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Incidence and consequences of risk-taking behavior in tournaments--evidence from the NBA.

I. INTRODUCTION

The performance of individuals or teams is assessed in relation to peer performance in many situations. The better employee will be promoted, the more successful salesperson will receive a bonus, the best R&D team will be able to get a patent, or the better sports team in a final will win the championship. In tournament theory, these situations are analyzed theoretically as rank-order tournaments. In their seminal contribution, Lazear and Rosen (1981) explore the incentive and selection effects of tournaments.

In rank-order tournament-type situations as well as in other kinds of incentive structures, individuals may not only choose an effort level but also affect the outcome by adopting a certain risk strategy. A portfolio manager can take more or less risky assets into account, a salesperson may concentrate on traditional or new products, or a general manager can invest in traditional or innovative markets. Risk-taking may then be fostered by incentive structures that focus on rewarding very good results and do not penalize bad results. Stock options for managers can be quoted as a well-known and broadly discussed example (e.g., DeFusco, Johnson, and Zorn 1990; Coles, Naveen, and Naveen 2006; Rajgopal and Shevlin 2002; Sanders and Hambrick 2007). Besides, employed managers usually face limited liability and do not have to bear extensive losses personally (e.g., Gollier, Koehl, and Rochet 1997).

An important strand of the literature on risk-taking in tournaments examines the impact of revealed heterogeneity of agents on risk-taking (either right from the start of the tournament or at an intermediate stage). A main theoretical prediction is that less able agents usually choose riskier strategies (Bronars 1987; Hvide 2002; Krakel and Sliwka 2004). Nieken and Sliwka (2010) explicitly model the role of interim performance information for subsequent risk-taking. They consider a risk-taking tournament between two agents who decide between a risky and a safe strategy with regard to the variance of the outcome. They allow for uncorrelated outcomes of strategies and derive a Nash equilibrium, in which the leading agent sticks to the safe strategy, whereas the trailing agent opts for the risky strategy.

Some information during the tournament can reveal heterogeneity also in practice: portfolio managers realize that their portfolio performs rather weakly, or a sports team is behind during a game, for example. Agents can then react by adjusting their risk strategies. Brown, Harlow, and Starks (1996), as well as Chevalier and Ellison (1997) and Taylor (2003) show that portfolio managers with relative performance contracts, who realize that their own intermediate performance is weak, usually switch to riskier portfolios. However, Kempf, Ruenzi, and Thiele (2009) reveal that this effect may change if managers are afraid of losing their jobs. (1)

A few empirical studies analyze the incidence of risk-taking behavior with sports and gaming data. Lee (2004) examines poker tournaments and confirms that the incentive for risk-taking is strengthened by a larger expected gain, and bottom-ranked players take more risk. Genakos and Pagliero (2009) analyze the risk-taking in weightlifting competitions measured by the chosen weight for the decisive attempt and find an inverted-U relationship between risk-taking and rank. Stock car races are examined by Becker and Huselid (1992) and Bothner, Kang, and Stuart (2007). The results include that drivers (as a group) take more risks if both prizes and prize spreads are large, and individual drivers take more risks in a race when facing the possibility of a displacement by lower-ranked counterparts.

A typical assumption in the theoretical economic literature is that people fully and rationally anticipate the expected outcome of their behaviors and adjust their risk strategies to maximize individual utility. However, whether this happens in practice is difficult to establish. In the case of car races, for example, risk is measured by observing its consequences (accidents) so that the benefits of risk-taking cannot be separated from its incidence. Additionally, sabotage intentions may be an integral part of risk-taking in car racing tournaments if a driver negligently or even intentionally takes a certain opponent out of the race. Under these circumstances, it is difficult to disentangle pure risk-taking from a sabotage effect.

Separate evidence for the incidence and the consequences of risk-taking in tournaments is rare. Grund and Gurtler (2005) use data from the German soccer Bundesliga and argue that team managers may change the risk strategy through substitutions of players (e.g., a striker is exchanged for a defender) during a game. (2) They find that managers of trailing teams tend to substitute defenders for midfielders or strikers. On average, this risk-taking behavior is counterproductive in terms of the probability of improving the result in the residual time. It cannot be observed, however, whether defenders or even goal keepers act as strikers at the end of a game, so the risk measure is somewhat crude because actual behavior is not explicitly observed.

Apart from studies on tournaments, Bromiley (1991) as well as Wiseman and Bromiley (1996) explore how firms' past performance (e.g., return on assets or return on equity) affects corporate risk-taking and how risk affects future firm performance. They use the firms' variance in security analysts' forecasts of income as their measure of risk-taking. Palmer and Wiseman (1999) argue that these organizational risks in terms of volatile income streams have to be distinguished from direct managerial risk-taking. The amount of R&D investments may then be used as a proxy for managerial risk-taking (see also Sanders and Hambrick 2007).

