Copyright is owned by the Author of the thesis.  Permission is given for 
a copy to be downloaded by an individual for the purpose of research and 
private study only.  The thesis may not be reproduced elsewhere without 
the permission of the Author. 
 
iPrice formation in parimutuel
markets
A thesis presented in partial fulfilment of the requirements for the degree of
Master of Management
in
Economics
at Massey University, Albany, New Zealand
Paul G Geertsema
2010
Copyright (C) 2010 Paul G Geertsema. The author asserts the moral right to be
identified as the author of this work.
ii
Abstract
Two types of betting are common in sports betting: fixed odds betting and parimutuel
betting. In fixed odds betting, the payout conditional on winning is fixed once the
bet is placed and is not affected by the placing of subsequent bets. By contrast,
winning bettors in a parimutuel contest share pro-rata in the total betting pool.
This means that the payout to winning bettors in a parimutuel contest depends not
only on selecting the winning outcome, but also on the amounts bet by other bettors
(which cannot be observed at the time a bet is placed). Therefore a parimutuel con-
test can be viewed as a game at the level of individual bettors. Existing models in
the parimutuel literature explain the data by either assuming a single, representative
bettor with certain risk preferences or by assuming that a number of risk neutral
bettors compete strategically within a game theoretic framework. Our contribution
is to construct a novel theoretical framework of parimutuel markets in which we
model both strategic interaction and risk preferences at the level of individual in-
siders, in the presence of exogenous outsiders. We solve this model analytically for
the optimal insider betting amount in a static symmetric Nash equilibrium. Using
a new dataset of 1.6 million individual horse race bets in New Zealand from 2006 to
2009, we document a strong inverse linear relationship between our model-implied
insider risk preferences and the strength of insider beliefs relative to outsiders. That
is, as the strength of insiders’ beliefs relative to that of outsiders decrease, implied
risk sensitivity moves from risk averse to risk loving. At a level of insider beliefs
congruent with actual performance in the data, average implied risk preferences are
close to zero, that is, insiders are effectively risk neutral. While risk neutrality is
a standard assumption in strategic interaction models of parimutuel betting, our
study is the first to provide empirical support for this assumption. Finally, we doc-
ument a strong relationship (not previously reported in the literature) between the
average bet size and the average payout ratio, suggesting that bettors with inside
information self-select by placing larger bets.
Contents
1 Summary 1
2 Literature review 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Why are betting markets important? . . . . . . . . . . . . . . . . . . 7
2.3 Parimutuel betting markets . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Empirical regularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.1 Market efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.2 The favourite longshot bias . . . . . . . . . . . . . . . . . . . 10
2.4.3 Factors correlated with the favourite longshot bias . . . . . . . 12
2.4.4 Early betting . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.5 Late betting is smart betting . . . . . . . . . . . . . . . . . . 15
2.4.6 Rounding of odds . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Theories of price formation . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.2 Representative agent models . . . . . . . . . . . . . . . . . . . 16
2.5.3 Market structure . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.4 Strategic interaction . . . . . . . . . . . . . . . . . . . . . . . 22
2.5.5 The current state of theory . . . . . . . . . . . . . . . . . . . . 28
2.6 Areas for further work . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6.1 Extending strategic interaction models to encompass risk sen-
sitive bettors . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6.2 Endogenising the distinction between inside and outside bettors 28
iii
CONTENTS iv
2.6.3 Deriving additional testable implications from existing theo-
retical frameworks . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6.4 Creating a richer set of empirical stylised facts . . . . . . . . . 29
2.7 Next steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Strategic interaction with risk preferences 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.3 Risk neutral model . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Payoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.2 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.3 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.4 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 Best response function (first order conditions) . . . . . . . . . 39
3.3.2 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.3 Admissibility and second order conditions . . . . . . . . . . . 41
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.2 No equilibrium zones . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.3 Insider beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.4 Equilibrium final market probabilities . . . . . . . . . . . . . . 46
3.4.5 Special case 1: Risk neutral insiders . . . . . . . . . . . . . . . 48
3.4.6 Special case 2: Risk sensitive representative insider . . . . . . 49
3.4.7 Special case 3: A simplified model . . . . . . . . . . . . . . . . 49
3.4.8 Empirical statics . . . . . . . . . . . . . . . . . . . . . . . . . 51
CONTENTS v
4 Empirical tests 52
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Data description and analysis . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Insiders vs outsiders . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1 Cut-off point . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.2 Payout ratio and average bet size . . . . . . . . . . . . . . . . 53
4.4 Race level data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5 Operational definition of model parameters . . . . . . . . . . . . . . . 57
4.6 Empirical tests and results . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6.2 Direct estimation of risk sensitivity . . . . . . . . . . . . . . . 59
4.6.3 Estimation conditional on specified insider beliefs . . . . . . . 60
4.6.4 Model vs empirical final market probabilities . . . . . . . . . . 64
5 Conclusion 68
6 Appendix 73
6.1 Maple code to verify derivation of equilibrium insider betting amounts 73
6.2 Average payout rate and average bet size by percentile . . . . . . . . 74
6.3 Statics based on an empirical approximation . . . . . . . . . . . . . . 76
6.4 Stata code for estimating reduced form final market probability (mˆ) . 78
6.5 Stata code for direct empirical testing of predicted final market prob-
ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.6 Stata code for indirect empirical testing of predicted final market
probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
List of Figures
2.5.1 Final market probability m for p = 0.6 and p = 0.8 as a function of N 25
3.1.1 Amount bet by individual insiders for given q and N . . . . . . . . . 35
3.1.2 Net payout of individual insiders for given q and N . . . . . . . . . . 36
3.4.1 Critical values of α as a function of q, assuming different values of b0 44
3.4.2 Risk loving insider betting amounts on either horse as a function of
q, with associated utility (utility scaled 20x) . . . . . . . . . . . . . . 45
3.4.3 Risk averse insider betting amounts on either horse as a function of
q, with associated utility (utility scaled 20x) . . . . . . . . . . . . . . 46
3.4.4 Final market probabilities where N →∞ and τ = 0.2 . . . . . . . . . 47
3.4.5 Insider betting with N = 4, τ = 0 and q = 12 . . . . . . . . . . . . . . 50
4.3.1 Payout ratio (per percentile) against average bet (per percentile) . . . 55
4.6.1 Histogram of implied α, at an insider/outsider cut-off point of c = 100
(Model compliant observations only) . . . . . . . . . . . . . . . . . . 59
4.6.2 Mean implied risk sensitivity αc vs insider/outsider cut-off point c . . 60
4.6.3 Mean of model implied risk sensitivity αq as a function of ∆q . . . . 63
4.6.4 Estimated risk sensitivity αq against ∆q for different insider bet cut-
off points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6.5 Regression coefficient γ against risk sensitivity α for different q . . . 66
vi
List of Tables
4.1 Data summary – horse racing bets from 1 Aug 2006 to 31 July 2009 52
4.2 Data summary – insiders vs outsiders . . . . . . . . . . . . . . . . . 54
4.3 Race level data summary . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Implied risk sensitivity α∆q for a range of ∆q . . . . . . . . . . . . . 62
4.5 Best fit regression results for α and q . . . . . . . . . . . . . . . . . . 67
6.1 Bets and payouts by percentile . . . . . . . . . . . . . . . . . . . . . 75
6.2 Empirical approximation of model final market probabilities . . . . . 77
vii
Chapter 1
Summary
Betting markets are of interest to economists because they are real-world markets
which resemble simplified financial markets in many important ways. The similari-
ties include uncertain future outcomes, observable trading and pricing, economically
significant financial risk and the presence of potentially complex information and
belief structures. At the same time, the outcomes in betting markets are directly
observable and those outcomes are exogenous to the price formation process – sim-
plifying features that are absent from most financial markets. As a consequence
betting markets have become a popular venue for testing economic theories of price
formation under conditions of uncertainty.
Sports betting markets can be classified as fixed odds markets, in which a fixed pay-
out is made conditional on a specified outcome occurring; and parimutuel markets,
in which the sum of all bets relating to an event is paid out pro-rata to successful
bettors. It is common for bets on the outcome of team sports (such as football
games) to be conducted in fixed odds betting markets. By contrast, bets on events
in which the outcome is an ordinal ranking are often structured as parimutuel bets;
horse racing being the prime example. While the payout on fixed odds betting is
unaffected by other bets placed subsequently, this is not true for parimutuel bets
– the payoff of a parimutuel bet depends also on the amounts bet on the event by
other bettors (whether bet on the winning outcome or not). Crucially, at the time
they place their own bets, bettors in a parimutuel contest are not able to observe the
amounts simultaneously or subsequently bet by other bettors. Hence, at the level of
individual bettors, parimutuel markets exhibit a strategic interaction element which
is absent from fixed odds betting markets.
The early empirical literature dealing with parimutuel markets did not explicitly
incorporate this strategic interaction element; instead, the focus was on the rela-
tionship between the probability distribution of outcomes implied by dollar betting
1
CHAPTER 1. SUMMARY 2
volumes and the actual empirical distribution of those outcomes. The genesis of this
strand of the literature can be traced to Griffith (1949), in which it is shown that
bettors in horse races exhibit a bias towards outcomes with low ex-ante probabilities
and against outcomes with high ex-ante probabilities. The favourite longshot bias,
as this has come to be known, is one of the more enduring anomalies in economics
(as noted in Walls and Busche (2003)). The favourite longshot bias is by no means
restricted to parimutuel markets, as it has also been widely documented in fixed
odds betting markets. A large number of theories have been put forward to explain
the favourite longshot bias; recent surveys can be found in Coleman (2004) and
Ottaviani and Sørensen (2008).
Initial attempts to explain the favourite longshot bias theoretically made use of a
representative agent framework in which a single representative agent is assumed
to have certain risk preferences (Ali (1977) is a typical example of this approach).
More recently, the focus has shifted towards modelling the strategic interaction in
parimutuel betting markets explicitly as a game between many risk neutral bet-
tors. A recent example of this approach is that of Ottaviani and Sørensen (2010).
However, both approaches appear to be consistent with aggregate betting data.
Our contribution is to structure a theoretical model of parimutuel betting markets
in which we incorporate both strategic interaction and bettor risk preferences at the
individual bettor level. We believe this is a novel contribution to the literature1.
Our model postulates an arbitrary number of strategic insiders that compete in a
simultaneous-move2 Nash game in the presence of exogenously specified outsiders.
Insiders have utility that allows for an arbitrary degree of risk sensitivity. We solve
this model for the static symmetric Nash equilibrium in continuous strategies and
obtain a closed-form solution to the optimal individual insider betting amount as
well as the resulting implied “market prices” of event outcomes.
To test our model, we turn to a unique dataset of 1.6 million individual horse race
bets in New Zealand from 2006 to 2009. In applying our model to the data, we
1The only other work combining risk aversion and strategic interaction in a parimutuel setting
(as far as we are aware) is that of Qiu (2007), in which the author demonstrates the existence of
equilibrium in a sequential move framework with agents subject of prospect theory utility. In our
model, by contrast, we consider a simultaneous move game between a number of insider bettors
with a common arbitrary degree of individual risk aversion. We also explicitly solve for a closed
form equilibrium solution, while Qiu is focused on demonstrating the existence and properties of
equilibrium in his model.
2Betting normally opens a day or two before the start of the race. Bets can be placed at any
time until closing time, and the total betting volume on each horse is publicly reported every 5
minutes or so throughout this time. Our decision to model insiders in a simultaneous game is
motivated by a previous finding in the literature that around 40% of bets are placed in the last
minute prior to closing and that last minute betting is a bettor predictor of final outcomes than
earlier bets (Gramm and McKinney (2009)). These regularities are discussed more fully in our
literature section.
CHAPTER 1. SUMMARY 3
need to operationalise our distinction between insiders and outsiders. We do so by
classifying as insiders all bettors that bet NZD 1003 or more in a single race; the
remaining bettors are classified as outsiders. Our classification rule is motivated by
the observation that insiders as defined above outperform outsiders by 9.2% ex-post
in our dataset, suggesting that bettors that bet more than NZD 100 do in fact bet
on the basis of information not available to outsiders.
Based on this classification rule we are able to calculate proxy variables for all of our
theoretical model parameters, save for inside information and insider risk preference.
(Note that in the context of our model inside information denotes the difference
in probabilities that insiders and outsiders assign to the winning outcome.) We
then proceed to calculate the model implied risk sensitivity of insiders at different
levels of inside information. The results reveal a very strong linear relationship
between inside information and risk sensitivity – in effect, inside information and
risk sensitivity could be considered close substitutes within our model, since the same
insider betting behaviour can be induced by either assuming a sufficiently high level
of risk sensitivity or a sufficiently high degree of inside information on the part of
insider bettors. At a level of inside information consistent with the outperformance
of insiders over outsiders observed in the data, the average implied risk sensitivity is
very close to zero. This can be interpreted as evidence that insiders are risk neutral
in deciding their bets on a race-by-race basis.
Previous literature applying game theory to parimutuel markets has assumed risk
neutrality without any empirical justification. Given that in financial economics
the assumption of individual risk aversion in decision making is pervasive, this is
not a minor assumption. Our work suggests that, at a minimum, an assumption
of risk neutrality on the part of insiders is not inconsistent with the data. This
finding provides significant theoretical and empirical support to the growing body
of literature seeking to explain parimutuel markets within a game theoretic risk
neutral agent framework.
In addition, we find a strong positive linear relationship between the log of the
average individual amount bet and the average payout ratio; a result not previously
documented in the literature4. We find that the log of the average amount bet
explains 74% of the variation in average payout rates when aggregated in percentiles
by bet size. One plausible explanation for this regularity is that bettors who have
inside information self-select by placing larger bets – an intriguing hypothesis that
could form the basis for further research.
3As of 22 July 2010 1 NZD = 0.72 USD.
4This finding provides additional support for the classification rule we use to distinguish between
insiders and outsiders.
CHAPTER 1. SUMMARY 4
The rest of the thesis is set out as follows: in chapter 2 we conduct a broad sur-
vey of the literature pertaining to parimutuel betting markets. Chapter 3 presents
our theoretical model of parimutuel markets in which we combine individual risk
preferences with strategic interaction. Chapter 4 introduces a new bet-level dataset
and demonstrates that success in betting is systematically related to the size of the
bet placed. We then proceed to test our hypothesis that insider bettors are risk
neutral against the data using the theoretical results obtained in chapter 3. We
find that individual bettors are effectively risk neutral conditional on a degree of
inside information that is consistent with the observed data. Chapter 5 concludes.
Relevant computer code and data tables are collected in the Appendix in chapter 6.
A summary of commonly used variables follows; readers may find it useful to have
this summary to hand while reading the remainder of the document.
CHAPTER 1. SUMMARY 5
Summary of commonly used
variables
Variable Explanation
N Number of insider bettors
b0 Fraction bet on horse 1 by exogenous outsiders
τ Takeout rate
q Belief of insiders regarding the probability of horse 1 winning
p Objective probability of horse 1 winning
∆q = q − b0, or the amount of inside information available to insiders
α Risk preference parameter (from insider utility function)
αL Lower bound on insider risk preference
αH Upper bound on insider risk preference
α∆q Model implied risk preference conditional on q = b0 + ∆q
b1 Amount bet by insider bettor 1
bn Amount bet by insider bettor n, for n > 1
b∗ The optimal amount for insider bettors to bet in equilibrium
pi Winning bet payoff for horse 1
w Final period wealth of bettor
m Final market probability of horse 1 winning
mˆ OLS estimate of final market probability
m¯ Empirical (observed) final market probability
m(q,α) Model implied final market probability for given q and α
u Utility of insider bettor
c Cut-off point (in NZD) between insider and outsider bets
γ Coefficient of OLS regression of m¯ on m(q,α):γ = 1⇒ α = 0
Chapter 2
Literature review
2.1 Introduction
Parimutuel betting markets, of which betting on horse races is the prime example,
make up only a small fraction of total global economic activity. By way of example,
the total amount bet on horse races in the US in 2008 amounted to c. $14bn, of
which around 80% was returned to bettors, resulting in gross revenue to the racing
industry of slightly less than $3bn1. While this is not a small number, it pales
into insignificance when considered in the context of US GDP of $14.4 trillion2 for
the same year. This begs the question why these markets should be of interest to
economists. It turns out that betting markets have a number of intrinsic features
that make them particularly suited to testing economic theories of price formation,
as have been noted by several contributors to the literature on betting markets (see
for instance Sauer (2005) and Thaler and Ziemba (1988)) 3.
The purpose of this chapter is to conduct a broad review of the literature associated
with price formation in parimutuel markets, with a view to identifying areas that
might benefit from further theoretical development. To motivate this line of inquiry,
in Section 2.2 we elaborate on the features of betting markets that make them
particularly amenable to the study of price formation. Section 2.3 explains the
operation of parimutuel markets before we turn to a review of empirical regularities
that have been uncovered by researchers in Section 2.4. Attempts to explain these
empirical regularities form the substance of Section 2.5, which is an introduction to
1International Federation of horse racing Authorities, retrieved at 22 April 2010 from
http://www.horseracingintfed.com. Assumes an exchange rate of 1 US dollar to 0.71 euros.
2US Bureau of Economic Accounts, Table 1.1.5, retrieved from http://www.bea.gov at 22 April
2010.
3Of course there is always the possibility that some economists may find horse racing intrinsically
interesting. This author does not number among them. Instead, this author finds general theories
of price formation to be intrinsically interesting.
6
CHAPTER 2. LITERATURE REVIEW 7
the main theories of price formation in parimutuel markets. Section 2.6 presents an
analysis of current gaps in the theoretical literature that could benefit from further
work.
2.2 Why are betting markets important?
Understanding the way in which markets determine prices is of central importance
in economic theory, since prices are inextricably linked to the production of goods
and the allocation of resources. While not particularly important as a driver of
general economic activity, betting markets do have significant potential to help
economists form a clearer understanding of how markets work (a point that has been
made by many authors – see for instance Thaler and Ziemba (1988), Sauer (1998),
Vaughan Williams (1999), Coleman (2004) and Ottaviani and Sørensen (2008) for
further elaborations). In particular, the following features of betting markets noted
by the aforementioned authors are relevant:
1. Bets have a well-defined termination point which is fixed in advance and pub-
licly known.
2. The outcomes can be ascertained with certainty and is publicly observable.
3. The payout conditional on winning is known in advance in the case of fixed-
odds betting (but not in the case of parimutuel betting).
4. Outcomes are exogenous to the trading process and the prices that result from
it. Therefore econometric problems with endogeneity can be avoided.
5. Parimutuel markets do not require the presence of a price-setting market
maker, thereby eliminating a potential source of price distortion.
6. Though simpler, betting markets share important features with financial mar-
kets – potentially complex information structures, large numbers of (possibly
heterogeneous) participants and the potential for differential information sets
(aka insider trading). Therefore insights derived from examining betting mar-
kets may also have applicability to wider financial markets.
Also, in contrast to most experimental studies, betting market “experiments” are
repeated on a daily basis, are large scale, involves significant amounts of money and
are free from the Hawthorne effect since bettors do not generally consider themselves
to be participants in an experiment.
CHAPTER 2. LITERATURE REVIEW 8
However, betting markets do suffer from potential drawbacks as a test-bed for the-
ories of price formation. First, as noted by Ottaviani and Sørensen (2008) and
others, bettors may be motivated by recreational utility as opposed to monetary
payoff, which makes it more difficult to model behaviour within the standard eco-
nomic frameworks. Second, as discussed in the next section, parimutuel betting
markets at the level of the individual bettor constitute a game between bettors in
that payoffs depend not only on the decision of the individual bettor, but also on
decisions by other bettors, who are in turn similarly affected. Hence the strategic
interaction of bettors adds a layer of complexity to bettor’s expectations that are not
necessarily present in more conventional financial markets (or, indeed, in fixed-odds
betting markets).
