Lecture 6: Belief Distortions in Gambling
ECOS3997 Stream 1, Semester Two 2023
ECOS3997 | Dr Stephen L. Cheung
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Gambler’s Fallacy
The gambler’s fallacy was first described by the mathematician Laplace in 1825:
when one number has not been drawn in the French lottery, the mob is eager to
bet on it. They fancy that, because the number has not been drawn for a long
time, it, rather than the others, ought to be drawn on the next draw.
The most famous instance occurred in Monte Carlo in 1913 when black came up 26
times in a row in roulette. As the patrons became increasingly convinced that red was
“due”, they bet increasing amounts on red, and the casino made a fortune.
The longest recorded streak in roulette is said to have been 32 reds in 1943.
ECOS3997 | Dr Stephen L. Cheung
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Gambler’s Fallacy
Make sure you are clear on why it’s a fallacy:
The probability of 26 blacks in European roulette is around one in 137 million.
But the probability of the next black is always 18/37.
More recently, number 53 went 153 times without being drawn in the Venice lottery.
When it finally came up, the winners shared around €600m between them.
But by that time, more than €3.5bn had been bet on it.
ECOS3997 | Dr Stephen L. Cheung
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Fixed-Odds versus Parimutuel Betting
Roulette is an example of a fixed-odds game. This means that the size of the payout to
a winning bet on red or black is fixed, and known at the time of placing the bet.
It doesn’t matter how many other people are betting on red or black.
So while it’s irrational to expect red to be more likely, it’s at least no more costly to
act on the fallacy by betting on red, as opposed to the alternative of black.
By contrast, most lotteries have a parimutuel structure. This means that the prize pool
is shared among all winners, and payouts are not known until the outcome is decided.
So 53 is no more likely to win, but if it does more people have to share the prize.
It would be more profitable (less unprofitable) to bet on a less popular number.
ECOS3997 | Dr Stephen L. Cheung
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Gambler’s Fallacy in Roulette
Croson and Sundali (2005) studied 18 hours of security footage of a roulette table in
Nevada, and coded all bets placed by 139 individuals.
Focusing on the “outside” bets of Red/Black, Odd/Even, and Low/High, they classify
bets against a streak as consistent with the gambler’s fallacy.
For streaks of length five or more, there is statistically significant evidence of the
gambler’s fallacy (compared to a benchmark of random betting).
They also claim to find evidence of hot-hand beliefs, in that gamblers are more likely to
keep playing, and to place more bets, after winning compared to when they have lost.
But there may be other explanations for this (such as wealth or house money effects).
ECOS3997 | Dr Stephen L. Cheung
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Gambler’s Fallacy in Parimutuel Betting
In a study of racetrack betting, Metzger (1985) found that when the previous two races
were won by favourites, punters bet less on the favourite in the next race.
Notice that in each race, the favourite is a different horse!
In the New Jersey Lottery, Terrell (1994) found that 25% fewer players bet on a number
that won in the past week than on numbers that had not won for more than 8 weeks.
ECOS3997 | Dr Stephen L. Cheung
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Gambler’s Fallacy and Hot Numbers in Lotto
Suetens et al. (2016) study administrative data from Danish (“7 from 36”) Lotto. This lets
them track individual gamblers’ number choices from week to week.
The other studies we discussed only looked at aggregate behaviour.
While all 36 numbers are equally likely to be drawn each week, they are not all equally
likely to be chosen by players (next slide).
Why do you think that bigger numbers are chosen less often than smaller ones?
The “hotness” of a number is defined as the number of times it was drawn as part of a
winning combination in weeks to (past five weeks, not counting last week).
ECOS3997 | Dr Stephen L. Cheung
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Gambler’s Fallacy and Hot Numbers in Lotto
Players are less likely to choose a number that was drawn last week ( ), holding its
“hotness” constant.
This is evidence of a gambler’s fallacy in the short run.
