Gender Differences in Risk-Taking: Evidence from
Professional Basketball ∗
René Böheima,b , Christoph Freudenthalera , and Mario Lacknera
a
Department of Economics, Johannes Kepler University Linz (JKU)
b
Austrian Institute of Economic Research, Vienna
January 29, 2016
PRELIMINARY VERSION for submission to ESPE 2016
Abstract
We analyze risk taking behavior of women and men in high-pressure situations using
a large data-set from professional basketball tournaments. We find that women
exhibit less risky behavior in situations where the importance of success is most
important. Our results indicate that men take more risk in situations that are
associated with high costs of failed risk taking. As the associated cost increase,
the gender gap widens. We do find that the incentive structure of the observed
tournament has a strong influence on this specific gender gap in risk taking.
JEL Classification: D81
Keywords: Risk taking, gender differences, tournament incentives
∗
Corresponding author: Mario Lackner, Department of Economics, Altenberger Str. 69, 4040 Linz;
Tel.:+43/732/2468-7390; Email: [email protected]. Thanks to Martina Zweimüller, Rudof WinterEbmer, Jochen Güntner, and Alexander Ahammer for valuable comments.
1
1
Introduction
Gneezy and Rustichini (2004), among others, have pointed to gender differences in risktaking as a potential cause for differences in other outcomes, for example, wages and
promotions. In the labor market, there are persistent differences between men and women.
For example, empirical studies find large and persistent differences in pay (Weichselbaumer
and Winter-Ebmer, 2005). Additionally, women are underrepresented in top-tier jobs
(OECD, 2012). Systematic differences in risk taking between men and women could
explain these findings as jobs with high salaries, e.g., managerial jobs or jobs in the
financial sector, typically reward risk taking. If women take fewer risky choices than
men, they would earn less than men and would have lower chances to be promoted to
top positions. Risk-taking is typically not observed and standard wage decomposition
techniques might be biased due to this omitted variable problem.
More generally, it is difficult to analyze risk-taking in the field as available options
are typically not restricted to a well-defined set of options and realized choices are often
the result of a combination of effort, information, and of risk preferences. Apart from
laboratory or field experiments where the manipulation of environmental parameters is
possible, there is limited evidence on the risk-taking of men and women in the field.
Professional sport is an area in which rules clearly define the set of options. In
addition, in some situations the available options can be clearly distinguished by the
level of their risk, for example, passing a height in high jumping competitions (Böheim
and Lackner, 2015).1 Moreover, professional sports have strong hierarchical incentive
structures where athletes or teams who come win receive sizeable rewards, which limits
concerns about the pursue of different objectives.
We analyze differences in risk-taking in professional basketball, using data from male
and female teams from the 2002/3 to 2013/14 seasons. We measure risk-taking by the ratio
of two-points versus three-points attempts, following Grund, Höcker and Zimmermann
(2013). By analyzing male and female teams, we contribute to a deeper understanding of
risk-taking by professional men and women. In contrast to other studies, which investigate
single athletes (e.g., Gerdes and Gränsmark (2010) or Böheim and Lackner (2015), we
1
Another area which arguably has similar advantages is the management of funds, e.g., Barber and
Odean (2001) or Atkinson, Baird and Frye (2003).
2
analyze teams of athletes who need to cooperate in order to win. This coordination, and
the cooperation of the team with their coaches, might eliminate any gender differences
that might be observed in more individually focused events. This research, to the best
of our knowledge, is the first to examine gender differences in risk-taking by examining
teams, using field data from a highly competitive professional setting.
As a first step, we extend the analysis of Grund et al. (2013) and examine whether
female basketball player are more likely to use risky strategies when they are trailing,
a pattern which Grund et al. (2013) document for male basketball players. However,
the structure of a match allows to analyze risk-taking in more detail. Since a basketball
match–without overtime–lasts a limited time (48 minutes without overtime), the relative
costs (and benefits) of risky choices change over the course of a match. We investigate if
the change in the willingness to incur risk in highly critical situations towards the end of
matches, holding all other factors constant, is different for male and female teams.
The analyzes of matches from playoffs also allows us to investigate if risk-taking of
female and male teams is affected by the associated prize of winning a game. For example,
Milgrom and Roberts (1988) argue that the incentive systems of tournaments may also
lead to excessive risk-taking. Becker and Huselid (1992) find evidence that drivers engage
in more reckless behavior in car races when the spread of the prizes is greater.
Our research thus contributes to the growing literature that investigates gender differences in risk-taking by examining field data from professional settings. Because of the
cooperation, which is essential for a team’s performance, each basketball team resembles
an organization led by a manager and his support staff. Our results on the reaction of
risk-taking to the (incentive) structure of tournaments and towards the incentive structure
within a match is thus related to work on team performance and compensation schemes.
(Delfgaauw, Dur, Sol and Verbeke, 2013)
Our key findings allow the conclusion that there is a sizeable and significant gender
gap in risk taking in crucial situations that decide the outcome of games. The higher
the importance to perform in these situations, the larger is the gap we are measuring.
Results from a detailed investigation of the underlying tournament structure show that
this gender gap is coming from increased risk taking by male teams in situations where
the prize of winning a game is a win or lead in the series.
3
The remainder of the paper is structured as follows: Section 2 provides a selective
review of previous literature, section 3 will introduce the underlying data. Section 4 will
describe the empirical strategy and present the results, while section 5 will discuss the
findings and conclude.
2
Theory and Related Literature
Risk taking behavior in competitive settings or contests has been the focus of several
theoretical contributions. Cabral (2003) analyzes how two decision makers decide to opt
for risky strategies depending on whether they are lagging behind or leading in a contest.
In this model, which is an infinite R&D contest with costly risk taking, effort is held
constant, while risk is modeled as a mean preserving spread. In equilibrium, the decision
maker who is lagging behind chooses a risky strategy, while the leading decision maker
decides on a safe strategy.
In contrast, in Taylor’s (2003) model of mutual fund tournaments, the fund manager
who leads in a tournament chooses risky strategies, especially when the lead in the contest
is large. Empirical analyses of mutual funds data from 1984 to 1996 support this view.
