**ON THE
OPTIMAL DESIGN OF LOTTERIES**

**John
Quiggin**

**University
of Sydney**

Quiggin, J. (1991), ŒOn the
optimal design of lotteries¹, *Economica* 58(1), 116.

Abstract

In recent
years the Expected Utility model of choice under risk has been generalized to
cope with phenomena such as probability weighting. In the present paper, one such generalized approach, the
Rank-Dependent Expected Utility model, is applied to the problem of lottery
gambling. The model is used
to derive an optimal prize structure for lotteries, involving a few large
prizes and a large number of small prizes. Other forms of gambling, such as racetrack betting, are
discussed in the light of this result.

On The
Optimal Design Of Lotteries

Introduction

The expected
utility (EU) theory developed by von Neumann and Morgenstern (1944) has been
one of the success stories of modern economics. It has permitted formal analysis of a large and important
area of economic behavior which had previously been largely outside the realm
of economic theory. Problems such
as insurance, price stabilization and portfolio choice have been subjected to
rigorous analysis. Meanwhile, the
main competing approach, the mean-variance analysis advocated by Markowitz
(1959) has been largely abandoned as a tool of economic analysis, although it
is still used as a rough and ready guide to portfolio choice. The advantages of the EU approach
included a clear axiomatic basis, greater generality and the availability of tools
such as the Arrow-Pratt coefficients of risk-aversion.

The EU
approach has been fruitful as a source of novel and unexpected hypotheses. These include some ideas which are
meaningful only in the context of EU theory, such as the hypothesis of
declining absolute risk-aversion, as well as others, such as the idea that full
insurance is sub-optimal, which
reflect the analytical power of the approach.

Finally, the
EU approach has permitted the development of new and powerful analytical tools
and concepts. The Arrow-Pratt
coefficients of relative and absolute risk-aversion have already been
mentioned. The concepts of
stochastic dominance (Hadar and Russell 1969) and increasing risk (Rothschild
and Stiglitz 1970,1971) are also closely related to the EU framework.

This very
real success has, however, been achieved despite the existence of a growing
body of theoretical and empirical objections to EU theory. These date back as far as the work of
Allais (1953, translated in Allais and Hagen 1979) who rejected the EU
independence axiom and produced as a counter-example the well-known Allais
paradox. Major theoretical
objections were also raised, at
the outset, by post-Keynesian writers such as Shackle (1952).

The body of
evidence conflicting with EU theory has grown over time. Three main groups of problems may be
distinguished. First, there are
problems observed in experiments such as those of Ellsberg (1961) and
MacCrimmon and Larsson (1979).
Second, there are difficulties associated with attempts to construct
utility functions using questionnaires.
Finally, there are behavioral choices, such as those relating to
gambling and the widespread preference for full insurance which appear to
conflict with the theory. It is
the problem of gambling behavior which will be examined in the present paper.

Despite a
number of attempts, most notably that of Friedman and Savage (1948), it has not
been possible to produce an adequate explanation of observed gambling behavior
within the EU framework. In
addition to a discussion of these approaches and the objections which have been
made against them, this paper includes a new argument against the
Friedman-Savage approach.

The problems
associated with EU have led to many attempts to construct alternative theories
of behavior under risk such as those of
Allais (1953), Handa (1977), Karmarkar (1978, 1979) and Kahneman and
Tversky (1979). Until recently,
however, none of these approaches has been able to match the sharp predictions
yielded by EU theory. More
recently, however, generalizations of EU theory have been developed by Machina
(1982), Quiggin (1982) and Chew (1983).
These models account for many of the observed violations of the
predictions of EU theory, such as the Allais paradox (Machina 1983, Quiggin 1985,
Segal 1987) while retaining its analytic power in that many of the standard EU
analytic concepts, such as risk-aversion can be generalized. The primary approach used here is the
rank-dependent expected utility (RDEU) model (Quiggin 1982, Yaari 1987). A number of the results can be
carried over to the generalized expected utility framework of Machina (1982).

