Individual Responsibility in Team Contests
Antoine Chapsal and Jean-Baptiste Vilain
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, L83, M54.
Keywords Team Economics, Multiple-Pairwise Battles, Individual Responsibility.
∗ Chapsal: Department of Economics, Sciences Po, 28, rue des Saints Pères, 75007 Paris, France (e-mail:
[email protected]); Vilain: Department of Economics, Sciences Po; 28, rue des Saints Pères, 75007
Paris, France (e-mail: [email protected]). 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
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 (e.g., 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,3
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, i.e., 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.
3 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 U.S. 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 (i.e., payoff from winning a single battle even if the match is lost) reduces the “discouragement
effect.”
2
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 60.5 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 (2013) 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 (2013), p. 2833.
3
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 3 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
2.1
Neutrality in multiple pairwise battles
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
2.1.1
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 3 risk-neutral players in both
teams. Each player only plays one individual battle. Xi (respectively Yi ) is the player of team
X (respectively Y ) that plays the ith battle, i = {1, 3}; (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 pi be the probability that Xi wins his battle against Yi ,
xi
,
pi =
xi + yi
where xi is the level of effort of Xi and yi is the level of effort of Yi . 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
xi
,
C(Xi ) =
θXi
where θXi is the innate ability of Xi . 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.7
2.1.2
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 Xi wins a confrontation is given
by
θXi ∆UXi
p∗i =
,
θXi ∆UXi + θY i ∆UY i
where ∆UXi = (UXi |W inXi ) − (UXi |LossXi ), respectively ∆UY i = (UYi |LossXi ) − (UYi |W inXi ),
is the prize spread of player Xi , respectively Yi .
Proof. Let UJi |W inKi (respectively UJi |LossKi ) be the utility of player Ji , Ji = {Xi , Yi }, when
Ki wins (respectively looses) battle i, Ki = {Xi , Yi }.
Both players choose their level of effort to maximize their expected utility:
yi
xi
xi
,
(UXi |W inXi ) +
(UXi |LossXi ) −
max
xi
xi + yi
xi + yi
θXi
yi
xi
yi
max
(UYi |LossXi ) +
(UYi |W inXi ) −
.
yi
xi + yi
xi + yi
θ Yi
Assuming that U s are independent of xi and yi , the first order conditions yield the following
optimal levels of effort and equilibrium probability of winning p∗i :
x∗i =
(θXi ∆UXi )2 θY i ∆UY i
(θXi ∆UXi + θY i ∆UY i )2
6 See
Buchanan, Tollison & Tullock (1980)
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.
7 Fu,
5
yi∗ =
θXi ∆UXi (θY i ∆UY i )2
(θXi ∆UXi + θY i ∆UY i )2
Finally,
p∗i =
θXi ∆UXi
θXi ∆UXi + θY i ∆UY i
where ∆UXi = (UXi |W inXi ) − (UXi |LossXi ) is the prize spread of player Xi and ∆UY i =
(UYi |LossXi ) − (UYi |W inXi ) is the prize spread of player Yi .
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 (θXi = θYi ), 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 Xi , 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 (i.e., ∆UXi = ∆UY i ), and (ii) no psychological effects, players choose
their optimal level of effort such that the probability that player Xi wins a confrontation is given
by
θXi
.
p∗i =
θXi + θY 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 (i.e., 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.
Xi
(i) Common prize spreads We observe neutrality (i.e., p∗i = θXiθ+θ
) if and only if players
Yi
have common prize spreads (i.e., ∆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 3,8 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∗3 V if he loses (as he can still get the collective reward V if his teammate wins battle 3,
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 3 is non-trivial if and only if each team has won one battle in the two previous rounds.
6
which occurs with a probability p∗3 ), so his prize spread is V + v − p∗3 V = v + (1 − p∗3 )V . The
player in the lagging team gets v +(1−p∗3 )V if he wins, as he gets the battle reward for sure
and the collective reward if his teammate wins battle 3, which occurs with a probability
(1 − p∗3 ). If he loses, the contest ends and he gets a payoff 0, so his prize spread is also
v + (1 − p∗3 )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 3. 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.
2.2
2.2.1
Empirical setting and data
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 4.1.
7
(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: 1-2-3,11 1-3-2, 2-1-3 and 3-1-2. This
ex-ante randomly-drawn order of play ensures that teams cannot manipulate in any way
the sequence of games to be played.
2.2.2
Data
Our data12 include 2.039 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.13 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.
2.3
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 (i.e., 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.
2.3.1
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 3rd play the third game.
12 The data comes from the website http://www.squashinfo.com.
13 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.
14 2015 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.
11 1-2-3
8
(i) ratio < 1.5 – variant 1, (ii) ratio < 1.4 – variant 2, and finally (iii) ratio < 1.3 – variant 3.
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 1,13). We note X1 the player who won the first battle against Y1 , and X2
the player who belongs to the leading team involved in battle 2 against Y2 . Table 1 displays the
empirical probability that X2 wins the second battle for each of the variants considered. It also
specifies the frequency of the cases where the ranking of X2 is smaller than the ranking of Y2 ,
i.e., situations where X2 is slightly better than his opponent, as a high observed probability that
X2 wins could be simply caused by the fact that X2 is better skilled than his direct opponent.
Table 1: Satistical evidence against neutrality
Variant 1
X2 wins battle 2
59.7%
Ranking of X2 < Ranking of Y2
53.5%
Number of observations
Statistically different from 50% at
∗∗
211
∗
p < 0.05,
∗∗
Variant 2
∗
Variant 3
59.1%
60.4%∗
53%
50.4%
181
139
p < 0.01
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 X2 is better ranked than Y2 is not significantly different from 50% in any of the three
variants.
2.3.2
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 / 16-30 / 31-50 / 51-75 /
76-105 / 106-400, 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 A1 ,
A2 , A3 , B1 , B2 and B3 where the subscript indicates the battle in which the player is engaged.
