aharoni sarig - hot hands in basketball & equilibrium

8/3/2019 Aharoni Sarig - Hot Hands in Basketball & Equilibrium

1/58Electronic copy available at: http://ssrn.com/abstract=883324

Hot Hands in Basketball and Equilibrium

By

Gil Aharoni

University of Melbourne

Oded H. Sarig

IDC, Herzeliya

and

The Wharton School

First version: September 2005

This version: August 2008

Part of this research was conducted while Aharoni visited IDC. We thank Jacob Boudoukh, Christine Brown, Bruce

Grundy, seminar participants at IDC, Tel Aviv University, University of Melbourne, University of New South Wales,

the Wharton School, and participants in the EFA meetings in Zurich, the Australian Finance and Banking meetings in

Sydney, and the AFFI in Bordeaux for helpful suggestions and comments. We also thank Gilad Katz and Gilad Shoor

for their excellent research assistance.


2/58Electronic copy available at: http://ssrn.com/abstract=883324


Abstract

Past literature suggests that success rates in professional basketball are independent of past

performance and has been interpreted as evidence that the commonly-shared belief in Hot Hands

(HH) is a cognitive illusion. This is often cited as evidence of biased decision-making even when

financial stakes are high. We argue that this interpretation ignores changes in both teams behavior

after the detection of anHHplayer. We derive testable hypotheses that differentiate betweenHH

as a real phenomenon and a cognitive illusion. Analyzing an entire NBA season, our results are

consistent withHHbeing a real phenomenon.


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Introduction

Traditional economic analysis assumes rational, optimizing behavior on the part of economic

agents. Behavioral economics casts doubt on this traditional view by pointing out limitations and

biases in the way people behave. In particular, when agents face uncertainty, behavioral economic

studies document a limited ability of people to properly judge probabilities. For example, it is

argued that people expect small samples to look like the properties of the generating distribution,

even though probability theory does not imply such a deduction (e.g., Tversky and Kahnemann

(1971), Wagenaar (1972)). Similarly, people incorrectly generalize patterns in small samples to

the properties of the generating distribution, expecting small samples to represent the

population. Behavioral economists argue that this bias, called the Representative Heuristic,

illustrates the inability of people to properly infer probabilities from small samples.

Like many other behavioral biases, the Representative Heuristic is documented in

laboratory experiments. Some researchers question whether these laboratory results extend to real-

world situations. The main reason to doubt the general applicability of such experimental results

is that there are more significant consequences to real-world decisions than to laboratory

decisions. Indeed, Slonim and Roth (1998) show that deviations from optimal strategy decline

when financial stakes and experience increase, even in laboratory experiments. Hence, behavioral

studies attempt to show that such biases impact actual decisions, resulting in significant

consequences. For example, several studies such as DeBondt and Thaler (1985), and Barberis,

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Shleifer, and Vishny (1998) use the Representative Heuristic to explain patterns in securities

returns. However, Fama and French (1992), Chordia and Shivakumar (2002), and Cochrane et al.

(2008) argue that these patterns in securities are consistent with rationality.

A notable exception to the reliance on laboratory data in the study of biases is the body of

studies that examine beliefs regarding hot hands (HH) in professional sports, where financial

stakes are high. According to the HH belief, players enjoy periods in which they perform

significantly better than their own respective averages. Coaches, players, and fans identify these

HHperiods via strings of successes: strings of successful basketball shots, tennis serves, baseball

hits, etc. Studies of theHHbelief examine whetherHHtruly exists or is a cognitive illusion that

illustrates the Representative Heuristic, in that people see patterns in small sports samples where

no pattern truly exists. In these real-life settings, misperceptions of the chance of success (i.e.,

perceiving anHH where none exists) may affect player behavior and result in significant

monetary consequences.

One of the first, and most cited, studies ofHHis Gilovich, Vallone, and Tversky (1985)

(GVT).1

GVT test for the existence ofHH in basketball. In particular, GVT test whether a

players chances of scoring a basket (Field Goal Percentage orFG%) after consecutive

successful shots are higher than after consecutive misses. Their main result is that the actualFG%

of NBA players is independent of past performance. GVT argue that the data demonstrate the

operation of a powerful and widely shared cognitive illusion. More importantly, as argued by

1 To date, the GVT paper has been cited in more than 300 papers in various academics fields, including psychology,

Medicine, statistics, and sport. Economics and finance papers that quote GVT include: Barber and Odean (2001),

Barberis and Thaler (2002), Bloomfield and Hales (2002), Durham et al. (2005), Hirshleifer (2001), Li (2005), Odean

(1998), Rabin (2002), and Romer (2006).

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GVT and others, the false belief in HH is likely to result in suboptimal decision making by

basketball teams.2

Since professional basketball is a multi-billion dollar business involving

experienced professionals, these results suggest that the Representative Heuristic operates not just

in laboratories, but also where decision making carries significant monetary consequences. This

result, therefore, casts doubt on the universal rationality assumed by classical economic analysis.

In this paper, we revisit the results of GVT and others and their interpretations. In

particular, we examine the potential behavior changes entailed by anHH, and their implications

for the empirical tests of theHHphenomenon as either a real phenomenon or a cognitive illusion.

Consider, for example, the behavior of the defensive team once it identifies an offensive player as

anHHplayer. The mere belief inHHshould result in the defensive team shifting its efforts from

non-HHplayers to theHHplayer. As a result, non-HHplayers (henceforth average hand AH)

are afforded easier shots and theirFG% improves. The effect of the defensive shift on theHH

player himself depends on the validity of theHHbelief. IfHHis a cognitive illusion then a player

who is perceived to beHHwill shoot with the same offensive abilities but face a tougher defense.

This will entail a decline in theFG% of the player who is erroneously perceived to beHH. If the

HHbelief is real then theFG% of theHHplayers should change little as they net two opposite

effects. The improvement in the shooting ability of theHHplayer and the higher defensive efforts

allocated to him.

We develop a simple model of game shots. Our model predictions yield two testable hypotheses

that can differentiate between HHas a real phenomenon and cognitive illusion. a) IfHHis a

2 Various researchers argue that deviations from rationality do not necessary lead to suboptimal decision making (for

example, Kyle and Wang (1997), Bernardo and Welch (2001)).

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cognitive illusion then the increase in theFG% of theAHplayers should be accompanied by a

precipitous decrease in theFG% of theHHplayers. IfHHis real phenomenon then the increase in

FG% of theAHplayers should be accompanied by little change in theFG% of theHHplayer.

b) IfHHis a cognitive illusion then theFG% of the team with the HHplayer are likely to be

similar to their normalFG%.However, ifHHis a real phenomenon then the FG% of the team

should increase substantially.3

Using an extensive database of more than 1,200 games, we show that equilibrium

predictions regarding player behavior and game characteristics are consistent withHHbeing a

real phenomenon. Specifically, we examine game characteristics after identification of an HH

player, as in GVT and others when a player has a string of successful shots. Our empirical

investigation consists of three main parts. First, we test whether game participants react to a string

of successful shots by a single player. We find that after identification of anHHplayer, there are

significant changes in both defensive and offensive strategies. This is evident from the number of

shots attempted byHHplayers, in the fraction of difficult shots they attempt, in the number of

three-point shots allowed, and in the number of fouls committed. These changes are largely

consistent with the argument that defensive efforts are shifted fromAHplayers toHHplayers.

The second part of the empirical investigation tests the validity of the HHbelief by

examining the change inFG% of theHHplayers, theAHplayers and the team as a whole. Our

findings suggest a large and significant increase in theFG% forAHplayers after a teammate is

3 The intuition behind this test is that ifHHis a cognitive illusion, then both the offensive and defensive teams err in

their strategies. The defending team allocates too much defensive efforts to the player erroneously perceived to be

HH. The offensive team errs in relying too heavily on the shooting ability of the players perceived to be HH.

However, ifHHis real, then the improved shooting ability of theHHplayer should lead to a higherFG% for the team.

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identified asHH. Consistent with previous research, we find that the success rate ofHHplayers is

insignificantly different from their normal shooting ability. The ability ofHHplayers to maintain

their normalFG% in the face of intensified defensive efforts shows that their shooting ability is

indeed improved when they are identified asHH, and thatHHis not a costly cognitive illusion.

