Learning When It Counts: Evidence from. Professional Bowling Tournaments

Transcription

1 Learning When It Counts: Evidence from Professional Bowling Tournaments Choon Sung Lim June 2017 Abstract While demand for high-skill workers has been increasing, high-skill jobs often require workers to make a decision, facing uncertainty underlying their tasks. Highly skilled professionals have deep insights to pick up meaningful patterns of information. Therefore, if they are in an environment that allows them to learn additional information from co-workers, their productivity can improve. In this paper, I examine the productivity effects of learning among high-skill peers about uncertain conditions underlying their tasks with variations in the space of ideas, exploiting a unique, novel dataset from professional bowling competitions. Specifically, a bowler learns about lane conditions in part by watching his competitor bowl on the same lane. A right-handed bowler learns more relevant (to his task) information from competing with another right-hander than with a left-hander, as the used part of the lanes (the proximate space of ideas) varies with handedness. I compare the probabilities of bowling a strike of bowlers matched with like-handed competitors versus opposite-handed competitors. I find a large impact of the same ideas space on learning, e.g, being paired with a like-handed bowler increases strike probability by 14 percentage points. This finding adds evidence for the existence of peer effects in high-skill jobs. I also show that learning curves exist only when bowlers are in same-handed match-ups, by examining how these differences change from one frame to the next over a game. Another calculation is determining how much total scores could be increased by pairing bowlers to raise the proximity in the space of ideas. These results are suggestive of how much workplaces might increase productivity by optimally pairing workers based on the proximity of the space of ideas. JEL Classification: D83, J24, J44, L83, M54 Keywords: Peer Effects, Space of Ideas, Learning, Professional Bowling Research Department, Bank of Korea. Mailing Address: 39 Namdaemun-ro, Jung-gu, Seoul 04531, South Korea; chnsng.lim@gmail.com. I am deeply grateful to Andrew Beauchamp, Arthur Lewbel and Mathis Wagner for their guidance and advice. I also thank Anant Nyshadham, Tracy Regan, Utku Unver and participants in the Boston College Dissertation Seminar and Applied Micro Workshop for valuable comments and suggestions. I am solely responsible for any mistakes or omissions.

2 1 Introduction Technology progress has driven the job polarization, the phenomenon in which the demands for high- and low-skill workers are rising, while the labor demand in middle-class jobs declines (Autor, Katz, and Kearney, 2006; Goos and Manning, 2007; Goos, Manning, and Salomons, 2009). The high-skill jobs intensively require workers to work on abstract tasks 1 such as problem solving and coordination. In such a task, workers often need to make a decision, facing uncertainty underlying their tasks. If they had more or better information about the uncertain conditions related to their tasks, they would be more productive. 2 Experts notice meaningful patterns of information that are not noticed by novices. (National Research Council, 2000) So the marginal benefit of additional information on the productivity can be increasing with their skills. If such a highly-skilled worker can learn additional information from co-workers and therefore improve his productivity, it would be beneficial for the economy as well as firms or himself. For example, internalizing such a learning effect from peer workers in management can help improve the aggregate productivity of the firm, given the same labor demand level. Reflecting the importance, a large body of research explores the peer learning in the highskill jobs such as scientists (Azoulay, Graff Zivin, and Wang, 2010; Waldinger, 2012), teachers (Jackson and Bruegmann, 2009) 3 and professional golfers (Guryan, Kroft, and Notowidigdo, 2009). Unlike the confirming evidence about the peer effects on motivation in low-skill jobs, 4 1 See Levy, Murnane et al. (2003) and Autor, Katz, and Kearney (2006). 2 For example, doctors can make a better treatment if they have more information on the conditions of a patient, and professional golfers can perform better if they know more about green conditions. 3 Jackson and Bruegmann (2009) find peer effects among elementary school teachers. They show that higher ability of peer teachers contributes to improve students math and reading scores beyond the value added by their teacher and the size of value-added of replacing a peer teacher with a higher productive teacher is averagely one fifth to one tenth of the the value of replacing student s teacher. 4 For low-skill jobs, empirical evidence is well documented. Falk and Ichino (2006) experiment the peer effect in the task of stuffing letters into envelopes in laboratory settings. They show the evidence of peer effects that an individual raises one s effort by 1.4% in the response to 10% increase in peer s output. Beyond a laboratory, Mas and Moretti (2009) study about the scanning job in grocery stores and find peer effects that 10% increase in peers average productivity helps a worker to put 1.7% more effort. Bandiera, Barankay, and Rasul (2010) focus on fruit pickers and find that there are peer effects between friend workers such that workers who are more (less) productive than their friends are willing to reduce (increase) 10% of their efforts. 1

3 however, the empirical evidence about the peer learning effect, mainly explored in high-skill jobs (Cornelissen, Dustmann, and Schönberg, 2017), is not convincing. One of the main challenges in this literature is the limited data. Peer effect itself is hard to empirically identify. 5 In addition, economists can usually observe little information about peers. As a result, most studies should rely on the broad categories to identify the relevant peers. In this paper, I examine the productivity effects of learning from high-skill peers (competitors in the context of the paper) about uncertain conditions underlying their tasks, exploiting a unique, novel dataset collected from a professional bowling competition (the U.S. Open). To overcome the shortcomings of the data from the workplace, especially selection bias, there are already many papers using professional sports data such as golf (Guryan, Kroft, and Notowidigdo, 2009; Brown, 2011), baseball (Gould and Winter, 2009) or basketball (Kendall, 2003; Arcidiacono, Kinsler, and Price, 2017). In accordance with those studies, the data at hand features a quasi-experiment from a non-selective pairing, a clear measure of productivity/outcome, and detailed information about pairing and the play order. Importantly, the professional bowling also offers exogenous shock in the space of ideas 6. Specifically, this paper tests whether a bowler can bowl more strikes by learning from a competitor s ball movement when the two bowlers are same-handed. The quote below from a professional bowler suggests how bowlers try to collect information from a competitor for updating their beliefs on the uncertain lane conditions. Make a mental note that after two or three frames begin to study your teammates and opponents ball reactions. Their ball will also give you a great indication of what could happen to your ball. On tour, I am making sure that I watch a few balls down the lane to see how their reaction has changed frame to frame, if at all. Ultimately it s a guess, and every week is different, so don t fall into the trap of accepting it s going to be every four frames. 7 5 See Manski (1993). Also, see Angrist (2014) for difficulties in the causal inference. 6 Azoulay, Graff Zivin, and Wang (2010) and Borjas and Doran (2015) used the terminology, space of ideas, to address the proximity of specializations among scientists or mathematicians. 7 You Asked, We Answered: Can You Predict Transition?, Jason Belmonte, International Art of Bowling, accessed February 19, 2016, 2

4 The bowler wants to extract information about the lane from the competitor s ball movement. By integrating the information with his priors, he can improve his expectation, and thereby improve his productivity. However, bowlers use different parts of the lane, depending on their handedness. For bowling a strike, right-handers roll a ball on the right part of the lane, whereas lefties use the opposite part. Hence, the ball reactions of left-handers cannot inform relevant information to the righties. This suggests that the coincidence of handedness can proxy the proximate space of ideas. So if there is learning effect, it is expected to be larger in same-handed match-ups. I divide the sample into treatment and control groups by the coincidence of handedness in the match-up; the treatment group contains the observations in same-handed match-ups, while the control group contains the others. As noted, the coincidence of handedness in a match-up can be a proxy for the proximate space of ideas between two bowlers. Hence, the treatment group represents those who play against a competitor in the proximate space of ideas. The variation of matched handedness is exogenous because each bowler s handedness is predetermined and each unique pair of bowlers plays exactly one match-up in the round-robin tournament 8. Hence, the concern on the selection of the treatment can also be minimized. In this study, I first estimate the average treatment effect (ATE) on the probability of bowling a strike. In addition, employing a functional form of learning curve (Benkard, 2000; Levitt, List, and Syverson, 2013), I parametrically estimate the learning curves for both groups to confirm that the ATE is mainly driven by learning from a competitor. Next, I implement a simulation to show how much the aggregate productivity can increase when match-ups are constructed to raises the proximity of the space of ideas among workers. An iteration simulates a match-up (frames 1-9) and calculate the total score for all the possible pairs, given a bowler pool. Then, I find an (ex-post) optimal pairing 9 that produces the maximum total score and use a random pairing as a benchmark, given the pair scores. After you-asked-we-answered-can-you-predict-transition/ 8 Each bowler play match-ups against all other bowlers exclusively exhaustively. 9 A pairing is to pair all bowlers into 2-bowler pairs 3

5 200 iterations, I compare the average scores and other characteristics, such as the ratio of the same-handed pairs, between the optimal and random pairings. I find that the strike rate can increase by 13.9 percentage points on average when the space of ideas are proximate between two competitors. The size of the effect is 3 standard deviations of year-bowler s strike rate. The effect become significant since frame 2 and is concavely increasing over frames. A parametric estimation of learning curves suggests that there exists learning from a competitor only when the space of ideas is proximate. In addition, I find suggestive evidence that the learning effect could be amplified by 2.6 percentage points through a motivation channel by which a bowler can be motivated when the match-up is tight ( percentage points per 1 point difference in scores). This finding reflects that learning from a competitor is, in effect, sharing information, which makes the information common. Common information can lead to convergence in performances and thereby tighten the competition. Furthermore, I find that the lower-ability peers reveal information more easily (probably through mistakes) and higher-ability bowlers can have better learning abilities. From the simulation, I find that an optimal pairing can increase the total productivity, compared to the random pairing. Unsurprisingly, the ratio of the same-handed pairs in the optimal pairings is higher than random pairings. The results suggest that by raising the proximity of the space of ideas among workers, learning from peers can be proliferated. This means that a firm can increase its productivity by pairing workers based on the space of ideas. My findings contribute to many strands of economics literature. Firstly, this research bridges the gaps in the literature of peer effects in the workplace by presenting evidence for the existence of peer effects in high-skill jobs. Considering the peer effects in team production can vary with the production technology (Gould and Winter, 2009) 10, bowling 10 Reflecting this concern, the results from two researches about co-authorship in science are contrary to each other. Azoulay, Graff Zivin, and Wang (2010) document that the unexpected death of a superstar reduces the publications of collaborating scientists 5 to 8%. On the contrary, Waldinger (2012) finds little evidence of peer effects among scientists working in the same department, using unexpected dismissal of 4