We can test for neutrality by assessing whether the probability that A2 wins against B2 is
higher when A1 won against B1 in the previous battle, controlling for A2 ’s and B2 ’s modality of
ranking. Thus we regress the dummy variable indicating whether A2 wins or loses battle 2 on a
dummy variable indicating whether A1 won or lost battle 1, on dummies indicating the ranking
modalities of A2 and B2 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 3 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
reference being neutral-field).18
A2 wins battle 2 = β0 + βN on−neutrality × A1 won battle 1
+
6
X
r=1
βr × Rankingr,A2 −
6
X
βr × Rankingr,B2 + βhome × HomeA + βaway × AwayA + AB2
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 0.1% level
and the magnitude of the effect is very strong: 0.14. This means that in a battle involving two
players with similar rankings, the probability that the player in the leading team wins is 0.57
while the probability that the player in the lagging team wins is only 0.43.
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 0.1% level once teams’ rankings are introduced and the magnitude of the effect does not change much (0.11). 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 12 so as to adjust standard errors correctly.
19 We obtain very similar results with probit and logit estimations.
10
Table 2: Evidence against neutrality (rankings’ categories)
Dep. var: A2 wins battle 2
(1)
(2)
A1 won battle 1
0.139∗∗∗ (0.029)
0.114∗∗∗ (0.031)
A2 ’s ranking: Top 5
0.723∗∗∗ (0.057)
0.651∗∗∗ (0.070)
A2 ’s ranking: 6-15
0.577∗∗∗ (0.046)
0.521∗∗∗ (0.057)
A2 ’s ranking: 16-30
0.467∗∗∗ (0.043)
0.420∗∗∗ (0.051)
A2 ’s ranking: 31-50
0.293∗∗∗ (0.043)
0.261∗∗∗ (0.047)
A2 ’s ranking: 51-75
0.185∗∗∗ (0.047)
0.156∗∗ (0.049)
A2 ’s ranking: 76-105
0.093∗ (0.047)
0.074 (0.049)
∗∗∗
B2 ’s ranking: Top 5
-0.723
(0.057)
-0.651∗∗∗ (0.070)
B2 ’s ranking: 6-15
-0.577∗∗∗ (0.046)
-0.521∗∗∗ (0.057)
B2 ’s ranking: 16-30
-0.467∗∗∗ (0.043)
-0.420∗∗∗ (0.051)
B2 ’s ranking: 31-50
-0.293∗∗∗ (0.043)
-0.261∗∗∗ (0.047)
B2 ’s ranking: 51-75
-0.185∗∗∗ (0.047)
-0.156∗∗ (0.049)
B2 ’s ranking: 76-105
-0.093∗ (0.047)
-0.074 (0.049)
A2 at home
0.022 (0.051)
0.009 (0.052)
A2 away
-0.022 (0.051)
-0.009 (0.052)
A2 ’s team seeding
-0.007 (0.004)
B2 ’s team seeding
0.007 (0.004)
Constant
Observations
R2
0.430∗∗∗ (0.038)
0.443∗∗∗ (0.058)
931
0.42
893
0.42
Standard errors in parentheses
∗
p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
Reading note (column 1): the probability that A2 wins increases by 0.139 when his teammate A1 won battle 1.
Reading note (column 1): the probability of winning of a player ranked in the Top 5 is 0.723 higer than
the probability of winning of a player ranked below 105 (reference category).
2.3.3
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 3, 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 (0.09 vs 0.11). This additional result confirms that the outcome of battle 1
11
affects the probability of winning battle 2. This result contradicts neutrality.
Table 3: Evidence against neutrality (ratio of rankings)
Dep. var: A2 wins battle 2
0.087∗∗ (0.030)
A1 won battle 1
RankingA2
RankingB2
-0.522∗∗∗ (0.054)
(< 1)
A2 at home
0.085 (0.049)
A2 away
0.048 (0.050)
A2 ’s team seeding
-0.001 (0.004)
B2 ’s team seeding
0.004 (0.003)
0.916∗∗∗ (0.042)
Constant
Observations
R2
893
0.17
Standard errors in parentheses
∗
3
p < 0.05,
∗∗
p < 0.01,
∗∗∗
p < 0.001
The role of individual contribution to team success
In section 2.3, 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 2.1 to describe the potential drivers of non-neutrality: psychological
effects and individual responsibility.
3.1
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 X2 and player X3 win. In that case,20
p∗3P M =
20 See
θ X3
,
θ X3 + θ Y 3 ψ
Appendix for detailed computations.
12
where team X is defined as the team that won battle 1 and lost battle 2.
p∗2P M =
θX2 ψ
,
θX2 ψ + θY2
where team X is defined as the team that won battle 1.
θ
2
As p∗2P M > θX X+θ
and p∗3P M <
Y2
2
our theoretical setting.
θX3
θX3 +θY3
, the following empirical test can be derived from
Test 3. 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 3, 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 3
(i.e., 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 A1 lost battle 1 and A2 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 A3
wins battle 3. The model we test is therefore given by:
A3 wins battle 3 = β0 + βP M × A1 lost battle 1 and A2 won battle 2
+
6
X
r=1
βr × Rankingr,A3 −
6
X
βr × Rankingr,B3 + βcontrols × Controls + AB3
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 A1 lost battle 1 and A2 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).