The object of
the present paper is to extend analysis to an area not handled successfully by
EU theory, namely the demand for gambling opportunities such as lottery
tickets. The Friedman-Savage
model of simultaneous gambling and insurance is discussed and criticisms of it
reviewed. An even more
fundamental flaw in the model, not previously observed, is pointed out. The RDEU model is outlined and compared
with Machina's (1982) model.
Optimality conditions for a lottery are derived and interpreted for both
models. It is shown that the
RDEU model provides a good explanation of observed lottery design.

I. Gambling and
insurance

One of the
first major problems to emerge in EU theory arose from observed behavior in
markets involving uncertainty such as those for insurance and gambling. Gambling, as well as being the
inspiration for the development of probability theory, is an economically
significant aspects of uncertainty.
Yet gambling is inconsistent with EU theory unless the very plausible
hypothesis of risk-aversion is abandoned, and the coexistence of gambling and
insurance purchase is very difficult to fit into the EU framework.

Some types of
gambling can be explained easily enough.
For example, given the virtual certainty of losing on slot machines in
the medium run, it seems reasonable to assume that this activity is undertaken
largely for entertainment. Similar
explanations may be advanced for participation in various time-consuming
gambling games such as bingo.
Racetrack betting provides a more complex case. Some betting on horse races may reflect
divergent beliefs about the outcome.
This is reflected in the large amounts of effort devoted to collecting
and analyzing information about the quality of horses, jockeys, tracks
etc. On the other hand, there are
a large number of bettors who collect no information at all, and bet on an
essentially random basis. The
problem of racetrack betting will be discussed further below.

In respect of
lottery tickets, however, there is no acceptable explanation consistent with
risk aversion. The only plausible
reason for betting is the chance of winning a large amount of money. The predominance of this motive is
confirmed by psychological studies (Walker 1984). Thus, the purchase of lottery tickets by people who are
generally risk-averse constitutes a significant problem for EU theory, which is
very difficult to explain without resorting to the content-decreasing measure
of excluding gambling from the realm of rational behavior.

Friedman and
Savage (1948) attempted to explain the coexistence of gambling and insurance
using the concept of an *S*-shaped utility function (see Figure 1). The basic idea is that people will be
risk averse with respect to changes in a neighborhood of their current wealth
level but may be risk-seeking with respect to prospects which may take them
into a higher social class. Thus,
the utility function may be regarded as concave for low income levels and
convex for some higher incomes.
The function initially suggested by Friedman and Savage had this
form. The third concave
segment (at still higher income levels) is a modification introduced as a
response to the observation that lotteries typically have multiple prizes
whereas convexity throughout the upper range would imply that a lottery with a
single large prize would be preferred.

It may be
noted that this modification does not in fact account for the observed
distribution of prizes in lotteries. Preferences of this kind would imply a desire for a
lottery with several prizes of the same value, a pattern which is rarely
observed in practice. These
preferences cannot account for the typical prize distribution having a few
large prizes and a large number of small ones.

FIGURE 1 NEAR
HERE

A variation
on this theme, expressed in terms of indivisibilities in expenditure, is
developed by Ng (1975). He argues
that even if the underlying utility function is concave, the existence of large
indivisible items of expenditure (eg the purchase of a university education)
will lead people to buy lottery tickets.
Broadly similar approaches to the problem have been taken by Flemming
(1969) and Kim (1973) . None of
these approaches appear adequate as explanations of gambling in developed
countries with well-developed credit markets where a large proportion of people
buy lottery tickets. Even allowing
for considerable costs associated with imperfections in credit markets, it is
unlikely that these could justify the purchase of lottery tickets with an expected return of about 60 cents
for each dollar in outlays.

A number of
objections to the Friedman-Savage approach have been made by Machina
(1982). The most important is that
the observed gambling behavior of individuals does not appear to change
radically in response to changes in their initial wealth. However, the utility function in Figure
1 suggests that behavior will be highly sensitive to changes in initial wealth,
with only those individuals near the inflexion points displaying propensities
to both gamble and insure.