6 has an advantage in empirical research, as it is a single-player sport 11. Given that the research using a single-player sport, golf, shows only insignificant or adverse peer effects (Guryan, Kroft, and Notowidigdo, 2009; Brown, 2011) 12, the findings in this paper add new evidence that there also exists a positive peer effect. Furthermore, this paper can also contribute to other peer-effect literature, as the findings suggest that the space of ideas can be a factor to reconcile the mixed evidence. While peer effects have attracted increasing attention in the economics literature, empirical evidence for the existence of peer effects is often inconclusive 13. Some attribute such mixed evidence to the context-specific existence of peer effects. But less is known about which context matters for peer effects. This paper suggests that the space of ideas can be a candidate. For example, it is possible that scientists may benefit from knowledge spillover only from peers in the same specialized field. 14 Estimated peer effects can vary due to unobserved variation in the specializations. That is, by controlling for the space of ideas, we may identify the peer effect more clearly. One possible expansion from the findings in this paper is peer effects in schools. In the education literature, evidence for the existence of peer effects is mixed, too. (Sacerdote, 2011) Recent research shows that peer-student group composition matters in peer effects; higherability students are more affected by higher-ability students (Lavy, Paserman, and Schlosser, 2012; Imberman, Kugler, and Sacerdote, 2012; Burke and Sass, 2013). Integrating Akerlof scientists in Nazi Germany as an exogenous variation of peer. In particular, it also shows that superstar s effect found in Azoulay, Graff Zivin, and Wang (2010) doesn t exist in their sample. 11 Even in the literature for peer effects in sports, most researches study the team sports such as professional basketball (Kendall, 2003; Arcidiacono, Kinsler, and Price, 2017), baseball (Gould and Winter, 2009) and college relay teams (Depken and Haglund, 2010). For single-player sports, only the golf is used. Note that professional bowling data was used once by Dorsey-Palmateer and Smith (2004) in sports psychology to test the hypothesis of hot hands. (For more details on the hot hands literature, refer to Bar-Eli, Avugos, and Raab (2006). 12 Guryan, Kroft, and Notowidigdo (2009) find less evidence for both learning and emotional channels of peer effects between professional golfers. Brown (2011) shows that presence of a star golfer in a competition makes other golfers give up. 13 For instance, mixed evidence for the peer effects among students and scientists is well documented in, respectively, Sacerdote (2011) and Waldinger (2012) 14 Waldinger (2012) examines this possibility in a robust check but can not find an interaction between specialization and knowledge spillover among scientists. 5

7 and Kranton (2002) s theoretical model of identities 15, those who score high in academics can be considered as nerds. They are probably in the distant space of ideas against low-ability students (burnouts), since their interest is likely to be in high academic achievements, which is less relevant to burnouts. In this sense, the coincidence of identities among students can indicate the proximate space of ideas. Hence, this research can support the recent education literature addressing the peer group composition. Another contribution is to introduce a new type of a quasi-experiment. Existing literature exploits random assignments to avoid a selection bias (Sacerdote, 2001; Zimmerman, 2003; Guryan, Kroft, and Notowidigdo, 2009) 16. Instead, I avoid selection bias by exploiting the round-robin tournament, in which each unique pair of bowlers are matched exactly once. Since all bowlers match each other once in a pair, the concern on the selection disappears. The schedule for the whole tournament is based on pre-determined rankings in the previous rounds. In other words, the schedule is not endogenously determined. 17 Additionally, this novel dataset can also minimize the reflection problem, pointed out by Manski (1993), and further developed in the social interactions literature. In a linear specification, it is hard to identify the endogenous effects caused by a peer s behavior or outcome separately from the contextual effect by a peer s characteristics. This is mainly because economists can usually observe only the final outcome. 18 As only the final outcomes are usually observable to economists, it is hard to separate an effect from the combined effects estimated. The data in this paper includes frame-by-frame performances, and bowlers 15 Akerlof and Kranton (2002) s model implies that there are different social categories (leaders, nerds, burnouts) and a student behaves differently to fit in the category, depending on the student s category. 16 A rigorous, exhaustive literature review in education can be found in Sacerdote (2014). 17 In addition, the schedule distributes bowlers equally in terms of locations (lanes) and play orders so that it does not have any special pattern for a certain bowler. Another possible bias source can be the order of match-ups. For example, bowlers can be eventually concentrated through the schedule. However, the estimation adding the fixed effects for the order of match-ups does not affect the ATE significantly. 18 Suppose that there are two bowlers (say, A and B) and they play sequentially. If there is contextual effect, then B s characteristic can affect A s play. Then, A s affected performance can affect B s performance through the endogenous effect and B s affected performance can affect A s performance again and so on so forth. If economists can observe only the final scores (like many other observables from the workplace), they can estimate the combined effects but cannot identify what fraction of them were caused by either endogenous or contextual (exogenous) effect. 6

8 alternate their turns every two frames in a match-up. 19 Hence, the analysis can be made on a sequence of the outcomes in a panel. That is, we can avoid the limitation of identification caused by the fact that final outcomes are only available. This paper contributes to the learning-by-doing literature, as well. I modify the learning curve (Benkard, 2000; Levitt, List, and Syverson, 2013) by adding indirect experience. The estimation shows that indirect experiences effectively increase productivity only when the space of ideas are proximate between two bowlers. 20 The findings in this paper can also be related with personnel economics. Competitions are fostered to increase motivation. Clearly identifying externalities can be important for effective competition design. 21 This paper shows that the proximity of the space of ideas should be controlled for when estimating peer effects in a competition. In addition, a simulation shows that, by using the relationship between the space of ideas and learning from peers, a firm can improve its productivity. 22 The results in this paper have implications to personnel economics. First, strategic deployment (in this paper, internalizing the learning) can increase productivity in addition to the motivational effect of a competition. Second, the finding that the proximate space of ideas can raise the tightness of the competition through social learning provides a policy implication. Considering the adverse effects under pressure, the proximity of the space of ideas can be a policy instrument to maintain appropriate competitiveness. The remainder of the paper is structured as follows. Section 2 describes the backgrounds for bowling and the conceptual framework. Section 3 summarizes the data. Section 4 specifies empirical models and section 5 shows their results. Section 6 shows the details and results 19 Exceptions are first and last frames of the first player. For details, see figure Due to the information structure, the personal experience does not affect the productivity significantly. Because of the play order, the personal experience on a lane is always out-dated, whereas indirect experience is fresh information. 21 Lazear and Oyer (2012) points out that there is an optimal reward spread to maximize the motivation. If externality exists but is not internalized correctly in the optimization problem, a wrong decision can be made. 22 In fact, considering the proximity of the space of ideas is not new in businesses. For example, a trading room in an investment company was reorganized by forcing teams to be closely aligned based on the proximity of asset classes (Beunza and Stark, 2004). 7

9 of a simulation. Finally, section 7 concludes. 2 Backgrounds and Conceptual Framework 2.1 Backgrounds Uncertainty on Lane Conditions: Oil Pattern Bowling lanes are oiled and each professional competition has its formal oil patterns. The problem is that the oil pattern on a lane continuously changes as the competition proceeds. Given that the oil is liquid, a rolling ball can break or carry the oil down the lane 23. Since the oil on the lane is not visible, substantial uncertainty is created regarding the lane conditions for any given ball, which impacts bowlers in at least two ways. First, when a bowler starts a match-up, he likely chooses an optimal strategy conditional on the initial state of the lane. Every bowler changes lanes every match-up but starts the match-up without practice on the lanes previously used by other pairs of bowlers. Hence, the initial lane conditions are uncertain to the bowler 24. Experiencing the lanes through play, he should figure out the lane conditions by observing the specific reaction of the ball, allowing him to update his beliefs. Second, through a match-up, a bowler usually adjusts his release point 25 in response to the changed condition 26. Since the change in lane conditions is unobservable, he should modify his belief on the lane conditions with the expectation on the changes. 23 For detail, refer to the column, Breakdown and Carrydown - By The Numbers (Ted Thompson, BowlingDigital, accessed February 20, 2016, 24 There are two exceptions: 9th and 17th games. Lanes are re-oiled every eight games. Hence, bowlers in 9th or 17th games can play on fresh oil, whose patterns are known. 25 For more details, refer to Appendix A. 26 While there could be many other ways to adjust for the lane conditions such as using different strategies or different balls, we will focus on the release point for simplicity in this paper. Our interest is on learning about the lane condition per se but not on how to adjust for it. Therefore, this simplified story is sufficient for this paper. 8

10 Learning Channel: Information Revealed from Competitor s Ball Movement Throughout a match-up, the motion of balls on a lane can be a source of information about the lane conditions. For instance, depending on the lane conditions, a rolling ball can start to hook early or late and deeply or narrowly. Hence, the timing and the depth of the hook can give a hint as to the lane condition. As the match-up advances, a bowler can watch the motions of more balls. According to learning by doing, he can accumulate knowledge about the lane conditions over time. In particular, if a bowler can collect information from competitor s play in addition to his own play, he can accumulate knowledge about the lane conditions faster. In the tournament examined in this paper, a bowler can closely watch the competitor s ball thanks to the sequential order of play. Figure 1 depicts the play order in a match-up, in which there are two competing bowlers and two contiguous lanes. The order follows the black line with arrows from the left to the right. The first bowler (P1) starts the match-up by playing on the left lane of two lanes at frame 1. After that, two bowlers alternate their turns every two frames from the second bowler (P2) until P1 finishes frame 10. Due to changing turns every two frames, a bowler can watch the competitor s play for two frames before his own two frames. In addition, a bowler plays the first frame on the right lane and the second one on the left at every turn. So in the second frame or later, a bowler could have observed a peer s performing on each lane and so had an opportunity to update his beliefs using his observations on each lane s condition with the newly acquired information. The same-colored boxes in the figure indicate the same lanes; the lane in his own frame and the peer s previous frame (e.g. P1 s frame 3 and P2 s frame 2) always coincide. That is, a bowler can learn about the condition of the lane to play by referencing the peer s previous frame. 9

