their chance on the superior distance to try to get something thanks to their superior speed. Indeed, tactical races, which are usually the rule of finals, are races that are usually ran slowly in the first two thirds of the race, with a strong acceleration in the last third to the last quarter of the race. This slow pace does not tire off completely the runners from shorter distances such as would have done the pace in a meeting. Therefore, they are still «fresh enough» to use with full capacity their final kick in the last part of the race and this kind of effort is much closer to the distance they are used to run which gives them a significant advantage. To represent this aspect of championships races, I decided to take the PB of the runner on the distance just below the studied distance. This PB was taken in an area of 3 years before and after the studied race because the idea is not to get the level of the runner on the distance but his potential to run fast, which is generally constant over time. When the data was not available, I used the following rules to fill the gaps:

For the explanations on the estimation of the PB on the shorter distance, I need to explain a concept first. For every distance there are two profile of runners, the short profile and the long profile. The short profile is the runner who has abilities on the distance and the shorter distance. The long profile is the runner who has abilities on the distance and the distance above. The problem with long profile runners is that they often do not run on the distance below, therefore their PB on the shorter distance has to be estimated. The estimation of the final speed of the runner in a race for the 800m is the trickiest one. Indeed, the 800m is the distance that is at the limit between the sprint and the middle-distance race. Therefore, to estimate the final speed of an 800m with his level on 400m is

probably less relevant compared to 1500m and 5000m. therefore, and to limit the impact of the 400m on the results, I decided to take the same value of PB for every runner with lacking data, which are runners that have a long profile rather than a short profile or that have an insufficient PB on the distance because they do not run 400m at their best shape or often enough have been given a time on 400m which is the minimum level they need to have in order to qualify for a world class competition. For the men for example, the minimum level to get into a world final is of 1'45»50 on 800m. To do this time, runners need to pass at least 51» at the 400m because 800m is a positive split race, i.e. a race where the second part of the race is slower than the first part, because of the energetic sector that is used in this event (lactic anaerobia). Therefore, to be able to pass in 51sec-onds at the 400 with sufficient margin on their 400m PB to finish the race correctly, this margin needs to be of 3.5 seconds, which gives an estimated PB on 400m of 47.5 seconds for men. The same calculation was made for women and gave the estimated PB on 400m of 55.0 seconds. For the 1500m, the calculation method is based on a very popular model in the world of athletics that stipulates that a 1500m runner with a longer profile can run his 1500m in a time which is twice is time on 800m. This method of calculation of is usually very accurate and I decided to use it. Moreover, there is no risk that a 1500m runner with a short profile might get this estimation because these runners usually run 800m in reality since the two races are quite close and do not necessitate different preparations in the training. For the 5000m, I took the 3000m into consideration since if a 5000m runner has not run a 1500m, it is because he has a long profile and the short distance for 5000m long profile is 3000m. Then I used the fact that an elite athlete can run a 3000m at his maximum aerobic speed (MAS). Then as it has been proven that an elite runner can run 1500m at 106.5% of his MAS (Scherrer & Monod, 1960; Ettema, 1966), I estimated this way the PB of the 5000m runner over 1500m.

- the experience and the tactical sense of the runner, for every contests, it seems an evidence that the experience of a runner is a substantial help to his claim for victory. Indeed, it will help him to deal with pressure, to feel when to make his effort, etc. Moreover, and as the parameter that we try to determinate in this part is the perception of how favorite a runner is and how this parameter influences his strategy but also the relative strategy of other runners, it is important to note that a runner who has a lot of experience and who has one titles in the past is usually perceived as favorite by other runners. Indeed, even if every runner in a World final is a fierce competitor, it is easier to frighten your opponents if you are the reigning world champion and world record holder than if you

are a runner who has never won anything in his life and for who this race is the first final in an international championship. Therefore, the champion has more chances to influence the strategies of the other runners, because being afraid, they will try to surprise him in order to have a chance to beat him, while the newbie will not be calculated. In order to represent this impact, I will take the previous honors that the runner has won in his career to represent his experience. The advantage of taking the previous honors of a runner is that it will also be a good representation of the tactical intelligence of the runner. Indeed, to be able to win or to finish in the first places of a world championships or a continental championship is a good indicator of the tactical intelligence of a runner, since without tactical sense, it is very complicated to get in these places, unless you are much stronger than other, but this will be compensated by the PB and SB data in the frame of my underdog-favorite variable proxy. To represent the effect of the experience over a runner I decided to use the system of a multiplier. Each level of experience will be given a coefficient that will multiply the score that the runners will have obtained with the other gathered data. The multipliers follow the rule in the Table 9:

Continental Indoor Champion, Continental Championships Top 5, World Cham 0.02
pionships Top 8

The logic of this settings is the following, a World Championships honor is superior to a Continental honor but also to a World Indoor Championships honor because usually, the indoor season is not as dense as the summer season, because a lot of middle distance

runners prefer either to run in cross-country running or to train a lot in order to be better in the summer. Therefore, the level of these indoor competition is not as high as the outdoor competition and are therefore similar to an outdoor continental level. The bonuses and maluses are here to affine the analysis. Indeed, in the perception of a lot of runners, to be the world record holder makes of you a kind of a legend of your sport, therefore, a big bonus is given to the runner who is the holder of this record on this race. Anyway, the case happens just once in the study for the Kenyan David Lekuta Rudisha on the men 800m, so its impact is not very significant. A more significant bonus is the last global championships winner. In terms of perception of who is a favorite of the race, the memory of runners who have usually witnessed or participated to last final of a global championships reminds them a lot of who won the last race and therefore they usually adapt their strategy to this parameter. Eventually, I added the running at home bonus to represent the fact that some runners are inspired by the fact of running in his country in front of his friend and family and this inspiration can lead to big surprises since it helps these runners to surpass themselves. The maluses are here to represent the fact that the relevance of honors diminish rapidly with the time passing, and that an honor obtained on another distance is not as good a predictor as an honor on the same distance for.

