Memorandum submitted by Professor Ian
Walker and Ms Juliet Young
THE DUMMIES' GUIDE TO LOTTERY DESIGN
ABSTRACT
This paper[8]
outlines the issues relevant to the operation of lottery games:
it attempts to put some science into the art of lottery design.
Our research suggests that lottery ticket sales depend positively
on: the average return (ie the proportion of revenue returned
as prizes) because punters like better bets; the skewness in the
prize distribution (eg how much of the prize money goes to the
jackpot) and we find that the more the better, and negatively
on the variance in the prize distribution (which is a measure
of the riskiness of the return—so the less the better).
The sizes of these effects are important: our statistics suggest
that the effect of the mean is small, as is the effect of the
skewness, while the (negative) variance effect is quite important.
The work suggests that good causes revenue might
be higher if: the game were meaner (less of the stakes used as
prize money) because, although sales would fall a little, the
good causes would get a larger share of the smaller revenue; more
of the prize money was used for the jackpot, or the variance in
the expected prizes were reduced. BUT, in practice, it is difficult
to change one aspect of the design of the game without having
a counterveiling effect on another aspect. Thus, it is difficult
to make judgements about the merits of alternative game designs
without looking at ALL of the parameters being proposed.
However, the research suggests that there is
no obvious case for not increasing the take-out (the revenue that
is not returned as prizes) if the current game design is kept.
If the game were changed to make the odds longer then our research
suggests that other parameters of the design of the game may have
to be changed to stop sales falling.
While we feel that there should be a lottery
(because people enjoy playing and it does little harm), we also
feel that lotteries are not good vehicles for taxation because
they are a larger part of the spending of the poor than the rich.
Moreover, we find no compelling empirical evidence to suggest
that there is any merit in having much of the take-out dedicated
to good causes—hypothecation is bad for sound investment
decision-making, the best causes are already the recipients of
taxpayer largesse, and adding lottery funds to these causes simply
displaces Treasury dollars. That is not to say that (some, perhaps
most) of the good causes are deserving—rather that they
should be funded in some other way.
Finally, the current "beauty contest"
process of choosing an operator is fraught with risk (for the
Commissioners but not the bidders) and we suggest that, if the
aim is to raise good causes funds, then the license should be
auctioned.
INTRODUCTION
1. The UK National Lottery has been in existence
for almost 6 years.[9]
It is operated by Camelot PLC, a consortium which was awarded
the license in competition against other bidders, including a
consortium bid headed by Richard Branson's company, Virgin. The
license is due to expire in one year and both Virgin and Camelot
have been bidding for this second license which gives the right
to operate lottery games in the UK for a further seven years.
2. While both the current and the 1993/4
competitions between Virgin and Camelot have been the subject
of some controversy, the intention of this paper is to step back
from the controversy and to consider how the UK game ought to
be designed, operated, taxed and regulated. Little attention has
so far been given to the considerations raised here and yet they
are central to both the objectives that government have set for
the operator and for wider objectives such as the welfare of society
as a whole. Unfortunately, there has also been little analytical
research into how lottery games work, or how they should be operated
and designed. This paper attempts to draw together the little
analytical work that has been done and uses it to address the
UK lottery market.
3. There are statistical issues concerned
with how to structure the game to generate sales and this involves
choosing the number of combinations of numbers that are bought
so as to make the game attractive to players. The statistical
design affects how hard it is to win, and this affects how attractive
the game is on a draw-by-draw basis, and so also affects how sales
might be expected to behave in the long run. Our evidence suggests
that the current Camelot online game may be "too easy"
but it may be difficult to design a harder game that is not also
"meaner".