2.3 Parimutuel betting markets
Parimutuel (from the French pari mutuel, meaning mutual stake4) betting is a form
of betting in which two or more bettors place bets on a number of mutually exclusive
outcomes, such as the winning horse in a horse race. When the winning outcome
is realised, the total amount originally bet - the pool - is returned to the winning
bettors in proportion to their original bets, after deducting a certain percentage -
the takeout rate - for taxes and expenses (typically in the range 15%-25%).
Mathematically, let h ∈ H be one of a set of horses in a race. Denoting by b(h) the
total amount bet on horse h and by B the total amount bet on all horses, the net
payoff after deduction of the takeout rate τ to a unit bet on a winning horse is:
pi(h) = (1− τ)
B
b(h)
− 1 = (1− τ)
∑
H b(h)
b(h)
− 1
The invention of the parimutuel method of betting is credited to the French busi-
nessman Pierre Oller around 18705. Since then it has become the dominant form
of betting on the outcome of horse races in the US, Western Europe, the UK,
Australasia and the Far East. In some jurisdictions, such as the UK and Australia,
parimutuel betting on horse races coexists with fixed-odds betting, as noted in Sauer
(1998). In other jurisdictions, parimutuel betting is mandated by law.
One notable feature of parimutuel betting is that the payoff to a bet depends not
only on the outcome of the contest, but also on the bets placed by other players,
either on the winning horse or on any of the other horses. Recall that winning
bettors share the total pool of money bet (after subtracting the takeout), the size
4Merrian-Webster Online Dictionary (from http://www.merriam-webster.com/dictionary/pari-
mutuel at 21 April 2010.
5Encyclopedia Britannica
CHAPTER 2. LITERATURE REVIEW 9
of which depends on the total money bet on all of the horses. Furthermore, the
proportion of the total pool that a winning bettor is entitled to depends not only on
the size of her own bet, but also on the size of bets made by other winning bettors.
This is because the total pool is distributed to winning bettors in proportion to their
winning original bets.
This implies that payoffs to bettors in a parimutuel contest is strategically interde-
pendent, being contingent not only on betting on the winning outcome, but also on
the other bettor’s bets. This is in contrast to fixed-odds betting where the outcome
contingent on winning is known with certainty.
Note that, once the final distribution of bets is known, the payout on a winning
parimutuel bet can be computed with ease. The key point, however, is that no
bettor is privy to the final distribution of bets at the time she has to decide her bet
- there is no practicable way to know with certainty how much other bettors will bet
subsequently (or simultaneously, if our bettor is holding out until the last moment
before betting closes).
2.4 Empirical regularities
2.4.1 Market efficiency
The concepts and methodologies of financial market efficiency (Fama, 1991) has also
been applied to betting markets. Many researchers have found evidence inconsistent
with market efficiency – Sauer (1998) and Vaughan Williams (1999) are surveys
that discuss the topic in some depth. The favourite longshot bias, which we discuss
in detail below, is often cited as evidence that sporting markets are not efficient.
Despite this, there is a consensus in the literature that final market probabilities in
parimutuel markets are generally good first-order predictors of actual outcomes.
For instance, Thaler and Ziemba (1988) writes:
The racetrack betting market is surprisingly efficient. Market odds
are remarkably good estimates of winning probabilities. This implies
that racetrack bettors have considerable expertise, and that the markets
should be taken seriously.
but then goes on to admit the existence of apparent biases.
More recently, Sauer (1998) comments:
CHAPTER 2. LITERATURE REVIEW 10
Although there are many published exceptions that reject efficient pric-
ing, it is the benchmark result that prices are approximately efficient
that yields insights to which economics should be proud to lay claim.
For example, odds at the racetrack imbed subjective estimates of the
probability of winning that are (a) quite close to their empirical counter-
parts, (b) similarly close to those obtained using sophisticated statistical
methods, and (c) very difficult to exploit on a systematic basis.
Empirical testing of market efficiency in sports betting markets have been ap-
proached from a number of different angles. In discussing the literature, Sauer
(1998) considers three approaches to testing for market efficiency:
1. Constant returns: Bets should all yield the same expected return.
2. Absence of profit opportunity: No way to make money in expectation.
3. Equilibrium pricing: Observed prices should match theoretical prices.
Thaler and Ziemba (1988) organises their discussion of market efficiency around the
financial markets benchmarks of weak vs strong market efficiency. Weak market
efficiency implies that bets should not have positive expected values while strong
market efficiency implies that all bets should have expected values equal to the
amount bet less the takeout.
Most of the literature on market efficiency in sports betting assumes that in an
efficient market the market probability of an outcome (which can be inferred from
the final market odds) should be an unbiased predictor of the observed outcome.
This assumption will be true for some models of price formation and but not true
for others. This leads us to an inescapable problem when testing for any form of
market efficiency: any such test can only test the joint hypothesis that the market is
inefficient and that the benchmark model used to define efficiency is correct. There-
fore, any empirical rejection of efficiency may simply indicate that the benchmark
model used is incorrectly specified, rather than proving that a particular market is
inefficient (Cochrane, 2001 p128 citing Fama (1970)).
2.4.2 The favourite longshot bias
If one assumes that there is a single representative bettor which is risk neutral and
a maximiser of expected utility over wealth, then it follows that market prices of
CHAPTER 2. LITERATURE REVIEW 11
outcomes will be unbiased predictors of expected outcomes.6 To see why this must
be the case, assume that there exists an outcome for which the market price is lower
than the expected outcome. This implies that the representative bettor can increase
his expected utility by betting a larger fraction of his money on this outcome. This
process will continue until the market price of this outcome is equal to the expected
outcome.
Therefore one might expect the fraction of money bet on each horse (its “price”) to
be an unbiased predictor of the frequency of actual wins for that horse. However
this is not the case. Griffith (1949) is credited with the discovery that horses with
low odds (that is, favourites to win the race) are under-bet and horses with high
odds (that is, longshots) are over-bet. In other words, favourites win more often
than the final market odds suggest they should, while longshots win less often than
the market odds suggest they should. This empirical regularity has been coined
the favourite longshot bias (or sometimes the longshot bias or the FLB) and it
holds a pride of place as one of the most robust anomalies in the literature on
betting markets. It has even been described as “one of the most robust anomalous
empirical regularities in economics” (Walls and Busche (2003) citing Thaler and
Ziemba (1988)). The tendency to overestimate the likelihood of rare events also
has support in the psychology literature where it is considered a subcategory of
optimism bias.
In a survey of empirical papers Coleman (2004) found 19 studies addressing the
longshot bias of which 15 found a statistically significant favourite longshot bias
while a further 4 studies did not.7 So while the favourite longshot bias has been
observed in several markets and over a time-frame of more than 50 years, it cannot
be claimed to be universal. However, in a recent study employing a very large data
set of 678,729 US races spanning 1992 to 2001 Snowberg and Wolfers (2010) continue
to find strong evidence of the favourite longshot bias.8
More recently, empirical research has focused on identifying factors associated with
the favourite longshot bias in the hope that it might shed more light on the under-
lying causes. (Sauer (1998) notes that work in this direction “would be particularly
useful”.) Some of this work is examined below. It is perhaps also worth pointing
out that the term favourite longshot bias presupposes that the discrepancy between
6The Law of Large Numbers ensures that the average empirical outcome will approach the mean
of the distribution from which outcomes are drawn (in the limit). Therefore, over a sufficiently
large number of trials the average observed outcome will be an unbiased estimate of the expected
outcome.
7As will be discussed later, three of these studies considered races from Far East Asian countries,
while the fourth used a small sample of 288 races.
8Just to confound matters further, some authors have documented a reverse favourite longshot
bias in lotteries which are also commonly organised on a parimutuel basis.
CHAPTER 2. LITERATURE REVIEW 12
market odds and empirical outcomes are the result of a bias. While early authors
such as Griffith (1949) postulated bias on the part of bettors as an explanation for
the discrepancy, there is no reason to suppose that the discrepancy must necessarily
be the result of bettor bias – it could equally be the result of other factors such
as loss aversion, strategic interaction or private information. We therefore use the
term favourite longshot bias without prejudice as a shorthand for the discrepancy
between market probabilities and empirical outcomes, as has become the practice
in this literature.
2.4.3 Factors correlated with the favourite longshot bias
2.4.3.1 Pool size and number of horses
In a study of 10,000 races from North America, Japan and Hong Kong Busche and
Walls (2000) find that the favourite longshot bias is inversely related to the size of
the betting pool. For betting pools of the order of $200 million to $300 million, they
do not find any favourite long shot bias - in other words, the market probabilities
are unbiased predictors of empirical race outcomes. However, they do find a positive
favourite longshot bias in races where the betting pool is in the range $300,000 to
$1.8 million. They interpret this as evidence supporting decision cost theory, on
the assumption that only large betting pools are able to generate sufficient profit
opportunities to warrant the expense of obtaining superior information by sophis-
ticated bettors. However, this evidence is also consistent with other explanations,
such as inside information (it may be more difficult to exploit inside information
in larger races) or the number of informed players (larger races may proxy for a
greater number of sophisticated players, which forces the market price closer to the
expected price).
In a similar paper that controls for additional race factors, Gramm and Owens
(2005) also find support for the notion that larger betting pools are associated with
a smaller favourite longshot bias. In addition, they find that the favourite longshot
bias is decreasing in the number of horses in the race and for lower quality fields.
These conclusions are robust to fixed track effects, for which they control.
On balance, the evidence does seem to indicate that “large” races (ie more dollars
bet, more bettors, more horses) result in a smaller associated favourite longshot
bias, even to the point where the bias becomes statistically insignificant.
CHAPTER 2. LITERATURE REVIEW 13
2.4.3.2 Late races and capital constraints
In a paper examining 728 races held in 1978 in Atlantic City, Asch, Malkiel and
Quandt (1982) finds that the favourite longshot bias is higher in races that are
held later in the day.910 They hypothesise that late races may proxy for capital
constrained bettors – since bettors will, on average, lose the equivalent of the takeout
rate on each race, there will be a large number of bettors who have lost a significant
portion of their “race day money”. The idea is that these bettors would much prefer
to end the day with a profit, but given the depleted state of their capital, the only way
they can achieve this is by betting on a longshot horse with a high payout relative
to the amount bet. Similarly, Ali (1977) found that a (representative agent) bettor
at the last race of the day exhibited higher risk love than at the corresponding first
two races of the day. However, Snowberg and Wolfers (2010) fail to find a last-race
effect in their large scale study of US horse races from 1992 to 2001.
It appears that some bettors may be motivated by risk love or extreme loss aversion
linked in some sense to the amount of money they bring to the races. This points to
a potential Kahneman and Tversky (1979) prospect theory type loss aversion on the
part of bettors, perhaps combined with mental accounting. (Also see Qiu (2007) for
an attempt to incorporate prospect theory with mental accounting in a sequential
parimutuel setting.)
2.4.3.3 Racing in East Asia
A number of studies have failed to find a favourite longshot bias. In his survey
of previous empirical work, Coleman (2004) found four studies where a favourite
longshot bias was not found, one of which (Swindler and Shaw, 1995) uses a small
sample of 288 early season races in Texas. The remainder, which consists of c. 6000
races in Hong Kong ((Hausch and Ziemba, 1995) and (Busche and Hall, 1988)) and
c. 13000 races in Japan (Walls and Busche, 2003), are both based on data from Far
East Asian localities, which has prompted speculation that there may be cultural
element at work.
9The effect is strong - for bets on favourites, the expected loss is 4%, while for bets on longshots,
the expected loss is 69%. This can be compared to a takeout rate typically in the range 15%-25%,
which is by definition equal to the dollar weighted average loss across all bets. The presence of
such a high takeout rate means that it is usually not possible to profit from the observed favourite
longshot bias despite the strength of the apparent bias.
10Coleman (2004) attributes the suggestion that the favourite longshot bias strengthens during
the racing day to McGlothlin (1956).
CHAPTER 2. LITERATURE REVIEW 14
2.4.4 Early betting
Most studies find that slightly over half of total betting dollars are wagered before
closing time. In a study of 1644 races Gramm and McKinney (2009) find that
40% of all dollars are bet in the last minute before closing time, which implies
around 60% must be wagered before then, often a day or more beforehand. In
the case of fixed-odds betting, there may be a benefit to betting early if the odds
offered by the bookmaker are sufficiently far away from the bettor’s own estimate.
If the odds offered by the bookmaker later converge to the bettor’s own estimate,
the bettor is still entitled to a payout at the fixed odds he originally contracted
with the bookmaker in case the bet pays off. Therefore, it may be advantageous
for an informed or sophisticated bettor to bet early if the fixed odds on offer from
bookmakers are sufficiently different from his own estimate. The situation is slightly
different in the case of parimutuel betting. A bettor in a parimutuel contest does
not face a bookmaker with whom he can contract for a fixed payout – instead, the
payout on a bet in the parimutuel system depends on the final amount bet on the
winning horse as well as all the other horses. An informed or sophisticated bettor
may therefore prefer to postpone betting to the last minute because
1. He cannot be sure of “locking in” what appear to be favourable odds based on
current betting pools.
2. He gives up the opportunity of benefiting from future information that might
become available.11
3. He runs the risk of revealing his information to other bettors – and if other
bettors also “get on the bandwagon” the effect of their additional bets would
be to reduce the payout to the first bettor.
That is not to say that there are no advantages to betting early in a parimutuel race:
Ottaviani and Sørensen (2006) shows in detail how sequential parimutuel betting
corresponds to a Cournot competition in quantities, such that a Stackelberg first-
mover advantage obtains, a correspondence also noted in passing by Adams, Rusco
and Walls (2002).
There are other potential explanations of early betting, the most obvious being the
possibility that bettors are motivated by non-monetary recreational utility, in which
case they would be indifferent as to the exact timing of their bets.
11This is mitigated by the fact that such future information is likely to alter the payoffs implicit
in his bet so as to equalise returns across all bets.
CHAPTER 2. LITERATURE REVIEW 15
2.4.5 Late betting is smart betting
If late betting were simply an implication of early betting as discussed above, then
it would not warrant further discussion. But there is more to it: several studies
have found that late betting moves market probabilities closer to the empirical
probabilities. In other words, late betting eliminates some of the favourite longshot
bias and results in final market probabilities that are closer to observed empirical
outcomes – late betting is smart betting.
In one of the early studies, Asch, Malkiel and Quandt (1982) found that final odds
were substantially better at predicting outcomes than the opening odds. In a study
of 10,332 races in New Zealand, Gandar, Zuber and Dare (2000) finds that a favourite
longshot bias is apparent in the early betting pools, but that the bias is mostly elim-
inated by late betting. Gramm and McKinney (2009) performs a similar analysis
and finds that late betting volumes (in the last minute before closing) is the best
predictor of empirical outcomes, even outperforming total betting volumes. The
implication appears to be that late betting is largely free of bias and motivated to
correctly assess winning probabilities, while early betting appears to be more suscep-
tible to the favourite longshot bias and perhaps also more motivated by recreation
than expected monetary reward.
2.4.6 Rounding of odds
A feature of most parimutuel markets is that the odds used to calculate payouts
(which are based on the ratio between the total betting pool and the amount bet on
the winning horse) are rounded downwards. The net effect of this is to reduce the
amount of money paid out to winning bettors and to increase the amount of money
retained by the organisers of the parimutuel market. Odds are typically rounded
down to the nearest 5 or 10 cents. Walls and Busche (2003) point out that using
rounded odds to calculate the favourite longshot bias results in a statistical bias
towards rejecting market efficiency12 – using exact betting volumes, they find the
favourite longshot bias fails to materialise in their sample of c. 10,000 races. Since
a number of the early studies used published (ie rounded) odds, it is possible that
at least some of the reported favourite longshot bias might be attributable to the
effect of rounding. However, as Coleman (2004) points out, the rounding effect has
the greatest impact on favourites with short odds and the least impact on longshots
with high odds. So while rounding biases published odds, it does so in a way that
weakens the overall favourite longshot bias.
12This is because on favourites the payout odds will be less than suggested by the betting pools
themselves. This will make it look like favourites were over-bet based on the payout odds.
CHAPTER 2. LITERATURE REVIEW 16
2.5 Theories of price formation
2.5.1 Overview
The theoretical literature on parimutuel betting markets has developed in three
strands. The first strand attempts to explain the favourite longshot bias as a result
of certain attributes collectively inherent in the bettors themselves (within a repre-
sentative agent framework). This includes bias or misperception, as well as risk-love
or skew-love. We consider this literature in sub-section 2.5.2. The second strand,
which we cover in sub-section 2.5.3, explains the favourite longshot bias by way
of various market structure features. Examples include limited arbitrage and het-
erogeneous beliefs. Finally, the third strand explicitly models strategic interaction
between individual bettors using the tools of game theory. In some of the models
information is assumed to be public, while in other models, it is private to individual
bettors. Sub-section 2.5.4 reviews this literature.
2.5.2 Representative agent models
Most of the early literature explained the favourite longshot bias by appealing to
some aggregate characteristics of bettors, such as misperception or risk-love. As such
we consider this segment of the literature under the heading of representative agent
models. A weakness of this class of models is that a representative agent framework
cannot incorporate strategic interaction within the context of parimutuel betting.
Therefore, such models may confound the impact of strategic interaction (which is
not modelled) with representative agent characteristics (which are).
2.5.2.1 Bettor bias
The early papers interpreted the discrepancy between market probabilities and em-
pirical probabilities as evidence that bettors collectively suffer from bias (hence the
choice of the term “favourite longshot bias” to describe the discrepancy). In the
first paper to identify the favourite longshot bias, Griffith (1949) noted that, while
bettors collectively managed to predict outcomes with a remarkable degree of ac-
curacy, there remained a systematic under-estimation of the winning probabilities
for favourite entries along with an over-estimation of the winning probabilities for
longshot horses.
Remarkably, the debate whether the favourite longshot bias should be seen as evi-
dence of collective bias or collective risk-love on the part of bettors rages on more
CHAPTER 2. LITERATURE REVIEW 17
than 50 years later. Recently, Snowberg and Wolfers (2010) used a very large sam-
ple of North American races (> 600,000) to try and discriminate between these
two competing explanations. Their approach centered around using straight win-
bets to calibrate two simple models of decision making, the first corresponding to
risk-love and the second corresponding to misestimation of probabilities. They then
tested these calibrated models against data on compound bets13, and found that the
compound bet data was more consistent with a misestimation of probabilities than
risk-love.
2.5.2.2 Risk preference
In contrast with sub-section 2.5.2.1 (“Bettor bias”), Weitzman (1965) postulated a
model of an unbiased representative bettor. While the representative bettor is not
subject to bias, Weitzman postulates a utility function over betting outcomes for
this bettor. Using data from 12,000 races during the period 1954 to 1963 at four
New York racetracks, he estimates a utility curve which fits the observed aggregate
betting data and concludes that the representative bettor is a risk-lover. Weitzman
is careful to note that this does not imply that individual bettors are risk-loving
(even in an average sense), only that bettors in a collective sense can be modelled
closely by assuming they were represented by a hypothetical individual bettor with
a risk-seeking utility over betting outcomes.
