Hot Hand Fallacy and the impact of perceived streakiness on human behaviour

Table of contents I

TABLE OF  CONTENTS  

Table of  contents  I

  List of figures  II

  List of tables  III

  List of abbreviations  IV

1  Introduction  1

1.1  Motivation and basic background  1

1.2  Methodology and organization  2

2  The Hot Hand: A literature overview  3

2.1  Introducing the Hot Hand  3

2.2  Hot Hand and Gambler’s Fallacy  6

2.3  Streakiness in the financial market  11

2.4  The Hot Hand effect in sports  15

2.4.1  Theoretical Issues  19

2.4.2  Problematic issues of Hot Hand research in sports  29

2.4.3  More research regarding the Hot Hand in sports  31

3  Empirical analysis of professional bowling data  35

3.1  Basics of bowling  36

3.2  Collection of data  37

3.3  Analysis of professional bowlers’ performance  39

3.3.1  General analysis of bowling data  39

3.3.2  Individual analysis of bowling data  42

3.3.3  Analysis utilizing conventional tests  45

4  The Hot Hand in the betting market  49

5  Conclusion and suggestions for future research  55

6  Bibliography  58

7  Appendix  64

  List of figures  

  LIST OF FIGURES  

  Figure 1 : Distribution of the number of heads  

  Figure 2 : Perception of sequences depending on alternation rate  

  Figure 3 : Development of the MSCI EM Eastern Europe index  

  Figure 4 : Advantage scores in favour of the Hot Hand simulation  

  Figure 5 : Arrangement of bowling pins  

  Figure 6 : The AC 2 test  

  Figure 7 : Formula for the AC 2 test  

  List of tables  III

  LIST OF TABLES  

Table 1  : Subgroups of participants  8

Table 2  : Proportion of outcomes rejected by runs test varying persistence  23

Table 3  : Top 10 average annual strike rates  38

Table 4  : Actual and expected streaks of all bowlers in the 2004 2005 season  40

Table 5  : Conditional strike probabilities of all bowlers in the 2004 2005 season  41

Table 6  : Comparison of actual and expected patterns in individual bowlers’ performance  42

List of abbreviations IV

LIST OF ABBREVIATIONS

$ U.S. Dollar Avg. Average F Failure H Heads (of a coin) H A Alternative hypothesis H 0 Null hypothesis MLB Major League Baseball NBA National Basketball Association NHST Null hypothesis significance testing PBA Professional Bowling Association PGA Professional Golf Association S Success T Tails (of a coin)

Introduction 1

1 INTRODUCTION

1.1 Motivation and basic background

The discussion about the existence of the Hot Hand, a widespread belief that success breeds success, has been unsettled since it was initially mentioned in 1985. The authors of the study originally concluded that any observed streakiness was simply a misperception and that the sequences did not significantly differ from the outcome of a randomly generated process.

The reasons why the discussion became this extensive may only be assumed. One reason definitely is that the main application areas for the Hot Hand belief are sports and the financial market, two areas that a vast number of people get in touch with. A large number of men and women at least occasionally watch sports on television or attend a live game. More and more people also get involved in actively managing their savings by a growing number of different financial instruments like shares and bonds.

Another reason for the amount of existing analyses and studies about topics related to the Hot Hand effect is that arguments for and against the Hot Hand range from statistical problems, e.g. which statistical test to apply, to deep psychological issues like the perception of random processes.

Knowing whether the believers in the Hot Hand or the sceptics are right is essential to everybody because in many everyday situations, people act according to which of the two groups they belong to. This is especially important in situations where money is at stake. Therefore, the aim of this paper is to answer the following research question: Is the belief in the Hot Hand effect a fallacy and what is its impact on human behaviour in an economic environment as the betting market?

While a lot of research has been done in the search for the existence of the Hot Hand, it has rarely been attempted to take into account a wide range of suggestions concerning the way of examination and inference from data, and to conduct an empirical study that satisfies most of the criticism. Thus, the analysis of data will be conducted in a way that avoids errors and uncertainties that have been encountered in more than 20 years of discussion about the Hot Hand effect.

Introduction 2

The observations of gambling behaviour in the sports betting market shall indicate how a belief in the Hot Hand influences economic decisions. In the betting market, as well as in every other economic environment it is essential that decisions are made on a rational basis.

1.2 Methodology and organization

If performance of players in sports (success or failure) was simply due to their individual ability and day to day deviations of performance, the player’s seasonal record of performance should resemble the flips of a coin with the corresponding probability to come up heads or tails.