Consistent with the improved shooting ability ofHHplayers, we find that the overallFG% of a

team improves materially after one of its players is identified asHH. We note that intertemporal

substitution of defensive efforts can affect the improved performance of a team with an HH

player. This effect is clearly limited in the fourth quarter of a close basketball game. It is in this

period that we find the greatest improvement of the team. We estimate theHHeffect in the fourth

quarter of close games to be roughly ten points on a game basis.4

Finally, we examine whether alternative explanations can account for our results. First, we

test whether the improvement inAHFG% is because they are encouraged by the successes ofHH

players and put more effort into their own offensive game. Second, we examine whether the

documented changes are not directly related to the identification of theHHplayers but rather to

the quality of the players that are on court when HHplayers are identified. Third, our results may

be attributable to concavity in the relation between defensive efforts andFG%. We examine these

alternative explanations and find that their predictions are not consistent with the data.

Prior research, including GVT, recognizes that team behavior might change when a player

is identified to beHH. Thus, prior research attempts to overcome this difficulty by examining the

HHphenomenon in cases where defensive behavior cannot change when a player is perceived to

beHH. For example, GVT examine two cases of shots without defense: free throws and practice

5

4 A team scores in a basketball game roughly 100 points.


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throws.5

GVT find no serial correlation in free throws or practice shots, which they interpret as

supporting the conclusion that the belief in HH in basketball is a cognitive illusion. We re-

examine the findings of GVT regarding free throws. As in GVT, we find no simple dependency

between the outcomes of first free throws and second free throws. However, definingHHin free

throws as a string of consecutive shots (as we do for field goal attempts) reveals thatHHdoes

exist in free throws as well.6

This paper proceeds as follows. In Section II, we present our model and derive testable

hypotheses. Section III presents our data. In Section IV, we present our results regardingHHin

field goal attempts. Section V tests the existence ofHHin free throws. Section VI points out

several economic implications from both the model and the empirical results, and Section VII

details our conclusion.

I. MethodologyWhile a complete model of basketball games is a daunting task, for our purposes, it is enough to

model an individual play within a game. Within an individual play, we analyze offensive shot

selections and defensive decisions. In each play, the objective of the offensive team is to

maximize the probability of scoring, while the objective of the defensive team is to minimize this

probability. Our model is simple enough to be tractable, yet rich enough to yield testable results.

5 Note that unlike field goals and free throws, GVT (and other research) has not documented a widespread belief in

HHfor practice shots.6 Several studies examine the existence ofHHin other sports and report mixed results. The evidence suggests thatHH

does not exist or is minimal in baseball (Albright (1993)) and tennis (Klassen and Magnus (2001)). Evidence

supportive ofHH is reported in billiards (Adams (1995)), bowling (Dorsy-Palmateer and Smith (2004)), and

horseshoe pitching (Smith (2004)).

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In our model, all players begin with equal abilities, both defensively and offensively.7

At

some point, both teams identify an offensive player as an HHplayer. Later, we discuss the

empirical process by which we identifyHHin our study. For modeling purposes, however, we

abstract from the identification process and assume that the identification is correct in the

following sense: the temporaloffensive ability, HHiO , of anHHplayer isperceivedto be higher

than his normaloffensive abilities by . Non-HHplayers are called Average Hand players,AH.

Thus:

+= AHi

HH

i OO

Note that ifHHis a real phenomenon, will be both perceived as and actually positive, while if

HHis a cognitive illusion, the actual is zero, but both teamsperceive it to be positive.

Our focus is on the differences between the five players that are on the field of play at the

same time. The model examines a setting in which there is at most oneHHplayer on the offensive

team.8

In particular, we do not distinguish between the fourAHplayers in our model. Therefore,

there is no loss of generality in assuming that the offensive team is comprised of two players:

player 1, who can be eitherHHorAH, and player 2, who isAHand represents fourAHplayers.

Accordingly, in our model, there are two defensive players, who allocate their defensive efforts

between the two offensive players.

7 While this simplifying assumption may seem untenable to anyone even mildly familiar with the game, as we show in

Table 2 below, prudent allocations of defensive efforts counteract natural differences among players. Therefore, this

assumption should be viewed as depicting games afterteams adjust to the differential abilities of players.8 Consistent with this setting, in our empirical investigation we examine shots in which up to one player is identified

asHH. The proportion of shots with more than one player identified asHHis less than 1%.

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We consider each shot as a sequence of three events: identification of player 1 as AHor

HH, defensive effort allocation, and shot selection and realization. Accordingly, there are three

times in the model:

0 1 2

|________________________|____________________________|

HH or AH Defense Allocation Shot Selection& Realization

To model defensive decisions, we simplify the complexity of possible shots by considering two

types of shots: difficult shots, denoted by d, and easy shots, denoted by e. An AH playersFG%

of an e shot is Oe, and theFG% of a dshot is O

d. Defensive effortsaffect the mix of shots: the

more defensive efforts allocated to a player, the lower the probability that the shot will be an easy

one. In our model, when i defensive effort is allocated to playeri, the probability of a difficult

shot is i, and the probability of an easy shot is (1 - i). Since Od


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1 < =+ 21 < 2 (2)

The first assumption imposes the condition that none of the NBA players is so weak that the

defense will allow him easy shots in all offensive plays. The second restriction requires that the

defense is not strong enough to eliminate all easy shots from the game and not weak enough that

the majority of shots that each player faces are e shots.

Next, we analyze shot selection and defensive effort allocation in the two possible states of

the game: when both player 1 and player 2 areAHand when player 1 isHHwhile player 2 isAH.

Since our analysis focuses on player 1 being eitherAHorHH, we call the case where both players

areAHtheAHcase. Similarly, the case where player 1 isHHwhile player 2 isAHis called the

HHcase. In both cases, there are four possible realizations of shot difficulty at time 2, namely

{ } { } { } { }, , , , , ,d d d e e d and e e

212121 )1(),1(,

.The probabilities of these shot combinations

are , and )1)(1( 21 , respectively.

A. The AH Case

To maximize the probability of making the shot, at time 2, the offensive team selects the player

with the maximal probability of success to take the shot. Since both players have equal ability in

this case, if one player has an e shot and the other a dshot, the shooting player will be the one

with the e shot. If both players have equal shotse ordthe shooting player will be chosen

randomly.

The defensive team allocates its defensive efforts to minimize theFG% of the offensive

team.

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Result 1: In theAHcase, the defensive team divides equally between the offensive players:

221

==

Proof: See the proof for this result and proofs for all other results in the appendix.

Since Result 1 shows that, when both players areAH, the optimal allocation of defensive

efforts is /2 to each player, in this case the probability of a dshot is (/2)2, the probability of an e

shot is 1-(/2)2, and the resultingFG% is:

( ) ( )ed

OOFG

+=

22

212% (3)

B. The HH Case

While theHHcase has the same four possible realizations of shot difficulty as in theAHcase, the

choice of the shooting player is less obvious. When both players have the same shot type e ord

the HHplayer will shoot, as the offensive team attempts to exploit his perceived improved

probability. TheHHplayer will also shoot when he has an e shot, while theAHplayer has a d

shot. Potential ambiguity arises when theHHplayer has a dshot, while theAHplayer has an e

shot. This is because, in this shot combination, the difficulty of the shot theHHplayer faces is

offset by his perceived improved offensive ability. We assume that theFG% of an e shot is higher

than theFG% of a dshot by anHHplayer:

+> de OO (4)

Note that if this assumption is not true, theHHplayer will shoot in all plays, which is counter

factual.

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Result 2a: Regardless of whetherHHis a cognitive illusion or not, the defensive team allocates

greater defensive efforts to the HHplayer than to the AHplayer, the FG% of the AHplayer

improves compare to their normal abilities.

In the Appendix, we show that the defensive efforts allocated to theHHplayer are:

22)(221

+

+

=

de OO(5)

Accordingly, the defensive efforts allocated to theAHplayer are:

222

= (5')

Result 2a reflects the main thrust of our argument: the optimalbehavior of the defensive team

changes when a player is identified to be HH. This means that one cannot analyze the HH

phenomenon by simply comparing theFG% ofHHplayers with their normalFG%. A correct

analysis of this problem accounts for the changes in the behavior of all players. As we show

below, the resulting characteristics of the game, when all changes are accounted for, differ from

the direct impact of the improved shooting ability of theHHplayer. This is analogous to the

difference between a partial equilibrium analysis and a general equilibrium analysis.

Result 2b: Regardless of whetherHHis a cognitive illusion or not, theHHplayer takes a greater

proportion ofdshots in comparison with theAHcase.