In a model with heterogeneous risk-loving bettors (in the mean-variance sense)
Quandt (1986) constructs a model that exhibits the favourite longshot bias. His
argument is based on the fact that, in contrast to standard mean-variance frame-
works in the finance literature where risk aversion is assumed, risk-loving bettors do
not achieve any benefit from diversification. Since they are risk-lovers, they would
prefer to bet on a single horse, which would generally result in a higher variance
than if they bet on a combination of horses. Quandt argues that, if market proba-
bilities equalled empirical probabilities for each horse, this would result in a set of
betting opportunities with different variances, but equal expected returns.14 How-
ever, such a set of betting opportunities cannot represent an equilibrium, since each
bettor would prefer to bet on the single horse with the highest variance. Using a
fixed-point argument, Quandt shows that in equilibrium, market probabilities will be
above the empirical probabilities for favourites and below the empirical probabilities
for longshots, hence giving rise to the favourite longshot bias.
13Compound (or exotic) bets are bets dependent on the performance of more than one horse.
For instance, a trifecta bet pays off if the bettor correctly predicts the first, second and third placed
horses in that order.
14This follows from the fact that market probabilities equal empirical probabilities
CHAPTER 2. LITERATURE REVIEW 18
It is worth noting that while Quandt allows for an unspecified number of mean-
variance optimising bettors, the way in which he solves for equilibrium effectively
assumes a representative bettor and stops short of modelling the full strategic in-
teraction between bettors in a parimutuel betting contest.
2.5.2.3 Skew preference
The presence of a positive track takeout rate implies that it is not possible for all
bettors to be motivated by the average payout only. In sub-section 2.5.2.2 above,
an attempt was made to explain bettor behaviour by linking their utility to the
variance of payouts, the second moment. Golec and Tamarkin (1998) goes further by
arguing that it is an affinity not for variance, but the third moment, skewness, that
motivates bettors. They support their argument with a regression study showing
that a specification that includes skewness provides a better description of the data
than one based on variance only.
2.5.2.4 Prospect theory
A number of authors (see for instance Thaler and Ziemba (1988)) has noted that
the behaviour of bettors appear to be consistent with the prospect theory of Kah-
neman and Tversky (1979). The key components of prospect theory is that people
1) overweight small probabilities and underweight large probabilities, 2) utility is
defined separately over losses and gains, rather than total wealth and 3) the utility
function is specified so that people exhibit risk-aversion for gains and risk-love for
losses, with the added constraint that the utility function is flatter for gains than
for losses, implying that people are loss-averse.
A recent attempt to incorporate prospect theory is due to Qiu (2007), which we dis-
cuss more fully in sub-section 2.5.4.4 since it also incorporates strategic interaction.
2.5.2.5 Recreational utility
Many papers implicitly acknowledge the existence of a class of bettors motivated
by non-monetary or recreational utility. Typically such bettors are modelled as
exogenous. For instance Isaacs (1953) considers a situation where a single informed
bettor have to bet optimally given previous betting by exogenous bettors, while
Ottaviani and Sørensen (2006) is a more recent example.
Hausch, Ziemba and Rubinstein (1981) considers a model in which some bettors
derive recreational utility from making successful bets on longshots (ie “bragging
CHAPTER 2. LITERATURE REVIEW 19
rights”), and show that the presence of such bettors implies a favourite longshot
bias.
In contrast, Conlisk (1993) models recreational bettors endogenously by append-
ing a “tiny utility of gambling” to an otherwise standard expected utility model,
which allows him to explain the apparent risk seeking behaviour of gamblers and
bettors within an otherwise risk averse framework. Ottaviani and Sørensen (2010)
incorporates recreational utility within the framework of a simultaneous move game
with private information. Participation of bettors is endogenous in this model, and
bettors only obtain recreational utility if they choose to bet. The authors show that
even bettors without strong private beliefs will bet on a horse if recreational utility
is sufficiently high and that the favourite longshot bias becomes more pronounced
as recreational utility decreases.
2.5.3 Market structure
A number of theoretical models of parimutuel betting has been inspired by assuming
different market features. This stands in contrast to the earlier literature, which
tried to explain observed betting probabilities by appealing to bettor misperception
or attitudes towards risk within a representative agent setting. The key argument
of the market structure literature is that the favourite longshot bias can arise as a
result of specific market microstructure, without having to make assumptions about
the characteristics of a representative bettor.
2.5.3.1 Single insider
The first explicit consideration of inside information is due to Isaacs (1953), who
postulated a model with exogenous uninformed bettors (outsiders) and a single risk
neutral informed bettor who bets optimally in response to the previous bets of the
outsiders. In this model the insider knows the objective (true) win-probabilities of
a set of H horses and also observes the amount bet by the outsiders. Assuming that
no other bettors will bet subsequently, Isaacs solves for the insider’s optimal betting
schedule. He also shows that if one ranks all horses by the ratio of their objective
win probability to their outsider market probability (that is, by the ratio of p(h) to
b0(h)
B0
) then if the insider bets on any horse j he will also bet on all the other horses
ranked above j.
A consequence of equilibrium for a single informed bettor is that the final market
probabilities (which include bets from both the outsiders and the insider) will fall
short of the objective market probabilities for those horses that the insider bets
CHAPTER 2. LITERATURE REVIEW 20
on, giving rise to the favourite longshot bias. The reason is that if the insider bets
money until market probabilities equal objective probabilities, then the insider will
make zero profits in expectation. Therefore, in order to profit from her inside infor-
mation, the insider has to bet less than would be required to equate the final market
probabilities with the objective probabilities. Since the final market probabilities
are less than the objective probabilities in equilibrium, the favourite longshot bias
obtains15.
2.5.3.2 Heterogeneous beliefs
It is reasonable to believe that bettors may have different subjective estimates re-
garding the winning probabilities of different horses. In a model with two horses and
a continuum of risk neutral infinitesimally small bettors with heterogeneous beliefs,
Ali (1977) shows that the mean of the distribution of bettors beliefs (which are equal
the objective probability by assumption), is above the final market probability for
the favourite horse and below the final market probability for the longshot, which
is in accordance with the favourite longshot bias.
2.5.3.3 Endogenous bookmaker
A separate strand of literature investigates the impact of bookmakers on the price
formation process. This line of inquiry is not directly applicable to parimutuel
betting markets, since in a parimutuel market there is no bookmaker – bettors
effectively bet against each other only and the parimutuel betting market operator
(often referred to as the track operator) takes as compensation a fixed proportion of
the total amount bet without having to take any risk on the outcome of the race. In
contrast, bookmakers offering fixed-odds betting do take risk, at least to the degree
that they are unable to balance betting interests evenly. We include a discussion of
bookmakers because it show clearly how the presence of a bookmaker can complicate
the price setting process, and therefore further motivates the study of parimutuel
markets in which this distortion is not present.
Shin (1991) and Shin (1992) provide a model in which a bookmaker posts fixed-odds
prices against an uninformed public as well as an unknown insider. The uninformed
public has heterogeneous beliefs distributed uniformly from zero to one, which are
not correlated to the objective winning probabilities. The bookmaker has knowledge
of the objective winning probabilities of all the horses. However, the insider has
15Interestingly, Isaacs (1953) fails to make any mention of the favourite longshot bias in his paper
and appears to have been unaware of it. The fact that his equilibrium for a monopolist insider
implies a favourite longshot bias nonetheless is therefore telling.
CHAPTER 2. LITERATURE REVIEW 21
knowledge of the actual winning horse and bets her total wealth on this horse. The
bookmaker always losses the bet contracted with the insider, but can generate profit
by exploiting the mispricing of the uninformed public. Shin shows that in order to
maximise expected profit in the presence of inside information, the bookmakers will
post odds incorporating market probabilities that are different from the objective
probabilities that the market maker is privy to. This relationship is given by a
“square root rule”:
m(1)
m(2)
=
√
p(1)
p(2)
The favourite longshot bias is inherent in this pricing rule as the favourite horse is
quoted at a market probability below its objective probability, while the longshot
horse is quoted at a market probability above its objective probability.
In a parimutuel race the track operator does not set prices, but it does have control
over a choice variable that impacts its profitability - the track takeout rate16. Watan-
abe, Nonoyama and Mori (1994) puts forward a model in which the track takeout
rate is endogenised. They consider a fixed number of bettors with heterogeneous
a-priori beliefs which are public knowledge. The bettors play a simultaneous move
game parameterised by the takeout rate in stage 2. In stage 1, the track operator
chooses a takeout rate that will maximise his profit conditional on the beliefs of the
bettors. By solving simultaneously for the outcome of the bettor’s game and the
optimal choice of takeout rate for the track operator, Watanabe, Nonoyama and
Mori are able to numerically characterise the dependence of the track takeout rate
on bettor’s beliefs.
In a setting with a fixed-odds bookmaker, Levitt (2004) shows that an optimising
bookmaker has an incentive to quote odds inconsistent with objective probabilities
if faced with a betting public prone to predictable bias. The quoted odds results
in betting volumes which are not balanced across outcomes, so that the bookmaker
takes a net betting position against the public in a direction that exploits bettor
biases. Levitt provides empirical evidence that bookmakers do in fact deviate from
the “balanced books” model and that such a deviation earns them profits in excess
of what they would have earned if they were seeking a balanced book.
2.5.3.4 Transaction costs and limits to arbitrage
In real-life parimutuel betting transaction costs (in the form of the takeout rate and
downwards rounding of odds) is large, typically ranging from 15% to 25% (Thaler
16This is not always the case, as in some jurisdictions the takeout rate is fixed by law while other
jurisdictions impose upper limits.
CHAPTER 2. LITERATURE REVIEW 22
and Ziemba, 1988). Nor is it possible to directly “short” a horse if you think it is
over-bet (ie overpriced)17. Taken together, this implies that arbitrage by sophisti-
cated bettors may not be able to eliminate biases altogether, even if sophisticated
bettors exhaust all profitable betting opportunities. It is notable, that despite the
widespread occurrence of the favourite longshot bias, it has generally not been of a
sufficient magnitude to present consistently profitable betting opportunities. Since
the favourite longshot bias is economically significant, the lack of profitable arbitrage
opportunities seems to be driven at least in part by high takeout rates.
Hurley and McDonough (1995) construct a model of insiders betting competitively
to exploit the mispricing of a group of exogenous outsiders. While the insiders have
access to inside information that should allow them to benefit at the expense of
outsiders, they face high transaction costs in the form of a takeout rate and the
generation of costly inside information. If transaction and information costs are
equal to zero and the number of insiders approach infinity, final market probabilities
approach objective probabilities. In the presence of positive transaction or infor-
mation costs, this becomes an inequality with the objective probability forming the
upper bound on the final market probability for the favourite horse, consistent with
the favourite longshot bias. Hurley and McDonough then proceed to test their hy-
pothesis that the bias is generated by the takeout rate in a laboratory experiment,
in which the hypothesis is rejected. One is left with the conclusion that either the
laboratory experiment does not generalise to the race track or the track takeout rate
is not in itself an explanation for the existence of the longshot bias.
2.5.4 Strategic interaction
As noted in section 2.3, parimutuel betting comprises a game from the point of view
of individual bettors in the sense that the optimal strategies of individual bettors
depend on the optimal strategies of other bettors and vice versa. Despite the fact
that the game theoretic nature of parimutuel betting has long been recognised18,
the bulk of early theoretical models were constructed within a representative agent
framework. This precluded any modelling of the game-theoretic interaction between
bettors (see sub-section 2.5.2).
Another, more recent, literature models individual bettors but assume the number
of bettors is so large and their individual bets so small that the impact of individual
17Although it is possible to create a position that might mimic a short interest in a particular
horse by betting (probability weighted) amounts on all the other horses.
18(Ottaviani and Sørensen, 2008) cites a paper by Borel (1938) in which several key game-
theoretic ideas relating to parimutuel betting are presented. This is even more remarkable con-
sidering that Borel’s paper pre-dates the development of modern game theory (formalised by von
Neumann and Nash) by several decades.
CHAPTER 2. LITERATURE REVIEW 23
decisions can be ignored. However, this approach is not without criticism, as Feeney
and King (2001) point out:
Much of the theoretical work analysing parimutuel systems, however,
makes a critical ‘small player’ assumption that one individual cannot in-
fluence the actions of others. In other words, there are always enough
players in the system so that the effect of one player’s actions on the
information and returns of other players can be ignored. For example,
Potters and Wit (1995) examine a parimutuel system where players in-
dependently choose actions after receiving an individual signal that is
drawn from a common distribution. But each player ignores the conse-
quences of their action on the odds of the gamble. Similarly, Watanabe
(1997) analyses a parimutuel system with a continuum of players. The
small player assumption greatly simplifies the modelling of parimutuel
systems, but it is very strong. For example, if there are few players or if
some players wager relatively large sums of money then the interdepen-
dency of returns in a parimutuel system means that the ‘small player’
assumption is likely to be violated.
In this section we consider the strand of the literature that explicitly models the
strategic interdependence between bettors inherent in the parimutuel method of
betting. Our discussion follows the standard distinction in the game theoretical
literature between simultaneous games and sequential games.19 We also make a
distinction between games where bettors posses private information and those where
information are common to bettors.
2.5.4.1 A note regarding rational risk neutral bettors
In a simultaneous parimutuel betting model that consists only of rational risk neutral
bettors, no bets will take place if the takeout rate is non-zero. This follows from
the fact that all bets lose money in expectation in such a market, and therefore
rational risk neutral bettors will not place bets in the first place. Betting by rational
risk neutral bettors can nonetheless be motivated if we assume some bettors are
motivated by recreational utility, or if we assume some bettors have beliefs or inside
information that differs from the implied market probabilities implicit in the existing
betting odds they face. Most models proceed by postulating two groups of bettors -
19A simultaneous game is a game where the players have to decide their actions without having
the benefit of observing the actions of other players. Therefore, it is not required that actions take
place at the same time, only that those actions are not observable by other players. In a sequential
move game, by contrast, players have the opportunity to observe the actions of previous players
before having to decide their own actions.
CHAPTER 2. LITERATURE REVIEW 24
outsiders that bet for recreational reasons and insiders that bet on the basis of inside
information in order to take advantage of the mispricing of recreational bettors. In
these models outsiders are for the most part modelled exogenously, so any strategic
interaction takes place between insiders.
2.5.4.2 Simultaneous move – symmetric information
There are two ways to think about insiders’ private information: 1) information
that is private to the group of insiders but are common knowledge among those
insiders (symmetric information) and 2) information that is strictly private to each
individual in a group of insiders (asymmetric information). The first is addressed in
this sub-section, while the second is addressed in the next sub-section.
In an early paper Eisenberg and Gale (1959) presents a model of m bettors and n
horses, with each bettor’s win-probability beliefs over horses modelled as am×nma-
trix. They show that, if bettors are constrained by an arbitrary budget – which may
differ between bettors – then there exists a unique Nash equilibrium of final market
probabilities (but not necessarily betting amounts). In their own words “Thus, at
equilibrium the bettors, as a group, maximize a weighted sum of logarithms of sub-
jective expectations, the weights being the bettor’s budgets. As noted previously,
equilibrium probabilities turn out to be unique, although equilibrium bets need not
be unique.” (Eisenberg and Gale, 1959, p168)
Hurley and McDonough (1995), previously discussed in some detail in sub-section
2.5.3.4 in the context of transaction costs, is an early paper explicitly seeking to
explain the favourite longshot bias within a game theoretic framework using two
horses and N insiders. They show that, as the number of insiders N →∞, the final
market probability of the favourite horse becomes m = (1 − τ)p. In other words,
the market probability of the favourite will be less than the objective probability –
the favourite longshot bias.
Although Hurley and McDonough neglects to mention this, their model is also able
to explain the favourite longshot bias without an appeal to transaction costs. We
take as a starting point the model set out in sub-section 3.1.3, which is equivalent
to the Hurley and McDonough model in which we assume a zero takeout rate.
Consistent with Hurley and McDonough we further assume b0 = 0.5. Making use
of the simplifying assumptions above and substituting the optimal insider betting
amount (equation 3.1.4) into the formula for the final market probability of the
winning horse (equation 3.1.6), we obtain a closed form solution to the final market
probability
CHAPTER 2. LITERATURE REVIEW 25
m =
√
p (pN2 − 6pN + 4N + p) + pN − p
√
p (pN2 − 6pN + 4N + p)− pN + 2N − p
(2.5.1)
In the absence of a takeout rate, we have p ≥ b0 = 0.5. It can be shown that m = p
in equation 2.5.1 above is solved by only three values of p, that is, [0, 0.5, 1]. The
first solution can be disregarded since it violates the inequality above, while the
remaining two are corner solutions for the feasible range of p.
It can be shown that m < p for all p in the range (0.5..1) for finite N , which is
again consistent with the favourite longshot bias, since m and p refer to the final
market implied probability and objective probability respectively of the favourite
horse. For a small number of insiders N , the difference between the final market
probability m and the objective probability p is substantial. As shown earlier, for
large N , the final market probability converges to the objective probability. To give
a sense of the relationship between m and p as N becomes larger, we plotted the
curves corresponding to equation 2.5.1 for p = 0.6 and p = 0.8 in Figure 2.5.1 below.
Figure 2.5.1: Final market probability m for p = 0.6 and p = 0.8 as a function of N
It can therefore be argued that the favourite longshot bias may equally well be
the result of a small number of insiders able to bet sizeable amounts of money.20
The above exposition can be viewed as an extension of the Isaacs (1953) model of
a single monopolistic insider to a model of finitely many insiders, and the same
20We are assuming a zero takeout rate. More realistically, in the presence of a positive takeout
rate, the difference between insiders assessment of winning probabilities and the probabilities im-
plicit in existing betting odds needs to be sufficient to generate profits in expectation for insiders,
otherwise insiders won’t bet.
CHAPTER 2. LITERATURE REVIEW 26
general results obtain. Chadha and Quandt (1996) constructs a more general model
along the same lines in which they model a finite number of insiders betting on
a finite number of horses in the presence of exogenously specified outsiders. The
insiders have access to the objective win-probabilities of all the horses. The insiders
attempts to bet optimally, given the expectation that other insiders will also bet
optimally, gives rise to a Nash equilibrium. Chadha and Quandt are only able
to solve for equilibrium final market probabilities numerically and arrive at similar
results. Another paper taking a similar approach is that of Adams, Rusco and Walls
(2002), again arriving at similar conclusions.
2.5.4.3 Simultaneous move – private information
Situations where insiders have asymmetric information are somewhat more compli-
cated than situations where the inside information is symmetric to insiders. In the
case of symmetric information, insiders can fairly easily form expectations regarding
other insiders’ actions in equilibrium, since all actions are conditioned on the same
information set, which is known to all insiders. When insiders have asymmetric infor-
mation, it is necessary for each insider to consider the distribution from which other
insiders obtain their private information. This is to enable the formation of expec-
tations regarding the private information of those other insiders. Koessler, Noussair
and Ziegelmeyer (2008) is credited21 with being the first to integrate asymmetric
information into a game theoretic formulation of parimutuel betting. The Koessler,
Noussair and Ziegelmeyer model postulates N bettors that bet based on private
binary signals, first in a simultaneous move setting and then in a predetermined
sequential move setting. Among other results, Koessler, Noussair and Ziegelmeyer
proves the existence of separating equilibria in which all bettors follow their private
signals within a Bayesian-Nash framework, which yields final market probabilities
consistent with the favourite longshot bias. They also show that such an equilibrium
is unique if the number of bettors is large enough. However, separating equilibria
are shown to disappear in a sequential move setup if the number of sequential bet-
ting periods grows large enough. In a simultaneous move model with a continuum of
insiders and real valued private signals Ottaviani and Sørensen (2005) and Ottaviani
and Sørensen (2009) also derives a favourite longshot bias, which is driven to zero
as the number of insiders grows large.