All of the
approaches cited above have focused on the value of the outcomes. This is a natural consequence of
the use of the EU framework, but it seems far more reasonable to suppose that
participation in lotteries has to do with attitudes to probabilities and, in
particular, with the placing of a high weight on extremely favorable, low
probability events. Adam
Smith (1776, pp 164-165) stated:

That the chance of gain
is naturally over-valued, we may learn from the universal success of
lotteries. The world neither
ever saw, nor ever will see, a perfectly fair lottery; or one in which the
whole gain compensated the whole loss; because the undertaker could make
nothing by it. In the state
lotteries, the tickets are not worth the price which is paid by the original
subscribers, and yet commonly sell in the market for twenty, thirty and
sometimes forty per cent advance.
The vain hope of getting some of the great prizes is the sole cause of
this demand. The soberest people scarce look upon it
as folly to pay a small sum for the chance of gaining ten or twenty thousand
pounds; though they know that even that small sum is perhaps twenty or thirty
per cent more than the chance is worth. In a lottery in which no prize exceeded twenty pounds,
though in other respects it approached nearer to a perfectly fair one than the
common state lotteries, there would not be the same demand for tickets. In order to have a better chance
for some of the great prizes, some people purchase several tickets, and others,
small shares in a still greater number. There is not, however, a more certain proposition in
mathematicks, than that the more tickets you adventure upon, the more likely
you are to be a loser.
Adventure upon all the tickets in the lottery, and you lose for certain;
and the greater the number of your tickets the nearer you approach to this
certainty

This passage
encapsulates a number of objections which have been made to the EU explanations
of gambling and a number of requirements for a successful explanation. In particular, it is necessary
that the theory should explain preference for lotteries with a few large
prizes, that it should not suggest that people are unaware that the game in
which they are participating is not (actuarially) fair, and that it should
explain the purchase of one or a few tickets.

Even with the
abandonment of the plausible postulate of risk-aversion, EU theory cannot
account for the observed behavior of a large segment of the population in
undertaking both gambling and insurance.
What is needed is a model of choice under uncertainty which takes
explicit account of phenomena such as probability weighting which have
frequently been recognized in psychological studies of the problem such as
those of Edwards (1962). Such a
model is developed in the following section.

II. An outline of the Rank-Dependent
model

RDEU theory
deals with individual preferences over a set *W* of outcomes and an
associated set *Y* of prospects, or random variables, taking
values in *W*. For the
purposes of this article, *W* will be assumed to be a connected subset
of Â, and may be interpreted in terms of wealth levels.
A random variable *y* Î *Y* may be regarded as a measurable function from
some s-field of events, W, to *W*, and characterised by
its cumulative distribution function, *F** _{y}* :

In this
paper, attention will be confined to discrete random variables or 'prospects'
(those with finite support), although the model is quite applicable to
continuous distributions. These
may be represented in the form

(1) {** w**;

where *p _{j}*

(2) *V*({*w***; p**}) = iSU(

The RDEU
model includes EU as a special case.
In addition to the utility function, it employs a probability weighting
function *q*: [0,1] ® [0,1]. The function *q* is
continuous, monotonically increasing and such that *q*(0) = 0,*q*(1) = 1. This
function may be composed with a cumulative distribution function *F* to yield a
weighting function H = *q**°**F*, which
possesses the usual properties of a cumulative distribution function. In the present context, where we
are mainly concerned with the upper tail of the distribution, it is also
convenient to define *q**(*p*) = 1 - *q*(1-*p*).

The RDEU functional is defined as

(3) *V*({*w*;** p**}) = iSU(

where

(4) *h _{i}*(

It will be
assumed throughout that the utility function *U* is globally concave,
since the arguments against the plausibility of a convex utility function,
advanced in the previous section in relation to EU theory, apply with equal
force to the RDEU model.
By contrast, as is shown below, these arguments do not apply to
risk-seeking behavior associated with the possibility that *q**(*p*) > *p* for small *p*.

Assuming that
extreme low-probability events are overweighted, the function *q* will
take the shape illustrated in
Figure 2, which is clearly reminiscent of the Friedman-Savage utility
function. The basic
requirement for this condition is that
*q*'(*p*) should be a *U*-shaped function convex
in *p*. In order to avoid
St. Petersburg paradoxes, it is
also necessary to assume that *q*'(*p*), and hence *q*(*p*)/*p* , are
bounded.