In case the examined data sample is small, a large deviation from the expected outcome is likely. Therefore, the analyzed sample will consist of frame-by-frame data of all professional bowlers who participated in official tournaments during the 2004-2005 season.

Supposing that performance is not correlated and does not depend on previous performance, e.g. the last or some of the last games, the analyzed data should not exhibit more patterns of success and failure than a corresponding coin. The result of this analysis and findings about the behaviour of people in an economic environment can then be used to evaluate if and to what degree human perception of streakiness deviates from reality.

The deterministic approach that every outcome could be predicted by knowing all influencing factors, and that thus randomness was not existent, will be disregarded in this paper. The main task is to find an answer to the research question.

The following chapter will summarize a number of studies about streakiness in the financial market and various articles about the existence or non-existence of the Hot Hand in sports. The methodology of the analysis of professional bowling data, which is conducted in order to examine the extent of the Hot Hand effect, will be explained in detail in Chapter 3. Chapter 4 will provide an overview of the behaviour of people in the betting market, which represents an economic environment. In the final chapter, the most important findings will be summarized and it will be proposed what the correct perception of streaks in skilled performance and the according behaviour should be.

The Hot Hand: A literature overview 3

2 THE HOT HAND: A LITERATURE OVERVIEW

2.1 Introducing the Hot Hand

One of the main issues in the discussion about the Hot Hand is the perception of randomness. In terms related to this paper, a process generating sequences without memory and any bias towards one result can be regarded as random ¹ . If an absolutely fair coin was

tossed an infinite number of times, half of the tosses would come up heads and half would come up tails.

The bars in the following graph show the distribution of the number of heads in 1,040 sets of 100 coin flips. The black curve represents the expected results of an infinite number of tosses (binomial distribution).

Source: Wetzel, 1998, http://wetzel.psych.rhodes.edu/random/images/nheadsdist.gif

A number of 1,040 sets of flips is still far from infinite, therefore the actual distribution of outcomes only resembles but not exactly matches the expected curve.

¹ B. Burns, B. Corpus (2004): “Randomness and inductions from streaks: ‘Gambler’s Fallacy’ versus ‘Hot Hand'”.

The Hot Hand: A literature overview 4

The expected probability of obtaining between 40 and 60 heads on 100 coin flips is 95.4%, given that the coin is fairly balanced. This can be calculated with the following formula ² :

Substituting the variables,

lies two standard deviations above the mean of 50, which in turn means that this interval comprises 47.7% of the data in a normal distribution. Accordingly, approximately 95.4% of all sets of 100 coin tosses yield 40 to 60 heads.

However, the probability that a fair coin by chance produces less than 40 or more than 60 heads is still 4.6%. Supposing the task is to determine whether a coin is biased or not, and that a result of less than 40 or more than 60 heads indicates a biased coin, on average 4.6% of the times the (false) conclusion will be drawn that the coin is biased even though it actually is a fair coin.

This error margin can be reduced by widening the interval of the number of heads to between 35 and 65 for instance. The probability of a fair coin to come up less than 35 or more than 65 heads, which is equal to an interval of three standard deviations above and below the mean, is only 0.2% ³ . Accordingly, this percentage is the chance of rejecting the

hypothesis that the coin is fair even though it actually was a fair coin.

As randomness means absence of a system, a requirement of randomness is that there are few patterns and that they do not repeat often. Additionally, the probability of flipping a head on the next coin toss should be independent of the outcome of the previous flip or flips ⁴ . A related phenomenon is the misperception of the number and length of runs in data.

A run is a series of uninterrupted outcomes, e.g. a row of heads in coin tossing. As soon as a different outcome occurs, a new run begins. The amount of symbols that one run contains is the run length ⁵ .

² W. Mendenhall, J. Reinmuth, R. Beaver (1993): „Statistics for management and economics“.

³ W. Mendenhall, J. Reinmuth, R. Beaver (1993): „Statistics for management and economics“.

⁴ C. Wetzel (1998): “Can you behave randomly?”.

⁵ P. Ayton, I. Fischer (2004): „The Hot-Hand Fallacy and the Gambler’s Fallacy: Two faces of Subjective Randomness?”.