In an ambitious undertaking, Ottaviani and Sørensen (2010) presents a model of
parimutuel betting that incorporates k outcomes, N ex-ante identical bettors with a
common prior belief distribution over all outcomes (the symmetric information com-
ponent), independently drawn private signals (the asymmetric information compo-
21See Ottaviani and Sørensen, 2006, p5
CHAPTER 2. LITERATURE REVIEW 27
nent) and fixed recreational utility which is dependent on participation. Each bettor
can chose to bet an amount normalised to unity on a single outcome, or to abstain,
thereby forfeiting their recreational utility. Unlike other contributions to the litera-
ture, the distinction between recreational bettors and insiders is endogenised – some
bettors with poor private signal realisations will nonetheless bet in order to realise
recreational utility, while other bettors with strong private signal realisations will
bet in order to monetise the value of their private information. The model is able
to simultaneously account for a number of known empirical regularities
• The favourite longshot bias in the presence of private information and the
reverse favourite longshot bias as documented in lotteries (where private in-
formation is assumed lacking).
• Late betting being more informative
• The occurrence of the favourite longshot bias in both parimutuel and fixed-
odds betting markets.
In addition the theory posits a number of empirically testable observations; a stronger
favourite longshot bias is predicted when there are many bettors, bettors have more
private information, there are fewer outcomes, recreational utility decreases or if the
takeout rate increases.
2.5.4.4 Sequential move
In sequential move games, players commit to their actions in a predefined order.
Previous actions are observable by players. This means that, unlike a simultaneous
move game, players reveal information via their actions to subsequent players and
need to incorporate this effect into their decision making process.
Feeney and King (2001) shows that there is an early mover advantage in a (symmet-
ric information) sequential parimutuel game with two outcomes. Furthermore, early
players may choose outcomes with low ex-ante probabilities in rational anticipation
of later players opting for the more likely outcomes. Ottaviani and Sørensen (2006)
considers a sequential parimutuel betting model and finds that large bettors rely-
ing on common (symmetric) information are incentivised to bet early, while small
bettors with private (asymmetric) information are incentivised to bet late. They mo-
tivate their finding of early betting for large bettors with common information by
drawing an explicit analogy between parimutuel betting and Cournot competition in
quantities, such that there exists a Stackelberg first-mover advantage. Effectively,
by moving early on the basis of common information, a large bettor moves the
CHAPTER 2. LITERATURE REVIEW 28
odds so as to discourage subsequent bettors from placing similarly large bets, and
thereby appropriates to himself a larger share of the total profit accruing to those
in possession of the common information.
In a novel combination of strategic interaction and loss aversion, Qiu (2007) pos-
tulates a sequential move parimutuel betting model in which participants evaluate
outcomes by reference to the prospect theory of Kahneman and Tversky (1979) and
demonstrates the existence of a sub-game perfect Nash equilibrium in such a setting.
2.5.5 The current state of theory
In conclusion, the literature surrounding price formation in parimutuel markets in
general, and the favourite longshot bias in particular, has matured to the point
where there are now dozens of candidate theories of the favourite longshot bias. In
order to progress our understanding in this area it is necessary to focus more effort
on work that has the potential to falsify existing theories.
2.6 Areas for further work
2.6.1 Extending strategic interaction models to encompass
risk sensitive bettors
Virtually all of the recent literature which models bettors within a game theoretic
framework makes the assumption that bettors are risk neutral.22 While this as-
sumption may hold for outsiders betting small amounts for recreational reasons, is
it still plausible in the context of sophisticated insiders betting large amounts? In
the financial literature, it is standard to assume some type of risk aversion on the
part of agents that put a substantial part of their wealth at risk, and is seems natural
to extend this approach also to the sophisticated insiders populating game theoretic
models of parimutuel betting.
2.6.2 Endogenising the distinction between inside and out-
side bettors
There is a growing consensus in the literature that bettors tend to be of two types
– either recreational outsiders or sophisticated insiders. However, this distinction
22The only exception we are aware of is a paper by J. Qiu, ‘Loss aversion and mental accounting:
the favorite longshot bias in parimutuel betting’, Jena Economic Research Papers, (2007), in which
bettors have risk aversion in accordance with prospect theory within the context of a sequential
parimutuel game.
CHAPTER 2. LITERATURE REVIEW 29
is typically exogenously imposed (Ottaviani and Sørensen (2010) is an important
exception that points the way). In many ways, it might be preferable to consider
the distinction between insiders and outsiders in a more nuanced fashion. It is
plausible to assume heterogeneity in the degree of utility different people derive
from the recreational and monetary aspects of betting. Extending theories that
endogenise this distinction would be a step towards a more realistic description of
bettor behaviour.
2.6.3 Deriving additional testable implications from existing
theoretical frameworks
The large number of theoretical contributions that lay claim to explaining the
favourite longshot bias makes it all the more important to bring empirical evidence
to bear on these theories. However, most of the empirical work to date has focused
on aggregate prices determined via parimutuel betting. In order to improve our abil-
ity to test various theories against the data, it is necessary to derive more detailed
empirically testable implications from existing theories. This could be achieved by
elaborating in more detail the various relationships between elements such as the
distribution of bet sizes, the variability of implied market prices and the composition
of bettors implied by different theories.
2.6.4 Creating a richer set of empirical stylised facts
Being able to generate the favourite longshot bias is no longer a sufficient criterion
for deeming a theory plausible. In order to guide the development of theory in the
right direction it will be necessary to rely on a richer set of empirical “stylised facts”.
Documenting additional empirical regularities would help both in falsifying existing
theories and in guiding the development of new theories by providing hints as to the
underlying mechanisms that are at work in parimutuel betting contests.
2.7 Next steps
Some areas were identified for further work in the previous section; in the following
chapters we will address a number of these issues. In chapter 3 we formulate a theory
of parimutuel markets in which we combine strategic interaction with individual
risk preferences. The motivation for this endeavour is more fully explained in the
introduction to the next chapter, but draws on the arguments set out above. Chapter
4 introduces a new dataset containing data at the individual bet level, which we use
CHAPTER 2. LITERATURE REVIEW 30
to investigate the relationship between the amount bet and the level of betting
“success”. We also use this data to test the theory expounded in chapter 3.
Chapter 3
Strategic interaction with risk
preferences
3.1 Introduction
3.1.1 Motivation
Both risk preferences and strategic interaction has been employed to describe price
formation in parimutuel markets (and empirical phenomena such as the favourite
longshot bias). Generally, risk preferences have been framed within a representative
agent framework, which obviously precludes strategic interaction since there can be
no strategic interaction with a single agent. On the other hand, almost all strategic
interaction models assume risk neutrality on the part of bettors. Our contribution
to the literature is to formulate a theoretical model of parimutuel markets that
combine both strategic interaction and risk preferences.
Combining risk aversion and strategic interaction in a single model comes at some
cost in terms of complexity, therefore it is worth discussing the motivation behind
this effort. We are motivated by both theoretical and empirical considerations. From
a theoretical point of view, combining risk preferences and strategic interaction in
a single model of parimutuel markets is a novel endeavour from which new insights
into the interaction of these two effects may emerge.
From an empirical point of view, we note that both risk loving representative agent
models and models of strategic interaction with inside information are capable of
explaining empirically observed regularities such as the favourite longshot bias. How-
ever, these models are based on mutually inconsistent assumptions. To disentangle
simultaneously the empirical implications of risk preferences and strategic interac-
tion requires a theoretical framework that is capable of modelling both. For in-
31
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 32
stance, it may well be that both strategic interaction and risk preferences play a
role in parimutuel price formation. Current theoretical frameworks offer no basis for
testing such an hypothesis, since none combine both strategic interaction and risk
preferences. However a model such as the one we propose holds out the prospect of
testable implications which would allow empirical data to reject this hypothesis, or
offer support for it if not rejected.
A strength of our model is that it not only incorporates risk preferences and strategic
interaction, but that it parameterises those features. Risk preference is modelled by
a real variable α such that negative values of α correspond to progressively higher
levels of risk aversion, while positive values of α implies correspondingly higher levels
of risk love. Setting α equal to zero is equivalent to assuming risk neutrality. This
implies that this model encompass existing models of strategic interaction between
risk neutral insiders as a special case.1 Likewise, the degree of strategic interaction
is modelled by the number of insiders N . Therefore the case of a representative
bettor with risk preferences is a special case of our model with N = 1. Low values
of N > 1 correspond to situations where game theoretical strategic interaction is
an important driver of parimutuel price equilibrium. In the limit as N becomes
large, the model reduces to a framework that assumes infinitesimally small inside
bettors, such that the impact of each insider’s bet on the parimutuel price becomes
negligible. In addition to the above, we also model the common belief of insiders
regarding the probability of horse 1 winning, q, which differs from that implied by the
total amount bet by exogenous recreational outsiders, b0. Finally, we incorporate the
track takeout rate τ . Solving for the symmetric static Nash equilibrium, we obtain
an analytic closed form solution for the optimal insider betting amount. Since the
takeout rate τ is directly observable and the number of insiders N and outsider total
betting b0 may be estimated by a classification rule, this holds out the promise of
being able to jointly estimate the degree of insider risk preference α and the degree
of inside information q directly from the individual bet level data.
The chapter is set out as follows: the remainder of this section introduces our nota-
tion and derives, for exposition, a simple strategic interaction model of risk neutral
insiders corresponding to that of Hurley and McDonough (1995) with zero takeout
rates. In section 3.2 we present our formal model combining strategic interaction
with risk preferences, followed by section 3.3 in which we solve for the static Nash
equilibrium of insider betting. Section 3.4 considers the results obtained in more
detail using a number of simplified models.
1At the cost of a simpler model in other respects. We assume a parimutuel contest with only
two outcomes (ie horses), and we do not model information asymmetries between insiders.
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 33
3.1.2 Notation
Our notation closely follows that of Ottaviani and Sørensen (2008), while also bor-
rowing from the earlier literature, notably Sauer (1998). We assume a parimutuel
contest with two outcomes h ∈ H = {1, 2}. The set of bettors are indexed by
n ∈ N = {0, 1, 2, ..., N}. We use the convention that n > 0 indexes strategic insiders
with common private belief q = Pr(h = 1) not shared with a single representative
outsider bettor denoted by n = 0. A representative exogenous outsider bets a total
of b0(1) > 0 in the first period on horse h = 1 and b0(2) > 0 on horse h = 2, followed
by N ≥ 1 identical insiders betting bn(1) on horse h = 1 in period two. Without loss
of generality, we label the outcomes such that q ≥ b0. The total amount bet in the
race is B =
∑
h∈H
∑
n∈N bn(h). We normalise all amounts so that the total amount
bet by outsiders equal unity, implying b0(1) + b0(2) = 1. For convenience, we drop
the outcome index when referring to horse h = 1, so that b0 = b0(1), bn = bn(1) and
so forth. Insiders are assumed to be financially unconstrained. The track takeout
rate is denoted by τ ∈ [0, 1). The gross payoff of a winning bet is denoted by pi.
The final market probability of horse h winning is given by m(h) = b(h)B .
When we explicitly take into account risk preferences, insiders are assumed to have
utility over net final period wealth2 wn = −bn + pin given by un(wn) = E[wn] +
α(
√
V AR[wn]) with risk preference parameter α ∈ R.3 Insider expectations and
variance are taken with regards to their horse h = 1 winning probability belief q.
3.1.3 Risk neutral model
By way of introducing above the notation, we construct a strategic interaction model
of a horse race with risk neutral insiders which corresponds to that developed by
Hurley and McDonough (1995), but with the track takeout rate τ = 0. We assume
insiders are risk neutral maximisers of expected payoffs. Since all insiders are ex-ante
identical the game is symmetrical, so that in equilibrium b1 = b2 = ... = bN . We
denote by bn the amount individually bet by other insiders for n ≥ 2. The expected
net payoff pi1 of a bet b1 on horse 1 by bettor 1 is then given by
pi1(b1) = −b1 + q
(
b1
b0 + b1 + bn(N − 1)
)
(1 + b1 + bn(N − 1)) (3.1.1)
The first term represents the amount bet by bettor 1. The last term represents the
expected winning payout on horse 1, which is equal to the insiders’ belief regarding
2Since the time between placing a bet and receiving a payout is a matter of days at most, we
ignore the time value of money in our analysis.
3However we will need to impose restrictions on the range of α in order to guarantee strictly
positive betting amounts.
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 34
the probability of horse 1 winning (q = Pr(h = 1)) times the winning payout. The
amount due to insider 1 in case horse 1 wins is equal to the product of insider 1’s
share of all bets laid on horse 1, being b1/(b0 + b1 + bn(N − 1)) and the total betting
pool, which is equal to 1 + b1 + bn(N − 1).
First order conditions for insider 1 maximising her expected net payoff is given by
∂pi1(b1)
∂b1
= 0 (3.1.2)
Which yields
−1 +
q (bn (N − 1) + b1 + 1)
bn (N − 1) + b1 + b0
+
b1q
bn (N − 1) + b1 + b0
−
b1q (bn (N − 1) + b1 + 1)
(bn (N − 1) + b1 + b0)
2 = 0
(3.1.3)
Solving simultaneously for the above and b1 = bn (symmetry condition), we obtain
the unique positive solution to b1
b1 =
√
b20q2N2 − 2b0q2N2 + q2N2 + 2b
2
0q2N − 2q2N − 4b
2
0qN + 4b0qN + b
2
0q2 − 2b0q2 + q2
−2 (q − 1)N2
+
b0qN + qN − 2b0N + b0q − q
−2 (q − 1)N2
(3.1.4)
which by symmetry is also the Nash equilibrium betting amount for all the other
insiders. Note that the optimal insider betting amount in this model is a function
of only three variables:
1. The fraction b0 bet by the outsider on horse 1,
2. The insiders’ belief q regarding the probability of horse 1 winning the race,
and
3. The number of insiders N .
To gain further insight into the result, we let b0 = 0.5 and then plot the amount
bet, b1 against q ranging from 0.51 to 0.99 (on the x-axis) and N in the sequence
{1, 2, ..., 10} in Figure 3.1.1.
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 35
Figure 3.1.1: Amount bet by individual insiders for given q and N
Using the same variables, we also plot the net payout (ie profit) for an individual
insider in Figure 3.1.2. Note that a single monopolistic insider à la Isaacs (1953)
corresponding to N = 1 bets less than would be the case in a duopoly corresponding
to N = 2. This is a consequence of strategic interaction in the case of N > 1.
Competing insiders bet more in anticipation of other insiders’ bets diluting the
payoff on their own bets. As N becomes larger, however, the optimal individual
insider betting amount decreases in order to avoid collective over-betting in the
final pool.
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 36
Figure 3.1.2: Net payout of individual insiders for given q and N
It can be seen that the profitability of insider betting is increasing in the extent
of outsider mispricing (which is proportional to q given that b0 is fixed at 0.5) and
decreasing in the number of insiders N . As the number of insiders N tends towards
infinity, the optimal bet for each insider tends towards zero. This does not mean
that total insider betting disappear; on the contrary, we can show that as N tends
towards infinity, total insider betting is bounded by
lim
N→∞
Nb =
q − b0
1− q
(3.1.5)
In other words, the total amount of insider money bet depends only on the distance
between the their assessment of the winning probability and the winning probability
implied by outsider betting (q − b0) relative to the insiders’ assessment regarding
the probability of their chosen horse losing (1− q).
The final market probability which corresponds to this limit is
m =
b(1)
B
=
b0 + (
q−b0
1−q )
1 + ( q−b01−q )
= q (3.1.6)
In other words, as the number of insiders tends towards infinity, the final market
probability approaches the true probability. The above results are equivalent to
those derived by Hurley and McDonough (1995) (except that we assume a takeout
rate of zero).
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 37
3.2 Model
The model presented here extends that of Hurley and McDonough (1995) by ex-
plicitly incorporating an arbitrary risk preference (denoted by α) on the part of
insiders, while retaining the strategic interaction element. The Hurley and Mc-
Donough (1995) model is therefore a special case of our model with a risk preference
parameter α = 0, indicating risk neutral insiders. The model follows the basic setup
introduced in sub-section 3.1.2 (“Notation”).
3.2.1 Payoff
Let b1 be the amount bet by insider n = 1 on horse h = 1, and let bn be the
amount bet by all the other insiders n > 1 on horse 1. As before, outsider bettors
bet a fraction b0 on horse 1 and 1 − b0 on horse 2, so that total outsider betting
is normalised to unity. We calculate the best response function for insider n = 1.
Since bettors are ex-ante identical, symmetry dictates that all the other insiders will
face the same best response function mutatis mutandis.
All variables are real, with the exception of the number of bettors N , which is a
strictly positive integer. The variables are subject to the constraints below:
b0 > 0, b0 < 1, q > b0, q < 1, τ >= 0, τ < 1, b1 >= 0, bn >= 0 andN ≥ 1
It follows that the total amount B1 bet on horse h = 1 should satisfy
B1 ≡ bn(N − 1) + b1 + b0 > 0
and similarly, the total amount B bet on all horses should satisfy
B ≡ bn(N − 1) + b1 + 1 > 0
The gross winning bet payoff pi is given by
pi ≡



b1
B1 (B) (1− τ) h = 1
0 h = 2
or in expanded form
pi =



b1
bn(N−1)+b1+b0
(bn (N − 1) + b1 + 1) (1− τ) h = 1
0 h = 2
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 38
Net final wealth w is defined as the gross winning bet payoff less the initial bet
w ≡ pi − b1
3.2.2 Expectation
Expected final wealth is taken with respect to the insider’s belief in the winning
probability of horse h = 1, q ≡ Pr(h = 1).
E[w] = q (pi − b1) + (1− q) (−b1)
Substituting in the payoff pi, the expression can be written as
E[w] = q
((
b1
bn (N − 1) + b1 + b0
(bn (N − 1) + b1 + 1) (1− τ)
)
− b1
)
− (1− q) (b1)
3.2.3 Variance
The variance of final wealth is calculated by reference to
V AR[w] = E[w2]− (E[w])2
Substituting in the expressions for w and E[w] we obtain
V AR[w] = q (pi − b1)
2 + (1− q) (−b1)
2 − (q (pi − b1) + (1− q) (−b1))
2
Replacing payoff pi with its definition, this becomes
V AR[w] = q
((
b1
bn(N−1)+b1+b0
(bn (N − 1) + b1 + 1) (1− τ)
)
− b1
)2
+(1− q) (−b1)
2−
(
q
((
b1
bn(N−1)+b1+b0
(bn (N − 1) + b1 + 1) (1− τ)
)
− b1
)
+ (1− q) (−b1)
)2
3.2.4 Utility
We define utility u as a linear combination of expected final wealth and the standard
deviation of final wealth, with the parameter α indicating risk attitude – risk neutral
if α = 0, risk averse if α < 0 and risk loving if α > 0.
u = u1 ≡ E[w] + α
√
V AR[w]
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 39
Substituting in E[w] and V AR[w] we obtain
u = q (pi − b1) + (1− q) (−b1)
+α
√
q (pi − b1)
2 + (1− q) (−b1)
2 − (q (pi − b1) + (1− q) (−b1))
2
Substituting in pi and simplifying yields
u =
b1 (q − qτ − qb1τ − qbnNτ + qbnτ − b0 − b1 − bnN + bn + b1q + qbnN − qbn)
b0 + b1 + bnN − bn
+
b1
(
α (1 + b1 + bnN − bn) (1− τ)
√
q (1− q)
)
b0 + b1 + bnN − bn
3.3 Equilibrium
3.3.1 Best response function (first order conditions)
To find insider 1’s best response (choice of betting amount b1) to other insiders’
best responses (choice of betting amount bn), we formulate first order conditions for
utility maximisation.