11 Proximate Space of Ideas: Coincidence of Handedness in a Match-up Even after observing a peer s ball movement, the learning may not be effective if the incoming data is less informative to his play. In bowling, the coincidence of handedness in a match-up can be a proxy for the proximate space of ideas. While there are pocket zones a ball should hit to make a strike; they differ, depending on bowler s handedness. The pocket for a right-handed bowler is typically between right front two pins (Pins 1 and 3), while the one for a left-handed bowler is between left front two pins (Pins 1 and 2). In other words, for a right-handed bowler to hit a strike, the ball should approach to the pocket from the right side of the lane. For the left-hander, it should be from the left. Hence, a righty s task for bowling a strike to is to send the ball to the pocket on the right part of the lane, while a lefty s task is on the left part of the lane. This suggests that the tasks are different between the right- and left-handed bowlers. Since the motion of the ball rolled by a right-handed bowler can give information on the conditions on the part of the lane where left-handed bowlers usually do not use for making a strike, it would rarely be relevant to left-handers, and vice versa. Hence, the knowledge on a lane may not be effectively updated by a peer s performing when the handedness in a match-up does not coincide. In this way, the learning channel of peer effects should be more effective. Quasi-Experiment: Non-selective schedule from a Round-Robin Tournament In a round-robin tournament, every bowler plays all other bowlers exclusively and exhaustively 27, like the group stage of the World Cup in soccer. Since all bowlers match each other once in a pair, the concern on the selection disappears. The schedule that determines the order of opponents to play against is based on pre-determined rankings in the previous (qualifying) stage. In other words, the schedule is not endogenously determined in the sample periods. In addition, the schedule distributes bowlers unbiasedly in terms of locations 27 For more details, please refer to Appendix A 10

12 (lanes) and play orders so that it does not have any special pattern for a certain bowler. Specifically, a bowler plays on the one of 12 sets (locations) of lanes at most twice: once as P1 and once as P2. Each bowler is assigned as P1 for either 11 or 12 of 23 match-ups; whether to have 12 match-ups as P1 is not correlated with rankings. While there could still remain a minor problem from the schedule of match-ups such as behavioral effects from the structure of the tournament (e.g. the order of match-ups), such an impact is controllable by adding corresponding fixed effects. For example, bowlers in the first a few match-ups can tend to not be warmed up yet or bowlers can get tired after several match-ups. By adding dummy variables for the order of match-ups, the estimation can control for such effects. As noted, the coincidence of the handedness can proxy the the proximity of the space of ideas in bowling. In fact, bowlers who play their match-up with the same-handed competitor can be considered as the treatment group in which the subjects can observe the performance of competitors in the proximate space of ideas, while the others are the control group in which the subjects can also observe competitor s performing, but their spaces of ideas are not proximate. Given the exhaustive, exclusive, and non-selective pairing, the control group is determined non-selectively, which provides an experimental environment. 2.2 Conceptual Framework The outcome of interest in this paper is rolling a strike. Let Y mif be the indicator for a strike, such that it is one if the performance is a strike and zero otherwise. m, i and f respectively index match-ups, bowlers and frames. Y mif can be considered as a realization of a Bernoulli trial with its probability of a strike (hereafter, strike rate), P mif, i.e. Y mif B(P mif ) (1) where B(P mif ) denotes the Bernoulli distribution with P mif. The strike rate, P mif, can be affected by the bowler s knowledge level about the lane conditions, K mif, as well as many 11

13 other factors, such as the bowler s innate ability. Let p(k mif, X p mif ) denote a function for the strike rate, so that the strike rate can be represented P mif = P (Y mif = 1 K mif, X p mif ) = p(k mif, X p mif ) (2) where naturally p K 0, and X p mif denotes a set of all other factors that can impact the strike rate. Note that it is assumed that the strike rate is a non-decreasing function with respect to the knowledge level; that is, better information can only improve outcome. The knowledge of a lane is a set of expectations on the state of the relevant parts of the lane through which a bowler s ball will travel 28 and updated by a learning process through a match-up. The learning process filters out incoming data to relevant information based on prior expectations (Boisot, MacMillan, and Han, 2007). Figure 2 depicts the process. The knowledge base is the information set from which the bowler s expectations are formulated. The information filtered from data is added in the knowledge base. If the new information improves the expectation, the knowledge level can rise, vice versa. For example, suppose K mif is a measure of accuracy of the expectations. If data from a competitor help a bowler understand the lane condition, K mif can rise. Formally, let Y mj,f 1 be a set of data generated by competitor j s play in the previous frame of match-up m. The knowledge level after incorporating Y mj,f 1 can be expressed K mif = δk(y mj,f 1 H m, K mi,f 1, X k mif) (3) where δ [0, 1], H m is the indicator for the same-handed match-up, i.e. H m = I(H m = h mj ), and X k mif is a set of other conditions or factors that can affect the learning process. Note that a discount factor δ is included in the knowledge equation (3), for the updated knowledge is in fact outdated 29. The prior expectations and data from peer s play are for the lane 28 This is consistent with Arrow (1984) s view that knowledge is a set of expectations (Boisot, MacMillan, and Han, 2007). 29 The state to have an expectation is after the ball rolled down but the information obtained is about 12

14 condition facing the peer, while K mif is for the lane condition facing the bowler in the current frame after the competitor s play. So whether the knowledge level grows over a match-up is ambiguous. However, as it is assumed for the discount factor to be identical in both cases of same- and different-handed competitors, whether the knowledge level for the same-handed bowler (H m = 1) is greater than the one for the different-handed bowler is determined by the knowledge function k( ). As noted in the section 2.1, when H m = 1, a bowler can glean more information relevant to his play. Hence, if the learning from a peer is available, we reach an intuitive, yet powerful hypothesis, given other variables (Y mj,f 1, K mi,f 1, Xmif) k equal, k(h m = 1) > k(h m = 0). (4) By equation (3), it suggests that if the discount factor is not zero, K mif H=1 > K mif H=0. (5) That is, the knowledge level in the same-handed case (H m = 1) is greater than when H m = 0. Recall that the sample can be divided into a treatment and a control groups by the coincidence of handedness, H m. I can estimate the average treatment effect (ATE) on the strike rate as follows. β = E (Y mif H m = 1, X mif ) E (Y mif H m = 0, X mif ) (6) where X mif = {K mi,f 1, Y mj,f 1, X p mif, X k mif}. the state before the ball rolled down. 13

15 3 Data The data for the U.S. Open (the bowling tournament) is publicly available from to season (6 seasons) on the Professional Bowlers Association (PBA) official website 30. This paper uses the second (round-robin tournament) stage only; there are 1,656 match-ups for the whole sample 31. The total number of observations is 29,808, as there are two bowlers per match-up and the first 9 frames are used. The reason for excluding the tenth frame is to rule out the concern that the different play format in the tenth frame 32 may yield different strategies and incentives from other frames. Indeed, the strike rate of the first roll in the tenth frame is much higher than other frames. Figure 3 depicts the strike rate and its 95% confidence interval for each frame. As seen, the strike rates are between 45% and 50% through frame 9 but it jumps up to 60% in frame 10. Finally, after removing missing observations, the sample size is 28,548. Table 1 represents descriptive statistics for the data after removing missing observations 33. Panel A shows the statistics for a panel with 145 year-bowlers 34. In the sample, 79 bowlers appeared in the second stage and the average number of appearances within the sample periods is 2.6. Average score in the first stage is 212, greater than the average of the second stage. The average strike rate is 48%, while the average open rate is 12% 35. Also note that 90% of year-bowlers are right-handed. Finally, 90% of bowlers who appeared in the second stage are professional bowlers registered in PBA. Considering the fact that there are some foreign professional bowlers who did not register in PBA, this suggests that even though U.S. Open is open to non-professional bowlers, most of the bowlers who can advance In each season, there are 276 match-ups for the second stage. 32 Bowlers can have at most three shots in the tenth frame, while through frame 9, bowlers can have at most two shots in each frame. The scoring formula is also different. 33 Compared to the data including missing variables, the descriptive statistics are not significantly different. 34 Note that the number of year-bowlers in six seasons is in principle 144 but 4 year-bowlers left out the tournament before the end of the second stage. As a result, additional 4 bowlers appeared in the second stage to fill the vacancies. After removing missing observations, 145 year-bowlers remain. 35 Because of tricky oil patterns, the strike rate tends to be lower in professional tournaments than at regular bowling alleys. 14

16 to the second stage are in fact professional bowlers. Panel B shows the for frame-by-frame observations. As seen previously, the share of bowler-frames with a strike is 48%. The second row indicates the share of frames in which the bowling hands of the two bowlers coincide. 80% of the observations are from the match-ups with matched bowling hands: That is, we have 5,580 observations (310 match-ups) for the control group, and around 22,900 observations for the treatment group. Finally, the third row shows the distance of a bowler from the competitor in terms of scores at the start of a frame. The shorter the distance is, the tighter the game is. The average distance in frames 2 to 9 is Since we use the variation of competitor s bowling hand for estimation, it is also important to see whether there is difference in bowler s characteristics between right-handed and left-handed bowlers (table 2). Except for the appearance, the characteristics are not statistically different. One thing to note is that there is no significant difference in the strike rate. If the learning effect exists, the right-handed bowlers can have advantage in terms of strike rate, compared to the left-handed bowlers. This is because the number of right-handed bowlers is much larger, so that the share of match-ups against competitors with the same bowling hand can be greater for the right-handed than the left-handed competitor. This suggests that there are possibly compounding factors to the strike rate. When the data is divided to matched- and unmatched-hands samples, there is not a significant difference between two samples, either. This also suggests that there are possibly compounding factors. 15

17 4 Empirical Specifications 4.1 Average Treatment Effect on Strike Rate The baseline specification 36 is Y mif = α + βh m + θ s + θ f + θ si + θ ml + ɛ mif (7) where m,i,f,s and l index respectively match-ups, bowlers, frames, years and lanes. The dependent variable is the binary variable for strike. H m is the indicator for whether the match-up includes same-handed competitors, so the coefficient β represents the ATE as explained in section 2.2. For unobservable heterogeneity, θ s and θ f control for year- and frame-specific effects, respectively, that commonly affect bowlers in a year and a frame. In addition, θ si captures individual (year-bowler) fixed effects. Finally, θ sgl controls for the common factors, such as a lane-specific difficulty that jointly affects both bowlers in a match-up. The second specification is Y mif = α + β f H m + θ s + θ f + θ si + θ ml + ɛ mif (8) This specification estimates the ATE for each frame. β f = E (Y sigf H sig = 1) E (Y sigf H sig = 0). (9) The difference from equation (6) is to allow the ATE to vary with frames. 36 This specification is a linear probability model (LPM). For comparison, the estimates of non-linear probability models (e.g. logit and probit) are reported in table A.8. 16

