Individual Responsibility in Team Contests

Transcription

1 Individual Responsibility in Team Contests Antoine Chapsal and Jean-Baptiste Vilain

2 Individual Responsibility in Team Contests Antoine Chapsal & Jean-Baptiste Vilain March 7, 2017 Abstract This paper empirically analyzes team effects in multiple-pairwise battles, where players from two rival teams compete sequentially Using international squash tournaments as a randomized natural experiment, we show that winning the first battle significantly increases the probability of winning the second battle This result contradicts recent theoretical literature on multi-battle team contests, according to which outcomes of past confrontations should not affect the present ones Furthermore, we implement an estimation strategy in order to identify the effect at stake It allows us to rule out psychological effects such as psychological momentum or choking under pressure, and to show evidence of an individual contribution effect: players not only benefit from their team s win, but also value the fact of being individually - even partly - responsible for the collective success Such an effect is of first importance to understand why, despite free riding, individuals can make a significant effort when facing collective-based incentives JEL Classification C72, D79, L8, M54 Keywords Team Economics, Multiple-Pairwise Battles, Individual Responsibility Chapsal: Department of Economics, Sciences Po, 28, rue des Saints Pères, Paris, France antoinechapsal@sciencespofr); Vilain: Department of Economics, Sciences Po; 28, rue des Saints Pères, Paris, France jeanbaptistevilain@sciencespofr) We are grateful to Romain Aeberhardt, Ghazala Azmat, Gabrielle Fack, Philippe Février, Emeric Henry, Guy Laroque, Gael Le Mens, and seminar participants at Sciences Po The usual disclaimer applies 1

3 1 Introduction Multiple pairwise battles, 1 refer to extremely common situations where players from two rival teams compete sequentially Such situations include for example sport events or competition between firms for winning different markets A famous example of multiple pairwise battles is the tennis Davis Cup, where the players from two national teams compete sequentially in a best-of-five contest Fu, Lu & Pan 2015) present a benchmark theoretical analysis of multiple pairwise battles They show, under standard assumptions, that the outcome of a battle is independent from the outcome of previous and subsequent confrontations Such a result, which they refer to as neutrality, implies that confrontations can be considered independent, and there is not any dynamic linkage between subsequent battles The order of play does not affect the final result Neutrality comes from the fact that players do not internalize the cost of effort of the next battles, simply because it is borne by their teammates The opposite phenomenon namely, the discouragement effect, arises in multi-battle individual contests where the same players sequentially fight one against the other 2 In such individual contests eg, a two-set tennis match), winning the first confrontation or the first set) positively affects the probability of winning the next one: the remaining effort to win the contest is lower for the front runner than it is for the laggard The former is therefore more likely to win than the latter Such a discouragement effect, which has been intensively studied, cannot occur in multiple pairwise battles, as the remaining effort to obtain the final payoff after a non-definitive battle is not to be supported by the current player From a theoretical point of view, neutrality comes from the fact that the prize spread, ie, the difference in expected payoff from winning and from losing a battle, is the same for the two opposing players Therefore, the outcome of a battle does not depend on previous outcomes On the contrary, the discouragement effect in individual contests is caused by asymmetric prize spreads: the front runner has a higher prize spread than the laggard and accordingly more incentives to win As stressed by Fu, Lu & Pan 2015), the neutrality result sharply contrasts with the usual wisdom according to which battles are not independent in a team contest Even when assuming that a player does not bear the cost of his teammates, three kinds of effects would explain that winning a first battle should affect the outcome of the subsequent one in multiple pairwise battles: i) environment effects; ii) psychological effects; and iii) individual responsibility First, neutrality implies that team environment does not affect players performance For instance, the neutrality result holds as long as there is no peer effects Peer effects would occur if the average level of a team or the average level of the players that are not part of the confrontation under scrutiny) positively affects each individual performance Being in a more 1 The expression Multiple pairwise battles is used by Fu, Lu & Pan 2015) The alternative expression Multi-battle team contest is also used by some authors 2 See Dechenaux, Kovenock & Sheremeta 2015) for a survey Contrary to multiple pairwise battles, there is an abundant literature on individual contests, which finds evidence of the dependence of outcomes in subsequent individual confrontations and confirms the discouragement effect For instance, Klumpp & Polborn 2006) model US presidential primaries as a best-of-n contest between two candidates and show that winning the early districts strongly affects the probability of winning later districts Malueg & Yates 2010) find empirical evidence of strategic effects in individual tennis matches Taking a sample of equally skilled players same betting odds), they show that the winner of the first set exerts more effort in the second set than the loser They rule out the fact that such an effect is psychological Mago, Sheremeta & Yates 2010) provide experimental evidence of a discouragement effect in a best of three Tullock contest They also show that this effect is strategic, not psychological Harris & Vickers 1987) show that in a two-firm R&D race model, an early lead yields easy wins in subsequent battles because of a discouragement effect of the lagging opponent Konrad & Kovenock 2009) show in a theoretical framework that the introduction of intermediate prizes for component battles ie, payoff from winning a single battle even if the match is lost) reduces the discouragement effect 2

