Field Evidence on the Law of Small Numbers
Claus Bjørn Jørgensen∗
31st October 2006
Abstract
The law of small numbers is the fallacious belief that even small samples should
closely resemble the parent distribution from which the sample is drawn. It is
expressed through two opposite behaviors; the hot hand fallacy and the gambler’s
fallacy. Both have been demonstrated to exist in many different settings in previous
research. Little empirical attention, however, has been given to study the link
between the biases. The new data set acquired for this study allows a considerably
more thorough analysis than seen in earlier studies. The results confirm earlier
findings that hot hand- and gambler’s fallacy behavior is prevalent among Lotto
players; although it appears to be concentrated within a relatively small group. For
the first time in field data this study shows evidence indicating that the two biases
are not mutually exclusive, but depend on the time-horizon in the way predicted
by recent behavioral theory. Further, I find that biased players gamble more, and
there are weak indications that women and the elderly are more often biased.
1
Introduction
Imagine sitting at the roulette table at a big casino in Las Vegas. After four spins of
the wheel, you have observed the outcome to be red every time. The following spin also
results in red, and the sixth time the outcome is red again. Now where do you place your
bet? Do you feel that black is ‘due’ and therefore must show up now? Or do you bet on
red in faith that its streak will continue? In the first case, you suffer from the gambler’s
fallacy; an erroneous belief that due numbers has to ‘catch up’, such that the random
appearance of the process is restored. In the second case you are subject to the hot hand
∗
This paper is based on my master’s thesis at University of Copenhagen (Jørgensen 2006), and is prepared for the Zeuthen Prize; awarded by The Society of Social Economics (SØS). The thesis additionally
features an analysis of WednesdayLotto and time-invariant players, an analysis of the selection bias in
the internet sample, more econometric details and more references. I thank my advisor, professor JeanRobert Tyran, and fellow students for excellent comments and suggestions. The author can be reached
at clausbjorn@gmail.com.
1
1 INTRODUCTION
fallacy; believing that the observed streak of reds reflects a higher probability of red than
its true rate.1
On the surface these two fallacies may appear mutually exclusive, but it has been
argued that they both stem from an underlying belief in the so-called law of small numbers
(Tversky & Kahneman 1971). The law of large numbers from statistics tells us that when
the number of observations is sufficiently large, the moments of a random sample becomes
arbitrarily close to the moments of the parent population from which it is drawn. Many
people, however, grossly underestimate the number of observations needed and believe
that even small samples should closely reflect the underlying distribution. They therefore
expect deviations in one direction to soon be canceled out by deviations in the other
direction (the gambler’s fallacy). If it does not happen, they find it very hard to believe
that the outcomes are indeed equally likely and they thus switch beliefs into thinking that
the streak will continue (the hot hand fallacy) (Rabin 2002). This mechanism will be
discussed in more detail in section 2; followed by a brief review of the previous empirical
research in section 3.
This paper asks the following:
• Is the belief in the law of small numbers widespread, and does it appear in a field
setting?
• Are the opposite behaviors of the hot hand fallacy and the gambler’s fallacy mutually
exclusive, or are they related in the way predicted by recent theory?
• Are some demographic groups more prone to hold biased beliefs, and do biased
individuals spend more money on (Lotto) gambling?
To answer these questions, I make use of an exciting and unique new field data set acquired
specifically for the purpose of this study. It describes internet Lotto-players’ individual
choice of Lotto numbers over a period of six months. Earlier studies have found evidence
of both the gambler’s fallacy and the hot hand on the aggregate level. Since the law of
small numbers works in opposite directions depending on the circumstances, the effect may
partially be canceled out in the aggregate. If constrained from studying micro behavior,
important patterns may therefore be missed, especially regarding the interaction between
the two biases. The rich data material in this study allows me to investigate these issues
in far more detail than seen before in field data. Specifically, I look into how people react
to different time horizons of the game’s history.
1
As a fruitful reality check, the reader is encouraged to Google the web for phrases like “lotto predictions”. An example of what shows up is the web site LotteryPost.com. It features an advanced prediction
page where players can compete against other members to make the ‘best’ predictions of future drawings.
In the very active forum people eagerly defend their predictions, typically with arguments resembling
the hot hand- or the gambler’s fallacy. A casual web search also reveals a myriad of dubious, but costly,
number-prediction software packages.
2
1 INTRODUCTION
Section 4 describes the Lotto game and the internet retailing platform of Danske Spil.
Lotto is a very well-suited setup for studying the law of small numbers: It is close to
being a controlled experiment, but the artificial environment of the laboratory is avoided.
The players do not know they are being studied; eliminating all concerns of “Pygmalion
Effects”.2 Compared to other markets of choices under risk such as insurance-, financialand sports betting markets, lotteries are very clean and the researcher thus have no doubt
about the underlying statistical properties. The Lotto game has a widespread popularity
with more than sixty percent of the Danish population having played it within the past
year; hence it is not just an obscure minority who participate in this market. Almost
200,000 unique individuals, close to five percent of the adult population, play Lotto on
the internet and are thus included in the data set. Since players are obliged to set up
individual accounts I am able to track them over time. The game has similar stakes
to experiments, although the possible upside gain is much higher. Finally, a trivial but
necessary condition is satisfied: The data is available; electronically and in large amounts,
as described in section 5.
The principal part of this paper is the empirical investigation in section 6. It can be
categorized as a weekly repeated natural experiment, in the taxonomy of Harrison & List
(2004). Every week, nature randomly draws seven winning Lotto numbers. There is no
control group since every player is subject to the same treatment (the same winning numbers apply to all players). Each Lotto number, however, is treated differently according
to the random process of balls being drawn from an urn. In the empirical investigation, I
make use of these truly exogenous treatments of Lotto numbers to make inferences about
the players’ beliefs. Being a whole new source of data there is no ‘common practice’ to
guide the empirical investigation. In result some of the econometric challenges are handled
here, while others are left for future research.
A fundamental assumption underlying the empirical investigation is that the players’
number selection can be used as a proxy for their subjective beliefs about the numbers’
probability of being drawn. Using this assumption, I compare the players’ choice of
numbers with the drawing outcomes from previous weeks to see if their beliefs are affected
by the history. The rich data material allows me to estimate individual belief-parameters
for each player in the data set to account for heterogeneous rationality. The results show
that hot hand- and gambler’s fallacies behavior are prevalent among Danish Lotto players,
although they appear to be concentrated within a relatively small group. A correlation
analysis show that the gambler’s fallacy and the hot hand are not mutually exclusive, and
that they are related in the way predicted by recent behavioral theory. Biased players
spend more money on gambling than unbiased players, suggesting that the law of small
2
The notion that agents may alter their behavior when being observed by others. It is related to
the well-known Heisenberg Uncertainty Principle from physics telling that the act of measurement and
observation alters what we are observing (Harrison & List 2004, p. 1034).
3
1 INTRODUCTION
numbers may partly explain problem gambling.
The assumption above requires that most players fail to understand that Lotto is not
a game about guessing which numbers will be drawn—that is impossible. Rather, Lotto
is about guessing which numbers the other players pick, and stay clear of those to reduce
the risk of having to share a prize. Therefore the number selection process is a game of
strategic substitutes, in the sense of Fehr & Tyran (2005). The aggregate effect of having
even a substantial share of biased players should thus be approximately canceled out by
rational players pursuing the opposite strategy for purely strategic reasons. Nevertheless,
the present study confirms earlier findings that the aggregate distribution of bets is far
from uniform. An obvious explanation is that the best response is not to play at all! The
rational and strategic types who would otherwise have dragged the aggregate distribution
towards being uniform do therefore not participate.
Fallacious understanding of randomness may affect peoples’ behavior in all situations
involving choices under risk. This includes economically important areas as insurance and
financial markets, but also social scientists making too strong conclusions based on limited
data material. In fact, this was what originally led Kahneman & Tversky to develop the
theory of the law of small numbers. Other applications are discussed in section 7; along
with suggestions for future research.
This study is the first to apply individual-level field data of lottery number selection
and is thus breaking new ground. The improvement works along at least two dimensions:
First, the very large pool of subjects increases the credibility of the findings. Second, the
detailed panel data structure with the ability to track customers over time enables me to
conduct a considerable more in-depth analysis than seen in earlier studies. Specifically, I
show that the gambler’s fallacy and the hot hand fallacy are not mutually exclusive, but
depend on the time horizon in the way predicted by theory. To the author’s knowledge,
this has not previously been shown in the field.
