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

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.

Services	L	1622
	R	1404
	Total	3026
Services (in %)	L	54
Services (in %)	R	46
Won points	L	1040
Won points	R	918
Won points rate (in %)	L	64
	R	65

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:

		Goalkeeper
		L	R
Striker	L	(58.30, 41.70)	(94.97, 5.03)
	R	(92.91, 7.09)	(69.92, 30.08)

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.

	Striker		Goalkeeper
	L (%)	R (%)	L (%)	R (%)
Nash Equilibrium	38.54	61.46	41.99	58.01
Observations	39.98	60.02	42.31	57.69

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.

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

2.1 Why sport is a good field to study economics?