18 4.2 Parametric Estimation of Learning Curve If β f in the previous model (equation (8)) increases concavely through frames, the learning-by-doing can be supported. The third model specifically introduces a functional form of the learning curve, close to the specification of Benkard (2000) and Levitt, List, and Syverson (2013) 37. Y mif =α + η 1 log(e 1lf ) + η 2 log(e 2lf ) + η 3 log(e 2f )H m (10) + β f H m + θ s + θ f + θ si + θ ml + ɛ mif where E 1lf and E 2lf represent the accumulated experiences from, respectively, own and competitor s plays on lane l in frame f. The experience increases by 1 as the number of own or competitor s turns on the lane increases. Following Benkard (2000), the initial value of the experience variable is unity: E 1lf = E 2lf = 1. η 1 captures the learning rate from own experience, and η 2 captures the learning rate from indirect experience when the handedness does not coincide. Finally, η 3 captures the difference in learning rate from indirect experience between the treatment and the control. 4.3 Control for Behavioral Effects This section adds four psychological effects in the baseline model (equation (7)) 38. Y mif = α + βh m + γd mif + φ 1 D 1,mif + φ 2 D 2,mif + πd 3,mi + θ s + θ f + θ si + θ ml + ɛ mif (11) 37 These two papers also address forgetting. Including forgetting does not change the results and its coefficient is insignificant. 38 For the sake of brevity, I use the baseline model. The result is not significantly affected by the choice of one of 3 previously defined specifications. 17

19 where d sigf is the difference in scores 39 such that d mif = score mi,f 1 score mj,f 1. D 1,mif, D 2,mif are dummies for when the number of streak strikes is, respectively, 1 or 2 or greater, i.e. D 1,mif = I(# strikes = 1), D 2,mif = I(# strikes 2). D 3,mi is the indicator for whether the bowler i in match-up m is P2. First, I introduce a motivation channel of peer effects. As a match-up becomes tight, bowlers can be more motivated to defend or upset the match-up 40. Since d mif represents the looseness of the game, the negative coefficient can support such a hypothesis. Second, I test the hot hand hypothesis, a widespread belief in sports that a player can have a better chance to succeed after a streak of successes. To test the belief, I introduce a two dummy variables (D 1,mif, D 2,mif ) for the number of strikes in a row. Third, while the scientific support for hot hand belief is not particularly convincing (Bar- Eli, Avugos, and Raab, 2006), the idea of choking under pressure studied in Neuropsychology (Yu, 2014) can be an alternative hypothesis for the effect of the streak of strikes. In bowling, the play after a streak of more strikes up to three strikes can have a higher stake, for a frame with a strike can earn bonus scores from the following two rolls 41. By replacing D 1,mif, D 2,mif with dummies for that the stake is, respectively, 20 and 30, the choking can be tested. Note that the base stake is 10 points. Fourth, the order of play itself might have a psychological impact. Apesteguia and Palacios-Huerta (2010) show that there is a second mover disadvantage (first mover advantage) in a penalty shoot-out of a football game. To control for the effect from the order, the indicator, D 3,mi, for whether the bowler is the second bowler is added. 39 Bowling scores depend on both previous and following frames as well as the current frame. Hence, the actual score for the previous frame cannot be determined in a frame. Instead, the score for previous frame is calculated by assuming next frames performance is a strike. 40 One possible source of motivation is the bonus pins given depending on the match-up result. The bonus pins are 30, 15 and 0 for, respectively, win, draw and lose. Hence, if the match-up were sufficiently tight so that a bowler believes that the game result can change as a result of his actions, a bowler would probably put more effort to either prevent the result from changing if it is a leader or change the result if it is a follower. 41 When a frame is open such that all pins could not be cleared by either one or two balls, the stake of the first shot in the following frame is 10 points. On the other hand, if the frame is spare or strike, the stake increases to 20 points. When there is a streak of two or more strikes, the stake increases to 30 points. 18

20 5 Results 5.1 Main Results Baseline Specification and Common Factors Table 3 shows the estimates of the ATE in the baseline specification. In all estimations, standard errors are robust and clustered by year-bowlers 42. From the left to the right columns, the fixed effects to be controlled expand. The last column (5) is the same specification as the baseline model (equation (7)). Looking at the last column, the estimated ATE is 13.9 percentage points and statistically significant at the 5% level. This suggests that productivity, measured by the strike rate, is higher by 13.9 percentage points when the space of ideas are proximate between paired competitors. Considering the mean of the rate for the whole sample is 48%, the scale of the ATE represents a meaningful impact. By maintaining the higher strike rate by 13.9 percentage points through frames 1 to 9, the final score can be larger by 16 points of 300 (Perfect score) 43. Comparing this result to other columns (1)-(4), the importance of controlling common shocks emerges. The estimate is -1 percentage points with no fixed effects and -0.4 percentage points with year and frame fixed effects. Adding year-bowler fixed effects, the estimate rises back to zero. Finally, by controlling for all of the fixed effects including the matchup-lane fixed effects, the estimate jumps to 13.9 percentage points. Unless the common factors in each matchup-lane are controlled for, the coefficient can be downward-biased. One plausible explanation for the bias is the correlation between matched hands and 42 The results for alternative clusters are reported in the robustness checks. 43 For the calculation, I estimate a simple regression model where Ȳmi = 1 9 Final Score mi = a + bȳmi + ɛ mi 9 Y mif. The estimates of a and b are, respectively, and 115.2, and both are statistically f=1 significant at 5% level. The change of the score caused by the change of 13.9 percentage points in strike rate can be predicted by b

21 difficulty of the lane 44. Suppose that the matchup-lane fixed effect θ ml is correlated with difficulty of the lane. For simplicity, let it have a linear relationship such that θ ml = θ 0 + θ 1 Z ml where Z ml is the level of difficulty on the lane l in the match-up m and θ 1 < 0 naturally since the difficulty of the lane will negatively affect the strike rate. When two bowlers share the same handedness, i.e. H m = 1, the difficulty of the lane, Z ml, can increase endogenously throughout the match-up. Since both bowlers use the same part of the lane, the lane condition changes faster than on the lane used in a different-handed match-up (H m = 0). So, cov(h m, Z ml ) > 0. This suggests that if we fail to control for the matchup-lane fixed effect, θ ml, the regressor, H m, can be correlated with the error terms, and the sign of the bias is negative since θ 1 < 0. As a result, the estimate of the ATE can be downward biased. Dynamics of ATE over Frames Figure 4 shows the estimated ATE in the second specification (equation (8)). The black line is the point estimate for the ATE in the corresponding frames and a vertical line represents a 95% confidence interval of the estimate. The effect in frame 1 is not different from zero at the 5% level 45. However, the ATE increases concavely by 5.1 percentage points between frames 1 and 5 and peaks in frame 7 whose effect is 7 percentage points higher than frame 1, while fluctuating around the 5th frame level. The concave increasing effect suggests the learning-by-doing hypothesis. A bowler does not know the state of lanes when he starts a match-up, but tries to learn about the lanes by acquiring information. As frames go by, he can accumulate information on the lane by observations, although some information acquired in the later frames might be already known. Hence, room for incoming data to improve expectations at the margin is getting smaller in later frames. Furthermore, 44 Here, I mean the factors that commonly make it difficult for both bowlers to roll a strike. 45 In principle, the learning from the competitor likely not happens in frame 1 as no information is available from the competitor. 20

22 continuous change in the lane condition can accelerate the rate the value of new information diminish because it can make some of the acquired information obsolete 46. Learning Curves for Treatment and Control Groups Figure 5 depicts the learning curves estimated by the specification seen in equation (10). The black and orange dots represent the strike rate for the treatment and control groups, respectively. They are estimated by allowing the experiences from both personal and competitor s play to vary by the frame, while others are fixed at the means of the sample. The black and orange dashed lines are the 83% confidence intervals for the treatment and control groups, respectively. The predicted strike rate for treatment group increases concavely from 41% to 56% by learning, while the one for the control group stagnates around 40%. The result supports that the learning channel of peer effects can increase the strike rate from below to above 50%, when the space of ideas are proximate for each other. The gap in the effect of the learning is statistically significant from frame 4 and concavely increases over frames 47. Behavioral Effects Table 4 reports the test results for behavioral effects added to the baseline model. Column (1) shows the estimation for the specification including motivational effect on strike rate. A negative coefficient for distance means that as the difference in score grows, a bowler gets less motivated. A standard deviation (14.6) decrease in distance results in around a 3 percentage points higher strike rate. Column (2) does not support the hot hand belief. The coefficient of the dummy for 46 This is partly addressed by the discount factor δ in the conceptual framework. 47 The estimated gap in learning rate (η 3 ) between two groups is and statistically significant at 1% level. The estimate can be interpreted that 1% increase in the indirect experience can raise the effect of the learning on the strike rate by percentage points, when the handedness coincides. For example, suppose a bowler is in frame 7. Comparing to frame 5, his experience increases approximately 29% (=log 4 log 3). If the handedness coincides, the learning can raise his strike rate more by 2.5 percentage points than the different-handed match-up. 21

23 whether the number of strikes in a row is greater than or equal to 2 is negative, while the one for a non-continuous strike can have a positive effect on the strike rate. If the hot hand existed, the coefficient of the dummy for that the number of strikes in a streak is equal or greater than 2 should have been positive. Rather, choking under pressure better fits the data, and column (3) shows that result. The base stake is 10 points and there is not significant difference in the impact between the stakes of 10 and 20 points. To the contrary, when the stake increases to the maximum, 30 points, the strike rate tends to be lower by 7.6 percentage points. It suggests that as marginal loss of a failure increases, a bowler can be distracted by fear of failure. Column (4) shows that there is not a second mover disadvantage in bowling. It could be partly because there is symmetry between the first and second movers in bowling, unlike football. In football, when the first kicker in the first team succeeds in scoring, the second team always starts the shoot-out under pressure caused by the feeling of being behind. Looking at figure 1, P2 in frame 1 can be pressured as in the same fashion as in football. But if P2 has two continuous strikes in the first two frames, P1 can be situated under the same pressure. The difference comes from the fact that bowlers alternate their turns every two frames, whereas the football players alternate their turn every kick. Finally, column (5) shows the estimation for the full specification in equation (11). The estimates are almost same as those in left columns. ATE, Controlled for Behavioral Effects The first row in table 4 shows the ATE after controlling for behavioral effects. All columns show the estimates for ATE are statistically significant, while adding explanatory variables reduces the effect. Including all the behavioral effects in column (5), the ATE decreases by 4 percentage points in total. In particular, column (1) suggests that the motivational effect lowers the ATE by 2.6 percentage points alone. The lower ATE, along with negative motivational effects, suggests that there would 22

