Having the Lead vs. Lagging Behind : The Incentive Effect of Handicaps in Tournaments

Transcription

1 Having the Lead vs. Lagging Behind : The Incentive Effect of Handicaps in Tournaments Andreas Steinmayr a, Rudi Stracke b, and Dainis Zegners c a LMU Munich b Ministry of Finance, NRW c University of Cologne February 4, 2018 Abstract This article seeks to answer whether handicaps can restore optimal effort provision by agents in heterogeneous contests. To this end, we study data from swimming relay competitions where swimmers with heterogeneous abilities inherit leads or lags from their previous team members. Inheriting a lead or lag corresponds to a head start or handicap affecting the effective level of heterogeneity in races. Our results suggest that as predicted by theory appropriately chosen handicaps can restore optimal effort provision: In our data, swimmers who receive a lead (lag) that corresponds to a third of their ability disadvantage (advantage) exert the same effort as swimmers in balanced races. We thank Oliver Gürtler and audiences at Technical-University Munich, LMU Munich, and the LEOH Workshop on Organizational Economics in 2017 at the University of Cologne for their valuable comments. Corresponding author: Dainis Zegners, University of Cologne, Faculty of Management, Economics and Social Sciences, Digital Transformation and Value Creation Group, Universitätsstraße 91, D Köln, dainis.zegners@uni-koeln.de.

2 1 Introduction Tournaments are widely used to motivate agents in the workplace and in contexts such as sports, science and innovation, or elections. However, a disadvantage of using tournaments to motivate agents is that heterogeneity in ability may reduce agents effort provision. Lowerability agents have a decreased incentive to exert high effort as their probability to win the tournament against a higher-ability agent is low even if they exert high effort. Higher-ability agents also have a decreased incentive to exert high effort as their probability to win the tournament against a lower-ability agent is high even if they exert low effort. Therefore, the larger the heterogeneity in agents abilities in a tournament, the lower the effort they exert. This prediction originally made by Lazear and Rosen (1981) has been confirmed both in the lab (Schotter and Weigelt, 1992) and in the field (Brown, 2011; Sunde, 2009). A natural remedy to the problem of decreased effort provision in heterogeneous tournaments already studied theoretically by Lazear and Rosen is to reduce the effective level of heterogeneity by introducing head starts or handicaps to level the playing field. An example of such heterogeneity-reducing interventions are affirmative action policies that have the goal to remedy the effects of discrimination (Feinberg, 2003). Although such policies are widely used, empirical evidence on their impact on effort provision is limited. Existing studies either focus on tournaments without monetary incentives (Franke, 2012), on particular types of agents such as children or adolescents (Calsamiglia et al., 2013; Sutter et al., 2016), or are confined to lab-environments (Schotter and Weigelt, 1992; Balafoutas and Sutter, 2012; Leibbrandt et al., 2017). In this paper, we seek to fill this gap in the literature by using data from amateur and professional swimming team competitions. We study the performance of swimmers of heterogeneous abilities who inherit leads or lags from their previous team members. We observe races where inherited leads or lags partly or perfectly compensate for ability differences, as well as races where they increases the effective level of heterogeneity. Since inherited leads or lags affect the chances of swimmers to win a race, theory predicts that effort is maximized 2