4 stimulating environment might increase each teammate s probability of winning The existence of peer effects is still debated in the literature For instance, Mas & Moretti 2009) show, using high-frequency data from a field experiment, that worker effort is positively related to the productivity of workers who see him On the contrary, Guryan, Kroft & Notowidigdo 2009) find no evidence of peer effects in a high-skill professional labor market: neither the ability nor the current performance of playing partners affect the performance of professional golfers Second, two kinds of psychological effects namely psychological momentum and chokingunder-pressure effects, may explain the absence of neutrality More precisely, a psychological momentum refers to the fact that winning a battle provides the player with extra confidence and helps him winning the next one It therefore implies that initial success in a contest produces momentum that leads to future success The choking-under-pressure effect may occur when a player faces more pressure than his opponent This pressure might have a detrimental effect on performance and might thus explain that winning the first battle affects the probability of winning the next one For instance, Apesteguia & Palacios-Huerta 2010) use the random nature of the order of soccer penalty shoot-outs to provide evidence of such psychological pressure They show that teams that take the first kick in the sequence win the penalty shoot-out 605 percent of the time Given the characteristics of the setting, they attribute this high difference in performance to psychological effects resulting from the consequences of the kicking order Kocher, Lenz & Sutter 2012) find different results using a larger sample of penalty shoot-outs Third, individual responsibility may explain outcome dependence in multiple pairwise battles Two kinds of opposite individual responsibility effects may exist A player may dread being partly) responsible for his team s defeat guilt aversion ) This effect has been developed in theoretical and experimental literature For instance, Charness & Dufwenberg 2006) examine experimentally the impact of communication on trust and cooperation Their design admits observation of promises, lies, and beliefs They found evidence of guilt aversion showing that people strive to live up to others expectations Furthermore, Chen & Lim 201) develops a behavioral economics model in order to analyze whether managers should organize employees to compete in teams or as individuals They consider that their main conclusion, according to which team-based contests yield higher effort than an individual-based contest, is driven by guilt aversion However, their experiment does not allow them to rule out an alternative interpretation of their results: players may value not only the reward yielded from their team s win but also the fact of being partly responsible for the collective success what we refer to as individual contribution ) They indeed specify that within-group comparisons may exist among team members that contributes the most to the team s victory 4 In such a case, the current player of the leading team has a higher probability of being partly) responsible for the collective success than his opponent This higher probability increases his incentive to make a high costly effort, thereby increasing his probability of winning If players value contributing to their team s success, then the prize spread is no longer the same for the front runner and the laggard In such a case, winning the first battle endogenously creates asymmetry in prize spreads and may therefore lead to outcome dependence between subsequent battles These potentially strong effects motivate to empirically test for neutrality Fu, Ke & Tan 2015) provide an experimental test by conducting a simple best-of-three team contest experiment In their setting, teams compete by counting the number of zeros in a series of 10-digit number strings composed of 0s and 1s Players from rival teams are pairwise matched Fu, Ke & Tan 2015) conclude that players from both teams remain equally motivated after observing the outcome of the first component contest, and therefore a team tournament is equally likely to end after two or three component contests However, their results are based on a limited number of observations, which may affect their conclusions Moreover, Huang 2016) uses team squash 4 Chen & Lim 201), p 28

5 data and does not find evidence against neutrality However, his findings are robust neither to a more extensive dataset on team squash championships nor to alternative measures of players relative ability than the ratio of their rankings Our main contribution is to show strong evidence, based on field data, of a dynamic linkage between confrontations in multiple pairwise battles Following Huang 2016), we use international squash team championships as a randomized natural experiment to test the neutrality result in multiple pairwise battles International squash team confrontations offer a perfect empirical setting, as they consist in best-of-three team contests, where players of the rival teams compete sequentially, each player only playing once Furthermore, there is no selection bias that might affect team compositions as only the best national players are selected to participate in such events More importantly, the sequence of battles in a team confrontation is randomly drawn and cannot be manipulated Based on an extensive dataset of international squash team confrontations from 1998 to 2016, we find evidence of a dynamic linkage between subsequent battles More precisely, we show that winning the first battle significantly increases the probability of winning the second battle Such a team effect contradicts neutrality We also derive testable predictions from a simple theoretical model to further explain outcome dependence in this multiple pairwise battles setting We find strong evidence against psychological effects: psychological momentum and choking-under-pressure effects do not explain the dynamic linkage between the first two squash battles Our empirical findings suggest that outcome dependence among battles results from individual contribution Players value being at least partly) responsible for the collective success The remainder of this paper is organized as follows Section 2 provides empirical evidence against neutrality in multiple pairwise battles After presenting the neutrality results from Fu, Lu & Pan 2015), we introduce the empirical setting and the available field data We show evidence against neutrality in this setting: winning the first battle strongly increases the probability of winning the second battle, which implies that battles are not independent Section focuses on the identification of the mechanism driving non-neutrality: we show that outcome dependence is explained by the fact that players value not only the final reward yielded by their team s win but also being actively part of the collective success Section 4 further discusses the individual responsibility result and show that individual contribution can mitigate the effect of free-riding behaviors Section 5 concludes 2 Neutrality in multiple pairwise battles 21 Theoretical framework This section theoretically presents the neutrality result from Fu, Lu & Pan 2015) For the sake of simplicity, we consider a simple case developed by Fu, Lu & Pan 2015), which is a best-ofthree team contest with complete information where the contest success function is a lottery 5 This basic framework allows us to present the neutrality result and the assumptions on which it relies 5 Fu, Lu & Pan 2015) s setting encompasses any best of 2n+1 multiple-pairwise battles for which the contest technology is homogeneous of degree-zero in players efforts and induces a unique bidding equilibrium in any two-player one-shot contest with a fixed prize They identify seven contest functions satisfying these two requirements: all-pay auction with two-sided continuous incomplete information, generalized Tullock contest with two-sided continuous incomplete information, all-pay auction with discretely distributed marginal costs and two-sided continuous incomplete information, all-pay auction with one-sided continuous incomplete information, generalized Tullock contest with complete information presented in this paper) and all-pay auction with complete information 4