In this empirical study, we examine both the incidences and the consequences of risk-taking behaviors in tournaments. We use data from the National Basketball Association (NBA) and analyze teams' tactical orientations. We argue that the fraction of three-point shot attempts acts as a measure of the chosen risk. By observing this risk measure for different periods toward the end of games, we are able to examine the following research questions:

1. Do teams respond to intermediate scores by adjusting their risk strategies?

2. Do teams benefit by increasing the riskiness of their strategies?

In contrast to the vast majority of previous studies, we therefore do not consider only the incidence of risk-taking, but also investigate its consequences. By using data from basketball games, we have many observations in a controlled setting. (3) We indeed find evidence for increased risk-taking behavior by teams that are behind. It turns out, however, that increased risk-taking is counterproductive in the vast majority of situations.

We contribute to the existing literature in several respects: First, we add to the evidence for the relevance of past performance information on subsequent risk-taking. Second, and more important, we provide novel evidence on the consequences of changes to risk-taking behavior during tournaments.

The paper proceeds as follows: in Section II, we explore risk-taking behavior in the NBA in more detail and derive our hypothesis. The data and variables are presented in Section III, followed by the results in Section IV. Section V concludes the paper.

II. RISK-TAKING BEHAVIOR IN THE NBA

The NBA is a 1-year tournament. Thirty teams compete during the regular season and in subsequent playoff games for the championship. Because the ranking of a team at the end of the regular season depends solely on its number of wins, the margin by which a team wins or loses a single game does not matter. The teams with the highest ranking proceed to the playoffs, where each round is organized in a best-of-seven format so that a team moves on to the next round once it reaches four wins.

A game lasts 48 minutes and is divided into four quarters. The time for the offensive team until a scoring attempt has to be made is limited to 24 seconds. Therefore, there are many situations in which players have to make decisions on how to score. Coaches explicitly or implicitly affect these decisions by establishing a strategy before the game, through time-outs, by instructions throughout the game, or substitutions. Considering the intermediate score, they may change the tactical orientation of their team during a game.

In general, there are two possibilities to score in basketball. First, players can shoot from a distance of less than 7.24 m (23'9") to get two points when successful. Second, they can try to hit the basket from a farther distance (three-point field goal area), getting three points when making the shot. In both cases, one additional point can be scored when a player is fouled during a successful shot and he hits the following free shot. In cases where a foul prevents a successful shot, players have the chance to hit two or three free shots depending on whether they got fouled during a two- or three-point attempt.

In the 2007/2008 season, players hit 0.457 (0.362) of two- (three-) point shots and three out of four free throws (0.755). Fouls of the defending team occur much more often on two-point shots (0.119) than they do on three-point attempts (0.007). When getting fouled during a shot attempt, more than one-fourth of the attempts are successful for both two-point and three-point shots. These numbers result in the following distribution of points for two-point and three-point shots (see Table 1).

The standard deviation of points for a three-point shot is much higher than that for a two-point shot, indicating a considerably higher amount of risk (the variance is twice as high). The comparison of the means indicates a slight advantage of three-point shots (1.10) over two-point attempts (1.07). However, an additional benefit from two-point shots can be found in the higher probability of being fouled, which harms the opposing player (who will be disqualified after six personal fouls) and the opposing team as a whole (because of free throws after every foul from the fifth foul of the team in a quarter onward). We can therefore argue that the expected outcome of two- and three-point attempts is comparable, whereas both strategies differ considerably in their variances. In this sense, we speak of increased (decreased) risk-taking behavior when teams increase (decrease) the fraction of three-point shots during a game. This risk measure is ideal because it is an ex ante measure of risk instead of the type of ex post measures of outcome such as performance variance used in previous studies, which confound risk-taking behavior with other exogenous factors. Besides, it has an advantage over other examples of risk-taking, such as accidents in car races, which carry a possible issue of sabotage, whereas we want to concentrate solely on risk-taking.

In sports, intermediate scores reveal current heterogeneity. We measure this heterogeneity by defining a point difference variable as the points of the team in question less the opponent's points. We want to examine whether this increased risk-taking behavior is relevant in the NBA, and formulate the following hypothesis:

Hypothesis: The amount of risk-taking of a team (the fraction of three-point shots) decreases in the point difference during a basketball game.

Increasing the risk changes the random distribution of results in a way that the probability mass is shifted from the mean to the tails. Hence, extreme results become more likely. As the higher likelihood of a poor value is not harmful because damage is limited to losing the game, the increasing probability for a high result makes it more likely for the trailing team to win the game after all.

However, the number of points by which the team is behind at a certain point of time may also affect the relative advantage of changing the risk strategy. Figure 1 illustrates the cumulative distribution of the pure strategies of taking two-point and three-point shots, respectively. The two curves show the distribution function to receive at least a certain number of points attempting 10 and 20 two-point shots and three-point shots using the shooting percentages of the NBA as described in Table 1. For instance, the probability of receiving at least 27 points when attempting 20 shots is more than twice as high for three-point attempts (0.28) as it is for two-point attempts (0.13). In our example, the threshold at which the advantage in terms of the likelihood of a high result switches from two-point to three-point shots is at 23 points for 20 shots and at 12 points for 10 shots.

These considerations lead to the following corollary of our hypothesis:

(a) A change to a riskier strategy for the trailing team should increase its chances of winning the game.

(b) Increasing risk is more likely to be associated with a higher winning probability the more negative the absolute point difference is.