The Hot Hand: A literature overview 5

Rapoport and Budescu (1997) ⁶ conducted a study in which the participants were asked to

create a sequence of 150 draws with replacement from a card deck containing five red and five black cards. The number of times when the testees switched the colour, i.e. two subsequent draws did not consist of the same colour, was on average 58.5%, compared to approximately 50% when the outcome was generated by a random process such as coin flipping. Because of this elevated switching rate, the number of longer streaks was too small.

Accordingly, when dividing the 150 outcomes in 50 triplets, only 15% were either only black or only red compared to an average of 25% when the generating process was random. 15% of triplets with the same colour are expected if the switching rate is 61.3% ⁷ .

Similarly, only 4% of the quadruplets consisted of one colour compared to an expected 12,5%, which implies a colour switching rate of 65.8%. These findings indicate that people generate too many and too short runs when the task is to create a random sequence.

This irrationality may be one reason why the belief in the Hot Hand is substantial. The more biased a spectator is, the more persistent will be his belief in the Hot Hand because even random sequences will be perceived to be streaky.

Gilovich, Vallone and Tversky (1985) ⁸ have found the proof that people not only fail to set

up random series of data, but that also a randomly generated sequence is actually perceived as non-random. 100 basketball fans were shown six different sequences of hits and misses and were asked to decide if the data looked like streak shooting, chance shooting or alternate shooting. Each of the sequences consisted of 11 hits and 10 misses, they only differed in the probability of alternation (0.4, 0.5,…,0.9).

Streaks are likely to be produced when the alternation rate is low, i.e. when the probability that the symbol changes is low, while short streaks and a large number of runs are due to a high alternation rate.

⁶ A. Rapoport, D. Budescu (1997): “Randomization in Individual Choice Behavior”. = − = − × − × ^{2 7} The first outcome does not have to be taken into account: 15 . 0 ) 613 . 0 1 ( ) 1 ( ) 1 ( 1 p p .

⁸ T. Gilovich, R. Vallone, A. Tversky (1985): “The Hot Hand in Basketball: On the Misperception of Random Sequences”.

The Hot Hand: A literature overview 6

The result of this study is shown in the following graph, which has been extended by the values for alternate shooting that are indicated by dots.

The remarkable finding is that people were most likely to classify a sequence as chance shooting when the probability of alternation was 0.7 and 0.8, rather than 0.5. At an alternation rate of 0.5, which is similar to tossing a coin, only 32% of the participants thought that it was chance shooting but 62% classified the sequence as streak shooting ⁹ .

This fact suggests that a hypothetical basketball player with a hit rate of around 50% and scoring in a perfectly random manner is actually perceived as streaky by a majority of spectators.

2.2 Hot Hand and Gambler’s Fallacy

When people see a series of data or performance and have to make a prediction for the next event or events, they are generally biased towards the Hot Hand or the Gambler’s Fallacy 10 . The former is the main focus of this paper and stands for people’s belief in

streakiness or momentum, which means that after a success on the previous trial, the chance of a subsequent success is greater than after a miss or failure on the last trial.

⁹ T. Gilovich, R. Vallone, A. Tversky (1985): “The Hot Hand in Basketball: On the Misperception of Random Sequences”.

¹⁰ P. Ayton, I. Fischer (2004): „The Hot Hand Fallacy and the Gambler’s Fallacy: Two faces of Subjective Randomness?”.

The Hot Hand: A literature overview 7

The latter is also called Law of Small Numbers ¹¹ and refers to the common gambling misperception that a relatively small amount of observations reflects the true underlying probability of all possible outcomes. Accordingly, one evening at the roulette table should be representative of the randomness of a roulette wheel and each number would appear approximately the same amount of times. Thus, the likelihood of a number showing up a

5 ^th time should be smaller than that of a number which has not shown up all night.

In fact, the chance is the same for each number and on every spin - 1 out of 37 or 0.027 in European roulette and 1 out of 38 or 0.026 in American roulette, which, apart from 36 numbers and zero, additionally features a double-zero.

Similarly, people believe that a series of events, such as a streak of heads on tosses of a fair

coin, has to be evened out by a subsequently increased appearance of other events (tails) ¹² . The likelihood that a small set of data reflects the true probability of an event is overestimated.

This belief can be compared to drawing numbers without replacement from a bowl that

originally contained the same amount of copies of each number ¹³ . Under these circumstances, drawing a number reduces the likelihood that the same number will be drawn again at the next attempt.

Burns and Corpus (2004) ¹⁴ found out that the less random a process was supposed to be, the more likely people were to continue a streak. The experiment consisted of three scenarios: outcomes of a roulette wheel, basketball free-throws and two salespersons competing for higher week sales.