On the methodological note, this study exemplifies how researchers in the social sciences can make use of the astronomical amounts of data collected every day about our
behavior. In that manner, it resembles DellaVigna & Malmendier (2006) who use data
on health club visits to make inferences about the customers’ time preferences and selfcontrol problems. With the rapid progress in information technology and the continuing
adaption of CRM3 databases even in smaller businesses, this source of data is becoming
increasingly promising for future research.
3
Customer Relationship Management
4
2 THEORETICAL BACKGROUND
2
Theoretical Background
Neoclassic economic theory has trouble explaining why people buy insurance, thereby
exhibiting risk-averseness, while they at the same time spend money on gambling; suggesting risk-lovingness. One answer has been to assume special forms of individuals’
utility-of-wealth function (Friedman & Savage 1948); this however only explains specific
combinations of lottery and insurance purchases. Prospect theory offers a different explanation that suffer from the same weaknesses (Kahneman & Tversky 1979, Kőszegi
& Rabin 2005); while a third approach has been to add an intrinsic utility of gambling
(Conlisk 1993).
It may be that the players simply misunderstand the chances of winning, and therefore
play in the false belief that their expected pay-off from the lottery is positive. In a
series of now classic papers, Tversky and Kahneman suggested a number of heuristics
and biases that affect individuals’ decision making. According to the representativeness
heuristic, people evaluate the probability of whether A originates from process B by
the degree to which they resemble each other (Tversky & Kahneman 1974, p. 4). In
the case of a sequence of independent random events, such as drawing lottery numbers,
the representativeness heuristic gives rise to a biased dubbed the law of small numbers;
according to which people believe that even small samples are highly representative of
the population of which they are drawn (Tversky & Kahneman 1971). In Lotto, players
would thus expect a very even distribution of draws across the 36 numbers after only a
few drawings. However, if every segment of the sample is believed to closely reflect the
underlying distribution that will lead players to assume that a deviation in one direction
must soon be followed by a deviation in the opposite direction. This belief is called the
gambler’s fallacy.
The gambler’s fallacy is the belief that random sequences are active self-correcting
processes. People fail to understand that “deviations are not canceled out as the sampling
proceeds, they are merely diluted” (Tversky & Kahneman 1971, p. 106). Rabin (2002)
proposes a formal theoretical model of the law of small numbers.4 In his model, the agents
erroneously believe that a random, independent process is generated by draws from an
urn without replacement. As the size of the urn N approaches infinity the agents become
perfect baysian. It should be immediately clear that his model directly captures the
gambler’s fallacy. Consider for instance the roulette, and assume for simplicity that the
wheel only has red and black outcomes. If the urn in the player’s mind initially contained
ten balls of which half was red and half was black, and if the two previous drawings
resulted in black, the urn now still contains five red balls but only three black balls. In
that case the agent now believes that the next draw will be red with probability 5/8;
4
The model has been further developed in Rabin & Vayanos (2005). Related theoretical approaches
are found in in Mullainathan (2002) and Epstein & Sandroni (2003).
5
3 REVIEW OF PREVIOUS EMPIRICAL EVIDENCE
instead of the true fifty percent. That is, he believes that red is ‘due’.
When the agents are uncertain about the underlying probability, Rabin (2002) shows
that the belief in the law of small numbers can result in the hot hand bias: an erroneous
belief that numbers that have been drawn frequently in the near past are ‘hot’; i.e. they are
more likely to be drawn again (opposite to the gambler’s fallacy). The argument goes as
follows: Suppose the agent has observed five black outcomes in a row at the roulette table.
If he knows that the underlying rate is 0.50, he is convinced that the next ball will be red
so the balance between black and red can be restored (the gambler’s fallacy). Suppose
now that also the next outcome is black. Since he believes that even small samples should
represent the underlying distribution, he finds it increasingly difficult to believe that six
black balls in a row could stem from a fifty-fifty process. Instead, he ‘switches belief’ and
infer that black is more likely than red—that the color is ‘hot’. Hence, believers in the
law of small numbers overinfer from short sequences.
The idea of an urn in the players mind may seem a bit abstract. However, psychologists
Altmann & Burns (2005)“put cognitive processing flesh”on Rabin’s theory and explain the
urn size as working memory capacity. A related study in the field of neuroscience suggests
that the fallacy is rooted in the prefrontal cortex (Huettel, Mack & McCarthy 2002).
3
Review of Previous Empirical Evidence
Several experimental studies have demonstrated behavior consistent with the hot hand and
the gambler’s fallacy in the laboratory; see e.g. Offerman & Sonnemans (2004), Papon
(2005), and the reviews in Bruce & Johnson (2003) and Rabin (2002). Questionnaire
surveys from the psychology literature show that frequent lottery players are more likely
to show some misunderstanding of probabilities (Coups, Haddock & Webley 1998, Rogers
& Webley 2001).
A large literature uses field data from lotteries to study other aspects of gambling.
One branch has studied the demand for gambling; finding that sales clearly increase
as the effective price of the lottery falls due to rollovers (e.g. Clotfelder & Cook 1990,
Paton, Siegel & Williams 2003). A few recent papers have obtained more disaggregate
sales data that allow them to investigate differences in behavior between socio-economic
groups. Using sales by zip code in the Connecticut lottery, Oster (2004) find that richer
areas respond much more to increases in the jackpot than poorer areas. One possible
explanation is that the presumably higher educated individuals in the richer areas have
a better understanding probabilities and odds. Hence, they save their money until the
jackpot goes up and the expected pay-off is increased. The hypothesis is substantiated by
the somewhat similar findings of Guryan & Kearney (2005); using store-level sales data.
6
3 REVIEW OF PREVIOUS EMPIRICAL EVIDENCE
Another branch of the literature has investigated whether players pick their numbers
randomly, as predicted by game theory (see section 6.2.1). The studies unequivocally
conclude that this is not the case (e.g. Chernoff 1981, Thaler & Ziemba 1988, Simon 1999,
Hauser-Rethaller & König 2002); suggesting that few players respond to the lotteries’
strategic element.
Studies of how players use the history of draws to ‘predict’ future outcomes is much
more scarce. Using data from the Maryland 3-digit lottery, Clotfelder & Cook (1991,
1993) find that after a number is drawn, the amount bet on that particular number falls
sharply, indicating gambler’s fallacy beliefs. However, since the Maryland lottery pays a
fixed amount to winners, this behavior is not irrational. The expected pay-off is the same
no matter how many players pick a particular number. In contrast, the New Jersey 3-digit
numbers game studied by Terrell (1994) is parimutuel, such that the lucky winners split the
pot. As in the Lotto, it is then costly to follow a common heuristic, so people responding
rationally to economic incentives should try to avoid them (i.e. strategic substitutes). In
spite of this, Terrell finds clear evidence of the gambler’s fallacy; suggesting that strategic
considerations play a minor role when picking numbers.
Henze (1997) and Papachristou (2004) investigate the gambler’s fallacy and the hot
hand in the modern Lotto; both using aggregate data. In contrast to the above, they only
find weak indications that players react to the previous outcomes of the game. Belief in
the law of small numbers has also been shown to exist in other settings such as basketball
(Gilovich, Vallone & Tversky 1985, Camerer 1989), greyhound racetrack betting (Terrell
1998), and in finance (Odean 1998).
Only one previous study has managed to obtain individual-level field data.5 Croson
& Sundali (2005) study the behavior of 139 players at the roulette table; using 18 hours
of security videotape recordings supplied by a large casino in Nevada. As in Lotto, the
outcome of a roulette wheel is completely random and serially uncorrelated. However,
unlike Lotto the odds at the roulette table are fixed, so it has no costs to follow a common
heuristic in terms of expected pay-off of the individual game. The authors find clear signs
of gambler’s fallacy. As seen from figure 1, the proportion of players betting against the
streak (e.g. betting on black after N occurrences of red) increases considerably as the
length N of the streak grows. When N ≥ 5, about two thirds of the bets are against the
streak, and the difference is significant.
In a companion piece (Sundali & Croson 2006), the authors look at heterogeneity
between players. They find that about 18 percent of the players show significant gambler’s
fallacy behavior while 20 percent behave in consistence with the hot hand fallacy.6
5
With the possible exception of Walker & Wooders (2001) who study tennis players’ serve strategies
in the Wimbledon tournament. Although their results can be interpreted as supporting the law of small
numbers, their focus is on a different issue and they do not study responses to the history of the game.