3 when leads or lags compensate for ability differences between swimmers. Our results show that swimmers strongly react to ability-compensating head starts and handicaps: Both favorites and underdogs increase their effort when inherited leads or lags reduce the effective level of heterogeneity in races. In terms of size, our estimations suggest that inheriting a lead or lag that reduces heterogeneity by a third restores optimal effort provision. This suggests that heterogeneous contests with an appropriately designed handicap structure can be as effective as balanced contests in incentivizing agents to exert maximal effort. We contribute mainly to the literature studying contests as a device to motivate agents. The theoretical literature on contests was initiated by Lazear and Rosen (1981), who already proposed handicaps as a remedy to restore effort provision in heterogeneous contests. Other early theoretical contest papers are Green and Stokey (1983), Nalebuff and Stiglitz (1983) and O Keeffe et al. (1984). See Konrad (2009) for an overview of the literature on contests. Theoretical papers that deal more specifically with heterogeneity in contests are Baik (1994), Moldovanu and Sela (2001) and Szymanski and Valletti (2005). There is also a small theoretical literature on handicaps in tournaments (e.g. Denter and Sisak, 2016) and on biased or unbalanced contests, where one of the contestants has an advantage over other contestants (Imhof and Kräkel, 2016; Drugov and Ryvkin, 2017). The effects of heterogeneity in contests has also been studied empirically both in the lab (Bull et al., 1987; Schotter and Weigelt, 1992) and in the field using data from sports competitions (Brown, 2011; Sunde, 2009; Bach et al., 2009; Nieken and Stegh, 2010; Deutscher et al., 2013; Berger and Nieken, 2016; Bamieh, 2016), internal firm records (Cowgill, 2015) or data from online platforms (Boudreau et al., 2016; Lemus and Marshall, 2017). Particular close to our paper is Franke (2012), who studies handicaps in Amateur Golf competitions, and Brown and Chowdhury (2017), who study handicaps in horse-racing focusing on the incidence of sabotage. 3

4 Figure 1: The Effect of Head Starts and Handicaps on Effort by Relative Ability (Theory) Effort 1.0 upper envelope a 1 a 2 5 a 1 a 2 0 a 1 a Note: The figure plots equation (3) for a 1 a 2 { 5, 0, 5} as a function of the head start h, assuming M = 1 and c(e) = 39 8 e2. 2 Model In this section, we present a simple theoretical model to guide our empirical analysis. Consider the simplest Lazear and Rosen (1981) tournament with two agents 1 and 2 who compete for a prize M that is awarded to the agent whose performance is relatively higher. The performance y i of agent i {1; 2} reads y i (e i ) = a i + e i + u i, (1) where a i is the ability, e i is the effort, and u i is the random performance component of agent i. Consider the optimization problem of agent i who maximizes her expected pay-off Π i (e i, e j ) = p[y i (e i ) > y j (e j ) + h] M c(e i ), (2) where i j, p[ ] is the probability that the prize is awarded to agent i, h is a head start (or handicap when negative) awarded to agent j, and c( ) is the cost of effort function. Assuming that the random performance components of both agents are independent draws from a normal distribution with mean zero and that the cost function is sufficiently convex, 4

5 equilibrium efforts in the unique symmetric pure strategy Nash equilibrium satisfy e e 1 = e 2 = c [g(a 1 a 2 h) M] 1, (3) where g( ) is the density of the normal distribution, and c [ ] 1 is the inverse function of the first derivative of the cost function. As the density is symmetric and has its maximum at zero, the equilibrium effort is strictly decreasing in the absolute value of a 1 a 2 h for a given prize M. Figure 1 shows this relation graphically: Effort is highest whenever the head start or handicap just compensates for the difference in ability between both agents, and the upper envelope is a line with slope zero through e max = c [g(0) M] 1. The model provides the following predictions that we will take to the data: 1. Agents with an ability disadvantage (underdogs) increase their effort when a head start increases their probability of winning the contest Agents with an ability advantage (favorites) increase their effort when a handicap decreases their probability of winning the contest. 3. Combined effort of agents is highest whenever contests are balanced in terms of ability or handicaps and head starts fully compensate for differences in ability. Note that there is an interaction between ability differences and head starts / handicaps: Whether a head start or handicap increases or decreases agents effort depends on whether the agent is a favorite or an underdog. 3 Empirical Setting and Data We use data from amateur and professional swimming competitions collected by the website swimrankings.net that are based on information from the European Swimming Federation 1 As long as it does not overcompensate the underdog, i.e. making it the favorite. 5