There were two possible outcomes for every scenario; red or black in roulette, hit or miss in basketball and one of the two salespersons. Additionally, the participants had to evaluate their belief regarding the degree of randomness of every scenario. The most interesting conclusions can be drawn from that group of testees who were shown a sequence of 100 outcomes of each scenario that contained a run of four similar outcomes

¹¹ A. Tversky, D. Kahnemann (1971): “Belief in the law of small numbers“.

¹² A. Tversky, D. Kahnemann (1971): “Belief in the law of small numbers“.

¹³ M. Rabin (2002): “Inference by believers in the law of small numbers”.

¹⁴ B. Burns, B. Corpus (2004): “Randomness and inductions from streaks: ‘Gambler’s Fallacy’ versus ‘Hot Hand'”.

The Hot Hand: A literature overview 8

on the last four events. According to what had been expected, the Gambler’s Fallacy was present and only 12% of the 109 students said that the roulette wheel would continue the streak on the 101 ^st event. 56% stated that the streak of free-throws would continue and

66% said that the salesperson would achieve higher sales a fifth time. On a scale ranging from 1 (completely random) to 6 (completely non-random), the average degree of randomness was 3.6 for the salespersons scenario, 3.2 for the free-throws and 2.2 for the roulette wheel ¹⁵ . It is evident that the perceived degree of randomness and the continuation

of the streak were connected.

Roney and Trick (2005) ¹⁶ conducted a coin-tossing experiment and asked the participants

after seven (manipulated) coin tosses with the result HTHTTTT (and the opposite outcome of heads and tails for the second half of the group), whether the streak would continue or not. The two groups were subdivided into a total of six groups and each group was tested separately. In three groups of testees, the same person tossed all trials while in the other three groups, the person tossing the coin changed at the critical trial. In both sets of three groups, the experimenter either mentioned that the coin was very likely to come up tails (heads), made a statement to draw the focus on the person tossing the coin (himself) or did not influence the participants at all (control group). The following table shows the subgroups of participants:

¹⁵ B. Burns, B. Corpus (2004): “Randomness and inductions from streaks: ‘Gambler’s Fallacy’ versus ‘Hot Hand'”.

¹⁶ C. Roney, L. Trick (2005): “Heuristic processes in perceptions of event sequences: The Gambler’s Fallacy and the Hot Hand”.

The Hot Hand: A literature overview 9

A significant majority of participants in the three groups where the person changed at the critical trial (group 4, 5 and 6) and the group where the same person tossed all trials but no influencing statement was made (group 1), believed the streak of four tails (heads) would end at the subsequent trial.

Only in the two groups where the same experimenter tossed all trials and the focus was drawn on him (group 2) or the coin (group 3), the effect of the Gambler’s Fallacy was not existent.

This indicates that the two beliefs (Gambler’s Fallacy and Hot Hand) compensate each other when attention is focused on a hot person tossing a coin or a person tossing a hot coin 17 .

A surprising finding is that the Hot Hand and the Gambler’s Fallacy do not necessarily compensate each other but that one person can actually exhibit both contrary phenomena at the same time. Ayton and Fischer (2004) ¹⁸ conducted an experiment and proved that people betting on the outcome of random results (red or blue) tended to pick the alternative when one colour had appeared several times in a row (Gambler’s Fallacy) but that after a streak of right guesses they became more confident to repeat the right guess at the next attempt (Hot Hand).

In the second experiment, the experimenters showed 28 sequences with signs of 11 @ and 10 # to the participants; four sequences with each of the following seven alternation rates: 0.2, 0.3,…,0.8. For every single sequence, the testees had to decide whether it had been generated by one of two processes: basketball throws or coin tossing; scoring in soccer or roulette; serving in tennis or rolling a die. Therefore, each of the pairs consisted of one process of human performance and one chance process. The idea was to test whether the size of alternation rate influenced people’s perception of how the sequence had been generated.

The results showed that the lower the alternation rate was, which is similar to longer streaks and less runs, the more likely the participants were to attribute the sequence to human performance. Taking into account the results of the two studies, this caused the

¹⁷ C. Roney, L. Trick (2005): “Heuristic processes in perceptions of event sequences: The Gambler’s Fallacy and the Hot Hand”.

¹⁸ P. Ayton, I. Fischer (2004): „The Hot-Hand Fallacy and the Gambler’s Fallacy: Two faces of Subjective Randomness?”.