Eventually there are some other data that are relevant such as the weather preferences, the previous results of runners in direct confrontation. But this kind of data is too complex to be collected and will be therefore let out of the model.

Now that all the data that need to be required has been defined it is time to build our underdog-favorite parameter. Note that all the data concerning athletes have been collected from the respective athlete profile on the website of World Athletics. If we define ?? as the year of the race, ???? the year in which the runner realized his PB. If ???? is the PB of the runner over the distance and ???? his SB over the distance. Then we can do a weighted average of the PB and the SB of the runner. This way, the older is the PB of a runner the less its influence will be strong on the time that will be implemented in the variable. We therefore define the estimated time over the distance ?????? as:

Once this variable representing the estimated time is obtained, we want to incorporate into it the variable that represent the final kick of the runner, which is his time over the distance just below, ???????. The problem is that this two data are times that represent

different distances; therefore, it is difficult to aggregate them simply by adding the times or by combining them.

Luckily we have a very practical tool in the world of athletics that help us to compare the level of performance of athletes in events that have nothing in common such as the hammer throw and 50km Walk Race: the Scoring Table of Athletics (Spiriev & Spiriev, 2017). This table enables the comparison between all events of athletics by scoring the possible performances for each event on a 1400 points table. This table is built statistically from the World Record and the global level of performance over the years. This table is modified every 3 years and I took the latest version (2017) for my study. In this table, a World Record is usually around 1300 points, an international level runner is around 1200 points, a continental level runner is around 1100 points, and a national level runner is around 1000 points in France. This way, if we take the barrier of the 1200 points, we can say that 20»13 on Men 200m is equivalent to 13'02»56 on Men 5000m, 17.33m in Men Triple Jump but also 66.55 m for Women Javelin Throw since the table is built with the same method for women. Thanks to this table, it is possible to combine the estimated time on the distance and the time on the shorter distance by converting the times of each distance in the scoring table. In order to represent the fact that, the time of the runner on the distance is still more important that his final kick, I decided to put a 2/3; 1/3 balance in favor of _TDist. Then to implement the experience and the tactical sense of the runner, I combine the raw score obtained by the combination of the two times with the relative multiplier m calculated by the analysis of the athlete profile on World Athletics Database. I obtain this way the Favor_iteindex which is:

This index is built such as the greater is Favor_iteindex_, the more chances he will have to be considered as a favorite of the race in the empirical model. This way I hope to have built a variable relying on data that will give a strong indication of who is perceived as the favorite before the race when applied to the empirical model, and therefore, that I will have improved Boyd and Boyd experiment.

The empirical model that I am going to use is largely inspired by the empirical model used by Boyd and Boyd (1995) but with the slight modifications explained in the 3.1 and

3.2. For a race with T successive intervals. Let's _Rtt be the ranking of the runner i over the interval t, with t = 1, 2, ..., T. My interest is to analyze the manner in which Rtt evolves over the course of the race. As for Boyd and Boyd, this is modelized with the relationship:

Like for Boyd and Boyd experiment if the theoretical results of Baik and Shogren are correct, underdogs have an incentive to overexert effort early in a race whereas favorites have an incentive to let them do, saving energy for the end of the race. In this case, fi_o is negative, fi₁ is positive, and both are statistically significant.

Like Boyd and Boyd, I use fixed effects (Mundlak, 1978; Greene, 1993) to separate individual characteristics that affect the relative position over time (at) from interval-to-

interval strategic behavior stemming from the individual status as a pre-race favorite or underdog (Ni) and which represents unobservable individual characteristics such as physical attributes that may be a form of misspecification of the model and may bias the estimates _No and Ni. Each ai is unobservable, but constant over time, therefore its effect can be removed by taking into consideration deviations from the mean. Furthermore, by specifying Ni as a function of pre-race status as given in Equation 2, what seemed to be individual-specific slope characteristics in Equation 1 is estimated by _No and N1. This way, I am able to obtain unbiased estimators for Equation 3 using fixed effects model, just like Boyd and Boyd did in their experiment:

the event, f is the mean interval of the event and the error terms vi_tare again independent and identically distributed. I pooled the data for each athlete like Boyd and Boyd did within an event which enables me to estimate Equation 5 using ordinary least squares, which provides me consistent estimates of _No and N_i for each race.

The result section will be organised in the following way: in a first time, I will verify that my perception of who is favorite of the race has a good correlation with the final result. Then, in a second time will be presented the results of regression race by race classified by event. The only aggregation of data that will be done at this stage will be done by event and by sex. In a third time, I will present the general results of the regression by aggregating both sexes. Eventually, I will present the results obtained by dividing the studied population in order to study the difference in strategic behavior in function of the characteristics of this population. The results analyzed are the Student statistics (T-Stat), the critical probability (Crit-P) and the R² since they were the results analyzed by Boyd and Boyd.

Of primary concern for this study is to know if the favorite index is a good one. In order to verify it, the solution, in my opinion, is to verify if the expected result of runners that is predicted by the Favor_iteindex is a good explainer of the final result. For this, I did a

regression of the final result by the proxy that is given for each race by the ?????????????????????. Therefore, I tried to explain the final result with the predicted position of the proxy. For example, if a runner had the third best ????????????????????? of his race, his proxy value was 3. I tried to figure out with this regression how far is the final result for runners from this proxy value. If the ?????????????????????. model is a good estimator of the underdog-favorite variable, then the coefficient ??? of the regression should be close to 1, for a model, of regression such as:

where ??? is the final position of the runner ?? at the end of the race, ??? ? is the predicted final position calculated thanks to the favorite index, and ?? is the error term. The results for this regression are the following Table 10:

The number of runners studied is of 279, this means that we compared the final ranking with the predicted ranking built with the ????????????????????? 279 times. As expected, the coefficient of the regression ??? is close to 1. Moreover, it is highly significant and the R² is of 0.862. These observations enable us to assert that the ranking predicted with the use of the ????????????????????? is a good explainer of the final result of the race. This result validates the use of the ????????????????????? as a method to determine who is the pre-race favorite.