III. DATA AND VARIABLES

We make use of data from the 2007/2008 NBA season. (4) Each of the 30 teams plays 82 games during the regular season. Additionally, we use information on 86 playoff games, which results in a total sample size of 2,632 observations from 1,316 games. The data were collected from the official website of the NBA (www.nba.com), where information on every single-shot attempt during every game is provided.

We also examine subsamples of games. It may well be the case that in certain situations incentives to win are no longer existent. First, teams without any chance of reaching the playoffs may even have an incentive to lose to benefit from an earlier pick in the upcoming draft (Taylor and Trogdon 2002). We therefore also consider the subgroup of teams that have qualified for the playoffs in the 2007/2008 season, ensuring that we only observe teams that actually benefit from winning. Second, it is obvious that the playoff games are particularly important to teams so that we also analyze this subsample. Last, but not least, the intermediate score of each game is obviously public information at any point of time. This information may destroy incentives because the trailing team may at some point of time realize that the point difference is simply too large to have any chance to catch up. In this case, it is quite common in the NBA that the team lying hopelessly behind substitutes their best players, with the opponent following suit. This leads to a somewhat different game, which is hardly comparable to the serious game before. We therefore also examine the subgroup of games without give-ups at certain points of time. We take four points of time into account and exclude observations for the subgroups, in which a team is lying behind with (1) at least 24 points after the third quarter, (2) at least 21 points 9 minutes before the end, (3) 17 points 6 minutes before the end, and (4) 12 points 3 minutes before the end. There is no team that won the game being behind by more points than these mentioned thresholds. We will also refer to these four points of time with respect to the risk-taking behavior in our empirical analysis. The sample size considerably decreases by applying these limitations. For instance, in 40% of the games, the winner is already established 3 minutes before the end using our defined threshold. Table 2 provides an overview of the subgroups' sample sizes.

In the subsequent parts of the paper, we will refer to the following notation: points in time are noted in brackets, so that [12] means 12 minutes before the end of a game, and the time is captured from the start of a game [48] to the end [0]. A period of time from m minutes to play until n minutes to play is written as [m - n].

In our empirical investigation, we examine the risk-taking behavior at the end of games and its consequences with respect to the probability of winning a game. We calculate the fraction of three-point attempts during the last 12 (9, 6, and 3) minutes for every team and game (Risk [12-0], Risk [9-0], Risk [6-0], and Risk [3-0]). Our hypothesis is that teams that are behind choose a particularly high-risk action to increase their chance of catching up. We therefore examine whether the intermediate score explains the risk-taking behavior, and expect a negative effect for the point difference (team's score--opponent's score) at a certain point in time on the risk strategy for the remainder of the game. Hence, we observe the point difference 12, 9, 6, and 3 minutes before the end of the game. We also take into account the risk-taking behavior in the first three quarters of the game (Risk [48-12]) as a benchmark for the previous strategic orientation of a team.

Other control variables include information on the relative quality of a team revealed in previous games, measured by the difference in games won by a team and its opponent during the regular season (Difference in wins), whether a team plays at home or away (Home game), and whether a team also played the day before (Consecutive game day). The latter variable is included to capture possible modifications to the strategy due to a somewhat exhausted team. Nutting (2010) finds that NBA teams with fewer days since their last game produce fewer wins. We also include team dummies in our analysis.

It is important to note that we have two observations per game (one for each team). Strictly speaking, the two observations of a particular game are not independent from each other. For example, examining an observation of a team trailing by ten points automatically causes another observation for the opponent being ten points ahead. We checked that there is hardly any correlation for the extent of risk-taking and for the change in the risk-taking behavior between teams in a game. (5) We therefore assume independent decisions in regard to the risk strategies and make use of both observations per game while controlling for the point difference as well as other possible determinants of risk-taking and winning probability. The results do not change if we only take trailing teams into consideration. Table 3 provides an overview of the descriptive statistics of our variables.

IV. RESULTS

As a first result, it can be established that both teams increase their risk during the fourth quarter. Figure 2 illustrates that teams that are behind go for more three-point attempts, especially in the last 3 minutes, whereas the leading teams increase their risk-taking only slightly. This result is even more apparent if give-ups are excluded. The increase in risk-taking by the trailing teams is in line with our first hypothesis. However, the teams that were ahead before the last quarter also increase the fraction of three-point shots significantly in the last 12 minutes of the game (t-test on two dependent samples; p < .001 for both teams behind and ahead). (6)

In our multivariate analysis, we test whether the increase in risk is more pronounced for teams that are behind by larger margins, that is, whether the risk at the end of the game is decreasing in the intermediate point difference. The dependent variable is the fraction of three-point shots during the remaining time slots (see Table 4).

The ordinary least squares (OLS) regression analysis reveals that the intermediate point difference (which is negative for the trailing team) has a significant negative impact on risk-taking. (7) The more a team is behind, the more three-point attempts are taken by that team. The result is robust across the different points of time and the coefficient rather increases as it gets closer to the end of the game. A coefficient of 0.1 indicates an increase in the fraction of three-point attempts in the amount of 1 percentage point for every ten points of deficit. (8)

As expected, the risk-taking in the first three quarters, as a measure of the initial strategic orientation of the game, is highly and significantly positively related to the risk-taking behavior in the last minutes of the game. There is no significant effect to be found for any of the other control variables, such as playing at home, the overall performance difference of teams, or playing the day before. There is also no increased risk-taking in the last minutes for playoff games in general.