However, it is likely that most of the presented evidence is coming from male decision
makers. In a two-player tournament model, Nieken and Sliwka (2010) provide a model
where they analyze the consequence of risk-taking on subsequent outcomes when the
outcomes are correlated, or not. While the trailing player should always play the risky
strategy, the leading player should choose the risky strategy under perfectly correlated
outcomes. If the outcomes are not correlated, the leading player will choose the safe
strategy. If the outcomes are weakly correlated, the model predicts mixed strategies.
Empirical research typically finds that men take more risk than women (Croson and
Gneezy, 2009; Eckel and Grossman, 2008). However, most evidence is obtained from laboratory experiments and we have little evidence from the field. Evidence coming from
the field is less comprehensive. In a seminal paper Barber and Odean (2001) present
evidence for gender differences in risk taking using a large data set on household investment decisions. They find that men do trade more frequently (i.e. take more risk) than
women, however at the cost of lower returns. Atkinson, Baird and Frye (2003) analyze the
4
behavior of fund managers: the analyzed female and male fund managers did not differ
in the taken risk for their funds. A possible for this finding explanation is that education
and professional experience rule out gender differences in risk-taking. Results in Dwyer,
Gilkeson and List (2002) support this hypothesis.
Recently, several authors use sports data to study gender differences in risk-taking.
Gerdes and Gränsmark (2010) use data from chess competitions to assess risk-taking
behavior. The authors find that female chess players choose risky opening strategies less
often than men, even after controlling for differences in skills or age. Böheim and Lackner
(2015) analyze results from high jump and pole vault competitions. Similarly, they find
that female athletes choose less often the risky strategy of passing a height than male
athletes.
Focusing on risk taking behavior of individuals, Ozbeklik and Smith (2014) present
evidence from top-level match play golf tournaments. They show that trailing players
take more risk, while leading ones reduce risk taking. Interestingly, they find that ranking
differences between contestants do play a role early in contest, while gradually decreasing
in importance.
Most empirical studies analyze risk taking behavior of teams. Grund and Gürtler
(2005) analyze different types of substitutions in professional soccer in Germany. Their
findings suggest that teams which are leading reduce effort, while trailing teams will take
more risk—however at the cost of further increasing the scoring difference. Genakos and
Pagliero (2012) use data from top-level weight lifting contests and relate past performance
to risk taking and success. They find an inverted-U shaped relationship between intermediate ranking and the willingness to take risks. However, neither paper does focus on any
potential gender differences.
Grund et al. (2013) use data for the 2007/2008 NBA regular season and playoffs and
identify risk taking by the willingness to take three-point shots. Their findings suggest
that NBA players are willing to incur more risk in order to close deficits in a game.
Moreover, they argue that this increased risk taking is mostly ineffective and attribute
this to coaching decisions.
In contrast to the existing empirical literature, we compare risk taking behavior of
female and male teams. By using detailed play-by-play data from professional basketball
5
we try to identify situations where the costs associated with risk taking increase and the
subsequent effect on potential gender differences in risk taking. Potential failure in highly
important situations at the end of close games will significantly decrease the probability
to win the game and therefore increase the cost of risk taking. By making use of the
elimination tournament structure of professional basketball playoff tournaments, we can
also investigate in how far changes in the size of the expected prize (i.e. probability to win
the current stage) influence risk taking behavior for both genders in critical situations.
Additionally, our rich data set allows to address potential selection issues that arise from
comparing female and male teams of different abilities over time.
3
Data and Descriptive Analysis
We use play-by-play data for 12 seasons2 from the two major professional basketball
leagues in the USA: the Women’s National Basketball Association (WNBA) and the
National Basketball Association (NBA). Both leagues operate under similar rules and
represent the top-level of basketball competition in the USA.
For all but one season during this period, there were 12 teams competing in the
WNBA and 30 teams in the NBA. In both leagues, a season consists of three phases —
pre-season, a regular season, and the playoffs. During the pre-season, the teams compete
in friendly matches, often against teams outside their league. After the pre-season, the
regular season is the first part in the championship. Each team in the NBA plays 82
matches against the other teams in the league, in the WNBA teams currently play only
34 games during the regular season. The best 8 teams in the WNBA and the best 16
teams in the NBA enter the playoffs.
The playoffs are pairwise elimination tournaments, consisting of three (four) rounds
in the WNBA (NBA). The winner of the final round is the champion. In the WNBA,
in the first two rounds a team needs to win two matches and as soon as one team has
won twice, the round is over (best-of-three). Since the 2005 season, the WNBA finals are
decided in a best-of-five series, in earlier seasons, also the finals were best-of-three. In
the NBA, all rounds are won in best-of-seven contests; only the first round in 2002/3 was
2
Due to rule changes introduced for the 2013/14 season, we restrict our analyses to data from the
2002/3 to 2013/14 seasons.
6
best-of-five. Until season 2013, the WNBA had a three point distance of approximately
6.25 meters. Since season 2013, this distance increased to approximately 6.75 meters.
The distance of the three point line in NBA matches is anyway larger as it is between
6.7 and 7.23 meters for all NBA matches in the dataset. So, women have an advantage,
particularly in seasons 2002/3-2012/13. Another advantage for WNBA player is that the
circumference of the basketball is 2.54 cm smaller than in the NBA while the basket has
the same diameter.
Following Grund et al. (2013), we use the option between two-point and three-point
attempts to identify risk-taking. A two-point attempt is the attempt of a player to score
(throw the ball through the opponents’ hoop) from inside a dedicated area. A three-points
attempt is a similar attempt but from the outside of this area.3
Both types shooting attempts are contested by specific defensive tactics and, ceteris
paribus, difficulty of any shooting attempt will increase with the distance to the basket.
Consequently, every attempt beyond the three-point line will be more difficult than any
attempt inside the line. Consulting table 1, we see that the probability of a successful
three-point attempt is significantly lower than a successful two-point attempt–for female
and male teams. Both ways of scoring yield a similar mean for both gender, however, the
standard deviation of a three-point attempt is larger than the the st. dev. for a two-point
shooting attempt.
The likelyhood of a three-point play resulting from a leass risky two-point attempt
is equal for men and women. This is evidence against the possibility that men could
substitute risk taking by a more physical style of play and decide for a shooting attempt
close to the basket with the intention to score two points plus draw a foul and get an
additional point from the following penalty free-throw. The only obvious difference is
that women have a lower probability to convert a two-point attempt than men4 . If we
restrict the analysis to the shooting attempts within the last 60 seconds of the games, the
general picture is the same as in table 15
3
Other possibility to score is presented by free throws which are awarded after personal fouls or certain
breaches of the rules. A successful free throw yields only one point.