6
Croson & Sundali apply a somewhat different definition of the hot hand fallacy than most of the
7
4 THE DANISH LOTTO
Figure 1: Proportion of gambler’s fallacy bets following a streak of length N
Source: Figure 4 from Croson & Sundali (2005).
Note: The number over the bars indicate the number of occurrences.
4
The Danish Lotto
Since its introduction in 1989, Lotto has become the most popular game of Danske Spil;
a government-controlled corporation with a legal monopoly of large-scale number games
in Denmark. Three out of four adult Danes have tried the Lotto game, and more than
sixty percent have played it within the past year, cf. table 1.
Table 1: Experience with games from Danske Spil (percent)
Game
Lotto
WednesdayLotto
Sports betting
Scratch-card games
Other games
Total
Ever played
75
39
37
54
31
90
Played within
the past year
62
23
14
37
6
78
Played within
the past week
29
6
5
4
1
37
Source: AC Nielsen AIM (2005)
Note: Percent of population over age 15. Third quarter 2005. Based on 2,613
phone interviews.
The Danish Lotto is a 7/36 lottery.7 For each row he or she wishes to play, the player
literature, including this paper. In their terminology, a hot hand fallacy-player believes that his personal
luck is positively autocorrelated, irrespective of which numbers he gambles on. Instead they dub believers
in positive autocorrelation of specific numbers ‘hot outcome’ players.
7
The source for this section is the web page of Danske Spil, http://tips.dk. It describes the Lotto
at the time the data was collected (fall 2005). Danske Spil has recently adjusted Lotto’s prize structure
slightly.
8
4.1 The Internet Retailing Platform
4 THE DANISH LOTTO
picks seven numbers from 1 to 36. The price per row is 3 DKK, or about 0.50 USD, and
the player must buy at least two rows. The player wins the first prize if all the seven
numbers on a single row match the seven winning numbers. With 36 different numbers,
there are
36
36!
=
= 8, 347, 680
(1)
7
7!(36 − 7)!
different combinations, so the chance of winning the first prize is a mere 8.3 million to
one. There are four different ways of picking the numbers, and they are all available for
purchase either online or in physical outlets:
• Manual Selection (Rækkespil). The player marks 7 numbers of his or her own choice
on a 4 x 9 grid (online) or a 6 x 6 grid (offline); one grid for each row.
• QuickLotto (LynLotto). A computer system picks 10 rows with random numbers.
• LuckyLotto (LykkeLotto). The player picks from 1 to 6 lucky numbers and choose
how many rows to play. The computer system puts all his lucky numbers on all the
rows and fills out the remaining spaces randomly.
• SystemLotto. Danske Spil offer a variety of ‘mathematical systems’ in which the
player picks for instance 9 or 10 numbers, and lets the computer generate rows
based on these numbers using some pre-defined algorithm.
4.1
The Internet Retailing Platform
Danske Spil launched its internet retailing platform at http://tips.dk in April 2002, and
in December the same year it became possible to play Lotto online. The internet platform
generated a revenue of 573 million DKK in 2005, making up six percent of Danske Spil’s
total revenue, and at the end of 2005, about 240,000 players had set up an account (Danske
Spil 2005). No other vendors are allowed to sell Lotto tickets online. To open an account
at tips.dk, the players must be at least 18 years old and have a Danish CPR-number
(social security number). The players must transfer at least DKK 200 from their debit
card to open an account, but there are no startup- or maintenance fees. The players can
gamble for no more than DKK 5,000 per day, and there is an annual limit at DKK 100,000
for transfers to the account. As described in section 4, the same number-picking options
are available online as in physical outlets, although the numbers are displayed on a 9x4
grid instead of a 6x6 grid; see figure 2.
9
4.2 Prizes and Expected Pay-Off
4 THE DANISH LOTTO
Figure 2: Screenshot of Lotto number picking procedure at tips.dk
4.2
Prizes and Expected Pay-Off
The winning numbers of the Lotto game are drawn by a mechanical device in a TVtransmitted session every Saturday. The payout rate is set by law at 0.45. In addition,
all prizes above 200 DKK are subject to a 15 percent tax; but the prizes are exempt from
all income taxes.8 This tax is automatically deducted by Danske Spil before prizes are
paid out; and all advertised prizes are net of the tax. One quarter of the winning pool is
used for the first prize. The average winnings from getting all seven numbers right is 2.4
million DKK after taxes, but it varies considerably and has been as high as 15.9 million
DKK. The second prize averages 61,500 DKK, while the other prizes are much smaller.
The parimutuel structure of the game implies that the prize pool is split between all
the winners in the prize category. This adds a clear strategic element to the game: If
other players pick the same numbers as you, your expected pay-off is reduced since you
risk sharing the prize with more people. Hence, the true nature of the game is not the
impossible task of guessing the winning numbers, but to guess what numbers other players
are likely not to pick, and then bet on those.
If there are no first prize winners a given week, the jackpot rolls over to the subsequent
drawing; generating a higher payout-rate. Since the 0.45 pay-out rate must be fulfilled
in the long run, this implies that the expected pay-off is lower than 0.45 minus taxes in
non-rollover weeks. In a typical non-rollover week, the expected pay-off from betting on
8
Source: Bekendtgørelse af lov om visse spil, lotterier og væddemål, Consolidated Act no. 1077, 11th
December 2003. Available on http://retsinfo.dk.
10
4.3 Do Hot Numbers Exist?
4 THE DANISH LOTTO
random numbers can be calculated to be 41 percent net of taxes; rising to 47 percent in
rollover weeks.9
4.3
Do Hot Numbers Exist?
It is indeed possible that the Lotto drawing device could be biased towards certain numbers. Even though the drawings are performed by a highly sophisticated drawing device,
and one must assume that the machinery is carefully calibrated; it can not a priori be
ruled out that slight differences in the physical properties between the numbered balls,
or the procedure of how the balls is loaded into the machine could imply that some numbers are favored. In this case, it would be perfectly rational for players to prefer often
drawn numbers, although gambler’s fallacy behavior could not be explained by this. As
expected, however, a χ2 goodness-of-fit test does not reject the H0 hypothesis of a fair
drawing device in the Danish lotto.10
If one of the numbers indeed was positively biased, how many observations would be
needed to identify it? Keren & Lewis (1994) answer this question in the case of a roulette
wheel with 37 numbers; a situation quite similar to the Lotto drawing procedure. With
a perfectly calibrated wheel, the probability of any number being drawn is 1/37. If one
imagines a biased wheel with a single favored number, and the probability of that number
is 1/35; Keren & Lewis (1994) show that it requires a whopping 133,435 trials to identify
the number with 90 percent probability!
All of the above assumes that a bias is constant over a long period of time. It is
more likely that a bias would be changing over time or only occur in a shorter period; for
instance due to regular replacement of the balls or re-calibration of the drawing device.
In fact Danske Spil operates with three sets of Lotto balls and two identical drawing
machines. Before each drawing, a preliminary lottery is held to determine which set of
balls to use. The machines are switched every six months. The public is not informed
of which set of balls and which machine is being used, and it is therefore not possible
to track different biases in the different machines and balls. In addition, the balls are
replaced when they have been used 52 times.11 It is thus in practice impossible to detect
a small bias and identify the favored number(s), given the huge number of trials required.
Any attempt to do so can thus be labeled irrational.
9
These figures do not appear on http://tips.dk but have been calculated using the methodology in
Cook & Clotfelder (1993). See the details
P in Jørgensen (2006).
10
The usual goodness-of-fit statistic (Observed − Expected )2 /Expected needs to be scaled by a factor
(36 − 1)/(36 − 6) to account for sampling without replacement (Stern & Cover 1989). Using data from
the past 10 years of drawings, the test statistic thus becomes 32.54. Compared with a critical value of
χ20.95 (36 − 1) = 49.80, the null hypothesis of a fair drawing device is not rejected.
11
Source: Email correspondence with Danske Spil’s Customer Service.
11
5 DATA
5
Data
The main data material was generously supplied by Danske Spil. It encompasses all Lotto
tickets purchased online during the last six months of 2005, covering 189,531 unique players
who have purchased a total of 5,375,916 tickets; each with 10.3 rows of seven numbers on
average, cf. table 2 below.12 The reliability of the data set is judged to be almost perfect,
being register data generated directly from Danske Spil’s central computer system. In
the raw data set, each observation represents a lotto ticket and contains the number of
rows played, the numbers picked, the selection mode (QuickLotto vs. manual pick) etc.