13
Table 4: Evidence against psychological momentum
Dep. var: A3 wins battle 3
A1 lost battle 1 and A2 won battle 2
RankingA3
RankingB3
(1)
(2)
-0.026 (0.067)
-0.002 (0.063)
-0.364∗∗ (0.134)
(< 1)
A3 ’s ranking: Top 5
0.765∗∗ (0.261)
A3 ’s ranking: 6-15
0.548∗∗∗ (0.153)
A3 ’s ranking: 16-30
0.376∗∗ (0.141)
A3 ’s ranking: 31-50
0.261∗ (0.127)
A3 ’s ranking: 51-75
0.185 (0.114)
A3 ’s ranking: 76-105
0.006 (0.116)
B3 ’s ranking: Top 5
-0.765∗∗ (0.261)
B3 ’s ranking: 6-15
-0.548∗∗∗ (0.153)
B3 ’s ranking: 16-30
-0.376∗∗ (0.141)
B3 ’s ranking: 31-50
-0.261∗ (0.127)
B3 ’s ranking: 51-75
-0.185 (0.114)
B3 ’s ranking: 76-105
-0.006 (0.116)
A3 at home
-0.018 (0.127)
-0.126 (0.114)
A3 away
0.018 (0.127)
-0.071 (0.123)
A3 ’s team seeding
-0.020 (0.011)
-0.021∗ (0.011)
B3 ’s team seeding
0.020 (0.011)
0.015 (0.010)
Constant
0.513
Observations
R2
∗∗∗
(0.142)
191
0.26
0.971∗∗∗ (0.106)
191
0.10
Standard errors in parentheses
∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
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) X1 defeated
Y1 in the first confrontation, and (ii) X3 has a extremely low equilibrium probability of winning
the third match, if any, (i.e., p∗3 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 X3 has a extremely high equilibrium probability
of winning the third match (i.e., p∗3 goes to 1), the second battle is a confrontation with nothing
14
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 X1 won battle 1, the choking-under-pressure effect yields the
following prediction:22
p∗2CU P =
θX2 f (p∗3 )
,
θX2 f (p∗3 ) + θY2 g(p∗3 )
∂f
where f (.) and g(.) are any function such that ∂p
∗ > 0,
3
(with 0 < η < 1) and limx→1 f (x) = limx→1 g(x) = 1.
We therefore have
∂g
∂p∗
3
> 0, limx→0 f (x) = limx→0 g(x) = η
lim (p∗2CU P |p∗3 = x) = lim (p∗2CU P |p∗3 = x) =
x→0
x→1
θX2
= p∗2
θ X 2 + 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, i.e., 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 3 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 3 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 3 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
A1 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 3 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
Table 5: Evidence against choking under pressure
Dep. var: A2 wins battle 2
A1 won battle 1
RankingA2
RankingB2
(1)
(2)
0.215∗∗∗ (0.061)
0.137∗ (0.060)
-0.417∗∗∗ (0.098)
(< 1)
A2 ’s ranking: Top 5
0.503∗∗∗ (0.112)
A2 ’s ranking: 6-15
0.500∗∗∗ (0.096)
A2 ’s ranking: 16-30
0.297∗∗ (0.092)
A2 ’s ranking: 31-50
0.183∗ (0.087)
A2 ’s ranking: 51-75
0.163 (0.095)
A2 ’s ranking: 76-105
-0.003 (0.097)
B2 ’s ranking: Top 5
-0.503∗∗∗ (0.112)
B2 ’s ranking: 6-15
-0.500∗∗∗ (0.096)
B2 ’s ranking: 16-30
-0.297∗∗ (0.092)
B2 ’s ranking: 31-50
-0.183∗ (0.087)
B2 ’s ranking: 51-75
-0.163 (0.095)
B2 ’s ranking: 76-105
0.003 (0.097)
A2 at home
0.000 (0.084)
0.044 (0.073)
A2 away
-0.000 (0.084)
0.034 (0.080)
A2 ’s team seeding
-0.005 (0.006)
0.002 (0.007)
B2 ’s team seeding
0.005 (0.006)
0.002 (0.005)
Constant
0.392
Observations
R2
∗∗∗
(0.113)
224
0.63
0.867∗∗∗ (0.076)
224
0.17
Standard errors in parentheses
∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
3.2
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” (i.e., the player dreads being
partly responsible for the collective failure) or an “individual contribution effect” (i.e., 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
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:23
p∗2GA =
θX2 (v + (1 − p3 )V + (1 − p3 )s)
θX2 (v + (1 − p3 )V + (1 − p3 )s) + θY2 (v + (1 − p3 )V + s)
where team X is defined as the team that won battle 1. p∗2GA <
following empirical test.
θ X2
θX2 +θY2
, which yields the
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 3). 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 3. For example, if X1 won
battle 1 and X3 has a extremely low equilibrium probability of winning the third match (i.e.,
p∗3 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 X1 won battle 1 and X3 has a extremely high
equilibrium probability of winning the third match (i.e., p∗3 goes to 1), the asymmetry between
the two players reaches its maximum: X2 will get the contribution reward for sure if he wins
while Y2 has no chance to get it if he wins. Formally, we obtain the following predictions, which
23 See
Appendix for detailed computations.
17
confirm the role played by p∗3 in the individual contribution scenario:24
p∗2IC =
θX2 (v + (1 − p3 )V + c)
θX2 (v + (1 − p3 )V + c) + θY2 (v + (1 − p3 )V + (1 − p3 )c)
where team X won battle 1.
∂p∗
2IC
∂p3
∂(1−p∗
)
> 0 and ∂(1−p2IC
> 0, so the probability of winning battle 2 increases with the
3)
teammate’s probability of winning battle 3. 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 3 is
favorite
We can test for the second condition of test 6 by assessing whether the probability that A2
wins against B2 increases when A3 is favorite in battle 3 (i.e., when A3 has a better ranking than
B3 25 ). In specification (1) of table 8, we regress A2 ’s victory on a dummy variable indicating
whether A3 has a better ranking than B3 , on dummies indicating the ranking modalities of A2
and B2 and on the control variables used previously (playing home/away and teams’ seedings).
A2 wins battle 2 = β0 + βIC × A3 favorite in battle 3
+
6
X
r=1
βr × Rankingr,A2 −
6
X
βr × Rankingr,B2 + βcontrols × Controls + AB2
r=1
As predicted in the individual contribution scenario, the variable A3 favorite in battle 3 is
positive and significant at the 1% level. Being in the team that is favorite in battle 3 increases
the probability of winning battle 2 by about 0.13. 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
3. In specification (3), we use the ratio of rankings of A2 and B2 instead of the modalities of
rankings. The variable A3 favorite in battle 3 remains significant at the 5% level and quite strong
in magnitude (about 0.08).