Table 4 indicates statistically significant results for point difference, but the effect is rather small. However, this may very well be due to the fact that give-ups are still included in Table 4. Once we consider the subsamples in Table 5, a more distinct picture emerges. As expected, the influence of the point difference on the three-point shots becomes larger when give-ups are excluded. Furthermore, the correlation becomes stronger depending on how many minutes are left to be played, and how strong the incentives to win are. Thus, in playoff games with 3 minutes left, each additional point increasing the deficit raises the fraction of three-point attempts by 1.6 percentage points if the game is not yet decided. The results are therefore not only highly statistically significant, but also meaningful in terms of a noticeable change in behavior. (9)

   Overall, the findings show clear evidence for our
   hypothesis. Teams that are behind react by going
   for more risk in terms of an increased fraction of
   three-point attempts.


One may expect that teams choose risk strategies contingent upon a number of moderating factors. Introducing an additional interaction term between Home game and Point difference, we find that teams playing at home react more sensitively to intermediate results. Additionally, we examine whether ex ante favorites and underdogs behave differently. We can control for relative team quality by the Difference in wins during the regular season. By doing this, we attach the interaction term between Difference in wins and Point difference in the base regression shown in Table 4. We find that favorites alter the fraction of three-point shots to a higher extent when trailing. (10) One explanation for this may be that utility functions differ between underdogs and favorites with respect to the point difference. While underdogs may benefit to some extent from losing a game by a small margin, for example, because of praise from the press, this might not be the case for favorites. Then the higher marginal return of catching up for favorites may lead to greater risk adoption.

As formulated in the corollary of our hypothesis, we now analyze whether increasing risk is beneficial. Figure 3 shows how the probability of winning is affected by the change in risk-taking behavior classified by different sizes of point differences and for different lengths of the time remaining in the game. The riskiness of a strategy is regarded as larger (reduced) if the proportion of three-point shots is increased (lowered) by at least 5 percentage points compared to the first three quarters. A variation of less than 5 percentage points is treated as steady. As Figure 3 shows, the increase of risk-taking may be appropriate only for very high deficits because the probability of winning a game is higher for a reduced risk in most other cases. For example, 42% of the teams trailing by four to seven points after the third quarter and reducing the risk in the fourth quarter still manage to win. This is only the case for 21% of the corresponding teams that increase the fraction of three-point shots. However, more than half of the teams choose a risk-enhancing strategy in this situation, while only one out of six teams reduces risk.

The results of a binary logit estimation for winning the game are presented in Table 6. Obviously, the point difference has a highly significant and strong impact: the larger the lead is, the larger the winning probability is as well. The change in risk-taking (dRisk) is measured as the difference in the fraction of three-point shots during the analyzed time period (last 12, 9, 6, or 3, respectively) and the fraction in the first three quarters. (11) The results reveal a clearly negative and highly significant influence of a risk enlargement on the winning probability. For example, based on model (1) an increase in the fraction of three-point shots by 5 percentage points in the last quarter reduces the odds of winning by about 14% on average.

The difference in wins throughout the season has, as expected, a positive and significant impact on the winning probability, while there is no effect of the team having played the day before. The influence of playing at home on the probability of winning is positive and significant.

Additional estimations that are not presented in tables show that the results are robust to considering only trailing teams and other subsamples. There is almost no difference if give-ups are excluded. The negative effect of increasing the risk becomes even stronger for playoff teams or playoff games, while the impact becomes smaller when fewer minutes are left.

Because the effect of a change in risk-taking may depend on the point difference, interaction terms for point difference categories and the change of risk are taken into account in models (2) to (5). The point difference is considered in the form of categories and the category "Even" acts as the reference point)2 First, it can be seen that the categories for point differences have the expected and significant impacts on the probability of winning. The main effects of increasing the risk are very similar to the result in the regression without the interaction terms. For moderate point differences, there are no significant interaction effects. Only if a team lies clearly behind, does an increased risk-taking action in the last 12 minutes (and with limited extent in the last 9 minutes) raise the probability of winning. This corresponds to the few cases of the higher likelihoods of winning for teams lying considerably behind and increasing their risk, as shown in Figure 3. Increasing risk jeopardizes the chances of winning for teams that are ahead. Except for the case in which a team is clearly leading and there are 3 minutes left, the interaction effects become weaker when fewer minutes are remaining. All of the qualitative effects remain stable for other sensible thresholds. Focusing on the subsample without give-ups does not affect the results. Additionally, we calculated marginal effects for a change in risk for each category of point differences considering all other variables at their means. While these effects differ depending on the situation, they turn out to be significantly negative in most cases. For example, being slightly behind and having 12 minutes to play, an increase of the fraction of three-point shots by 10 percentage points reduces the winning probability by 4.7 percentage points.