∂
∂b1
u = u′(b1) = 0
where
u′(b1) = qG− 1 + q +
α(2q(−b1+pi)G+2(1−q)b1−2(q(−b1+pi)−(1−q)b1)(qG−1+q))
2
√
q(−b1+pi)
2+(1−q)b21−(q(−b1+pi)−(1−q)b1)
2
and G equals the first derivative of final period wealth w = pi − b1 with regards to
b1
G = −1 +
pi
b1
−
pi
B1
+
pi
B
(Recall that pi denotes the gross winning bet payoff such that pi ≡ b1B1 (B) (1− τ) if
horse h = 1 wins, while B denotes the total amount bet B ≡ bn(N − 1) + b1 + 1 > 0
and B1 denotes the total amount bet on horse h = 1 such that B1 ≡ bn(N − 1) +
b1 + b0 > 0).
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 40
Hence a choice of b∗1 = f(bn, N, q, α, τ) such that u
′
1(b
∗
1) = 0 is a best response for
insider 1 to the amount bn individually bet by the other insiders for n ∈ [2..N ].4
3.3.2 Nash equilibrium
We are seeking insider betting amounts in a Nash equilibrium. Since all insiders
are ex-ante identical, the amount they bet will be identical in equilibrium, that is
b1 = bn, where b1 denotes the amount bet by the first insider and bn denotes the
(equal) amount bet by each of the remaining insiders indexed by n ∈ {2..N}.
The symmetric Nash equilibrium is found by simultaneously solving for u′(b1) = 0
and b1 = bn. (Since the insiders are ex-ante identical, the above automatically
implies that u′n(bn) = 0 also for all n ∈ {2..N}. Hence each insider play their best
response to every other insider’s best response, as is required for a Nash equilibrium
to hold.)
We obtain two candidate solutions (denoted s1 and s2) from the above:
s1 = A (C +B)
s2 = A(C −B)
where
A = 1
2N2(
√
1−q(1−τ)q3/2−α(1−τ)q2+α (1−τ)q−
√
q
√
1−q)
B =
(
(1− b0) (1− τ) q
3
2 (1− q)D
) 1
2
C = (1− τ) q
(
α (q − 1)−
√
q
√
1− q
)
(N + b0 − 1 + b0N) + 2
√
q
√
1− qb0N
D = (α2 − 1)Kq
3
2 − 2
√
1− qαKq + (4b0N −Kα2)
√
q + 4α
√
1− qb0N
and
K = (2N − 1 + 2bN + b0 +N2b0 −N2) (1− τ)
However, to be an admissible solution, a candidate solution should be consistent
with a) the initial inequality assumptions and b) second order conditions for utility
maximisation. We show (below) that s2 is the unique positive solution and that it
satisfies second order conditions, so that the optimal insider betting amount is given
by
b∗ = b1 = bn = s2 = A(C −B)
4Subject to second order conditions and the inequalities assumed initially.
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 41
provided that αL < α < αH as shown in sub-section 3.3.3 below.
Note that optimal insider betting b∗ can be expressed more compactly as an implicit
equation in terms of the risk preference parameter α
α =
N2 (1− q (1− τ)) b∗2 + (2Nb0 − (1− τ) (b0 +Nb0 − 1 +N) q) b∗ + b20 − qb0 + qτb0
(1− τ)
√
(1− q)(q) (b∗2N2 + (b0 +Nb0 − 1 +N) b∗ + b0)
The final market probability of horse 1 winning (which we can also interpret as the
final market price of horse 1) is given by
m =
b0 +Nb∗
1 +Nb∗
=
b0 +NA(C −B)
1 +NA(C −B)
where A, B, and C retain their earlier definitions.
3.3.3 Admissibility and second order conditions
By initial assumption it is required that any solution to b1 must be positive. (Bettors
cannot bet a negative amount or “go short”). We now show that s2 > 0 and s1 < 0
subject to restrictions on the range of α.
3.3.3.1 Positive betting amount
First we solve for critical values of α such that b∗ = s2 = 0, obtaining
αL =
b0 − q (1− τ)
√
q
√
1− q (1− τ)
and
αH =
1− q (1− τ)
√
q
√
1− q (1− τ)
Since the denominators are always positive, and because q (1− τ) < 1 (each of the
factors are strictly less than 1), we have αH > 0 and since b0 < 1, we have αH > αL.
Closer examination shows that s2 is positive in the interval (αL, αH) and negative
outside it. Since s2 is continuous in α, that means s2 > 0 for all αL < α < αH .
The critical values of α for s1 are identical to that of s2, but s1 is negative in the
interval (αL, αH) and positive outside it. Analogous to the proof above, that means
s1 < 0 for all αL < α < αH , since s1 is also continuous in α.
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 42
3.3.3.2 Second order conditions
In order to ascertain that b∗ = b1 = s2 is a maximum, we have to verify second order
conditions.
∂2
∂b21
u = u′′(b1) < 0
The second derivative of utility u with respect to b1 is given by
u′′(b1) =
2 (1− τ) (1− b0)
(
q3/2α−
√
qα− q
√
1− q
)
(b0 − bn + bnN)
√
1− q (b0 + b1 + bnN − bn)
3
We are required to show that u′′(b∗) < 0 as required for second order conditions,
subject to the constraints on α to ensure a positive betting amount (αL < α < αH).
Recall that we initially assumed
b0 > 0, b0 < 1, q > b0, q < 1, τ >= 0, τ < 1, b1 >= 0, bn >= 0 andN ≥ 1
Additionally, we have shown that optimal b∗ is positive for α in the range αL < α <
αH . Substituting b∗ for b1 and bn, we obtain
u′′(b∗) =
2 (1− τ) (1− b0)
(
q3/2α−
√
qα− q
√
1− q
)
(b0 + b∗(N − 1))
√
1− q (b0 + b∗N)
3
It follows directly from the initial assumptions above that all factors of u′′(b1) are
strictly positive save for
J =
(
q3/2α−
√
qα− q
√
1− q
)
Hence we need to prove J < 0 in order to establish u′′(b∗) < 0.
Note that
∂
∂α
J = q3/2 + q1/2 > 0 since 0 < b0 < q
Therefore J is strictly increasing in α for all q. Now let JH be J with αH substituted
for α, then
JH = q
3/2αH −
√
qαH − q
√
1− q >
(
q3/2α−
√
qα− q
√
1− q
)
= J
which follows from αH > α and the fact that J is strictly increasing in α.
Substituting αH =
1−q(1−τ)
√
q
√
1−q(1−τ) into JH and simplifying, we obtain
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 43
JH = −
(
2q (1− τ) + 1 + q
√
1− q (1− τ)
)
Since all the factors in 2q(1−τ)+1+q√1−q(1−τ) are strictly positive from our initial assumptions,
it follows that JH < 0 and since J < JH we have J < 0. As we have already
established that J < 0 implies u′′(b∗) < 0 as required, our proof is complete.
3.4 Discussion
3.4.1 Symmetry
This model is inherently symmetrical as there are only two mutually exclusive out-
comes. To illustrate this symmetry, consider an equilibrium solution for insider
betting amounts b∗ = f(N, q, τ, α, b0(1)) where b0(1) < q is assumed. Now, how
do we cope with a situation where b0(1) > q instead? We can deal with this by
re-writing the equation so that b∗ = f(N, 1 − q, τ, α, 1 − b0(1)) is re-interpreted to
mean the amount an insider should bet on horse 2, rather than horse 1. We can
do this since 1 − b0(1) < 1 − q as required if b0(1) > q. But this is really just
equivalent to switching the labels on the horses around, so that horse 1 becomes
horse 2 and vice versa. If we do that, b∗ = f(N, 1 − q, τ, α, 1 − b0(1)) becomes
b∗ = f(N, q, τ, α, b0(1)) as before with b0(1) < q as originally required. By way of
example, let N = 5,q = 0.3, α = 0.1 and τ = 0.15 with b0(1) = 0.7 and q = 0.3.
Since b0(1) = 0.7 ≮ q = 0.3, we cannot directly use the equilibrium solution for the
optimal amount b∗ for insiders to bet on horse 1, which requires b0(1) < q. However,
we can use the formula to calculate the amount that would be bet by insiders on
horse 2. Note that b0(2) = 1− b0(1) = 1− 0.7 = 0.3 and insiders’ belief that horse 2
will win is given by 1− q = 1− 0.3 = 0.7. Now we have b0(2) = 0.3 < (1− q) = 0.7
as is required for the equilibrium insider betting amount, but with the understand-
ing that this insider betting amount now relates to horse 2 rather than horse 1.
We then calculate the equilibrium amount insiders will bet individually on horse 2
as b∗ = f(N, (1 − q), τ, α, b0(2)) = f(5, 0.7, 0.15, 0.1, 0.3) ≈ 0.1480. (Recall that all
amounts are expressed as a fraction of total outsider betting so that b0(1)+b0(2) = 1.
So that means each of the 5 insiders will bet 14.80% of total outsider betting, or
73.98% of total outsider betting for all 5 insiders together.)
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 44
3.4.2 No equilibrium zones
Another feature of our model is that it places restrictions on the range of the risk
preference parameter α in order to enforce positive betting amounts by insiders. We
showed that the amount bet by insiders will be positive only if αL =
b0−q(1−τ)√
q
√
1−q(1−τ) <
α < 1−q(1−τ)√q√1−q(1−τ) = αH . The two critical values are not necessarily symmetric. This
is illustrated in the Figure 3.4.1 below where we plot the critical bounds of α as a
function of q assuming that the takeout rate is zero (τ = 0).
Figure 3.4.1: Critical values of α as a function of q, assuming different values of b0
Note that in plotting the lower bounds, the plots are only shown for q(1− τ) > b0,
which is one of the initial model assumptions q > b0, adjusted for the takeout rate.
As the insider’s belief in the winning probability of horse 1 approaches certainty,
the higher and lower bound on alpha tends towards positive and negative infinity
respectively. In other words, the “better” the quality of the insiders’ inside infor-
mation, the greater the range of risk preferences across which they are willing to
participate. On the other hand, if q is close to b0 indicating that insiders’ beliefs
do not diverge much from outsiders, the lower bound of α approaches zero. This
suggest that when there is no inside information, only risk loving insiders have an
incentive to participate.
We can obtain further intuition regarding this result by solving αL = α =
b0−q(1−τ)√
q
√
1−q(1−τ)
for q. This allows us to frame the lower bound constraint on α as a constraint on
insider’s beliefs q instead. The critical value of q thus obtained is
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 45
qcr =
α2(1− τ) + |α|α
√
α2 (α2τ 2 − 2α2τ − 4b0τ + α2 − 4b20 + 4b0)− 2b0
2(α2 + 1)(1− τ)
Hence, enforcing a positive betting amount requires q > qcr. (Note that the formula
is sensitive to the sign of α, with the square root term being positive for positive
α and negative for negative α due to the presence of the sign function |α|α .) In the
special case of α = 0 (risk neutrality), this constraint simplifies to
qcr =
b0
1− τ
which is the same as the constraint imposed in Ottaviani and Sørensen (2008) dis-
cussing a simplified version of the model set forth in Isaacs (1953).
3.4.3 Insider beliefs
Combining the ideas explored in sub-section 3.4.1 (“Symmetry”) and subs-section
3.4.2 (“No equilibrium zones”), we now consider the individual insider betting amount
on both horses as the winning belief regarding horse 1 ranges from 0 to 1. First, we
consider the case of risk-loving insiders where N = 5, τ = 0.1, b0 = 0.5 and α = 0.5.
Figure 3.4.2: Risk loving insider betting amounts on either horse as a function of q,
with associated utility (utility scaled 20x)
Note that the cross-over point, where insiders change the horse they should opti-
mally bet on, corresponds to b0 = 0.5, the winning probability that outside bettors
associate with horse 1. The switch happens because at q = 0.5 the utility of betting
on horse 1 becomes higher than betting on horse 2 as q increases. Because insiders
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 46
are risk lovers in this example, they never abstain from betting. Now consider the
same example, but with the risk aversion parameter α set to −0.5, so that insiders
are risk averse.
Figure 3.4.3: Risk averse insider betting amounts on either horse as a function of q,
with associated utility (utility scaled 20x)
This paints a very different picture – there is a large range of beliefs where insiders
are not prepared to bet. Due to their risk aversion, insiders require a large amount of
risk adjusted return, in the form of a large difference between the beliefs of outsiders
represented by b0 and their own beliefs, denoted by q, before they are willing to bet.
The examples above suggest that in considering participation by insiders, we can
view inside information and risk preference as substitutes, in the sense that better
inside information can compensate for higher risk aversion and vice versa. Similarly,
with lower risk aversion insiders are willing to participate on the basis of weaker
inside information.
3.4.4 Equilibrium final market probabilities
In equilibrium, combined insider and outsider betting gives rise to a total betting
pool in which B1 = b0 + Nb∗ is bet on horse 1 and B2 = 1 − b0 is bet on horse 2.
Total betting is then simply B = B1 +B2 = 1 +Nb∗. As noted earlier, the implied
market probability of a horse winning is simply the ratio of the total amount bet on
that horse to the total amount bet on all horses. Thus the final market probability
of horse 1 is given by
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 47
m =
b0 +Nb∗
1 +Nb∗
We could substitute in our solution for b∗ in the general case to frame the total
market probability in terms of the original model parameters, but the result is
unwieldy. Instead, we let τ = 0.2 which is close to the typical takeout typically
observed and let N →∞ . The expression we obtain for m then becomes a function
of only two model parameters – α and q.
m =
0.8
(√
q
√
1− q − αq + α +
√
1− q
√
−α2q + q + 2α
√
q
√
1− q + α2
)
q
αq2 − αq − q3/2
√
1− q + q
√
1− q
√
−α2q + q + 2α
√
q
√
1− q + α2 + 2
√
q
√
1− q
Graphically, this is represented in Figure 3.4.4.
Figure 3.4.4: Final market probabilities where N →∞ and τ = 0.2
Interestingly, the amount b0 bet by outsiders does not enter into this formula – this
suggests that inside information (the difference between q and b0) is not the cause
of the divergence between insider beliefs and final market probabilities when there
are a large number of insiders. It would also seem that, with a large number of
insiders, some degree of risk love on the part of insiders are required in order to
induce over-betting on horse 1.
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 48
If instead we consider the special case where N →∞ and q = 12 (a fair game), then
m reduces to
lim
N→∞
m =



1
2 (1− τ) (1 + α) α > −1
0 otherwise
Ignoring the degenerate case where α ≤ −1, we note that the final market probability
of horse 1 depends crucially on the take out rate τ and the risk aversion parameter
α. That is to say that, if the takeout rate is zero and insiders are risk averse, then a
large number of insider bettors will drive the final market probability to their ex-ante
belief in the winning probability of horse 1 of q = 12 , ie there would be no favourite
longshot bias. Inverting the above, we can say that in a race where insiders believe
either horse is equally likely to win, horse 1 will be over-bet if (1− τ) (1 + α) > 1.
Solving for α, this is equivalent to α > 11−τ − 1. Given that the typical takeout rate
is not much above 20%, over-betting on horse 1 would require a risk preference of
α > 11−0.2 − 1 = 0.25; in other words, substantial risk love on the part of insiders.
3.4.5 Special case 1: Risk neutral insiders
We can recover the equilibrium betting amounts for risk neutral insiders as a special
case of the model with α = 0. The equilibrium betting amount is then given by
b∗ =
(1− τ) q3/2 (−b0 + 1−N (b0 + 1)) + 2 b0N
√
q
2N2
(
(1− τ) q3/2 −
√
q
)
−
√
(1− b0) (1− τ) q3/2 (1− q)
((
(N − 1)2 − b0 (N + 1)
2) (1− τ) q3/2 + 4 b0N
√
q
)
2N2
√
1− q
(
(1− τ) q3/2 −
√
q
)
And the final market probability of the winning horse is then given, as before, by
m =
b0 +Nb∗
1 +Nb∗
It can be shown that, in the limit as N →∞, m = (1−τ)q. (This result corresponds
to Proposition 1 of Hurley and McDonough (1995)). In other words, with a large
number of risk neutral insiders the only source of mispricing (relative to insider’s
beliefs) is due to the track takeout rate. With a zero track takeout rate, we obtain
m = q, so that final market probabilities are equal to the common private beliefs of
insiders.
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 49
Interestingly, risk neutral insiders collectively bet a finite amount Nb∗ as N →∞.
lim
N→∞
Nb∗ =
q(1− τ)− b0
1− q(1− τ)
This is despite the fact that each individual insider bet approaches zero as the
number of insiders tends towards infinity.
3.4.6 Special case 2: Risk sensitive representative insider
Similarly, we can reduce the model to a single risk sensitive insider parameterised
by α by letting N = 1 in our model. We find that a single risk sensitive insider will
optimally bet
b∗ = A (C −B)
where
A = 1
2(
√
1−q(1−τ)q3/2−α(1−τ)q2+α (1−τ)q−
√
q
√
1−q)
B =
(
(1− b0) (1− τ) q
3
2 (1− q)D
) 1
2
C = (1− τ) q
(
α (q − 1)−
√
q
√
1− q
)
(2b0) + 2
√
q
√
1− qb0
D = (α2 − 1)Kq
3
2 − 2
√
1− qαKq + (4b0 −Kα2)
√
q + 4α
√
1− qb0
and
K = (4b) (1− τ)
With α = 0 this further simplifies to
b∗ =
q3/2b0 − q3/2b0τ − b0
√
q + q
√
1− b0
√
1− τ
√
b0
√
1− q + qτ
√
q (1− q + qτ)
3.4.7 Special case 3: A simplified model
In order to illustrate the model we now consider a special case in which the number
of free parameters is reduced to two. We do this by fixing
• the number of insiders N = 4. This allows us to preserve the strategic inter-
action element of the model
• the takeout rate τ = 0. This allows us to isolate the impact of risk preferences
and strategic interaction from the impact of taxes and costs.
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 50
• the insider’s common private belief regarding the winning probability of horse
1 is set to q = 12 (a fair game).
Our free parameters are then the risk preference α and the amount of outsider
betting b0. Since q is fixed, we can use b0 to mimic the degree of insider information,
since the difference q − b0 represents the quantum by which the insider’s beliefs
regarding the winning probability of horse 1 differs from that of the outsiders.
Despite adding a number of simplifying assumptions, this model retains strategic
interaction (since N = 4 > 1), parameterised risk preferences (α) and inside infor-
mation (represented by q − b0).
In our simplified model the amount bet by insiders reduces to
b∗ =
b0(11− 5α)− 3α− 3−
√
(1− b0) (α + 1) (9α− 25αb0 + 7b0 + 9)
32(α− 1)
which we plot in Figure 3.4.5 below.