One potential concern with specifications (1) and (3) is the confounding peer-effects story:
being favorite in battle 3 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 3. In specifications (2) and (4), we include a dummy variable indicating whether A1 is
favorite in battle 1 as a control to test for peer effects. The variable A1 favorite in battle 1 is
not significant and its inclusion does not affect the coefficient associated to A3 favorite in battle
24 See
Appendix for detailed computations.
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 3 (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
893 to 759 in table 8.
25 Some
18
3. This result rules out the confounding peer effects story and shows that the effect at stake is
individual contribution.
Table 8: Evidence for individual contribution
Dep. var: A2 wins battle 2
A3 favorite in battle 3
(1)
0.131
∗∗∗
(2)
(0.035)
A1 favorite in battle 1
RankingA2
RankingB2
0.133
∗∗∗
(0.036)
(3)
∗
0.082 (0.034)
-0.012 (0.037)
(< 1)
A2 ’s ranking: Top 5
0.641∗∗∗ (0.076)
0.644∗∗∗ (0.076)
A2 ’s ranking: 6-15
0.524∗∗∗ (0.062)
0.525∗∗∗ (0.063)
A2 ’s ranking: 16-30
0.411∗∗∗ (0.057)
0.412∗∗∗ (0.057)
A2 ’s ranking: 31-50
0.253∗∗∗ (0.052)
0.254∗∗∗ (0.053)
A2 ’s ranking: 51-75
0.172∗∗ (0.056)
0.172∗∗ (0.056)
A2 ’s ranking: 76-105
0.046 (0.056)
0.046 (0.056)
0.092
(0.034)
-0.066 (0.036)
-0.514∗∗∗ (0.059)
-0.522∗∗∗ (0.059)
B2 ’s ranking: Top 5
-0.641
(0.076)
-0.644∗∗∗ (0.076)
B2 ’s ranking: 6-15
-0.524∗∗∗ (0.062)
-0.525∗∗∗ (0.063)
B2 ’s ranking: 16-30
-0.411∗∗∗ (0.057)
-0.412∗∗∗ (0.057)
B2 ’s ranking: 31-50
-0.253∗∗∗ (0.052)
-0.254∗∗∗ (0.053)
B2 ’s ranking: 51-75
-0.172∗∗ (0.056)
-0.172∗∗ (0.056)
B2 ’s ranking: 76-105
-0.046 (0.056)
-0.046 (0.056)
A2 at home
0.026 (0.055)
0.026 (0.055)
0.068 (0.050)
0.065 (0.050)
A2 away
-0.026 (0.055)
-0.026 (0.055)
0.008 (0.053)
0.001 (0.054)
A2 ’s team seeding
-0.005 (0.004)
-0.005 (0.004)
-0.002 (0.004)
-0.004 (0.004)
B2 ’s team seeding
0.005 (0.004)
0.005 (0.004)
0.002 (0.003)
0.004 (0.003)
0.435∗∗∗ (0.066)
0.439∗∗∗ (0.068)
0.941∗∗∗ (0.046)
0.987∗∗∗ (0.053)
759
0.43
759
0.43
759
0.17
759
0.18
Constant
∗∗∗
(4)
∗∗
Observations
R2
Standard errors in parentheses
∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
Thus, our results suggest that the absence of neutrality is driven by the fact that players
individually value the fact of being responsible for their team’s success and we interpret the
dynamic linkage between subsequent battles as an individual contribution effect.
3.3
Summary
Our analysis of the link between subsequent pairwise battles relies on potential effects that
directly affects players’ utility, based on the assumption – which seems rather plausible, but is
further discussed in the next section, that players do no internalize their teammates’ efforts. We
19
first ruled out negative effects. The fact of winning the first battle increases the probability
of winning the second one. Furthermore, we ruled out environment-driven effects: the fact
of evolving in a specific environment (playing in a specially good team or playing home) does
not explain the observed link. Finally, we showed evidence that psychological positive effects
reviewed in the literature (i.e., psychological momentum and choking under pressure) were not
at play. The last kind of effects that we tested so as to understand the observed link between two
subsequent games is guilt aversion and individual contribution. In the former case, teammates
integrate in their utility function the fact of not being responsible for their team’s defeat. We
ruled out guilt aversion as it would imply a negative effect. We find evidence that players value
the fact of being at least partly responsible for the success of their team.
4
Discussion
This section further discusses our main result according to which battle dependence is driven by
an individual contribution effect.
4.1
Altruism
So far, our result is based on the assumption that players do not take into account their teammates’ costs of effort. However, one would argue that the observed link between the first two
battles of the contest is driven by the fact that individuals internalize their teammates’ costs of
effort. This effect could be referred to as “altruism”. Each player within a team would maximize
his utility function taking into account not only his own effort cost but also his teammates’.
For instance, the player in the leading team involved in battle 2 could make an additional effort in order to win, thereby preventing his teammate from playing battle 3 and incurring the
corresponding effort cost.