6 211 Setting We consider a best-of-three team contest with complete information A team X is opposed to a team Y The contest presents the following features: i) there are risk-neutral players in both teams Each player only plays one individual battle X i respectively Y i ) is the player of team X respectively Y ) that plays the i th battle, i = {1, }; ii) team X wins as soon as it wins two battles and loses as soon at it loses two battles; and iii) the third individual battle takes place only if team X and team Y have both won one battle Let p i be the probability that X i wins his battle against Y i, p i = x i x i + y i, where x i is the level of effort of X i and y i is the level of effort of Y i This function is the simplest version of the Tullock contest success function, 6 also referred to as a lottery contest Players do not have the same ability This is reflected in a linear cost function, given by CX i ) = x i θ Xi, where θ Xi is the innate ability of X i The cost of effort is thus a decreasing function of the innate ability of a player The payoff associated to the collective win denoted V ) is the same for every player Players also get a battle reward v when they win their individual battle independently of their team s outcome) V and v are strictly positive Theoretical results Result 1 Equilibrium probability of winning In a multiple pairwise battle, players choose their optimal level of effort such that the probability that player X i wins a confrontation is given by p θ Xi UXi i =, θ Xi UXi + θ Y i UY i where UXi = U Xi W in Xi ) U Xi Loss Xi ), respectively UY i = U Yi Loss Xi ) U Yi W in Xi ), is the prize spread of player X i, respectively Y i Proof Let U Ji W in Ki respectively U Ji Loss Ki ) be the utility of player J i, J i = {X i, Y i }, when K i wins respectively looses) battle i, K i = {X i, Y i } Both players choose their level of effort to maximize their expected utility: xi max U Xi W in Xi ) + y i U Xi Loss Xi ) x ) i, x i x i + y i x i + y i max y i yi x i + y i U Yi Loss Xi ) + θ Xi x i x i + y i U Yi W in Xi ) y i θ Yi Assuming that Us are independent of x i and y i, the first order conditions yield the following optimal levels of effort and equilibrium probability of winning p i : x i = θ Xi UXi ) 2 θ Y i UY i θ Xi UXi + θ Y i UY i ) 2 6 See Buchanan, Tollison & Tullock 1980) 7 Fu, Lu & Pan 2015) also consider the case where v = 0 As it does not affect their predictions, we only consider the case where v > 0 in this paper ) 5

7 Finally, y i = θ Xi UXi θ Y i UY i ) 2 θ Xi UXi + θ Y i UY i ) 2 p θ Xi UXi i = θ Xi UXi + θ Y i UY i where UXi = U Xi W in Xi ) U Xi Loss Xi ) is the prize spread of player X i and UY i = U Yi Loss Xi ) U Yi W in Xi ) is the prize spread of player Y i This first result shows that the outcome of a battle depends on two parameters only, which are i) players relative ability or cost of effort), and ii) players relative prize spreads When players are symmetric θ Xi = θ Yi ), the outcome of the battle will only depend on players incentives: if UXi > UY i, the prize spread between winning and losing will be higher for player X i, who will be therefore more likely to win than his opponent Result 2 Neutrality Fu, Lu & Pan 2015) In a multiple pairwise battle where there are i) common prize spreads ie, UXi = UY i ), and ii) no psychological effects, players choose their optimal level of effort such that the probability that player X i wins a confrontation is given by p θ Xi i = θ Xi + θ Y i Proof This result is directly derived from Result 1 The neutrality result comes from the fact that, when there are common prize spreads and no psychological effects, the equilibrium probability of winning is only determined by each player s ability The team contest therefore boils down to a series of independent lotteries Fu, Lu & Pan 2015) derives the three following results from neutrality: History Independence Players winning odds in each battle are independent of the state of the contest ie, leading or lagging behind has no effect) Sequence Independence Reshuffling the sequence of battles will not affect the probability of each team s winning the contest Temporal-Structure Independence The temporal structure of the contest sequential or simultaneous) will not affect the probability of each team s winning the contest These theoretical results show that neutrality is based on two crucial assumptions, which are i) common prize spreads, and ii) the absence of psychological phenomena affecting performance i) Common prize spreads We observe neutrality ie, p i = θ Xi θ Xi +θ Y i ) if and only if players have common prize spreads ie, UXi = UY i ) This condition is satisfied in the case where players only value the collective win payoff V ) and the battle reward payoff v) In a non-trivial battle, 8 both players have a prize spread of V +v, as they get both the collective and the battle rewards if they win and a payoff of 0 if they lose In battle 2, the two players also have the same prize spread: the player in the leading team gets V + v, if he wins, and p V if he loses as he can still get the collective reward V if his teammate wins battle, 8 A non-trivial battle is a battle for which the winning team has not been determined yet In a best of three team contest, battle is non-trivial if and only if each team has won one battle in the two previous rounds 6