(FIGURE 2
NEAR HERE)

It is
intuitively apparent that the pattern of probability weighting illustrated in
Figure 2 is consistent with both gambling and insurance. The overweighting of outcomes at the
lower tail of the distribution reinforces the effects of a concave utility
function (risk-aversion in the EU sense) in promoting a demand for insurance
against adverse low-probability events.
It may be noted, however, that the effects of probability weighting
discourage insurance against high probability events. By contrast, the effects of overweighting on events at the
upper tail of the distribution counteract the effects of EU risk aversion which
discourage low-probability, high-payoff gambles. The extent to which gambles are accepted depends on the
relative curvature of the *U* and *q* functions. However, it is straightforward to show
that for any person with a weighting function of the kind illustrated in Figure
2, there are some actuarially unfair gambles which will increase utility.

III. Lotteries with a Single Prize

The
discussion above has indicated that the RDEU model is capable of explaining
simultaneous gambling and insurance.
However, such an explanation is only valuable if the model can generate
sharp and (at least in principle) testable predictions of behavior. While some individual testing has been
carried out (see Quizon, Binswanger and Machina 1984 and Quiggin 1981), the
most useful tests are those which relate to behavior in actual gambling and
insurance markets. Gambling
markets are of particular interest because they have previously been outside
the scope of effective economic analysis.
Thus it is important to see whether the observed structures of gambling
markets and, in particular, the gambles offered and accepted are consistent
with the predictions of the theory.

In this
section, the RDEU model will be applied to the problem of determining the
optimal prize structure for a lottery, given that all consumers have identical
RDEU preferences. Thus, if the
RDEU model is correct, we would expect that the prize structure actually
prevailing will be fairly similar to the optimal solution derived here. The consequences of relaxing the
assumption of identical consumers will be considered below.

It will be
assumed that the operator's return is predetermined, either by government
regulation or by a perfectly competitive market, and that the problem is to
maximize the perceived net value of the ticket to the potential purchaser. This would appear to correspond fairly
closely to the actual situation for lottery-type gambles. Attention is therefore
confined to actuarially fair gambles, although the analysis extends fairly
easily to the case of a fixed 'rake-off'.

Before
proceeding to the general solution of the problem, it is useful to consider two
subproblems. An individual's
demand for any commodity is determined by their demand curve and the market
price. In this context, it is
useful to think of the demand curve being determined by the anticipated utility
of the (net) prize payouts on the winning tickets, and the price being the
anticipated utility loss associated with the losing tickets. Thus, in order to optimize lottery
design, the operator should maximize the anticipated gains and minimize the
anticipated losses. Given
the assumption of actuarial fairness, these problems may be analysed
separately.

The simpler
of these two subproblems is that of minimizing anticipated losses. Since ticket prices are normally small
in relation to total wealth, the utility function *U* can be
approximated by a function linear in the ticket charge *c*. Units can be chosen so that the required average revenue is 1, and
the decision problem becomes

(5)
Min *c q*(*p*)

subject to *cp* = 1

where *p* is the
probability of a losing ticket.
This problem is equivalent to minimizing *q*(*p*)/*p* and the
first and second order conditions on *p* are

(6) *pq*'(*p*) - *q*(*p*) = 0

and

(7) * pq*"(*p*) > 0

respectively. It is easy to show that these
conditions will be satisfied at point A in Figure 2, where a ray drawn from the
origin is tangent (from below) to the weighting curve.

While data
are scanty and not very reliable, the evidence which is available, such as
Edwards (1962), suggests that this point will lie somewhere between 0. 75 and 0. 95. The solution derived here has direct
application to one class of gambles, namely promotions for a product in which
purchasers have some chance of having the purchase price refunded, that is, of
getting the product for nothing.
Obviously, this involves an effective increase in price, since the
promotional funds could equally be applied to an across the board price
reduction. The analysis given here
suggests that promotions of this kind will be most effective in increasing
demand if the proportion of refunds lies between 1 in 4 and 1 in 20. In general, however, the gamble
offered in a promotion of this kind will be dominated by a lottery ticket
offering a chance of a large
positive prize. Hence such gambles
will appeal only to consumers who are unable (or perhaps unwilling) to
participate in lotteries, or who exhibit an 'isolation' effect, in which the
gamble offered by the promotion is assessed without taking account of other
available gambles. The fact that
promotions of this kind are offered only occasionally supports the view that
they are attractive only when some kind of isolation effect comes into play.