We develop a test to address the fact that individuals may internalize their teammates’
costs. It allows us to distinguish between individual contribution and altruism. Intuitively, our
identification strategy is based on the fact that, in a best-of-three contest, players involved in
the first battle cannot prevent their teammates from playing the second match, and can only
internalize the cost of effort of the players involved in the third battle. Accordingly, we focus on
battle 1 and keep in our sample only the contests where the favorites in battles 2 and 326 do not
belong to the same team. Hence, there are only two possible scenarios regarding future battles:
either i) A2 favorite, A3 underdog or ii) A2 underdog, A3 favorite. According to the individual
contribution effect, these two scenarios are equivalent, as player A1 is equally likely to get the
contribution reward in the two settings. On the contrary, if players were altruistic, the scenario
A2 favorite, A3 underdog would be much more motivating for player A1 . Indeed, when A2 is
favorite, A1 knows that winning battle 1 implies that his teammate A3 will probably not have to
play battle 3 and to make any costly effort. On the other hand, when A2 is underdog, A1 knows
that winning battle 1 implies that his teammate A3 is very likely to play battle 3 and to exert
a costly effort. Thus, altruism implies that A1 has more incentives to win when A2 is favorite
and A3 is underdog than in the symmetric situation.27 Accordingly, we regress A1 ’s victory on
a dummy variable indicating the situation regarding battles 2 and 3 (which equals 1 when A2
is favorite and A3 is underdog, and 0 when A2 is underdog and A3 is favorite), on the ranking
26 Like
27 See
in the previous section, a player is defined as the favorite when he has a better ranking than his opponent.
Appendix for more formal details on this test.
20
modalities of A1 and B1 as well as control variables.
A1 wins battle 1 = β0 + βaltruism × A2 favorite, A3 underdog
+
6
X
βr × Rankingr,A1 −
r=1
6
X
βr × Rankingr,B1 + βcontrols × Controls + AB1
r=1
Individual contribution predicts that the variable A2 favorite, A3 underdog has no significant
effect on the probability that A1 wins battle 1, whereas altruism predicts that this effect should be
significant and positive. Results are displayed in column (1) of table 9. The coefficient associated
to the variable of interest is not statistically significant, which rules out altruism and put forward
individual contribution. Taking the ratio of rankings instead of the rankings’ categories leads to
the same conclusion (see specification (2)).
Table 9: Evidence against altruism
Dep. var: A1 wins battle 1
(1)
(2)
A2 favorite, A3 underdog
-0.086 (0.065)
-0.106 (0.061)
RankingA1
RankingB1
-0.522∗∗∗ (0.129)
(< 1)
A1 ’s ranking: Top 5
0.837∗∗∗ (0.188)
A1 ’s ranking: 6-15
0.695∗∗∗ (0.153)
A1 ’s ranking: 16-30
0.462∗∗∗ (0.134)
A1 ’s ranking: 31-50
0.361∗ (0.141)
A1 ’s ranking: 51-75
0.301∗ (0.124)
A1 ’s ranking: 76-105
0.167 (0.128)
B1 ’s ranking: Top 5
-0.837∗∗∗ (0.188)
B1 ’s ranking: 6-15
-0.695∗∗∗ (0.153)
B1 ’s ranking: 16-30
-0.462∗∗∗ (0.134)
B1 ’s ranking: 31-50
-0.361∗ (0.141)
B1 ’s ranking: 51-75
-0.301∗ (0.124)
B1 ’s ranking: 76-105
-0.167 (0.128)
A1 at home
0.080 (0.130)
0.137 (0.125)
A1 away
-0.080 (0.130)
0.002 (0.121)
A1 ’s team seeding
-0.007 (0.010)
-0.001 (0.010)
B1 ’s team seeding
0.007 (0.010)
0.002 (0.009)
Constant
∗∗∗
0.543
Observations
R2
(0.159)
208
0.25
Standard errors in parentheses
∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
21
1.027∗∗∗ (0.102)
208
0.11
This confirms that the observed link between two subsequent battles is driven by individual
responsibility. Individuals do not take into account the cost of effort of their teammates, but value
the fact of being individually responsible for their team success. To the best of our knowledge, our
paper is the first empirical study showing that individuals want to directly participate in collective
success. This result is of first importance to understand the gap between theoretical predictions
according to which individuals free ride when they are offered collective-based incentives and
empirical or experimental results that show that free-riding is not at play. This important point
is further discussed below.
4.2
Individual contribution and free riding
Economic models on teams predict that individuals have an incentive to free-ride, as they do not
internalize the benefits that accrue to other members of the team when making effort decisions.28
Hence, the optimal level of effort achieved in individual contests should be higher than the
effort observed in team-based incentive contests when the individual reward is based on team
production.
However, such a theoretical result seems to contradict both experimental and behavioral
literature on teams. As a matter of fact, organizing work in teams has progressively become
the linchpin of most organizations. Many firms, like partnerships, use collective-based incentives
and profit-sharing plans. The detrimental free-riding incentive is therefore outweighed by some
positive incentive for team members. In a very influential paper, Kandel & Lazear (1992) analyze
the role of peer pressure to explain the success of collective-based incentives. Peer pressure is
even more efficient when firms or teams create norms or develop mutual monitoring to establish
an expected level of effort. Kandel & Lazear (1992) also show that shame (which requires
observability) and guilt (which does not) can explain why collective-based incentive contracts
are efficient and may yield a higher equilibrium level of effort than individually-based incentives.
Chen & Lim (2013) show, using an experiment, that guilt is at play in teams and can explain
why team-based incentives are more efficient that individually-based contracts. Interestingly,
their experiment does not allow them to distinguish between guilt and individual contribution.
Individual contribution is an additional mechanism that mitigates free-riding behaviors. Each
teammate values the fact of being responsible for the collective success and therefore makes a
significant effort, which could be higher than the effort that he would make in an individual
contest.
5
Conclusion
Using team squash championships as a randomized natural experiment, we find strong empirical
evidence against neutrality in multiple pairwise battles: in a best of three team contest, winning
the first battle increases the probability of winning the second battle. We show that psychological
momentum, choking under pressure and guilt aversion are not the drivers of non-neutrality.
We provide evidence of an individual contribution effect: people derive utility from contributing to their team’s success. This effect is of first importance to understand team-based contests
or contracts. This is why it needs to be further analyzed. One important pending question is
whether individual contribution depends on the observability of each teammate’s performance.
This has important implications concerning management practices and contest design. If individual contribution is based on the observability of individual performance, team-based contracts
28 See
Prendergast (1999) for a survey on team production.