24 possibly be a negative correlation between matched hands, H m, and distance in scores, d mif. Despite the exogeneity of H m, a possible source for the correlation is that the distance in the same-handed match-up can decrease endogenously through the learning from a competitor. In fact, if additional information were revealed from a competitor, the competitor could have also acquired the information from his own play: social learning is, in effect, sharing information. As a result, the two bowlers performances can converge, so the distance between scores is lower, compared to different-handed match-ups. Considering that the estimated coefficient on the distance d mif is negative, the motivational effect can be one of the channels through which the learning effect can increase the strike rate. In other words, the learning effect can be amplified by 2.6 percentage points through the motivation channel. Regarding the motivation channel, a reasonable concern is the possibility that the motivational effect varies with frames. The same distance can give different motivational effect, depending on which frame a bowler is in. For example, bowlers can think 20 points of difference in frame 2 is reversible but irreversible in frame 9. Considering that the inclusion of the motivational effect can lower the estimated learning effect, it is possible that different motivational effects per frame can change the estimated ATE per frame so that its dynamics are no longer concavely increasing. To check this possibility, I allow both the ATE and motivational effect vary with frames. Y mif = α + β f H m + γ f d mif + θ s + θ f + θ si + θ ml + ɛ mif (12) This specification adds the motivational effect per frame (γ f d mif ) to the second specification (equation (8)). Table A.3 shows the result of the estimation. The first panel shows the estimates of the ATE per frame and the second panel gives the estimates of the motivational effect per frame. The second panel suggests that the motivational effect does indeed vary by frame. In particular, bowlers appear to be concerned with the score gap from frame 4 and react to 23

25 the marginal change in the gap sensitively until frame 7. As the possibility to change the match-up result becomes lower in the late frames (frames 8 and 9), bowlers are less motivated to the same marginal change in the gap. However, the varied motivational effect does not significantly affect the dynamics of the learning effect, while the size is lowered. Figure 6 compares the estimated ATEs from specifications both with and without the motivational effects. Both curves look almost identical, except that the gaps become narrower between frames 6 and Heterogeneous Learning Curves This section explores heterogeneity of the ATE. Recall the learning process described in figure 2. The learning process filters out irrelevant information from the stream of data. From the process, two possible channels heterogeneous incoming data and heterogeneous filters are explored with ability of bowlers. For the first channel, while some bowlers may reveal more information through higher quality or the larger amount of data. It is ambiguous whether higher-ability peer spill more information. On the one hand, if knowledge spillover exists, incoming data from peers with higher ability can be better in terms of either quality or amount or both. On the other hand, if a bowler with low ability are more vulnerable to the change in lane condition, peers with lower ability can reveal information with higher chance by the failure of a strike. In terms of the second channel, even though the same data arrive from a peer, some bowlers can glean more information from them with a better filter. It is possible that higher ability reflects the ability of extracting information. For ability, bowlers are divided into two groups: top 25% 49 and bottom 75% based on the rankings 50 in the first stage. Since the rankings are predetermined in the first stage, this 48 Inclusion of other behavioral effects does not significantly change the dynamics of the ATE significantly. 49 The reason to select top 25% is because top 6 of 24 bowlers in the second stage usually advance to the final rounds. For comparison, however, the results for top 9 (37.5%) and 12 (50%) are also reported in tables A.5a-A.6a. 50 Average scores in the first stage, from which the rankings are determined, are also available for the ability but the rankings are preferable since the average scores may contain the year-specific effects which can confound the effect of bowler s ability. 24

26 division is not endogenously changed in the sample. For the remaining, let G si represent the top group such that it is one if the year bowler si s ranking is within top 6 and zero otherwise. Heterogeneity in Incoming Data For heterogeneity in incoming data, the specification (12) is modified to Y mif = α + β 0f H m + β 1f G sj H m + γ f d mif + ηg sj + θ s + θ f + θ si + θ ml + ɛ mif (13) where G sj is the indicator for peers in the top group. β 0f can capture the ATE from a peer in the bottom group, while β 0f + β 1f represents the one from a peer in the top group. β 1f shows the difference of average treatment effect between two peer-ability groups. Note that G sj is also added so that η capture the composite peer effect 51 from a dimension of peer s characteristics. Interestingly, figure 7 shows that the ATE from the top group is less than the one from the bottom group 52. In particular, the gap between two groups is large in the first half of the match-up and narrows down after frame 6. This suggests that rather than a knowledge spillover, peers with lower ability can reveal the state with a higher chance by the failure of bowling a strike. For example, suppose a lane condition changes so that a bowler should adjust his play. Higher ability can reflect that the bowler can adjust such a situation better than a bowler with lower ability. In other words, a lower-ability bowler can be more vulnerable to a change in lane conditions. In terms of the learning effect, a bowler wants to catch a change in lane conditions from peer s ball movement. In that sense, peer s failure of bowling a strike can be an indicator of a significant change. Hence, a lower-ability peer s ball can reveal such change with a higher chance than from a peer with higher ability. This 51 Recall the reflection problem (Manski, 1993). In the linear-mean model like ours, the estimated coefficient on peer s characteristics cannot be entangled to endogenous and contextual effects. Hence, the estimate should be interpreted as the composite effect. 52 For the full results, refer to table A.5a. 25

27 suggests that the performance of peers with low ability can be more informative. As a result, a bowler facing a peer in the top group tends to get less information. One thing to note from the result is that the coefficient on the indicator of the top group is insignificant (table A.5ab). The coefficient can capture all but the learning effect caused by a dimension of peer s characteristics, ability. Hence, the result shows that whether the peer is in the top group can affect the strike rate only through the learning channel. 53 Heterogeneity in Learning Ability Another source for heterogeneity in learning is bowler s ability in learning. The specification (12) is modified to Y mif = α + β 2f H m + β 3f G si H m + γ f d mif + ψg si + θ s + θ 0f + θ 1f G si + θ si + θ ml + ɛ mif (14) where G si denotes the indicator of the top group for year bowler si. β 2f can capture the ATE for a bowler in the bottom group, while β 2f + β 3f represents the effect for the top group. β 3f shows the difference in ATE between two groups. Inclusion of G si will control a dimension of bowler s characteristics. Note that the frame fixed effects are also allowed to vary by G si. This is because the heterogeneous learning ability can affect the learning process of the data from the bowler s own play as well as the data from a peer. Since the learning effect from personal experience is included in the frame fixed effects as a base effect, adding the interaction terms can control the change in fixed effects caused by the learning ability. Figure 8 shows the ATEs are heterogeneous depending on the ability of the bowler 54. The ATE for the top group tends to be higher overall than the one for the bottom group. In particular, the gaps in frames 4 and 8 are sizeable However, one should be cautious when interpreting that there is not any other effect than the learning effect. It is because we cannot rule out the possibility of cancelling out of various effects as the estimate can be a composite effect of endogenous and contextual effects (Manski, 1993). 54 For the full results, refer to table A.6a. 55 Recall Jason Belmonte s advice quoted in the introduction. It suggests that a significant change in the 26

28 6 Simulation This section changes my point of view to a managerial perspective and implements a counter-factual simulation to show an example how much an optimal pairing with consideration of the space of ideas can elevate the total productivity. The empirical finding suggests that there is a positive effect of learning from a competitor on productivity, but it is less effective when the space of ideas between competitors is proximate. If the effect exists in workplace, a firm should include the proximity of the space of ideas in their consideration for a strategic deployment of workers to promote its productivity. To show how much the productivity can be improved by considering the space of ideas, a simulation is implemented. In the simulation, 200 iterations will be implemented. In each iteration, a match-up is simulated for every possible pair from a pool of bowlers. Given the simulated scores, an ex-post optimal matching that produces the maximum total scores can be found. Compared to a matching randomly selected in each iteration, characteristics of the optimal assignments can be drawn. 6.1 Primitives Bowler Pool For the simulation, a pool of 145 year-bowlers is constructed along with the set of their abilities, standard deviations of strike rate disturbance and handedness. The abilities are collected from the estimates of corresponding fixed effects in the estimation for estimation of Equation (12). As the estimation clusters the standard error by year bowlers, the standard deviation of the disturbance is also estimated from the residuals for each year bowler. lane condition on average arrives every four frames. Hence, knowing better about lane condition can be more advantageous in those frames. 27

29 Spare rate While the empirical model in this paper focuses on the strike rate, in bowling there are two exclusive sub-events for the non-strike: spare and open. To determine the sub-event, a conditional probability of spare (q = 77.03%), is calculated by a mean of an indicator of spare from observations conditional on non-strike in the data. Number of Pins Knocked For scores, the number of pins should be counted for non-strike events 56. Considering the goal of the simulation is not to imitate the bowling score closely, the number of pins fallen in the first roll for the non-strike is pinned down to the average number of pins knocked down in the first roll conditional on the sub-event of the non-strike in the data. The average numbers of pins knocked down in the first roll are for spare and for open. Similar to the first roll, the number of pins fallen in the second roll for open is pinned down to , the average number of pins knocked down in the second roll conditional on the event of open in the data. Distribution of Matchup-Lane Fixed Effects In estimation, the matchup-lane fixed effects are included to control for any common factors on each lane of a match-up. For simulation, I draw the fixed effects randomly from the distribution of the estimated matchup-lane fixed effects with the assumption of Normal distribution 57. The mean and standard deviation are estimated with the estimated fixed effects. 56 When the event is strike, the number of pins knocked down in the first roll is always The correlation of common shocks and the coincidence of handedness was discussed previously. Considering the correlation, the endogenous common factor may cancel out LTC effect significantly. However, since my interest is not the common factors but the LTC effect, I assume that the common factors are determined randomly and exogenously to let the LTC effect emerge. 28