The problem
of maximizing the anticipated value of the payout is somewhat more complex because of the need to take both
probability and risk attitudes into account. If only probability attitudes were considered, the problem
would be one of maximizing *q**(*p**)/*p**, where *p** is the
probability a winning ticket.
However, the assumptions behind the function illustrated in Figure 2
mean that this value increases as *p** decreases, converging
to (*q**)'(0) as *p** converges to zero. Thus, the optimal solution would be an infinitely large
prize with an infinitesimally small chance of winning. Risk aversion offsets this and implies
that a finite optimal prize will exist for a lottery with a single prize and a
fixed average return.

Expressing
the utility function in terms of deviations from initial wealth, the problem
may be expressed as

(8)
Max *U*(*w*)*q**(*p**) + *l*(1 - *wp**)

Deriving the
initial conditions and solving for the Lagrangean multiplier *l* yields

(9) f(*wU*'(*w*) , *U*(*w*)) =
f(*p***q**'(*p**), *q**(*p**) )

The LHS of
(9) may be interpreted as a measure of relative risk aversion. It is different from *r _{r}*

(10) *
r _{r}*

A similar
interpretation may be given to the RHS.
It is a measure of the weight given to a small change in *p* compared to
the weight on the total probability of winning *q**(*p**).

Before considering
the properties of this solution, it is necessary to establish that a solution
does in fact exist. This may be
shown as follows, subject to fairly weak conditions. First, in order for an increase in the prize to be
profitable, it is necessary that the LHS of (9) be greater than the RHS. Given the shape of the weighting
function in Figure 2, some actuarially fair gambles are regarded as favorable,
so this condition must be satisfied somewhere. However, the concavity of *U* implies that the LHS is
always less than one, and, provided *r** _{r}*(

This
discussion also gives a good indication of the comparative static properties of
the solution. In general the
optimal prize will be larger, the larger is the overweighting of small
probabilities and the lower is the level of relative risk-aversion.

IV. The General Case

The most
important point to note from the analysis of the previous section is that there
is no necessary relationship between the probability of winning which maximizes
the anticipated value of the prize and the probability of losing which
minimizes the anticipated loss.
For plausible values of the coefficients, the former will be very close
to zero, while the latter is likely to be significantly less than 1. This provides an intuitive
justification for the existence of multiple prizes in lotteries, and, in
particular, for the frequently adopted practice of giving a large number of
fairly small prizes which do not contribute much to the expected value of the
ticket. By reducing the
probability of losing from near-certainty to a value of *p* for which
significant underweighting applies, this practice reduces the anticipated loss
associated with the ticket.

The rationale
for multiple prizes given here must be distinguished from the discussion of
this issue by Friedman and Savage (1948). Friedman and Savage justified the final concave
segment in their utility function by the observation that lotteries have
multiple prizes and hence that people are presumably not risk-preferrers for
unbounded wealth levels. However,
what they were really concerned about here was the fact that there is a finite
optimal bet, and this can be inferred more satisfactorily from the observation
that otherwise there would be only one big lottery in any given time-period. The Friedman-Savage theory does not
predict multiple prizes unless it is assumed that there is a fixed minimum size
for the lottery pool, and even then it would predict a number of identical
prizes rather than the observed pattern of a maximum prize and a series of
smaller prizes.

With these
considerations in mind, it is possible to analyze the general problem of
optimal lottery design. It will be
assumed that there is a fixed number of tickets *N*. Provided *N* is large,
this is not a very severe restriction on the generality of the results. For any lottery with say, 1000 tickets
at $10 each, a fairly close substitute can be designed having 10,000 tickets at
$1 each. There are thus *N* outcomes
each occurring with probability 1/*N* . These will be denoted *w*_{1}, . . . *w** _{N}* , and
ordered as in Section 3, from worst to best, so that

(11) Max *i*SU*(**i*=1,*N*, ) *h*_{i}*U*(*w** _{i}*) + l(

where the *m** _{i}* are
Kuhn-Tucker multipliers.