Further, a customer ID number allows me to identify multiple tickets played by the same
customer a given week, and to track individuals over time. Since players are required
to submit their CPR-numbers to Danske Spil it is not possible for an individual to open
multiple accounts.
Table 2: Summary Statistics of Data Set
Average number per week (28 weeks)
Unique customers
Tickets
Rows per ticket
Rows per customer
Percent
Share of tickets ManualPick
Share of tickets SystemPlay
Share of tickets QuickLotto
Average number of drawings participated
per customer
119,340
191,997
10.3
17.6
37.8
7.2
55.0
17.6
Source: Own calculations on data from Danske Spil A/S.
In addition, the data contains basic demographic information about each player’s age,
gender and place of residence (postal code). Again this information is judged to be very
reliable since both age and gender are derived directly from the CPR-numbers.13 The
outcome of past drawings has been extracted from Danske Spil’s public web site.
5.1
Player Categories
The Lotto players are divided into three distinct subsets depending on how they pick their
numbers, as illustrated in figure 3. Players that have only participated in one drawing
12
The raw data set also includes WednesdayLotto; a game quite similar to the regular Lotto. WednesdayLotto is analyzed in Jørgensen (2006).
13
The Danish CPR-numbers (‘Social Security numbers’) consist of your date of birth followed by four
control digits. The last digit indicates your gender (odd for male, even for female). The actual CPRnumbers are naturally not in the data set given to me.
12
5.1 Player Categories
5 DATA
are excluded as they can not be categorized unequivocally, leaving back 171,272 unique
customers:
Figure 3: Player categories
Source: Own calculations on data from Danske Spil A/S.
a) QuickLotto players (83,850 players, 49 percent). Almost half the players in the
data set only buy QuickLotto tickets. There are several reasons why people choose
this option (Simon 1999). Some players realize that they can not affect the chance
of winning so why waste time filling out numbers?14 Others may recognize how
difficult it is to produce random numbers on their own. They therefore use the
QuickLotto option to reduce the risk of hitting popular combinations from using
a common number-picking heuristic. Finally, some players may believe that they
are more likely to win if their numbers are picked in a way that mimic the random
process by which the winning numbers are chosen (a representativeness heuristic).
b) Time-invariant players (48,040 players, 28 percent). About a quarter of the internet players pick the exact same numbers every week. One obvious explanation of
this behavior is it requires a minimal effort, since the internet store offers a functionality that makes it very easy to repurchase previously picked numbers. Another
motive is to minimize regret. A US state lottery once used the slogan “Don’t let
your numbers win without you”, reminding players how bad they would feel if they
missed a drawing and ‘their’ numbers hit (Clotfelder & Cook 1991).
c) Time-varying players (39,382 players, 23 percent). The remaining quarter of the
players also pick their numbers manually (or by using SystemLotto), but they change
their bets over time. This option clearly requires more effort than a) and b). One
possible explanation of this behavior is that the players continuously update their
14
They can however affect the expected payoff by choosing unpopular combinations, although it is quite
difficult. See the discussion in section 6.2.1.
13
6 EMPIRICAL INVESTIGATION
beliefs and want to bet on ‘due’ or ‘hot’ numbers as implied by the law of small
numbers.
The following analysis will focus on the latter group. The time-variation within each
customer in this group allows an elaborate investigation. In contrast, the behavior of
group b is time-invariant so the within-estimator cannot be calculated and it is thus not
possible to isolate individual ‘lucky number’ effects. For an analysis of this group, the
reader may consult Jørgensen (2006).
6
Empirical Investigation
How can an individual’s belief in the law of small numbers be identified by studying her
lottery gambling? In contrast to a controlled lab experiment, there are no randomly assigned control group of players to facilitate ceteris paribus comparisons in this field study.
Instead, I will utilize the truly exogenous random treatment of each lotto number that
takes place every week when the seven winning numbers are drawn. The players’ choice
of numbers will be used as a proxy for their belief in the numbers’ drawing probabilities.
The question posed in the beginning of this section can thus be answered by investigating
if the number choices are affected by the drawing history.
The analysis rests on one fundamental assumption: That the players’ number selection
must not be dominated by strategic considerations of avoiding popular numbers. This
could be problematic, but as will be discussed in section 6.2.1 there are strong arguments
that the assumption is indeed valid. A second assumption is that other factors influencing
the players’ beliefs should be uncorrelated with the drawing history. However, since the
drawing history is fully random (except the special cases of lotto numbers 35 and 36; see
below), this assumption should not be a problem.
The present section is the crux of the paper. First, in section 6.1, I present some
descriptive statistics demonstrating that the biases show up even in the aggregate. Section
6.2 sets up the analytical framework, while the results from the regressions are presented
in section 6.3.
6.1
Descriptive Statistics
Before turning to a more formal analysis, it is fruitful to take a look at the raw data.
Figure 4 shows the aggregate popularity of the 36 Lotto numbers in the Saturday game
as an average over the 28 weeks in the data set (the shares are quite persistent over time).
QuickLotto tickets are excluded to eliminate the noise from these computer-generated
14
6.1 Descriptive Statistics
6 EMPIRICAL INVESTIGATION
rows and focus on the number choices made deliberately by the players. The figure also
displays the all-time historical drawing frequency up until mid 2005.
A number of ‘stylized facts’ can be derived directly from figure 4: First, the number
selection is clearly non-random, given the huge number of underlying observations. A
standard chi-square test overwhelmingly rejects the H0 hypothesis of equal popularity
across numbers. This supports that most players do not put much weight on the strategic
element of avoiding popular combinations. Second, note the very low drawing frequency
and popularity of numbers 35 and 36. The low drawing frequencies has the natural
explanation that the numbers were not introduced until several years after Lotto’s launch
in 1989,15 . It appears that some players interpret the low drawing frequencies of 35 and
36 as if the numbers are less likely to be drawn, i.e. the hot hand fallacy. An alternative
explanation is that some players have picked the exact same rows ever since the Lotto
was introduced, and therefore does not include these ‘new’ numbers on their tickets. The
pattern however also shows up among the time-varying players, suggesting this is not the
sole explanation.
Figure 4: Aggregate popularity of Saturday Lotto numbers and result from
past drawings. Internet players excluding QuickLotto
Source: Own calculations on data from Danske Spil A/S.
Note: The numbers in the graph are the Lotto numbers (1-36). The figure covers the 25.6 million rows purchased
at http://tips.dk from week 25 to 52, 2005; excluding QuickLotto tickets. Number 35 and 36 were not in the game
from the beginning which explains their low drawing frequencies.
15
Number 35 was added in week 47, 1992 and number 36 was added in week 46, 1994. Source: Telephone
conversation with Danske Spil Customer Service
15
6.2 Analytical Concept
6 EMPIRICAL INVESTIGATION
Third, just by ‘eye balling’ figure 4 it is clear that there is a remarkable co-movement
between the two series. The correlation coefficient is 0.57 and is significantly different
from zero. Even when disregarding the special cases 35 and 36 there is still a clear
positive correlation between past drawings and popularity (r = 0.25). The correlation
is not perfect; for instance is number 7 the most popular number despite it has been
drawn considerably less than average. One explanation is that seven by many individuals
is perceived to be a ‘lucky number’. It also appears that large numbers are less popular
than small numbers. Nevertheless, the figure suggests that some players try to predict
the outcome by using the drawing history. As argued in section 4.3 it is unlikely that
the drawing device is biased, and even if there were a small bias it would be practically
impossible to detect. The players’ behavior is thus irrational and consistent with a belief
in the law of small numbers. They appear to overinfer the drawing probabilities from a
‘small’ sample, thereby exhibiting the hot hand bias.
A comparison of the recent drawing history of year 2005 with the aggregate popularity
shows no clear patterns. Looking at the very short run, the popularity of a winning
number drops 0.09 percentage points on average the week following the draw; indicating
a gambler’s fallacy pattern. Compared to a base level of 19.4 percent, the difference is
significant but admittedly very small.16
In conclusion, the results until now suggest that one type of bias, the hot hand,
clearly dominates when looking at the full history, even enough to show up clearly in the
aggregate. When looking at the short history there may be opposing biases approximately
canceling each other out in the aggregate although the gambler’s fallacy slightly dominates.