Result 2b describes changes in the behavior of theHHplayer that are driven by his own

beliefs regarding his own improved shooting ability. That is,HHplayers shoot when facing the

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same shot difficulty asAHplayers. Result 2b shows that HH players increase the proportion ofd

shots compared to theAHcase. Since in actualbasketball games the majority of shots are dshots,

this implies that the proportion of shots taken by theHHplayer increases as well.

Results 2c: Regardless of whetherHHis a cognitive illusion or not,HHplayers would have lower

FG% thanAHplayers.

Note that Result 2c does not characterize theFG% ofHHplayers relative to their normal

performance; rather, the comparison is to non-HHplayers. This is because the success rate ofHH

players nets two offsetting effects: improved personal ability and increased defensive efforts

allocated to them. Result 2c is a special case of the Simpson (1951) Paradox: Because of the

improved shooting ability of theHHplayer, he takes all the difficult shots (while sharing with the

AHplayer the easy shots) and theFG% of theHHplayer is actually lowerin equilibrium than the

FG% of theAHplayer. Next, we examine the success rate ofHH, dependant upon the validity of

theHHbelief.

Result 3: IfHHis a cognitive illusion,HHplayers experience a decline in theirFG% relative to

their normalFG%. IfHHis not a cognitive illusion,HHplayers may experience an improvement

or a decline in theirFG% relative to their normalFG%. Whether theirFG% improves or declines

relative to their normal FG% depends on the values of the parameters , Od

, Oe

and

parameters.

Result 3 is the basis for our tests of theHHhypothesis and, indeed, the reason our results

allow us to conclude that HHis not a cognitive illusion. This is because Result 3 provides a

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nd are forced to take difficult shots but despite this, are able to attain

virtuall

at unequivocally improves and the

testable hypothesis regarding theFG% ofHHplayers ifHHis a cognitive illusion: in this case,

theFG% ofHHplayers should materially decline once they are erroneously perceived to have

temporarily improved shooting ability. As we show, this is not the case: HHplayers do face

intensified defensive efforts a

y their normalFG%.

Results 2 and 3 illustrate the difference between a ceteris paribus analysis and an

equilibrium analysis of theHHphenomenon. Specifically, the ceteris paribus analysis, which

underlies the hypotheses tested by GVT, leads us to expect that a temporarily improved shooting

ability ofHHplayers entails a higherFG%. In contrast, when team reactions are accounted for in

an equilibrium analysis, it is the FG% of theAHplayer th

performance ofHHplayers can either improve or worsen.

Result 4: The FG% of teams with players perceived to be HHexceed their normal shooting

abilities. The improvement inFG% of a team is greater whenHHis a real phenomenon than when

HHis a

player will increase the offensive ability of

his own

cognitive illusion.

The intuition behind Result 4 is straightforward. Defensive efforts are shifted from AH

players toHHplayers. IfHHis a cognitive illusion, teams withHHplayers will improve their

overallFG% because of a suboptimal allocation of defensive efforts (cf., Result 1). IfHHis a real

phenomenon, the improved shooting ability of anHH

team and further improve the teamsFG%.

Note that, in our model, ifHHis a cognitive illusion, offensive teams that shift throws to

players that are erroneously perceived to beHHwill not bear a cost. This is because, for modeling

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e efforts and the offensive team overly relies on the player erroneously

o beHH.

simplicity, we assume that there are only two types of throws difficult and easy throws. As

offensive teams shift throws toHHplayers only when both players face the same difficulty, this

shift will not affectFG% ifHHis a cognitive illusion. In reality however, shot difficulty is not

dichotomous, so that players perceived to be HHmay shoot even when their teammates face

easier shots. IfHHis a cognitive illusion, this erroneous shift of throws will be costly. Thus, in

reality, ifHHis a cognitive illusion, theFG% of a team with a player erroneously perceived to be

HH may be little changed from normal levels, since both teams err: the defensive team

misallocates defensiv

perceived t

II. DataThe predictions of the above simple model of shot selections and defense allocations are tested

using the complete data for all NBA games in the 20042005 regular season. We collect all game

records from the official website of the NBA. There are 30 teams, each playing 82 games, for a

total of 1230 games. Due to recording problems of the NBA, we have detailed records for only

1,218 (99.0%) of these games. A typical game record is as follows:

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Since players rest for 15 minutes between halves, during which time temporarily improved

shooting ability HH may vanish, our basic unit of analysis is not a game but a half. Like GVT,

we define a player as anHHplayer if he successfully shoots two or three times in a row. Since

there are no material differences in the results, we report the results of the latter definition only. A

player is regarded asHHif he makes three successful shots in a row in a half. Similarly, a player

is regarded Cold Hand (CH) if he misses three shots in a row in a half.

By definition,HHand CHare transitory phenomena, affecting the shooting abilities of

players temporarily. Therefore, we examine the impact of improved or reduced shooting ability

HHor CH respectively within a short horizon. We do this by focusing on the next shot

following the identification of a player asHHorCH, which can be made either by the identified

player himself or by a teammate. This is different from the analysis of GVT. GVT examine the

next shot of the identified player himself, which may happen much later than the time the player is

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identified, even after the player is substituted for a while. Moreover, to ensure that we uniquely

identify the status of the game, we consider only time intervals in which there is at most one

player of the offensive team identified as eitherHHor as CH.9

In the model, we consider two types of shots: e and d. The recording of the games by the

NBA does not include all the necessary parameters to correctly determine shot difficulty, such as

defense intensity, angle, and shot distance, etc. Therefore, we are unable to control fully for shot

difficulty. Instead, we use the type of shot, which is recoded, to proxy for shot difficulty. We

consider jump shots (roughly 60% of all game shots) to be dshots and all other shots (mainly lay-

ups) to be e shots. As explained where relevant, this partial control for shot difficulty adds some

noise to our estimates, and we account for this in our interpretations of the results.

In some of our analysis, we separate player data by ability. We use two measures of player

ability: player salary and player field goal attempts (FGA). We report our results using both

measures of player ability but note that the two measures are highly correlated (=0.562).

Moreover, some game participants suggested in private discussions that superstar players are

treated differently by coaches and behave differently than other players. Therefore, we also use

these measures to identify superstar players on each team and to examine theHHeffect on them

separately. The results do not differ from the results for other players.

There are 528 signed players in the 2004-2005 season who can potentially be included in

our sample. We have shooting and salary data for 503 players. These players make up 95.2% of

9 Additionally we required that the Next Shot attempted in the same half as the Identify shot that the HH player is not

replaced until the Next Shot and that the Next Shot will be taken within 180 second of the Identify shot. These

restrictions cause the number of Next Shots to be slightly lower than the number of Identify Shots (4,993 to 5,009

respectively).

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the total player pool and account for 98.7% of allFGA. Table I reports the main characteristics of

players included in our analysis. The average NBA player makes about 3.6 million dollars, with

stars making on average more than three times that. Note that stars earn their pay by playing about

60% more time than the average player and by taking more than twice the number of shots. The

FG% of star players, however, are similar to the FG% of average players, a point which is

amplified in Table II. This suggests that teams adjust their allocation of defensive effort to

counteract the differential abilities of players. This point is reflected in the cross-player

distribution ofFG% and the cross-player distribution of successful free throws (FT%). The cross-

player differences inFT%, as measured by the standard deviation and by the inter-quartile spread,

are much higher than in FG%. This is probably because free throws are not defended, while

regular shots are defended, allowing teams to strategically allocate more defensive efforts to better

shooters in field goal attempts, but not in free throws.

Table II further illustrates the allocation of strategic defensive efforts by teams. In Table

II, we report the averageFG% of the top five players of each team. To ensure that a player that

transfers from one team to another during the season is not counted twice, the sample includes

only the 150 players who played at least half of the games.10

These players account for about 58%

of total game time and shoot about 65% of all FGA. The table shows no ostensible relation

between player ability, as measured either by salary or byFGA, andFG%. For example, the

highestFG% is that of players ranked #5 by salary and #2 byFGA, and the difference between the

best and the worstFG% is less than two percentage point. This means that, despite the differential

abilities of players, defensive efforts are successfully allocated in a strategic way to offset these

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differences. Thus, when coaches have time to plan their plays, they are able to correctly allocate

defensive efforts across players. This, by itself however, does not rule out cognitive illusion with

respect to temporary changes in ability HHand CH, which is the subject of our research.

