Discussion Papers In Economics
And Business
Reference Dependence and Monetary Incentives:
Evidence from Major League BaseballReio Tanji
Discussion Paper 20-23

March 2021

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANReference Dependence and Monetary Incentives:
Evidence from Major League Baseball∗Reio Tanji†

Abstract
This paper discusses what determines the reference point in decision making, using an empirical dataset of performance stats in professional baseball. Previous literature has argued thatsome round-numbers may work as such points, and as a result, bunching occurs around thesenumbers in the distribution of the target outcomes. On the other hand, in the setting of workplace, the outcomes are observable both for the worker (players) and the evaluator (managers).
This paper shows that this bunching do NOT occur from the structure of the contracts, or how the
managers evaluate the players. Bunching seems to stem from the reference-point dependenceof the workers themselves, and so to avoid this economically ineﬃcient behavior, we have todesign contracts that incentivize players to do so.
JEL Classification: D91, J01
keywords: reference dependence, round-number eﬀect, bunching, monetary incentive, sports∗ The author would like to thank Professor Masaru Sasaki from Osaka University, our supervisor and the chief

examiner of the master’s thesis. The author is grateful for the advice of Professor Fumio Ohtake from Osaka
University, our sub-chief supervisors, and Associate professor Sohtaro Katsumata from Osaka University, our
sub-chief supervisors. The author is also grateful for the comments of the seminar participants at Ohtake-Sasaki
Seminar at Osaka University, Graduate School of Economics.
† Osaka University, Graduate Shcool of Economics. Email: u280537h@ecs.osaka-u.ac.jp

1 Introduction
Reference-point dependence is one of the most important concepts to evaluate outcomes, andit aﬀects agents’ economic behavior. Classical economic models assume that economic agents
evaluate their choices/prospects according to the absolute value of the (expected) return. On theother hand, Tversky and Kahneman (1992) introduced the behavioral assumption: agents withreference-point dependent references consider outcomes by the relative value to some target
value of the outcomes, or reference points. That is, the agents have the feeling of gain or lossby the extent to which the current outcome deviates from the target. For example, workers feel
happy if her/his wage goes up to $15 per hour, but unhappy if it goes down to $15, although theabsolute value of $15 remains the same. In this case, the previous wage works as the reference
point and aﬀects her/his utility.
In this paper, we explore whether there are bunching in the salary of the player whose
performance stats are above round-numbers, using data on individual performance stats in
Major League Baseball (MLB).
Prospect theory consists of two main characteristics: one is the probability weighting function,and the other is the reference-point dependence, mentioned above. They allow us to interpretphenomena that cannot be explained by the traditional microeconomic theory. To explore themechanism, we observe that much following research has been conducted in field and laboratory
settings.
Reference dependence also is observed in the behavior of athletes. Pope and Schweizer (2011)
found that professional golf players regarded “par,” the standard number of shots determinedaccording to the diﬃculty of each hole, as reference points. Additionally, Allen et al. (2016)
argued that marathon runners adjusted their finish times just before the round number times (justthree or four hours), the reference points. Similarly, Pope and Simonsohn (2011) showed theexistence of the reference point in the MLB, a professional baseball league of the U.S..
MLB position players evaluate themselves by the stats of their performance. Moreover, they
seem to have some reference points in their self-evaluation, for example, about their batting
performance stats: .300 of batting average. Pope and Simonsohn (2011) have shown that there
exists bunching just above .300 of the distribution of these stats.
Pope and Simonsohn (2011) diﬀers from the former two papers. Golfers or marathon runners
do not receive any monetary incentives to achieve these goals, but MLB players receive monetary
rewards determined according to their performance.
Consider the case of professional golf players. Golf is essentially a competition of the total
number of shots they needed to finish the whole tour, regardless of that of every single hole,or whether she or he saves par or not in the hole. The final rank of order is determined1

according to the total number of shots, and those with better scores are rewarded. Then, there
appears a question as to what if there is some monetary incentive to make an eﬀort to save par.
Suppose when every time she or he saves par in each hole, she or he can get some additional
bonus separated from their total score. In this case, then, making an eﬀort to save par can beinterpreted as a suﬃciently “rational” choice for the player, although the observed behavior itself
appears to be evidence of the reference dependence. However, there usually does not exist any
additional monetary rewards such as “Save-Par-Bonus.”
On the contrary, in MLB, it might not be suﬃcient to prove that the bunching is caused byreference-dependent utility function about their performance stats: team managers may assign
some monetary incentives for the players to adjust their aspiration level to meet those points, as
we described in the example above.
Our most important contribution of this paper is to reveal the observed behavior of MLB
players to be in fact reference dependence. There are two things to be mentioned. First, wefound the evidence for the manipulation of the performance stats around the round numbers,
by using McCrary (2008)’s method to test manipulation. We additionally include analysis of
not only .300 of batting average but also other points of batting-average and other stats, suchas homerun or stolen-bases. Second, we explored whether this observed manipulation was
truly driven by the preference for reference dependence of the players. We applied a regression
analysis using the data of the players’ annual salary.
Our paper found three results. First, our examination for manipulation supported Pope andSimonsohn (2011). We observed there existed seemingly reference point dependent behavior,
where .300 of batting-average worked as a reference point. Similar results were obtained aboutother round numbers of batting-average, and other batting stats such as on-base percentage orhomerun. Second, we found that as a whole, there did not exist any monetary incentive forthem: for their fixed part of the salary contract, we conclude that their monetary rewards arecontinuous at just above each performance stats, such as .300 of batting-average. That is, they
behave as they consider these round numbers as reference points, even though they do not receive
any additional payment by achieving them. Furthermore, we reinforce our results from somealternative interpretations about other types of monetary incentives: the part of the incentivized
contract, and relation with contract length. Finally, there exist serial changes in the players’ stats
manipulation. .250 of batting-average was not considered as a reference point in recent years,
while 20 of homerun was only for the recent players. Among them, .300 of batting-average
seems to be a solid benchmark for the players.
Through the results above, this paper cotributes to the studies that argues how to design
the eﬃcient contract. As the same as other workplace, players (workers) are evaluated bytheir managers (evaluators), based on some performance outcomes. Observed bunching is2

economically ineﬃcient behavior for the managers, since reaching the round-numbers does notraise the expected winning-percentage of the teams (firm’s productivity). On the other hand, thepossible resource of this misallocation is not only the players themselves. In practice, paying
attention to which one (or other possible mechanisms) aﬀects to the observed ineﬃcient behavior
and adequately motivating individuals, such as designing contracts will enhance the workers
productivity.
This paper proceeds as follows. In Section 2, we review some literature and verify thestandpoint of my paper. Section 3 describes the data we availed. Section 4 presents thetheoretical framework and empirical specification, and make some conjecture. Section 5 shows
the results of the analysis. Discussion about some alternative interpretation and non-statistical
data are included in Section 6. Finally, Section 7 provides concluding remarks.

2 Literature Review
Tversky and Kahneman (1992) mentioned reference point dependence as one of the two
distinct respects of their prospect theory. The most primitive form of reference-point dependentutility function is:{u ( x |r ) =x−rλ( x − r )if x ≥ r
if x < rwhere x denotes a certain outcome, and r is one of the reference points (Figure 1). This agent
evaluates the outcome by a diﬀerence from the reference point. Besides, they assume

loss-

aversion of the individual, or λ > 1. The size of the disutility incurred when an individual loses
a certain amount of outcome is larger than the size of utility gained when she or he receives thesame amount of outcome.

Diminishing sensitivity,

which is concave in the phase of gain

and convex in that of loss is an advanced form of this specification (Figure 2).
Diecidue and Van de Ven (2008)‘s “aspiration level” model added discontinuity assumption:that is, a utility function that “jumps” at the reference point (Figure 3). When there exists a jumpin their utility function, then individuals try to manipulate their outcome level, paying additional
cost which is not incorporated into the model with the standard continuous utility function. As
a result, excess mass or bunching around or just above the reference point arises. We discuss
the required functional assumptions of them in Section 4.1.
Individuals with such reference-dependent utility aim to put their eﬀort to achieve theirinternal target or reference point. There is several empirical literature that specifies the existenceof reference dependence in the field or lab experimented studies. Farber (2008) applied thismodel to the labor supply of New York cab drivers to show that as soon as they reached daily
target sales, they stopped working, even when they reached it early. Jones (2018) analyzed the3system of American tax payment. He showed that individuals tried to manipulate their real
payment by substituting it by donation or other charitable action, and especially did so whenfacing losses. This observation is also caused by the feeling of loss-aversion, with the reference
point of the zero-payment threshold.
u( x )ru( x )u( x )

x

r

x

O

r

x

Figure 1: primitive gain-loss

Figure 2: diminishing sensi-

Figure 3: jump at the refer-functiontivityence point

Reference dependence also arises in cases of sports. One of the most well-known papers
among them is Pope and Schweizer (2011). They obtained the data of professional golf players,
to show that in each hole, players behave as they take “par” as the reference point. Specifically,
the probability of succeeding in their putts was significantly higher when the putt was one tosave par than when it was one to get “eagle” or “birdie.” Similarly, Allen et al (2016) specified
the existence of reference point dependence of marathon runners, using data of the finish timeof runners in an enormous number of races in the United States. In this case, the distribution offinishing time has an excess mass around every 30 minutes. Note that these cases are common
in that the outcomes themselves are nonmonetary ones, and even if they achieve their internalgoals, they do not receive any additional monetary reward for their success. Professional golf
players are awarded according to the total number of shots through the whole tour, not to thenumber of pars they saved.
Pope and Simonsohn (2011) mention a seemingly similar case. They picked up three empirical pieces of evidence of round numbers that were considered as reference points: SAT (astandardized test for college admission in the United States) scores, laboratory experiments, andbaseball. In their section of baseball, they picked up the evidence of Major League Baseball(MLB) players. They argue that the players have reference-dependent preferences for evaluating
themselves in their performance stats: batting-average (AVG). According to their paper, theposition players (batters) pay attention to their batting-average (AVG) especially to finish eachseason with their batting average of just above .300. They obtained MLB season individual AVG
data from 1975 to 2008 and observed position players (= players except for pitchers) with at least
200 at-bats in each season. Then, they found that their distribution of the batting-average had
4excess mass just above .300, which revealed the existence of manipulation there. Furthermore,they found that players with batting-average of just below .300 were more likely to hit a base-hit
and less likely to get a base-on-balls. Both base-hits and base-on-balls avoid the batter frombeing gotten out, so for the team which he belongs to, base-on-balls also has important value
to win the game. However, batting-average does not count base-on-balls as an element to raise
the number (For the definition of performance stats, see Appendix), so they prefer getting hit tobase-on-balls.
There is similar behavior to the cases of Allen et al (2016): bunching. However, one important
thing we have to take care of is that they are professional athletes, and so there exists a procedureof the contract between the player and the team managers: those who evaluate the player.
They propose contracts to the players for each season before it starts. In other words, theobserved excess mass may be derived by their monetary value function, not by the preferenceof themselves. We will distinguish them, which is the main determining factor of the bunching
behavior of this paper.
Pope and Simonsohn (2011) conducted analysis only for batting-average. To reinforce theirresearch, below are our following topics. We first search for the round numbers of various
batting stats manipulated by the players. Then, we test if there exist any monetary incentives forthe players. The team managers and the players agree to the contract that sets fixed additional
bonuses which are paid when certain performance stats reaches the point. For the players whosestats are around the cutoﬀ points, making a discontinuously large eﬀort to succeed in bunching
can be interpreted as an economically rational choice, under the above contract. For example,
there should be little diﬀerence in the skills as a player between players with .300 of battingaverage and .299. Thus, rational team managers contract them almost the same oﬀer to thesetwo players, controlling other factors than the batting-average.
Therefore, two possibilities are to be concerned. One is that team managers oﬀer contracts
that make players make additional eﬀorts to achieve the goals. In this case, we can conclude
that the managers show reference dependence about evaluating players. The other is that players
have reference-dependent preferences, which leads to observed excess mass around the cutoﬀ
points. In this case, contracts do not bunch at any round-numbers.
Research about performance and monetary rewards using Major League Baseball has been
conducted in the field of labor economics. One of the first papers is published by Scully (1974).
In the next section, we describe the data used in our paper.