But they are also more likely to choose a number the “hotter” it is (the more times it has
been drawn recently), holding constant whether or not it was drawn last week.
This is evidence of belief in a hot outcome in the long run. (It’s a hot outcome rather
than a hot hand, because it is the number not the player that is “hot”.)
ECOS3997 | Dr Stephen L. Cheung
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So, what about Hot Hands?
We said last week that the hot hand is usually a belief about human performance.
In situations that involve only chance, with no true element of skill (such as predict-
ing coin tosses or lotto numbers), any hot-hand belief is necessarily a fallacy.
But if outcomes depend also on skill (sporting performance, funds management),
hot-hand effects might be real. In that case, beliefs are only fallacious if they over-
estimate the strength of the true effect.
ECOS3997 | Dr Stephen L. Cheung
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The “Lucky Store” Effect
Guryan and Kearney (2008) showed that after selling a prize-winning lottery ticket, the
“winning” store experiences an increase in sales.
Customers behave as if they believe the store is “more lucky” than others, and thus
more likely to sell another winner.
But of course tickets sold from one store have the same chance of winning as all others,
so this is a case where the belief is clearly fallacious.
ECOS3997 | Dr Stephen L. Cheung
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Hot-Hand Beliefs in Sports Betting
Camerer (1989) studied “point spreads” in betting markets to test hot-hand beliefs over
the performance of professional basketball teams on winning (or losing) streaks.
A point spread of +6 means that a bet on the team that is favoured to win pays out
only if they win by 7 points or more.
Point spreads are set by the bookmaker to balance the amounts bet on the two teams.
Thus while they are set by the bookie, they reflect the weight of betting in the market.
So if bettors have well-calibrated beliefs, then point spreads should be unbiased
forecasts of teams’ winning margins.
Alternatively, if certain teams are too strongly favoured, their point spreads would
be too large relative to their actual results.
ECOS3997 | Dr Stephen L. Cheung
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Hot-Hand Beliefs in Sports Betting
Now suppose a team is on a winning streak. If bettors have hot-hand beliefs, they will
fancy it to also win the next game. This will be reflected in a positive point spread.
In sporting performance, it’s possible that a hot-hand effect might be real, in which case
a hot-hand belief is not necessarily irrational.
But remember that if beliefs are accurate then point spreads should be unbiased.
If bettors overestimate the strength of the hot hand, the spread will be too large.
Camerer found teams on winning streaks had negative forecast errors, indicating that
they performed worse than predicted by the market (and conversely for losing streaks).
However, the effects were minor (too small to be profitably exploited after allowing for
the bookie’s take). Camerer thus concludes that the hot-hand fallacy exists, but is slight.
ECOS3997 | Dr Stephen L. Cheung
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Illusion of Control
Many games of chance include arguably irrelevant elements of choice and personal
involvement that have no effect upon the probability of winning:
Choosing lottery numbers, throwing the dice in craps, stop buttons on pokies.
In the original study by Langer (1975):
Participants demanded a higher price to sell a lottery ticket that they had chosen
themselves, compared to one that was chosen for them randomly.
They were also unwilling to exchange a ticket that they had chosen themselves for
another one that had a higher objective chance of winning.
ECOS3997 | Dr Stephen L. Cheung
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Illusion of Control
Illusory control can mislead gamblers into believing that their actions can influence the
outcomes of chance events, in effect mistaking a game of chance for a game of skill.
For example, throwing the dice harder when trying to roll a larger number in craps.
Davis et al. (2000) found that craps players at a casino placed larger and riskier bets
on their own throws of the dice than when betting on another player’s rolls.
A gambler who believes in illusory control may also invest effort in honing their “skills”,
and respond to feedback suggesting that they are “mastering” the game.
ECOS3997 | Dr Stephen L. Cheung
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Illusion of Control in Pokie Machines
In a study of a poker machine stop device, Ladouceur and Sévigny (2005) found that:
57% of participants believed it could control the game outcome, 41% believed skill
affected the outcome, and 26% that the device could improve the chance of a win.