30 6.2 Simulating Match-Up Scores Given a 145-bowler pool, 20,880 pairs can be made if the order in a pair matters. Given the primitives and the strike rate equation, Equation (12), a match-up is simulated for every pair. Using the simulated scores, the pair-scores can be constructed by summing scores of two bowlers in each pair. Figure 9 shows the scoring process 58. Given primitives and a pair, (i, j), which also indicates a match-up, m, the strike rate, P mif, can be predicted for a bowler in a frame. With the strike rate, the outcome can be determined either strike or non-strike, following Bernoulli Distribution. For the non-strike, the outcome can be divided to spare or open with the spare rate (q). Hence, there are three possible outcomes for a frame. The score function can calculate the score for the frame with the outcome, given the score in the prior frame and the outcomes in the prior two frames. 6.3 Finding Maximum Score Pairing Given the pair-scores, a maximum score pairing, a matching that produces the maximum total-score 59, can be found among possible matchings. It can be considered as an ex-post optimal pairing strategy. Since the order matters in a pair, however, the number of possible matching 60 from a 145-bowler pool would be greater than Hence, it is not feasible to compare brutally all the possible pairings to find the maximum score pairing. For this matter, an efficient algorithm to find a maximum weighted matching can be employed, thanks to Galil (1986) 61 by assigning the pair-score for a weight of each pair. Still, there remains a problem: the algorithm is available only for (undirected) graphs, in which the order in a pair does not matter, while the context in this paper demands the algorithm 58 For more details, please refer to Appendix B.3 59 The total-score of a matching is the sum of the pair-scores of pairs in the matching. 60 Only near-perfect matchings are considered for the calculation. By perfect, all bowlers in a pool are matched in the matching. Similarly, for a pool whose number of bowlers is odd, near-perfectness is applied when all but a bowler in a pool are matched. This is because a bowler should remain unmatched in the odd pool. 61 A MATLAB function is available at weighted-maximum-matching-in-general-graphs. 29

31 for a directed graph (digraph). To deal with this problem, I conduct following steps, as suggested in Lim (2017), in each iteration: Step 1. Construct a weighted digraph D w = (V, A, w) with pair-scores as the weights, w; Step 2. Convert the digraph D w to an underlying graph G w = (V, E, w ) with the dominant weights 62, w ; Step 3. Find a maximum weighted matching for G w, using Galil (1986) s algorithm; Step 4. Recover the order of pairs in the matching. Lim (2017) shows that by conducting the 4-step procedure, a maximum weighted matching, which is a maximum score pairing in this paper s context, can be obtained. 6.4 Benchmark: Random Pairing For the benchmark, a random pairing is also created in each iteration. In the random pairing, bowlers are matched into 72 pairs, remaining a bowler unmatched, like an optimal pairing. 6.5 Simulation Results and Discussion Table 5 reports the characteristics of optimal and random pairings and their differences 63. The average 9th-frame score 64 in the optimal pairings is 30% (46 points) higher than the random pairings. The ratio of the same-handed pairs is 7.7 percentage points higher in the optimal pairings 65 and match-ups in optimal pairings is tighter than those in 62 The dominant weight means the weight of the dominant arc of two ordered pairs, the one with the higher weight. 63 For the distributions of characteristics for optimal and random pairings, please see figure A Total score of a pairing is the average score times 72 (pairs). 65 One thing to note is that the ratios of the matched hands in the optimal matchings are not 100%, even though the learning effect can increase the output. This is because of the large standard deviation of disturbance in the strike rate. In fact, the R 2 of the strike rate models in this paper is just around 12.5%. Hence, even after addressing the learning channel of peer effects, the strike rate can be largely affected by the disturbance terms. To removing the effect of disturbance terms, the expected match-up average scores are constructed by averaging out 200 simulated scores for each pair. The ex-ante optimal matching found by using the expected match-up average scores exhibits the 100% of the ratio of the matched hands. 30

32 random pairings. The results suggests that the learning and motivational effects increase the productivity. The results suggest that it is important to raise the proximity of the space of ideas between competitors to keep the learning effect effective. From a managerial perspective, a firm can improve its productivity through the learning channel by reducing the diversity of tasks among paired workers. Since such a strategic deployment can be made with given workforces, the increase in productivity can be considered expanding possible product frontier. Considering that the competition is introduced to motivate workers, the learning effect can be a value-added effect of a competition on their productivity. Furthermore, provided that the proximity of the space of ideas can be positively correlated with the tightness of the competition, the space of ideas can be an instrument to steer or amplify the motivation of workers in a competition. The simulation results also raise interesting questions to the economics literature. Considering the competitors are not cooperative, it is interesting to see how much the learning effect can be amplified in a cooperative environment such as inter-team competition where a cooperation among peers in a team production likely occurs. In fact, it is ambiguous if the learning effect in such a condition can be larger, since a freerider problem or other incentive issues can trade off such a positive externality. Finally, if a competitor knows the learning effect, he might try to hide or distort information by intended mistakes. Since this strategic behavior can give an adverse impact on the productivity, it could be meaningful to investigate such behaviors in a competition. 7 Conclusion Exploiting a novel dataset collected from professional bowling tournaments, this paper shows that there exists a learning channel of peer effects between professional bowlers, and the channel is effective when the bowlers are in the proximate space of ideas. Furthermore, a simulation shows suggestive evidence that the aggregate productivity can be lifted through 31

33 learning from competitors by keeping the space of ideas between them proximate. The findings in this paper suggest future research agendas. First, as competitors are not cooperative, it would be interesting to see how much the learning effect can be amplified in a cooperative environment, such as within a team. In fact, it is not clear whether the learning effect in such an environment is larger, since a free-rider problem or other incentive issues can counteract such positive externalities. Second, it should be also investigated whether competitors try to hide or distort information through intentional mistakes, which can affect outcomes adversely. In the meantime, as noted in Guryan, Kroft, and Notowidigdo (2009), one should be cautious when applying these findings in professional sports to other domains. Nonetheless, the similarity that many workplaces and schools exhibit, especially in terms of the competition-like incentive 66, suggests that the context in this paper can be relevant for a variety of domains. The simulation in this paper shows an example: For a firm to obtain more desirable outcome, the peer effect needs to be internalized by considering the proximity of the space of ideas in its workforce deployment. References Akerlof, George A and Rachel E Kranton Identity and Schooling: Some Lessons for the Economics of Education. Journal of Economic Literature 40 (4): Angrist, Joshua D The perils of Peer Effects. Labour Economics 30: Apesteguia, Jose and Ignacio Palacios-Huerta Psychological Pressure in Competitive 66 Competition is now pervasive in workplace. Marino and Zabojnik (2004) and Brown (2011) indicate that many large firms (e.g. DuPont, Fidelity, General Electric, 3M, Bloomingdale s, Procter & Gamble, IBM, Johnson & Johnson, General Motors, and Hewlett-Packard) use inter- or intra-team competition to allocate rewards such as promotion or profit sharing. For more rigorous explanation, refer to McLaughlin (1988), Eriksson (1999) and Gill et al. (2015). In education, competition is natural. GPA, admission and other contests situate students in competitions. 32

34 Environments: Evidence from a Randomized Natural Experiment. American Economic Review 100 (5): Arcidiacono, Peter, Josh Kinsler, and Joseph Price Productivity Spillovers in Team Production: Evidence from Professional Basketball. Journal of Labor Economics 35 (1): Arrow, Kenneth Joseph The Economics of Information, vol. 4. Harvard University Press. Autor, David H., Lawrence F. Katz, and Melissa S. Kearney The Polarization of the U.S. Labor Market. American Economic Review 96 (2): Azoulay, Pierre, Joshua Graff Zivin, and Jialan Wang Superstar Extinction. The Quarterly Journal of Economics 25: Bandiera, Oriana, Iwan Barankay, and Imran Rasul Social Incentives in the Workplace. The Review of Economic Studies 77 (2): Bar-Eli, Michael, Simcha Avugos, and Markus Raab Twenty Years of hot hand Research: Review and Critique. Psychology of Sport and Exercise 7 (6): Benkard, C. Lanier Learning and Forgetting: The Dynamics of Aircraft Production. American Economic Review 90 (4): URL id= /aer Beunza, Daniel and David Stark Tools of the Trade: The Socio-Technology of Arbitrage in a Wall Street Trading Room. Industrial and Corporate Change 13 (2): Boisot, Max H, Ian C MacMillan, and Kyeong Seok Han Explorations in Information Space: Knowledge, Agents, and Organization. OUP Oxford. 33

35 Borjas, George J and Kirk B Doran Which Peers Matter? The Relative Impacts of Collaborators, Colleagues, and Competitors. The Review of Economics and Statistics 97 (5): Brown, Jennifer Quitters Never Win: The (Adverse) Incentive Effects of Competing with Superstars. Journal of Political Economy 119 (5): Burke, Mary A and Tim R Sass Classroom Peer Effects and Student Achievement. Journal of Labor Economics 31 (1): Cornelissen, Thomas, Christian Dustmann, and Uta Schönberg Peer Effects in the Workplace. American Economic Review 107 (2): URL articles?id= /aer Depken, Craig A and Lisa E Haglund Peer Effects in Team Sports: Empirical Evidence from NCAA Relay Teams. Journal of Sports Economics. Dorsey-Palmateer, Reid and Gary Smith Bowlers Hot Hands. The American Statistician 58 (1): Eriksson, Tor Executive Compensation and Tournament Theory: Empirical Tests on Danish Data. Journal of Labor Economics 17 (2): Falk, Armin and Andrea Ichino Clean Evidence on Peer Effects. Journal of Labor Economics 24 (1): Galil, Zvi Efficient Algorithms for Finding Maximum Matching in Graphs. ACM Computing Surveys 18 (1): URL Gill, David, Zdenka Kissová, Jaesun Lee, and Victoria L Prowse First-Place Loving and Last-Place Loathing: How Rank in the Distribution of Performance Affects Effort Provision. Discussion Paper Series 9286, Institute for the Study of Labor (IZA). 34