3 Data Description
To make empirical research, we need information about the players’ performance, contracts,
and other details. Then, we obtained panel data that contain this specific information from some5

open data-source. Each sample consists of stats of a player at the end of a single season. We
explain the specific information about the dataset below.
First, Performance data are obtained from baseball fan website: FanGraphs and BaseballReferences. We collect information from the 1957 season, the year when the qualified numberof plate-appearances was regulated. It is the cutoﬀ point to be eligible for the title of leading
hitter, the player with the highest batting-average. Stats in each season cover only the regularseason, not Spring-Training or postseason games. The full-sample size is N = 54469.
Note that we then restrict with the subsample including the players who appear to the platein MLB games enough to be tested because those with a small number of plate-appearances
are likely to be evaluated by other elements than their batting skills . . . pitching or fielding,
performance at the minor leagues, or those who injured at the season. Especially the battingaverage and on-base percentage of players with few plate-appearances are not be regarded asreliable. Pope and Simonsohn (2011) set the cutoﬀ at 200 at-bats, but alternatively, we use
200 plate-appearances as the required number to be considered in our analysis. This is because
at-bat does not count the number of base-on-balls or sacrifice bunts in the denominator, eventhough they surely appear to the plate and made something to their teams. On the other hand,plate-appearance stands for the number of chances of batting their coaches gave him. Restricting
the sample reduces the sample size to N = 18143.
The dataset includes the players’ plate-appearance, batting-average, on-base percentage,
homerun, stolen-base, runs-batted-in, base-hit, all of which are the main stats of interest in
this paper. Besides, for the regression analysis, we obtain additional stats: batting, fielding,and BaseRun, the estimated contribution to the team expressed in the runs they produced, and
WPA, or winning percentage added. Further explanations used in our analysis are provided in
the Appendix.
We then describe the characteristics of the stats. Baseball batting stats are roughly divided into
two types. The first is the one that simply indicates the number of certain plays, and the second
is the one calculated using these numbers, which indicates the rate or the expected number ofthe plays. Here we call the former “count stats,” and the latter “rate stats.” count stats are forexample “base-hit” or “home run,” or “stolen-bases.” count stats are irreversible and so they
are monotonically increasing in their appearances. That is, once the players reached a certain
number of count stats such as 20 home runs, it does not matter how their performance goes
afterward. On the other hand, rate stats can fluctuate: even once they reached their internalgoals, it may fall if their performance becomes worse. Batting-average and on-base percentage
are categorized to this type.
Second, we mention the salary data. Salary data are obtained from USA TODAY and BaseballReferences. We collected annual salary data of the position players who registered into MLB

6

Table 1: Summary Statistics for Sample A
Statistic
PA
AVG
OBP
HR
RBI
SB
H
Age

N

Mean

St. Dev.

Min

Pctl(25)

Pctl(75)

Max

18, 143
18, 143
18, 143
18, 143
18, 143
18, 143
18, 143
18, 143456.477.264.331
11.81151.8827.846
108.94128.506152.836
.032
.039
9.747
26.912
10.86942.933
4.042

200

320

591

778

.135
.174

.242
.305

0
4
0
29
18

.285
.356

4
31
1
72
25

17
69
10
143
31

.394
.609
73
165
130
262
46

Table 2: Summary Statistics for Sample B
StatisticN

Mean

St. Dev.

Age
PA
AVG
OBP
HR
RBI
SB
H
+WPA
-WPA
Bat
Fld
BsR
SalaryFA8, 9158, 9158, 9158, 9158, 9158, 9158, 9158, 9158, 9158, 9158, 9158, 8708, 9158, 9158, 91528.714471.946.268.337
13.44656.3398.534
114.232
8.715−8.270
3.257
.304
.092
3, 487, 838
.168

3.901
150.890.031.038
10.21327.62110.85142.4813.471
2.610
16.1397.482
2.712
4, 487, 344
.374

Min

Pctl(25)

19
200

26
342
.146
.174

.248
.311

0
5
0
30
2.030
−15.050−44.200−36.100−12.600
62, 500
0

6
35
1
78
5.820
−10.420−7.300−4.000−1.200512, 750
0Pctl(75)
31
605.5
.289
.360
19
74
11
148
11.430−6.06011.1004.400
1.200
4, 658, 334
0

Max
46
778
.394
.609
73
165
109
262
19.160−2.740116.800
37.00014.30029, 200, 000
1Roster at the beginning of each season. They also contain information about other player-specificcharacteristics: age, position (such as catcher, 1st-baseman, left-fielder, designated hitter andso on), the teams they signed, and the possession of the free agency. We merged this datasetwith the player’s stats in the previous year, because the salary is usually determined based on theperformance of the previous season. Because of the lack of disclosed information, this dataset
contains data on salary only from 1987. Because we regard the players’ performance as reflectedin their annual reward in that of the next year, we cannot merge the stats dataset of 2018. So
the latest available season is 2017. The aggregated number of the panel is N = 13226, and therestriction of 200 plate-appearances reduces the number to N = 8915. Then, here we use twodatasets: one that contains salary data (we call this Sample B), while the other does not (Sample
A). As we explain in the next section, we use two main analyses: manipulation of performancestats (only use play stats) and design of the contracts (needs information about salary). In theformer, we use Sample A, and in the latter Sample B, The descriptive statistics of each sample
are described in Table 1 and Table 2, respectively.

7

4 Empirical Methodology
4.1 Test for BunchingFirst, we test if we can observe any behavior that seems to be related to reference-point
dependent preference. As we explained in the previous sections, manipulation is verified by theobservation of bunching or an excess mass around the possible reference point. To specify theexcess mass, we apply the method of regression discontinuity design (RDD).
RDD is a way to measure the eﬀect of a treatment, such that whether the treatment is assigned or
not depends on the threshold of a certain variable (called “running variable”). Then, comparing
the samples just above and just below the threshold is suﬃcient examination of the treatment,
since they are in almost the same states except for the existence of the treatment.
However, there is an important assumption for this specification is regarded to be valid (Leeand Lemieux, 2010): continuity of the running variable around the threshold. In other words,
individuals must not be able to manipulate the running variable to cross over the cutpoint. If
there exists manipulation, then a selection bias problem arises, that is, those who try to be
assigned the treatment can adjust their running variable. Therefore, there are some empiricalways to test the manipulation of a variable, which is the very method I apply in our analysis. Oneof the frequently applied methods for this specification is McCrary(2008)’s local linear density
estimation. We avail of this to our specification of bunching.
This test of the manipulation at the cutoﬀ point c proceeds in two steps. First, let d(.) denotethe density function of the variable x to be tested (here we use the performance stats). Then
we undersmooth the observed density: determine the binsize b of x, and obtain the histogram.
And finally, we conduct local linear approximation to both just above and below the cutoﬀ point.
Note that the optimal bandwidth is to be selected by the fourth-order polynomial approximation.
Then, we estimate the frequency at the cutoﬀ point, dˆ(r ), by fitting the estimated density function
from both below and above the cutoﬀ, dˆ+ and dˆ− , respectively. Finally, we make the diﬀerence
between ln dˆ+ and ln dˆ− to calculate the statistics θ. With θ and its estimator of standard
deviation, t-tests can be conducted to specifying manipulation.

4.2 Monetary Incentives
Then, we examine the existence of the monetary incentive. For this procedure we use thespecification of the two functional features mentioned in Section 4.1: notch and kink at certain
cutoﬀ points.
First, we specify the salary scheme f (.) without a notch at the cutoﬀ. Then the entire salary8scheme is expressed as follows:
wit = β 0 + f (PERFit ) + β 2 ABOVEit
If f (.) is linear in the performance stats PERFit , thenwit = β 0 + β 1 PERFit + β 2 ABOVEit
For each player i in the season t, wit is logarithm of their average annual salary in next season
t + 1. PERFit is the value of performance stats (batting-average, on-base percentage,. . . ).
ABOVEit is an indicator of the achievement of their goals for the stats. If the estimated valueof β 2 is supported to be positive and significant, then we consider there is an additional bonus
paid to the player that reached a certain target value of the performance stats, which leads themto bunch.
As the second possibility, the kink at the reference point is specified as follows.
wit = β 0 + β 1 PERFit + β 2 ABOVEit + β 3 PERFit × ABOVEit
The first three terms on the right-hand side are the same as the specification of a notch,
the linear salary scheme in the performance stats. In addition, we add the interaction term of
PERFit and ABOVEit . This term stands for the return to the performance stats for the playersthat achieve the cutoﬀ point. In words, players with .320 of batting-average, for example, areevaluated by β 1 until .300, and then β 1 + β 3 for the rest .020 (it is the case of that bunching at.300 is argued). If the estimated value of β 3 is negative and significant, then there exists kink in
the reward function, which again supports the bunching at the cutoﬀ point.
Then, we describe the empirical specification. In our paper, we specify the local polynomial
functions described above by the regression analysis, for each performance stats (batting-average,
for example) and cutoﬀ point (.300). We employ the methodology of regression discontinuity
design (RDD1 ). We restrict the sample to those who are around the cutoﬀ of each stats (forexample .025 around .300 for batting-average). The bandwidths in each analysis are selectedaccording to the optimization of Imbens and Kalyanaraman (2009).
We also use linear regression with the interaction term of PERFit and ABOVEit to see thekink at the reference point. In each analysis, we restrict the sample to the players with their statsaround the cutoﬀ. The window sizes are selected ad hoc, so we tried various cutoﬀs to checkthe robustness.
wit = β 0 + β 1 PERFit + β 2 ABOVEit + β 3 PERFit × ABOVEit
1 We have to note that there is a problem to apply RDD in this case:Section 5., we come back to this topic.