4. There are economic issues concerned with
the sizes of the prize pools for different winners. The bigger
the overall prize pool the better the bet being offered and the
more attractive the game will be. The market structure is also
important: online parimutuel games exhibit economies of scale—bigger
games are more "efficient" than small ones so regulating
entry into the market is likely to be very important. Moreover,
the stability of sales is likely to be adversely affected by a
competitive market structure. Of course, monopolistic supply may
imply a need for effective regulation—and this will be true
even if the licensee is operating on a not-for-profit basis. The
license competition is currently organised as a "beauty contest"
rather than as an auction and it is not obvious that the present
arrangements are optimal.
5. Further economic considerations arise
from the taxation of the games—the existing games are taxed
at a rate that is thought to be approximately the rate levied
on other forms of expenditure so as to be revenue neutral. But,
the good causes levy is also a "tax"—albeit one
which is "hypothecated" for particular forms of expenditure.
Thus, the overall tax rate is round 40 per cent at present. In
general taxes on goods and services should be designed with two
things in mind: to minimise the "distortions" to the
choices that people make, and to impose a smaller burden on the
poor than the rich. Our evidence is that lottery taxation fares
badly on both these counts.
6. Game design can be fine tuned to exploit
the psychological weaknesses of players so as to improve how the
game operates. For example, the distribution of the overall prize
pool between jackpot winners and lesser winners affects the attractiveness
of the game. Moreover, the ability for players to choose their
own numbers, as opposed to being forced to buy tickets with randomly
generated numbers is thought to be important for sales.
7. The distribution of play across individuals
may also be amenable to manipulation through intelligent game
design.
8. We expand on these issues below and summarise
our views as to what the UK game should look like in the concluding
section.[10]
HOW TO
DESIGN A
LOTTERY
Statistical Considerations in Online Parimutuel
Lottery Game Design
9. Online[11]
games usually feature players buying a ticket where they choose
n numbers from a possible N available numbers. Such
games are usually parimutuel in design—that is winners,
whose tickets match the winning combination (or some part of it),
receive a share of the prize pool with any other players who chose
the same numbers. The chances of winning depend on n and
N—the bigger is n the harder it is to match
the winning numbers since you have to match more of them, and
the bigger is N the more possible combinations there are
to be matched.
10. The problem of designing such a game
is one of choosing the right values of n and N for the market
circumstances. The bigger is N the less likely it is that anyone
will hold the winning combination of the n numbers drawn—so
if n=6 and N=49 then the probability of holding the winning combination
is (approximately) 1 in 14 million, while if n=6 and N=53 then
the chance of buying the winning combination is (approximately)
1 in 23 million.
11. Thus n and N affect the likely number
of winners: with n=6 and N=49 and 60 million tickets sold then
the number of jackpot winners to be expected is more than 4,[12]
but if N=53 then the expected number of winners is less than 3.
These are just the numbers of winners that we would expect on
average—there is a variance around these numbers and the
implication of N=53 rather than 49 is an increase in the chance
of there being NO winners, ie the rollover probability. In the
event of a rollover the jackpot is added to the jackpot of the
next draw.
12. The dynamic behaviour of sales over
time rests largely on the choice of n and N, which determine the
probabilities of winning the different prizes and the likelihood
of a rollover. If the game is easy to win, then rollovers are
infrequent and double rollovers very infrequent so each draw is
much the same as the next. The danger of the game being too easy
to win is that players become bored with the monotony of playing.
Estimates in an earlier paper[13]
suggested that the "half-life" of the UK game would
have been approximately 150 draws—sales would halve every
3 years (of weekly draws)—if there had been no rollovers.
What rollovers do is enhance the attractiveness of the next draw
so that players are enticed to then play more, come back to the
game, or join the game for the first time, and this effect takes
some time to decay. Thus, rollovers are an essential ingredient
of an attractive game to overcome its essential monotony.