Figure 3.4.5: Insider betting with N = 4, τ = 0 and q = 12
The horizontal grey plane in Figure 3.4.5 represents b∗ = 0, hence the surface above
this plane represents combinations of α and b0 which result in positive insider bet-
ting. Notably, there is a small area in the left hand corner where insider betting
is never positive – this area corresponds to situations where insiders are both risk
averse (α < 0) and have little inside information, that is, b0 is close to q = 12 . This
CHAPTER 3. STRATEGIC INTERACTION WITH RISK PREFERENCES 51
non-participation of insiders is not driven by the takeout rate, since we assumed the
takeout rate to be zero.
3.4.8 Empirical statics
The complexity of the general equilibrium solution in our model means that analysing
the model in terms of analytically derived statics is not feasible5. Instead we take
a more pragmatic approach, in which we use real data to calculate the final market
probabilities of each race according to our theoretical model. We then estimate a
reduced form representation of the final market probability by regressing the model
final market probability directly on the model parameters using OLS. The coeffi-
cients thus obtained can be interpreted as empirical aggregate statics for each model
parameter. The data that this analysis is based on is discussed in more detail in
sub-section 4.2 and sub-section 4.5. A more detailed discussion of our method and
the results we obtain is set out in the appendix in section6.3.
5Allthough statics can be calculated, their analytical complexity precludes meaningful interpre-
tation.
Chapter 4
Empirical tests
4.1 Introduction
In this section we analyse a dataset of 1.6 million horse race bets through the lens
of our theoretical model. In particular, we are interested in what the data has to
say about insider risk preferences in the context of strategic interaction. Virtually
all strategic interaction models of parimutuel betting assume risk neutrality – we
would like to test whether this assumption is borne out in the data and, if so, which
factors drive this result.
4.2 Data description and analysis
The dataset we use is sourced from the Totalisator Agency Board (TAB) of New
Zealand. Under the Racing Act of 2003 the TAB has a statutory monopoly on
all sports betting in New Zealand, including horse racing. The dataset records
the detail of 5,701,478 individual sport bets from 1 August 2006 to 31 July 2009,
including 1,634,808 straight win horse racing bets. All horse race bets administered
by the TAB pays out winnings according to the parimutuel method. The data is
summarised in Table 4.1 below.
Table 4.1: Data summary – horse racing bets from 1 Aug 2006 to 31 July 2009
Variable N Mean Std. Dev. Min Max Sum
Transaction Value 1,634,808 42.205 202.277 5 40,000 68,997,071
Odds 1,634,808 10.002 11.941 1.15 501 n/a
Bet Payout 255,511 249.852 970.026 5.5 75,000 63,839,934
52
CHAPTER 4. EMPIRICAL TESTS 53
A quick calculation shows that 15.63% of bets resulted in a winning payout. How-
ever, total payouts comprised 92.53% of total bets.
4.3 Insiders vs outsiders
4.3.1 Cut-off point
We operationalise the distinction between insiders and outsiders based on the dollar
amount bet. Insiders are defined as those bettors that bet an amount equal to
or above a specified cut-off point (denoted by c) in a single bet. Initially, we use
c = 100 or one hundred New Zealand dollars as our cut-off point. We recognise
that our choice of cut-off point is necessarily arbitrary – we have tried to choose
an amount that corresponds to the idea of “real money” to the average bettor1.
The data suggest that our classification rule may be capturing some of the essence
of the distinction between insiders and outsiders, as insiders are significantly more
successful in their betting – this can be seen from Table 4.2.
Although insiders, as we define them, account for slightly less than 10% of betting
volume by number of bets, they generate more than two-thirds of the betting by
dollar volume. As we can surmise from Table 4.2, they also tend to have a higher
winning ratio than outsiders, whether measured by number of bets or dollars. In
terms of dollar returns, insiders have a winning percentage that is 9.2% higher than
that of outsiders.
4.3.2 Payout ratio and average bet size
Such a finding naturally suggests that higher betting amounts may proxy for bet-
tors with better inside information, which in turn should lead to higher average
payouts. To investigate this idea, we partitioned our dataset into percentiles based
on individual betting amount. For each percentile, we calculated both the average
amount bet and the payout ratio (= total payouts/total amount bet). The results
are shown in Figure 4.3.1 below and appear to support our intuition that higher
betting proxies for better inside information.
1However, we conduct a robustness check in sub-section 4.6.3.2 where we consider three alter-
native cut-off points, namely NZD 50, NZD 200 and NZD 500.
CHAPTER 4. EMPIRICAL TESTS 54
T
ab
le
4.
2:
D
at
a
su
m
m
ar
y
–
in
si
de
rs
vs
ou
ts
id
er
s
V
ar
ia
b
le
N
M
ea
n
S
td
.
D
ev
.
M
in
M
ax
S
u
m
O
u
ts
id
er
s
T
ra
ns
ac
ti
on
V
al
ue
1,
47
4,
66
3
16
.1
1
15
.0
5
5.
00
99
.0
0
23
,7
56
,2
16
O
dd
s
1,
47
4,
66
3
10
.3
6
12
.3
5
1.
15
50
1.
00
n/
a
B
et
P
ay
ou
t
22
0,
84
5
93
.8
5
10
9.
68
5.
50
4,
08
0.
00
20
,7
27
,2
99
W
in
ni
ng
%
(b
et
s)
15
%
W
in
ni
ng
%
(d
ol
la
rs
)
87
%
In
si
d
er
s
T
ra
ns
ac
ti
on
V
al
ue
16
0,
14
5
28
2.
50
59
2.
95
10
0.
00
40
,0
00
.0
0
45
,2
40
,2
02
O
dd
s
16
0,
14
5
6.
68
6.
32
1.
15
20
1.
00
n/
a
B
et
P
ay
ou
t
34
,6
66
1,
24
3.
66
2,
39
0.
87
12
0.
00
75
,0
00
.0
0
43
,1
12
,5
79
W
in
ni
ng
%
(b
et
s)
22
%
W
in
ni
ng
%
(d
ol
la
rs
)
95
%
CHAPTER 4. EMPIRICAL TESTS 55
Figure 4.3.1: Payout ratio (per percentile) against average bet (per percentile)
y = 0.0275Ln(x) + 0.7907
R2 = 0.738
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1 10 100 1,000 10,000
Average bet (per percentile) - log scale
Pa
yo
ut 
ra
tio
 
(pe
r p
er
ce
nti
le)
average bet (per percentile)
Log. (average bet (per percentile))
Interestingly, the payout ratio appears to increase linearly with the log of the average
bet size. In other words, the payout rate is increasing in the average bet amount,
but at a decreasing rate, suggesting the omnipresent law of diminishing returns may
also be at work at the racetrack. An OLS regression in the form
PayoutRate = β0 + β1ln(Avg Bet) + ε
is fitted to the data points. We find that this simple specification explains 73.8%
of the variation in average payout rates. (A table containing the data that Figure
4.3.1 is based on is located in section 6.2 in the appendix)
The above analysis provides support for our notion that insiders self-select by wa-
gering larger amounts than outsiders, and in our subsequent analysis we use the
cut-off point c to classify bettors as insiders and outsiders respectively.
4.4 Race level data
In order to test the final market price predicted by our theory against the data, we
need to collapse the individual level data to race level data – this is because the
final market price is a race level metric. In aggregating the data, we keep track
separately of insider and outsider betting volumes (using our classification rule that
insiders are those betting NZD 100 or more). This results in a dataset with 9,543
individual races. The dataset is summarised in Table 4.3. Variable names suffixed
with “amt” refer to the sum of individual dollar bets – all other variables are counts
of individual bets.
CHAPTER 4. EMPIRICAL TESTS 56
T
ab
le
4.
3:
R
ac
e
le
ve
l
da
ta
su
m
m
ar
y
V
ar
ia
b
le
D
es
cr
ip
ti
on
N
M
ea
n
S
td
.
D
ev
.
M
in
M
ax
S
u
m
cb
et
s
C
ou
nt
of
b
et
s
9,
54
3
17
1.
31
86
2.
59
1
49
,7
40
1,
63
4,
80
7
sb
et
s
Su
m
of
b
et
am
ou
nt
s
9,
54
3
7,
23
0.
05
40
,6
74
.8
8
5
2,
44
8,
52
9
68
,9
96
,3
29
pa
yo
ut
C
ou
nt
of
pa
yo
ut
s
9,
54
3
6,
68
9.
71
39
,7
48
.1
5
0
2,
43
4,
71
6
63
,8
39
,8
74
in
si
de
r
C
ou
nt
of
in
si
de
rs
9,
54
3
16
.7
8
91
.9
5
0
5,
46
9
16
0,
14
4
w
in
ne
r
C
ou
nt
of
w
in
ne
rs
9,
54
3
26
.7
7
12
9.
50
0
6,
97
9
25
5,
51
1
in
w
in
C
ou
nt
of
in
si
de
r
w
in
ne
rs
9,
54
3
3.
63
18
.3
8
0
1,
05
9
34
,6
66
ou
tw
in
C
ou
nt
of
ou
ts
id
er
w
in
ne
rs
9,
54
3
23
.1
4
11
1.
43
0
5,
92
0
22
0,
84
5
in
lo
se
C
ou
nt
of
in
si
de
r
lo
se
rs
9,
54
3
13
.1
5
74
.2
7
0
4,
41
0
12
5,
47
8
ou
tl
os
e
C
ou
nt
of
ou
ts
id
er
lo
se
rs
9,
54
3
13
1.
39
66
4.
06
1
38
,3
51
1,
25
3,
81
8
in
w
in
am
t
Su
m
of
in
si
de
r
w
in
ne
r
b
et
am
ou
nt
s
9,
54
3
1,
18
8.
05
6,
58
3.
18
0
38
5,
46
8
11
,3
37
,5
52
ou
tw
in
am
t
Su
m
of
ou
ts
id
er
w
in
ne
r
b
et
am
ou
nt
s
9,
54
3
42
9.
67
2,
09
9.
96
0
11
3,
81
4
4,
10
0,
35
0
in
lo
se
am
t
Su
m
of
in
si
de
r
lo
se
r
b
et
am
ou
nt
s
9,
54
3
3,
55
2.
61
21
,8
27
.4
1
0
1,
31
8,
80
9
33
,9
02
,5
57
ou
tl
os
ea
m
t
Su
m
of
ou
ts
id
er
lo
se
r
b
et
am
ou
nt
s
9,
54
3
2,
05
9.
72
10
,7
41
.7
0
5
63
0,
43
8
19
,6
55
,8
70
in
pa
yo
ut
am
t
Su
m
of
in
si
de
r
pa
yo
ut
am
ou
nt
s
9,
54
3
4,
51
7.
72
28
,3
07
.2
6
0
1,
80
6,
49
5
43
,1
12
,5
73
ou
tp
ay
ou
ta
m
t
Su
m
of
ou
ts
id
er
pa
yo
ut
am
ou
nt
s
9,
54
3
2,
17
1.
99
11
,7
45
.6
5
0
62
8,
22
1
20
,7
27
,3
01
CHAPTER 4. EMPIRICAL TESTS 57
We will use this data to calculate proxy model parameters in order to formally test
the theoretical model we constructed in chapter 3.
4.5 Operational definition of model parameters
In order to apply our theory to the data, we need to operationally define our theo-
retical model parameters in terms of the race-level dataset in hand. For each race
we determine the theoretical model parameters as follows:
• Number of insiders, N : The count of insider bettors, where any bet of c or
above (c = 100 New Zealand dollars unless otherwise noted) is considered an
insider bet – smaller bets are considered outsider bets.
• Fraction of outsiders betting on horse 1, b0: The sum of outsider betting on the
winning horse expressed as a fraction of total outsider betting in that race.2
• The takeout rate, τ : The takeout rate is set at 15.5% prior to November 2006
and at 14.5% thereafter – this corresponds to the takeout rates applicable to
straight win horse race bets at the time.
• The belief of insiders regarding the probability of horse 1 winning the race, q:
We proxy insiders’ ex-ante belief by the implied market price of their betting
in the data; that is, by the ratio of the amount they bet on the winning horse
to the total amount they bet in the race.
• The final market probability, m¯: We calculate the empirical final market prob-
ability for horse h = 1 as the ratio of the total amount bet on horse 1 to the
total amount bet on all horses.
The remaining risk preference parameter α is our main variable of interest. Using
our theoretical model, we are able to derive an explicit analytical solution for α
based on the parameters above. The theoretical final market probability for horse 1
is given by m = b0+Nb
∗
1+Nb∗ . Substituting in the solution for the optimal insider betting
amount b∗in the above and solving for α yields an expression for the model implied
insider risk preference:
2Our dataset does not allow us to identify the particular horse that a bet relates to. We are
able, however, to identify those bets that relate to the winning horse as those are the bets for which
we observe a payout, which is recorded in our dataset. We therefore associate horse 1 of the theory
with the winning horse in the dataset. There is no additional assumptions implicit in this, as we
are concerned with the final market probability of the winning horse, which is determined before
it is known that a particular horse is in fact the winner. In other words, we are using the winning
payout to identify a single horse in the dataset which we identify with horse 1 in the theory.
CHAPTER 4. EMPIRICAL TESTS 58
α =
√
1− q ((b0N + q − qτ −N) m¯2 + (τ − 1) q (b0N + b0 −N + 1) m¯+ qb0 − qτb0)
√
q (τ − 1 ) (q − 1) (m¯b0 − b0 + m¯ −Nm¯ + m¯Nb0 − m¯2)
Thus we calculate the model implied risk preference α using the empirically observed
values of N, b0, τ, q and m¯ for each race.
4.6 Empirical tests and results
4.6.1 Overview
We wish to test the hypothesis that insiders are individually risk neutral while
allowing for strategic interaction within a parimutuel setting (our null hypothesis
– H0 : α = 0). The alternative hypothesis is that insiders are either risk loving or
risk averse (Ha : α 6= 0). It is important to note that our hypothesis concerns risk
preferences at the individual agent level – this is in contrast with earlier work (eg.
Ali (1977)) in which the risk preferences of a representative agent is the subject of
investigation. The second point to emphasise is that our hypothesis applies to a
strategic interaction setting – that is to say, a situation in which individual payoffs
are determined according to a game theoretic equilibrium. We take three different
approaches to analysing the data:
The first approach is also the most direct. We calculate the implied risk sensitivity
of insiders for each race, using c = 100 as the cut-off point between insiders and
outsiders. Realising that the cut-off point is somewhat ad-hoc, we then proceed
to calculate implied risk sensitivities for cut-off points ranging from NZD 10 to
NZD 500. This allows us to measure the effect of the cut-off point on implied risk
sensitivity.
In our second approach, we fix the beliefs of insiders at a fixed increment ∆q above
that of outsiders; that is, we let q = b0 + ∆q. We then proceed to calculate implied
risk sensitivities for ∆q ranging from 0 to 0.2 in increments of 0.01. This allows us
to measure how risk sensitivity changes as our assumptions regarding insider beliefs
relative to that of outsiders (ie ∆q) changes.
The third approach takes the final market probability m as the starting point. By
assuming different values for risk sensitivity (α) and insider beliefs relative to out-
siders (∆q), we can calculate the theoretical final market probability predicted by
our model, mα,q, as well as that observed directly from betting data, m¯. By re-
gressing m¯ on mα,q, we can identify those combinations of risk sensitivity (α) and
CHAPTER 4. EMPIRICAL TESTS 59
inside information (∆q) that yields theoretical predictions that best corresponds
with empirical observations.
4.6.2 Direct estimation of risk sensitivity
Implied risk sensitivity is calculated for each race as set out in sub-section 4.5 above,
assuming a cut-off point of c = 100 between insiders and outsiders. We restrict
ourselves to those races that conform with the assumptions of our theoretical model3
and find that our implied risk sensitivity α is approximately normally distributed
with a mean of 0.0188 and a standard deviation of 0.1894. A histogram of implied
risk sensitivities are shown in Figure 4.6.1 below.
Figure 4.6.1: Histogram of implied α, at an insider/outsider cut-off point of c = 100
(Model compliant observations only)
0
1
2
3
4
5
D
en
sit
y
-1 0 1 2
alphadata
The mean implied risk sensitivity of 0.0188 indicates slight risk love on the part
of insider bettors – a dollar of standard deviation is valued at 1.88% of a dollar of
expected return. Although the mean implied risk sensitivity is statistically signifi-
cant4, it is not economically significant given that it amounts to less than 2% of the
3In particular, the theoretical model requires q > b0, which holds for 3,646 of the 5,654 races
for which α can be calculated.
4The t-statistic for the mean risk sensitivity against a null hypothesis of zero is given by
α
stdev/
√
n =
0.0188
0.1894/
√
3646
= 5.998, which indicates that the mean is statistically significant at a
confidence level of 1% or above.
CHAPTER 4. EMPIRICAL TESTS 60
expected net payout of the insider.
We noted earlier the rather ad-hoc nature our decision to use a bet size of NZD
100 as the cut-off point between insiders and outsiders. It would be interesting to
see how implied risk sensitivity varies with different choices of c as the cut-off point
between insiders and outsiders. To address this, we recalculated our implied risk
sensitivities for different cut-off points, with c ranging from NZD 10 to NZD 500
in NZD 10 increments. The result of this analysis, which includes is illustrated in
Figure 4.6.2 below.
Figure 4.6.2: Mean implied risk sensitivity αc vs insider/outsider cut-off point c













        	 	



	




	





 





 
We can see that implied risk sensitivity is influenced by the cut-off point we select.
In particular, implied risk sensitivity tends to decrease as we impose higher cut-off
points, but at a decreasing level. This pattern is consistent with smaller, risk-loving
recreational bettors and larger, relatively more risk averse, sophisticated insider
bettors. As we move the cut-off point higher (and to the right in the graph), we
are capturing fewer recreational risk-loving bettors in our definition of insiders and
consequently the implied risk-love of our insiders starts to decrease.
4.6.3 Estimation conditional on specified insider beliefs
4.6.3.1 Method
As before, we wish to test the hypothesis that insiders are individually risk neutral.
In contrast with the approach taken in sub-section 4.6.2, we calculate implied risk
sensitivity based on an assumed level of insider beliefs. More exactly, we set the
CHAPTER 4. EMPIRICAL TESTS 61
level of insider beliefs equal to that of outsiders plus a positive adjustment ∆q, that
is q = b0 + ∆q. As before, b0 is calculated directly from the data. The quantity
∆q is formally the difference in belief between insiders and outsiders regarding the
probability of horse 1 winning the race. Informally, we can think of ∆q as an
indication of the inside information insiders have relative to outsiders; this is based
on the idea that if insiders have a stronger belief in the winning probability of horse
1, then this must be due to their having better information. We will often refer to ∆q
as “inside information” hereafter; however keep in mind that formally ∆q = q − b0.
This approach suffers from the drawback that one must assume some value of ∆q –
hence, each estimate of risk sensitivity α will be conditional on the assumed value of
∆q (to emphasise this point we will denote the implied risk sensitivity conditional
on ∆q as α∆q). The benefit of this approach is that it allows us to analyse more
directly the relationship between inside information and risk preferences. Another
benefit is that we avoid some of the data issues in the previous section – our analysis
can utilise a much larger set of races since we avoid problems posed by races where
q < b0. We assume a cut-off point of c = 100 in classifying insiders and outsiders,
unless otherwise noted.
4.6.3.2 Results
Using the formula for α set out in sub-section 4.5 above, we calculate the implied
risk sensitivity α∆q for each race in the race level dataset for a range of values of
q = b0 + ∆q where ∆q ∈ {0, 0.01, ..., 0.19, 0.20}. The results are shown in Table 4.4.