4
This is clearly a result of the fact that men are dunking the ball on a regular basis, while women
almost never do this.
5
All tables are available on request.
7
Table 1: Percentage of all possible outcomes by gender and attempt-type
WNBA
NBA
Outcome
2pt
3pt
2pt
3pt
0
1
2
3
4
47.0
5.7
45.2
2.1
0
63.8
0.1
0.2
35.7
0.2
44.2
6.2
47.5
2.0
0
64.9
0.1
0.3
34.6
0.2
Mean
St. Dev.
1.023
1.002
1.085
1.443
1.073
0.996
1.051
1.431
N
20329
5281
115890
28450
Notes: Percentages of all possible outcomes as scores resulting from either two- or three-point attempts.
Outcomes in bold letters are most relevant in terms of relative frequency and are achieved without points
from potential free throws resulting from a penalty.
Comparing the frequency of relatively safer and relatively risky attempts, i.e., twopoints and three-points attempts, we see that women attempted risky shots more often
than men. In the WNBA, 23.6 percent of all score attempts and 22.9 percent in the
NBA are three-point attempts. In Figure 1, we plot the fraction of three-points attempts
among all attempts over the match time, separately for the NBA and the WNBA. For
both leagues, three-points attempts vary over time and become more frequent towards the
end of matches. However, in the WNBA there are relatively more three-point attempts
than in the NBA during the first half of matches, but this is reversed in the second half,
where there are relatively more three-point attempts in the NBA than in the WBNA.
8
●
0.4
0.4
Figure 1: Three-points attempts as a fraction of all attempts in playoff games, by gender
and match time.
women
men
●
women
men
0.3
●
●
●
●
●
●
●
●
●
●
●
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●
0.0
0.0
0.1
0.2
●
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●
0.2
Fraction of 3−point attempts
●
●
●
0.1
Fraction of 3−point attempts
0.3
●
1
2
3
4
Q1.1
Current quarter
Q1.3
Q2.1
Q2.3
Q3.1
Q3.3
Q4.1
Q4.3
Current quarter and subquarter
Notes: Fraction of three-points attempts among all two-points and three-points attempts in playoff
matches, by match time and league. Potential overtime periods are included in Quarter 4.
Score differences should play an important role in risk taking. Figure 2 plots the
fraction of three-point attempts over two-point attempts for all observed gaps in scoring by
quarter. This simple descriptive approach highlights the fact that the higher the absolute
score difference gets the more likely men attempt a three-point shot. For women, the
results are similar if the attempts are grouped together in intervals of size 10 to improve
estimation accuracy. At the finer level, positive score differences seem to be an incentive
for women to decrease their three-point attempt rate. However, this might be a result
of the higher variance, as the conditional probabilities for intervals of score differences
show a different result. An interesting result at both granularity levels is that women
have a greater probability for a three-point attempt than men if they have to catch up.
In contrast to this, men have a higher probability for a three-point attempt than women
if they lead.
Interestingly in any quarter, the increase of the three-point attempt rate from the
third sub-quarter to the last sub-quarter is higher for men than for women. Thus, men
increase their three-point attempt rate more than women towards the end of any quarter. One explanation for this increase in three-point attempts towards the end may be
9
again risk-preferences. Another explanation may be physical and/or psychological exhaustion which triggers more three-point attempts. Tired players may want to avoid
one-against-one situations and thus opt for the three-point attempt. Alternatively, the
tactical discipline of the players may shrink with fatigue and/or the time since the last
coach intervention.
Figure 2: Probability of a shot being a three-point attempt for all observed scoring differences
9
15
22
29
[−60,−50]
●
●
●●
● ●●
●
● ●● ●●
● ●
●
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● ● ●●
● ●
●
●
●●
● ●
●
●
●
[−40,−30]
[−20,−10]
[0,0.5]
[11,21]
[31,41]
[51,61]
Score of shooting team before current attempt − Score of opponent team (bin size = 10)
●
Fraction of 3−point attempts in last quarter
●
●
women
men
1.0
1.0
0.8
0.6
0.4
●
0.2
Fraction of 3−point attempts in last quarter
●
●●
●
women
men
●
●
0.8
50
●
●
0.6
0.2
43
●
●
●
0.0
36
Score of shooting team before current attempt − Score of opponent team
●
●
0.8
4
●
0.6
−2
●
●
●●
−50 −42 −34 −26 −18 −10
●
●
●
0.4
●
women
men
1.0
●
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●
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●
● ●
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●● ●●●●● ●●
●
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● ● ●●●● ●
● ●
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●
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●
●
●
●
●
0.4
Fraction of 3−point attempts
0.4
●
●
●
●
●
●
0.2
0.6
●
●
0.0
●
●
0.8
●
women
men
●
0.2
Fraction of 3−point attempts
1.0
●
●
●
●
●
−50 −42 −34 −26 −18 −10
0.0
0.0
●
●
●
−2
4
9
15
22
29
36
43
50
[−60,−50]
Score of shooting team before current attempt − Score of opponent team
[−40,−30]
[−20,−10]
[0,0.5]
[11,21]
[31,41]
[51,61]
Score of shooting team before current attempt − Score of opponent team (bin size = 10)
Notes: Upper panel: all quarters. Lower panel: Fourth quarter only. Positive differences represent score
attempts of the leading team. Negative differences represent score attempts of the trailing team. Potential
overtime periods are included in Quarter 4.
10
The value of winning a certain game in the playoffs is defined by the overall standings
in the playoff series. If the series is tied, winning the game will give you a lead or even win
the series. When trailing, a win will make the series continue or tie it up. When a team
is in the lead, a win will either win the series or create an advantage in the series. Figure
3 illustrates the fraction of three- of all attempts stratified by current playoff standings
from the perspective of the observed team.
If a shooting team needs less matches to win the complete playoff than its opponent,
then the shooting team more likely attempts a three-point shooting attempt. If the playoff round is tight or if the shooting team has to catch up, the risk-taking behavior of the
shooting team is less pronounced. It seems as if the three-point attempt rate increases if
the costs for missed attempts is comparatively low. Even though this pattern is true both
for men and women, men increase their risk-taking behavior more than women in case of
a clear lead.