Building on the result that favorites enlarge their risks to a higher extent when they are trailing, we analyze if this reaction pays off. Integrating an interaction term between risk adoption and difference in wins in the base regression shown in Table 6 reveals no significant results. Trailing favorites suffer in terms of decreasing their winning probability; the same is true for underdogs.

   To sum up, our results show that increasing risk is not
   beneficial in most cases. Therefore, corollary (a) has
   to be rejected. Corollary (b)is supported in the
   regressions for situations with 12 and 9 minutes left,
   in terms of a positive interaction effect between point
   difference categories and changes in risk-taking.


At first glance, one may think that an increased fraction of three-point shots leads to rash attempts and reduces the shooting percentage; however, in general, this is not the case. The change in the fraction of three-point shots between the first three quarters and the last quarter with the change in shooting percentage even shows a slightly positive correlation. However, increasing the fraction of three-point shots does not only increase the number of the team's points but also increases those of the opponent (e.g., because of fast breaks subsequent to defensive rebounds). (13)

This raises the question of why teams react by enlarging risk even though this strategy is usually not appropriate. A possible explanation may be that teams systematically overestimate the benefit of increasing the fraction of three-point shots in the sense of the base-rate fallacy. They may give too much weight to the new information (point difference) and too little to the basic information that resulted in the originally chosen risk strategy. Besides, it may well be the case that public pressure from spectators, or the owner of the clubs, leads to a change in the risk strategy. If a deficit becomes obvious, retaining the strategy might be interpreted as ignoring the situation and not trying everything to catch up in the remaining time. Anticipating this, coaches may even take more risk to signal a kind of proactive behavior while being aware of a decreasing winning probability. (14)

In social psychology these decision errors are labeled as omission and commission biases. Spranca, Minsk, and Baron (1991) examine their general incidence and Moskowitz and Wertheim (2011, 7-30) applied the argument to specific situations in sports. The commission bias is defined as a tendency toward action rather than inaction (in our case: switching to a riskier strategy), although inaction would be beneficial. This expectancy is likely to be more relevant for teams playing at home, for ex ante favorites, and for important playoff games. Indeed, our results show that increased risk-taking is even more pronounced for these cases.

V. CONCLUSION

The strategy of individuals or teams with respect to risk-taking is highly relevant in many tournament-like situations when subjects are assessed in relation to one another. Making use of basketball data, we show that protagonists indeed adjust their risk strategy based on new information concerning the intermediate score in a game. Trailing teams increase the fraction of three-point shots in the last minutes of a game. Surprisingly, this reaction is not beneficial in terms of an increasing probability of winning in most cases. Therefore, trailing teams take too much risk too early in a game. This decision error may first be interpreted as the irrational behavior of the coaches who are in charge of the strategy.

Coaches may underestimate the risk of enabling more (easy) points for the opponent by fast breaks resulting from more of their own three-point shots. Even if the shot percentage remains constant, there are more unsuccessful attempts, and therefore more chances for fast breaks (triggered by defensive rebounds), which are connected with a higher success rate. As argued above, coaches may also be subject to a commission bias in the sense that expectations of spectators or club owners press them to signal proactive behavior.

Further research may address the question to what extent these phenomena affect behavior under different incentive structures and organizational systems in more detail. It could also be interesting to explore whether certain individuals or groups take the consequences of their risk-taking behavior sufficiently into account. There might be differences between different tournament or contest situations such as promotion decisions, sales contests, contests between R&D teams and portfolio managers, or competition about market shares in an oligopoly structure. In contrast to a single basketball game, some form of path dependency plays a more prominent role in many of these situations. For instance, investing in product A makes it difficult to switch to product B for the next period. Therefore, the commission bias will most likely be more relevant in non-path-dependent situations, which could be explored in future research.

Risk-taking does not have a strategic component in the basketball case which exceeds a single game. The coaches can determine a modified risk behavior from game to game. Chatterjee et al. (2003) argue that for private firms, however, continuous risk-taking may develop skills at these choices and, therefore, foster competitive advantages. By contrast, ad hoc risk-taking is argued to threaten firm survival. Applying this statement to the NBA case leads to the conclusion that the skills for three-point shots matter. If there are players with these skills on the team, a continuously high percentage of three-point shots is beneficial, but if there are no players with these skills on the team, an ad hoc increase of risk usually backfires.

Whether individuals take risks cautiously or distinctively depends on incentives for achieving very good results or avoiding very bad ones. A team trailing in a basketball game only benefits from a very good result in the remaining time that is sufficient to catch up and win the game. Stock option plans or the limited liability of managers induce similar effects. By contrast, managers may want to ensure their own jobs by avoiding very bad results. This is in line with experimental evidence by Gaba and Kalra (1999), who find that risk-taking behavior in tournaments is decreasing in the number of winners for a given number of participants.

The three-point rule was introduced by the NBA in the season 1979/1980 to make the game more interesting, and indeed we find that, for the case of a team trailing by a considerable margin, the three-point shot is an appropriate tool to catch up, although this strategy is used too early in many cases. Implementing an appropriate incentive system for risk-taking is a huge challenge for a principal, and the systems are supposed to differ substantially between portfolio managers and R&D teams, for instance. On the basis of our results, club owners (who are aware of the decision errors) may adopt incentives for coaches and/or players by introducing a fine for defeats with many points, for instance.