To explore the link between the biases and individual differences between players, I now
turn to a formal analysis of the data.
6.2
Analytical Concept
Believers in the law of small numbers erroneously believe that the outcome of an independent process such as lottery drawings depends on the history of the game. In addition, due
to ‘magical thinking’ they may feel that certain lucky numbers are more likely to be drawn
(such as seven), and others are less likely to be drawn (such as high numbers).17 Hence
the belief of individual j ∈ [1, . . . , 39 382] about the probability of number i ∈ [1, . . . , 36]
16
In the q
usual approximative U-test of equal means, the test statistic is (0.02278 −
2
2
(−0.09438))/ 0.29057
+ 0.32814
= 4.58. Evaluated in the normal distribution, this clearly rejects the
812
189
null hypothesis of equal means.
17
Some ‘magical thinkers’ believe that special numbers or combinations are only lucky in their own
hands, not in somebody else’s (Wagenaar 1988). Hence they do not believe that these numbers are more
likely to be drawn per se, but that they can increase their personal chances by betting on them.
16
6.2 Analytical Concept
6 EMPIRICAL INVESTIGATION
at time t ∈ [2005w25, . . . , 2005w52] can be stated as
Subj Pr(i)jt = p + f j (Drawing Historyit ) + αij
(2)
where p denotes the true probability of a number being drawn (7/36), f j (·) is some
function to be estimated and αij indicates the unobserved time-invariant perceived luckiness of the number. Previous research suggest that individuals differ considerably in
their degree of rationality, and that cognitive biases are heterogeneous (e.g. Croson &
Sundali 2005, Camerer, Ho & Chong 2004, Terrell & Farmer 1996). The rich data material allows me to explore these individual differences instead of merely studying the
average bias. Therefore the terms in equation (2) that indicate biased beliefs are allowed
to depend on the individual; as indicated by the j’s.
6.2.1
From Beliefs to Behavior
The nature of the data material does not allow me to measure people’s beliefs directly.
Rather I am constrained to studying their behavior. Being a strategic game the translation
from beliefs to behavior is not as straight-forward as it might seem. If all players were
fully rational, but nevertheless decided to play, the game-theoretic prediction would be
that every number should be picked equally many times: If one number turned out to
be picked less than the rest, players would gain from deviating and put the less popular
number on their tickets to reduce the risk of sharing a prize; hence it would not constitute
a Nash equilibrium. Suppose now that a player find some of the numbers more likely to
be drawn due to a susceptibility to the gambler’s fallacy. On the face of it, he would now
increase his bets on the “favored” number, even though this would marginally increase
his risk of sharing a first prize. However, it would all depend on his second- and higher
order beliefs: If the biased player believes that other players also want to bet on the
favored number, he might consider the price of following his instincts too high; i.e. that
the increased risk of sharing the jackpot outweighs the perceived higher chance of winning
the jackpot with that number. Therefore, he might not pick the number even though he
considers it more likely to be drawn.
Conversely, a Lotto-playing but otherwise fully rational agent might have second-order
beliefs suggesting that a substantial part of the other players have gambler’s fallacy beliefs;
and that they therefore bet more on numbers with a low historical drawing frequency. A
rational response would be to avoid these numbers, and instead pick numbers with a high
historical drawing frequency. Hence, on the surface he would appear to have hot-hand
beliefs even though he acts perfectly rationally. The bottom line is that it is very hard to
separate strategic thinking from fallacious beliefs.
Camerer et al. (2004) develop a formal model of limited strategic thinking. They set
17
6.2 Analytical Concept
6 EMPIRICAL INVESTIGATION
up a “cognitive hierarchy” in which each player assumes that he is the most sophisticated.
Agents are divided into different types, where k = 0 types fully randomize; step 1 types
best-respond assuming that all other players are 0-types; k = 2 thinkers best-respond
assuming all other players are distributed over step 0 and step 1 etc.18 Estimating their
model on data from a broad range economic experiments, assuming k is Poisson distributed across individuals, Camerer et al. find that the average number of cognitive
steps is typically around 1.5, but it is highly context specific and often much smaller
depending on the nature of the game.
It therefore seems reasonable to assume that the perceived-probability-of-number element dominates the strategic element. Three distinct elements of the Lotto further
substantiates this conjecture: First, the information and advertising from Danske Spil
does not mention that you can improve your pay-off by selecting unpopular combinations.
Instead it advocates that Lotto is a game about guessing the winning combination. Just
watch any commercial for the game, or take a look at official rules at http://tips.dk where
the first paragraph reads:
“1. Lotto is a number game about predicting the correct result of a drawing
lots, in which 7 numbers from 1 to 36 are drawn. [. . . ]”19
Second, it is difficult, albeit possible, to collect information about the other players’ actions. Danske Spil does not publish the popularity of different numbers and combinations;
but it can be inferred with some noise and significant effort from previous outcomes and
average winnings. Even if the player was somewhat up the cognitive ladder, he would thus
find it very costly to align his predictions with empirical evidence. Third, if the strategic
element dominated, we would find only small differences in the aggregate popularity of the
Lotto numbers. As we saw in figure 4, however, this is far from being the case. Finally,
fully rational players are likely not to participate in the Lotto at all, given the very low
expected pay-off of less than 0.50.
In conclusion, the players’ number selection behavior will therefore be used as a proxy
of their beliefs, but the problems outlined above will be kept in mind when interpreting the
results. Note that I am not interested in inferring the level of the subjective probabilities.
Rather, I want to investigate if the drawing history changes these beliefs, and if so in
which direction. Therefore I only need to assume that the players’ beliefs are positively
18
The recurring case used in Camerer et al. (2004) is the well-known pick-a-number game. In this
game, each player is asked to pick a number between 0 and 100 (non-integers allowed). The player that
comes closest to two thirds of the average is declared winner of the game. k = 0 types just pick a random
number between 0 and 100. Type 1 thinkers assume everybody else randomize, and therefore play 2/3 · 50
etc. The authors claim that their theory can explain why experimental evidence fail to reproduce the
standard game-theoretic prediction that all players should pick zero.
19
My translation and emphasis. The original Danish text goes “Lotto er betegnelsen på et talspil, som
går ud på at forudsige det rigtige lodtrækningsresultat, når der af 36 tal udtrækkes 7.”
18
6.2 Analytical Concept
6 EMPIRICAL INVESTIGATION
correlated with their playing frequencies.
6.2.2
Dependent Variable
The discussion in the preceding subsection leads directly to how to construct the dependent variable. As a proxy for individual beliefs about a number’s probability, I will use
the number of his or her rows on which the number appears in the present drawing. Naturally, some players buy far more rows than others. To facilitate comparisons of parameter
estimates between players, the measure is therefore normalized by the average number
rows played by the individual.20 The dependent variable Sharejit thus becomes
Sharejit =
6.2.3
Frequency of rows on which number appearsjit
Average number of rows playedj
(3)
Explanatory Variables
LongHist One can think of many ways of constructing indices to measure the outcome
of previous drawings. However, I focus on the information most easily available to the
internet Lotto players thus having the lowest search cost, namely the number statistics
page at http://tips.dk; see the screen shot in figure 5.21 Here, just one mouse click from
the number-pick page, players can see bar charts of the historical drawing frequencies;
either covering the full history (1989/1993 till present) or one year at a time. It seems
likely that the players will look at the all-time historical frequency, and in figure 4 above
we indeed saw a strong correlation between this measure and popularity of numbers.
Therefore this measure, denoted LongHisti , is included as an explanatory variable. To
avoid collinearity with the ShortHist-measure (see below), it has been cut off such that
it measures the drawing frequency up until the end of year 2004. In consequence this
measure is time-invariant, but it obviously varies across the different Lotto numbers.
ShortHist Players are likely to also study the most recent drawing statistic available,
that is the drawing outcomes of the present year (2005). The drawing frequencies increase
20
An alternative specification where the dependent variable is normalized with the number of rows
played in the present drawing was tried out on a random sample of 500 players. Evaluated in model
(4), it only slightly affected the results by increasing the share of significantly biased players with 0.6
percentage points and 1.1 percentage points on the LongHist and ShortHist-measures, respectively.
21
Other indices were tried out, including the due and streak measures mentioned in the text and the
drawing frequency the past X weeks. Except streak that has very little variation, the measures typically
turned out significant of the same magnitude as the variables included in the final regressions or slightly
less. The measures are however all highly correlated with, in particular, the ShortHist variable. Adding
them to the regressions therefore causes serious interpretation problems; and they were thus left out in
favor of ShortHist and LongHist for the reason given in the text. The very short history (∼ a dummy
indicating if the number was drawn last week) was significant on the five percent level for less than five
percent of the players when added to the baseline regression.