All of these beliefs are, of course, fallacious.
Participants who were randomised to have access to the stopping device played
twice as many games as those who did not have access it.
The stop device also makes it mechanically possible to play more rapidly, but
the participants were free to quit at any time.
ECOS3997 | Dr Stephen L. Cheung
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Near Misses, and “Losses Disguised as Wins”
Finally, gambling researchers suspect that poker machines have several other features
designed to “sugar coat” the experience of a loss.
A near miss occurs when the reels stop just short of a winning outcome.
A loss disguised as a win occurs when the player “wins” a prize smaller than the
amount they wagered. (It also triggers the same sensory reinforcements, such as
music and flashing lights, as a proper win.)
These features can mislead players into overestimating the frequency of actual wins.
If players suffer from illusion of control (they attribute the outcomes to skill not luck),
they may also misinterpret them as “better performance” than a full loss.
They interpret these outcomes, but not full losses, as evidence that they are getting
“better” at “mastering” the game. This in turn motivates continued play.
ECOS3997 | Dr Stephen L. Cheung
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Near Misses in the Brain
A neuroeconomics study by Clark et al. (2009) studied neural responses to near misses
that result from a choice by the participant, compared to ones chosen by a computer.
They study a simplified slot machine game that can be played inside the fMRI scanner.
The reel on the left is fixed. In half of the trials, its position is set by the player
(potential for illusory control), and in the other half it is set by the computer.
The reel on the right is spun, resulting in one of three possible types of outcomes:
Win: the symbol on the right matches up with the one on the left.
Near miss: the reel on the right stops one position either side of a win.
Full miss: the reel stops more than one position away from a win.
ECOS3997 | Dr Stephen L. Cheung
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How do neural responses differ for
wins, near misses, and full misses?
How do they differ between player-
chosen and computer-chosen trials?
ECOS3997 | Dr Stephen L. Cheung
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Near Misses in the Brain
Comparing wins to full misses identified activation in regions known to be associated
with the “reward system” of the brain.
These win responses did not differ discernibly between player and computer trials.
Moreover for near misses, the brain responded in a similar manner to an actual win.
But in this case, there was also an interaction with personal choice.
The authors interpret the near-miss response (in a chance situation) as the anomalous
response of a system that evolved for skill-based situations.
In skill situations it makes sense to assign value to near misses if they signal in-
creasing mastery of the game. But in chance situations they should have no value.
ECOS3997 | Dr Stephen L. Cheung
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Week 7 Tutorial
Two prominent forms of sports betting involve combinations of multiple bets:
“Multi bet”: a bet combining multiple outcomes in independent events.
Example: “Sydney to defeat Carlton, and St Kilda to defeat GWS Giants.”
“Same-game multi”: a bet combining multiple outcomes within the one event.
Example: “Manchester United to defeat Brighton by 2-1, and Marcus Rashford
to score the first goal.”
These complex bet types feature extensively in advertising, and are likely to be highly
lucrative for bookmakers.
You will be asked to write about belief biases that make these bet types attractive to
bettors, and how bookies exploit biases to offer bets that are particularly seductive.
ECOS3997 | Dr Stephen L. Cheung
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Week 7 Tutorial
Readings:
Newall (2015), “How bookies make your money”, Judgment and Decision Making,
10(3): 225-231.
Nilsson and Andersson (2010), “Making the seemingly impossible appear possible:
Effects of conjunction fallacies in evaluations of bets on football games”, Journal of
Economic Psychology, 31(2): 172-180.
A strong answer should go beyond simply describing conjunction bias, to discuss po-
tential mechanisms that have been proposed to underpin that bias (as detailed in the
readings), and how specific bet features might be exploited under each mechanism.
ECOS3997 | Dr Stephen L. Cheung