The constancy ofFG% reported in Table II is consistent with the simplifying assumption

of our model, that all players are of equal shooting ability at the outset. This is because, once

coaches allocate defensive efforts strategically, based on the natural abilities of opponent players,

all players do have roughly the same FG%. Given this defensive adjustment, star players are

distinguish mainly by the fact that they play longer and take more shots than average players. The

last two columns in Table II confirm this observation by showing that higher ranked players play

more minutes per game and attempt more shots per minute.

III.HHin Regular ThrowsMuch of the literature onHHtests for serial correlation in success in field goal attempts. Table III

presents summary statistics that are similar to those reported by prior studies. 11 Since our sample

is much larger than prior samples, we do not break down the results to the level of the individual

player. Rather, we report the results by the rank of players in their teams based on their annual

salary. The results are consistent with prior research in that we do not find a positive serial

correlation in shot success. Rather, as in prior studies, we find a slight negative correlation

between past and current success. This negative correlation is evident both in overall results and

10 Note that as a result the sample sizes of Tables I and II differ.11 In order to be consistent with previous literature, we do not require in this test that the shots of the identified player

occur in the same half. As a result, the number of cases in which a player succeeds or misses three consecutive times

is much greater than in Table IV.

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in results broken down by rank. Specifically, the overall results show that the FG% after three

consecutive misses is 45.87% vis--vis only 43.00% after three consecutive successes. The

similarity of our results to prior results suggests that, even though our sample is much larger and

is from a different season, the salient characteristics of shot serial correlation are present in our

data as well.

As our theoretical model suggests, temporal changes in shooting ability need not manifest

themselves in positive serial correlations in shooting records, because of changes in team

behavior. Therefore, we proceed to examine the impact ofHHon coach and player behavior. We

define a player as anHH(CH) player after he makes (misses) three consecutive shots in the same

half.12

We call the shot by which a player is identified as eitherHHorCH, the Identifying Shot,

and the player is called the Identified Player. Most of our analysis focuses on the filed goal

attempt that follows the Identifying Shot the Next Shot, which can be attempted by either an

Identified player or by a regular player.

A. Do Game Participants React to Sequential Shooting?

We begin our analysis by documenting, in Table IV, the behavior of the identified players

and their coaches when we identify a temporary change in shooting ability. Row 2 in Table IV

shows that coaches retain an HHplayer in the game 35 seconds longer than they retain a CH

player. This difference is both statistically and strategically significant since anHHplayer plays

more than 10% longer than a CHplayer. Rows 3 and 4 in Table IV show that players change their

behavior as well. Specifically,HHplayers take statistically significant less time to try another

shot, and they take a larger fraction of Next Shots than CHplayers. Moreover, as our model

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predicts, Row 5 shows that players also change their shot selection: HHplayers take substantially

more jump shots (which is our proxy for difficult shots) than CHplayers. Therefore, the evidence

presented in Table IV indicates that both coaches and players react to perceived temporary

changes in shooting ability.

As we show in Table IV, coaches react to temporary changes in the ability of their players

by substituting CHplayers quicker thanHHplayers. Coach reactions cause the CHsample to be

biased. The reason for this bias is that coaches are able to detect temporary changes in shooting

ability based on shot characteristics that are not in our data set: number of defenders and their

abilities, angle of shot, distance to basket, etc. Thus, our CHsample includes players that we

incorrectly identify as CHfor lack of better data, and it excludes data of true CHplayers who are

replaced by coaches. (To illustrate this bias, we note that players who miss consecutive lay-up

shots are replaced faster than players who miss jump shots.) Since this problem does not affect our

HHsample, in the remaining analysis, we focus on comparisons of the HHsample with the

unidentified-player sample.13

In the analysis of the impact ofHHon team behavior, we focus on the first shot of the

team following the Identifying Shot, which we call the Next Shot. To ensure that we analyze

clearly identified effects, we consider shots only where there is no more than oneHHPlayer on

the offensive team (more than 99% of all shots). Throughout, we exclude shots taken within 5

seconds of the previous shot of the same team (offensive rebounds), since these are not necessarily

characterized by the same strategies as normal, planned shots. Our final sample consists of

12 DefiningHHand CHplayers by two consecutive shots rather than three has little effect on our results.13 We note, however, that tests on the CHsample, unreported in the paper, yield results that are in line with the

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168,959 shots (88% of all attempts) of which 4,993 are Next Shots with anHHIdentified Player

on the offensive team.

In Table V, we report the characteristics of Next Shots taken byHHplayers,AHplayers,

and teams as a whole, relative to the same characteristics of all other shots, hereafter called

regular shots. The first line of Table V reports shot characteristics that allow us to examine

whether there is reallocation of defensive efforts from non-HHplayers to players perceived to be

HH. The results show an increase in the difficulty of shots taken byHHplayers a larger fraction

of the shots they take are jump shots. Conversely,AHplayers take fewer difficult Next Shots than

usual.

The next two rows show additional indications of changes in defensive efforts allocation.

Consider three-point shots first. Since three-point shots are very difficult shots, they are typically

taken when players are relatively less guarded. Thus, the significant increase in the number of

three-point shots made by AHplayers is another indication that they are less defended when

another player on the team isHH. Similarly, the change in the number of fouls committed shows

changes in the allocation of defensive efforts. Specifically, our results show an increase in the

number of fouls committed onAHplayers, and a decrease in fouls committed on HHplayers.

While this result may be counterintuitive, we note that in basketball, fouls are typically the last

result of the defensive team. This is because in many cases fouls lead to free throws attempts that

have a much higher success rate thanFGA.14

Hence, it is likely that the reduced number of fouls

predictions of the model. For example,FG% ofCHplayers are better than their season averages.14Additionally, if the foul occurs in the act of shooting and the shot is successful, the shooting player is then awarded

an additional free throw. Furthermore, each player is limited to six personal fouls per game before he is ejected from

the game.

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onHHplayers and the increased number of fouls onAHplayers may indeed indicate a shift of

defensive efforts fromAHplayers toHHplayers.

B. Testing the Validity of the HH Belief

The results indicate large changes in both defense and offence once a player is identified

asHH. Next, we examine the validity of theHHbelief directly. Row 4 in Table V compares the

FG% in regular shots to theFG% in Next Shots. The results show a significant improvement in

theFG% ofAHplayers of nearly four percentage points, as predicted by our model. This finding

is consistent with the argument that the defense is shifted away from theAHplayers. Since it is

unlikely that the defense reacts to the identification of an HHplayer by reducing the total

defensive effort, it is likely (and logical) that the defense is shifted to theHHplayer. However,

despite the significant increase in defensive efforts allocated to players perceived to beHH, they

are able to maintain their regularFG%. The difference between theFG% ofHHplayers in Next

Shot and theFG% in regular shots is less than two percentage points and is insignificant. Thus,

our results are consistent with the argument that players truly possess temporarily improved

shooting ability. This suggests that theHHbelief is nota cognitive illusion.

In our model, we assume that, except forHHplayers, the defensive and offensive abilities

of teams remain constant throughout a game. It is possible, however, that when a player is

perceived to beHH, both teams adjust their effort levels, effectively shifting efforts over time

within a game. This adjustment may increase or mitigate the effect ofHHon the performance of

the offensive team. To examine this potential effect on our estimates, we consider periods in

which there is little ability to shift efforts over time: when the game reaches the fourth quarter and

the score is close. In such periods, it is likely that both offensive and defensive efforts are close to

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their maximum. Therefore, examination ofHHin such periods is likely to be less affected by

effort shifts over time.

We examine shots taken in fourth quarters when the score difference between the teams is

10 points or less.15

Out of 4,993 Next Shots in our sample, 803 shots occur at such times when

effort shifting is not likely. Results show that during fourth quarter of close games the FG% of

Regular Shots decreases by nearly two percentage points. This decrease is likely to result from

greater defensive efforts and fatigue. The results for theAHplayers show an increase of more than

6.5 percentage point. To illustrate the magnitude of this increase, we note that the differences

between quartile cutoff points (cf. Table I) are roughly four percentage points. The large increase

inFG% ofAHplayers suggests that the defense is being shifted away from these players to HH

players. Yet, results for theHHplayers show that they are able to maintain the same success rate

as in regular shots (41.9% to 42.0% respectively). The ability ofHHplayers to maintain the same

success rate, while facing much tougher defensive efforts, is consistent withHHbeing a real

phenomenon and not a cognitive illusion.