8 which occurs with a probability p ), so his prize spread is V +v p V = v +1 p )V The player in the lagging team gets v +1 p )V if he wins, as he gets the battle reward for sure and the collective reward if his teammate wins battle, which occurs with a probability 1 p ) If he loses, the contest ends and he gets a payoff 0, so his prize spread is also v + 1 p )V A similar logic applies to battle 1 However, players might not only value the collective win and the battle reward but also the fact of being individually partly) responsible for the collective success If such motivation is at stake, prize spreads become asymmetric in battle 2: the player in the leading team has more incentives to win than his opponent because he is sure to contribute to the success of his team if he wins his battle, while his opponent will be success-responsible if and only if his teammate also wins in period Thus individual responsibility would invalidate the assumption of common prize spreads and lead to non-neutrality 9 ii) Absence of psychological effects affecting performance The second assumption on which neutrality holds is that players cost of effort are not affected by the circumstances of the contest: the outcome of a battle only depends on players relative innate abilities Nevertheless, players effort cost may be affected by the context through psychological factors A player might have a psychological momentum following the victory of his teammate, which would be equivalent to a decrease in his effort cost On the contrary, players cost of effort could increase when they face pressure This choking under pressure phenomenon could occur when stakes are high, for example when two players are opposed in a pivotal battle Incorporating such psychological effects in the cost function of players would also lead to non-neutrality There is a few phenomena that might explain why these two crucial assumptions do not necessarily hold The next section presents an empirical strategy to test for neutrality 22 Empirical setting and data 221 International squash championships as a randomized natural experiment Professional squash team data are particularly suited for analyzing multiple pairwise battles The structure of international squash confrontations exactly corresponds to a theoretical bestof-three team contest with complete information: the identity of the six players three in each rival team) involved and the order in which they play are determined before the beginning of the confrontation Battles are played sequentially; each player only plays one battle A team wins as soon as two of its players win International squash tournaments can be exploited as a randomized natural experiment to analyze potential team effects in multiple pairwise battles as they present the two following features: i) there is no selection bias; ii) the order of the battles is randomly drawn i) No selection bias Given the high stakes of these international championships, only the best players of each participating country are selected to compete Selection in the national team is driven by the results of each player on the various individual championships before the team event Players individual results, and, consequently, players national rankings, are not determined by the fact of being selected in the national team Therefore, there is no selection bias regarding the sample of players to be part of the team 9 A situation where players internalize part of their teammates cost of effort would also induce asymmetric prize spreads in battle 2 This is further discussed in section 41 7

9 ii) Ex-ante randomly-drawn order of play Each National Squash Association has to rank its players by descending order of strength and has to declare this order truthfully: a ranking that does not reflect the actual hierarchy amongst teammates can be ruled out by opponents or organizers 10 More importantly, the order of the three battles is randomly drawn among four possibilities for every confrontation: 1-2-, , 2-1- and -1-2 This ex-ante randomly-drawn order of play ensures that teams cannot manipulate in any way the sequence of games to be played 222 Data Our data 12 include 209 national team matches from 1998 to 2016 We consider 55 international team tournaments, including Men s and Women s World Team Championships, Men s and Women s Asian Team Championships and Women s European Team Championships 1 The World Team Championships are organized by the World Squash Federation WSF) The competition is held once every two years, with the venue changing each time The men s and women s events are held separately in different years 14 The Asian Team Championships are organized by the Asian Squash Federation ASF) and take place every two years Finally, the European Team Championships are organized by the European Squash Federation ESF) every year We also collected player s monthly world rankings, which are published by the Professional Squash Association PSA) These rankings are only based on players performance in individual tournaments, so they are not correlated with their performance in past team tournaments We use the PSA rankings as a proxy for players ability 2 Testing for neutrality in multiple pairwise battles According to Fu, Lu & Pan 2015) s model, the probability of winning a battle is not affected by the outcome of previous battles: only the relative ability of the players involved in a battle matters Neutrality comes from the fact that both players have the same incentive to win because they have the same prize spread ie, the same utility gap between winning and losing) Test 1 There is evidence in support of neutrality if winning the first battle does not affect the probability of winning the second one 21 The absence of neutrality: statistical evidence The simplest way to assess whether winning the first battle affects the probability of winning the second match is to construct a sample in which players from both teams involved in the second battle have similar rankings Based on this sample of equally skilled players, one would expect, if there were neutrality, half of the contests to be won by the player who belongs to the leading team 15 To do so, we compute the ratio of the rankings of both players involved in the second battle, and we restrict our sample to observations where this ratio is lower than some threshold values: 10 See section R of World Squash Championship Regulations for details meaning that players ranked 1st play the first game, players ranked 2nd play the second game and players ranked rd play the third game 12 The data comes from the website 1 We do not include Men s European Team Championships in our sample because this tournament adopts a best-of four structure with ties broken by points count back Men s World Team Championship, which was planned in Cairo, Egypt, has been canceled 15 Such an identification strategy is implemented by Malueg & Yates 2010) who use betting odds to construct a sample of tennis matches with equally skilled players 8