9

The running variable is manipulated. In

Furthermore, to check the robustness of the results, we include the terms about the playerspecific characteristics as well. Thus, the model to be estimated is:

wit = β 0 + β 1 PERFit + β 2 ABOVEit + bZit
wit = β 0 + β 1 PERFit + β 2 ABOVEit + β 3 PERFit × ABOVEit + bZit
Team and individual fixed eﬀect models are also included.
One important problem of this method is the lack of consideration of the possibility ofendogeneity about our results. We discuss this in Section 5.

5 Result
5.1 Excess Mass Around The Reference Point
In this section, we present our main results of the analysis. First, we show the results thatverify bunching. Table 3 includes the summary of McCrary (2008)’s manipulation tests about
the performance stats. Consistent with Pope and Simonsohn (2011), there exists excess massaround .300, and besides .250 of batting-average. Manipulation was also observed in some ofthe other round numbers of other stats: .350 of on-base percentage, 20 of home runs, 100 ofruns-batted-in, 30 and 40 of stolen-bases, and 200 of base-hits.
For precise estimation of bunching, we set the binsizes of undersmoothing artificially: .001 forbatting-average and on-base percentage, 1 for homerun (HR), stolen-base (SB), plate-appearance(PA), and base-hit (H), and 4 for runs-batted-in (RBI). Batting-average (AVG) and on-basepercentage (OBP) are usually shown by three decimal digits, rounding the fourth decimal digit,
so strictly batters with .2995 of batting-average are taken as .300. As we mentioned in Section 3,
home run, stolen-base, plate-appearance, and base-hit stand for the number of plays in interest,
and they, therefore, take an integer and earn one for each plate-appearance. Runs-batted-in isalso an integer-stats, but it can get at most 4 at one plate-appearance, so we set it 4. To confirm
the robustness of our results, we repeated this test with various binsize, but we yielded essentiallythe same results. Bandwidths are optimized by calculation, following McCrary (2007).
For .300 of batting-average, when we increased the sample size compared with Pope andSimonsohn (2011), we obtained similar results: Players put eﬀort to manipulate their battingaverage. A diﬀerence between the estimated frequency according to the approximation below.300 and that of above .300 was significant at the .1% (z = 7.442) level.
In addition, bunching occurs in .250 (z = 5.061, p < .1%) as well. It was not reported
in Pope and Simonsohn (2011): there was no bunching observed in any other round numbersof batting-average, so bunching may have occurred before 1973 (Pope and Simonsohn (2011)10

Figure 4: Histgram of Batting-Average

Figure 6: Discontinuity at .250 of AVG
Figure 5: Discontinuity at .300 of AVG

11

Table 3: Test for Manipulation, leastPA = 200
stats

type

cutpoint

binsize

bandwidth

θ

z

AVG

rate

.300

.001

.019

7.442***

.250

.001

.024

.499
(.067).212(.042).139(.049).259(.075).311(.094).529(.124).481(.174).160(.063).453(.178)

OBP

rate

.350

.001

.024

HR

count

20

1

5.309

RBI

count

100

4

15.423

SB

count

30

1

10.00040

111.505

PA

count

500

1

.003

H

count

200

1

18.922Note

5.061***
2.854**
3.465***
3.295***
4.274***
2.764**
2.515*
2.547 ****: p < 0.1%, **: p < 1%, *: p < 5%.
Bandwidth is optimized following the method of McCrary(2008).

includes dataset after 1974). We specifically analyzed this in Section 6.4. Replicating analysis
with larger binsizes: .002 and .005 also yielded similar results.
On-base percentage, on the other hand, showed a similar tendency in .350, at the significancelevel was the 5% level (z = 2.854). Pope and Simonsohn (2011) reported that when theirbatting-average was just below .300 at the last plate-appearance of the season, they tried to
avoid getting walks to get a base-hit. This implies that players regard the batting-average as
more important than the on-base percentage. Our results, however, suggest that to some extent,
players set goals about their on-base percentages as well.
Bunching also occurs in count stats. They showed similar results. As well as batting-average
and on-base percentages, bunching is observed only in some round-numbers, not in all of them.
For Homerun, bunching occurs only in 20 (z = 3.465, p < 0.1%): there may be diminishingsensitivity: 20 is located on above the 75 percentiles of the whole Sample A. We observe
bunching also at 30 (z = 4.274, p < 0.1%) and 40 (z = 2.764, p < 1%) of stolen-base and
200 in base-hit (z = 2.547, p < 5%). Stealing-bases require skills that are talented to a limited
number of the players but succeed with the probability of 60% to almost 100%, so those whoare evaluated by their number of stolen-bases can manipulate them. 30 and 40 of stolen-bases
12Figure 7: Histgram of On-Base Percentage

Figure 8: Discontinuity at .350 of OBP

Figure 10: Discontinuity at 20 of HR
Figure 9: Histgram of Homerun

are also far above the 75 percentiles of all the players.
Base-hit was also manipulated, but the confidence level of the discontinuity was lower thanthat of batting-average (z = 2.547, p < 5%). Base-hit is a stats close to batting-average because
both stats increase by getting base-hits. We obtain this result because the number of base-hit
is not regarded as important as batting-average (Most TV live on baseball display a player’s
batting-average, not the number of base-hits.) Alternatively, we can consider that this is thediﬀerence between the count stats and the rate ones. If a player reaches .300 of batting-average,
for example, then he can keep it by not attending the rest of the plate-appearances or games.
Pope and Simonsohn (2011) pointed out that players with just above .300 were more likely tobe replaced with their last scheduled plate-appearance. On the other hand, if he gets the 200thbase-hit, then he does not have to care about the number of base-hits and attend the games toget better performance.
Surprisingly, such manipulation occurs in runs-batted-in (z=3.295, p<0.1%), too. Compared13

Figure 12: Discontinuity at 100 of RBI

Figure 11: Histgram of Runs-Batted-In

Figure 14: Discontinuity at 200 of Base-Hit
Figure 13: Histgram of Base-Hit

to other stats, runs-batted-in is harder to be manipulated because the number of that depends on
the performance of his teammates, and the number they can earn at a single plate-appearance
varies from 1 to 4. As in Table 3, there occurs the evidence in plate-appearances. However, it
may not be convincing because the optimized binwidth was smaller than one, even though thenumber of PAs takes only integers. Setting binwidth larger than 1, then the result turned out tobe insignificant, so we regard it as not supportive results for bunching. (In fact, there certainly
exist monetary incentives for plate appearances. We come back to here in Section 6.1.)
Summarizing the results, there exists bunching at some round-numbers of the batting stats
and they are the possible reference point of the players. In the case of marathon runners, Allenet al. (2016), there occurred bunching in every round number of the goal time, although thesize of discontinuity decreased. That is, it should be considered that the reference points are
not determined only by round numbers as argued by Pope and Simonsohn (2011). That is, thenature of the reference points are likely to be close to “par” in Pope and Schweizer (2011), or
the well-known standards that are related to the image of “skilled players,” rather than round14

Figure 15: Histgram of Stolen-BaseFigure 16: Discontinuity at .300 of

Figure 17: Discontinuity at .250 of

AVG

AVG

numbers; or well, these numbers are monetarily incentivized goals by the team managers. So
next, we examine whether there is any monetary bonus in their contract.

5.2 Existence of Monetary Incentive
In Section 5.1, we confirmed the discontinuities at the cutoﬀ points of the representative
stats. Then in this section, we show whether they target these goals based on their reference
dependence or their design of the contracts with a monetary incentive.
Table 4 describes the results of local linear regressions on the logarithm of their salary nextyear, with the cutoﬀs of each possible reference point. Column “Other Control” indicates “Yes” if
the model includes covariates (Defense, BaseRun, player’s age, WPA and dummy for possessionof the right of free agency) and season dummies, but “No” otherwise. “bw type” indicates the15bandwidths used in the model: “Optimal” includes the sample that is in the optimal bandwidth
calculated by Imbens and Kalyanaraman (2009), while “Half-BW” and “Double-BW” are using
a half and a twice of the Optimal bandwidth, respectively. The local average treatment eﬀect ofachieving the goals are described in the sixth column.
As a whole, no evidence supports the existence of monetary incentive to make an eﬀort for
their observed goals. There is no essential diﬀerence between the count stats and rate stats. Thatis, the reward function f (.) does not discontinuously jump at each cutoﬀ point.
The other possibility that encourages players to manipulate stats is the kink of the monetary
incentives. Table 5 to 12 show results of local linear regression including the interaction termsof the dummy variable and those of the corresponding stats, or Xit and ABOVEit . "PA" isthe number of plate-appearance. The second row is the performance stats, and the third one
is ABOVEit for the corresponding possible reference points. The fourth one is the interaction
terms.
Results again rejected the hypothesis of the existence of monetary incentives for bunching.
There is no evidence of neither kink or jump of their reward functions. Here we consider eachstats respectively.
First, for batting-average, results rejected the hypothesis of the existence of any additionalmonetary bonus for achieving either .300 or .250. Although estimated jumps at each cutoﬀ
point were positive, their standard errors are large and so the diﬀerence was insignificant. The
same results were obtained in the model with interaction terms (Tables 5 and 6): the estimatedcoeﬃcients of the dummies for achieving their internal goals and the interaction term with thebatting-average are all insignificant. That is, the monetary reward does not discontinuously
jump or kink at .300 of .250. These findings are consistent with the hypothesis that preferencesof the players are reference-dependent about their batting-average and excess mass is caused byreference-dependent preferences, not by the monetary incentives.
On-base percentage seems similar. The estimated jump is negative, although statisticallyinsignificant. Table 7 shows that negative kink is not observed. On-base percentage is considered
as relatively more important stats: it is a closer correlation with the winning-average of the teamthan batting-average. Thus, it can be the case that team managers attach a higher value toon-base percentage than to batting-average and think of paying players with a higher number
more. However, our results are against this hypothesis.
For the count stats, observed results are similar: 20 of homerun, 100 of runs-batted-in, and200 base-hits work as the cutoﬀ points of bunching, but not as the marginal points for the players’salary. Homeruns produce at least one score to the team and take one of the most “attractive”
aspects of baseball, so there may exist an additional positive eﬀect for the team: it may bring a
lot of audiences, which profits them by stadium fees. Nevertheless, the discontinuous scheme16

of the salary was not observed. Regressions with the interaction term do not report kink at eachcutoﬀ point.
Stolen-base, however, shows diﬀerent results. Some of the results from Table 4 showed asignificant discontinuity of about 30 stolen-bases. According to Tables 11 and 12, some of theregressions (column (1) for 30 SB and (1) and (6) for 40 SB) support the jump and negative
kink at the cutoﬀ points. That is, not all the results are consistent with the analyses of otherstats. However, we do not regard them as suﬃcient support. First, in Table 4, controlling otherplayer-specific characteristics, their significance level drastically goes down. In Table 11, theestimated values are a mixture of significant ones and insignificant ones, and they fluctuate from.816 to 13.567. And finally, for 40 stolen-bases, the results of estimation showed a negative
jump, inconsistent with that of 30, even though these results argue the same stats. Thus, weconclude these results cannot support the hypothesis of no jump or kink of the salary contracts
f (.), but either vice versa. Especially at this point, we require further analysis.
One possible alternative interpretation is that there exit the players that sign the contracts that
include plural-year service. Such a player receives a fixed salary regardless of their single-year
performance. Thus, we restrict the sample to those who have the right of free agency, whichenables them to negotiate with any MLB or other professional baseball teams. These players
just have finished their contract in the season, and so cannot play for any MLB teams without
signing a new contract, which always reflects his performance of the previous year. In theanalysis above, we consider the possession of the right of free agency by adding the dummy
variable that indicates whether he holds the right or not.
Table 13 shows the results of local-linear regression, using the restricted sample of free-agent
players. This is consistent with the main results: there does not exist evidence that supports the
additional reward at each cutoﬀ point. Analysis of stolen-bases was not conducted because ofa lack of players around there. Table 14 shows the results about .300 of batting-average, also
for the free-agent players. The obtained implication is the same as the first analysis: there is nokink and notch in the reward function.