19
6.2 Analytical Concept
6 EMPIRICAL INVESTIGATION
Figure 5: Screenshot of number statistics page at http://tips.dk
over time, simply because new numbers are drawn every week. Therefore the measure
ShortHistit has been constructed as the deviation from expectation:
ShortHistit = Drawing frequency year 2005i(t−1) − p · (t − 1)
(4)
Hence the measure is positive if the number has been drawn more than expected in 2005,
and negative if it has been drawn less than expected. The expected sign of the LongHist
and ShortHist parameter estimates depends on the individual. For non-biased (and nonstrategic) individuals, the expected value is zero. Gambler’s fallacy behavior is consistent
with negative signs, while hot hand behavior is consistent with positive signs.
Rollovers and campaigns If no players won the first prize a given week, the jackpot
rolls over to the subsequent drawing. In non-rollover weeks, Danske Spil sometimes add
outside money to the first prize as part of campaigns, for instance during Christmas. The
amount is typically similar to that of a rollover. We therefore expect players to place
more bets in those weeks; not because they now have a stronger belief in certain numbers,
but simply in response to the lower price (or alternatively, to the possibility of winning
a higher jackpot). To control for this behavior, a dummy variable Rollovert to indicate
when rollover or campaign money is included.22 The expected value is positive.
22
An overview of Lotto campaigns in 2005 was kindly provided by Danske Spil’s Customer Service.
20
6.2 Analytical Concept
6.2.4
6 EMPIRICAL INVESTIGATION
Regression Analysis
To sum up, the panel data regression equation can be stated as
Sharejit = β0j + β1j LongHisti + β2j ShortHistit + β3j Rollovert + jit
(5)
where
jit = αij + ηitj
(6)
and ηitj ∼ N (0, σj2 ) (see the below). Notice that the β-estimates are allowed to vary
between the players, j. Hence we will obtain separate β-estimates for each customer
allowing us to asses the distribution of biases among the the internet-playing population.
The drawing history variables are included as simple linear terms.23 Individual timeinvariant preferences for specific numbers are controlled for by including number- and
player specific error terms (the αij ’s).
Estimation Procedure The simplest approach is to estimate equation (5) for each
customer in the data set using the standard linear estimator. Under the standard assumptions this will provide consistent and efficient estimates. For a large proportion of
the observations, however, the dependent variable is zero. Since it is prohibited from
being negative (it is not possible to buy fewer than zero rows), the error terms are clearly
non-normal in this range. Hence the OLS-assumptions are violated.
Alternatively, the problem can be viewed as a censored dependent variable problem;
also called a corner solution outcome problem.24 Recall that the dependent variable
Sharejit is merely a proxy for the unobserved variable of interest, that is player j’s belief in
Lotto number i, Subj Pr(i)jt . Consider a player who is almost certain that Lotto number
14 will not be drawn in the next round since it has been drawn far more than expected
in the recent past (the gambler’s fallacy); i.e. his subjective probability of number 14 is
close to zero. Suppose that he also has bad feelings about number 25, but believes that
it might show up anyway. Hence his subjective probability of number 25 might be 0.10.
How will this belief structure be reflected in his actions? Most likely he will bet on neither
number 14 nor number 25. Hence we will observe Sharejit = 0 for both number 14 and
25; despite the difference in his underlying beliefs. This is because it it is not possible to
buy fewer than zero rows with Lotto number 14. As a finance analogy, the players are
constrained from short selling Lotto numbers. Therefore, Subj Pr(ijt ) can be viewed as
23
Higher-order polynomials were included in test regressions on a random sample of 500 players. Although the higher-order terms turned out significant for a considerable portion of the players, typically
indicating that the marginal impact of the history is decreasing with the magnitude of its deviation
from expectation, it only affected the parameter estimates of the linear terms marginally. None of the
significant estimates changed signs, but about 1 percentage point more players turned out significant.
24
The discussion in this section draws on Johnston & DiNardo (1997, ch. 13).
21
6.3 Results
6 EMPIRICAL INVESTIGATION
the latent variable of interest that is only observed through the censored variable Sharejit
such that

0
if Subj Pr(i)jt ≤ k j
j
(7)
Shareit =
g j (Subj Pr(i)j )
if Subj Pr(i)j > k j ,
t
t
where g j (·) is some function translating beliefs into behavior and k j is a cut-off value,
for instance 0.10. In this view, observing a player making zero bets on a number should
be interpreted as if he finds the number’s probability to be in the interval [0, k j ]. Since
I am not interested in translating the observed behavior into levels of the subjective
probabilities it is not necessary to estimate k j and g j (·). I merely have to assume that
g j (·) is non-decreasing.
If the true model is as stated in equation (7) then ignoring the censoring and using the
standard linear model will often yield attenuated β-estimates (i.e. biased towards zero).
As a solution to this problem, I implement the Tobit model, named after Nobel laureate
James Tobin. The Tobit model consists of a probit part used to handle the discrete
choice between Sharejit = 0 and Sharejit > 0; put together with an OLS part to describe
the (approx.) continuous choice of how many bets to place on a number given that it is
larger than zero. Note that an underlying assumption is that the processes generating the
discrete and the continuous choices are the same; i.e. the specification in equation (5).
Random vs. Fixed effects The explanatory variables are all truly exogenous, being
the results from random drawings. I therefore assume that the number-specific error terms
are uncorrelated with explanatory variables, such that a random effects estimator is appropriate.25 This allows me to recover estimates on the time-invariant variable LongHistgi
by using the between-variation.
For technical reasons however, it was not possible to estimate the Tobit models with
random effects.26 Therefore, the model will be estimated with two different specifications:
First, using fixed effects Tobit that does not recover estimates on the LongHist variable;
and second, using the standard linear random effects model which produces estimates on
all the variables, but may be econometrically less suitable.
6.3
Results
Table 3 shows a summary of the results from the regressions. Keep in mind that when
estimating a huge number of parameters, the fraction of significant estimates has to be
25
A Wu-Hausman test carried out on a random sample of 500 players only rejected the random effects
hypothesis for about 7 percent of the players.
26
For more details, see Jørgensen (2006).
22
6.3 Results
6 EMPIRICAL INVESTIGATION
substantially higher than the expected share of type I-errors, i.e. the significance level.27
When operating on the usual five percent significance level, observing anything shorter
than five percent significant estimates cannot be interpreted as a truly significant relationship. Depending on the specification, between 2.4 and 5.9 percent of the players behave
significantly in accordance with the gambler’s fallacy over the short history. Hence they
tend to bet more on numbers that have been drawn less than expected during year 2005.
A bit more, between 2.6 and 6.5 percent, bet in accordance with the opposite bias, the hot
hand. Hence between 5.0 and 12.3 of the players can be identified as reacting significantly
to the short history.
Table 3: Results from regressions. Time-varying players
Model no.:
Estimation procedure:
Correction for heteroscedasticity:
Correction for AR(1)-autocorrelation:
No. of players
Average no. of observations per player:
Share of biased players (percent)
ShortHist
GF: Neg. significant (p≤0.01)
GF: Neg. significant (0.01<p≤0.05)
HHF: Pos. significant (0.01<p≤0.05)
HHF: Pos. significant (p≤0.01)
LongHist
GF: Neg. significant (p≤0.01)
GF: Neg. significant (0.01<p≤0.05)
HHF: Pos. significant (0.01<p≤0.05)
HHF: Pos. significant (p≤0.01)
Median parameter estimates
ShortHist
Among significant GF players
Among significant HHF players
LongHist
Among significant GF players
Among significant HHF players
(1)
Tobit
÷
÷
39,382
497
(2)
Tobit
X
÷
39,382
497
(3)
Linear
÷
÷
39,382
497
(4)
Linear
X
÷
39,381
497
(5)
Linear
X
X
37,228
521
3.0
2.7
3.1
3.4
3.0
2.9
3.0
3.4
2.6
2.7
2.6
3.2
2.5
2.5
2.5
3.0
0.8
1.6
1.6
1.0
.
.
.
.
.
.
.
.
0.4
1.2
4.1
1.7
0.6
1.4
3.8
1.4
0.8
1.4
3.7
1.4
-0.052
0.047
-0.050
0.048
-0.016
0.016
-0.016
0.017
-0.020
0.021
.