Future research may compare risk-taking behavior and its consequences under different incentive structures systematically.

doi: 10.1111/j.1465-7295.2012.00499.x

ABBREVIATIONS

NBA: National Basketball Association

OLS: Ordinary Least Squares

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(1.) Theoretical considerations (Gilpatric 2009) and experimental evidence (Gaba and Kalra 1999) indicate that the amount of risk-taking depends on the prize structure in rank-order tournaments in general.

(2.) Garicano and Palacios-Huerta (2005) also focus on substitutions next to fouls in soccer games, but interpret substitutions of forwards by a defender as sabotage rather than risk taking.

(3.) In some theoretical papers, the interaction between effort and risk choice is analyzed (Krakel 2008). In this study, we assume the effort level to be exogenous and concentrate on risk-taking behavior. Casas-Arce and Martinez-Jerez (2009) explore the effects of intermediate results on effort in detail.

(4.) Because of the availability of comprehensive data, the NBA case has already been used in previous studies on different topics such as the escalation of commitment effects and sunk costs (Camerer and Weber 1999; Staw and Hoang 1995) or compensation and discrimination (Bodvarsson and Brastow 1998; Hoang and Rascher 1999).

(5.) Note that we abstain from examining possible changes in defensive behavior because of a lack of data.

(6.) One may speculate that, in some situations, it is demanding for teams to push the ball near the basket. In this case, effort costs may increase for two-point attempts when players are exhausted at the end of the game. Therefore, a general increase in three-point shots may occur.

(7.) Except for 3 minutes left, there are only a few observations with fractions of three-point shots of 0% and 100%. The results are robust for a corresponding Tobit model. The results are also robust for another model with the single shot as the unit of observation. A logit model (three-point shot = 1) reveals the negative effect of point difference on the probability of a three-point shot for the fourth quarter, which is controlled for the minute of the shot next to the other independent variables. The detailed results are provided by the authors upon request.

(8.) Alternative specifications with several dummy variables for point difference categories confirm the results in general. The negative linear relation changes for high leads, however. This category, though, includes several observations in which the trailing team has no realistic chance of catching up.

(9.) As an alternative specification, we also included the square of point difference to control for a possible nonlinear effect. It shows an inverted U-shaped relationship between point difference and the fraction of three-point shots. As the slope is monotonically decreasing in the relevant interval of point differences, we continue with the linear specification. Additionally, we conducted a pooled regression of all of the points in time in all games to ensure that no time effect is skewing the results for the separate regressions. Because we obtained very similar coefficients, we continue with separate analyses for the different points in time.

(10.) Detailed regression results regarding the moderating effects of Home game and Difference in wins are provided by the authors upon request.

(11.) Calculating the risk with the traction of three-point shots in the first half as baseline (instead of the first three quarters) does not affect the results.

(12.) A difference of less than four points is attached to the category "even." We characterize a difference of more than six (8, 10, and 12) points 3 (6, 9, and 12) minutes before the end of a game as a severe difference. Point differences in between are labeled as "slightly behind" and "slightly ahead," respectively.

(13.) The detailed results of a regression of the change in risk-taking of the team and its opponent on the points made in the last minutes are provided by the authors upon request.

(14.) If this is true, the effect is similar to the phenomenon of coaches in professional team sports being dismissed quite often (due to public pressure), even though the benefits of such a proceeding are not evidenced (see Wirl and Sagmeister 2008).

CHRISTIAN GRUND, JAN HOCKER and STEFAN ZIMMERMANN *

* Many thanks to two anonymous referees and especially to Jeff Borland for helpful comments.

Grund: Department of Business and Economics, RWTH Aachen University, Templergraben 64, Aachen 52056, Germany. Phone 492418093355, Fax 492418093356, E-mail christian.grund@rwth-aachen.de

Hocker: Department of Business and Economics, University of Wuerzburg, Sanderring 2, Wuerzburg 97070, Germany. Phone 499313180222, Fax 499313182105, E-mail j.hoecker@uni-wuerzburg.de

Zimmermann: Department of Business and Economics, University of Wuerzburg, Sanderring 2, Wuerzburg 97070, Germany. Phone 499313182754, Fax 499313182105, E-mail stefan.zimmermann@uni-wuerzburg.de

TABLE 1
Distribution, Means, and Standard Deviation
of Points for Two-Point and Three-Point Shots

Points        Two-Point   Three-Point
              Shots       Shots

0             0.462 (a)   0.633 (e)
1             0.032 (b)   0.001 (f)
2             0.481 (c)   0.002 (g)
3             0.025 (d)   0.362 (h)
4             --          0.002 (i)
Mean          1.069       1.099
Standard      1.019       1.446
  deviation

Note: Numbers for the NBA 2007/2008
season. The above numbers are
calculated by adding up the
probabilities for all possible ways to
reach a certain number of points.

(a) The player missed his two-point shot without
getting fouled (0.457), or he got
fouled but missed both free throws
(0.005).

(b) He got fouled and made one of two
free throws.

(c) He made a two-point shot without
getting fouled (0.424), or made it
despite getting fouled but missed the
additional free throw (0.008), or he got
fouled, missed the shot but made two of
two free throws (0.049).