22
and team contests should be designed in such a way that individual outcomes are (at least partly)
observable and the link between individual outcome and team success is measurable.
23
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24
Appendix
Neutrality
T=3 Both team won one individual battle. Players X3 and Y3 are now opposed in a decisive
3
game. If X3 wins, which occurs with a probability x3x+y
, his team wins and he gets a payoff
3
3
V + v. If he loses, he gets a payoff 0. In any case he has to pay the cost of effort θxX3
. X3 ’s
maximization problem is therefore given by
x3
x3
(V + v) −
.
max
x3
x3 + y3
θX3
Symetrically, for player Y3 :
max
y3
y3
y3
(V + v) −
x3 + y3
θY 3
.
First-order conditions give the optimal levels of effort and p∗3 :
x∗3 = (V + v)
2
θ Y3 θ X
3
,
(θX3 + θY3 )2
y3∗ = (V + v)
θX3 θY2 3
,
(θX3 + θY3 )2
p∗3 =
θX3
.
θ X3 + θ Y 3
This yields the following prediction: the outcome of battle 3 only depends on the relative
ability of the players involved in battle 3.
T=2
Team X won the first battle (X1 won against Y1 ).
X2 chooses his level of effort x2 to maximize his utility. If he wins, which occurs with a
2
2
probability x2x+y
, he gets a payoff V + v. If he loses, which occurs with a probability x2y+y
,
2
2
he can still get V if his teammate X3 wins the third battle (which occurs with a probability
p3 ). Finally, whatever the outcome of the battle, he has to pay the cost of his effort θxX2 . X2 ’s
2
maximization problem is therefore given by:
x2
y2
x2
∗
max
(V + v) +
p V −
.
x2
x2 + y2
x2 + y2 3
θ X2
Y2 chooses his level of effort y2 to maximize his utility. If he wins, which occurs with a
2
, he will get a payoff v + (1 − p∗3 )V . If he loses, the match ends and he gets a
probability x2y+y
2
payoff 0. He has to pay the cost of effort θyY2 , whatever the outcome of the battle. Note that in
2
the neutrality model, the two players have the same prize spread (difference of utility between
winning and losing): v + (1 − p∗3 )V . Y2 ’s maximization problem is
y2
y2
∗
(v + (1 − p3 )V ) −
.
max
y2
x2 + y2
θ Y2
First-order conditions yield the optimal levels of effort and p∗2 :
25
x∗2 = (v + (1 − p∗3 )V )
2
θ Y2 θ X
2
;
(θX2 + θY2 )2
y2∗ = (v + (1 − p∗3 )V )
θX2 θY2 2
;
(θX2 + θY2 )2
p∗2 =
θX2
.
θ X2 + θ Y 2
This yields the two following predictions.
Prediction 1 Winning the first battle does not affect the probability of winning the second
battle.
Prediction 2 The outcome of battle 2 only depends on the two players involved in battle 2.
Psychological momentum
T=3 Team X won the first battle and lost the second battle (X1 won against Y1 and X2
lost against Y2 ). Y3 has a psychological momentum because his teammate won the previous
battle. It is conceptually equivalent to multiplying his ability by a factor ψ (with ψ > 1). The
maximization problem is
x3
x3
max
(V + v) −
;
x3
x3 + y3
θX3
y3
y3
.
max
(V + v) −
y3
x3 + y3
θY 3 ψ
Optimal levels of effort and p∗3P M are therefore given by:
x∗3P M = (V + v)
2
θX
(θY3 ψ)
3
;
(θX3 + θY3 ψ)2
∗
y3P
M = (V + v)
θX3 (θY3 ψ)2
;
(θX3 + θY3 ψ)2
p∗3P M =
θ X3
.
θ X3 + θ Y 3 ψ
This yields the following prediction: the player in the team that won the second battle is
more likely to win the third battle than the player in the team that won the first battle.
T=2 Team X won the first battle (X1 won against Y1 ). X2 has a psychological momentum
because his teammate won the previous battle. It is conceptually equivalent to multiplying his
ability by a factor ψ (with ψ > 1). We therefore have the following maximization problems.
y2
x2
x2
∗
max
(V + v) +
p
V −
;
x2
x2 + y2
x2 + y2 3P M
θ X2 ψ
y2
y2
∗
max
(v + (1 − p3P M )V ) −
.
y2
x2 + y2
θ Y2
26
Deriving the first order conditions yield the optimal levels of effort and p∗2P M :
x∗2P M = (v + (1 − p∗3P M )V )
(θX2 ψ)2 θY2
;
(θX2 ψ + θY2 )2
∗
∗
y2P
M = (v + (1 − p3P M )V )
(θX2 ψ)θY2 2
;
(θX2 ψ + θY2 )2
p∗2P M =
θX2 ψ
.
θX2 ψ + θY2
Accordingly, the model predicts that winning the first battle increases the probability of
winning the second battle.
Choking under pressure
T=3 Both team won one individual battle. Players X3 and Y3 are now opposed in a decisive
game. Since battle 3 is pivotal, players X3 and Y3 might both choke under pressure, which is
conceptually equivalent to multiplying their ability by η with 0 < η < 1. Therefore, we have
x3
x3
max
;
(V + v) −
x3
x3 + y3
θX3 η
y3
y3
max
.
(V + v) −
y3
x3 + y3
θY 3 η
This gives
x∗3CU P = (V + v)η
2
θ Y3 θ X
3
;
(θX3 + θY3 )2
∗
y3CU
P = (V + v)η
θX3 θY2 3
;
(θX3 + θY3 )2
p∗3CU P = p∗3 =
θ X3
.
θX3 + θY3
T=2 Team X won the first battle (X1 won against Y1 ). The degree of pressure faced by the
two players decreases with p∗3 . If p∗3 = 0, battle 2 becomes decisive for both players and their
level of pressure reaches its maximum (the situation is the same as the one presented for T=3).