Concerning the 800m, the results of Boyd and Boyd were interesting in the way that it was the distance for which their results had been the least significant. For the 800m, four women finals and four men finals were studied. Each of these finals had 8 runners that contested over a race sliced in four intervals which gives 32 estimations of ???? for these events. The results are presented in the following Table 11. Every time, before the detailed results for each race, which is the equivalent of the result presented by Boyd and Boyd, I aggregated the results for all the races below. This way, I am able to evaluate the general behavior on 800m races for women and for men and see if there is a significative difference. Moreover, by aggregating the data, I improve the significance of my regressions since the number of position ???? is greater, as the compilation of all the aggregate

races studied is analyzing the evolution of ???? on 128 different positions. The results for the 800m event are the following:

Race	???	t-stat	Sign.	Crit-P	???	t-stat	Sign.	Crit-P	N	R²
Women	-0.238	-0.695		0.489	0.053	0.779		0.437	128	0.005
2013 W	-0.300	-0.749		0.692	0.067	0.449		0.657	32	0.007
2015 W	-0.311	-0.407		0.687	0.069	0.457		0.651	32	0.007
2017 W	-0.654	-1.136		0.265	0.145	1.275		0.213	32	0.053
2019 W	0.311	0.476		0.674	-0.069	-0.476		0.637	32	0.008
Men	-0.541	-1.547		0.124	0.126	1.826	*	0.070	128	0.026
2013 M	-0.921	-1.236		0.226	0.205	1.387		0.176	32	0.062
2015 M	-1.596	-2.559	**	0.016	0.380	3.074	***	0.005	32	0.249
2017 M	-0.782	-1.005		0.323	0.174	1.128		0.269	32	0.042
2019 M	1.135	1.875	*	0.071	-0.252	-2.104	**	0.044	32	0.132

* Significant at the level of 10% ** Significant at the level of 5% *** Significant at the level of 1%

As for Boyd and Boyd experiment, the results for the 800m are not very significant and are rather ambiguous. For women, no race has significant estimates of ??? and ??? at the critical probability of 10% and the aggregation is not significant. However, we can note that for the women races, at the exception of the 2019 final, the signs are such that they validate the Baik and Shogren hypothesis, which is that ??? is negative and ??? is positive. The estimates for ??? and ??? for the aggregation of women races is not significant either, but their signs seem to confirm Baik and Shogren hypothesis. For men the results are a little bit better but more ambiguous, the estimates of the races from 2013 to 2017 have signs that confirm the hypothesis of Baik and Shogren and the race of 2015 is even significant at the level of 5%. But the 2019 final estimates are significant at the level of 10% with signs that are opposite to Baik and Shogren hypothesis. Despite this data opposite to the expected result, the aggregation of the men 800m races data gives estimates with a small significance at the level of 10% for ??? and with a critical probability of 12,4% for ???. This means that if the 2019 race was ejected out from the sample, the results would be probably significant for the aggregated races. As a conclusion of these results, we can say that the results for the 800m are as ambiguous as the results of Boyd and Boyd experiment and it is complicated to validate Baik and Shogren hypothesis.

Concerning the 1500m, the results of Boyd and Boyd results had been for the validation of Baik and Shogren hypothesis with significant estimates for men's races and non-significant estimates for women races. As for the 800m, four women finals and four men finals were studied. Each of these finals had 12 runners that contested over a race sliced in four intervals which gives 48 estimations of ???? for each of these events. The results are presented in the following Table 12. As for 800m, before the detailed results for each race, I aggregated the results for all the races below. This way, I am able to evaluate the general behavior on 1500m races for women and for men and see if there is a significative difference. Moreover, by aggregating the data, I improve the significance of my regressions since the number of position ???? is greater, as the compilation of all the aggregate races studied is analyzing the evolution of ???? on 192 different positions. The results for the 1500m event are the following:

Race	???	t-stat	Sign.	Crit-P	???	t-stat	Sign.	Crit-P	N	R²
Women	-0.468	-1.192		0.235	0.072	1.349		0.179	192	0.010
2013 W	-0.936	-1.189		0.241	0.144	1.346		0.185	48	0.039
2015 W	0.514	0.586		0.561	-0.079	-0.663		0.511	48	0.010
2017 W	-1.009	-1.175		0.246	0.155	1.331		0.190	48	0.038
2019 W	-0.441	-0.676		0.503	0.068	0.765		0.448	48	0.013
Men	-0.656	-1.643		0.102	0.101	1.86	*	0.064	192	0.018
2013 M	-0.441	-0.512		0.611	0.068	0.580		0.565	48	0.007
2015 M	-1.845	-2.366	**	0.022	0.284	2.679	**	0.010	48	0.138
2017 M	0.218	0.279		0.782	-0.034	-0.316		0.754	48	0.002
2019 M	-0.554	-0.687		0.496	0.085	0.778		0.441	48	0.013

For women, no race has significant estimates of ??? and ??? at the critical probability of 10% and the aggregation is not significant. However, we can note that for the women races, at the exception of the 2015 final, the signs are such that they validate the Baik and Shogren hypothesis, which is that ??? is negative and ??? is positive. The estimates for ??? and ??? for the aggregation of women races is not significant either, but their signs seem to confirm Baik and Shogren hypothesis. For men the results are a little bit better, at the exception of the 2017 final, the estimates of the races have signs that confirm the hypothesis of Baik and Shogren and the race of 2015 is even significant at the level of 5%. The

aggregation of the men 1500m races data gives estimates with a small significance at the level of 10% for ??? and with a critical probability of 10,2% for ??? which is very close of being significant. Therefore, we can conclude from this that Baik and Shogren hypothesis on strategic behavior are somewhat validated by the results of the men races.