The first
order conditions are

(12) *h*_{i}_{ }*U*'(*w** _{i}*) +

(13) *m** _{i}*(

The
interpretation of these conditions is straightforward for the upper tail of the
distribution, where (12) can be satisfied with the *m*_{i}_{ }equal to
zero. There will be a declining
sequence of prizes starting at *w** _{N}* and
satisfying the relationship

(14) *U*'(*w** _{i}*) /

That is,
the declining marginal utility of income is exactl,y offset by the increasing
weight placed on events at the upper tail of the distribution. This relationship is particularly
tractable for the constant relative risk-aversion functions of the form *U*(*w*) = *w** ^{a}*, discussed
above. The relationship reduces to

(15)
*w** _{i}*/

While the
ordering of the *w** _{i}* is essential for a tractable solution to the
problem, there are some difficulties in determining the precise nature of the
solution over the range where the

**Proposition
1: **Let the utility function be *U* be concave and let *q* be such that
*q*'(*p*) is convex in *p*. Then the optimal prize structure is as
described above.

Proof: See
Appendix.

V. Evidence from
observed lottery designs

The prize
structure described above seems to correspond fairly well to that prevailing in
actual lotteries. There are,
however, a number of qualifications which should be taken into account. First, in most lotteries, the
small prizes do not form a strictly increasing sequence but are bunched
together, with, say one set of $10 prizes, another of $50 prizes and so
on. This prize structure is
presumably adopted for simplicity.
The precise structure of the set of small positive prizes does not make
much difference to either the statistical expectation or the anticipated value
of the lottery. These will be
determined primarily by the probability of losing and by the value of the main
prizes at the extreme upper tail of the distribution.

Second, the
solution above is derived on the assumption that all lottery purchasers have
identical preferences. The obvious
market response to heterogeneous consumers is product differentiation. Consumers with limited risk-aversion
and heavy overweighting of extreme low probability events will prefer lotteries
dominated by one large prize while those with higher risk-aversion and
more weight on events not quite so
close to the tail will prefer a more even distribution of prizes. Obviously, a large group will simply
abstain, including both global risk-averters and those whose anticipated
utility gain from actuarially fair lotteries is insufficient to offset the
operator's margin in an unfair lottery.
If product differentiation is not possible, the solution would involve a
compromise between different groups, with the weight depending both on numbers
in different groups and the elasticity of demand with respect to deviations
from the preferred prize structure.

Third, the
analysis presented here assumes that the individual's income is known with
certainty apart from a single lottery purchase. In practice there are numerous other sources of
uncertainty and most people typically make a sequence of lottery purchases. As regards the first point, the upper tail of the wealth
probability distribution will, for many people, be contributed primarily by the
possibility of winning the lottery, since there is no prospect of attaining
substantial wealth other than that associated with events, such as a random
gift from a passing billionaire, which are even less probable than a win in the
lottery. As regards the
second, it is useful to consider the position of someone who is presented with
a sequence of lotteries being resolved at regular intervals, for example, weekly,
but whose time-horizon is longer, say a year. A somewhat more complex version of the analysis presented
above is applicable, in which the individual's anticipated utility is
determined by the probability distribution yielded by the sum of the different
lotteries in which they participate.
Thus the sum of the lotteries will yield a distribution similar to that
of the optimal single lottery derived above. This implies that the optimal winning probability for any
one lottery is smaller as is the optimal ticket price. The rational strategy in this case will
be one in which a small number of tickets is purchased each week. The most obvious alternative would be one in which a large
number of tickets were purchased in the first available lottery and no more
subsequently. However, this
strategy would be dynamically inconsistent, since unless a major prize was won,
the position after the lottery was resolved would again be one in which it was
rational to buy tickets.

Fourth,
lotteries are only one of a number of gambling forms and others do not
obviously fit the pattern predicted here.
This reflects the fact that lotteries are a very pure form of gambling
with no element of skill and little of the entertainment available from
time-consuming forms of gambling such as bingo and poker machines. The analysis of lotteries presented
here forms a useful basis for the treatment of more complex cases such as that
of race-track betting. An analysis
of this problem is presented as an indication of the way in which the insights
derived form the simple model presented here may be applied to more complex
cases.