0.35
Figure 3: Three-points attempts as a fraction of all attempts over current playoff standings.
women
men
0.25
●
●
●
●
●
●
●
0.20
●
0.10
0.15
●
0.00
0.05
Fraction of 3−point attempts
0.30
●
1:4
2:4
2:3
1:1
3:3
2:1
4:3
4:2
Ordered list of intermediate playoff results
Figure 4: Fraction of three-point attempts of all attempts by current standing in playoff series. The
first part of the score gives the number of still required wins for the team of the shooting player to win
the playoff round. The second number gives the same number for the opponent team.
Figure 5 plots the fraction of successful three-point attempt of all such attempts over
the remaining time on the match clock by quarters and sub-quarters where a sub-quarter
11
denotes a quarter of quarter. Overall, female teams are only slightly more successful with
three-point attempts (35.4) than male teams (34.7). Regarding 2-point attempts, male
have a higher success rate than women. In our data, WNBA players have a two-point
success probability of 44.5 percent whereas NBA player have a significantly higher probability of 47.1 percent.
●
0.5
0.5
Figure 5: Fraction of a successful three-point shot by quarter and sub-quarter.
women
men
●
women
men
●
●
●
●
●
●
0.4
●
0.3
0.3
0.2
●
●
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●
0.0
0.1
●
0.2
●
●
Fraction of successful 3−point attempts
●
●
0.1
0.4
●
0.0
Fraction of successful 3−point attempts
●
●
●
1
2
3
4
Q1.1
Q1.3
Current quarter
Q2.1
Q2.3
Q3.1
Q3.3
Q4.1
Q4.3
Current quarter and subquarter
Figure 6: Left panel: fraction of successful three-point attempts of total attempts in a
given quarter. Right panel: fraction of successful three-point attempts of total attempts
for given sub-quarters of quarters. Potential overtime periods are included in Quarter 4.
4
Empirical Strategy and Results
The main focus of this section is to analyze how pressure–associated with increased importance to perform and higher potential cost of risk taking–influences the willingness to
takes risks for men and women. We associate a higher propensity to bypass a less risky
two-point attempt for a riskier three-point attempt with a risky strategy. Risk taking is
associated as a way to cover gaps and erase scoring differences. The further behind teams
are, the more risk they should take to have a chance to come back into the game. In the
first step of our analysis we will focus on risk taking behavior of female and male teams
in crucial situations.
12
Crucial situations are defined as situations where performance is of high importance,
as a negative outcome will significantly reduce the probability to win the game. Consequently, depending on the scoring differences, risk taking will be associated with very
high costs. It is reasonable to assume that the potential costs of risk taking are inverse-U
shaped in terms of the scoring differences. A team which is far behind in the closing
minutes of the game has little to lose. The same is true for a team with a sizeable lead. In
close games during the final moments, the cost of risk taking are very high, as a potential
miss will likely lead to a loss.
A similar relationship between score difference and the time remaining until the end
of the game is true for the potential gains of risk taking. A three-point score at the end
of a close game is clearly superior to a two-point score. Teams with a substantial lead or
lag in the score will not benefit much from making the more risky shot, as there is too
little time left for the score to get even.
Consequently, we will analyze how scoring differences and situations at the end of
basketball games interact and influence risk taking of teams. The focus will be on crucial
situations when the cost and potential gains from risk taking are largest. In order to
measure the influence of time and score-difference on the probability of attempting a
three-point shot versus a two-point shot, we estimate the following linear probability
model
three-point attempt = β0 +
14
X
γk S +
14
X
δk S · ψ + X0it π + ξi + i ,
(1)
k=1
k=2
where S is a set of dummies measuring the score difference from the perspective
of team i and X is a a vector of control variables. These control variables consist of a
dummy for home/away matches, a yearly trend variable, the number of remaining wins
(defeats) to win (lose) the playoff, and the three-point and two-point attempt success
rates previous to the corresponding shot attempt. To control for team-season potential
unobserved heterogeneity and control for team-specific playing styles, we include teamround-season dummies.
13
The main focus of this analysis lies on the coefficients δk , which measure the additional
probability of a throw being a three-point attempt when the score difference is within the
bracket k and in the time interval ψ at the end of the game. In order to capture the
effect of a shrinking time period at the end of games, we chose ψ = 1 if time left ∈ [8, 3, 1]
measured as minutes remaining to be played, expecting an increased pressure to score the
closer we get towards the end.6 . This specification of our linear probability model allows
us to directly measure the additional probability of a three-point shooting attempt on the
omitted base category of a score diff of 0 or +1.
Table 2 tabulates all coefficients δk for all variations of ψ. We see that the probability
of a throw being a three-point attempt is increasing if team, male or female, are trailing.
The focal point is the situation where teams are trailing by 1 or two points: male players
increase the propensity to take a three-pointer in the final 8 minutes of the game by 5
percentage points, while women do not behave differently than when the game is tied
before the final 8 minutes. This gap in risk taking increase, the shorter the remaining
time becomes. Within the last minute, women even reduce risk taking at that particular
score difference, while male player still take higher risk. This is unexpected and a clear
indication that male teams are risk-seeking: they bypass the safer two-point shot which
would either tie the game or give them a 1 or even 2 point lead and take the much riskier
shot in order to get a lead in any case if the shot is successful. Within the final minute
this is most surprising, as a missed shot dramatically increases the probability to lose the
game. Figure 7 plots the estimated coefficients for the general score difference dummies
(top panel) and the interaction of the score difference dummies and a binary variable
indicating the final moments of the game (bottom panel) ψ.
The shorter the time to the end of the game, the higher the importance of success, as
it will get more difficult to cover the scoring gap. Consequently, the amount of pressure
will increase as the remaining time decreases. Our results indicate, that as pressure
increases the gender gap in risk taking increases as well. We can conclude that men tend
to increase risk taking as pressure increases, while women actually decrease risk taking.
Consequently, the gender gap in the willingness to take risks increases.
6
Table 6 in the appendix tabulates the absolute number of observed scoring attempts for all relevant
time intervals
14
Concerning situations where teams hold a lead in a game, we find that there is little
evidence that they increase risk taking under pressure to perform in the final minutes:
Male and female players both take three point shots at almost the same rate as when
the game is tied in the finale minute(s). Both sexes play conservatively when defending
a lead during the final minutes. This is expected, as the correlation of outcomes in the
case of three-point shooting is most likely close to zero, confirming the theoretical results
presented by Nieken and Sliwka (2010).7 We find little evidence that an increase of the
lead leads to increased risk taking of male and female players in the deciding minutes of
the game.