6 (LEN). 2 Relay competitions in swimming are perfectly suited to analyze the effect of handicaps on effort provision for several reasons: First, the performance of swimmers in individual competition races provides us with a proxy for ability, since we know how much time any athlete needs to swim a certain distance within the same environment. Second, abilities of competitors are likely to be common knowledge among swimmers. In particular, a swimmer in a relay competition is likely to know the ability of her immediate competitors. In addition, a swimmer in a relay competition can assess the lead or lag, respectively, to competing teams before choosing effort. 3 Our data set comprises 6,193 relay races from the years In a relay competition, a team consists of four swimmers that race consecutively 4 Types of relays in our data set include 4 x 50m freestyle competitions (36 percent), 4 x 50m medley 5 competitions (34 percent), 4 x 100m medley competitions (19 percent) and 4 x 100m freestyle competitions (11 percent). For each relay competition, our data set includes the individual swimming times of each swimmer, their gender and birth year, and the length of the pool (25m or 50m). To concentrate on cases where winning a competition is the main goal of a team, we only look at final races in our analysis. We also exclude timed finals, where teams compete in separate races and the team with the fastest racing time across races is declared the overall winner. Additional to the relay competitions, we also have data on swimming times of swimmers in individual competitions in the same and other meetings throughout a year. We will use these individual swimming times to compute the ability of each swimmer within a swim style on a certain distance. To mimic most closely the set-up of the theoretical model in the previous section, we focus 2 For a detailed data description, see Neugart and Richiardi (2013), who use the same data to investigate free-riding in sequential team production settings. Another recent paper using data from swimming competitions is Jiang (2016), who studies peer effects. 3 Imhof and Kräkel (2016) distinguish between ex-ante unbalanced tournaments, where ability differences are known to agents at the start of the tournament, and ex-post unbalanced tournaments, where agents find out about ability differences at the end of the tournament. We argue that swimming relay competitions are best thought of as ex-ante unbalanced. 4 We exclude relay competitions with more than four swimmers from our analysis. 5 In a medley swimming competition, each consecutive swimmer swims the distance in a different swimming style: butterfly stroke, backstroke, breaststroke or freestyle. 6

7 only on the two leading swimmers in a race in our analysis. Distance to and the ability of the swimmer placed third will be included as controls in our regression analyses. We exclude the starting leg of each relay, i.e. the first swimmers, as neither swimmer inherits a lead or lag and it is not clear which two swimmers are leading. 3.1 Computing Effort, Ability and Handicaps Even though effort is not directly observable in our data, we observe the performance of all swimmers in the relay competition. In what follows, we compute effort using the time taken by a swimmer to complete his or her leg of the race compared to the average time of the same swimmer within the same year in individual competitions for the same swim style, same distance and pool type. To make the measure comparable across different types of races, effort is normalized by the average time of all swimmers in relay competitions in the same year, swimming the same distance within the same swim style, gender, pool type and age-group. Formally, we calculate effort for swimmer i in race k as effort i,k = average individual time i,same year relay time i,j average relay time all swimmers, same year. (4) By subtracting the relay time from the average time in individual races, our measure increases the less time a swimmer needs to complete the relay race i.e. the faster he or she swims. To compute relative ability of two swimmers competing in a relay, we calculate the difference in time taken by both swimmers to complete the same distance within the same swim style within the same meeting as the focal relay. We normalize the difference between the times taken by both swimmers again using the average time of all swimmers in relay competitions in the same year, swimming the same distance within the same swim style, gender, pool type and age-group. Formally, we calculate relative ability of swimmer i competing 7

8 against swimmer j in race k as relative ability i,j,k = individual time j,same meeting individual time i,same meeting average relay time all swimmers, same year (5) By using the average individual time within the same year excluding the current meeting to compute effort, and the individual time from the current meeting to compute relative ability, we avoid any spurious correlation between both measures e.g. induced by the form of a swimmer. The lead or lag of swimmer i competing against swimmer j in race k is computed as: lead/lag i,j,k = starting time j,k starting time i,k average relay time all swimmers, same year. (6) The starting time is the sum of the times taken by previous swimmers of the same team, therefore our measure is positive when a swimmer inherits a lead and negative when he or she inherits a lag. Using the same normalization, we also compute the lead over and the ability of the swimmer positioned third in a race, which we will include as controls in our regression analysis Outliers The distribution of our three measures for effort, relative ability and lead/lag shown in figure 2 indicates the presence of outliers in our data. Inspecting several of these outliers showed that they are likely to be the result of measurement problems such as entry errors. Therefore, we chose to truncate the data by excluding observations that are outside the 5 and 95 percentiles of the empirical distributions of the effort, relative ability and lead/lag measures. 6 We use the absolute ability but not the ability difference to the third swimmer as the ability difference would be strongly correlated with our main measure of the difference in ability between the first and the second swimmer. The reason is that a swimmer with a high (low) ability often also has an ability advantage (disadvantage) over the third swimmer. 8