36 Goos, Maarten and Alan Manning Lousy and Lovely Jobs: The Rising Polarization of Work in Britain. The Review of Economics and Statistics 89 (1): Goos, Maarten, Alan Manning, and Anna Salomons Job Polarization in Europe. American Economic Review 99 (2): Gould, Eric D and Eyal Winter Interactions between Workers and the Technology of Production: Evidence from Professional Baseball. The Review of Economics and Statistics 91 (1): Guryan, Jonathan, Kory Kroft, and Matthew J. Notowidigdo Peer Effects in the Workplace: Evidence from Random Groupings in Professional Golf Tournaments. American Economic Journal: Applied Economics 1 (4):pp URL org/stable/ Imberman, Scott A, Adriana D Kugler, and Bruce I Sacerdote Katrina s Children: Evidence on the Structure of Peer Effects from Hurricane Evacuees. American Economic Review 102 (5): Jackson, Clement Kirabo and Elias Bruegmann Teaching Students and Teaching Each Other: The Importance of Peer Learning for Teachers. American Economic Journal: Applied Economics 1 (4): Kendall, Todd D Spillovers, Complementarities, and Sorting in Labor Markets with an Application to Professional Sports. Southern Economic Journal : Lavy, Victor, M Daniele Paserman, and Analia Schlosser Inside the Black Box of Ability Peer Effects: Evidence from Variation in the Proportion of Low Achievers in the Classroom. The Economic Journal 122 (559): Lazear, Edward P. and Paul Oyer Personnel Economics. The Handbook of Organizational Economics : URL 35

37 Levitt, Steven D., John A. List, and Chad Syverson Toward an Understanding of Learning by Doing: Evidence from an Automobile Assembly Plant. Journal of Political Economy 121 (4): URL Levy, Frank, Richard J Murnane et al The Skill Content of Recent Technological Change: An Empirical Exploration. The Quarterly Journal of Economics 118 (4): Lim, Choon Sung Simple Transformation for Finding a Maximum Weighted Matching in General Digraphs. Working Paper. Manski, Charles F Identification of Endogenous Social Effects: The Reflection Problem. The Review of Economic Studies 60 (3): Marino, Anthony M and Jan Zabojnik Internal Competition for Corporate Resources and Incentives in Teams. RAND Journal of Economics : Mas, Alexandre and Enrico Moretti Peers at Work. American Economic Review 99 (1):pp URL McLaughlin, Kenneth J Aspects of Tournament Models: A Survey. Research in Labor Economics 9 (1): National Research Council How People Learn: Brain, Mind, Experience, and School: Expanded Edition. The National Academies Press. Sacerdote, Bruce Peer Effects with Random Assignment: Results for Dartmouth Roommates. The Quarterly Journal of Economics 116 (2):pp URL http: // Peer Effects in Education: How Might They Work, How Big Are They and How Much Do We Know Thus Far? Handbook of the Economics of Education 3:

38 Experimental and Quasi-Experimental Analysis of Peer Effects: Two Steps Forward? Annual Review of Economics 6 (1): Waldinger, Fabian Peer Effects in Science: Evidence from the Dismissal of Scientists in Nazi Germany. The Review of Economic Studies 79 (2): Yu, Rongjun Choking under Pressure: The Neuropsychological Mechanisms of Incentive-Induced Performance Decrements. Frontiers in Behavioral Neuroscience 9: Zimmerman, David J Peer Effects in Academic Outcomes: Evidence from a Natural Experiment. The Review of Economics and Statistics 85 (1):

39 Appendices A Details of the U.S. Open Tournament Structure In the U.S. Open, there are three stages: qualifying, match-play, and final stages. In the qualifying stage, more than 200 players play games. Top 24 players advance to the next stage and the rankings are based on the total scores. In the second stage, players compete for four to six seeds of the final stage. The seeds are determined by the rankings based on the total points which is the sum of total scores of first and second stages and bonus pins 67. In the final stage, the tournament follows a step-ladder format. In the format, the lowest seed plays against the second-lowest seed and then the winner of that match-up plays against the third-lowest seed, and so on. Hence, the seed number is very critical for the final results. Cash Rewards While 25% of bowlers can earn a cash reward, only top 24 bowlers who advance to the second stage can increase cash reward convexly. By convexly, we refer to the increasing marginal benefit with respect to a ranking. Figure A.1 represents the cash reward curve of rankings in the 2013 U.S. Open. In the season, only top 65 bowlers could earn some cash rewards but the slope of the curve increases only from the 24th rank. This suggests that through the second stage, bowlers can be motivated to increase their rankings by aiming the higher score, which is highly correlated with the number of strikes. 67 The bonus pins are based on the result of the match-up: 30, 15 and 0 for, respectively, win, tie and lose. 38

40 Match-Play Stage The second stage of U.S. Open is match-play rounds. They consist of three rounds; each round takes a half day so that the second stage takes place for one and a half days. Each round has 8 games (7 for the third round) and there are 12 match-ups (pairs of two bowlers) in each game. Since the number of bowlers in the second stage is 24, each game can accommodate all bowlers to play their match-up simultaneously on 12 different sets of lanes. The first 23 games are for round-robin tournaments, while the last game is for a position round which finalizes the rankings for the seeds of the final stage. Note that the position round is not included in the sample because it is not part of round-robin tournament and its pairing is selective based on the latest rankings so that adjacent two bowlers in terms of rankings compete. Round-Robin Tournament The round-robin tournament is a structure of competition wherein each player plays against all other bowlers in turn. That is, each of 24 bowlers plays 23 different match-ups exclusively and exhaustively. The schedule (an opponent and the location) of the match-ups is predetermined based on the final rankings in the (previous) qualifying stage. As noted, the whole schedule takes one and a half days and 12 sets of 2 lanes are used so that 12 pairs play at the same time and no one has a break. The lanes are also assigned such that each bowler plays on the same set of lanes at most twice. Each set of lanes consists of two contiguous lanes. Adjustment for Transition of Oil Condition Figure A.2 is the heat map indicating how much the oil was depleted on some parts of a lane after 12 men s games in an European professional tournament. 68 The boxed areas 68 Note that re-oiling is every 8 games in U.S. Open. Hence, the depletion rate can be lower than the figure. 39

41 indicate highly depleted areas. Many parts of the lane are depleted over 40%. In particular, the boxes in the fifth column suggests that most bowlers fix the position where their balls exit from the oiled area: R4-11 for right-handers and L5-11 for left-handers. On the other hand, the large box in the second column indicates right-handed bowlers eventually move their release points to the right, whereas left-handers adjustment is smaller than them The spin of the ball creates a hook more effectively as oil depletes. Hence, the adjustment is made for the higher angular hook. 40

42 B Details of Simulation B.1 Bowler Pool For the simulation, a pool of 145 year-bowlers V = {1,..., 145} is constructed along with the set Ω of their estimated ability (fixed effects), standard deviation of strike rate disturbance and main bowling hands h v such that Ω = {ˆθ v, ˆσ v, h v } v V. The bowler fixed effects ˆθ v are collected from the estimates of year-bowler fixed effects in the estimation of Equation (12). As the estimation clusters the standard error by year bowlers, the standard deviation of the disturbance ˆσ v is also estimated from the residuals for each year bowler. B.2 Pairs A pair consists of two bowlers, e.g. (u, v) for u, v V. The order in a pair matters, e.g. (u, v) (v, u). Hence, 20,880 pairs are possible from the pool V. In each iteration, scores for all of the possible pairs are calculated. The scoring process will be explained in the following section. B.3 Scores For the calculation of scores, frame performance should be necessarily determined. The frame performance can be divided largely to strike, spare or open 70. The score is basically the number of pins knocked down. For strikes and spares, however, there are bonuses. When the frame performance is strike, the bowler gets to add the total number of pins knocked down from his following two rolls in addition to the number of pins knocked down in the frame. For the spare, the bonus is to add the number of pins knocked down from his next roll only. This suggests that the frame performance can affect scores retrospectively. In particular, if the performances in the previous two frames were strikes in a row, the number of pins 70 Strike is to knock down all ten pins with a roll in a frame. In the non-strike events, a bowler has another chance to roll the ball. If the second roll knocks down all the remaining pins, it is called the spare. Otherwise, the frame is remain open. 41

43 knocked down in the first roll of the current frame would not be added only to the score in the two previous frame but would be also added to the score in the previous frame. This suggests that a score in a frame is not deterministic when the frame performance is either strike or spare. To deal with this issue, the score in a frame, if it is not deterministic yet, is calculated by assuming that the performance in the following frame is a strike and ignoring the effect of the performance of two frames later 71. As mentioned, when the performances were two strikes in a row, the assumption would affect the score twice: the score for the last frame and the one for the current frame. Note that the scores are calculated only up to 9th frame. As discussed in the section 3, the play format and the scoring formula in 10th frame is different from those in other frames. Hence, the 10th frame was excluded in the analysis. Since the purpose of the simulation is not to imitate the real bowling score closely, for the parsimony, the simulation runs 9 frames only and takes the score in the 9th frame as the final score. B.3.1 Frame Performance As noted, the frame performance can be divided largely to three events. However, spare and open can be considered as sub-events after the bowler could not make a strike in the first roll of the frame. Hence, the frame performance is structured into two steps: in the first roll, the event is determined to either strike or non-strike and then either spare or open is determined in the second roll for non-strike event. Step 1: Strike vs Non-Strike Let p mif, to be calculated by the estimated strike rate equation, be the strike rate of the bowler i in the frame f of the match-up with the bowler j. Then, following Bernoulli distribution with p mif, we can randomly select the event of either strike or non-strike. 71 This is a common method used in TV casting for showing the difference between two bowlers. 42

44 Step 2: Spare vs Open When it comes to a non-strike, there are two exclusive sub-events that can happen: spare and open. To determine the sub-event, a conditional probability of spare (77.03%), i.e. q mif = P r(spare non-strike), is calculated by a mean of an indicator of spare from observations conditional on non-strike in the data. Given q mif, the sub-event can be randomly chosen either spare or open, again following Bernoulli distribution with the spare probability q mif. Number of Pins Knocked For scores, the number of pins should be counted. When the event is strike, the number of pins knocked down in the first roll is always 10, which suggests that the effect of the strike on the scores in previous frames is invariant. Hence, the calculation of the score is not complicated. However, the calculation of score can be complicated for non-strikes, since the score can vary, depending on the number of pins knocked down in the first roll. Considering the goal of the simulation is not to imitate the bowling score closely, the number of pins fallen in the first roll for non-strike is pinned down to the average number of pins knocked down in the first roll conditional on the sub-event of the non-strike in the data. The average numbers of pins knocked down in the first roll are for spare and for open. Now that the number of pins fallen in the first roll is fixed, the spare has a unique impact on score, which facilitates to calculate the score easily. However, when the performance is open, the number of pins fallen in the second roll matters, too. Similar to the first roll, the number of pins fallen in the second roll for open is pinned down to , the average number of pins knocked down in the second roll conditional on the event of open in the data. 43