10 i) ratio < 15 variant 1, ii) ratio < 14 variant 2, and finally iii) ratio < 1 variant According to this definition, a match between a player ranked 15 and a player ranked 25 will not be included in any variant the ratio of these rankings being 1,66), while a match between a player ranked 15 and a player ranked 17 will be included in the three variants the ratio of these rankings being 1,1) We note X 1 the player who won the first battle against Y 1, and X 2 the player who belongs to the leading team involved in battle 2 against Y 2 Table 1 displays the empirical probability that X 2 wins the second battle for each of the variants considered It also specifies the frequency of the cases where the ranking of X 2 is smaller than the ranking of Y 2, ie, situations where X 2 is slightly better than his opponent, as a high observed probability that X 2 wins could be simply caused by the fact that X 2 is better skilled than his direct opponent Table 1: Satistical evidence against neutrality Variant 1 Variant 2 Variant X 2 wins battle 2 597% 591% 604% Ranking of X 2 < Ranking of Y 2 55% 5% 504% Number of observations Statistically different from 50% at p < 005, p < 001 The results presented in table 1 show that the probability that the player who belongs to the leading team wins is larger than 50% significant at the 5% level in the three variants) In other terms, winning the first battle significantly increases the probability of winning the second one This first finding against neutrality is not driven by a sample bias, as the proportion of matches where X 2 is better ranked than Y 2 is not significantly different from 50% in any of the three variants 22 Evidence against neutrality: main specification Restricting the sample to players who have similar rankings is a convincing way to control for players relative ability but it considerably reduces the number of observations In order to use our entire sample, we need to integrate a measure of players ability as a control variable We use rankings as a categorical variable with 7 modalities Top 5 / 6-15 / 16-0 / 1-50 / / / , which is the reference category) to control for players ability 16 We label the two opposing teams as Team A and Team B 17 and their players as A 1, A 2, A, B 1, B 2 and B where the subscript indicates the battle in which the player is engaged We can test for neutrality by assessing whether the probability that A 2 wins against B 2 is higher when A 1 won against B 1 in the previous battle, controlling for A 2 s and B 2 s modality of ranking Thus we regress the dummy variable indicating whether A 2 wins or loses battle 2 on a dummy variable indicating whether A 1 won or lost battle 1, on dummies indicating the ranking modalities of A 2 and B 2 and on dummies indicating whether team A is at home or away the 16 We chose these modalities of ranking because they provide a very good fit to predict the winner on individual squash championships data Indeed, increasing the size of the ranking range by 5 from one modality to the next except for the last one) allows us to get a very good trade-off between an accurate measure of players ability and a sufficient number of observations in each modality Furthermore, we show in table that taking the ratio of players rankings instead of ranking modalities does not significantly affect the results 17 In the remainder of this paper, we label Team A and Team B each of both opposed teams in a given confrontation, with no further condition on the outcome of the first battle When we deliberately choose the team that won the first battle, we refer to it as Team X, or X 9

11 reference being neutral-field) 18 A 2 wins battle 2 = β 0 + β Non neutrality A 1 won battle β r Ranking r,a2 β r Ranking r,b2 + β home Home A + β away Away A + ɛ AB2 r=1 r=1 We use a linear probability model so as to interpret the coefficient easily 19 The results are displayed in column 1) of table 2 The coefficient of interest is significant at the 01% level and the magnitude of the effect is very strong: 014 This means that in a battle involving two players with similar rankings, the probability that the player in the leading team wins is 057 while the probability that the player in the lagging team wins is only 04 Neutrality implies that there is no environment effect, including peer effects Being in a team with high-performing teammates may increase the productivity of a player, as a more stimulating environment may increase performances Since high-performing players tend to win their battle, the player in the leading team is likely to be surrounded by more talented teammates than the player in the led team Therefore, peer effects might be a confounding factor for sequence dependence We take into account environment effects and other unobservables such as the relative quality of teams managers or the cohesiveness between players by including teams ranking each team is seeded) as additional continuous control variables in specification 2) Teams ranking reflects the extent to which teams are favorite and are determined before the beginning of the competition by specialists, who base their judgment on all available information Such a ranking therefore encompasses most of the environment effects that may be at stake, including the current physical condition of each player Test 2 There is evidence in support of environment effects as the only driver of non-neutrality if we do not observe any dynamic linkage once taken into account team characteristics in particular the overall team level) Winning the first battle remains significant at the 01% level once teams rankings are introduced and the magnitude of the effect does not change much 011) This is clear evidence that sequence dependence is not caused by confounding peer effects 18 Note that the structure of our dataset is very particular because it is symmetric: if a player wins, his opponent loses Since we want to use rankings modalities, we need to decompose every battle into two observations We then weight each observation by 1 so as to adjust standard errors correctly 19 2 We obtain very similar results with probit and logit estimations 10