(d) He made his two-point shot despite
getting fouled and made the additional
free throw.

(e) He missed a three-point shot without
getting fouled (0.633), or he got
fouled but missed all three free throws
(0.00007).

(f) He got fouled during a three-point
shot, missed it, and made one of three
free throws.

(g) He got fouled during a three-point
shot, missed it, and made two of three
free throws.

(h) He made a three-point shot without
getting fouled (0.360), or made a
three-point shot despite getting fouled
but missed the additional free throw
(0.00049), or he got  fouled, missed the
shot but made three of three free throws
(0.002).

(i) He made a three-point shot despite
getting fouled and made the additional
free throw.

Source: www.nba.com.

TABLE 2
Number of Observations

                              Give-Ups Excluded

                 All    t = 12   t = 9   t = 6   t = 3

All             2,632   2,472    2,334   2,082   1,634
Playoff teams   1,484   1,384    1,312   1,158     904
Playoff games     172     162      154     130     100

Note: t = x: No give-up x minutes before the end of the
game. Observation is excluded if one team trails by 24 or
more points for t = 12, 21 or more points for t = 9, 17
or more points for t = 6, and 12 or more points for t = 3.

TABLE 3
Descriptive Statistics (Whole Sample)

               N           Mean     SD      Min.      Max.

Risk [48-12]   2,632       20.83    7.585     1.587    49.09
Risk [12-0]    2,632       26.11    12.36     0        70.00
Risk [9-0]     2,632       26.89    13.72     0        80.00
Risk [6-01     2,632       27.94    16.92     0        87.50
Risk [3-0]     2,631 (a)   29.85    23.33     0       100.0
Point
  difference
  [12]         2,632        0       12.81   -45        45
Point
  difference
  [9]          2,632        0       13.24   -47        47
Point
  difference
  [6]          2,632        0       13.64   -48        48
Point
  difference
  [3]          2,632        0       14.15   -49        49
Home game      2,632        0.500   0.500     0         1
Consecutive
  game day     2,632        0.220   0.414     0         1
Difference
  in wins      2,632        0.000   19.20   -51        51

Note: Risk [m-n] stands for the fraction of three-point
shots during the period of time from m minutes to play until
n  minutes to play. Point difference [t] represents the
team's score minus the opponent's score with t minutes left
to play.

(a) There is one team that did not shoot at all in the last
3 minutes of one game, so the risk figure cannot be
determined.

TABLE 4
Determinants of Risk-Taking

                  (1) Risk        (2) Risk       (3) Risk    (4) Risk
                   [12-0]           [9-0]         [6-0]       [3-0]

Home game          -0.256          -0.139       -0.586       -1.717 *
                   (0.466)         (0.524)      (0.669)      (0.969)
Difference          0.001          -0.006       -0.034       -0.024
  in wins          (0.018)         (0.020)      (0.025)      (0.037)
Playoff game       -0.030          -0.257       -0.074       -1.390
                   (1.061)         (1.159)      (1.5185)     (2.066)
Consecutive        -0.038          -0.025       -0.444       -1.259
  game day         (0.574)         (0.006)      (0.777)      (1.128)
Risk [48-12]        0.271 ***       0.288 ***   0.239 ***    0.135 *
                   (0.037)         (0.041)      (0.051)      (0.073)
Point              -0.064 ***
  difference       (0.020)
  [12]
Point                              -0.094 ***
  difference                       (0.022)
  [9]
Point                                           -0.113 ***
  difference                                    (0.028)
  [6]
Point                                                        -0.100
  difference                                                 (0.039)
  [3]

Team dummies      Yes             Yes           Yes          Yes
Constant           15.321 ***      15.410 ***   16.965 ***   21.053
                   (1.332)         (1.426)      (1.729)      (2.456)
Observations    2,632           2,632           2,632        2,631 (a)
[R.sup.2]           0.16            0.14        0.10         0.05

Notes: Robust standard errors in parentheses. Risk [m - n]
stands for the fraction of three-point shots during the
period of time from m minutes to play until n minutes to
play. Point difference [t] represents the team's score minus
the opponent's score with t minutes left to play.

(a) There is one team that did not shoot at all in the last
3 minutes of one game, so the risk figure cannot be
determined.

* Significant at 10%; ** significant at 5%; *** significant
at 1%.

TABLE 5
Effect of Point Difference on Risk-Taking for Subsamples

Dependent Variable        Risk [12-0]       Risk [9-0]
Independent Variable         Point            Point
                        Difference [12]   Difference [9]

Whole      All            -0.064 ***        -0.094 ***
  sample   Give-ups       -0.116 ***        -0.183 ***
             excluded

Playoff    All            -0.076 ***        -0.110 ***
  teams    Give-ups       -0.144 ***        -0.203 ***
             excluded

Playoff    All              -0.082          -0.190 **
  games    Give-ups         -0.084          -0.273 **
             excluded

Dependent Variable        Risk [6-0]       Risk [3-0]
Independent Variable        Point            Point
                        Difference [6]   Difference [3]

Whole      All            -0.113 ***       -0.100 ***
  sample   Give-ups       -0.302 ***       -0.699 ***
             excluded

Playoff    All            -0.182 ***         -0.171
  teams    Give-ups       -0.427 ***       -0.830 ***
             excluded

Playoff    All            -0.383 ***       -0.374 **
  games    Give-ups       -0.738 ***       -1.565 ***
             excluded

Note: Least square regression with robust standard errors.
Further independent variables accounted for: Home game,
Difference in wins, Playoff game, Consecutive game day, Risk
[48-12], and team dummies (see Table 4). Risk [m-n] stands
for the fraction of three-point shots during the period of
time from m minutes to play until n minutes to play. Point
difference [t] represents the team's score minus the
opponent's score with t minutes left to play. The complete
output of the  20 regressions is available from the authors
by request.