Concerning the 5000m, the results of Boyd and Boyd results had been for the validation of Baik and Shogren hypothesis with significant estimates for men's races and no results for women since in 1992, women were not allowed to run this distance, and were running the 3000m. As for 800m and 1500m, four women finals and four men finals were studied. Each of these finals had 15 runners that contested over a race sliced in five intervals which gives 75 estimations of ???? for each of these events. However, for the 2017 men final, a runner did not start the race and was therefore not accounted in the results. The results are presented in the following Table 13. As for 800m and 1500m, before the detailed results for each race, I aggregated the results for all the races below. This way, I am able to evaluate the general behavior on 1500m races for women and for men and see if there is a significative difference. Moreover, by aggregating the data, I improve the significance of my regressions since the number of position ???? is greater, as the compilation of all the aggregate races studied is analyzing the evolution of ???? on 300 different positions for women and 295 positions for men . The results for the 5000m event are the following:

Race	???	t-stat	Sign.	Crit-P	???	t-stat	Sign.	Crit-P	N	R²
Women	-1.049	-4.305	***	0.001	0.131	4.571	***	0.001	300	0.066
2013 W	-1.583	-3.041	***	0.003	0.198	3.456	***	0.001	75	0.142
2015 W	-1.414	-2.421	**	0.018	0.177	2.752	***	0.008	75	0.095
2017 W	-1.222	-2.322	**	0.023	0.151	2.610	**	0.011	75	0.086
2019 W	0.023	0.051		0.959	-0.003	-0.058		0.9591	75	0.000
Men	-0.815	-2.537	**	0.012	0.105	2.932	***	0.004	295	0.029
2013 M	-0.863	-1.217		0.228	0.108	1.383		0.171	75	0.026
2015 M	-1.991	-3.159	**	0.002	0.249	2.752	***	0.001	75	0.152
2017 M	-0.388	-0.592		0.556	0.059	0.771		0.443	70	0.009
2019 M	0.017	0.030		0.977	-0.002	-0.034		0.973	75	0.000

For women, the results are globally very significant. The estimates of ??? and ??? for the finals of 2013, 2015 and 2017 are significant at the 5% level, and their signs validate the hypothesis of Baik and Shogren. The only contradictory result is for the 2019 final for which the signs of the estimates are opposite to Baik and Shogren hypothesis, but this result is not significant and therefore can be not overlooked at. The estimates for ??? and ??? for the aggregation of women races is very significant in the validation of Baik and Shogren hypothesis since their critical probabilities are at the level of 1%. For men the results are a little bit shallower, the estimates of the races have signs that confirm the hypothesis of Baik and Shogren but not at the significant level of 10% at the exception of the final of 2015 which is significant at the 5% level. As for women, the 2019 final is not validating Baik and Shogren hypothesis but as the estimates are pretty insignificant, the result does not have much impact on the results of the study. The aggregation of the men 5000m races data gives however significant estimates at the level of 5% for both estimates. Therefore, we can conclude from this that Baik and Shogren hypothesis on strategic behavior are validated by the results of the 5000m races.

For this part, I did something that Boyd and Boyd did not do in their experiment, which is to aggregate all the data of both sexes for each event in order to see if the behavior is different from one race to another. Moreover, I have aggregated the data of all races in order to see if there is a general strategic behavior that is significant and verifiable for all middle-distance races. The results are presented in the following Table 14:

Race	???	t-stat	Sign.	Crit-P	???	t-stat	Sign.	Crit-P	N	R²
800m	-0.390	-1.598		0.111	0.090	1.858	*	0.064	256	0.014
1500m	-0.562	-2.014	**	0.045	0.086	2.281	**	0.023	384	0.013
5000m	-0.933	-4.535	***	0.001	0.118	5.182	***	0.001	595	0.043
General	-0.570	-4.240	***	0.001	0.081	4.914	***	0.001	1235	0.019

The results of the aggregated data are really interesting; indeed, it increases the significance of the estimates for every event. For example, once the data of men and women are aggregated for 800m, the estimates are almost significant while it was very difficult to obtain any significance for this distance. The estimate of ??? is significant at the 10%

level and the estimate of f.?_?is almost significant at the 10% level since its critical probability. And the interesting point is that the signs of the estimates agree with the hypothesis of Baik and Shogren which enables to affirm that on 800m, if by taking the case of a particular race, it is not possible to assert that underdogs overexert their effort in the first stages of the race. But if a sufficiently big number of races is aggregated in the database, then it can be affirmed that this tendency of underdog's behavior is real, even for 800m. This conclusion is strengthened by the results obtained by the 1500m and the 5000m. Indeed, the 1500m estimates are significant at the level of 5% and the 5000m estimates are significant at the level of 1%. The signs of the estimates for 1500m and 5000m all confirm the hypothesis of Baik and Shogren. Eventually, if the data for all races is aggregated, the results are highly significant and in the sense of Baik and Shogren hypothesis since the signs of the estimates support their hypothesis and that they are significant at the level of 1%. Since the races are different, we can therefore conclude with a small margin of possible error, that in the context of middle-distance races, the hypothesis of Baik and Shogren are correct. This mean that in a situation of a competition where there is a clear favorite to the race, the more a runner is an underdog, the more is level of effort will be high at the beginning of the race, while the more a runner is a favorite, the more is level of effort will be high at the end of the race.

Another interesting point observation is that the magnitude of the f.?_?coefficients is proportional to the race distance and it seems, even if the progression is not perfect, that the magnitude of the f.?_?coefficients is also proportional to the race distance. This is opposite to what Baik and Shogren had found in their results, and as this time, there is no bias due to the greater number of laps. This mean that the gradient of runners is much steeper in the long-distance races than in the short races, therefore, this means that the pre-race favourites tend to move to the front of the peloton much faster in the longer races. This can be explained by the facts that in longer races, favorites may have an incentive to start tiring off the runners from shorter distance with a medium intensity pace.