CHAPTER 4. EMPIRICAL TESTS 62
Table 4.4: Implied risk sensitivity α∆q for a range of ∆q
Implied risk sensitivity α∆q
∆q N Mean Std. Dev. Min Max t-Statistic*
0.0000 8577 0.2338 0.6450 -1.4557 8.6528 33.57
0.0100 8685 0.2243 0.7689 -1.5395 11.6543 27.19
0.0200 8685 0.1743 0.6543 -1.6335 8.2114 24.83
0.0300 8685 0.1361 0.6040 -1.7403 6.6804 21.00
0.0400 8684 0.1029 0.5739 -2.3319 5.7644 16.71
0.0500 8682 0.0737 0.5536 -2.6068 5.1370 12.40
0.0600 8678 0.0449 0.5430 -7.6602 4.6722 7.70
0.0700 8674 0.0197 0.5262 -2.3983 4.3096 3.49
0.0800 8672 -0.0039 0.5316 -2.6770 11.5234 -0.68
0.0900 8666 -0.0293 0.5164 -3.2297 8.2036 -5.28
0.1000 8657 -0.0537 0.5012 -3.6322 3.5653 -9.97
0.1100 8653 -0.0770 0.5004 -4.6551 3.3865 -14.31
0.1200 8647 -0.1006 0.5150 -13.1717 3.2299 -18.16
0.1300 8633 -0.1206 0.4867 -3.9475 3.0912 -23.02
0.1400 8627 -0.1449 0.5359 -21.0616 3.2980 -25.11
0.1500 8617 -0.1641 0.4808 -6.7713 2.8554 -31.68
0.1600 8615 -0.1849 0.4812 -5.6823 2.7539 -35.66
0.1700 8607 -0.2034 0.4737 -3.2428 2.7528 -39.84
0.1800 8599 -0.2251 0.4940 -11.2598 6.4871 -42.25
0.1900 8582 -0.2477 0.4840 -6.1525 5.2286 -47.41
0.2000 8569 -0.2687 0.4988 -14.4026 2.4240 -49.87
* Mean of α∆q vs null hypothesis of α∆q = 0; t-stat=
α∆q
stdev/
√
n
From the table above we can see that risk sensitivity is strictly decreasing in inside
information (∆q), with risk neutrality (α∆q = 0) corresponding to ∆q = 0.08. This
suggests that inside information and risk preference are in a sense substitutes (an
intuition we first hinted at in sub-section 3.4.3 when we discussed insider beliefs).
The actual values of α∆q for individual races are quite widely distributed, as can be
seen from the associated standard deviations. The last column in the table shows
the t-statistic calculated for the mean of α∆q against the null hypothesis of α∆q = 0
– this indicates how statistically different from zero our estimates of the mean of
α∆q are. The mean risk sensitivity α∆q is statistically different from zero at the 5%
confidence level for each level of assumed inside information ∆q save one: conditional
on ∆q = 0.08, the implied risk sensitivity is statistically indistinguishable from
zero. In other words, we cannot reject the hypothesis that insiders are risk neutral,
conditional on ∆q = 0.08. It is intriguing to note the close correspondence between
∆q = 0.08 = 8% and 9.2%, the degree by which insiders outperform outsiders in the
data (see sub-section 4.3.1). We may interpret this as providing tentative support
CHAPTER 4. EMPIRICAL TESTS 63
for the notion that insiders are close to risk neutral on average.
In Figure 4.6.3 we plot the mean of model implied risk preferences αq against ∆q
based on the data presented in 4.4.
Figure 4.6.3: Mean of model implied risk sensitivity αq as a function of ∆q
The strong linear relationship between the mean risk sensitivity α∆q and ∆q is
striking. The association between risk aversion and inside information suggested by
the data is given by the linear relationship
α∆q = −2.4911(∆q) + 0.2103
We could interpret this line as an indifference curve between higher risk love (α∆q)
and better inside information (∆q). A cursory examination of the formula we use
to calculate α∆q does not immediately suggest that we should expect such a linear
relationship only on the basis of the theoretical model. This suggests that this
relationship is a feature of the data, rather than of our theoretical model.
It is reasonable to ask whether our results are driven by our choice of NZD 100 as
the cut-off point between insiders and outsiders. To address this concern, we re-did
the above analysis using three additional cut-off points, namely NZD 50, NZD 200
and NZD 500, in addition to our assumed cut-off point of NZD 100. We plot the
results in Figure 4.6.4.
CHAPTER 4. EMPIRICAL TESTS 64
Figure 4.6.4: Estimated risk sensitivity αq against ∆q for different insider bet cut-off
points












    	     	 






	
















We find that the linear relationship described above remains intact. We also find that
for the lower cut-off point (NZD 50) risk neutrality is associated with a slightly higher
∆q of 0.085, and similarly, for the higher cut-off points (NZD 200 and NZD 500) risk
neutrality is associated with a slightly lower ∆q of 0.07. Hence the actual level of
inside information (∆q) that corresponds to implied risk neutrality is sensitive to the
cut-off point between insiders and outsiders. The pattern we observe suggests that
raising the cut-off point lowers the degree of inside information that is consistent
with risk neutrality. We can only speculate why this appears to be the case –
one possible explanation might be that recreational bettors bet smaller amounts of
money and are also on average risk loving. If we lower the cut-off point from NZD
100 to NZD 50, we may be capturing a larger number of recreational risk-loving
bettors in our definition of insiders, thereby resulting in a higher (more risk loving)
implied risk sensitivity α at each level of inside information ∆q for such insiders.
We leave this intriguing possibility as a topic for further research.
4.6.4 Model vs empirical final market probabilities
4.6.4.1 Method
We first calculate the empirical (observed) final market probability, which we denote
by m¯. We then proceed to calculate the theoretical final market probability accord-
ing to our model using the model parameters N, b0, τ as determined above, along
CHAPTER 4. EMPIRICAL TESTS 65
with assumed values of α and q to arrive at a theoretical final market probability
m(α,q). (We explicitly subscript m(α,q) to remind ourselves that this theoretically
calculated value is dependent on the assumed values for α and q). By choosing
a suitable range of assumed values for α and q, we proceed to generate a set of
candidate theoretical solutions such that:
M =
{
m(α,q=b0+∆q)| α ∈ [−0.5,−0.48, .. , 0.48, 0.50], ∆q ∈ {0, 0.05, 0.1, 0.15, 0.2}
}
.
We would like to test which combinations of α and q generate a theoretical final
market probability m(α,q) which is most congruent with the empirically observed
final market probability m¯. To do this we estimate an OLS equation in the form
m¯ = γm(α,q) + ε
which yields an estimate of the empirical final market probability mˆ = γm(α,q) for
each element of M . (Note that we do not include an intercept in our model). If
some m(α,q) is congruent with m¯ then we would expect to obtain γ = 1 in the
above regression. This allows us to test all of the candidate theoretical solutions in
M and to identify for each q the corresponding α for which γ = 1 in a statistical
sense, or equivalently, m¯ = m(α,q) + ε. Let Mf ⊂ M be those elements of M for
which the regression coefficient γ is statistically closest to 1 for a given choice of
q. We can think of Mf as the set of feasible theoretical solutions, in the sense
that they accord most closely with the observed final market probability m¯ in a
statistical sense. Equivalently, we can interpret Mf as a set of combinations of α
and q that corresponds most closely with m¯. That is, {(α, q) |m(α,q) ∈Mf} is the set
of combinations of α and q which are statistically most congruent with the observed
pattern of final market probabilities.
4.6.4.2 Results
The results of the above analysis is presented in Figure 4.6.5 below. The graph
plots the regression coefficient γ against different assumed values of risk sensitivity
α. Each of the different lines in the graph is associated with a level of q determined
according to the key shown in the graph. Each point at which a line intercepts the
horizontal grey line y = 1 shows a combination of risk preference α (according to
the x-axis) and insider beliefs q (according to the specific line) which resulted in a
regression coefficient of γ = 1. The intersection points are, in a sense, combinations
of α and q jointly estimated from the data.
CHAPTER 4. EMPIRICAL TESTS 66
Figure 4.6.5: Regression coefficient γ against risk sensitivity α for different q
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
-
0.5
0
-
0.4
0
-
0.3
0
-
0.2
0
-
0.1
0
0.0
0
0.1
0
0.2
0
0.3
0
0.4
0
0.5
0
risk  sensitivity - alpha
Re
gre
ss
ion
 
co
eff
icie
nt
q = b[0]
q = b[0] + 0.05
q = b[0] + 0.10
q = b[0] + 0.15
q = b[0] + 0.20
coefficient = 1
It is intriguing to note that the middle line, which corresponds to q = b0 + 0.1,
intersects with the grey line (γ = 1) very close to the point where α = 0. We could
interpret this as evidence that the data fails to reject our hypothesis that insiders
are risk neutral, or α = 0. This interpretation depends, of course, on the admittedly
strong assumption that q = b0 + 0.1. More generally, this would suggest that where
insiders have better inside information (say q = b0 +0.2) they are willing to be more
risk loving (intercept with y = 1 at α = 0.3) and conversely, where insiders have
little inside information (say q = b0) they will tend to be more risk averse (intercept
with y = 1 at α = −0.2).
For each choice of q we present below in Table 4.5 the choice of α for which the
regression did not reject γ = 1. We present both in-sample results (based on 80% of
the data randomly chosen) and out-of-sample results (based on the remaining 20%
of the data); the out-of-sample results show the regression of predicted final market
probabilities using in-sample estimated coefficients (mˆIS = γISm(α,q)) against the
out-of-sample empirical final market probabilities (m¯).
CHAPTER 4. EMPIRICAL TESTS 67
Table 4.5: Best fit regression results for α and q
q b0 b0 + 0.05 b0 + 0.10 b0 + 0.15 b0 + 0.20
α 0.28 0.14 0.02 -0.1 -0.2
In-sample (m¯ = γISm(α,q) + ε)
γIS 0.999 1.002 0.997 1.003 1.009
t-Stat 176 175 170 167 166
Lower CI (5%) 0.988 0.990 0.985 0.991 0.997
Upper CI (5%) 1.010 1.013 1.008 1.015 1.020
Adj R-square 0.8191 0.8048 0.7962 0.7900 0.7893
N 6815 7440 7423 7391 7350
Out-of-sample (m¯ = ηOSmˆIS + ε, where mˆIS = γISm(α,q))
ηOS 0.997 0.997 0.994 1.003 1.002
t-Stat 88 88 84 83 83
Lower CI (5%) 0.975 0.974 0.971 0.979 0.978
Upper CI (5%) 1.020 1.019 1.018 1.027 1.025
Adj R-square 0.8131 0.7992 0.7868 0.7839 0.7847
N 1760 1924 1916 1908 1901
The out-of-sample results provide some comfort that the results we obtained are not
the result of data mining. Again, the above results are consistent with the idea that
risk preference and inside information are “substitute goods” from the perspective
of insiders.
Chapter 5
Conclusion
A broad survey of the literature dealing with parimutuel betting was conducted.
Based on this survey, we identified a gap in the existing literature – the lack of
any theoretical models that combine both individual risk preferences on the part of
bettors and the strategic interaction between bettors. We construct a theoretical
model of parimutuel betting which includes both strategic interaction between in-
siders and arbitrary common risk preferences of individual insiders. By solving this
model for the static symmetric Nash equilibrium in continuous strategies, we obtain
a closed-form solution to the optimal insider betting amount and the associated fi-
nal market probabilities. Within the context of this model, we find that individual
insider risk preferences and insider inside information operate as close substitutes,
in the sense that a decrease in risk appetite can be compensated for by an increase
in inside information and vice versa.
Using a unique dataset of 1.6 million individual horse race bets, we document a
positive empirical association (not previously reported in the literature) between
the log of average bet size and average payout rates in bet level data aggregated in
percentiles by bet size. On the basis of this regularity, we operationalise the dis-
tinction between insider and outsider bettors by classifying all bets of NZD 100 or
above as insider bets and the remainder as outsider bets. In so doing, we are able
to construct a race-level dataset containing all of the model parameters with the ex-
ception of risk preference and inside information. This allows us to formally test our
hypothesis that insiders are risk neutral within the context of our theoretical model.
We are unable to reject the hypothesis that insiders are risk neutral, conditional
on assuming a degree of inside information that corresponds to the degree to which
insiders actually outperform outsiders in our dataset. This provides some empirical
support to existing strategic interaction models of parimutuel markets that assume
risk neutrality on the part of informed bettors.
68
Bibliography
Adams, B.R., Rusco, F.W. and Walls, W.D., ‘Profesional bettors, odds arbitrage
competition and betting market equilibrium’, Journal of Singapore Economic
Review , 47 (2002), pp. 11–27.
Ali, M.M., ‘Probability and utility estimates for racetrack bettors’, The Journal of
Political Economy , 85 (1977):4, pp. 803–815.
Asch, P., Malkiel, B.G. and Quandt, R.E., ‘Racetrack betting and informed behav-
ior’, Journal of Financial Economics , 10 (1982):2, pp. 187–194.
Busche, K. and Hall, C.D., ‘An exception to the risk preference anomaly’, Journal
of business , 61 (1988):3, pp. 337–346.
Busche, K. and Walls, W.D., ‘Decision costs and betting market efficiency’, Ratio-
nality and Society , 12 (2000):4, p. 477.
Chadha, S. and Quandt, R.E., ‘Betting bias and market equilibrium in racetrack
betting’, Applied Financial Economics , 6 (1996):3, pp. 287–292.
Cochrane, J.H., Asset pricing, (Princeton University Press Princeton, NJ, 2001).
Coleman, L., ‘New light on the longshot bias’, Applied Economics , 36 (2004):4,
pp. 315–326.
Conlisk, J., ‘The utility of gambling’, Journal of Risk and Uncertainty , 6 (1993):3,
pp. 255–275.
Eisenberg, E. and Gale, D., ‘Consensus of subjective probabilities: The pari-mutuel
method’, The Annals of Mathematical Statistics , 30 (1959):1, pp. 165–168.
Fama, E.F., ‘Efficient capital markets: A review of theory and empirical work’,
Journal of finance, (1970), pp. 383–417.
Fama, E.F., ‘Efficient capital markets: II’, Journal of finance, 46 (1991):5, pp. 1575–
1617.
69
Bibliography 70
Feeney, R. and King, S.P., ‘Sequential parimutuel games’, Economics Letters , 72
(2001):2, pp. 165–173.
Gandar, J.M., Zuber, R.A. and Dare, W.H., ‘The search for informed traders in the
totals betting market for national basketball association games’, Journal of
Sports Economics , 1 (2000):2, p. 177.
Golec, J. and Tamarkin, M., ‘Bettors love skewness, not risk, at the horse track’,
Journal of Political Economy , 106 (1998):1, pp. 205–225.
Gramm, M. and McKinney, C.N., ‘The effect of late money on betting market
efficiency’, Applied Economics Letters , 16 (2009):4, pp. 369–372.
Gramm, M. and Owens, D.H., ‘Determinants of betting market efficiency’, Applied
Economics Letters , 12 (2005):3, pp. 181–185.
Griffith, RM, ‘Odds adjustments by American horse-race bettors’, The American
Journal of Psychology , 62 (1949):2, pp. 290–294.
Hausch, D.B. and Ziemba, W.T., Efficiency of sports and lottery betting markets,
(Elsevier, 1995), pp. 545–580.
Hausch, D.B., Ziemba, W.T. and Rubinstein, M., ‘Efficiency of the market for race-
track betting’, Management science, (1981), pp. 1435–1452.
Hurley, W. and McDonough, L., ‘A note on the Hayek hypothesis and the favorite-
longshot bias in parimutuel betting’, The American economic review , (1995),
pp. 949–955.
Isaacs, R., ‘Optimal horse race bets’, American Mathematical Monthly , 60 (1953):5,
pp. 310–315.
Kahneman, D. and Tversky, A., ‘Prospect theory: An analysis of decision under risk’,
Econometrica: Journal of the Econometric Society , 47 (1979):2, pp. 263–291.
Koessler, F., Noussair, C. and Ziegelmeyer, A., ‘Parimutuel betting under asymmet-
ric information’, Journal of mathematical Economics , 44 (2008):7-8, pp. 733–
744.
Levitt, S.D., ‘Why are gambling markets organised so differently from financial
markets?’ Economic Journal , 114 (2004), pp. 223–246.
McGlothlin, W.H., ‘Stability of choices among uncertain alternatives’, The American
Journal of Psychology , 69 (1956):4, pp. 604–615.
Ottaviani, M. and Sørensen, P.N., Parimutuel versus fixed-odds markets, 2005.
Bibliography 71
Ottaviani, M. and Sørensen, P.N., The Timing of Parimutuel Bets, 2006.
Ottaviani, M. and Sørensen, P.N., The Favorite-Longshot Bias: An Overview of the
Main Explanations, (Citeseer, 2008), pp. 83–101.
Ottaviani, M. and Sørensen, P.N., ‘Surprised by the Parimutuel Odds?’ American
Economic Review , 99 (2009):5, pp. 2129–2134.
Ottaviani, M. and Sørensen, P.N., ‘Noise, Information, and the Favorite-Longshot
Bias in Parimutuel Predictions’, American Economic Journal: Microeco-
nomics , 2 (2010):1, pp. 58–85.
Qiu, J., ‘Loss aversion and mental accounting: the favorite longshot bias in
parimutuel betting’, Jena Economic Research Papers , (2007), p. 017.
Quandt, R.E., ‘Betting and equilibrium’, The Quarterly Journal of Economics , 101
(1986):1, pp. 201–207.
Sauer, R.D., ‘The economics of wagering markets’, Journal of Economic Literature,
36 (1998):4, pp. 2021–2064.
Sauer, R.D., ‘The state of research on markets for sports betting and suggested
future directions’, Journal of Economics and Finance, 29 (2005):3, pp. 416–
426.
Shin, H.S., ‘Optimal betting odds against insider traders’, The Economic Journal ,
101 (1991):408, pp. 1179–1185.
Shin, H.S., ‘Prices of state contingent claims with insider traders, and the favourite-
longshot bias’, The Economic Journal , 102 (1992):411, pp. 426–435.
Snowberg, E. and Wolfers, J., ‘Explaining the favourite-longshot bias: is it risk-love
or misperceptions?’ NBER Working Paper April (2010).
Thaler, R.H. and Ziemba, W.T., ‘Anomalies: Parimutuel betting markets: Race-
tracks and lotteries’, The Journal of Economic Perspectives , 2 (1988):2,
pp. 161–174.
VaughanWilliams, L., ‘Information efficiency in betting markets: A survey’, Bulletin
of Economic Research, 51 (1999):1, pp. 1–39.
Walls, W.D. and Busche, K., ‘Broken odds and the favourite-longshot bias in
parimutuel betting: a direct test’, Applied Economics Letters , 10 (2003):5,
pp. 311–314.
Bibliography 72
Watanabe, T., Nonoyama, H. and Mori, M., ‘A model of a general parimutuel
system: characterizations and equilibrium selection’, International Journal of
Game Theory , 23 (1994):3, pp. 237–260.
Weitzman, M., ‘Utility analysis and group behavior: An empirical study’, The Jour-
nal of Political Economy , 73 (1965):1, pp. 18–26.