* Significant at 10%; ** significant at 5%; *** significant
at 1%.

TABLE 6
Determinants of Success-with Interaction Terms

                             (1)                (2)
                           [t = 12]           [t = 12]

Home game                  0.458 ***          0.521 ***
                          (0.122)            (0.118)
Difference in wins         0.030 ***          0.031 ***
                          (0.004)            (0.004)
Consecutive game day       0.058              0.073
                          (0.145)            (0.139)
Point difference [t]       0.200
                          (0.008)
Behind [t]                                   -5.501 ***
                                             (0.795)
Slightly behind [t]                          -1.195 ***
                                             (0.156)
Slightly ahead [t]                            1.383 ***
                                             (0.154)
Ahead [t]                                     5.086 ***
                                             (0.638)
dRisk [t-0]               -0.029 ***         -0.025 ***
                          (0.005)            (0.008)
Behind [t] x                                  0.099 ***
  dRisk [t-0]                                (0.031)
Slightly behind [t]                          -0.005
  x dRisk [t-0]                              (0.011)
Slightly ahead [t]                           -0.011
  x dRisk [t-0]                              (0.012)
Ahead [t] x                                  -0.052 *
  dRisk [t-0]                                (0.027)
Team dummies             Yes                Yes
Constant                  -0.134             -0.250
                          (0.370)            (0.358)
Observations           2,632              2,632
Pseudo-[R.sup.2]           0.486              0.480

                             (3)                (4)
                           [t = 9]            [t = 6]

Home game                  0.297 **           0.155
                          (0.130)            (0.141)
Difference in wins         0.029 ***          0.022 ***
                          (0.005)            (0.005)
Consecutive game day      -0.009             -0.052
                          (0.151)            (0.162)
Point difference [t]
Behind [t]                -4.397 ***         -4.579 ***
                          (0.456)            (0.510)
Slightly behind [t]       -1.573 ***         -1.547 ***
                          (0.191)            (0.217)
Slightly ahead [t]         1.442 ***          1.528 ***
                          (0.168)            (0.179)
Ahead [t]                  4.416 ***          4.404 ***
                          (0.417)            (0.377)
dRisk [t-0]               -0.026 **          -0.029 ***
                          (0.007)            (0.006)
Behind [t] x               0.042 **           0.017
  dRisk [t-0]             (0.015)            (0.032)
Slightly behind [t]       -0.001             -0.008
  x dRisk [t-0]           (0.013)            (0.012)
Slightly ahead [t]         0.012              0.023 **
  x dRisk [t-0]           (0.012)            (0.010)
Ahead [t] x               -0.024             -0.002
  dRisk [t-0]             (0.018)            (0.014)
Team dummies             Yes                Yes
Constant                  -0.072             -0.055
                          (0.390)            (0.398)
Observations           2,632              2,632
Pseudo-[R.sup.2]           0.550              0.610

                             (5)
                           [t = 3]

Home game                  0.083
                          (0.162)
Difference in wins         0.016
                          (0.006)
Consecutive game day      -0.032
                          (0.190)
Point difference [t]
Behind [t]                -4.559 ***
                          (0.429)
Slightly behind [t]       -2.458 ***
                          (0.363)
Slightly ahead [t]         1.769
                          (0.227)
Ahead [t]                  4.285
                          (0.323)
dRisk [t-0]               -0.027 ***
                          (0.005)
Behind [t] x               0.003
  dRisk [t-0]             (0.020)
Slightly behind [t]        0.020
  x dRisk [t-0]           (0.015)
Slightly ahead [t]         0.016
  x dRisk [t-0]           (0.010)
Ahead [t] x                0.037
  dRisk [t-0]             (0.012)
Team dummies             Yes
Constant                  -0.009
                          (0.401)
Observations           2,631 (a)
Pseudo-[R.sup.2]           0.690

Notes: Logistic regression for game won with
robust standard errors in parentheses.
dRisk [t-0] stands for the change of
Risk [t-0] = Risk [t-0] = Risk [48-12].

(a) There is one team that did not shoot at
all in the last 3 minutes of one game, so
the risk figure cannot be determined.

* Significant at 10%; ** significant at 5%;
*** significant at 1%.
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Title Annotation:National Basketball Association
Author:Grund, Christian; Hocker, Jan; Zimmermann, Stefan
Publication:Economic Inquiry
Article Type:Abstract
Geographic Code:1USA
Date:Apr 1, 2013
Words:8720
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