12 Table 2: Evidence against neutrality rankings categories) Dep var: A 2 wins battle 2 1) 2) A 1 won battle ) ) A 2 s ranking: Top ) ) A 2 s ranking: ) ) A 2 s ranking: ) ) A 2 s ranking: ) ) A 2 s ranking: ) ) A 2 s ranking: ) ) B 2 s ranking: Top ) ) B 2 s ranking: ) ) B 2 s ranking: ) ) B 2 s ranking: ) ) B 2 s ranking: ) ) B 2 s ranking: ) ) A 2 at home ) ) A 2 away ) ) A 2 s team seeding ) B 2 s team seeding ) Constant ) ) Observations R Standard errors in parentheses p < 005, p < 001, p < 0001 Reading note column 1): the probability that A 2 wins increases by 019 when his teammate A 1 won battle 1 Reading note column 1): the probability of winning of a player ranked in the Top 5 is 072 higer than the probability of winning of a player ranked below 105 reference category) 2 Robustness check In this section, we test an alternative specification in order to check the robustness of the results displayed in table 2 One potential concern would be that the categories of rankings do not correctly reflect players ability In such a measurement error case, winning the first battle might not have any causal impact on winning the second battle We thus perform the same regression using an alternative measure of players relative ability In table, we use the ratio of rankings instead of the categories of rankings to control for players relative ability The reference player is defined as the player with the better ranking in battle 2, so that the ratio of rankings is strictly smaller than 1 Winning the first battle remains significant at the 1% level and the magnitude of the effect is close to the one estimated with ranking categories 009 vs 011) This additional result confirms that the outcome of battle 1 11

13 affects the probability of winning battle 2 This result contradicts neutrality Table : Evidence against neutrality ratio of rankings) Dep var: A 2 wins battle 2 A 1 won battle ) Ranking A2 Ranking B2 < 1) ) A 2 at home ) A 2 away ) A 2 s team seeding ) B 2 s team seeding ) Constant ) Observations 89 R Standard errors in parentheses p < 005, p < 001, p < 0001 The role of individual contribution to team success In section 2, we presented empirical evidence against neutrality in multiple pairwise battles We further showed that the dynamic linkage between battles does not only result from team environment effects Other effects should explain why the outcomes of subsequent battles are dependent Based on our theoretical results, such a dynamic linkage may be caused by psychological effects or changes in prize spreads, which affect players incentives to win We adapt the theoretical framework from section 21 to describe the potential drivers of non-neutrality: psychological effects and individual responsibility 1 Psychological effects Psychological effects imply an increase or a decrease in players performance under certain circumstances, without any change in players incentives prize spreads) Two main psychological effects, which are discussed in recent economic literature, might be at stake in multiple pairwise battles: psychological momentum and choking under pressure Psychological momentum Psychological momentum implies that winning a battle increases a player s confidence and makes him more likely to win the next one success breeds success ) We integrate psychological momentum in our theoretical setting by multiplying by ψ ψ > 1) the ability of the player whose team won the last battle This changes the probabilities that player X 2 and player X win In that case, See Appendix for detailed computations p P M = θ X θ X + θ Y ψ, 12

14 where team X is defined as the team that won battle 1 and lost battle 2 p 2P M = θ X2 ψ θ X2 ψ + θ Y2, where team X is defined as the team that won battle 1 As p 2P M > θ X 2 θ X2 +θ Y2 and p P M < θ X θ X +θ Y, the following empirical test can be derived from our theoretical setting Test There is evidence in support of psychological momentum if: 1 Winning the first battle increases the probability of winning the second battle 2 In a non-trivial battle, the player in the team that won battle 2 is more likely to win than the player in the team that won battle 1 21 In order to test for the second condition, we focus on the subsample of non-trivial battles ie, the 191 matches of the sample for which the winning team has not been determined after the first two battles) For these matches, there are only two possible scenarios regarding the outcome of the two previous battles: either Team A won battle 1 and lost battle 2, or Team A lost battle 1 and won battle 2 We create a dummy variable labeled A 1 lost battle 1 and A 2 won battle 2, that is equal to 0 in the first scenario and to 1 in the second scenario Psychological momentum would imply that this variable has a positive and statistically significant effect on A wins battle The model we test is therefore given by: A wins battle = β 0 + β P M A 1 lost battle 1 and A 2 won battle β r Ranking r,a β r Ranking r,b + β controls Controls + ɛ AB r=1 r=1 and the results are displayed in table 4 column 1) As a robustness check, we use the ratio of rankings as an alternative measure of players relative ability column 2) The effect of the sequence variable A 1 lost battle 1 and A 2 won battle 2 is not statistically significant in any of the two specifications: psychological momentum does not explain non-neutrality 21 This identification strategy is also used by Malueg & Yates 2010) and Mago et al 2010) 1