.
.
.
-0.0029
0.0027
-0.0031
0.0028
-0.0034
0.0028
Source: Own calculations on data from Danske Spil A/S.
Note: Estimations of equation (5). GF abbreviates the gambler’s fallacy, HHF abbreviates the hot
hand fallacy. Only players that have participated at least twice and do not pick the same numbers
every week are included. The linear models are estimated with Lotto number specific-random effects,
while the Tobit models are estimated with fixed effects. Therefore the time-invariant LongHist variable
drops out of the Tobit regressions. Heteroscedasticity is modeled on ShortHist using WLS in the linear
models and with an exponential specification in the Tobit models. For technical reasons, it has not
been possible to correct for autocorrelation in the Tobit models (see Jørgensen (2006) for details). The
rollover dummy showed significant among about half the players.
The magnitude of the estimates seems reasonable: When a number was not drawn the
past week the ShortHist-measure is reduced by 0.2. The median gambler’s fallacy player
therefore increases his bets on the number by 1.0 percentage point according to the Tobit
27
A type I-error is the risk of rejecting a true hypothesis, such as to declare that a parameter estimate
is significantly different from zero when its true value is zero.
23
6.3 Results
6 EMPIRICAL INVESTIGATION
models and about 0.3 percentage points according to the linear models; compared with a
base level of 19.4 percent. The magnitude of the hot hand players’ bias is almost identical.
The apparent symmetry is a bit concerning and could suggest that some of the significant
short-bias players are just noise that happen to be significant; i.e. type I errors.
Compared with the short-history variable, considerably fewer players, around 7.3, react
significantly to the long history. One explanation may be that this time-invariant variable
is only estimated using the between-variation, and the significance test therefore has much
less statistical power. The hot hand fallacy is here the dominant bias, as expected. The
median estimate among the hot hand players is about 0.0028 in all the specifications,
implying that a Lotto number with a historical frequency of, say, 169 will be placed on
2.8 percentage points more rows, ceteris paribus, than a number that have been drawn
the expected number of times, 159.
Substantially fewer players show significant in the linear regressions compared with
the Tobit and the size of the parameter estimates are smaller; suggesting that the linear
estimators are indeed attenuated, as expected. The heteroscedasticity correction only
affects the results marginally, but the correction for autocorrelated error terms brings
down the share of significant estimates to the critical five percent. This underpins that
autocorrelation is a serious problem.
The fraction of biased players found in this study is considerably smaller than the 25
to 80 percent found in earlier studies; in particular when recalling that the percentages
above only cover one quarter of the players. One explanation may be the design of the
Lotto game, for instance that it requires some effort to display the bias in Lotto. A second
explanation is that the pool of subjects in this sample is more representative compared
to most other studies. Finally, it may be a statistical artifact stemming from the low
statistical power and the technical problems discussed above.
Correlation of biases Rabin (2002) argues that the gambler’s fallacy and the hot hand
fallacy both stem from an underlying belief in the law of small numbers. The theory predicts that agents showing hot hand beliefs over the short horizon will switch to hot hand
behavior over the longer horizon. A plausible switching point could be between ShortHist
and LongHist. The empirical evidence does indeed support this. Among the short-horizon
gambler’s fallacy players that can be classified on the long-horizon scale, more than ninety
percent show hot hand beliefs on the long run; cf. table 4 below. Conversely, 81 percent
of the long-horizon hot hand players that can be classified on the short horizon show
gambler’s fallacy beliefs. A look beyond the relatively few players that are safely categorized in both dimensions confirms the pattern. 83 percent of the 1,970 significant
short-horizon gambler’s fallacy players have a positive, but not necessarily significant sign
on the LongHist variable. In the same way 67 percent of the 2,058 significant long-horizon
24
6.3 Results
6 EMPIRICAL INVESTIGATION
hot hand players have the opposite sign on the ShortHist parameter.
Table 4: Correlation between biases. Contingency table
ShortHist
Significant GF
Insignificant
Significant HHF
LongHist
Significant GF
18
679
84
Insignificant
1,683
32,846
2,013
Significant HHF
269
1,727
62
Source: Own calculations on data from Danske Spil A/S.
Note: The table shows the number of players in the different categories, based
on model (4). Saturday Lotto only. The pattern is the same in the other models.
The players showing hot hand beliefs on ShortHist might have have a switching point
with a shorter horizon than considered above. To investigate this, an additional regression
was estimated for these players with an extra explanatory variable for the very short
history, indicating if number i was drawn the preceding week. Indeed, it turns out that
among the 2,159 ShortHist hot hand fallacy players, more than 21 percent show significant
gambler’s fallacy behavior over the very short horizon, while less than 6 percent show hot
hand behavior.
Summing up, these important results clearly support that the gambler’s fallacy and the
hot hand are not opposite biases ruling each other out, but manifestations of an underlying
belief in the law of small numbers. The correlation between the biases is as predicted by
theory. To the author’s knowledge, this has not previously been demonstrated in field
data.
6.3.1
Characteristics of Biased Players
Playing behavior etc. The use of mathematical systems to pick Lotto combinations
in itself suggests a limited understanding of the laws of probabilities. It is therefore not
surprising to find an overweight of SystemLotto-players among the significantly biased
players, see table 5 below. The difference is less than 3 percentage points but is clearly
significant.28 Players demonstrating biased beliefs are likely to think that the expected
pay-off from the Lotto is higher than the true 0.45 since they may feel that they can ‘beat
the system’. We therefore expect biased individuals to buy more rows than unbiased
players. As seen from the table, this is indeed the case. Biased players purchase 22.4
rows on average, compared with 19.0 for the other group, and the difference is clearly
significant.29 This supports the findings by Hardoon, Baboushkin, Derevensky & Gupta
(2001), Rogers & Webley (2001) and Coups et al. (1998).
28
The usual chi-square test of independence yields a test statistic of 17.0, compared to a critical value
of χ20.95 (1) = 3.84.
29
The test statistic to be evaluated in the normal distribution is 6.50. The results are equally strong
when restricting the sample to the players who have participated in all the 28 drawings.
25
6.4 Conclusion
6 EMPIRICAL INVESTIGATION
Table 5: Characteristics of biased players
Significantly
biased players
42.1%
Insignificant/
unbiased players
39.4%
No. of
observations
39,381
Average no. of Lotto rows
purchased per drawing
22.4
19.0
39,381
Average age
45.1
43.0
39,381
16.1%
18.1%
83.9%
81.9%
30,280
9,101
6,535
32,846
39,381
Share of players using SystemLotto†
Share of players biased
Males
Females
No. of observations
Source: Own calculations on data from Danske Spil A/S.
Note: Based on model (4). The qualitative results are the same in the other models. A player
is categorized as biased if either ShortHist or LongHist is significantly different from zero.
† Players that have used SystemLotto at least once..
Demographics Women are identified as biased slightly more often than men, cf. table
5 below.30 The average age is two years higher in the biased group, and the difference is
clearly significant.31 One possible explanation could of the latter be that the educational
level has increased over time, and that more young people therefore have learned the basic
laws of probabilities in high school.
6.4
Conclusion
There is a surprisingly clear positive correlation between Lotto numbers’ all-time drawing
frequency, LongHist, and the numbers’ popularity among Lotto players; even when looking
at the aggregate distribution. This behavior is consistent with the hot hand fallacy. A
comparison of the numbers’ popularity with the short history shows weak signs of the
opposite behavior, consistent with the gambler’s fallacy. This is despite it being costly in
terms of expected pay-off to follow a common heuristic, suggesting that players do not put
much weight on the strategic element of the game. Taken together, the patterns confirm
Rabin (2002) that the hot hand and the gambler’s fallacy are not mutually exclusive, but
stem from an underlying belief in the law of small numbers.
The picture is supported by a rigorous regression analysis of the individual players.
A correlation analysis show that players with hot hand beliefs over the longer horizon
typically show gambler’s fallacy beliefs over the shorter horizon. The average age is
higher among the biased players, and females are overrepresented in this group. Biased
players more often use SystemLotto and play more rows on average.
30
31
The difference is significant with a χ2 test statistic of 19.9, cf. the methodology in footnote 28.
The test statistic to be evaluated in the normal distribution is 13.9, cf. the methodology in footnote
16.