9 Figure 2: Histograms of Untruncated Distributions of Effort, Relative Ability and Lead/Lag. Density effort Density Relative Ability Density Lag or Lead Figure 3: Histograms of Truncated Distributions of Effort, Relative Ability and Lead/Lag. Density effort Density Relative Ability Density Lag or Lead Such truncating of data is a common strategy for dealing with outliers and measurement error (e.g. Angrist and Krueger, 1999; Acemoglu et al., 2012). Figure 3 shows histograms of the truncated empirical distributions. The empirical distribution of the lead/lag measure appears to be more skewed, probably due to leads and lags accumulating over consecutive swimmers in a race. 3.3 Summary Statistics Table 1 shows the summary statistics for the 15,097 observations in our final data set. Effort has a positive mean of suggesting that swimmers exert higher effort when competing in a relay as compared to individual competitions. This effect, however, is driven by the fact that the second, third, and fourth swimmer in a relay start upon observing their previous team member touching the wall, which is called a flying start. This decreases their reaction 9

10 Table 1: Summary Statistics N Mean Sd Min Max Effort Relative Ability Lead or Lag Lead over 3rd Swimmer Relative Ability 3rd Swimmer Age Group Adult 7208 Below Above Total Swim Style Breaststroke 3196 Butterfly 2527 Freestyle 9374 Total Type Relay 4 x 100m Freestyle x 100m Medley x 50m Freestyle x 50m Medley 4668 Total Order Swimmer Total time as compared to starts in individual competitions, where swimmers start upon hearing the starting signal. 4 Results In this section, we present the results of our empirical analysis. We start by examining the effects of heterogeneity in relative ability, head starts and handicaps separately, ignoring their theoretically predicted interaction. In a next step, we examine their interaction by looking at the dichotomized effects, i.e. the effects of head starts and handicaps on favorites and underdogs, ignoring additional heterogeneity in these measures. Then, we examine the effects of heterogeneity in our measures. Finally, we present results from semi-parametric 10

11 estimations of the effects of relative ability and head starts, providing a clearer picture of their interaction. 4.1 Main Effects of Heterogeneity in Ability and Head Starts Figure 4 shows means of effort by relative ability (upper panel) and leads or lags (lower panel). Inspecting these graphs hints at effort being decreasing in both the heterogeneity in relative ability and the size of the lead or lag that a swimmer inherits. In both cases, effort seems to be highest close to zero i.e. in balanced contests. Table 2 shows the results of linear panel regressions including either athlete-level random or fixed effects. For 6986 out of 9908 athletes our data includes only one observation, limiting the amount of within-athlete variation and statistical power of the fixed effects estimation. Throughout the paper, we therefore prefer to present results using both random and fixed effects. As can be seen in table 2 in columns (1) and (2), heterogeneity in ability has a statistical significant negative effect on effort when included as a main effect (p < 0.01 and p < 0.1). In columns (3) and (4), we examine the effect of heterogeneity in ability separately for favorites and underdogs by including an interaction with a dummy indicating whether an athlete is a favorite. The results indicate that heterogeneity in ability has only an effect for favorites., although the result is only statistically significant in case of the random effects regression (p < 0.01). 7 Table 3 shows regression results adding the distance between competitors as a measure of the absolute size of head starts and handicaps. It has a statistically significant negative effect in all specifications (p < 0.01), indicating that a larger distance between competitors decreases effort provision. In columns (3) and (4), an interaction between the distance 7 There are papers in experimental economics showing that underdogs exert higher effort in heterogeneous contests than favorites (Bull et al., 1987). In the field, Bach et al. (2009), Berger and Nieken (2016) and Nieken and Stegh (2010) also find that favorites reduces effort when facing underdogs but not vice-versa. Results seem, however, to be mixed in literature: According to Nieken and Stegh: While Bull et al. (1987) report that in their experiment the mean effort level of disadvantaged subjects was higher than equilibrium effort, Fershtman and Gneezy (2011) show that quitting may depend on the relation of tournament incentives and social costs of quitting. Regarding our setting in hockey the social costs of reducing effort might be higher for underdogs as those teams are the presumable losers of the match. 11