5.3 Endogeneity Problem
One important factor to explain the results is the possible endogeneity problem. This is causedby the methodology we applied in the analysis above.
As we mentioned above, we partly used the method of RDD to specify monetary incentives.
The required condition for this analysis is that the running variable is not manipulated by theindividuals. RDD design assumes that individuals just above and just below the cutoﬀ are almost
the same except for that one is reaching the cutoﬀ. If some players know the jump or kink inf (.) at the .300 of batting-average, they try to keep their batting-averages above thereby avoiding
17

Table 4: Linear Regression on Monetary Incentives
stats,cutpoint
AVG, .300

Other Control
No
Yes

AVG, .250

No
Yes

HR, 20No
Yes

OBP, .350

No
Yes

RBI, 100

No
Yes

H, 200No
Yes

SB, 30No
Yes

SB, 40

No
Yes

Note:

Observations Estimate Std. Error
z
8514
.047
.061
.773
5599
.088
.075
1.174
8915
.067
.056
1.184
5930
.034
.056
.615
3005
.061
.077
.788
8605
.016
.045
.354
6110
.019
.068
.286
3496
.015
.092
.161
8539
.034
.054
.636
7271
.070
.052
1.340
4402
.066
.069
.953
8810
.075
.044
1.713
1315
.071
.175
.406
562
.073
.127
.576
2582
-.004
.109
-.034
1307
-.002
.141
-.015
560
.030
.102
.299
2558
-.032
.088
-.364
6440
-.038
.065
-.592
3542
-.076
.089
-.849
8656
-.029
.051
-.570
6525
-.013
.049
-.272
3673
-.055
.069
-.807
8637
.004
.039
.107
393
.072
.289
.250
228
.282
.400
.707
714
.008
.185
.043
390
.209
.018
.086
227
-.042
.324
.130
708
.056
.127
.435
75
-.786
.396
-1.985*
35
.386
.271
-1.421
137
-.061
.309
-.199
75
-.420
1.042
-.403
35
-4.779.576-8.288**
137
-.109
.413
-.265
282
.962
.372
2.585**
134
.920
.263
3.492***
714
.008
.185
2.941**
282
.379
.297
1.271
134
.290
.249
1.163
533
.408
.180
2.260*
134-1.276.453-2.818**
56
-.736
.383
-1.924
245
-.712
.313
-2.274*
134
-.346
.396
-.875
56
-.313
.429
-.730
245
-.115
.244
-.472
***: p < 0.1%, **: p < 1%, *: p < 5%.
Bandwidth is optimized following the method of Imbens-Kalyanaraman (2009).

bw type
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW

bandwidth
.084
.042
.170
.045
.023
.090
.036
.018
.072
.048
.024
.096
3.32
1.66
6.64
3.30
1.65
6.61
.044
.021
.087
.045
.022
.089
4.08
2.04
8.16
4.04
2.02
8.07
3.173
1.587
6.347
3.175
1.587
6.349
3.39
1.70
8.16
3.40
1.70
6.79
3.16
1.58
6.32
3.16
1.58
6.33

Table 5: Local Linear Regression on Monetary Incentives
(1)

(2)

(3)

(Intercept)

11.269 ***-3.374 ***-2.122 ***(0.386)(0.631)(0.635)PA

0.004 ***

0.004 ***

0.004 ***

0.002 ***

(0.000)(0.000)(0.000)(0.000)3.794 **

3.677 **

0.172

0.387

(1.418)(1.209)(1.219)(1.010)

AVG
AVG_300(4)-1.081-0.949-0.394-0.708(0.920)(0.815)(0.799)(0.647)

AVG * AVG_300

3.851

3.312

1.414

2.477

(3.009)(2.663)(2.611)(2.110)0.885 ***

0.874 ***

0.901 ***

(0.037)(0.036)(0.031)-0.013 ***-0.013 ***-0.013 ***(0.001)(0.001)(0.001)Age
Age^2Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

5960

5960

5930

5930

R squared

0.237

0.431

0.453

0.642

F statistic

519.892855.842520.717

P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

18

Table 6: Local Linear Regression on Monetary Incentives, .250 of Batting-Average
(1)

(2)

(3)

(Intercept)

12.923 ***-2.363 ***-1.746 **(0.465)(0.639)(0.644)PA

0.004 ***

0.004 ***

0.003 ***

(0.000)(0.000)(0.000)(0.000)AVG-2.561-3.677 *-5.514 **-3.425 *(2.004)(1.736)(1.719)(1.344)

AVG_250-1.404 *-1.753 ***-1.329 **-0.904 *(0.566)(0.487)(0.483)(0.385)

AVG * AVG_250

5.698 *

7.180 ***

5.480 **

3.783 *

(2.315)(4)0.002 ***

(1.993)(1.974)(1.568)Age

0.942 ***

0.937 ***

0.952 ***

(0.034)(0.034)(0.028)Age^2-0.014 ***-0.013 ***-0.014 ***(0.001)(0.001)(0.000)

Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

7307

7307

7271

7271

R squared

0.213

0.424

0.444

0.643

F statistic

544.8421040.701

621.448

P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2.01e+03).

Table 7: Local Linear Regression on Monetary Incentives, .350 of On-Base Percentage
(1)

(2)

(3)

(Intercept)

10.861 ***-4.325 ***-3.212 ***(0.457)(0.639)(0.644)PA

0.004 ***

0.004 ***

0.004 ***

(0.000)(0.000)(0.000)(0.000)4.720 ***

3.504 **

0.488

2.218 *

(1.407)(1.227)(1.249)(1.026)-0.824-0.1120.288-0.275(0.796)(0.685)(0.680)(0.558)

OBP
OBP_350
OBP * OBP_350(4)0.001 ***

2.274

0.295

-0.8670.745(2.254)(1.943)(1.930)(1.581)

0.944 ***

0.938 ***

0.962 ***

(0.035)(0.035)(0.029)-0.014 ***-0.013 ***-0.014 ***(0.001)(0.001)(0.000)Age
Age^2Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

6656

6656

6620

6620

R squared

0.220

0.419

0.435

0.632

F statistic

528.545927.252

540.942P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

the scheduled plate-appearances, for example. If the team managers that oﬀer the contract can
detect those who manipulate stats, the contracts seem worse than the observed number of eachstats. That is, the eﬀect of achieving the goals might be underestimated.

19

Table 8: Local Linear Regression on Monetary Incentives, 20 Homeruns
(1)

(2)

(3)

(Intercept)

13.540 ***-5.255 ***-4.906 **(1.094)(1.570)(1.545)PA

0.003 ***

0.003 ***

0.005 ***

0.003 ***

(0.000)(0.000)(0.001)(0.001)-0.024

0.015

0.018

0.032

(0.061)(0.052)(0.052)(0.049)HR
HR_20
HR * HR_20(4)-1.422-0.918-0.863-0.367(1.360)(1.157)(1.143)(1.059)

0.074

0.043

0.040

0.013

(0.072)(0.061)(0.060)(0.056)1.110 ***

1.091 ***

1.113 ***

(0.081)(0.079)(0.070)-0.016 ***-0.016 ***-0.016 ***(0.001)(0.001)(0.001)Age
Age^2Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

1315

1315

1307

1307

R squared

0.075

0.356

0.386

0.570

F statistic

26.693125.158

80.250P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

Table 9: Local Linear Regression on Monetary Incentives, 100 Runs-Batted-in(Intercept)
PA
RBI
RBI_100
RBI * RBI_100

(1)

(2)

(3)

14.542 ***-5.262-5.563(4)(3.858)(3.786)(3.762)0.002 *

0.003 ***

0.005 ***

0.003 *

(0.001)(0.001)(0.001)(0.001)-0.005-0.007-0.006-0.005(0.040)(0.034)(0.034)(0.028)-9.535-6.213-5.312-0.929(6.248)(5.703)(5.720)(5.132)

0.096

0.063

0.054

0.010

(0.063)(0.057)(0.057)(0.051)1.217 ***

1.253 ***

1.122 ***

(0.135)(0.146)(0.146)-0.019 ***-0.019 ***-0.017 ***(0.002)(0.002)(0.002)Age
Age^2Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

476

476

470

470

R squared

0.025

0.286

0.337

0.655

F statistic

2.914

27.13822.314

P value

0.021

0.000

0.000

Linear regressions on loggarithm of players’ salary next
season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust
standard errors are in the parentheses. The bandwidth ineach sample is chosen following to Imbens andKalyamanaran (2009).