Chapter 6
Appendix
6.1 Maple code to verify derivation of equilibrium
insider betting amounts
r e s t a r t ; _EnvExplicit := true ; p r i n t l e v e l :=1;_EnvTry := normal ;
assume (b [ 0 ] > 0 , b [ 0 ] < 1 , b [ 0 ] : : r ea l , q > b [ 0 ] , q < 1 , q : : r ea l ,
tau >= 0 , tau < 1 , tau : : r ea l , b [ 1 ] >= 0 , b [ 1 ] : : r ea l ,
b [ n ] >= 0 , b [ n ] : : r ea l , N >= 2 , N : : i n t ege r ,
b [0 ]+b [1 ]+b [ n ]∗N−b [ n]>0 , 1+b [1 ]+b [ n ]∗N−b [ n]>0) ;
p i := ( b [ 1 ] / ( b [0 ]+b [1 ]+(N−1)∗b [ n ] ) ) ∗
(1+(b [ 1 ] ) +(N−1)∗(b [ n ] ) ) ∗ (1−tau ) ;
E_u := q∗(−b [1 ]+ pi )+(1−q )∗(−b [ 1 ] ) ;
VAR_u := q∗((−b [1 ]+ pi ) ^2)+(1−q ) ∗((−b [ 1 ] ) ^2) − ( (E_u) ^2) ;
u := E_u + alpha ∗(VAR_u^(1/2) ) ;
d_u := d i f f (u , b [ 1 ] ) ;
foc_b:=d_u=0; foc_b:= s imp l i f y ( foc_b ) ;
d2_u := d i f f (u , b [ 1 ] $2 ) ;
soc_b:=d2_u ; soc_b:= s imp l i f y ( soc_b ) ;
s o l := so l v e ( [ foc_b , b [ n]=b [ 1 ] ] , [ b [ 1 ] , b [ n ] ] ) ;
b_star1 := s imp l i f y ( rhs ( s o l [ 1 ] [ 1 ] ) ) ;
b_star2 := s imp l i f y ( rhs ( s o l [ 2 ] [ 1 ] ) ) ;
b_star :=b_star2 ;
# s o l v e s f o c ( b_star )=0, meets b_star > 0 and soc ( b_star ) < 0
m:=(b [0 ]+N∗b_star ) /(1+N∗b_star ) ; m:= s imp l i f y (m) ;
m i sp r i c i ng := s imp l i f y (m−q ) ;
73
CHAPTER 6. APPENDIX 74
6.2 Average payout rate and average bet size by
percentile
Table 6.1 below is a result of the analysis described in sub-section 4.3. Note that
although the analysis is based on a percentile partition of the bet amount data, we
do not have exactly 100 data points as one might expect – this is because of the large
amount of small, “round” bets such as NZD 5 and NZD 10, each of which accounts
for much more than one per cent of the total betting volume. Since there is no point
in having multiple data points relating to NZD 5 bets, we group bets of the same
size in the same percentile grouping, even if this results in fewer data points.
CHAPTER 6. APPENDIX 75
T
ab
le
6.
1:
B
et
s
an
d
pa
yo
ut
s
by
p
er
ce
nt
ile
P
er
ce
n
ti
le
N
T
ot
al
b
et
am
t
T
ot
al
p
ay
ou
t
am
t
M
in
im
u
m
b
et
M
ax
im
u
m
b
et
A
ve
ra
ge
b
et
P
ay
ou
t
ra
ti
o
1
46
1,
07
6
2,
30
5,
38
0
1,
89
3,
97
9
5
5
5.
00
0.
82
15
29
34
,8
53
20
9,
11
8
18
1,
40
4
6
6
6.
00
0.
86
75
31
22
,0
85
15
4,
59
5
13
0,
70
5
7
7
7.
00
0.
84
55
32
21
,6
64
17
3,
31
2
14
6,
40
9
8
8
8.
00
0.
84
48
34
36
6,
84
4
3,
65
8,
18
8
3,
10
3,
87
9
9
10
9.
97
0.
84
85
56
16
,9
34
19
6,
80
2
16
5,
54
2
11
12
11
.6
2
0.
84
12
57
11
,6
84
15
7,
91
4
13
8,
30
0
13
14
13
.5
2
0.
87
58
58
40
,1
45
60
2,
17
5
53
8,
57
4
15
15
15
.0
0
0.
89
44
60
8,
03
8
13
1,
92
4
11
6,
13
1
16
17
16
.4
1
0.
88
03
61
20
0,
46
9
3,
99
8,
54
0
3,
47
0,
05
5
18
20
19
.9
5
0.
86
78
73
10
,1
30
22
8,
14
5
19
5,
78
8
21
24
22
.5
2
0.
85
82
74
27
,1
68
67
9,
20
0
60
4,
34
1
25
25
25
.0
0
0.
88
98
75
5,
44
1
14
6,
73
6
12
6,
88
5
26
28
26
.9
7
0.
86
47
76
69
,0
09
2,
06
9,
18
0
1,
79
9,
63
7
29
30
29
.9
8
0.
86
97
80
12
,7
01
43
5,
78
1
38
4,
66
9
31
37
34
.3
1
0.
88
27
81
38
,5
23
1,
53
8,
30
0
1,
36
9,
29
9
38
40
39
.9
3
0.
89
01
83
89
,7
64
4,
45
4,
15
8
3,
94
4,
86
4
41
50
49
.6
2
0.
88
57
88
2,
87
0
15
4,
26
3
13
0,
91
5
51
55
53
.7
5
0.
84
87
89
15
,9
29
96
0,
11
9
85
4,
43
7
56
65
60
.2
7
0.
88
99
90
16
,0
93
1,
21
0,
08
9
1,
14
9,
74
8
66
84
75
.1
9
0.
95
01
91
72
,1
96
7,
18
7,
60
3
6,
58
5,
17
9
85
10
0
99
.5
6
0.
91
62
95
15
,8
97
2,
14
6,
17
2
2,
06
2,
61
8
10
1
15
0
13
5.
00
0.
96
11
96
27
,0
55
5,
32
4,
04
2
4,
86
2,
17
3
15
1
20
0
19
6.
79
0.
91
32
98
16
,9
54
4,
68
3,
92
9
4,
56
3,
62
6
20
1
30
0
27
6.
27
0.
97
43
99
18
,3
52
8,
22
1,
39
9
8,
16
4,
42
0
30
1
50
0
44
7.
98
0.
99
31
10
0
12
,9
34
18
,0
00
,0
00
17
,2
00
,0
00
50
1
40
00
0
13
91
.6
8
0.
95
56
Su
m
1,
63
4,
80
8
69
,0
27
,0
64
63
,8
83
,5
77
A
ve
ra
ge
0.
92
55
CHAPTER 6. APPENDIX 76
6.3 Statics based on an empirical approximation
In chapter 3 we obtained an analytic solution to our model. Unfortunately, the com-
plexity of this solution renders the calculation of analytic model statics infeasible.
Instead, we pursue an alternative approach – applying an OLS regression to our
race level dataset to estimate a reduced form approximation to the analytic solu-
tion. Such an approximate solution can then be used to gain an understanding of
aggregate model statics. In constructing the race level dataset, we operationalised
the model parameters as set out in detail in sub-section 4.5. Using these parameters
we calculated a “model implied” final market probability m based on the forego-
ing parameters (as detailed more fully in sub-section 4.6.3). We then proceeded
to estimate two specifications. The first is simply a linear combination of model
parameters (excluding q, since it is a linear function of b0).
mˆ1 = β1N + β2b0 + β3τ + β4α + ε
The second includes squared model parameter terms in order to capture any second
order effects in the data.
mˆ2 = β1N + β2b0 + β3τ + β4α + β5N
2 + β6b
2
0 + β7τ
2 + β8α
2 + ε
The results of these regressions are set out in in Table 6.2 below. Notably, these
simple specifications are able to account for 96.75% and 98.42% of the variation
in model final market probabilities respectively. This compares with 91.98% when
regressing the model implied final market probability on outsider betting implied
market probability, that is, mˆ0 = β1b0+ε. As one would expect, the betting patterns
of outsiders have a substantial impact on the model implied final market probability;
that said, the additional model parameters clearly add further explanatory power
to the specification.
CHAPTER 6. APPENDIX 77
Table 6.2: Empirical approximation of model final market probabilities
mˆ0 mˆ1 mˆ2
Variable Coefficient t-Stat Coefficient t-Stat Coefficient t-Stat
N 0.000071 1.75 0.000245 2.63
b0 1.087512 248.91 1.018508 254.26 1.041681 132.82
τ -0.059058 -7.85 -0.320130 -1.21
α 0.123297 89.08 0.209928 139.20
N2 -0.0000003 -2.51
b20 -0.120900 -11.20
τ 2 1.693191 0.93
α2 -0.025373 -74.34
N 5,402 5,402 5,402
Adj R2 91.98% 96.75% 98.42%
Based on the regression results above, we could draw the following conclusions re-
garding the aggregate statics of our theoretical model based on the race level dataset:
1. Final market probability is increasing in outsider betting b0, but at a decreasing
rate (negative coefficient of b20). (This effect is statistically insignificant in the
first specification which does not include squared terms)
2. Final market probability is increasing the number of insiders N , but at a slowly
decreasing rate (small negative coefficient of N2)
3. Final market probability is decreasing in the takeout rate τ , however, this
effect is statistically insignificant when including τ 2 in the specification.
4. Final market probability is increasing in insider risk preference α at a decreas-
ing rate (negative coefficient of α2)
CHAPTER 6. APPENDIX 78
6.4 Stata code for estimating reduced form final
market probability (mˆ)
s e t mats ize 500
cd D:\ Paul\Masters \Data
use horse rac ing_co l lapsed , c l e a r
// ∗∗ tau [ t ] ∗∗
// takeout ra t e was 15.5% be f o r e Nov 2006 , and 14.5% a f t e r
gen t =0.155 i f tdate_e < date ("1/11/2006" ,"dmy")
r ep l a c e t =0.145 i f tdate_e >= date ("1/11/2006" ,"dmy")
// ∗∗ M DATA [mdata ] ∗∗
gen mdata = ( inwinamt+outwinamt ) /( sbe t s )
// ∗∗ i n s i d e r m ∗∗
gen mdata ins ider = ( inwinamt ) /( inwinamt+in loseamt )
// ∗∗ N [ n ] ∗∗
gen n = i n s i d e r
// ∗∗ b0 [ b ] ∗∗
gen b = ( outwinamt ) /( outwinamt+outloseamt )
// ∗∗ alpha [ a ] ∗∗
gen a = (1−q ) ^(1/2) ∗ ( ( b∗n+q−q∗ t−n) ∗mdata^2+(−1+t ) ∗q∗(b∗n+b−n+1)∗mdata+q
∗b−q∗ t ∗b) /q^(1/2) /
(−1+t )/(−1+q) /(mdata∗b−b+mdata−n∗mdata+mdata∗n∗b−mdata^2)
// ∗∗ b_star [ b_star ] ∗∗
gen K = (b+2∗b∗n+n^2∗b−1−n^2+2∗n)∗(1− t )
gen A = 1/(2∗n^2∗(−a∗(1− t ) ∗q^2+(1−q ) ^(1/2)∗(1− t ) ∗q^(3/2)+a∗(1− t ) ∗q−q
^(1/2)∗(1−q ) ^(1/2) ) )
gen BB = ((1−b)∗(1− t ) ∗q^(3/2)∗(1−q )∗((−1+a^2)∗K∗q^(3/2)−2∗(1−q ) ^(1/2) ∗a
∗K∗q+(−K∗a^2+4∗b∗n)
∗q^(1/2)+4∗a∗(1−q ) ^(1/2) ∗b∗n) ) ^(1/2)
gen C = (1− t ) ∗q∗( a∗(−1+q)−q^(1/2)∗(1−q ) ^(1/2) )∗(−1+b+b∗n+n)+2∗q^(1/2)
∗(1−q ) ^(1/2) ∗b∗n
// optimal i n s i d e r be t t i ng amount = bstar = bstar2
gen b_star = A∗(C−BB)
// drop 5% o u t l i e r s
_pct i l e b_star , p e r c e n t i l e s ( 5 ( 5 ) 95)
CHAPTER 6. APPENDIX 79
drop i f b_star<r ( r1 )
drop i f b_star>r ( r19 )
// ∗∗ THEORETICAL M ∗∗
gen m=(b+b_star∗n) /(1+b_star∗n)
gen n_n = n∗n
gen b_b = b∗b
gen t_t = t ∗ t
gen a_a = a∗a
// approximate t h e o r e t i c a l m us ing model parameters
reg m n b t a , noconst
reg m n b t a n_n b_b t_t a_a , noconst
CHAPTER 6. APPENDIX 80
6.5 Stata code for direct empirical testing of pre-
dicted final market probability
// STATA s c r i p t to t e s t data
s e t mats ize 500
cd D:\ Paul\Masters \Data
use horse rac ing_co l lapsed , c l e a r
// add takeout ra t e " tau"
// takeout ra t e was 15.5% be f o r e Nov 2006 , and 14.5% a f t e r
gen tau=0.155 i f tdate_e < date ("1/11/2006" ,"dmy")
r ep l a c e tau=0.145 i f tdate_e >= date ("1/11/2006" ,"dmy")
// generate emp i r i c a l f i n a l market p r i c e
gen mdata = ( inwinamt+outwinamt ) /( sbe t s )
f o r each qde l ta o f numl ist 0 ( 0 . 0 1 ) 0 .2 {
// # Generate parameters (N, b0 , tau , q ) and t h e o r e t i c a l va lue s
( b1t , mt −− po s t f i x ed w " t " f o r theory )
// N i s the number o f i n s i d e r s − s i n c e CAPITAL N i s r e s e rved by
s ta ta we use smal l n
gen n = i n s i d e r
// b0 i s the r a t i o o f winning horse bet s to a l l bet s f o r
ou t s i d e r s only
gen b0 = ( outwinamt ) /( outwinamt+outloseamt )
// Adds qde l ta to q in order to ensure that q > c r i t i c a l_q as
r equ i r ed
gen q = min (1 , b0+‘ qdelta ’ )
l o c a l q d e l t a l a b e l = round ( ( ‘ qde lta ’ ) ∗100 ,1)
gen alphadata ‘ qde l t a l ab e l ’ = (1−q ) ^(1/2) ∗ ( ( b0∗n+q−q∗ tau−n) ∗
mdata^2+(−1+tau ) ∗q∗( b0∗n+b0−n+1)∗mdata+q∗b0−q∗ tau∗b0 ) /q
^(1/2)/(−1+tau )/(−1+q) /(mdata∗b0−b0+mdata−n∗mdata+mdata∗n∗
b0−mdata^2)
drop n b0 q
} // end fo r each qde l ta
// save r e g r e s s i o n e s t imate s to f i l e
summarize alphadata0−alphadata100
save horse racing_direct_completed , r ep l a c e
CHAPTER 6. APPENDIX 81
6.6 Stata code for indirect empirical testing of pre-
dicted final market probability
// STATA s c r i p t to t e s t data
s e t mats ize 500
cd D:\ Paul\Masters \Data
use horse rac ing_co l lapsed , c l e a r
// add takeout ra t e " tau"
// takeout ra t e was 15.5% be f o r e Nov 2006 , and 14.5% a f t e r
gen tau=0.155 i f tdate_e < date ("1/11/2006" ,"dmy")
r ep l a c e tau=0.145 i f tdate_e >= date ("1/11/2006" ,"dmy")
// generate emp i r i c a l f i n a l market p r i c e
gen mdata = ( inwinamt+outwinamt ) /( sbe t s )
// f l a g hold out sample
s e t seed 85559683
gen holdout = ( uniform ( ) >0.8)
f o r each qde l ta o f numl ist 0 ( 0 . 0 5 ) 0 .2 {
e s t s t o c l e a r
f o r each alpha o f numl ist −0.5(0 .02) 0 . 5 {
// # Generate parameters (N, b0 , tau , q ) and t h e o r e t i c a l va lue s (
b1t , mt −− po s t f i x ed w " t " f o r theory )
// N i s the number o f i n s i d e r s − s i n c e CAPITAL N i s r e s e rved by
s ta ta we use smal l n
gen n = i n s i d e r
// b0 i s the r a t i o o f winning horse bet s to a l l bet s f o r
ou t s i d e r s only
gen b0 = ( outwinamt ) /( outwinamt+outloseamt )
// Use c r i t i c a l_q formula from maple f i l e " S t r a t e g i c with Risk −
Non−core "
// Adds qde l ta to q in order to ensure that q > c r i t i c a l_q as
r equ i r ed
gen q = min (1 , b0+‘ qdelta ’ )
// bs ta r i s c a l c u l a t ed us ing the formula in maple f i l e " S t r a t e g i c
with Risk − Formulae"
// r ep l a c e "b [ 0 ] " with "b0 " , "N" with "n" and "alpha " with " ‘
alpha ’"− r e s t should work
gen K = (b0+2∗b0∗n+n^2∗b0−1−n^2+2∗n)∗(1− tau )
gen A = 1/(2∗n^2∗(− ‘ alpha ’∗(1− tau ) ∗q^2+(1−q ) ^(1/2)∗(1− tau ) ∗q
^(3/2)+‘alpha ’∗(1− tau ) ∗q−q^(1/2)∗(1−q ) ^(1/2) ) )
gen BB = ((1−b0 )∗(1− tau ) ∗q^(3/2)∗(1−q ) ∗((−1+‘ alpha ’^2) ∗K∗q^(3/2)
−2∗(1−q ) ^(1/2) ∗ ‘ alpha ’∗K∗q+(−K∗ ‘ alpha ’^2+4∗b0∗n) ∗q^(1/2)+4∗ ‘
alpha ’∗(1−q ) ^(1/2) ∗b0∗n) ) ^(1/2)
gen C = (1−tau ) ∗q ∗ ( ‘ alpha ’∗(−1+q)−q^(1/2)∗(1−q ) ^(1/2) )∗(−1+b0+b0∗
n+n)+2∗q^(1/2)∗(1−q ) ^(1/2) ∗b0∗n
CHAPTER 6. APPENDIX 82
// optimal i n s i d e r be t t i ng amount = bstar = bstar2
gen bs tar = A∗(C−BB)
// generate a l a b e l s to save r e s u l t s
l o c a l a l pha l ab e l = ( ‘ alpha ’ ) ∗50+25
l o c a l q d e l t a l a b e l = ( ‘ qde lta ’ ) ∗20
// t h e o r e t i c a l f i n a l market p r i c e
gen mt = (b0+n∗ bstar ) /(1+n∗ bstar )
// r e g r e s s emp i r i c a l on t h e o r e t i c a l f o r non−holdout sample
reg mdata mt i f ! holdout , noconst
p r ed i c t pred_m_‘ a lpha labe l ’_‘ qde l t a l ab e l ’
e s t s t o alpha ‘ a lpha labe l ’ // record r e g r e s s i o n e s t imate s
// drop old v a r i a b l e s at s t a r t to be ready f o r new loop
drop n b0 q K A BB C bstar mt
} // end fo r each alpha
// save r e g r e s s i o n e s t imate s to f i l e
e s tout us ing q ‘ qdelta ’ . txt , c e l l s (b t c i_l ci_u ) s t a t s ( r2_a N)
r ep l a c e
} // end fo r each qde l ta
save horse racing_completed , r ep l a c e
// out o f sample r e g r e s s i o n s
e s t s t o c l e a r
f o r each opt imal reg in 39_0 32_1 26_2 20_3 15_4 {
reg mdata pred_m_‘ opt imalreg ’ i f holdout , noconst
e s t s t o osalpha ‘ opt imalreg ’
}
e s tout us ing outofsample . txt , c e l l s (b t c i_l ci_u ) s t a t s ( r2_a N)
r ep l a c e