12 Figure 4: Means of Effort by Relative Ability and Handicaps or Head Starts. Effort (Mean) Relative Ability Effort (Mean) Handicap (Lag) or Head Start (Lead) Notes: Error bars show 95 percent confidence bands. between competitors and whether a swimmer is leading is added, therefore measuring whether a head start has a different effect than a handicap. The coefficient on the interaction is positive and statistically significant in case of the random effects model (p < 0.05), indicating that a larger head start reduces effort provision to a lower degree than a handicap. Overall, we conclude that heterogeneity in ability and the distance between competitors has the predicted effects of decreasing effort provision. 4.2 Interaction Between Ability and Head Starts Now we turn to examining the interaction between ability, head starts and handicaps. This is the central test of the theory: Head starts should increase the effort of underdogs as they compensate for their disadvantage in ability, reducing the effective level of heterogeneity in a contest. By the same argument, handicaps should increase the effort of favorites Our analyses in the previous section, looking at ability and head starts separately, ignored this interaction. We test these predictions by examining how swimmers react to leads or lags inherited from previous team members depending on whether they compete against a faster or slower swimmer, i.e. whether they are the favorite or the underdog in a given match-up. Figure 5 shows a comparison of average effort between swimmers with ability advantages and disadvantages inheriting leads or lags. The main effects of head starts and handicaps 12

13 Table 2: Impact of Heterogeneity in Relative Ability Dependent variable: Effort (1) (2) (3) (4) Heterogeneity Ability (0.006) (0.010) (0.009) (0.015) Favorite Heterogeneity Ability (0.008) (0.015) Athlete RE X X Athlete FE X X Number Athletes R 2 Overall Observations Clustered standard errors in parentheses p < 0.1, p < 0.05, p < 0.01 Note: This table shows results from linear random and fixed effects regressions. Each individual observation is an athlete observed within a race. Additional controls include the distance to the swimmer placed third, the ability of the third swimmer, athletes age group, the swim style, the type of relay and the order of the swimmer. Only swimmers placed first and second in a relay are included. Standard errors are clustered on the athlete-level. Table 3: Impact of Head starts and Handicaps Dependent variable: Effort (1) (2) (3) (4) Heterogeneity Ability (0.009) (0.015) (0.009) (0.015) Favorite Heterogeneity Ability (0.008) (0.015) (0.008) (0.015) Distance btw. Competitors (0.005) (0.007) (0.006) (0.010) Head Start (Lead) Distance btw. Competitors (0.006) (0.010) Athlete RE X X Athlete FE X X Number Athletes R 2 Overall Observations Clustered standard errors in parentheses p < 0.1, p < 0.05, p < 0.01 Note: This table shows results from linear random and fixed effects regressions. Each individual observation is an athlete observed within a race. Additional controls include the distance to the swimmer placed third, the ability of the third swimmer, athletes age group, the swim style, the type of relay and the order of the swimmer. Only swimmers placed first and second in a relay are included. Standard errors are clustered on the athlete-level. 13

14 are as predicted: Underdogs increase their effort by 0.39 units (p < 0.01) when inheriting a lead as compared to underdogs who inherit a lag. Favorites increase their effort by 0.20 units (p < 0.01) when inheriting a lag as compared to favorites inheriting a lead. As all measures are normalized using the average relay time of all swimmers, and leading swimmers lead on average by 5.8 percent of the average relay time, the effect sizes correspond to 4 to 7 percent of an average lead, a sizable effect. Examining the same comparisons within a regression analysis in table 4 confirms these main effects. Figure 6 shows means of effort plotted against how large a head start or handicap of a favorite or underdogs is. Visually inspecting these graphs also indicates that effort of favorites is higher when they inherit a handicap in the form of lag, and effort of underdogs is higher when they inherit a head start in the form of a lead. Table 5 shows regression results of the impact of handicaps on effort for favorites only. The estimated coefficient on the dummy indicating whether a favorite inherits a handicap in the form of a lag is positive and statistically significant (p < 0.05) in columns (1) and (2). This, however, is no longer the case when adding a continuous measure of the distance between competitors and an interaction between this measure and whether a swimmer is lagging in columns (3) and (4). Here, the results are not conclusive, as both the dummy and the interaction are statistically insignificant. Table 6 shows regression results of the impact of head starts on effort for underdogs only. The estimated coefficient on a dummy indicating whether an underdog is leading is positive and statistically significant (p < 0.01) in the random effects model (column 1) although it is not statistically significant in the fixed effects model (column 2). The coefficient remains statistically significant (p < 0.05) when adding the continuous measure of the distance between competitors into the random effects model in column (3). In the fixed effects model in column (4), both the main effect and the interaction between the distance between competitors remain statistically insignificant. Overall, we find evidence that there is an effect of head starts and handicaps on effort as 14