7
It is a plausible assumption that outcomes are not correlated, as an increased number of three point
attempts of one team will not make it easier for the opposing team to convert three point attempts.
15
Table 2: Effect of score difference in pressure situations - stratified by last minutes
ψ = 1 if attempt within
[ -60 -20 )·ψ
[ -20 -10 )·ψ
[ -10 -8 )·ψ
[ -8 -6 )·ψ
[ -6 -4 )·ψ
[ -4 -2 )·ψ
[ -2 0 )·ψ
[ 0 2 )·ψ
[ 2 4 )·ψ
[ 4 6 )·ψ
[ 6 8 )·ψ
[ 8 10 )·ψ
[ 10 20 )·ψ
[ 20 60 )·ψ
N
last 8 minutes
last 3 minutes
last minute
WNBA
NBA
WNBA
NBA
WNBA
NBA
0.042
(0.046)
0.064***
(0.023)
0.113***
(0.032)
0.104***
(0.034)
0.094***
(0.028)
0.066**
(0.027)
0.019
(0.022)
0.007
(0.015)
0.086***
(0.009)
0.106***
(0.016)
0.139***
(0.014)
0.147***
(0.014)
0.104***
(0.013)
0.05***
(0.012)
-0.029
(0.058)
0.139***
(0.034)
0.199***
(0.05)
0.171***
(0.052)
0.143***
(0.044)
0.126***
(0.035)
-0.007
(0.033)
0.008
(0.02)
0.113***
(0.015)
0.184***
(0.024)
0.229***
(0.021)
0.257***
(0.022)
0.154***
(0.019)
0.082***
(0.019)
-0.024
(0.095)
0.198***
(0.049)
0.294***
(0.06)
0.322***
(0.078)
0.238***
(0.069)
0.294***
(0.059)
-0.094**
(0.045)
0.012
(0.031)
0.123***
(0.021)
0.306***
(0.034)
0.335***
(0.03)
0.401***
(0.027)
0.24***
(0.031)
0.071**
(0.029)
0.04
(0.026)
-0.047*
(0.024)
0.089***
(0.033)
-0.027
(0.035)
-0.009
(0.03)
-0.006
(0.016)
-0.01
(0.027)
0.023**
(0.011)
0.033***
(0.012)
-0.008
(0.012)
0.012
(0.014)
-0.006
(0.016)
0.022**
(0.009)
0.01
(0.016)
0.035
(0.032)
0.02
(0.042)
0.129**
(0.058)
-0.02
(0.069)
-0.05
(0.045)
-0.015
(0.028)
0.03
(0.063)
0.027
(0.017)
-0.008
(0.017)
-0.032*
(0.018)
-0.027
(0.021)
0.018
(0.026)
0.038**
(0.015)
-0.003
(0.021)
-0.056
(0.054)
-0.111*
(0.059)
0.073
(0.103)
-0.151**
(0.07)
-0.045
(0.098)
0.06
(0.055)
0.038
(0.136)
0.022
(0.028)
0.006
(0.029)
-0.033
(0.036)
-0.021
(0.044)
0.007
(0.054)
0.103***
(0.026)
0.03
(0.036)
25610
144340
25610
144340
25610
144340
The Dependent variable is equal to 1 if throwing attempt is a three-point attempt. Coefficients for time dummies and additional control
variables are not reported due to space limitations. All specifications include home match dummy, a yearly trend variable, the number
of remaining wins (defeats) to win (lose) the playoff series, the three-point and two-point success rates previous to the corresponding shot
attempt and team-round-season dummies. *, ** and *** indicate statistical significance at the 10-percent level, 5-percent level, and 1-percent
level, respectively. Clustered standard errors (clustered for team-round-year) in round parentheses.
16
Changing score differences interacted with the time remaining in a game represent
dynamic in-game incentives for risk taking, holding initial ability difference constant.
Data from professional basketball playoffs provides an additional source of changes in
potential outcome, as we can also observe playoff standings. Any team can face three
potential situations in the playoffs: a lead, a tied series as well as a standing where they
are trailing.
The ultimate prize of winning a playoff tournament is winning the championship.
In both leagues, the WNBA and the NBA, winning individual games within a playoff
round is not compensated. Any financial compensation is directly linked to winning a
playoff round. Each possible standing in a playoff series (elimination tournament with
multiple interactions) is associated with an expected prize E(P ). Holding ability of both
contestants constant, any team leading a playoff series has a higher expected prize than
a team facing a tie. When down, there is an increased possibility to lose the series and
E(P ) is even lower.8 Put differently, while winning a game when already leading the
series increases the likelyhood to win the overall prize. Winning when trailing in the
series, however, on average will avoid losing the series.
In order to analyze potential differences in the reaction of female and male teams in
crucial situations for different playoffs situations, we split up the overall sample in three
sub-samples: trailing, tied and lead. Table 3 presents the results for estimating model 1
for all three different sub-samples and for ψ = 1 if shooting attempt is observed during
the final minute of the game. From the results it emerges, that the different willingness
to incur risk for men and women is mainly coming from games in a tied playoff series
as well as games when the observed team is in the lead. Figure 8 plots the estimated
coefficients for the general score difference dummies (top panel) and the interaction of the
score difference dummies and a binary variable indicating the final moments of the game
(bottom panel) ψ for the three different standings in the playoff series.
8
obviously, all three possible situations have the possibility for pivotal games. In such games a win in
the game either loses the series or wins it. Obviously, at both, trailing and tied, losing a game can also
lose the series immediately, if the current game is a pivotal game. In a best-of-5 (best-of-7 series), losing
at 1:2, 0:2 or 2:2 (0:3, 1:3, 2:3 or 3:3) will effectively lose the series. In a best-of-3 series losing when down
0:1 will also lose the series. There are no pivotal games where the team can lose the series when it leas,
any pivotal game at a tie is one where the team can lose and win the series and all pivotal games when
trailing are games where it can only lose the series. So the ordering is also consistent with an increasing
E(P ).