On the contrary, if p∗3 = 1, the two players know that team X will win the contest anyway, so
battle 2 becomes stakeless and neither of them faces pressure. Apart from these two extreme
cases, the impact of a change in p∗3 on players’ pressure is not necessarily the same. Therefore,
we can model the choking under pressure scenario as follows:
x2
y2
x2
max
(V + v) +
p∗3 V −
;
x2
x2 + y2
x2 + y2
θX2 f (p∗3 )
y2
y2
max
(v + (1 − p∗3 )V ) −
;
y2
x2 + y2
θY2 g(p∗3 )
27
∂f
with ∂p
∗ > 0,
3
limx→1 g(x) = 1.
∂g
∂p∗
3
> 0, limx→0 f (x) = limx→0 g(x) = η (with 0 < η < 1) and limx→1 f (x) =
First-order conditions give the optimal levels of effort and p∗2choking :
x∗2CU P = (v + (1 − p∗3 )V )
(θX2 f (p∗3 ))2 θY2 g(p∗3 )
(θX2 f (p∗3 ) + θY2 g(p∗3 ))2
∗
∗
y2CU
P = (v + (1 − p3 )V )
θX2 f (p∗3 )(θY2 g(p∗3 ))2
(θX2 f (p∗3 ) + θY2 g(p∗3 ))2
p∗2CU P =
θX2 f (p∗3 )
θX2 f (p∗3 ) + θY2 g(p∗3 )
This yields the following prediction: limx→0 (p∗2CU P |p∗3 = x) = limx→1 (p∗2CU P |p∗3 = x) =
, so winning the first battle does not affect the probability of winning the second battle
when the outcome of battle 3 is almost certain.
θ X2
θX2 +θY2
Guilt aversion
When a player loses and his team loses, he gets a negative payoff payoff −s.
Player’s team wins
Player’s team loses
Player wins
V +v
v
Player loses
V
−s
T=3 Both teams won one individual battle. Players X3 and Y3 are now opposed in a decisive
3
game. If X3 wins, which occurs with a probability x3x+y
, his team wins and he gets a payoff
3
V + v. If he loses, he gets a payoff −s because he is “scapegoat-averse”: being partly responsible
3
for the failure of his team is costly for him. In any case, he has to pay the cost of effort θxX3
and
faces the following maximization problem.
x3
y3
x3
max
(V + v) +
(−s) −
.
x3
x3 + y3
x3 + y3
θX3
Symetrically for player Y3 :
max
y3
y3
x3
y3
(V + v) +
(−s) −
x3 + y3
x3 + y3
θY 3
The optimal levels of effort and p∗3scapegoat are given by:
x∗3GA = (V + v + s)
2
θX
θ
3 Y3
;
(θX3 + θY3 )2
∗
y3GA
= (V + v + s)
θX3 θY2 3
;
(θX3 + θY3 )2
28
.
θ X3
.
θX3 + θY3
The model thus predicts that the outcome of battle 3 only depends on the relative ability of
the players involved in battle 3.
p∗3GA = p∗3 =
T=2 Team X won the first battle (X1 won against Y1 ).
X2 chooses his level of effort x2 to maximize his utility. If he wins, which occurs with a probability
y2
x2
x2 +y2 , he gets a payoff V + v. If he loses, which occurs with a probability x2 +y2 , he gets a payoff
V with a probability p∗3 and −s with a probability 1 − p∗3 . Finally, whatever the outcome of the
battle, he has to pay the cost of his effort θxX2 . His maximization problem is:
2
x2
y2
x2
max
(V + v) +
(p∗3 V + (1 − p∗3 )(−s)) −
x2
x2 + y2
x2 + y2
θX2
Y2 chooses his level of effort y2 to maximize his utility. If he wins, which occurs with a
2
, he will get a payoff v + (1 − p∗3 )V . If Y2 loses, the match ends and he gets a
probability x2y+y
2
payoff −s. He has to pay the cost of effort θyY2 , whatever the outcome of the battle.
2
y2
x2
y2
∗
max
(v + (1 − p3 )V ) +
(−s) −
y2
x2 + y2
x2 + y2
θ Y2
Thus, the two players face asymmetric prize spreads with ∆UX2 < ∆UY 2 and Y2 has more
incentives to win than X2 because Y2 is sure to be partly “defeat-responsible” if he loses his
battle while X2 will be “defeat-responsible” if and only if his teammate also loses in T=3.
Deriving the first-order conditions yield the optimal levels of effort and p∗2scapegoat :
x∗2GA =
2
θX
(v + (1 − p∗3 )V + (1 − p∗3 )s)2 θY2 (v + (1 − p∗3 )V + s)
2
[θX2 (v + (1 − p∗3 )V + (1 − p∗3 )s) + θY2 (v + (1 − p∗3 )V + s)]2
∗
y2GA
=
θX2 (v + (1 − p∗3 )V + (1 − p∗3 )s)θY2 2 (v + (1 − p∗3 )V + s)2
[θX2 (v + (1 − p∗3 )V + (1 − p∗3 )s) + θY2 (v + (1 − p∗3 )V + s)]2
θX2 (v + (1 − p∗3 )V + (1 − p∗3 )s)
θX2 (v + (1 − p∗3 )V + (1 − p∗3 )s) + θY2 (v + (1 − p∗3 )V + s)
This yields the two following predictions.
p∗2GA =
Prediction 1 Winning the first battle decreases the probability of winning the second battle.
Prediction 2
in battle
∂p2SG
<
∂p∗
3
∂p2SG
3 ( ∂θX
3
0 so the outcome of battle 2 depends on the ability of players involved
< 0 and
∂p2SG
∂θY3
> 0).
Individual contribution
When a player wins and his team wins, he gets an additional payoff of individual contribution c.