The overall performance of the team is another indication thatHHis a real phenomenon. If

HHwere a cognitive illusion, there should be little change in the overallFG% of the team. This is

because the overallFG% nets the errors of both teams: defensive teams allocate too much effort

to players perceived to be HHand offensive teams rely too heavily on the shooting ability of

players erroneously perceived to beHH. Our results show a significant improvement in the team

FG% of almost 2.5% on average, and 5.2% during the fourth quarter of close games. This

improvement is statistically significant and similar to the difference between the averageFG%

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Def

allowed by the best and worst defensive teams in the NBA. To illustrate the importance of theHH

phenomenon, we note that there are roughly 80 field goal attempts (FGA) in a game. Since each

field goal is worth 2.2 points (the weighted average between two and three points), the

improvement of 5.2% inFG% equates roughly to ten points on a game basis.

The simple comparison of the FG% ofHH (AH) players to regularFG% implicitly

assumes a matching of players in the subsamples. To improve our control of player ability, we

estimate a Probit regression that accounts for several potential determinants ofFG%. In this

regression, we control for the mix of shots jump or non-jump. Additionally, we improve our

control for player ability in three ways. First, we include in the equation a control variable for the

normal ability of a player his seasonFG% relative to the NBAs overallFG%. Second, we

include two measures of fatigue: the total time that the player played in the half, and a dummy

variable that takes the value of one if the shot is in the second half, and zero otherwise. Finally,

we control for the defensive ability of the opposing team. This is because it is possible that the

probability of three consecutive successful shots is related to the defensive ability of the opponent

team. Hence, we include as a control variable the difference between theFG%allowedby the

defensive team in the season, and the average overall FG% in the NBA, as a measure of the

defensive ability of the defensive team.

Table VI presents the probit estimates of the following equation:

0 1 2 3 4 5 6 7%Success AH HH Jump Half I I I SFG I I Time = + + + + + + + +

Where:

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15 Changing the definition of a close game to a 15-point difference has no significant effect on our results.


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ISuccessis a dummy variable that takes the value of one when the shot is successful and

zero otherwise;

IAHis a dummy variable that takes the value of one when the Next Shot is taken by anAH

player and zero otherwise;

IHHis a dummy variable that takes the value of one when the Next Shot is taken by anHH

player and zero otherwise;

SFG% is the fraction of successful field-goal attempts of the shooting player in the entire

season relative to the NBAs overallFG%;

IJumpis a dummy variable that takes the value of one when the field goal attempt is a jump

shot, which proxies for difficult shots, and zero otherwise;

IHalfis a dummy variable that takes the value of one when the field goal attempt is in the

second half of a game and zero otherwise;

Time is the number of minutes the shooting player played in the half prior to the shot;

Defis the difference between theFG% allowed by the defensive team in the season and

the overallFG% in the NBA.

We also report Probit estimates of the equation for two sub-samples: one sub-sample is comprised

of jump shots only and the other sub-sample of non-jump shots. In these sub-sample regressions

we do not include theIJump control variable.

The results reported in Table VI confirm the results reported in previous tables. First, even

after we control for player ability and shot mix, the improvedFG% of theAHplayers remains

significantly different from zero. The improvement of theFG% of theAHplayers shows that the

jump / non-jump classification does not fully capture the reduced defensive efforts allocated to

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AHplayers inHHperiods. The improvement demonstrates the transfer of defensive efforts away

from them toHHplayers. Nonetheless, despite the increased defensive efforts allocated toHH

players, the estimated coefficient ofIHHis small in magnitude and insignificantly different from

zero. The results presented in Table VI also indicate that the improvement inFG% of theAH

players is greater during the fourth quarter of close games. Specifically, the coefficient of theAH

player doubles to 0.156, corresponding to a temporary improvement of 6.2% in hisFG%.

Importantly, despite the shift in defensive efforts, there is no reduction inFG% ofHHplayers: the

coefficient ofIHHispositive, albeit still insignificantly different from zero. Thus,HHplayers are

able to attain their normal FG% (or better) even though they face intensified defense, which

reflects their temporarily improved shooting ability. Other coefficients are of their expected signs.

The estimated impact of shot difficulty, proxied by jump shots,IJump, corresponds to a decrease of

roughly 20% inFG% when players attempt jump shots. The impact of fatigue is also significantly

different from zero: for every minute played, theFG% of the player is reduced by roughly 0.1%

and FG% in second-halves are about 1% lower than in first halves. As a robustness test, we

estimate the above regression for jump and non-jump shots separately. Results in the last two

rows of Table VI show that the coefficients of both the AHplayers and HHplayers remain

virtually the same.

Our results are consistent with defensive efforts being shifted within games. When a

player is perceived to beHHduring the first three quarters of a game, the defending team reacts

by both increasing total defensive efforts and by shifting it toward theHHplayers. The increase in

total defensive effort leads to mitigation of theHHphenomenon in these periods. However, in the

fourth quarter of close games, the defensive effort is most likely to be close to its maximum level,

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and further increase is impossible. Thus, the team can react toHHidentification only by shifting

defense fromAHplayers toHHplayers. The shift in defense is evident from the large increase in

FG% of theAHplayers in these periods. Importantly, this shift does not result in a lowerFG% of

HHplayers, suggesting thatHHin regular throws is a real phenomenon. (Recall that ifHHwere a

cognitive illusion, the FG% of players incorrectly identified to beHHshould decline as they face

an increased defense while possessing no improved ability).

C. Alternative Explanations

(i) Team spirit

Our model assumes that except for the perceived improved ability of theHHplayers, all other

players possess the same offensive abilities after the identification of theHHplayer. However,

this assumption may not hold in real-world conditions. Specifically, it has been suggested to us

that a possible explanation for improved teamFG% is thatAHplayers may be encouraged by the

perceived improved ability ofHHplayers and decide to put more efforts into their offense.

Although this team spirit argument, does not explain the documented significant shifts in

defensive efforts following the identification of anHHplayer, it can explain the improvement in

AHplayersFG%.

We argue that the team spirit argument is unlikely to be the driving force behind our

results, for two main reasons. First, as previously noted, during the fourth quarter of close games

both defensive and offensive efforts are close to their maximum. Hence, team spirit is less likely

to affect performance during these periods. However, it is in these periods that we find the

greatest improvement inAHsuccess rate (cf. Table V and Table VI). Secondly, improved team

spirits should be observed afterany string of successful shots (i.e., not just by HHplayers).

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Accordingly, we examine theFG% of teams following strings of success for the whole team

from three to seven consecutive successful throws. We find virtually identical FG% after any

duration of successes.16

Therefore, our findings indicate that team momentum does not exist in

basketball games, suggesting that improved shooting ability for the team is unique to periods in

which a specific player is identified asHH.

(ii) Hot hand or hot period?

An alternative explanation for our empirical results is that it is not the Next Shot that is special,

but the period in which it is attempted. Such explanations suggest that the FG%of teams fluctuate

within games. In those periods whenFG% are high (low) the probability ofHHdetection is high

(low), leading to a spurious relation betweenHHdetection andFG% in Next Shot.17

We account

for this possibility in two ways. First, we examine the shot closest to Next Shot before the

identification of the HH player. Accordingly, we select the shot that is one shot before the

Identifying Shot (i.e., two shots before Next Shot) and examine the performance ofAHplayers.18

Consistent with our argument, we find no changes inFG% ofAHplayers compared with their

normal shooting ability (IAH= -0.011 and insignificant). Secondly, compare between the quality of

the players in Next Shot and regular shot. Our findings show that the average number of starter in

Next Shots and regular shots is almost identical (3.21 to 3.25 respectively). Similarly, the

proportion of shots that the star player is on court is similar between Next Shots and regular shots.

Finally, we note that if the basketball game is indeed characterized by periods of high and low

16 Table is available upon request.17 For example, it may be that we documented a superstar effect. In periods of the game when the superstar is playing

(not playing) the FG% of the team are relatively high (low), leading to both higher (lower) probability of HH

detection and higher (lower)FG% in periods (including Next Shot) when the superstar is playing.18 Note that if theHHplayer attempts the shot, then it must be successful as it is part of the three consecutive shots by

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FG%, then team momentum should also exist. However, our previous results show that team

momentum does not exist.