17
Again, the main focus is on the situation where teams are trailing by 1 or 2 points:
when observing a tied playoff series, men increase risk taking dramatically, as the probability of a shooting attempt at score [-2 0) in the last minute is increased by 10.6 percentage
points. In the same situation for women, this additional probability is actually decreased
by 14 percentage points. In games where male and female teams are behind in the playoff
series there is no gender gap, as both genders do not have a different risk taking behavior
compared to the base group. When holding a lead in the playoffs, male teams still increase
risk taking (10.6 percentage points) while the point estimate for women is positive but
small and insignificant (0.025 percentage points).
These findings suggest that in a situation where risk taking does not only endanger
the outcome in terms of risking a loss in the current game but the overall playoff series
(trailing and tied), women take less risk in highly critical situations. When trailing in
the series, men act similarly to women, as the also decrease risk taking when the game is
close in the final minute of the game. However, in contrast to women, the probability of
a three-point shot is still high when the game is close at the end and the playoff series is
tied. In a situation where the probability to lose the series is increasing by the on average
smallest amount, men take risky shots and women also show a small insignificant increase
in risk taking. We can conclude, that men only take less risk when the danger of losing
the series is high, while women mostly stay conservative regardless of the standings in the
playoff series.
Table 4 plots the estimated coefficients for the interaction of ψ and the dummy for
the scoring difference interval of [ -2 0 ). The large gender gap in the willingness to incur
risk at this crucial situation of the game is found at all three time intervals at the end of
games when the playoff series is tied. In these situations, male teams increase risk taking,
regardless if at the finals minutes or the final 3 or 8 minutes. In contrast, female teams
actually decrease risk taking in the last minute, while showing no significant difference to
the base group when looking at the final 3 or 8 minutes of the game.
Male teams also increase risk taking when they are leading in the series, but the level
of risk taking decreases the more time is still left in the game. When trailing, both genders
act more conservatively, as they do not show any increases in risk taking compared to the
base group.
18
Table 3: Effect of score difference in pressure situations - stratified by playoff series standing: last minute.
Series:
[ -60 -20 )·ψ
[ -20 -10 )·ψ
[ -10 -8 )·ψ
[ -8 -6 )·ψ
[ -6 -4 )·ψ
[ -4 -2 )·ψ
[ -2 0 )·ψ
[ 0 2 )·ψ
[ 2 4 )·ψ
[ 4 6 )·ψ
[ 6 8 )·ψ
[ 8 10 )·ψ
[ 10 20 )·ψ
[ 20 60 )·ψ
N
trailing
tied
in lead
WNBA
NBA
WNBA
NBA
WNBA
NBA
-0.103
(0.16)
0.156*
(0.082)
0.361***
(0.091)
0.401***
(0.114)
0.205*
(0.124)
0.27**
(0.091)
-0.03
(0.091)
0.122*
(0.063)
0.140***
(0.027)
0.330***
(0.05)
0.296***
(0.049)
0.38***
(0.054)
0.198***
(0.043)
-0.002
(0.043)
-0.018
(0.194)
0.165**
(0.066)
0.292***
(0.092)
0.302***
(0.096)
0.252***
(0.088)
0.364***
(0.053)
-0.14***
(0.053)
-0.048
(0.043)
0.122***
(0.036)
0.33***
(0.061)
0.294***
(0.054)
0.405***
(0.048)
0.272***
(0.045)
0.106**
(0.045)
0.008
(0.158)
0.265***
(0.099)
0.168
(0.12)
0.28
(0.236)
0.323
(0.26)
-0.029
(0.158)
0.025
(0.158)
-0.014
(0.054)
0.094**
(0.043)
0.224***
(0.074)
0.407***
(0.049)
0.41***
(0.051)
0.236***
(0.056)
0.102*
(0.056)
-0.212***
(0.048)
0.021
(0.134)
0.385**
(0.168)
-0.303***
(0.027)
0.05
(0.159)
-0.069
(0.092)
-0.163
(0.217)
-0.04
(0.052)
0.011
(0.054)
-0.078
(0.06)
-0.037
(0.081)
0.075
(0.11)
0.135***
(0.049)
-0.077
(0.06)
-0.05
(0.054)
-0.114
(0.086)
0.051
(0.164)
-0.045
(0.15)
-0.038
(0.135)
0.080
(0.075)
0.220
(0.176)
0.046
(0.046)
0.014
(0.051)
-0.008
(0.059)
-0.014
(0.071)
-0.173***
(0.059)
0.076*
(0.044)
0.004
(0.058)
0.022
(0.222)
-0.231***
(0.029)
-0.123
(0.136)
-0.252***
(0.044)
-0.077**
(0.032)
0.100
(0.132)
0.14***
(0.024)
0.059
(0.048)
0.000
(0.046)
-0.007
(0.068)
-0.009
(0.076)
0.132
(0.099)
0.104**
(0.042)
0.143**
(0.071)
5281
28450
5281
28450
5281
28450
The Dependent variable is equal to 1 if throwing attempt is a three-point attempt. ψ = 1 if shooting attempt is observed within
the final minute of the game. Coefficients for time dummies and additional control variables are not reported due to space
limitations. All specifications include home match dummy, a yearly trend variable, the number of remaining wins (defeats) to
win (lose) the playoff, the three-point and two-point success rates previous to the corresponding shot attempt and team-season
dummies. *, ** and *** indicate statistical significance at the 10-percent level, 5-percent level, and 1-percent level, respectively.
Clustered standard errors (clustered for team-year) in round parentheses.
19
Table 4: Additional probability of a three-point attempt by playoff standing for different
time intervals.
trailing
tied
in lead
ψ = 1 if
WNBA
NBA
WNBA
NBA
WNBA
NBA
t<1
-0.03
(0.091)
-0.002
(0.043)
-0.14***
(0.053)
0.106**
(0.045)
0.025
(0.158)
0.102*
(0.056)
t<3
0.056
(0.058)
0.036
(0.032)
-0.046
(0.043)
0.118***
(0.032)
0.066
(0.09)
0.082***
(0.03)
t<8
0.033
(0.041)
0.02
(0.019)
-0.003
(0.03)
0.075***
(0.022)
0.061
(0.055)
0.051**
(0.021)
The Dependent variable is equal to 1 if throwing attempt is a 3-pint attempt. ψ = 1 if shooting attempt is observed
within the final minute of the game. Coefficients for time dummies and additional control variables are not reported
due to space limitations. All specifications include home match dummy, a yearly trend variable, the number of
remaining wins (defeats) to win (lose) the playoff series, the three-point and two-point success rates previous to
the corresponding shot attempt and team-round-season dummies. *, ** and *** indicate statistical significance
at the 10-percent level, 5-percent level, and 1-percent level, respectively. Clustered standard errors (clustered for
team-round-year) in round parentheses.