15 Figure 5: Effort for Favorites and Underdogs by Head Start (Lead) and Handicap (Lags) Mean of Effort Underdog Favorite Handicap (Lag) Head Start (Lead) Notes: Error bars show 95 percent confidence intervals. predicted by the theory, however, only in the form of a main effect of leading or lagging, but not in the form of differential impacts of the continuous measures of the lead or lag. 4.3 Semi-Parametric Estimation In this section, we turn to a semi-parametric estimation of the effects of heterogeneity in ability and head starts and handicaps. This is a more appropriate approach compared to the previous parametric regression analyses as it allows for a more flexible estimation of the interaction between ability, head starts and handicaps. According to the model, head starts and handicaps increase effort when they reduce the effective level of heterogeneity in a contest but decrease effort when they increase heterogeneity. However, when this will be actually the case in our empirical setting depends also on how swimmers react to ability differences, which also has to be estimated from the data. 15

16 Table 4: Impact of Head Starts and Handicaps - Dichotomous Effects Dependent variable: Effort (1) (2) (3) (4) Favorite (0.000) (0.001) (0.001) (0.001) Handicap x Favorite (0.001) (0.001) Headstart x Underdog (0.001) (0.001) Athlete RE X X Athlete FE X X Number Athletes R 2 Overall Observations Clustered standard errors in parentheses p < 0.1, p < 0.05, p < 0.01 Note: This table shows results from linear random and fixed effects regressions. Each individual observation is an athlete observed within a race. Additional controls include the distance to the swimmer placed third, the ability of the third swimmer, athletes age group, the swim style, the type of relay and the order of the swimmer. Only swimmers placed first and second in a relay are included. Standard errors are clustered on the athlete-level. Figure 6: Means of Effort by Handicaps or Head Starts - Favorites and Underdogs (a) Favorites (b) Underdogs Effort (Mean) - Only Favorites Handicap (Lag) or Head Start (Lead) Effort (Mean) - Only Underdogs Handicap (Lag) or Head Start (Lead) Notes: Error bars show 95 percent confidence intervals. 16

17 Table 5: Impact of Heterogeneity in Ability and Handicaps - Favorites Only Dependent variable: Effort (1) (2) (3) (4) Heterogeneity Ability (0.008) (0.014) (0.008) (0.014) Handicap (Lag) (0.001) (0.001) (0.001) (0.001) Distance btw. Competitors (0.007) (0.012) Handicap (Lag)=1 Distance btw. Competitors (0.012) (0.022) Athlete RE X X Athlete FE X X Number Athletes R 2 Overall Observations Clustered standard errors in parentheses p < 0.1, p < 0.05, p < 0.01 Note: This table shows results from linear random and fixed effects regressions. Each individual observation is an athlete observed within a race. Additional controls include the distance to the swimmer placed third, the ability of the third swimmer, athletes age group, the swim style, the type of relay and the order of the swimmer. Only swimmers placed first and second in a relay are included. Standard errors are clustered on the athlete-level. Table 6: Impact of Heterogeneity in Ability and Head Starts - Underdogs Only Dependent variable: Effort (1) (2) (3) (4) Heterogeneity Ability (0.011) (0.024) (0.011) (0.024) Head Start (Lead) (0.001) (0.001) (0.001) (0.002) Distance btw. Competitors (0.009) (0.020) Head Start (Lead) Distance btw. Competitors (0.014) (0.027) Athlete RE X X Athlete FE X X Number Athletes R 2 Overall Observations Clustered standard errors in parentheses p < 0.1, p < 0.05, p <