15 Table 4: Evidence against psychological momentum Dep var: A wins battle 1) 2) A 1 lost battle 1 and A 2 won battle ) ) Ranking A Ranking B < 1) ) A s ranking: Top ) A s ranking: ) A s ranking: ) A s ranking: ) A s ranking: ) A s ranking: ) B s ranking: Top ) B s ranking: ) B s ranking: ) B s ranking: ) B s ranking: ) B s ranking: ) A at home ) ) A away ) ) A s team seeding ) ) B s team seeding ) ) Constant ) ) Observations R Standard errors in parentheses p < 005, p < 001, p < 0001 Choking under pressure Dynamic competitive settings may create psychological pressure on competitors, thereby affecting their performances The player who belongs to the lagging team might all other things being equal, face more pressure than the player in the leading team as he needs to win to ensure that his team remains in the contest Such a phenomenon might therefore explain why we observe a positive effect of winning the first game on the probability of winning the next However, the difference in competitive pressure faced by each player in battle 2 also depends on the expected result of a possible third game For instance, in a situation where i) X 1 defeated Y 1 in the first confrontation, and ii) X has a extremely low equilibrium probability of winning the third match, if any, ie, p goes to 0) the second battle is decisive for both players Both players therefore face maximal pressure as winning the second battle would give the final victory to the team they belong to On the contrary, if X has a extremely high equilibrium probability of winning the third match ie, p goes to 1), the second battle is a confrontation with nothing 14

16 at stake, its outcome will not affect team X s victory, and none of the players faces psychological pressure In other terms, the expected outcome of the third battle allows one to assess the gap in competitive pressure that is faced by the players in the second game Formally, considering that X 1 won battle 1, the choking-under-pressure effect yields the following prediction: 22 p 2CUP = θ X2 fp ) θ X2 fp ) + θ Y 2 gp ), where f) and g) are any function such that f p with 0 < η < 1) and lim x 1 fx) = lim x 1 gx) = 1 We therefore have > 0, g p > 0, lim x 0 fx) = lim x 0 gx) = η lim x 0 p 2CUP p = x) = lim p 2CUP p θ X2 = x) = = p 2 x 1 θ X2 + θ Y2 This yields the following empirical test Test 4 There is evidence in support of a choking-under-pressure effect if winning the first battle affects the probability of winning the second one, except when the equilibrium probability of winning the third battle takes extreme values, ie, the outcome of the third battle is almost certain In order to test this prediction empirically, we focus on the subsample of contests for which the outcome of battle is almost certain We consider that the outcome of a battle is almost certain when one player is at least two categories of rankings ahead of his opponent For example, a player in the top 5 facing a player ranked beyond 15 is expected to win This provides a precise approximation of the extreme cases where the outcome of battle is almost certain, as 88% of the battles involving players with a minimum gap of two ranking categories are won by the odds-on favorite We perform the same estimations as in the non-neutrality section on this subsample of contests Choking under pressure predicts that winning battle 1 should not affect the probability of winning battle 2 on this subsample because the two players involved in battle 2 face symmetric pressure when the outcome of battle is almost certain Results obtained with both ranking categories and the ratio of rankings as a measure of players relative ability show that the variable A 1 won battle 1 remains significant at the 5% level on this subsample see table 5) Moreover the magnitude of the effect is slightly higher than on the overall sample These findings show that winning the first battle affects the probability of winning the second one, even when the equilibrium probability of winning battle is either very high or very low Therefore, choking under pressure does not explain the dynamic linkage between subsequent battles 22 See Appendix for detailed computations 15

17 Table 5: Evidence against choking under pressure Dep var: A 2 wins battle 2 1) 2) A 1 won battle ) ) Ranking A2 Ranking B2 < 1) ) A 2 s ranking: Top ) A 2 s ranking: ) A 2 s ranking: ) A 2 s ranking: ) A 2 s ranking: ) A 2 s ranking: ) B 2 s ranking: Top ) B 2 s ranking: ) B 2 s ranking: ) B 2 s ranking: ) B 2 s ranking: ) B 2 s ranking: ) A 2 at home ) ) A 2 away ) ) A 2 s team seeding ) ) B 2 s team seeding ) ) Constant ) ) Observations R Standard errors in parentheses p < 005, p < 001, p < Individual responsibility Psychological effects do not explain why we observe a dynamic linkage between subsequent battles in our empirical setting Non-neutrality may be explained by an asymmetry of players incentives, which is caused by the role played by individual responsibility in team performance Individual responsibility may consist either in guilt aversion ie, the player dreads being partly responsible for the collective failure) or an individual contribution effect ie, the player values the fact of being partly responsible for the collective success) Guilt aversion Players might suffer from being partly) responsible for the failure of their team In this case, a player who looses his battle and consequently contributes to the final defeat of his team, supports an additional loss s, s > 0) This additional loss asymmetrically affects players prize spreads and therefore may explain the absence of neutrality 16