VI. Racetrack Betting

Whereas the
main interest in lotteries is in bets at very long odds, racetrack betting
involves a wide range of odds from very short (odds-on) to very long (such as
doubles and trebles ). Clearly, a
'pure' gambling explanation of the kind advanced for lotteries will not work in
this case and it is necessary to introduce other explanatory factors. The central distinguishing feature of racetrack
betting is that there are no objectively given odds, and there is considerable
room for disagreement about the prospects of particular horses. The existence of such disagreements
would provide a reason for bets between risk-neutral or risk-averse
individuals. On the other hand,
the incentives for information collection and the well established markets for
information provision mean that it is unlikely that the market odds for bets
between risk neutral individuals could differ greatly from the true odds. Thus it would appear that an
explanation based on disagreements over odds must be integrated with an
analysis of risk preferences of the type presented for lotteries.

This argument
may be elaborated by assuming there are two groups of bettors. Members of the first group are
risk-neutral, and invest substantial effort in assessing the winning
probabilities of different horses.
Despite the efforts of this group, there remains some noise in the
system, leading to divergent estimates of winning probabilities. Thus, members of this group would be
willing to place at least some bets even under a totalisator or pari-mutuel
system in which it was certain that bettors as a group would lose. Members of
the second group ('mug punters') have RDEU preferences of the kind set out
above and are attracted by long-odds bets, even when they are actuarially
unfair. They have no information
about the merits of the horses and choose among the long-priced horses on a
frivolous basis, such as attractive names or lucky numbers. This group will normally have a lower
elasticity of demand for bets than the first group, for whom a small reduction
in odds is likely to render bets unacceptable. Hence a monopolistic bookmaker will choose to offer odds on
long-priced horses which are highly unfair. This will ensure that the bets of the first group are
concentrated on short-priced horses and, therefore, that the elasticity of
demand for bets on these horses is high and the bookmaker's margin
correspondingly low.

A number of
writers including Griffiths (1949) and Ali (1977) (see Bird and McCrae 1984 for
a summary of the literature) have examined the structure of returns to
racetrack bets. These writers
examine the average return per dollar invested on bets at particular odds or on
horses in order of favoritism.
Essentially the same features are found by all writers. First, all bets of this kind are
actuarially unfair. This is not
surprising, since otherwise there would exist a profitable betting strategy
requiring no information about the individual characteristics of the horse
involved. Second, the return to
punters is near unity for short-priced horses, but falls as low as 60 cents in
the dollar for long-priced horses.
This is consistent with the previous discussion. Bets on long-priced horses are a
substitute for lottery tickets, for people who know nothing about the prospects
of the horses concerned.
Thus, their rate of return should approach that of a lottery ticket as
the odds increase, As a result of taxes and operating expenses, this latter
return is typically of the order of 60 cents in the dollar. The general pattern of returns on
Œexotic¹ bets such as exactas and daily doubles is similar (Asch and Quandt
1987).

This suggests
that the returns structure may be regarded as being determined in the following
fashion. As a first approximation,
the return on a bet on a particular horse may be regarded as being determined
by the objective odds and the amount people are willing to pay for a bet at
those odds in the general gambling market. this would yield exactly fair returns for bets at evens with
a gradually decreasing rate of return as odds lengthen. Because of disagreements regarding the
appropriate odds for a particular horse, bookmakers are able to offer odds slightly
less favorable than those suggested by this rule, and in particular to lay bets
on short-priced horses which are, on average, slightly unfavorable to punters.

Thus, RDEU
theory permits a fairly straightforward explanation of the demand side of the
market for racetrack bets. There
are, however, a number of interesting problems on the supply side. The fact that the margins differ
substantially between the different bets suggests that bookmakers are able to
exercise a degree of price discrimination, which in turn implies the exercise
of market power. In the case
of totalisator or pari-mutuel betting, this pattern of payments is generated
automatically in response to bettor demand, but it is not clear precisely how
the same result is achieved by on-course bookmakers, since the market appears
to be fairly competitive. As with
other discriminating monopoly solutions, the existing pattern is profitable for
bookmakers as a group, but appears to offer incentives for any individual to
defect. At first sight, it would
appear profitable for an individual bookmaker to specialize in taking bets on
long-priced horses, and offer slightly better odds than those who took bets on
the field. A number of possible
explanations may be advanced including the existence of informational
difficulties or the operation of informal sanctions which act to preserve a
cartel, but none appears completely convincing at this stage.