18 To account for this interaction and to estimate it flexibly, we use the following semiparametric approach: In a first step, we estimate a parametric linear regression model using the same specification and control variables as in the previous section with athlete-level fixed effects, but without including ability and leads or lags as dependent variables. In a next step, we use the residuals from this model to estimate the relationship between effort and relative ability, and the lead/lag of swimmer i competing against swimmer j in race k as residual(effort i,j,k ) = f(relative ability i,j,k, lead/lag i,j,k ) (7) where f(.) is a flexible function estimated non-parametrically using the residuals. Figure 7 shows a contour plot of the surface of f(.) estimated using local polynomial regression smoothing (Cleveland and Loader, 1996). In general, the estimated function appears to be concave along the lag/lead dimension, with effort being maximal for a lead/lag between and The green line tracing the maxima of effort support our previous results and show that favorites exert their highest effort when receiving a lag, while underdogs exert their highest effort when receiving a lead. The slope of the green line indicates that receiving a handicap (head start) that corresponds to approximately a third of the ability advantage (disadvantage) restores maximum effort provision of swimmers in our sample. Although the estimated function appears to be double-peaked with two local maxima close to the extremes of the relative ability and lag/lead distribution, approximately 50 percent of peaks within 100 bootstrap samples of the data where located close to origin where contests are balanced in terms of relative ability and head starts / handicaps. Using the estimated function f(.), we can also compute the contest that maximizes combined effort of the favorite and the underdog. Figure 8 shows the contour plot of the surface of the estimated combined effort function as a function of the relative ability disadvantage of the underdog and the head start / handicap given to the underdog. Combined effort in general is maximized when the underdog receives a handicap corresponding to approximately half of his or her ability disadvantage. Although the global peak of combined effort lies close 18

19 Figure 7: Semi-parametrically Estimated Impact of Ability and Head Starts on Effort Relative Ability Maximum Effort for Fixed Ability Global Peak (Actual Data) Global Peak (Bootstrap Samples) Lag or Lead Notes: This figure shows a contour plot of the surface of semi-parametrically estimated impact of relative ability and head starts or handicaps on the effort of swimmers. The function is estimated using local polynomial regression smoothing (LOESS) on the residuals from a linear regression controlling for age-group, swim-style, relay-type, starting order, distance to and ability of third swimmer and athlete-level fixed effects. For each point on the (x, y) grid where the function is estimated, 90 percent of observations in its neighborhood are utilized for estimation using tricubic weighting The blue dotted line connects the maxima of the function for every fixed value of relative ability. Bootstrap samples for computing alternative peaks where simulated by resampling the data and re-estimating both the linear regression model and the non-parametric estimation on the residuals. 19

20 to the edge of the empirical distribution where contests are very unbalanced, about 50 percent of global peaks in a sample of 100 bootstrap samples lie close to the origin where contests are balanced. A conservative interpretation of this result is that an unbalanced contest is equally likely to induce optimal combined effort, given that head starts and handicaps are appropriately chosen to compensate for ability differences. 5 Conclusion Our empirical results show that handicaps restore incentives in heterogeneous tournaments as theory predicts. In our sample, giving a disadvantaged swimmer a head start that compensates for his or her ability differences restores maximal effort. In terms of optimizing combined effort of contestants, we find that unbalanced contests are equally likely to induce optimal combined effort, given that head starts and handicaps are appropriately chosen to compensate for ability differences between contestants. 20

21 Figure 8: Effort-Maximizing Contests Relative Ability of Underdog Global Peak (Actual Data) Global Peak (Bootstrap Samples) Lag or Lead of Underdog Notes: This figure shows a contour plot of the surface of the combined effort of the underdog and favorite as a function of relative ability of the underdog and the head start / handicap she receives. Effort is estimated using the same semi-parametric estimation as in figure 7. Bootstrap samples to compute alternative peaks where also resampled using the same procedure as previously. 21

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