20
Figure 7: Additional probability for a three-point shot during final moments of a game: final 8, 3 and final minutes.
Last 3
●
[−2, 0)
[2, 4)
●
●
●
[4, 6)
[6, 8)
[8, 10)
[−6, −4)
●
●
●
●
●
●
[−2, 0)
[2, 4)
[4, 6)
[6, 8)
●
1.0
●
●
●
●
●
[−10, −8)
[−6, −4)
[8, 10)
[−10, −8)
[−6, −4)
[−2, 0)
[2, 4)
[4, 6)
[6, 8)
[8, 10)
Last Min
●
●
●
●
●
●
●
●
[−2, 0)
[0, 2)
[2, 4)
Intervals of Score Differences
[4, 6)
[6, 8)
[8, 10)
1.0
women
men
0.5
1.0
0.5
●
−0.5
−0.5
●
●
●
●
●
●
●
0.0
●
●
women
men
●
●
●
●
●
−0.5
●
●
Additional Probability for a 3−point shot
women
men
●
●
●
[−6, −4)
●
Last 3
●
[−10, −8)
●
Last 8
1.0
●
●
Intervals of Score Differences
0.5
●
0.0
●
●
Intervals of Score Differences
●
women
men
0.5
1.0
0.5
●
Intervals of Score Differences
Additional Probability for a 3−point shot
21
[−10, −8)
Additional Probability for a 3−point shot
●
0.0
●
●
−0.5
●
0.0
●
Additional Probability for a 3−point shot
●
women
men
−0.5
●
●
0.0
0.0
0.5
women
men
−0.5
Additional Probability for a 3−point shot
1.0
●
Last Min
Additional Probability for a 3−point shot
Last 8
[−10, −8)
[−6, −4)
[−2, 0)
[0, 2)
[2, 4)
Intervals of Score Differences
[4, 6)
[6, 8)
[8, 10)
[−10, −8)
[−6, −4)
[−2, 0)
[0, 2)
[2, 4)
[4, 6)
[6, 8)
[8, 10)
Intervals of Score Differences
Additional probability for a three-point shot during final moments of a game. Coefficients and standard errors for score difference dummies (top panel)
and interaction of score difference dummies and ψ. Score difference [0 1) is the base group.
Figure 8: Additional probability for a three-point shot during final moments of a game: trailing, tied or leading in playoff series
Last Min
●
●
●
●
●
●
●
●
●
●
●
●
1.0
●
women
men
0.5
1.0
0.5
women
men
●
●
0.0
●
Additional Probability for a 3−point shot
●
●
●
●
●
●
●
0.0
women
men
0.5
●
●
0.0
Additional Probability for a 3−point shot
1.0
●
Last Min
Additional Probability for a 3−point shot
Last Min
●
●
●
●
●
●
[−10, −8)
[−6, −4)
[−2, 0)
[0, 2)
[2, 4)
22
Intervals of Score Differences
[4, 6)
[6, 8)
[8, 10)
−0.5
−0.5
−0.5
●
[−10, −8)
[−6, −4)
[−2, 0)
[0, 2)
[2, 4)
Intervals of Score Differences
[4, 6)
[6, 8)
[8, 10)
[−10, −8)
[−6, −4)
[−2, 0)
[0, 2)
[2, 4)
[4, 6)
[6, 8)
[8, 10)
Intervals of Score Differences
Notes: Additional probability for a three-point shot during final moments of a game. Score difference [0 1) is the base group. Playoff standings are grouped
as trailing (0:1;0:2;0:3;1:2;1:3;2:3), tied (0:0;1:1;2:2 and 3:3) and leading (1:0;2:0;2:1;3:0;3:1;3:2).
Individuals and teams incur risk because they can–with a certain probability–benefit
from positive returns. In our context, the direct positive consequences associated with
taking a three-point shot is a score, yielding 3 points. In order to link the decision to incur
a risky strategy with the observed returns, we estimate model 1 with a dummy equal 1 if
the observed three point shot is successful, 0 else, as the dependent variable.
Figure 9 plots the estimated coefficients for the interaction of the score difference
dummies and a binary variable indicating the final moments of the game.9 We see that
the changes in success rates for close scores in the final minutes of the game are not
differing for men and women.
Success can also be measured in the probability to win the game. In order to measure
the effect of risk taking in critical situations on the probability to win the game, we create
a sample of games that were within the range of [-2 2] when entering the finale 8 minutes,
3 minutes or last minute. We estimated a linear probability model where the dependent
variables is equal to 1, if the observed team is winning the game. Table 5 tabulates the
coefficients of interest, measuring the absolute number of two- and three-point attempts at
score interval [-2 0), as well as the respective success rates. We see that as the percentage
of successful three-point attempts at [-2 0) during the final minutes is positively associated
with a higher probability to win the game for female and male teams.
For male teams, we also find a positive effect of the success rate of two-point attempts
in the same situations. Consequently, we see that both genders have a higher probability
to win close games if they incur risk and are successful. On average, the positive returns
to successful risk taking are higher for female teams than for male teams.
The analysis of both success measures leads to the conclusion that there are no
systematic differences in the returns to risk taking in crucial situations for female and
male teams. Consequently, different success under high pressure to perform is likely not
the driving force behind the gender gap in risk taking we documented earlier.
9
Full estimation output is omitted due to space constraints. All results available upon request.