(iii)Concavity of the effect of defense onFG%

Another alternative explanation that is consistent with our empirical results is that the effect of

defense on FG% is highly concave. Figure 1a (solid line) demonstrates such a hypothetical

relation by plotting theFG% of offensive players as a function of defensive efforts allocated to

them. We refer to this line as the production function of the player. According to the concavity

explanation, the defense is able to push players into the flatter part of the curve, as demonstrated

by Point A in Figure 1a. When a player is perceived to beHH, the defending team coach

erroneously believes that the player has improved shooting ability, as demonstrated by Point B in

the upper dashed line in Figure 1a. As a result, the defensive coach shifts the defense from theAH

players to theHHplayers. Since theHHplayers indeed shoot at their normal ability (Point C on

Figure 1a), the coach erroneously believes that the tougher defense is responsible for the lack of

improvement in the success rate of theHHplayers. In reality, according to this explanation, the

HHplayer shoots at similarFG% to his normal abilities, simply because he is in the flatter part of

the production function.

However, such a solution suggests that the defensive ability of teams (and players) has

little effect on the success rate of offensive players. This is not consistent with the data that show

a difference of 5.4% in FG% allowed between the best and the worst defending teams. This

difference is highly significant and economically important, as it equates roughly to 10 points per

game. The fact that NBA teams are different in their defensive capabilities suggests that weaker

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which the player is identified as hot.


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defensive teams (lower defense intensity in Figure 1a) are unable to push opponent players into

the flatter part of the production curve. Therefore, if the concavity explanation is true then it is

among these teams that a drop in theFG% of theHHplayer is expected. We test this prediction

by examining the coefficient ofHHplayers (IHH) when the defending team is relatively weak

(allowing higherFG% than the median team on the NBA). Our unreported results do not confirm

this conjunction, showing a slight increase in the coefficient ofHHplayers compared to the

overall sample.

Importantly, the assumption that tighter defense does not lead to lowerFG% is in contrast

to basketballs core beliefs. In basketball, roughly 70% of all offensive plays end with a field goal

attempt. Decreasing theFG% of the opponent team is the most important aspect of the defensive

effort. Even a casual observation reveals the importance that all game participants attribute to

tougher defense. Thus, in assuming the above production curve, one argues that basketball

coaches not only fail to derive the right conclusions when analyzing random sequences of events,

but also that they are unable to learn the true nature of the production function. 19

IV. HHin Free Throws

While the focus of our analysis is on misperceptions of probabilities when statistical inference is

made over a short time span, we use our extensive data set to examine theHHphenomenon in free

throws (FT) as well. As discussed previously,FTare a form of controlled experiment of theHH

phenomenon since, unlikeHHin regular throws, inFTplayers and coaches cannot adjust their

19 Another possible explanation is that the defense is able to derive only star players into the flatter part of the

production curve. This explanation is inconsistent with our finding that theHHeffect is similar when theHHplayer is

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behavior once they identify a player as beingHH. Note, however, thatFTdiffer from field goal

attempts because they involve different settings and different player skills (e.g., ability to position,

jump, mislead, etc.). Presumably, these differences explain why basketball fans believe thatHH

has a larger effect in field goals than inFT, as the survey of GVT shows. Nonetheless, becauseFT

are not affected by changes in team strategy, in this section, we use FTdata to estimate an

equation similar to the one estimated for regular shots and test whetherHHexists inFT.

Our raw data are the same data that we use for the analysis of field goal attempts 1,218

games in the regular 2004-2005 NBA season. In these games, there are 60,556 regularFT(i.e., no

technical fouls or three-point attempts) from which we construct our sample. We begin by

replicating GVTs test of whether success in the secondFTis correlated with success in the first

FT. Consistent with their findings, we do not find such correlation. Note, however, that this test is

inconsistent with the tests ofHHin regular throws because it identifiesHHplayers inFTby a

single shot. This difference in identification is especially problematic inFTbecause of the high

success rate ofFTand their infrequency compared with FGA (c.f., Table I). Hence, in what

follows, we identify players to beHHinFTsimilar to the identification in field throws by a

series of successful shots in the same half. Since the probability of success inFT(75.6%) is much

higher than the probability of success in regular shots (44.3%), we consider a player to beHHin

FTwhen he succeeds infourconsecutiveFTin a half.20

We then compare the success rate ofHH

players inFTtoAHplayers who attempt at least fourFTwithin a half. This leaves us with a

sample of 9,659FT, of which 3,229 are attempted when the player isHHinFT.

the star player and when he is not the star player.20 The coefficient of the HHplayers remains significant when HH in FT is defined by two or three successful

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Similar to our analysis of regular shots, we estimate a Probit regression where the dependent

variable is a dummy variable that takes the value of one if a player succeeds in a FTand zero

otherwise. The explanatory variables are similar to those used in regular throws:

IHH_FT is a dummy variable that takes the value of one when the shooting player isHHinFTand zero otherwise

IHH_Regis a dummy variable that takes the value of one when the shooting player isHHinregular shots and zerootherwise

SFT% is success rate of the shooting player inFTin the entire season, normalized by theaverage SFT% in the NBA

ISecondis a dummy variable that takes the value of one when this is the secondFTin thevisit to theFTline and zero otherwise

IHalfis a dummy variable that takes the value of one when the field goal attempt is in thesecond half of a game and zero otherwise

Time is the number of minutes the shooting player played in the half prior to theFTThe first two variables indicate a potential temporary improvement in the shooting ability of the

player:IHH_FTindicates that the player is identified asHHinFT, andIHH_Regindicates that the

player is identified asHHin field goal shots in the previous three minutes before theFT. SFT%

controls for the ability of the player inFT.ISecondcontrols for potentially improved shooting ability

in the secondFT.IHalf and Time control for the potential impact of fatigue on players shooting

ability.

consecutiveFT.

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The estimated Probit coefficients are:

IHH_FT IHH_Reg SFT% ISecond IHalf Time

0.090**

(0.033)

-0.068

(0.059)

2.602**

(0.148)

0.112**

(0.029)

-0.007

(0.031)

0.002

(0.003)

** Significant at 1%.

Given that no strategic changes in player and coach behavior can be implemented when a player is

identified asHHinFT, it is not surprising that the HHphenomenon is clearly evident in these

throws. Specifically, the estimated impact of beingHHinFTis significantly positive, implying an

improvement of roughly 3% in the success rate inFTwhen a player isHHinFT. We note that

this improvement is similar to the estimate of basketball fans surveyed in GVT (4%). Consistent

with the hypothesis thatFTand regular throws entail different settings and skills, we find noHH

spill-over, that is beingHHin regularthrows does not improve success rate inFT.21

Controls that have a significant impact on the success rate in FTare the overall success

rate of the player inFTthroughout the season (SFT%), and the throw being the second one in the

same visit to theFTline (ISecond). (The latter result is consistent with the criticism and findings of

Wardrop (1995)).

To summarize, using a larger sample than GVT and the same methodology as in field

goals (i.e., controlling for various determinants of success in FT), we find that HHis a real

phenomenon inFT. Given that we document the validity of theHHphenomenon in throws where

no other player other than the one making the attempt is affected by a belief inHH, this represents

further evidence thatHHis a real phenomenon.

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V. Economic Implications

While the preceding theoretical and empirical analyses of basketball may naturally extend to other

sports, the results and their implications also apply to multiple economic situations. In this

section, we briefly point out a few of the areas in which the implications of our analysis are

applicable. The main implication we consider is that abnormal ability entails strategy changes,

which increases difficulty and reduces the improvement in the observedperformance of hot

agents. The strategy changes are more readily observed in the results of the otheragents, rather

than in the change in the results of the hot agent. Some of these areas have documented results

in the spirit of our analysis and some of the implications call for additional research.

One area where similar results have been modeled and documented is the field of money

management. It is well documented (e.g., Jensen (1969), Gruber (1996)) that the performance of

money managers is not serially correlated: superior investment results in one period are not

necessarily followed by superior results in subsequent periods. Rather, superior results in a given

period tend to be followed by average subsequent investment results. Yet, analysis of money

flows into and out of mutual funds suggests that investors tend to invest with money managers

who showed superior performance in a preceding period (e.g., Chevalier and Ellison (1997); Sirri

and Tufano (1998)).

The empirical evidence on investor money flows has been interpreted, similar to the

interpretation of the HH phenomenon, as an irrational investor reaction. Specifically, it is argued

that investors see patterns in investment results hot hands in investment management where

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21 Similarly, we find no spillover effect of beingHHinFTon the success rate in regular shots.