18 Player s team wins Player s team loses Player wins V + v v Player loses V s Table 6: Payoffs in the guilt aversion scenario Under this scenario, we derive from Result 1 the following predictions: 2 p 2GA = θ X2 v + 1 p )V + 1 p )s) θ X2 v + 1 p )V + 1 p )s) + θ Y2 v + 1 p )V + s) where team X is defined as the team that won battle 1 p 2GA < θ X 2 θ X2 +θ Y2, which yields the following empirical test Test 5 There is evidence in support of guilt aversion if winning the first battle decreases the probability of winning the second battle In out setting, winning the first battle increases the probability of winning the second battle see tables 2 and ) Non-neutrality is not driven by guilt aversion Individual contribution Players might value the fact of being partly) responsible for the success of their team If players individually value their contribution to the team, they get an additional reward c > 0) when their victory leads their team to success Table 7 displays players payoffs when there is individual contribution Player s team wins Player s team loses Player wins V + v + c v Player loses V 0 Table 7: Payoffs in the individual contribution scenario In such a case, the main intuition is that the player in the leading team would have more incentives to win the second battle than the player in the lagging team because he is sure to be part of his team s success if he wins while the player in the lagging team will be successresponsible if and only if his teammate also wins the third battle This asymmetry of incentives between the two players depends on the expected outcome of battle For example, if X 1 won battle 1 and X has a extremely low equilibrium probability of winning the third match ie, p goes to 0), both players can contribute to their team s victory winning battle 2, and both players would make a symmetrical positive effort to get the additional reward In this extreme case, winning the first battle should have no effect on the probability of winning the second one On the contrary, in the extreme case where X 1 won battle 1 and X has a extremely high equilibrium probability of winning the third match ie, p goes to 1), the asymmetry between the two players reaches its maximum: X 2 will get the contribution reward for sure if he wins while Y 2 has no chance to get it if he wins Formally, we obtain the following predictions, which 2 See Appendix for detailed computations 17

19 confirm the role played by p in the individual contribution scenario: 24 p 2IC = where team X won battle 1 θ X2 v + 1 p )V + c) θ X2 v + 1 p )V + c) + θ Y2 v + 1 p )V + 1 p )c) p 2IC p > 0 and 1 p 2IC ) 1 p ) > 0, so the probability of winning battle 2 increases with the teammate s probability of winning battle This allows us to derive the following empirical test for individual contribution Test 6 There is evidence in support of an individual contribution effect if Winning the first battle increases the probability of winning the second battle The probability of winning battle 2 is higher when the teammate involved in battle is favorite We can test for the second condition of test 6 by assessing whether the probability that A 2 wins against B 2 increases when A is favorite in battle ie, when A has a better ranking than B 25 ) In specification 1) of table 8, we regress A 2 s victory on a dummy variable indicating whether A has a better ranking than B, on dummies indicating the ranking modalities of A 2 and B 2 and on the control variables used previously playing home/away and teams seedings) A 2 wins battle 2 = β 0 + β IC A favorite in battle β r Ranking r,a2 β r Ranking r,b2 + β controls Controls + ɛ AB2 r=1 r=1 As predicted in the individual contribution scenario, the variable A favorite in battle is positive and significant at the 1% level Being in the team that is favorite in battle increases the probability of winning battle 2 by about 01 This effect is about as strong as the estimated effect of winning battle 1 on the probability of winning battle 2 see column 2) of table 6) This finding is perfectly consistent with the individual contribution effect, according to which winning battle 1 has no effect on battle 2 when the opposing team is expected to win battle In specification ), we use the ratio of rankings of A 2 and B 2 instead of the modalities of rankings The variable A favorite in battle remains significant at the 5% level and quite strong in magnitude about 008) One potential concern with specifications 1) and ) is the confounding peer-effects story: being favorite in battle might be significant because it might imply that the player is in a more stimulating environment with more performing teammates If such an effect were at stake, being favorite in battle 1 should have the same effect, as there is no reason to believe that the influence of the teammate playing battle 1 would be different from the influence of the teammate playing battle In specifications 2) and 4), we include a dummy variable indicating whether A 1 is favorite in battle 1 as a control to test for peer effects The variable A 1 favorite in battle 1 is not significant and its inclusion does not affect the coefficient associated to A favorite in battle 24 See Appendix for detailed computations 25 Some of the players in the sample do not have any PSA ranking because they are not professional players We consider that a player who has a PSA ranking is favorite when he is opposed to a non-professional player When two non-professional players are opposed in battle or in battle 1), we exclude the observation from our sample because we are not able to identify the favorite player This explains why the number of observations drops from 89 to 759 in table 8 18