20

Table 10: Local Linear Regression on Monetary Incentives, 200 Base-Hits(Intercept)
PA
H
H_200
H * H_200

(1)

(2)

(3)

-10.720-20.591-15.364(4)(41.586)(45.800)(52.998)

0.006

0.005

0.006

-0.004(0.004)(0.005)(0.006)(26.412)

0.111

0.083

0.046

0.584

(0.211)(0.227)(0.257)(3066.727)

47.35335.346

20.66494.406(48.323)(53.189)(55.472)(548282.725)-0.237-0.177-0.103-0.479(0.243)(0.267)(0.279)(2776.812)

Age
Age^2

1.070

1.121

2.640

(0.618)(0.717)(5465.018)-0.018-0.018-0.043(0.011)(0.013)(93.699)

Other Controls

X

X

Fixed eﬀects

Season, Team, Position#observations

75

75

75

75

R squared

0.042

0.119

0.207

0.936

F statistic

0.622

1.122

1.268

P value

0.648

0.359

0.264

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

Table 11: Local Linear Regression on Monetary Incentives, 30 Stolen-Bases
(1)

(2)

(3)

(Intercept)

12.892 ***-4.331-4.535(2.013)(2.910)(2.678)PA

0.005 ***

0.005 ***

0.003 *

0.002

(0.001)(0.000)(0.001)(0.001)SB
SB_30
SB * SB_30(4)-0.045-0.085-0.056-0.065(0.074)(0.065)(0.062)(0.069)6.364 *

1.687

2.209

0.632

(2.725)(2.444)(2.243)(2.531)-0.191 *-0.038-0.060-0.011(0.094)(0.084)(0.078)(0.088)1.154 ***

1.104 ***

1.132 ***

(0.164)(0.156)(0.170)-0.017 ***-0.017 ***-0.017 ***(0.003)(0.003)(0.003)

SeasonSeason, Team, Position

Age
Age^2
Other Controls

X

Fixed eﬀects#observations

361

361

361

361

R squared

0.232

0.456

0.547

0.751

F statistic

27.84358.66352.213

P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

5.3.1 Donut-RDD

To deal with this possible endogeneity, we repeat the analysis above without the players in the
bandwidth of .005 of batting-average. .005 is the obtained value by a base-hit for the player with
200 at-bats. Table 15 shows the results.
We report the same result: the coeﬃcients of the dummy variable and the interaction term
21

Table 12: Local Linear Regression on Monetary Incentives, 40 Stolen-Bases(Intercept)
PA
SB
SB_40
SB * SB_40

(1)

(2)

(3)

6.138

-4.8800.455(4.444)(4)(3.304)(4.399)0.004 ***

0.004 ***

0.003

0.005

(0.001)(0.001)(0.002)(0.004)

0.167

0.120

0.089

-0.007(0.087)(0.084)(0.081)(0.143)14.200 **12.501 *9.582 *

7.506

(5.387)(4.878)(4.666)(7.472)-0.370 **-0.321 **-0.252 *-0.186(0.135)(0.123)(0.118)(0.193)0.772 ***

0.466 *

0.935 **

(0.210)(0.227)(0.301)-0.011 **-0.006-0.014 **(0.003)(0.004)(0.005)Age
Age^2Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

167

167

167

167

R squared

0.215

0.365

0.520

0.781

F statistic

10.90114.65813.078

P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

Table 13: Local Linear Regression on Monetary Incentives for FA Players
stats,cutpoint
AVG, .300

Yes
AVG, .250

No
Yes

HR, 20No
Yes

OBP, .350

No
Yes

RBI, 100

No
Yes

H, 200

No
Yes

Note:

Observations Estimate Std. Error
z
503
-.175
.197
-.888
252
-.307
.302
-1.016
1043
-.180
.141
-1.271
509
-.253
.138
-1.832
266
-.209
.212
-.986
1038
.199
.102
-1.938
1366
.074
.102
.721
910
.147
.133
1.099
1501
.067
.090
.735
1367
.084
.082
1.020
923
.149
.107
.398
1480
.070
.072
.964
211
-.302
.300
-1.007
96
-.123
.226
-.543
387
.045
.203
.224
206
-.273
.296
-.924
95
-.156
.278
-.560
439
-.098
.174
-.565
1103
.034
.129
.262
597
-.106
.172
-.620
1469
.024
.105
.225
1044
.021
.107
.196
566
-.085
.153
-.558
1435
.016
.084
.194
50
-.100
.559
-.179
30
-.095
.949
-.101
102
.256
.333
.770
49
.195
.433
.449
30
-1.3601.295-1.050
100
.398
.160
2.481*
107
-.070
.447
-.156
61
-.439
.726
-.605
218
-.025
.293
-.086
106
.649
.355
1.824
61
1.084
.963
1.125
240
.264
.243
1.087
***: p < 0.1%, **: p < 1%, *: p < 5%.
Bandwidth is optimized following the method of Imbens-Kalyanaraman (2009).
For stolen-bases, it cannot be optimized because of lack of samples.

Other ControlNobw type
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW
OptimalHalf-BW
Double-BW

bandwidth
.025
.013
.052
.026
.013
.052
.056
.028
.114
.058
.029
.117
3.48
1.74
6.96
3.50
1.75
7.00
.045
.023
.092
.043
.021
.086
4.90
2.45
9.80
4.93
2.46
9.86
4.498
2.249
8.996
4.512
2.256
9.024

22

Table 14: For FA players: Local Linear Regression on Monetary Incentives
(1)

(2)

(3)

(Intercept)

6.803 ***

6.827 *

8.564 ***

(1.887)(3.000)(2.432)

PA

0.004 ***

0.004 ***

0.002

-0.001(0.000)(0.000)(0.001)(0.001)

20.329 **21.148 **12.552 *6.042(6.622)(6.699)(6.201)(4.675)

7.053 *

7.289 *

7.661 *

2.372

(3.266)(3.267)(3.100)(2.296)-23.807 *-24.652 *-25.787 *-7.886(10.864)(10.870)(10.311)(7.658)AVG
AVG_300
AVG * AVG_300
Age
Age^2

(4)

0.009

0.055

0.132

(0.150)(0.111)(0.101)-0.000-0.001-0.002(0.002)(0.002)(0.001)0.090 ***

0.090 ***

(0.012)(0.009)WAR3Other Controls

X

X

Fixed eﬀects

Season, Team, Position#observations

503

503

492

492

R squared

0.373

0.378

0.522

0.824

F statistic

82.74456.35347.385

0.000

0.000

0.000

P value

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

Table 15: Donuts RDD: .300 of Batting-Average
(1)

(2)

(3)

(Intercept)

11.640 ***-3.487 ***-2.397 ***(0.426)(0.683)(0.687)PA

0.004 ***

0.004 ***

0.004 ***

0.002 ***

(0.000)(0.000)(0.000)(0.000)2.412

2.904 *

-0.4640.526(1.575)(1.340)(1.349)(1.121)-3.049 *-2.262 *-1.597-1.109(1.249)(1.119)(1.086)(0.890)10.157 *7.501 *

5.233

3.721

(3.997)(3.574)(3.474)(2.841)0.908 ***

0.904 ***

0.923 ***

(0.039)(0.039)(0.033)-0.013 ***-0.013 ***-0.013 ***(0.001)(0.001)(0.001)

AVG
AVG_300
AVG * AVG_300
Age
Age^2
Other Controls(4)

X

Fixed eﬀects

X
Season, Team, Position#observations

5259

5259

5232

5232

R squared

0.234

0.430

0.452

0.642

F statistic

451.710744.160

455.735

P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

with the batting-average are statistically insignificant. That is, there is no jump or kink in theirreward functions. Analysis of other cutoﬀ points yields the same results.
5.3.2 Stats before Manipulation

Furthermore, we checked robustness by using sample before manipulation.
23

Table 16: For FA players, donuts RDD: .300 of Batting-Average
(1)

(2)

(Intercept)

7.929 ***

8.320 *

(2.338)(3.494)PA

0.004 ***

0.004 ***

-0.001-0.000(0.000)(0.000)(0.001)(0.001)

AVG(4)

16.17717.519 *12.830 *12.779 *(8.271)(8.418)(5.185)(5.851)

1.360

1.633

1.099

1.546

(5.017)(4.886)(3.368)(3.564)

AVG_300
AVG * AVG_300(3)-5.371-6.392-4.712-6.098(16.353)(15.962)(10.948)(11.645)

Age
Age^2

-0.018

0.114

0.169

(0.165)(0.105)(0.132)-0.000-0.002-0.003(0.002)(0.002)(0.002)

0.096 ***

0.089 ***

(0.011)(0.012)WAR3Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

397

397

389

389

R squared

0.372

0.379

0.795

0.823

F statistic

63.01343.812

0.000

0.000

P value

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

The objective of the players is to finish their season with their stats above the cutoﬀs, so they
start manipulation just before the end of each season.
Figure 17 shows the histogram of batting-average after 20022 , at the end of the games of
September 28th of each season. The regular seasons of MLB usually end by September, so on
28th, each team remain two or three games. Generally, players manipulate their barring-averages
by avoiding the last several plate-appearances at the end of each season: that is, this subsample
captures stats before manipulation.
Figure 18 displays the result of McCrary (2008)’s manipulation test.The figure in the left
shows the result on September 28th, while the other is that at the end of each season. No
bunching is found at .300 (z = 1.10, p = 27%) but at the end of the season, players manipulated
their batting-averages (z = 4.65, p < .1%).
Table 17 shows the result of local linear regression using the subsample above. Again, atleast, there are no statistically significant eﬀects found for achieving the round-number, .300.
Restricting the subsample to those with free agency yields essentially the same results.
5.3.3 Players Unable to Manipulate the Stats

Another point to consider the robustness is that manipulation is usually likely to take place
among players on the teams with their standings having been confirmed.

2 This is because of the limitation of data-availability.

24

Figure 18: Before-After: Manipulation Test

Table 17: Stats with “before” manipulation: .300 of batting-average
(1)

(2)

(3)

(Intercept)

13.037 ***-2.7000.058(2.835)(2.486)(2.414)PA

0.004 ***

0.004 ***

0.003 ***

(0.000)(0.000)(0.001)(0.001)-0.534-5.334-12.007-10.490(9.706)(7.592)(7.348)(7.189)0.717-0.117-0.027-0.054(4.424)(3.539)(3.445)(3.297)

AVG
AVG_300
AVG * AVG_300(4)0.002 *

-1.979

0.849

0.514

0.583

(14.727)(11.761)(11.436)(10.947)

1.038 ***

0.993 ***

1.005 ***

(0.073)(0.070)(0.071)-0.015 ***-0.014 ***-0.014 ***(0.001)(0.001)(0.001)Age
Age^2Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

1142

1142

1141

1141

R squared

0.193

0.460

0.508

0.598

F statistic

75.194183.063127.631

P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

The objectives of the MLB teams are to win the qualification for the postseason game, not tohelp players to achieve their internal goals, such as .300 of batting-averages. If teams contend
until the end of the seasons, it is natural that players are not allowed to manipulate their stats by
missing the games. Of course, the audience sees if players try to manipulate them, so there isno information asymmetry. In other words, players cannot secretly manipulate them.
Table 18 shows the results when restricting the sample to the players on the teams with closecompetition, which satisfies at least one of the following conditions. We define them as theplayers unable to manipulate their stats.
1. Teams that played 163 or more games at the season.