This part will be consisting in the analysis of the possible variations of strategic behavior in contests by analysis of the population through its characteristics. The analyzed characteristics will be the gender of participants, the cultural background of the participants, and the density of the races.

The first and most obvious impact to be tested is the impact of gender on the strategic behavior. Boyd and Boyd had found that the results for men were slightly more significant than the results for women. I divided therefore my population in two based on their gender and did the regression for men and the regression for women in order to see if there is a significant difference in the result. There are 139 men and 140 women in the sample which gives 615 rankings ???? to study for men and 620 for women. The results are the following:

Race	???	t-stat	Sign.	Crit-P	???	t-stat	Sign.	Crit-P	N	R²
Men	-0.406	-1.992	**	0.046	0.059	2.341	**	0.019	615	0.020
Women	-0.731	-4.144	***	0.001	0.103	4.766	***	0.001	620	0.035
General	-0.570	-4.240	***	0.001	0.081	4.914	***	0.001	1235	0.019

What we can say on a first basis is that results for women are more significant than the results for men. The estimates of women are also bigger than the estimates of men in absolute value with a ??? of -0.731 for women versus -0.406 for men, and a ???of 0.103 for women versus 0.059 for men. This means that their gradient is steeper, therefore that women tend to move to the front of the pack sooner than men in races. To see if the estimates are significantly different for men and women, I realised a test of Chow (Chow, 1960), which is a statistical and econometrical test in order to determine if the coefficients established by a linear regression of two linear series are equal. This test is based on the Fisher's law. This test realised with the software Gretl gives a F-Stat of 0.588 which gives a critical probability of 62%. Therefore, we can conclude that there is no significant difference in the strategic behavior of men and women in middle distance races.

Another impact that I would like to test is the impact of culture on strategic behavior. Indeed, it seems interesting to see in the cultural background of contestants can have an influence on their strategic behavior, since most of our intellectual constructions and way of thinking are actually based on our cultural background. Therefore, I believe there should be an impact between culture and strategic behavior. To test this hypothesis, I divided my population by nationality in four groups. Africa & Middle Orient, America, Europe, Asia and Oceania. The nationality taken was the nationality at the birth of the

runner and not the country they represent during the competition. Indeed, a lot of African runners are paid by countries in order to win medal for them in exchange of their nationality, these runners are considered as African & Middle Orient, not Asian or European. The population studied had 132 Africa & Middle Orient (AMO) runners, 59 America (AME) runners, 76 Europe (EUR) runners, and 12 Asia and Oceania (ASO) runners. The results are the following:

Race	???	t-stat	Sign.	Crit-P	???	t-stat	Sign.	Crit-P	N	R²
AMO	-0.774	-4.252	***	0.001	0.135	5.021	***	0.001	590	0.041
AME	0.731	2.524	**	0.012	-0.075	-2.451	**	0.015	258	0.025
ASO	-2.879	-1.616		0.112	0.256	1.834	*	0.072	57	0.071
EUR	-1.251	-4.201	***	0.001	0.155	4.228	***	0.001	330	0.054
General	-0.570	-4.240	***	0.001	0.081	4.914	***	0.001	1235	0.019

The results of this analysis are very interesting. Indeed, and to the exception of Asia & Oceania runners (but that can be explained by the small number of observations for this culture, and it is very close to significance, since the estimate for ??? is significant at the 10% level and the estimate for ??? has a critical probability of 11.2%), all cultures have significant estimates for ??? and ??? at the level of 5% which means that they all have a tendency in strategic behavior. However, we can notice that if Africa & Middle Orient runners and Europe runners estimates signs confirm the hypothesis of Baik and Shogren, the signs of the estimates for the America runners are opposite to this hypothesis, since ??? is positive and ??? is negative. This means that in a situation of contest where there is a favorite and an underdog, American runners who are pre-race favorites tend to overexert their level of effort at the beginning of the race, while underdogs tend to wait until the end of the race in order to exert their effort. By aggregating the three other populations, (AMO, ASO, EUR) in order to have two subset of the general regression, I realised a Chow test in order to see if the difference between the estimates for American runners is significantly different to the estimates for runners from other part of the World. The F-Stat for this test realised with Gretl is equal to 8.250 which gives a critical probability of 0%. We can therefore affirm that there is a significant difference in the strategic behavior in contests between Americans and the rest of the word.

The last phenomenon impact that I would like to analyze in this study is the impact of the density of competition on the strategic behavior in contests. Indeed, if we look at races from the point of view of the favorite, then we can think that the less dense are the races, the more interest the favorite has to exert his effort early in the race. For example, if we take the example of a runner who is 2 seconds ahead of any other runner on 800m, then why would I remain inside of the peloton with the risk of being jostled, the risk to fall, to be surprised by an unexpected attack while I am blocked by other runners, etc.? We can imagine that in such cases their may be an incentive for the favorite to overexert his level of effort in the first stages of the race, since he knows that he has such a margin over other runners that for whatever his margin may be, there is still a chance for him to win. This reasoning can be reproduced for the second and third biggest favorites who might want to eliminate potential rivals for the race to the podium. In order to represent this phenomenon, I decided to divide the 24 races that I studied in two classes, the low-density races and the high-density races. For this, I studied for each races the difference in percentage between the runner with the highest estimated time ?????? (see 3.2 for the explanations on the calculation of this variable) with the eighth runner with the highest estimated in order to avoid the bias of the number of runners in the races. Indeed, there is only 8 runners in an 800 m final while there are 15 of them in a 5000m final. This means that the probability of having the fifteenth runner of a 5000m final with a ??????being farther from the best one than the eighth runner in an 800m final is pretty high. Moreover, runners that are ranked ninth are farther have usually but a marginal influence on a race, therefore their impact on the density of the race can be considered as negligible. Once the density was determined for each race, I ranked them by level of density and separated the sample of race in two subsets: one with the races for which the density was above 2.5% and are categorized as Low-Density Races (LDR), one with the races for which the density was below 2.5% that I categorized as High-Density Races. There are 13 HDR in the studied sample and 11 LDR. The results of the regressions for the two subsets are the following:

Race	???	t-stat	Sign.	Crit-P	???	t-stat	Sign.	Crit-P	N	R²
LDR	-0.394	-2.056	**	0.040	0.059	2.429	**	0.015	631	0.009
HDR	-0.759	-4.016	***	0.001	0.104	4.595	***	0.001	604	0.034
General	-0.570	-4.240	***	0.001	0.081	4.914	***	0.001	1235	0.019

The results show that the two subsets follow the hypothesis of Baik and Shogren. However, we can notice that the estimates for f_?and f_?are more significant for the high-density races than for the low-density races which means that there is a bigger chance that the favorites do wait the end of the race to exert their effort in races with a high density than in races with a low density. Moreover, the absolute values of the estimates are bigger for low-density races than for high-density races which means that the gradient is steeper for the high-density races. This means that in high-density races, favorites tend to move to the front of the pack much later than in low-density races. However, we cannot say that the estimates of the low-density and high-density races have a significant difference, since if we realise the test of Chow for these two subsets, we obtain a F-Stat of 0.663 which gives a critical probability of 57.5%. We cannot therefore assert that there is a significant difference between the races with a high density and the races with a low density. This can be explained by the fact that we took only races which are world championships finals in the study. Therefore, the level of density is usually constant for all of these races. If we had taken races from a lower level, like a regional level, the runners would have been much more heterogenous in terms of level of performance, and we would have witnessed probably a much more significant result.

From a general perspective, the results are in favor and Baik and Shogren hypothesis that in a situation of contest where there is heterogeneity among the contestants with underdogs and a favorite, the underdogs have an incentive to overexert their level of effort in the first stages of the contest. Indeed, if the analysis of races taken individually may show cases of races that have contradictory results, as soon as a certain number of races are compiled in order to have enough data, the results follow the predictions of Baik and Shogren and are highly significant.

These results are highly interesting because if these strategic behaviors are applicable to athletic competition, it is a strong possibility that they also apply in other contests which happen in different contests such as R&D rivalry between corporations to get a lucrative or strategic innovation, bribery to assure a profitable license, patent or contract from the

government, war for a new global market that has been created by a new innovative product, a political election where candidates fights during long campaigns in order to get elected, or candidates who compete for a job or to win a promotion. This could represent the fact that the underdog in these situation, in order to have a chance to beat the favorite, is trying to anticipate the effort of the favorite in order to have a competitive advantage that might shake the balance of the game. This business strategy has been used for example by Microsoft (i.e. the underdog) with the release of the Xbox 360 in 2005 that anticipated the release of its rival the PlayStation 3 of Sony by one year. This strategy has enabled Microsoft to build the biggest possible installed base option for players (To-maselli, Di Serio, & De Oliveira, 2008) and therefore a strategic advantage that enabled Microsoft to catch up Sony in terms of sold home video game consoles. Another example can be the strategy of Small and Medium Enterprises (SME) that can have a more aggressive strategic behavior in their Human Resources Management (HRM) in their recruitment in order to get a chance to get the best candidates. (Brand & Bax, 2002) By overexerting their level of efforts in the early stage of the job research of the ideal candidate, with the promise of an higher pay, the fact that they contact the candidate instead of waiting the candidate to come to them, they indeed increase their chances to recruit a better candidate than if they had waited the candidate to come to them. As these results are complicated to find in the economic environment, we have obtained them by using the example of athletics and the result of the study seems to approve this hypothesis, at least for sports. However, there are still some little oddities that can be remarked in the results and that is not yet answered.

The results of this study about the impact of Gender, Culture and Density are also very interesting. Concerning gender, the results are a little bit different from what Boyd and Boyd had found in their study. Indeed, they had found that the results were more significant for men than for women, and in our experiment, it seems that the aggregated results are more significant for women than for men. However, the differences are very narrow and there is no significant difference between the estimates of the two subsets men and women, even if we can notice that the gradient of the women is higher, which means that women who are favorite of a race tend to make their move in the latest stages of the race. This result can be in part explained by the men can usually be more confident than women

in situations of competition (Gneezy et al., 2003; Gneezy & Rustichini, 2004). If a favorite has more confidence in his capacities, then his incentive will be to exert his effort soon in the first stages of the race to eliminate as many opponents as possible with an elevated pace. This observation is really interesting from a managerial point of view, because it would mean that a managerial environment based on competition could be more interesting for men in order to improve the immediate performance of employees. However, this observation is I believe in this study by the fact that the population studied is an elite that evolves in a very competitive environment, therefore, women in a world championships final are fierce competitors and maybe this is why this phenomenon is complicated to observe in this study. Maybe if we had observed a population with more «amateur» athletes, there would have been a stronger difference between men's and women's difference.

Concerning the impact of culture, the result have been quite surprising to me because if I expected the Africans to run differently from the rest of the World, eventually they had the same strategic behavior than others and it is the Americans who act differently. Not a lot of studies have studied the association between the dimensions of culture and strategic behavior. Some studies have studied the link between national culture and entrepreneurship characteristics and traits (Hayton, George, & Zahra, 2002; Müller & Thomas, 2000; Shane, Kolvereid, & Westhead, 1991) and some of these findings may explain the fact that American still overexert their level of effort at the beginning of races since it has been proved that entrepreneurial traits such as risk taking and high energy levels decrease as cultural distance from the US increases (Thomas & Müller, 2000). If risk taking and high energy levels are cultural traits closer to American culture, then it can explain why favorites may decide to overexert their level of effort at the beginning of races. This difference in behavior could be a possible explanation of why American corporations usually enter more easily foreign markets than corporation from other country do, because from the moment they enter in the competition, they set the level of effort required for the home compagnies at a level that the foreign companies can never reach. However, there is another explanation in the fact that the university athletics system is highly developed in the US and Canada. This means that almost all young American runners between 18 and 23 go through this systems of competitions which work only on a championships basis. While young Europeans and young Africans run in meetings where there are pacemakers, the Americans of the same age run in championships. Therefore, young Americans have learned from the beginning how to run fast in championships

races because they still have minimum time required in order to qualify for International championships. They have to do these minimum time required in the university competition where there are no pacemakers and therefore favorites are learned to start with a higher level of effort compared to the level of effort that favorites of middle distance races exert at the beginning of races, because if they still try to win the race, there is still a time to do for them, while in Europe, as soon as you have qualified for the championship, time doesn't matter any more at all. I believe this is the most plausible explanations of why Americans favorites have an opposite strategic behavior compared to the rest of the world. However, we can note that this behavior is also true for central and south American runners, even if these runners usually go to study in American or Canadian universities and therefore learn the same strategic patterns.