Player’s team wins
Player’s team loses
Player wins
V +v+c
v
Player loses
V
0
29
T=3 Both teams won one individual battle. Players X3 and Y3 are now opposed in a decisive
3
, his team wins and he gets a payoff
game. If X3 wins, which occurs with a probability x3x+y
3
3
V + v + c. If he loses, he gets a payoff 0. In any case he has to pay the cost of effort θxX3
.
x3
x3
max
(V + v + c) −
.
x3
x3 + y3
θX3
Symmetrically for player Y3 :
max
y3
y3
y3
(V + v + c) −
x3 + y3
θY 3
.
The first-order conditions give the optimal levels of effort and p∗3contribution :
x∗3IC = (V + v + c)
2
θ Y3 θ X
3
;
(θX3 + θY3 )2
∗
y3IC
= (V + v + c)
θX3 θY2 3
;
(θX3 + θY3 )2
p∗3IC = p∗3 =
θ X3
.
θX3 + θY3
This gives the following prediction: the outcome of battle 3 only depends on the relative
ability of the players involved in battle 3.
T=2 Team X won the first battle (X1 won against Y1 ). Contrary to T=3, the two players
do not face the same optimization problem. X2 chooses his level of effort x2 to maximize his
utility. If he wins, he gets a payoff of V + v + c. If he loses, he gets a payoff of p∗3 V (he will
get neither the private reward nor the “contribution reward” but he will get the collective win
reward if his teammate wins in T=3, which will occur with a probability p∗3 ), so his prize spread
is ∆UX2 = v + c + (1 − p∗3 )V . Therefore, x2 ’s maximization problem is given by
x2
y2
x2
∗
max
.
(V + v + c) +
p V −
x2
x2 + y2
x2 + y2 3
θ X2
Y2 chooses his level of effort y2 to maximize his utility. If he wins he gets a payoff of v + (1 −
p∗3 )V + (1 − p∗3 )c because he will get the collective win reward and the individual contribution
reward if his teammate wins, which will occur with a probability (1 − p∗3 ). If he loses he does
not get any reward and end up with a payoff 0, so his prize spread is v + (1 − p∗3 )V + (1 − p∗3 )c.
y2
y2
(v + (1 − p∗3 )V + (1 − p∗3 )c) −
max
y2
x2 + y2
θ Y2
Thus, both players face asymmetric prize spreads with ∆UX2 > ∆UY 2 and X2 has more
incentives to win than Y2 because X2 is sure to get the “contribution reward” if he wins his
battle while Y2 will get the “responsibility reward” if and only if his teammate also wins in T=3.
The first-order conditions yield the optimal levels of effort and p∗2contribution :
x∗2IC =
2
θX
(v + (1 − p∗3 )V + c)2 θY2 (v + (1 − p∗3 )V + (1 − p∗3 )c)
2
[θX2 (v + (1 − p∗3 )V + c) + θY2 (v + (1 − p∗3 )V + (1 − p∗3 )c)]2
30
∗
y2IC
=
θX2 (v + (1 − p∗3 )V + c)θY2 2 (v + (1 − p∗3 )V + (1 − p∗3 )c)2
[θX2 (v + (1 − p∗3 )V + c) + θY2 (v + (1 − p∗3 )V + (1 − p∗3 )c)]2
p∗2IC =
θX2 (v + (1 − p∗3 )V + c)
θX2 (v + (1 − p∗3 )V + c) + θY2 (v + (1 − p∗3 )V + (1 − p∗3 )c)
This yields the two following predictions.
Prediction 1 Winning the first battle increases the probability of winning the second battle.
Prediction 2
in battle
∂p2IC
>
∂p∗
3
2IC
3 ( ∂p
∂θX3
0 so the outcome of battle 2 depends on the ability of players involved
> 0 and
∂p2IC
∂θY3
< 0).
Disentangling between individual contribution and altruism
We want to compare the predictions of individual contribution and altruism in battle 1 in the
case where the favorites for battles 2 and 3 do not belong to the same team. Let X denote
the team whose players are favorite in battle 2 and underdog in battle 3. For simplicity, we
furthermore assume that X2 will win with certainty and X3 will lose with certainty.29
Individual contribution Player X1 gets the collective reward V , the individual reward v,
and the individual contribution reward c if he wins (as his teammate X2 will win battle 2 and
end the contest) and he gets a payoff 0 if he loses. Player Y1 faces the same prize spread as he
also gets V + v + c if he wins and 0 if he loses.
∆UX1 = ∆UY 1 = V + v + c
Altruism According to altruism, incentives of players X1 and Y1 are not symmetric anymore.
If X1 wins, he will get both the collective reward V and the individual reward v and he will
prevent his teammate X3 to make a costly effort in battle 3 (as the contest will end after battle
2 thanks to the victory of X2 ). On the contrary, if X1 loses, he will get neither the collective
reward nor the individual reward and he will force his teammate X3 to play which induces a
negative payoff −αC(X3 ) where C(X3 ) is the cost of effort of X3 and α reflects the degree to
which X1 internalizes it (0 < α < 1). Hence the prize spread of X1 will be V + v + αC(X3 ).
His opponent Y1 faces a different problem. If he wins, he gets both V and v as his team will win
the contest but he forces his teammate Y3 to play his battle, which is partly internalized by him
(−αC(Y3 )). If Y1 loses, he gets neither V nor v but he prevents Y3 from playing. Hence his prize
spread is V + v − αC(Y3 ).
∆UX1 = V + v + αC(X3 ) > ∆UY 1 = V + v − αC(Y3 )
29 Note that the logic would be the same with a more general framework where X is “as favorite as X is
2
3
underdog”.
31
Different predictions Thus, individual contribution predicts that X1 and Y1 have the same
prize spread while altruism predicts that X1 has a higher prize spread than Y1 . Since p∗1 =
θX1 ∆UX1
θX1 ∆UX1 +θY 1 ∆UY 1 , individual contribution predicts that being favorite in battle 2 is equivalent
to being favorite in battle 3 whereas altruism predicts that being favorite in battle 2 is more
preferable than being favorite in battle 3. This finding is the basis for our empirical test in
section 4.1.
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