45 B.4 Strike Rate Given the estimates from the results, the probability of the strike p mif can be calculated following the linear probability model defined as Equation 12 such that for f = 1,..., 9, p mif = ᾱ + ˆβ f Hm + ˆγ f dmif + ˆθ f + θ i + θ ml + ɛ mif (15) where the tildes over arguments indicate that their values were imputed or constructed in the simulation and the hats over arguments indicate that their values were estimated from the data in this paper (e.g. for any argument X, X and ˆX indicates respectively the imputed/constructed and estimated value of X). ᾱ = ˆα + 1 S ˆθ s. Comparing to Equation S s=1 12, the subscript s is dropped and the seasonal fixed effects θ s are averaged out in the constant term ᾱ since all the (year-)bowlers in the pool V are assumed to play in the same season. Still, to determine the strike rate, several variables should be imputed. Bowler-Specific Ability and Disturbance for Strike Rate Given a pair (i, j) of bowlers from the pool V, bowlers estimated fixed effects θ i can be found from the set Ω. In addition, ɛ mif can be drawn from Normal distribution with mean zero and variance σ i 2, where σ i is from Ω. Matched Hands Given a pair (i, j), bowling hands h i, h j for the bowlers can be also specified from the set Ω. Hence, an indicator H m for whether bowling hands match in the pair (i, j) can be constructed for each pair such that H m = I(h i = h j ). 44

46 Distribution of Matchup-lane Fixed Effects I draw the fixed effects randomly from Normal distribution, i.e. θ lmif N(ˆµ 2, ˆσ 2 2). The mean and standard deviation are calculated from the estimated fixed effects. Distance Given the last two frame performances of two paired bowlers, the distance between two bowlers can be calculated such that for f = 2,..., 9 d mif = z mi,f 1 z mj,f 1 (16) where z mi,f 1 and z mj,f 1 are scores of respectively bowlers i and j in the previous frame. Note that the distance for frame 1, d mi1, is zero, since there is not a previous frame. Recall that this model is the linear probability model. One of main concerns on the LPM is the predicted probability can be out of the range [0, 1]. In the simulation, the same problem remains. To deal with this problem, the strike rate p mif is forced to be within the range [0, 1] by replacing the negative values and the values greater than unity with respectively zero and the unity. In our simulation, 14.2% of observations are negative and 12.5% of observations are greater than the unity. B.5 Maximum Score Pairing A pairing is defined as a matching that matches 144 bowlers into 72 pairs 72. In each iteration, a maximum score pairing, a pairing that produces the maximum total score, will be found among possible pairings, based on the simulated scores. Since the order matters in a pair, however, the number of possible pairings from 145-bowler pool would be greater than (= 145! ). Hence, it is not feasible to compare brutally all the possible pairings to 72! find the maximum score pairing. 72 Hence, a bowler in the pool V is dropped off in a matching. 45

47 For this matter, an efficient algorithm to find a maximum weighted matching can be employed, thanks to Galil (1986) 73 by assigning a weight for each pair with the simulated pair total score. However, there still remains a problem: the algorithm is available only for (undirected) graphs, in which the order in a pair does not matter, while the context in this paper demands the algorithm for a directed graph (digraph). To deal with this problem, I will follow the algorithm suggested in Lim (2017). The details are followings: Step 1. Construct a weighted digraph For the simulation, a digraph D = (V, A) can be constructed. The vertex set V is a bowler pool containing 145 bowlers and the arc set A contains all the ordered pairs (arcs). Each arc can be assigned its weight. Let w be a weight function such that w : A R. Then, a weighted digraph D w = (D, w) can be constructed. In this study, the weight function assigns a simulated pair total score to each arc. Step 2. Convert the digraph to a underlying graph with dominant weights Step 2 is to convert the weighted digraph D w constructed in Step 1 to a underlying graph G w = (V, E, w ). Underlying graph is a graph showing the pairs in the original digraph without information on the order in each pair. In the conversion,, a problem can arise: if there are two orders for a couple of vertices, one of two weights should be selected. Following the procedures in Lim (2017), I select the dominant arc, which has the greater pair total score than its reverse arc. If there is a tie, I select the arc in which the first player has the higher seed (ranking in the (previous) qualifying rounds). In this way, I can construct the dominant orientation, A, which is the arc set containing only tie-broken dominant arc. A weighted sub-digraph can be constructed with the dominant orientation such that D w = (V, A, w). Since it is tie-broken, the dominant orientation includes only one arc for each pair of bowlers. Hence, the underlying graph with dominant weights G w = (V, E, w ) 73 A MATLAB function is available at weighted-maximum-matching-in-general-graphs. 46

48 can be obtained by converting the sub-digraph D w without the problem mentioned above. Step 3. Find a maximum weighted matching M G for G w Now that an underlying graph G w is constructed, a maximum weighted matching M G can be obtained using Galil (1986) s algorithm. The algorithm can be implemented by an existing MATLAB function. Note that the function finds only a matching, even if there are multiple maximum weighted matchings. Step 4. Recover the order of pairs in the M G Step 4 converts the edges (unordered pairs) in the maximum weighted matching to arcs (ordered pairs), using the orientation function φ, which projects each edge (pair of bowlers) to an arc in the dominant orientation, A. By the conversion from edge to arc in the matching M G, I can finally find a maximum weighted matching in the original digraph D w. Lim (2017) proves the matching found in this procedure should always be a maximum weighted matching in the original digraph. 47

49 C Figures Frame P1 L R L R L R L R L R P2 R L R L R L R L R L Figure 1: Play Order Note: P1 and P2 indicate respectively the first and second bowlers. L (light blue box) and R (dark red box) indicate respectively the left and right ones of two assigned lanes, on which P1 and P2 play. The first row indicates the frame. Finally, the solid line with arrows represents the order. P1 starts the first frame on the left lane. After that, P2 starts her first frame on the right lane and plays on the left lane for frame 2 and then P1 plays the frame 2 on the right lane and the frame 3 on the left and so on so forth. Prior Expectations World Turnable Data Filter Information Knowledge Base Data Action Source: Boisot, MacMillan, and Han (2007) Figure 2: Learning Process 48

50 Strike Rate Frames Figure 3: Strike Rate per Frame Note: The black dot represents a strike rate the mean of a binary variable for a strike for each frame and the line shows its 95% confidence interval. 49

51 30 25 ATE on strike rate (%p) Frames Figure 4: Dynamics of Average Treatment Effect Note: The bold line represents the estimated ATE for each frame and the vertical lines show its 95% confidence interval. 50

52 % CI, Same 83% CI, Different Same Hands Different Hands 55 Strike Rate (%) Frames Figure 5: Learning Curves for Treatment and Control Groups Note: The black and orange dots represent the strike rate for, respectively, the treatment and control groups, predicted by allowing the experiences from both own and competitor s play to vary by frame, while others are fixed at the sample means. The black and orange dashed lines are the 83% confidence intervals for the treatment and control groups, respectively. 51

53 18 16 ATE on strike rate (%p) w/o Mo0va0on Effects w/ Mo0va0on Effects Frames Figure 6: Dynamics of ATE, controlled for Motivational Effects Note: The bold red line represents the estimated ATE after including motivational effects, while the black line is the one estimated without motivational effects. 52

54 Top 25% Bo2om 75% 0.16 LTC Effects Frames Figure 7: Heterogeneity in Incoming Data Note: The bold red line represents the estimated ATE from peers in the top group, while the black dashed line is the one from those in the bottom group. The group is determined based on the rankings in the first rounds: Top 25% are in the top group and bottom 75% are in the bottom group. The full specification can be found in Table A.5a. 53

55 Top 25% Bo2om 75% 0.18 LTC Effects Frames Figure 8: Heterogeneity in Learning Ability Note: The bold red line represents the estimated ATE for bowlers in the top group, while the black dashed line is the one for those in the bottom group. The group is determined based on the rankings in the first rounds: Top 25% are in the top group and bottom 75% are in the bottom group. The full specification can be found in Table A.6a. 54

56 Primi%ves m=(i,j) Match-ups=Pairs d mif =Score mi,f-1 -Score mj,f-1 H m, θ i, σ i frame and lane fixed effects P mif Strike Y mif Non-strike Spare (q) Open (1-q) Score mif-1, Y mif-1, Y mif-2 Score mif d mjf+1, d mif+1, Score mif+1 Figure 9: Scoring Process in Simulation Note: The flow chart shows the simulation process of determining score for bowler i of match-up g in frame f. Primitives are a bowler pool with ability θ i, standard deviations of disturbance of strike rate σ i and handedness h i, spare rate q, the number of pins knocked for each outcome and the distribution of lane common factor s fixed effects. Given a pair (i, j), which indicates the match-up m, the matched handedness of the match-up (H m = I(h i = h j )) can be determined. Putting all together, the strike rate can be determined, which can determine the outcome to either strike or not. In non-strike, the outcome is divided to either spare or open with the given spare rate q. Using the outcome, the score can be determined, given the score in the prior frame and outcomes in prior two frames. 55

57 Prize (Thousand dolloars) Rankings 20 0 Figure A.1: 2013 U.S. Open Cash Rewards Note: The figure shows the cash rewards curve of rankings in 2013 U.S. Open. The vertical axis is the cash rewards in thousand dollars and the horizontal axis is rankings with ascending order to the right. Since only top 65 bowlers won the positive amount of cash rewards, the rankings are truncated at

58 Figure A.2: Oil Depletion If we look at the blocks which show where both the left-handed and right-handed players played, Source: Breakdown and Carrydown - By The Numbers, Ted Thompson, BowlingDigital, accessed February 20, you 2016, can plainly see how much carrydown is on the left side of the lane at the left-handers exit point Note: of the The pattern, figureyet shows not the so much heat map on indicating the right-handers how muchexit thepoint. oil waswhy depleted might on you someask? parts of a lane after 12 men s games in an European professional tournament. Balls roll from left to the right of the figure and the second row, Tape Dist., indicates the distance (feet) from the foul line. The top to the bottom of the figure It is actually represents very thesimple left to the once right we end think of the about laneit. andwe C20 know in the that firstmost column spare indicates balls thein center. use today Minusdo and plus in the second to fourth columns indicate respectively the percentage of depletion and accumulation not flare much, nor do they soak up oil like high flaring reactive resin strike balls that are in use from the fresh oil pattern. The number in the sixth column indicates accumulation on the dry area in the unit of thickness of oil. The length of bowling lane is 60 feet and the oiled area is up to 41 feet. Note that the lane of U.S. Open is oiled up to 40 feet. 57