25

Table 18: Monetary Incentives for .300 of Batting-Average: Players of the teams with closecompetitions
(1)

(2)

(3)

(Intercept)

12.583 ***-1.977-0.539(0.863)(1.461)(1.430)PA

0.004 ***

0.004 ***

0.005 ***

0.003 ***

(0.000)(0.000)(0.001)(0.001)-1.611

0.115-3.148-3.021(3.140)(2.623)(2.629)(2.417)

AVG
AVG_300(4)-3.562-1.998-1.022-1.451(2.169)(1.947)(1.927)(1.686)

AVG * AVG_300

12.815

7.425

4.122

5.251

(7.074)(6.342)(6.285)(5.497)0.826 ***

0.796 ***

0.799 ***

(0.083)(0.080)(0.075)-0.012 ***-0.011 ***-0.011 ***(0.001)(0.001)(0.001)Age
Age^2Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

1119

1119

1114

1114

R squared

0.267

0.466

0.491

0.669

F statistic

120.826

190.300

114.724

P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

The number of games is set at 162 since 1962. When some teams are lined up in winning
percentage, however, then they play an additional match to decide which team to advanceto the postseason games, called "One -game playoﬀ." Needless to say, such teams went
through too close competitions to adjust the players’ plate-appearances to manipulate
their batting-averages.
2. Teams that player 161 or less at the season.
When a game was not played due to weather or other reasons, it is to be rescheduled.
However, at the end of the season and when the standings of the team had been established,
it is the case that the game is not taken place and the team end their regular season.
3. Teams that won(missed out on) the playoﬀ berth by one game.
Again, they experience close competition: almost at the end of the season.
4. All teams in the 1994 season.
The regular season in 1994 was abruptly terminated on August 11th, because of the strikeby the Major League Baseball Players Association. It is natural to assume that players
would not predict when the season ended, so they could not manipulate their stats, either.
Figure 19 shows the result of McCrary(2008)’s manipulation test. As that of Section 5.3.3.,
no bunching was found (z = .91, p = 36%).
Results again show no significant diﬀerence with those of previous sections. The estimatedcoeﬃcients of interest are the opposite of what had been expected.

26

Figure 19: Manipulation Test for Players "Unable" to Manipulate Stats

In summary, we found that players did not have monetary incentives at their observed internalgoals. Observing the reward functions f (.) for each performance stats does not either jump orkink at the possible reference points that are reported in Section 5.1. That is, they adjust theireﬀort level to make their performance stats just above the reference points, because of theirreference-point dependent preferences. Although we do not specify which of the two characteristics that lead to bunching is more appropriate, the players’ utility function can be describedwith such characteristics. In the next section, we consider other alternative explanations byconducting additional analysis and empirical evidence.

6 Alternative Interpretation and Some Evidence
In section 5, our analysis presented that there in fact exists bunching in some of the battingstats, but no evidence was observed in their contracts, which supports the hypothesis that theseobservations are driven by reference dependent preference of the players. Here we consider
some possible alternatives and additional discussions about our results.

6.1 Incentivized Contract
One most possible interpretation of our results is the incentivized design of the contract. So
far, we tested monetary incentives for the player, analyzing only the fixed parts of the contract.
However, players often sign contracts with an additional bonus according to their performances.
Even though the results in Section 5.2 did not support the existence of incentives in the fixedsalary, it may occur as these additional rewards. Here we present that this hypothesis will berejected by showing the specific contracts of some players.
Table 19 shows the specific contents of the MLB position players’ contracts, quoted by Cot’s

27

Baseball Contracts from Baseball Prospectus, an open website that discloses information about
that. In addition to signing bonus, fixed payment (we analyzed this part), and other optional
bonus or service, players receive some monetary incentives according to their performance.
They are roughly grouped into two: award bonus and bonus for reaching a certain number of
their stats. While the former includes winning Gold Glove or All-Star Game selection (a match
between the two big leagues of MLB, each of which is composed by players selected by the
manager and the fan’s vote), the latter consists of only round numbers with plate-appearances
or games they attended, neither batting-average, on-base percentage nor home runs. Besides,
there are at most 2 or three position players who sign such contracts. Pitchers are more likely toagree to ones with performance bonuses, whose trigger stats are also related to attendance: thenumber of games appeared, or innings pitched. Therefore, we can conclude that in the additional
bonus parts of their contracts, there are no incentives that encourage them to manipulate theirbatting-average, on-base percentage or other batting-stats.
Team managers have to design contracts with limited budget constraints. Plate-appearances
given to single teams are on average constant throughout the year because they play the same
number of games, so the team manager distributes the fixed numbers of plate-appearances to theplayers. That is, managers can predict how many players at most achieve their goals. On the other
hand, it is hard for the managers to estimate the total numbers of the players that achieve someround numbers of batting-average or homerun. If almost all the players reached the benchmarks,
then even if they led the team to win, managers have to owe additional expenditure. This point
can be a supportive discussion of our results.

6.2 Contract Length
Skilled players often sign contracts with a plural-year duration. This is related to why weanalyzed using the sample of players who held the right of free agency. Furthermore, we should
also take account of their contract length, that is, until when the players are insured to play forthe team they signed because it can be considered as the additional monetary bonus.
Krautmann & Oppenheimer(2002) researched this point. They used the salary dataset of
MLB from the 1990 to 1994 seasons and regressed log salary on an interaction term of theperformance proxy and the contract years they signed.
ln(SALit ) = β 1 + β 2 PERFit + β 3 ( PERFit ∗ LENGTHit ) + β 4 LENGTHit
The model is employed by Krautmann & Oppenheimer(2002). According to their results, thecoeﬃcient of the interaction term, β 3 , was estimated to be negative at the 1% of significance.
In words, the longer the contract years at once stretch, the less the return to their performance
becomes. This is caused, they claimed, by the player’s risk-aversive preference that dislikes the28Table 19: Descriptions of the Contract of the Specific Players• Ichiro Suzuki, Outfielder, 4-year contract with Seattle Marinars (2004-’07)– signing bonus- $6M– fixed payment- 04:$5M, 05:$11M, 06:$11M, 07:$11M– performance bonuses- $1.25M in performance bonuses for plate appearances∗ $50,000 each for 400 PAs, 2004-06∗ $0.1M each for 500 & 600 PAs, 2004-06∗ $0.1M for 400 PAs, 2007
∗ $0.2M each for 500 & 600 PAs, 2007
– award bonuses: $50,000 each for Gold Glove, All Star selection– trade-protection (Veto for moving the team without his acceptance):
limited no-trade clause (may block deals to 10 clubs)– Other∗ housing allowance: $28,000 in 2004, $29,000 in 2005, $30,000 in 2006, $31,000 in2007∗ interpreter, trainer, transportation for spring & regular season∗ 4 annual round-trip airline tickets from Seattle to Japan• Eric Sogard, 2nd-baseman, single-year contract with Milwaukee Brewers (2018)– fixed Payment- $2.4M– performance bonuses- : $0.15M each for 30, 50, 70, 90 games. $50,000 for 120 games• Alex Avila, Catcher, two-year contract with Arizona Diamondbacks (2018, 2019)– Fixed Payment- 18:$4M, 19:$4.25M– annual performance bonuses: $25,000 each for 350, 400 plate appearances. $50,000 each for
450, 500 PA. $0.1M for 550 PA.
(Quoted from Cot’s Baseball Contracts)risk of being fired. From a viewpoint of our model, those who achieve their goals, receive anadditional bonus, but instead of getting higher baseline salary, and they, therefore, choose tosign the contract with a longer duration. For the team manager, it is profitable to propose suchcontracts, which may enable them to hold highly skilled players with relatively reasonable costs.
In these days, it is usual that players sign the plural-year package contracts with the right to
opt-out: the player or the manager nullify the contract while it is under duration, for the players
to get some better contract, or for the manager to modify the contract or release the player. So a
more complicated model might be required to describe this situation, but it helps us to consider
these nonmonetary bonuses.

6.3 By-Time Analysis
Our research used data from a wide range of time: 62 years for performance stats, 31 years
for salaries. Through such a long time, techniques of the players or the quality of instruments
have evolved, which leads to change in mean or the standard values of the stats: that is, unlike
29the reference point “par” of golf, the reference point of baseball might move over time. Besides,
there may have been a lot of changes in the design of the contract they agreed. Here we
consider time-variable elements in our analysis. Specifically, two main possible eﬀects change
the contract design: one is the relative market power of the players and another change in therelative importance of each performance stats.
Relative market power has a direct relation to the contract. Before the system of free agency
was introduced, players were forbidden to move to other teams without permission by the teamwhich they belong to. The 1994 strike by the Players Association of Major League Baseball,against the team owners to request improvement of their treatment, also may have a great
influence on their contracts (See Appendix about the specific information about free agency andthe Strike).
It is also important to capture the change in the evaluation of each stats. Through the historyof baseball, a lot of stats have been invented to measure the performance/ability of the player,
and it has been argued which stats are the most eﬃcient ones to evaluate them. One of the mostimportant revolutions was the publication of ’Moneyball’(2003), written by Michael Lewis, afinancial reporter. In this book, he described that batting-average was not appropriate to measure
the quality of the players: on-base percentage is more close correlation with total runs the teamearns in the season. In practice, Oakland Athletics applied this strategy to form the membersof the team, and won the playoﬀ. This story was widely spread and changed the sense of view
about the baseball stats.
The impact of this publication was so great that it was evaluated in an economic article.
Hakes and Sauer (2006) tested Lewis’s claim in the econometric specification. They reported
that on-base percentage gave us a better explanation about the winning percentage of the teamthan batting-average, but team managers had been taking batting-average of more importancewhen evaluating players. After Moneyball was published, however, their evaluation revolved.
In 2004, a year after its publication, the estimated return to on-base percentage for the players
increased, compared to the previous 4 years.
Then, one possible question occurs: “Does the tendency of manipulation/discontinuous contract design also change through the history of baseball?”In this section, we replicate the methodologies conducted in the previous sections, but sorting
the sample into periods below:
1. Before Free Agency (1957 - 1975)
2. After Free Agency and Before Strike (1976 - 1994)
3. After Strike and Before Moneyball (1995 - 2003)
4. After Moneyball (2004 - 2017)30

Sample B does not include data from 1957 to 1986, so in the section of monetary incentive,
We conducted tests for only three parts except for “Before Free Agency.” From here, we focus
on the three important batting stats: batting-average, on-base percentage, and homerun.
6.3.1 Bunching

Table 20 shows the results of the McCrary (2008)’s manipulation tests, for each grouped
samples. Compared to the full-sample analysis conducted in Section 5.1, we observed partly
diﬀerent results for each stats.
First of all, .300 of batting-average is the most solid benchmarks of the players. Each subsample
shows the bunching at the cutoﬀ point, at least at the 5% level of significance. Although there
exist skeptical discussions on the validity of batting-average as a performance stats, it still works
as an important status for the position players. There are no other stats that show such a consistenttendency among the samples.
On the other hand, .250 seems not to be regarded as important after Moneyball, as otherprevious days. In this term, the bunching at .250 becomes insignificant. Compared to thesamples of the old days, the average level of the batting-average has been increasing. The
mean value of each samples are .259 (samples of ’57-’75), .264 (’76-’94), .271 (’95-2003),.263 (2004-2018), respectively, median values of which are almost the same. Note that in fact,
restricting the sample to the years until 1965, then the manipulation test shows no statisticalsignificance at .300 (z = 1.577, p = 11.45%). So as we mentioned in Section 5.1, it may be
because we extended our data range to 1957, that .250 works as a reference point. For samples
before 1965, bunching at .250 was significant at 5% level (z = 2.12).
Excess mass of homerun was significant only in the latest subsample. After the publicationof Moneyball, SABRmetrics argue that extra-base-hits (double, triple and home run) have asignificant eﬀect on an increase in the runs their team earns (Information of SABRmetrics isprovided in Appendix). Besides, on-base percentage, surprisingly, showed no evidence forbunching in the subsample level. Thus, as we mentioned above, players consider these stats
as less important ones to evaluate players than .300 of batting-average. Otherwise, we have toconsider the possibility that there may have existed some diﬀerent design of contracts. In thenext section, we describe an analysis of these results, showing that there also did not exist any
monetary incentive to achieve these points.