26
7 DISCUSSION
The regressions carried out in this section suffer from two serious technical limitations:
The Tobit models could not be i) estimated with random effects, and ii) corrected for
serially correlated error terms. Standard linear models therefore also had to be applied,
despite being less appropriate and likely to yield attenuated estimates. Finally, some of
the players who show up biased in the regressions are likely not to have biased beliefs, but
simply act strategically to avoid popular combinations. On the other hand, many of the
non-significantly biased players are likely to hold biased beliefs but fail to be identified due
to lack of statistical power. The results should thus be interpreted with some caution.
7
7.1
Discussion
Applications
The empirical results have direct implications in relation to problem gambling. We saw
in section 6.3.1 that players identified as biased spend about 20 percent more on Lotto in
an average week; suggesting that a belief in the law of small numbers is one determinant
behind problem gambling. It is important to stress that pathological gambling is not a
big problem in Lotto (Nielsen & Røjskjær 2005). However, both Danish and international
research suggest that the same mechanisms are at work for many players addicted to slot
machines, poker etc. (Jørsel 2003, Dickerson 1984). A treatment strategy suggested in the
literature is cognitive restructuring, that is to unlearn the irrational beliefs and educate
the addicted players in the laws of probabilities (Nielsen & Røjskjær 2005, Walker 1992).
“Context is not a dirty word”, as noted by Harrison & List (2004, p. 1028). In Lotto
the statistical process is very clear-cut and communicated visually through a national
TV-transmission. In other more real-life settings, such as insurance or financial markets,
the properties of the underlying probability distribution are more muddy. Furthermore, it
requires effort to show a bias in Lotto, while in other markets you are often forced to make
an active choice (there is no Quickpick option). Fallacious beliefs therefore have much
more room to develop in these cases, and they are less likely to be corrected by experience.
On the other hand, people may here be more prone to seeking external advice since more
is at stake. Moreover, rational agents may participate in these markets and drive the
equilibrium towards the neoclassical prediction, in the case of strategic substitutes. It is
therefore not not clear to what extent the results found in Lotto play can be applied in
other contexts. Nevertheless, a few examples of applications are discussed below.
Finance The disposition effect in finance telling that investors tend to sell recent winners and keep recent losers may be explained by gambler’s fallacy beliefs (Odean 1998):
If a stock has lost value in the recent past it is ‘due’ to rise again. Similarly, if a stock has
27
7.1 Applications
7 DISCUSSION
recently won people find it unlikely that it will continue its streak. Note that the opposite
pattern would emerge under hot hands beliefs, and it is therefore crucial to improve our
understanding of under which circumstances the two fallacies prevail.
Rabin (2002) apply his theoretical model of the law of small numbers to the situation
in which a consumer has to choose a mutual fund manager. The consumer does not
know the managers’ abilities but are restrained to study their performance. Even when
the managers are equally good (or bad) at picking stocks, the consumer’s disbelief about
streaks of luck will lead him to overinfer that some fund managers are better than others.
Sooner or later he will inevitably be disappointed about their less stellarly performance,
and may therefore switch mutual fund too often. A similar reasoning can be applied in
other settings of limited information, such as in labor economics.
Health economics Related to the case of mutual fund managers; patients may put too
much confidence in a small number of reports from friends, families, physicians etc. when
considering which doctor to go to, or whether to participate in a new experimental treatment (Frank 2004). More generally, if patients are biased when making these decisions,
it may not make them better off to be offered more options. This observation questions
whether increasing free choice in health care is a good policy (Jensen 2005). Despite this
being a clearly paternalistic argument, similar considerations could be made in other areas
of where the paradigm of free choice is being introduced, such as investment profiles for
retirement savings.
Cost-benefit analyses in health economics, as well as other fields, typically need to
price the value of life. One commonly used method is to collect data on compensating
wages in risky jobs, for instance in the mining industry, and compare it with the job’s risk
to infer the workers’ valuation of their own life. If the workers are subject to the law of
small numbers, however, they may grossly over- or underestimate the risk of working in
the mine, depending on the recent history of accidents. The calculated value of life may
therefore be flawed.
Insurance According to the gambler’s fallacy, people that have not been hit by an
accident for some time will increase their level of insurance because they don’t want to
‘push their luck’ (Papon 2005). Similarly they will reduce their insurance if recently hit
since they find it highly unlikely that it should happen twice in a row. We all know
that lightning doesn’t strike twice! Hot hand beliefs imply the opposite response since
individuals would now overinfer the underlying risk from their past experience.
Research and superstition What originally led Daniel Kahneman and Amos Tversky
to develop the theory of the law of small numbers was the poor understanding of proba28
7.2 Extensions
7 DISCUSSION
bilities among their colleagues. Although researchers routinely apply statistical methods
when doing more formal empirical work, they may grossly overinfer from ‘anecdotal evidence’. The problem is likely to be more severe in the day-to-day work of practitioners,
such as doctors forming too strong an opinion of the effectiveness of different treatments
based on very limited data material. It may be exacerbated when the information collection is endogenous (Rabin (2002), section VII). For instance, the doctor may not seek
formal evidence from medical journals etc. if he has already formed a too strong opinion
based on experience with his own patients.
The catastrophic applications of superstitious treatments such as blood-letting in the
early history of medicine may similarly be partly attributable to overinference from small
samples (the hot hand fallacy). When modern statistical methods were introduced in
medical research these problems were partly resolved, but undoubtedly still occur.
Education To the extent that belief in the law of small numbers is a widespread fallacy,
an obvious policy implication is to improve the education in basic statistics in elementary school. Focus should be on unlearning the fallacious belief that random processes
are self-correcting, such that deviations in one direction are followed by deviations in
other direction. This is not very high tech and could be achieved by simply letting the
schoolchildren roll a dice and record the outcomes. However, recognizing that the belief
is deeply rooted in many of our minds, a second-best remedy is to insist that principals
should apply proper statistical methods instead of relying on intuition, in particular when
the stakes are high.
7.2
Extensions
This paper should be viewed as a first attempt of how this exciting new data set may shed
light on how well people understand the laws of probabilities in real life. In consequence
it raises a series of ideas of how the analysis could be improved and what other issues
could be investigated in future research. A few are discussed in this section.
First, more advanced statistical models could be applied. An example mentioned in
the text is the random effects Tobit model with correction for serially correlated error
terms. Robust standard errors could be calculated to make the significance tests more
reliable, but one should however be careful to ensure that the data’s asymptotic properties
are satisfied—recall that the number of cross sections per individual is not that large.
One possible path to improve the asymptotics might be to estimate all the customers in
one unified model; instead of estimating all the parameters individually for each player.
A clear advantage of a unified model is that it would allow a direct modeling of the
effects of player characteristics such as age, gender, and money spend on Lotto. If the
29
7.2 Extensions
7 DISCUSSION
individual estimation procedure is maintained, it could alternatively be interesting to run
a ‘second-stage’ multivariate analysis of how biased behavior can be explained by player
characteristics.
The data set can also be used to study other issues than the law of small numbers. An
interesting topic would be to study how players react to past winnings, and to see if there
is a ‘house money’ effect as shown in the lab by Thaler & Johnson (1990).32 One could
also estimate a ‘demand equation’ for each individual; investigating how players respond
to the price variation introduced by rollovers and campaigns. This would complement
Oster (2004) and Guryan & Kearney (2005), but could be done with much less noise and
more demographic details due to the superiority of the present data.
More data (a longer time-series) could improve the analysis. Even though the data set
covers a very large number of individuals, the number of observations per player and the
right-hand side variation is limited. Besides increasing the statistical power, a longer timeseries could be used to investigate what happens after New Year’s eve when the ShortHistmeasure abruptly changes (recall that ShortHist measures the drawing frequency in the
present year as it appears on http://tips.dk). This could prove an interesting test of the
appropriateness of this measure.
An exiting extension would be to link players’ CPR numbers to Statistics Denmark’s
comprehensive research database. Hereby the analysis could be augmented with variables
such as educational level, household income, or even high-school grade point average.
This could be achieved following official procedures that fully ensures the Lotto players’
privacy; without letting the researcher see the CPR numbers.
Finally, one could conduct follow-up experiments, for instance to isolate the strategic
effect from the perceived-probability effect, or to pinpoint the tipping point from gambler’s
fallacy- to hot hands beliefs.
32
Amount won does not appear directly in the data set, but it can be calculated straightforwardly by
combining the players’ bets with prizes and the results from past drawings; available at http://tips.dk.
30
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