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none exists. Investors investment funds to hot hand money managers but do not obtain superior

results. Berk and Green (2004), however, argue that these results are consistent with superior

investment ability of certain managers and with rational investors. Their model, while not quite

the same as ours, is based on the fact that managers with superior investment skills receive more

money to manage (for an increased fee). Analogous to the drawing of more defensive efforts when

a player isHH, as the sums managed by the managers with superior performance grow, the ability

of these managers to generate superior returns diminishes (for example, because of scale

limitations on attractive investment ideas). Thus, managing more money stretches the investment

skills of the managers to the point that they can generate only normal returns. The model of Berk

and Green (2004) is similar in spirit to our model and, like ours, is consistent with rationality and

with prior evidence on performance of managers and subsequent money flows. Note that our

analysis suggests additional testable hypothesis: as money is withdrawn from managers withpoor

performance their investment returns should improve, akin to the improved performance ofAH

players in our model.

One can transfer the logic of the money management industry to management in any other

industry and derive similar implications for industrial organization along the following lines.

While the quality of managers is typically little known at the outset of their tenure, as they prove

their superior ability they are put in charge of increasingly larger and more problematic assets. For

CEOs, increasing firm size is often achieved by acquisitions, which enable superior CEOs to

apply their superior skills to large firms and assets. As superior managers attempt to acquire

additional assets, however, they are increasingly required to pay larger sums for the acquired

assets that they identify as currently being under-managed (or under-priced). This is because

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sellers realize that the acquiring managers can enhance the value of the assets under their

management and bargain for a fraction of the value creation (unless they have no bargaining

power at all). Furthermore, as superior managers manage increasingly larger firms, their

managerial skills are stretched until they can no longer offer above-normal performance.

Extending the logic of our analysis of hot hands, therefore, implies that, analogous to the

observed performance ofHHbasketball players, the observedperformance of superior managers

should be little different from the performance of regular managers. The superior ability of

managers should manifest itself in the fact that they are put in charge of larger firms compared to

less able managers, which is analogous toHHplayers playing longer and shooting more thanAH

players. Superior managers should also pay larger premiums in acquisitions than regular

managers, which is analogous to the increased defense thatHHplayers face. Indeed, the empirical

evidence is consistent with these predictions. For example, while skill and pay are positively

correlated in general (e.g., OShaughnessy, Levine, and Cappelli (2001)), when it comes to top

managers, theirobservedperformance is uncorrelated with their pay (see, for example, the survey

in Bebchuk and Fried (2004)). Rather, managerial pay is correlated to assets under management

(Bebchuk and Grinstein (2005)). While Bebchuk and Fried (2004) interpret these results as pay

without performance, we interpret them as pay with performance, albeit when performance is

measured by the extent to which investors trust superior managers by placing them in control of

large sums of money. Indeed, we expect studies in labor economics and organizational behavior to

find the same relations between skill, observed performance, and pay of employees.

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VI. Conclusions

Behavioral economists use laboratory experiments to document limitations on individual ability to

process random data and to derive rational probability assessments. The relevance of such

limitation to actual decision making that involves substantial sums of money is often questioned.

Studies of hot hands (HH) in professional basketball are often cited as the prime proof of the

relevance of these laboratory results to real-life situations. Specifically, Gilovich, Vallone, and

Tversky (1985) and others argue that the chance of scoring is largely independent of past

performance, which they interpret to mean thatHHis a costly cognitive illusion.

We argue that prior empirical evidence regardingHHimplicitly assumes ceteris paribus,

which ignores concurrent changes in team behavior when a player isHH. In a simple model of

basketball, we show that changes in team behavior may reverse the predictions and conclusions of

the analysis of theHHphenomenon under the ceteris paribus assumption. Specifically, when a

player is identified asHH, defensive efforts are shifted toward that player and away from non-HH

players. This causes the player with the temporarily improved success rate theHHplayer to

take more difficult shots and ameliorates the temporary improvement in his ability. Conversely,

our analysis suggests that game realignment in HH periods is best revealed in the game

characteristics of the non-HHplayers.

We examine an extensive database of more than 1,200 games and show that the

equilibrium predictions regarding player behavior and game characteristics when a player isHH

are true in the data. Specifically, we find that defensive efforts are indeed shifted from non-HH

players toHHplayers, causingHHplayers to take more difficult shots whereas Non-HH players

are allowed easier shots. Consequently, the performance of non-HHplayers as well as the overall

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performance of teams improve when one of the teams players isHH. More importantly, we show

that, despite increased defensive efforts that are allocated toHHplayers, HHplayers achieve the

same success rate that they achieve normally. This means that the shooting abilities ofHHplayers

are indeed better than their normal abilities. Thus, our empirical findings are consistent with the

predictions of our model, and with the hot hands phenomenon being real, implying that team

behavior and beliefs are consistent with rationality. To complement our analysis of regular

throws, we examineHHin free throws as well. These shots are not affected by the behavior of

non-shooting players. We show thatHHexists in free throws as well.

One aspect that is highlighted by our model is the importance of distinguishing between

easy and difficult tasks in the analysis of performance, in our case the difference between easy

and difficult shots. A simple way to understand the importance of this distinction is to think of

Kobe Bryant and Allen Iverson, two of the NBAs most talented offensive players in the season

we analyze, whoseFG% are slightly below the NBA average. This is because their talents cause

defensive teams to allocate more defensive efforts to them, which forces them to take difficult

shots (yet they are able to attain nearly the averageFG% because of their superior talents).

In a different context, the ability of managers who manage large firms to attain the same

operating results that managers of small firms attain is not a proof that they are no better than

managers of small firms. Rather, it shows that they are better managers who can get results on par

with managers of small firms despite the more challenging task they face.

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References

Adams, R.M., 1995, Momentum in the performance of professional tournament pocket billiard

players,International Journal of Sport Psychology 26, 580-587.

Albright, S.C., 1993, A statistical analysis of hitting streaks in baseball,Journal of the

American Statistical Association 88, 1175-1183.

Barber, B.M., and T. Odean, 2001, The internet and investor,Journal of Economic

Perspectives 15(1), 41-54.

Barberis, N., A. Shleifer, and R. Vishny, 1998, A model of investor sentiment,Journal of

Financial Economics 49, 307-343.

Barberis, N., and R.H. Thaler, 2002, A survey of behavioural finance, NBER working paper

no. W9222.

Bebchuk, L., and J. Fried, 2004,Pay without Performance: The Unfulfilled Promise of

Executive Compensation, Harvard University Press.

Bebchuk, L., and Y. Grinstein, 2005, The Growth of Executive Pay, Oxford Review of

Economic Policy, 21, 283-303.

Berk, J., and R. Green, 2004, Mutual fund flows and performance in rational markets,Journal

of Political Economy 112, p.1269-1295.

Bernardo, A.E., and I. Welch, 2001, On the evolution of overconfidence and entrepreneurs,Journal of Economics & Management Strategy 10, 301330.

Bloomfield, R., and J. Hales, 2002, Predicting the next step of a random walk: Experimental

evidence of regime shifting beliefs,Journal of Financial Economics 65, 397-414.

Chevalier, J., and G. Ellison, 1997, Risk taking by mutual funds as a response to incentives,

Journal of Political Economy 105, 1167-1200.

Chordia, T., and L. Shivakumar, 2002, Earnings, business cycle, and time-varying expected

returns,Journal of Finance 57, 985-1019.

Cochrane J.H., F.A. Longstaff, and P. Santa-Clara, 2008, Two trees,Review of Financial

Studies 21, 347-385.

Daniel, K., D. Hirshleifer, and A. Subrahmanyam, 1998, Investor psychology and security

under- and over-reactions,Journal of Finance 53, 1839-1885.

39


42/58

DeBondt W., and R. Thaler, 1985, does the stock market overreact?Journal of Finance 40,

793-808.

Dorsey-Palmateer, R., and G. Smith, Bowlers hot hand, The American Statistician 58, 38-45.

Durham, G.R., M.G. Hertzel, and J.S. Martin, 2005, The market impact of trends and

sequences in performance: New evidence,Journal of Finance 60, 2551-2569.

Falk, W., 1981, The perception of randomness,Proceedings of the Fifth International

Conference for the Psychology of Mathematics Education, Grenoble, France.

Fama, E.F., and K.R. French, 1992, The cross-section of expected stock returns,Journal of

Finance 47, 427-465.