The last finding that could have interesting impact from a managerial point of view is the fact that it seems (even if results on this study are not significant because of the homogeneity of the population of races) that the more a race is dense, the more favorites wait before exerting their level of efforts. If we accept the fact that we do not obtain a significant result because of the homogeneity of the races which gives only races with a high level of density, this could explain for example why in the fight for innovation, the enterprises that deposit the patent first are rarely the ones that do the first macro-scale commercialization. (Golder, Shacham; Mitra, 2009). However, I fear that it is the only example that we may find an explanation in the economic world. Indeed, in the digitalized market world, the forces in high density contests have more incentive to overexert rapidly all their possible effort because these markets work in a fashion of «first mover wins» and «first wins all» (Liebowitz, 2002) and the strategy to let the underdogs do their first move is probably not the good one since by letting underdogs grow, you may end-up with a rival bigger than you that will make of you tomorrow's underdog. This logic can also be applied for example in politics, where if a favorite candidate, for example the mayor elected in the last election, has no real rival for the next election, he will start his campaign at the last moment because he will not have the need to campaign early and he will prefer to implement his policies as mayor instead. On the contrary, if his election is hanging by a thread, he will start his campaign much sooner and will exert his effort months before the election. This is why I believe the strategic impact of density in the way that a higher density leads to a lower exertion of effort at the beginning of the competition is only valid for the situations like athletics.

The results are not as significant for the aggregation of the 800m races than for the other studied events. Indeed, they are almost the only aggregated results with more than four races compiled that is not significant. But I believe an explanation to this insignificance can be found in the characteristics of this race.

800m races are the only one that are run in positive split in international championships, this means that the first half of an 800m final is usually faster that the second half of the 800m. A study by Hanon, Choffin, Descoux, Dupont, Gajer, Lacroix, Lamare, Ma-rajo, Mossant, Stéphan, Viale, and Vollmer (2015) who analyzed all the split times in all international finals since the Olympic Games of 1960 in Rome. He observes that on the 28 analyzed finals, 57% of them are run with a faster first half of the race. This means that on the contrary to 1500m and 5000m, the 800m in championships is a race that is run at a pace closer to the meeting pace in his first part than 1500m or 5000m. This can explain why favorites tend to be in front of the race much sooner in 800m than 1500m, because the pace is high enough during the race, therefore the chances for the favorite to be surprised by the underdog is much lower. The fact that the 800m in championships is run closer to its pace in meetings can also be explained by the fact that 800m is run with an intensity around 120% of the MAS of the runner (Spencer, Gastin, & Payne, 1996; Thomas, Hanon, Le Chevalier, Couturier, & Vandewalle, 2003), this is much higher than the value for 1500m which is 106.5% and 5000m which is 97% (Scherrer & Monod, 1960; Ettema, 1966). This means that even, when they are slow in the first half of the race, 800m runners run at a level that is already much above their MAS and production of lactic acid is high even with a slow start, which explains why it is possible to run a 800m in 1'44» with a first lap in 51 seconds and a second lap in 53 seconds, but it is not possible to do it with a first lap in 53 seconds and a second lap in 51 seconds. Indeed, to run 53 seconds on 400m for 800m elite runners, it is required a speed that is already much above their MAS. Therefore, they are usually not able to accelerate in the last stages of the race because their muscles are completely tetanized by the lactic acid produced by the muscles when he produces an effort above his MAS and the last 200m of an 800 meters in championships is often the slowest of the race while it is the fastest for 1500m and 5000m. Hanon et al. (2015) also show this phenomenon in their study and even show that the first and third 200m are usually the fastest ones in championships 800m because it is the moment where runners want to place themselves in the peloton for the first one, and

because it is the moment every runner accelerates to get the best position in the final «sprint» for the third one. In fact, the repositioning in the third 200m of 800m is more a question of getting a spot in the peloton where I have less chances to be jostled in the last 200m where runners know they will not be able to accelerate any more if they ever lose speed in a hustle because of the tetanizing effect of lactic acid.

This effects means that it is a little bit more an advantage in 800m to be in front of the pack a little bit sooner since you have less chances to be jostled in the last 200m, and therefore lose crucial km/h, if you are in front of the pack. It is as a consequence crucial to be able to run a strong third 200m without exploding in the last 200m in order to win a 800m race, this overexertion of the level of effort a bit sooner than in other events is for me a reasonable explanation of why the study does not work as well for 800m than for the other events.

As explained many times in this study, the main thing that could have been improved is the heterogeneity of the population of races. Indeed, because of the nature of the collected data, the split times for every runners for a number of intervals during the race, it was impossible to gather for me data of this quality on second-tier events such as national championships, or even continental championships because of the costs implied by such a technology. It was of course not available either for competitions of even lower importance such as regional races. However, such a data would be highly interesting in order to study the difference in behavior between elite athlete and more amateur ones, between genders, because amateur women may be less competitive than amateur men compared to elite women versus elite men. Therefore, a higher difference in competitive mindset may give a higher difference of impact concerning the gender. Such a data would be also interesting in order to see the strategic behavior in races where the density level is above 10% or even 20%. In such a case we could well imagine that favorites are absolutely not going to wait for the slowest underdogs and therefore exert their level of effort according their own level from the beginning of the race. Of course, those races would need to be championships races, i.e. races without a pacemaker.