VIi. Concluding Comments

Rank-Dependent
Expected Utility theory is a generalization of the Expected Utility approach
which is currently the basis of most economic analysis of behavior under
uncertainty. In order to show that
this increased generality is more than mere formalism, it is necessary to
demonstrate that RDEU theory not only resolves anomalies which have arisen in
EU analysis, but also that it permits the extension of economic analysis into
new areas of behavior. Gambling is
one of the most important aspects of economic behavior which has not until now
been amenable to analysis. At
least in the case of lotteries, RDEU theory appears to give a very satisfactory
account of gambling behavior and gambling markets.

The potential
scope of the theory is much wider.
A wide range of investments, such as shares and futures contracts
include low-probability high return events at the upper tail of the
distribution of possible outcomes.
It seems likely that use of an EU model with coefficients derived from
standard questionnaire methods will lead to underestimates of the
attractiveness of such investment.
Conversely, EU methods may give an excessively favorable evaluation of
private investments or public policy choices involving low probability events
with very severe losses attached.
Decisions associated with the siting of nuclear power plants provide an
obvious example.

Observed
behavior in the face of risk is dominated by risk aversion, but contains
elements which seem to imply risk preference. Under EU theory, this seems contradictory or
irrational. The RDEU
approach permits the development of a portfolio analysis which will include
elements of risk-seeking along with risk aversion.

Appendix

Proof of
Proposition 1:

Two lemmas
are required, before the main result is proved.

Lemma 1. Let *p** be the probability
which satisfies the condition (18) for minimization of the weighted probability
of loss, and let [i/*N*, (i+1)/*N*] be the interval of
measure 1/*N* containing *p**. Let *Z* be any prize
structure such that the probability,
o(*p*,* ^{~}*), of the lowest prize
does not lie within this interval.
Then there exists a preferred prize structure which does satisfy this
condition.

**Proof:** Assume
without loss of generality that the lowest prize is -1, and denote the next
prize by *z*_{1}. Suppose o(*p*,* ^{~}*) =

Hence, *q*(*k*/*N*) *U*(-1) + *h*_{k}_{+1}(** p**)

<
*q*((*k*+1)/*N*) *U* ((-*k*+*z*_{1}) / (*k*+1)),

by the
concavity of *U*.

Hence *Z* will be
dominated by a lottery with lowest prize
(-*k*+*z*_{1}) / (*k*+1) arising with
probability (*k*+1)/*N*. The converse case, where o(*p*,* ^{~}*) =

**Lemma 2:** Let *Z* be a lottery in which the probability of the minimum
prize satisfies the optimality condition derived in Lemma 1, and let *i* be such that

(i) *w** _{i}* >

(ii) *w*_{i}_{+1} = *w*_{i}* *.

then there
exists a preferred lottery with *w*_{i}_{+1} > *w** _{i}*.

**Proof:** By the conditions on the minimum prize *q*'(*p*) is
increasing over the relevant range and

*h*_{i+1} > *h*_{i} > *h*_{i-1}.

Hence there
exists some d such that

(i) *w*_{i }-_{ }*d* > *w*_{i-1}

(ii) *h*_{i}_{ }*U*(*w** _{i}*) +

If *w*_{i}_{+1}+ *d* < *w*_{i}_{+2} , then the
preferred lottery may be constructed by reducing prize *i* by *d*, and
increasing prize *i*+1 correspondingly. If this condition is not satisfied, then it is
necessary to spread the increased prize money over a range of higher valued
prizes to preserve the non-decreasing sequence of prizes, but a condition
analogous to (ii) will still hold.

The main
result may now be proved inductively.
After the sequence of minimum prizes satisfying lemma 1, consider the
two subsequent prizes in an optimal lottery design. By Lemma 2, they will form part of an increasing sequence. Lemma 2 may now be applied to each of
the subsequent prizes in turn.

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