31

Table 20: Bunching Test for the Grouped Sample by Timestats, cutpoint
AVG, .300

AVG, .250

OBP, .350

HR, 20Note’57-’75bw
θ’76-’94’95-2003

2004-

full sample

.023
.573
(.146)
3.934***.028.250(.080)
3.149**.031.137(.089)
1.538
6.313
.222
(.150)
1.479

.020
.022
.019
.019
.566
.310
.403
.499
(.120)(.130)(.120)(.067)
z4.732***
2.393*
3.376***
7.442***
bw
.028
.032
.027
.024
θ
.151
.306
.121
.212
(.069)(.094)(.076)(.042)
z2.188*
3.242**
1.595
5.061***
bw
.030
.036
.030
.024
θ
.149
-.035
.137
.139
(.081)(.093)(.082)(.049)
z
1.846
-.380
1.672
2.854**
bw
6.677
10.165
7.273
5.309
θ
.214
.145
.315
.259
(.123)(.129)(.112)(.075)
z
1.751
1.117
2.819**
3.465******: p < 0.1%, **: p < 1%, *: p < 5%.
Bandwidth is optimized following the method of McCrary(2008).

6.3.2 Monetary Incentive

Table 21 shows the results of the local-linear regression conducted in Section 5.2 for the
restricted samples. Each column shows the statistics for the subsamples, and the far-right edgecolumn displays the result using the full sample, the same one as Table 4. Each result includes theother controls (that is, results correspond to “Yes” in “other control” of Table 4). Results of theanalysis including the interaction terms are shown in Table 24 to 25 (for .300 of batting-average.
Analysis for other cutoﬀs show the same results). According to these results, they are consistent
with the analysis in Section 5.2 for all the stats: Any notch or kink in the players’ monetary
rewards was not observed. Additionally, we describe the same analysis for the players with free
agency in Table 22 that again shows the same results. All the statistics that stand for the notch
are insignificant to their fixed part of the salaries. Therefore, we repeat the same conclusion as
Section 5: there are no monetary incentives that verify the reason that players bunch their stats.
It implies that the players have preferences that lead to excess mass, or that the utility functionsof the players have kink or notch at each cutoﬀ point, which works as a reference point for theplayer.
In sum, we obtain the additional conclusion of this section: there may have some factorsthat cause a change in the players’ attitude to their stats about bunching. On the other hand,
it is consistent over time that team managers do not make the reward function f (.) that leads
players to bunching. That is, it is caused by the players’ reference point dependent preferences,32

Table 21: Local-Linear Regression for the Grouped Sample by Timestats, cutpoint

bw, type

AVG, .300

Optimal’87-’94

HR, 20

OBP, .350

OptimalOptimalOptimal2004-

full sample

bw

.024

.042

.030

.045

Obs.

697

1806

1872

5930

estimate

-.034

.064

.066

.034

(.137)(.092)(.103)(.056)

-.250

.697

.637

.615

z
AVG, .250’95-2003

bw

.036

.043

.075

.048

Obs.

1482

1806

3991

7271

estimate

.154

.064

.076

.070

(.084)(.092)(.060)(.052)

z

1.825

.697

1.277

1.340

bw

4.183

3.685

2.46

3.30

Obs.

341

371

475

1307

estimate

-.255

-.348

.343

-.002

(.228)(.218)(.264)(.141)

z-1.122-1.600

1.300

-.015

bw

.031

.025

.027

.045

Obs.

1098

1281

2042

6525

estimate

.109

-.151

-.030

-.013

(.106)(.120)(.093)(.049)

1.031-1.262

-.323

-.272

z

***: p < 0.1%, **: p < 1%, *: p < 5%.

Note:

Bandwidth is optimized following the method of Imbens-Kalyanaraman (2009).

supporting Pope and Simonsohn (2011)’s interpretation.

7 Conclusion
This paper has considered the possible existence of monetary incentives behind the behavior
that appears to be related to reference-point dependent preference.
Three important findings are obtained. First, MLB position players adjust their eﬀort levelso that their performance stats to achieve their internal goals. These internal goals are roundnumbers, but not all of them work as such benchmarks. As a result, around the cutoﬀ points,
excess mass or bunching are observed. Second, at least in the fixed part of their contracts, there
is no clear evidence that supports the existence of monetary incentives. Although there remains
a room for discussion in stolen-bases, the salary scheme did not kink or jump at the cutoﬀ points
where bunching was observed. That is, players try to reach the benchmark of each performancestats, even though there are not any monetary incentives that lead them to do so. Thus, theseobserved eﬀorts are likely to be caused by their reference-point dependent preferences, as Popeand Simonsohn argued. Finally, repeating the analysis with subsamples divided by time, theplayers’ attitude to the cutoﬀ points has been changing over time, and it results in the change in33

Table 22: Local-Linear Regression for the Grouped Sample Including FA Players
stats, cutpoint

bw, type

AVG, .300

OptimalAVG, .250

Optimal’87-’94’95-2003

2004-

full sample

.060

.032

.039

.026

bw
Obs.

218

229

354

509

estimate

-.026

-.309

-.186

-.253

(.247)(.182)(.182)(.138)-1.832z-.108-1.700-1.020

bw

.018

.023

.078

.058

Obs.

123

227

716

1367

estimate

HR, 20OptimalOptimal

.293

.047

.084

(.230)(.103)(.082)

z

1.512

1.272

-.448

1.020

bw

5.35

3.504

3.566

3.50

Obs.

47

70

102

206

estimate

OBP, .350.425(.281)

.004

-.701

-.337

-.273

(.284)(.492)(.513)(.296)

z-1.600-1.423

-.657

-.924

bw

.034

.042

.031

.043

Obs.

154

344

395

1044

estimate
z

.080

-.174

.115

.021

(.291)(.179)(.188)(.107)

.276

-.971

.616

.196

***: p < 0.1%, **: p < 1%, *: p < 5%.

Note:

Bandwidth is optimized following the method of Imbens-Kalyanaraman (2009).

Table 23: Before Strike
(1)

(2)

(3)

(Intercept)

12.995 ***-1.0500.524(1.807)(2.009)(2.032)

PA

0.003 ***

0.003 ***

0.002 **

0.003 ***

(0.000)(0.000)(0.001)(0.001)-3.0361.653-2.244-2.713(6.324)(5.534)(5.580)(5.109)AVG
AVG_300(4)-4.805-3.749-4.037-0.368(3.422)(2.961)(2.908)(2.499)AVG * AVG_300

16.75412.832

13.842

1.628(11.329)(9.808)(9.641)(8.317)0.769 ***

0.744 ***

0.683 ***

(0.088)(0.086)(0.080)-0.011 ***-0.011 ***-0.010 ***(0.001)(0.001)(0.001)Age
Age^2Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

703

703

697

697

R squared

0.214

0.409

0.432

0.631

F statistic

64.34699.54657.617

P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

34

Table 24: After Strike, Before ‘Moneyball’
(1)

(2)

(3)

(Intercept)

11.421 ***-4.372 ***-2.434 *(0.663)(1.091)(1.092)

PA

0.004 ***

0.004 ***

0.003 ***

0.002 ***

(0.000)(0.000)(0.001)(0.001)

2.574

2.412

-2.747-1.072(2.428)(2.078)(2.070)(2.021)

AVG
AVG_300(4)-1.630-0.863-0.684-0.492(1.482)(1.323)(1.304)(1.271)

AVG * AVG_300

5.779

3.288

2.691

1.895

(4.859)(4.332)(4.270)(4.157)0.968 ***

0.944 ***

0.972 ***

(0.060)(0.060)(0.059)-0.014 ***-0.014 ***-0.014 ***(0.001)(0.001)(0.001)Age
Age^2Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

1782

1782

1771

1771

R squared

0.320

0.515

0.554

0.606

F statistic

226.726

360.087

246.146

P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

Table 25: After ‘Moneyball’
(1)

(2)

(3)

(Intercept)

13.262 ***-2.952 *-1.635(1.056)(1.229)(1.213)PA

0.004 ***

0.004 ***

0.003 ***

0.002 ***

(0.000)(0.000)(0.001)(0.001)-1.601

1.583-2.049-0.004(3.742)(3.031)(2.981)(2.924)

AVG
AVG_300(4)-0.990-1.307

0.113

0.212

(2.142)(1.752)(1.703)(1.644)

AVG * AVG_300

3.990

4.638

-0.132-0.539(7.079)(5.784)(5.620)(5.425)0.922 ***

0.912 ***

0.942 ***

(0.059)(0.057)(0.056)-0.013 ***-0.013 ***-0.013 ***(0.001)(0.001)(0.001)Age
Age^2Other Controls

X

Fixed eﬀects

X
Season, Team, Position#observations

1880

1880

1872

1872

R squared

0.217

0.476

0.511

0.585

F statistic

143.016

338.800

224.065

P value

0.000

0.000

0.000

Linear regressions on loggarithm of players’ salary next season.
Note: *** p < 0.001; ** p < 0.01; * p < 0.05. Robust standard errors are in theparentheses. The bandwidth in each sample is chosen following to Imbens andKalyamanaran (2009).