Gilovich, T., R. Vallone, and A. Tversky, 1985, The hot hand in basketball: On themisperception of random sequences, Cognitive Psychology 17, 295-314.

Gruber, M., 1996, Another puzzle: The growth in actively managed mutual funds,Journal of

Finance 51, 783-810.

Hirshleifer, D., 2001, Investor psychology and asset pricing,Journal of Finance, 56(4), 1533-

1597.

Jensen, M., C., 1969, Risk, the pricing of capital assets, and the evolution of investment

portfolios,Journal of Business 42, 167-247.

Kaplan, J., 1990, More on the hot hand, Chance 3, 6-7.

Koehler, J.J., and C.A. Conley, 2003, The hot hand myth in professional basketball,Journal of

Sport and Exercise Psychology 25, 253-259.

Kyle, A.S., and W.F. Albert, 1997, Speculation duopoly with agreement to disagree: Can

overconfidence survive the market test?Journal of Finance 52, 2073-2090.

Li, X., 2005, The persistence of relative performance in stock recommendations of sell-side

financial analysts,Journal of Accounting and Economics, 40(1-3), 129-152.

Odean, T., 1998, Volume, volatility, price and profit when all traders are above average,

Journal of Finance 53, 1887-1934.

OShaughnessy, K., D. Levine, and P. Cappelli, 2001, Changes in managerial pay structures

1986-1992 and rising returns to skill, Oxford Economic Papers 53, 482-507.

40


43/58

41

Rabin, M., 2002, Inference by believers in the law of small numbers, Quarterly Journal of

Economics 117, 775-816.

Romer, D., 2006, Do firms maximize? Evidence from professional football,Journal ofPolitical Economy 114, 340-365.

Simpson, E.H., 1951, The interpretation of interaction in contingency tables,Journal of the

Royal Statistical Society, Series B 13, 238-241.

Sirri, E., and P. Tufano, 1998, Costly search and mutual fund flows, Journal of Finance 53,

1589-1622.

Slonim, R., and A.E. Roth, 1998, Learning in high stakes ultimatum games: An experiment in

the Slovak Republic,Econometrica 66, 569-596.

Smith, G., 2003, Horseshoe pitchers hot hand,Psychonomic Bulletin and Review 10, 753-

758.

Tversky, A., and D. Kahnemann, 1971, Belief in the law of small numbers,Psychology

Bulletin 76, 105-110.

Wagenaar, W., 1972, Generation of random sequences by human subjects: A critical survey of

literature,Psychological Bulletin77, 65-72.

Wardrop, R.L., 1995, Simpsons paradox and the hot hand in basketball, The American

Statistician 49, 24-28.

Wardrop, R.L., 1999, Statistical tests for the hot-hand in basketball in a controlled setting,

Working paper, University of Wisconsin.

http://scores.nba.com/games/20041114/ORLPHI/PlayByPlayPrint.html

http://www.hoopshype.com/salaries.htm

http://asp.usatoday.com/sports/basketball/nba/salaries/default.aspx
http://scores.nba.com/games/20041114/ORLPHI/PlayByPlayPrint.htmlhttp://scores.nba.com/games/20041114/ORLPHI/PlayByPlayPrint.htmlhttp://www.hoopshype.com/salaries.htmhttp://www.hoopshype.com/salaries.htmhttp://asp.usatoday.com/sports/basketball/nba/salaries/default.aspxhttp://asp.usatoday.com/sports/basketball/nba/salaries/default.aspxhttp://asp.usatoday.com/sports/basketball/nba/salaries/default.aspxhttp://www.hoopshype.com/salaries.htmhttp://scores.nba.com/games/20041114/ORLPHI/PlayByPlayPrint.html


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Table I

Player Summary Statistics

This table presents summary statistics for 503 players (95.3% of the 528 NBA players) who

attempted at least one field goal (FGA) in the regular season of 20042005, and for whom wehave salary data. The data reflect 1,218 games (99.0% of 1,230 regular games) and are taken from

the official website of the NBA. Salary information is taken from the HoopsHype and USA Today

websites. We report the means, standard deviations, and quartiles (25%, 50%, and 75%) for each

variable. Additionally, we report the characteristics of star players the top ranked player in each

NBA team. Star players are defined as either the player with the highest salary, or as the player

that has the highest number of field gold attempts (FGA) throughout the season. We report the

season's salary (Salary), average number of minutes played per game (Minute/game), fraction of

successful field-goal attempts (FG%), fraction of FGA that are jump shots (Jump%), free throw

attempts per game (FTA/game), visits to the free throw line per game (FTA/game), successful free

throws (FT%), and Hot-Hand periods a player enjoys per game (HH/game).

StarFGASalary

75%50%25%S.D.Mean

9.0913.955.021.820.813.993.58Salary

36.9833.3028.9519.1911.9410.3020.37Minutes/game16.5514.199.105.493.004.606.61FGA/game

0.440.420.370.300.240.100.31FGA/minute

45.245.146.942.938.79.6044.3FG%

61.256.372.961.946.720.460.1Jump%

6.254.732.781.420.811.912.09FTA/game

3.272.511.430.770.421.011.10Visits to line/game

78.474.081.275.066.614.475.6FT%0.740.640.370.160.050.240.23HH/game

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Table II

Shot Summary Statistics

This table presents average shooting statistics for the top five NBA players in each of the 30 NBA

teams who played at least half of their teams game 41 games in the regular season of 20042005. The sample is comprised of 150 players ranked in the top five in their teams (55% of all

NBA players in the 2004-2005 season). These players played 57.9% of total playtime and threw

64.9% of all throws. The data reflect 1,218 games (99.0% of 1,230 regular games) and are taken

from the official website of the NBA. Salary information is taken from HoopsHype and USA

Today websites. We report the fraction of successful field-goal attempts (FG%), minutes per

game (Min/Game) and shots per minute (FGA/Min) by player rank. Player rank is based on either

season salary or on the number of field goal attempts in the season relative to all team members.

The one-way ANOVA test is for the equality of all means. Numbers in parentheses arePvalues.

Ranked FG% Min/Game FGA/Min

By Salary By FGA By Salary By FGA By Salary By FGA

1 45.73 45.24 35.0 37.0 0.420 0.445

2 44.31 45.65 32.9 34.2 0.361 0.386

3 44.90 45.37 31.7 31.3 0.334 0.353

4 45.25 44.09 25.4 30.2 0.325 0.327

5 46.23 45.00 25.4 26.0 0.300 0.325

One way

ANOVA

1.120

(0.349)

0.730

(0.573)

11.501

(0.000)

30.446

(0.000)

6.400

(0.000)

28.759

(0.000)

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Table III

Serial Correlation in Shots

This table presents average shooting statistics of the top five NBA players in each of the 30 NBA

teams in the regular season of 2004-2005. The data reflect 1,218 games (99.0% of 1,230 regulargames) and are taken from the official website of the NBA. Ranking of players is by their season

salary relative to the salaries of all players in their own team. Salary information is taken from the

HoopsHype and USA Today websites. We report the fraction of successful field goal attempts

(FG%) by player rank and by performance in the preceding one, two, or three shots.

Unconditional FG% refers to the overallFG%.FG% 3 (or2, or1) missedrefers toFG% of a

player after he misses three (or two or one, respectively) consecutive shots. FG% 3 (or2, or1)

made refers toFG% after a player succeeds in three (or two or one, respectively) consecutive

shots. We exclude all shots taken within 5 seconds of the previous shot of the team ( offensive

rebounds). Numbers in parentheses are the total number of shots in each cell.

Player

rank

FG%3

missed

FG%2

missed

FG%1

missed

Uncon-ditional

FG%

FG%1

made

FG%2

made

FG%3

made

145.87%

(2,666)

45.78%

(6,144)

46.43%

(13,705)

45.78%

(29,390)

45.66%

(11,741)

44.64%

(4,621)

43.78%

(1,752)

246.42%

(1,775)

45.50%

(4,275)

45.37%

(9,861)

44.31%

(21,511)

43.26%

(7,995)

42.01%

(2, 859)

41.21%

(973)

347.20%

(1,373)

46.23%

(3,489)

46.81%

(8,520)

45.49%

(19,219)

44.78%

(7,334)

44.23%

(2,667)

42.11%

(938)

444.07%

(1,323)

43.91%

(3,352)

44.61%

aharoni sarig - hot hands in basketball & equilibrium

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