23
Figure 9: General score difference effect and additional probability for a successful 3-point shot during final moments of a game:
last 8, 3 and last minute
Last 3
●
●
●
●
●
●
●
●
●
24
[−10, −8)
[−6, −4)
[−2, 0)
[0, 2)
[2, 4)
Intervals of Score Differences
[4, 6)
[6, 8)
[8, 10)
3
2
1
0
3
2
1
●
−3
−2
●
●
women
men
●
●
●
●
●
●
●
●
−1
●
0
●
Additional Probability for successful 3−point shot
●
●
●
●
[6, 8)
[8, 10)
−3
●
women
men
−1
●
−1
●
Additional Probability for successful 3−point shot
0
●
●
●
−2
1
2
women
men
−3
Additional Probability for successful 3−point shot
3
●
Last Min
−2
Last 8
[−10, −8)
[−6, −4)
[−2, 0)
[0, 2)
[2, 4)
Intervals of Score Differences
[4, 6)
[6, 8)
[8, 10)
[−10, −8)
[−6, −4)
[−2, 0)
[0, 2)
[2, 4)
Intervals of Score Differences
[4, 6)
Table 5: Different strategies and win probabilities for close games.a
<8
<3
<1
WNBA
NBA
WNBA
NBA
WNBA
NBA
Number of three-point
attempts at [ -2 0 )b
0.015
(0.122)
-0.111***
(0.030)
-0.185*
(0.102)
-0.089*
(0.047)
-0.188
(0.116)
-0.221***
(0.068)
Percentage three-point
attempts successfulb
0.277
(0.181)
0.243***
(0.092
0.377**
(0.159)
0.197*
(0.108)
0.494***
(0.166)
0.196
(0.177)
Number of two-point
attempts at [ -2 0 )b
-0.050
(0.051)
-0.016
(0.014
-0.054*
(0.032)
-0.101***
(0.021)
-0.214***
(0.071)
-0.194***
(0.046)
Percentage two-point
attempts successfulb
0.279*
(0.167)
0.195***
(0.073)
0.147
(0.165)
0.376***
(0.073)
0.209
(0.132)
0.351***
(0.108)
N
52
308
74
292
80
286
No. of risky three-pointers
8
52
26
131
35
207
The Dependent variable is equal to 1 if observed team wins the game. Coefficients for time dummies and additional control variables are not
reported due to space limitations. All specifications include home match dummy, a yearly trend variable, the number of remaining wins (defeats)
to win (lose) the playoff series and team-season dummies. *, ** and *** indicate statistical significance at the 10-percent level, 5-percent level,
and 1-percent level, respectively. Clustered standard errors (clustered for team-round-year) in round parentheses.
a
Close games are defined as those where the first throwing attempt within the last 8, 3 or final minute(s) was observed at any score difference
in the interval [-2 2].
b
All variables measured as absolute number of attempts at score difference [ -2 0 ).
25
5
Discussion and Conclusion
Our empirical analysis provides evidence for the existence of a gender gap in the willingness
to to take risk. We find this gap to be large and significant in situations where pressure to
perform and potential costs of failure are high. The higher these costs are,–i.e. the worse
the potential negative outcome is–the larger is the gender gap in risk taking. In such
situations, men increase risk taking, while women act more conservatively. Our findings
confirm earlier theoretical and empirical findings: male and female professional basketball
teams use risky strategies to cover gaps. However, in very critical situations, men still
take high risk, while women are found to be rather risk averse.
We exploit the structure of basketball playoffs tournaments to investigate how the
prize associated with winning the current game is affecting risk taking in crucial situations.
We find that male teams increase risk taking when they are leading the playoff series or
when the series is tied, i.e. when the prize of a win in the current game is a high probability
to win the overall playoff series. For female teams, however, we do not find this behavior,
as they stay conservative in crucial situations, regardless of the overall playoff standings.
Consequently we measure a sizeable gender gap in risk taking when teams have a
higher probability to win the series than to lose it. This gap is largest when the playoff
series is tied in the last minute of the game, as female teams reduce and male teams
increase risk taking. In games that are associated with a potential prize of reducing the
overall deficit in games or not losing the series, both genders do not increase risk taking
in crucial situations.
In terms of consequences of risk taking, we provide evidence that success rates for
men and women in the final minutes of the game with small score differences are not
different for men and women. We do also find a positive relation between successful risk
taking at critical moments of a game and the chance to win the game for both genders.
Decisions to incur risk at during a game can be made by the players who then take
the risky shot, or by the coach standing on the sidelines. In our analysis we cannot
distinguish between decisions of coaches and players. As the observed gender gap in risk
taking could be explained by female players as well as female coaches making the decision,
future research on this issue is needed.
26
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28
6
DATA APPENDIX
The dataset was generated by downloading match data from websites
http://scores.espn.go.com/wnba/scoreboard for the WNBA and
http://scores.espn.go.com/nba/scoreboard for the NBA. Both websites provide detailed information of the match history at a score attempt level for each match since
2002 including pre-season matches. Thus, for each match every score attempt is available
including whether the attempt was successful, the person who made this attempt, and
when the attempt was made during the course of the match. The available data from
2002 onwards was downloaded.
In the subsequent preprocessing steps, we dropped all observations from the NBA
season 2013/2014 and the WNBA seasons 2013 and 2014. We excluded NBA season
2013/2014 as it had not been completed when the dataset was created. The removal of
the last two WNBA seasons 2013 and 2014 was necessary in order to exclude a confounding
effect of the change in the three point distance from season 2013 onwards.
In the final dataset, we restrict the analysis to WNBA and NBA playoff games. The
final playoff dataset consists of 177, 024 rows, i.e. shot attempts. 26, 033 rows represent
shot attempts from 203 WNBA playoff matches while the remaining 150, 991 rows stem
from 970 NBA matches.
29
Table 6: Absolute number of three-point attempts during the game and final minutes
> last 8
[
[
[
[
[
[
[
[
[
[
[
[
[
[
-60 -20 )
-20 -10 )
-10 -8 )
-8 -6 )
-6 -4 )
-4 -2 )
-2 0 )
02)
24)
46)
68)
8 10 )
10 20 )
20 60 )
last 8
last 3
last min
NBA
WNBA
NBA
WNBA
NBA
WNBA
NBA
WNBA
2226
11481
5023
6657
8229
10307
12249
13636
10607
8137
6631
5083
11229
2104
316
2185
952
1131
1598
1676
2075
2392
1897
1491
1177
943
2146
317
1590
3273
1136
1291
1387
1538
1553
1690
1436
1222
1042
921
3015
1534
214
799
281
274
333
380
361
369
302
271
215
212
677
243
669
1308
543
607
627
718
638
628
588
474
359
303
1086
601
86
336
113
127
141
183
173
150
113
77
60
56
223
96
226
494
247
277
291
305
245
214
196
123
84
62
344
178
26
127
55
59
55
71
78
54
22
20
11
14
68
21
Notes: Absolute number of observed throwing attempts by time interval during WNBA
and NBA games used in empirical analysis of section 4. Last 8, 3 or 1 minutes is either
observed in fourth quarter of any overtime period.
30