the tendency of bunching. Among the argued cutoﬀs, however, .300 of batting-average remained
consistent bunching over time. Thus, we conclude that it works as a solid benchmark for theplayers. On the other hand, there are no monetary incentives in the reward functions for all thesubsamples. This result supports the second finding.
Our results indicate that professional sports players seem to have preferences that yield utility
not only by the monetary rewards but also by their performance situationally. It may suggest
35that dealing with the cases of professional occupation requires some assumption that workers
do not consider the time for leisure and works as a tradeoﬀ. For the team managers, on theother hand, our analyses help make better contract packages that attract the skilled players more,or search the player relatively underestimated, to get the players more “eﬃciently.” We should
pay attention to the opposite-side approach: where kink or notch observed about the reward
function, even though any bunching does not exist. Although players do not feel importantas their reference points, there may be some cutoﬀ points that the team managers regard asreference points for evaluating players.
Simultaneously, as we described, we are required to continue the analysis of monetary incentives for performance stats. In Section 6.1, we suggested that contract durations be substituted
into direct monetary rewards. Rewards may work in the stage before signing contracts: players
who achieve the cutoﬀ points may be likely to be oﬀered by MLB teams than those who donot. Furthermore, although we do not include them in this paper, we might obtain another
implication by considering the rewards they may receive after their retirement: “reaching .300
of batting-average 4 times in his career,” should be some outstanding signals to get jobs as acoach of the baseball team, or a commentator in the TV show.
We can apply our results to design a more eﬃcient contract in the situation of labor economics.
The relative importance of the stats with expected runs or winning-percentages diﬀers from one
another. For example, On-base percentage is more closely correlated to them than battingaverage. Besides, the number of stolen-base has little correlation with them (the correlation isas weak as r = 0.07). For the team managers, then, players’ too much adherence to the statsthat do not aﬀect winning-percentage or ones that may be tradeoﬀ-relation with more importantstats (for example, batting-average and on-base percentage). To prevent such ineﬃciency, they
are sometimes required to make contracts that lead players to pay more attention to the priorstats. Of course, our analysis presents some implications for them. This viewpoint can also be
availed in other contract design.
As we mentioned above, in MLB, the performance stats have been recorded for almost 150
years, and thanks to the community of the fans, we can fairly compare the players played in thediﬀerent generations. As we used in this paper, rich information about their contracts has been
published. There exist only a few studies for international comparison: Many countries aroundthe world, Japan, Germany, Italy or Australia and so on, have their professional leagues. Of
course, there are several amateur players, those who do not receive any reward for their plays,
there also exists room for comparison among them.
To conclude our paper, we state that it is worth continuing analysis of baseball, both for the
sports itself and economics.

36

Appendix
A. Theoretical Frameworks
Many sports are so rich in performance stats that record plays in variable viewpoints. Amongthem, especially, baseball performance stats can distribute individual-separable evaluation to
the players, as well as units of the whole team. Thus, baseball performance stats are interpreted
as reliable proxies for the skills of the players. Here we assume that the players are trying tomaximize them, subject to their internal costs depending on their talented skills. After observing
the stats the players, team managers evaluate them and propose contracts of the next year (theyalso include the player-specific characteristics: age, position and so on). Then, the players’
monetary rewards are given by the following benefit function:
f it+1 = f ( Xit , Zit ),where F stands for the monetary reward function to the player i at time t + 1, and Xit and Zit
express the value of the stats and other player-specific characteristics, respectively.
We assume that Xit is determined by the eﬀort level of each player. To increase the eﬀort level,
they have to pay additional intrinsic costs. Let eit be an eﬀort level of the player i at season t,
and cit (.) be an continuous, increasing and convex intrinsic cost function. Note that the required
cost to achieve the same level of X varies by players, according to their skills. Therefore, themaximization problem of the players are described as follows:
max Uit = f ( X (eit ), Zit ) − cit (eit ).
eit

We assume the additive separable function of the utility.
An alternative specification is that each player evaluates himself as an athlete: he directly
yields benefit by his value of the performance stats. That is, players decide according to theirbenefit function
bit = b( Xit , Zit ).
In this model, the player no longer pays attention to his future monetary rewards. In this model,
we again assume the intrinsic cost determined by the player’s skill and eﬀort level. Anotherspecification of the maximization problem is described as follows:
max Uit = b( X (eit ), Zit ) − cit (eit ).
eit

These two ways of the specification are diﬀerent only in the first terms of the utility function
U (.).
37

Then, we consider the functional features of f (.) or u(.), which cause bunching of thehistograms of the stats. In this paper, we suppose f (.) or b(.) satisfy at least one of the two
assumptions quoted in Section 2: one is “kink” at the reference point, and the other is “notch”
of the function.
1. “notch” at r (Figure 20)
The first possible assumption is a discontinuity at a certain cutoﬀ point of the function,
described as follows:
lim br (r + ϵ) , lim br (r − ϵ)

ϵ →0

ϵ →0

lim f r (r + ϵ) , lim f r (r − ϵ)

ϵ →0

ϵ →0

This is the discontinuous form of the function at the cutoﬀ point r, introduced by Diecidue
and Van de Ven (2008).
2. “kink” at r (Figure 21)
lim br′ (r + ϵ) , lim br′ (r − ϵ)

ϵ →0

ϵ →0
′
lim f r (r + ϵ) , lim f r′ (r − ϵ)
ϵ →0
ϵ →0

u′ (.) and f ′ (.) stand for the first-order diﬀerential of u(.) and f (.), respectively (u(.)
and f (.) are assumed to be continuous diﬀerential). As we explained in Section 2, this is
the primitive form of reference dependence, introduced in Kahneman and Tversky (1979)
and Tversky and Kahneman (1992).
f ( X ), b ( X )f ( X ), b ( X )

r

O

r

X

X

Figure 20: "notch" at the reference point

Figure 21: "kink" at the reference point

Allen et al. (2016) also utilized these types of model, and specified bunching of the marathon
runners’ finish time around the round numbers.

We cannot determine which model is appropriate to interpret the observed behavior of theplayers: bunching around the possible reference points. Our interest is that which of the two
38

assumptions are observed. If the players’ reference-dependent preferences for the performancestats cause bunching, then b(.) has functional features that represent reference dependence. On
the other hand, if their contracts are designed to lead them to the bunching around some possiblereference point, now we can regard that f (.) has the features described.
Both two possible assumptions result in bunching at the cutoﬀ point: in our paper, we consider
that these cutoﬀs are the round numbers of the performance stats, such as .300 of batting-average.
3 In words, the players’ salary scheme might be designed such that players make an eﬀort tomeet their cutoﬀ points. In the rest of this subsection, we explain the possible design of thecontracts.
Let us consider the first case: the monetary reward “jumps” at the possible reference point.
Then, the salary function is decomposed into two terms as follows.
f it+1 (.) = a( Xit , Zit ) + Dit Bonus( Xit )f (.) is determined by the number of the performance stats X and other player-specific characteristics Z, note that this term is continuous in X. D is the dummy variable that depends
on X: to indicate if the player’s stats are above the cutoﬀ point, but otherwise zero. If theplayers achieve the cutoﬀ point like .300 of batting-average, then f discontinuously rises by theappearance of Bonus term. And as a result, the salary scheme with this form of function causes
the excess mass around the cutoﬀ, by the players who desire the additional monetary rewards.
The second case, on the other hand, expresses the “kink” of f as follows:
f it+1 (.) = (1 − Iit ) g( Xit , Zit ) + Iit h( Xit , Zit )I is the same indicator as D, which indicates 1 if X is larger than the cutoﬀ, but 0 otherwise.
Two salary schemes are depending on the players’ performance stats, represented by g and h.
g is applied for the players with the stats below the cutoﬀ, and h is for those with above there.
Furthermore, h consists of two terms: h(., .) = g(c, Zit) + k( Xit − c). c stands for the given
cutoﬀ. In words, players receive the continuous reward in their performance stats, but the returnto the stats changes after they achieve the cutoﬀ: in our assumption, decrease. More directly,
the marginal rate of return to their performance stats discontinuously diminishes: kinks at thereference point.
While we cannot observe bit directly, f it can be observed as the contracts that players agreed.
If f it has the discussed functional features, then we conclude that the contracts’ contribution tobunching. In the next section, we describe how to specify them.
3 Therefore, it is not so important to determine which of the two is appropriate to describe the bunching itself: thatis, our interest is whether f (.) has these types of functional features in the number of performance stats.

39

B. Reference to the stats
Here we describe the details of the stats used in the paper.
• Definition– Batting-Average (AVG)
AVG =Base-Hit
At-Bat

The rate of base-hit. At-Bat(AB) is calculated as AB = PA − Base-on-Balls − Hit-by Pitch −Sacrifice-Bunt − Catcher-Interferance.
AVG depends only on the number of base-hit, so when players intend to manipulate AVG,
they try to get base-hit, not base-on-balls. Moreover, AVG does not identify the number ofbases they get at one base-hit: single, double, triple, and a home run.
– On-Base Percentage (OBP)
OBP =Base-Hit + Base-on-Balls + Hit-by-Pitch
PA − Sacrifice-Bunt − Catcher-Interferance

OBP is diﬀerent from AVG in that it takes base-on-balls and hit-by-pitch into account. Moneyball and Society of American Baseball Research (SABR) argues that OBP is more correlated
to the runs the team earns than AVG.
– Batting, Fielding, BaseRun
SABRmetrics, which considers baseball by the scientific approaches, have argued that which
player is more important for the team. You measure Players’ contribution in the following
procedure:
1. Convert each play (base-hit, strike-out, sacrifice-bunt, . . . ) into runs-value.
2. Convert runs-value into a contribution to the number of wins the team obtains from thecontribution of the player.
Batting, Fielding, and BaseRun stand for the first step of the identification: the estimatednumber of runs the player produces by his batting, fielding, and base-running above the average.
For the team they play for, they are kinds of clear indicators of the players’ contribution.
– WPA (Win-Probability-Added)
By the statistics of the games, SABRmetrics have obtained the winning percentage at eachspecific situation of the game: scores (2-runs forward, 3-runs behind and so on) inning,
runners on-base. Then, we can define the change in the win-percentage generated by thesingle play. WPA is a cumulative stats that stands for the summation of this change that asingle-player produced.
Most studies by SABRmetrics evaluate only the result of each play, excluding the context
then: for example, if players hit a home run, then we treat it equally. That is, it does not matterthe home run was a walk-oﬀ homer occurred when the game was tied, or solo-homer when
the game had been broken. This is because this type of skill called “clutch” does not correlate
between a year to year.
However, it may be the case that team managers evaluate this type of skills in the contracts, so
our regression analysis for monetary incentives (Section 5.2 and 6.3.2), we employed WPA
as a regressor. We include the stats, distinguishing positive (raises the winning-percentages)

40

plays and negative (decline) ones, and each value divided by the plate-appearance. And
actually, the results of each regression supported that these terms aﬀect the monetary rewards.

C. Important Events Related to Section 6.4• Free Agency
Free agency, the right for players to negotiate and choose any teams he wants, was regulated in
1975. Until then, players were forced to play for the team serving, except for the manager fired
him. Thanks to this rule, now any players who have been serving in MLB for about 6 seasons, areeligible to get a free agent. By this procedure, they are free from the monopsony contract with theteam playing for. And so, players got more likely to be evaluated accurately with his actual skills.
The right to arbitration is the same kind of this.
• Strike by the Players Association (1994)
In 1994, the Players Association of MLB declared to make Strike, to stop the owners’ cartel. Thesedays, because of the increased salary of the skilled players, they are suﬀered from keeping their
payroll in their budget constraint. So to resist this, they are tried to make cartels that refrain fromproposing any or suﬃcient contract to the super-skilled players. This results in such cases that
players who have enough skills chose to go to Nippon Professional Baseball (NPB).

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