Another amelioration that we could make is to insert other variables in the study, by taking into account the biological characteristics of the runner such as his height. Indeed, a tall runner may have an incentive to run either more in front of the peloton because his

long legs need more place and he wants to avoid to trip over someone, and therefore lose energy, or at the back of it behind everybody to be like a tower guard watching the peloton and waiting his moment to attack from behind, surprising everybody. Therefore, if we succeeded to implement these variables in the model, it would be very interesting the impact they can have on the effort expenditure over the race. More variables would be needed in order to improve the R² of the aggregated data which is, I must admit it a little bit low. It could be the weather preferences, or the cultural background, since it seems to have an impact from what this studied have found. Maybe even the density of the race could be implemented in it.

However, I believe that it will always be complicated to have a strong R² for this model since if there can be found patterns based on quantitative and qualitative data, the problem of the level of effort during a middle-distance race is that the choice of exerting the level of effort is dynamic. Indeed, if a runner may choose before the race to take such or such a strategy, in reality all of them adapt their strategy to the living moment, except from the runner who chooses to go from the start and exert his level of effort from the beginning. The decisions have to be taken so quickly in these moments that they do not always really have the time to think their strategy during the race, they rather react by instinct and by experience to the behavior of other runners. It would be interesting I believe to measure the way runners react to other runners' actions and analyze it in terms of strategic choices in the context of a game. However, I must admit that my mathematics level is not sufficient to apply this theory to the model.

Overall, sport has proved once again that it is a very interesting field of study to test economics theories. The results of this study agree with Baik and Shogren hypothesis that in a contest with sequential moves and with heterogenous participants, the favorite has an incentive at the beginning of the race to let the underdog overexert his level of effort. If the fact to use the ranking of a runner over the interval instead of his position at the end of the interval like Boyd and Boyd had done in their experiment has reduced the significance of each race individually because of the increase of the volatility of the runners positions on the intervals, the aggregated results validate the hypothesis of Baik and Sho-gren with a strong level of significance.

This result seems to be true also for all distances, at least with the aggregated results, and for all sexes and cultures, to the exception of American runners who seem to choose an opposite strategy when they are favorite of the races. Moreover, the results do not show significant differences in the results of men and women and in terms of density even if the homogeneity of the sample (races of elite competitors with a very close level one to another) may have biased the observation of these phenomenon. We also found that it seems that the longer are the races, the sooner the favorite make their effort. This result is contradictory with the results of Boyd and Boyd. However, it is complicated to make real comparisons between my results and theirs results because they had not aggregated their data and they had not studied the races the way I did, therefore the difference of results probably comes from the different methodology. We obtain moreover strong level of significance for most of the results obtain and at least with one of the estimates with a significance at the level of 10% for all the results.

I also believe that the fact that American have an opposite strategic behavior compared to other runners is the most surprising result but also the most exciting one. Indeed, this result shows that there is a significant difference between Americans and other cultures and may be this should need further research in the future in order to see if it is just a particular case in athletics races or a more general result applicable to other types of contests, and possibly economic ones.

I believe that the fact that I found the same results than Boyd and Boyd despite of the fact that I changed two major important methodologies in their study, with the creation of a favorite index and the change from position to ranking, gives a strong support to Baik and Shogren hypothesis, because I believe these two changes are a considerable improvement of Boyd and Boyd experiment and makes the experiment of this study much more supportive for Baik and Shogren hypothesis.

These results are very interesting because if the hypothesis that underdogs usually overexert their level of efforts at the beginning of contests, then it could have implications for the numerous other situation where contests happen. This could apply to contests such as R&D rivalry for a lucrative or strategic innovation, bribery to assure a profitable patent or contract from the government, competition for a new global market that has been created by a new innovative product, a political election, or candidates who compete for a job or to win a promotion.

The biggest limitation of this study is I believe the low level of R² for all the results that I found, which means that the model that I used is not that good an explanation of the

evolution of effort in a race. However, the R² scores for each race are in the same dimension than the ones that Boyd and Boyd obtained in their study. Therefore, maybe this is just a characteristic of the model and this is not too relevant. Other limitations are that the data is not as various as it could have been, it would have been indeed interesting to study races from a lower level that a world championship final, for example a regional races. Moreover, the splits available were not always sufficient and it could have been interesting to study 1500 and 5000m with more intervals available in the data. Even if this narrowing of the number of laps have helped me getting results because of the increase of volatility of the positions due to the use of ranking over interval rather than position at the end of the interval.

Further research possible could be done by analyzing with a similar methodology races from a more heterogeneous sample, indeed, by only analyzing world championships finals, I analyzed only races with high-level competitors, i.e. people with extreme competitive mindset compared to other populations. Therefore, it would be really interesting to replicate the same study on more «amateur» races. Moreover, such a study could also help to observe differences in strategic behavior impacted by the gender of the runners or by the density of competition inside of the race. Therefore, further research may try to look in the direction of the differences of strategic behavior between genders and also between different contexts of competition density. Another possible way of further research that could improve this study would be to analyze the impact of the actions of runners on the runner actions. Such a study could possibly increase the level of R² for this particular case which is middle-distance races.

Eventually, I believe that the case of semi-finals could be really interesting in order to study the behavior of economic agents when they have secured their maximum utility. Indeed, the case of semi finals could be used to analyze the level of effort of an agent when he has already attained his objective. In general, a runner who knows that he will not be caught up and has therefore secured his position in the direct qualifying places for the final may have an incentive to save some energy and therefore reduce his level of effort. It could prove once again that sport is a relevant empirical background to study the behavior of economic agents, since this behavior, and the reduction of the level of effort of a employee for example that would have already reached his yearly objective before the end of the year.

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