13. However, a game that is too hard to
win will also be bad for sales in the long run. In the extreme
case imagine a game that was almost impossible to win, it would
rollover almost forever since sales would be very low and hence
few of the available combinations on sales in each draw would
be bought. But the size of the rollover would very slowly accumulate
and hence so would sales. This is an example of "intertemporal
substitution"—players sit on their hands waiting for
the jackpot to grow sufficiently large for the draw to become
attractive and only then play heavily. Even in less extreme cases,
rollovers give rise to intertemporal substitution since rollover
draws are more attractive than regular draws. While it is true
that extra sales occur when there is a rollover, this is, in part,
at the expense of sales in regular draws. Thus, designing the
game to maximise sales is a balancing act of making it hard enough
to win to overcome the tedium but easy enough to win to avoid
significant intertemporal substitution.[14]
14. It is important that the game design
matches the likely size of the market. A game that is sensible
for the UK is likely to be too hard for Israel whose population
is just 10 per cent of that of the UK.[15]
Economic Issues in Online Parimutuel Lottery Game
Design
15. The prize pool is defined by the take-out
rate, r, which is the proportion of sales (ie the stakes) that
is not returned as prizes. Thus, the overall prize pool is (l-r)S,
where S is sales revenue (in many games the cost of a ticket is
fixed at a unit of currency so S is both the number of tickets
sold and the level of sales revenue). It is common for the take-out
rate to be in the range of 40-50 per cent so that the pay-out
rate is 60-50 percent.[16]
16. Smaller prizes are usually awarded for
matching fewer than n numbers so it is common for the prize fund
to be split into separate pools. More complex designs are possible—for
example, in the Camelot game there is a seventh "bonus"
ball that is also used to define a prize pool for matching 5 of
the first 6 numbers drawn plus the bonus number. Thus, the overall
prize pool is usually divided into separate pools for funding
players who match all n numbers in the winning combination, match
n-1, n-2, etc. This set of prize pools might be characterised
by S=S1,S2,....,Sn. In the Camelot game S1=S2 = 0 and S3 is not
a share at all but a fixed payout and these match-3 prizes are
awarded first and the shares of the other prizes is defined out
of the residual.[17]
17. The odds of matching fewer than all
n numbers also depends on N: thus the odds of matching 3 in a
6/49 design is 1 in 57, while the odds of matching 3 in 6/53 is
1 in 71.[18]
Thus both n and N affect the number of prize winners for each
prize pool, and it is the shares that affect the amount of money
in each prize pool. Thus, the average amount won by each type
of prize winner depends on all of the design parameters of the
game.
18. So, for any specific n and N, the design
of the distribution of the prize money, through choice of r and
the si shares, affects the mean return from buying a ticket (which
is less than 1-r because of the rollover probability—the
higher the rollover probability the lower the return to the current
draw), the variance in returns around this mean, and the degree
to which the prizes are skewed towards large or small prizes.
The larger the share given to the jackpot the more positively
skewed is the distribution of prizes and the larger the share
given to the lower prizes the more negatively skewed is the distribution.[19]
The variance, depends of how much weight is given to middle as
opposed to extreme prizes. Note that, these "moments"
of the prize distribution are not independent of each other: for
example, re-allocating the prize money away from the easy-to-win
prizes and towards the hard-to-win prizes increases the skewness
but also increases the variance and lowers the mean (because there
is a chance that the jackpot rolls over).
19. One way of summarising the complications
of how many of the various aspects of game design impacts on sales
is through the mean, variance and skewness of the prize distribution.
To estimate the impact of these three moments of the prize distribution
on sales we would, ideally, like to conduct experiments where
the design features were changed and sales recorded—better
still, it would be useful to offer one group of individuals one
game design and a control group another design. In practice such
experiments are not available to us. In practice we observe either
no variation in game design over the history of sales or, at best,
changes in game design that the operator has chosen with a view
to increasing sales. That is, any variation in game design that
we may observe in ANY dataset is unlikely to tell us anything
useful about, for example, how sales would change if a policymaker
wanted to change the tax rate levied on the game.
20. The best we can do is to try to make
inferences about how sales would be affected by game design changes,
from the random variation in the terms on which people participate
that we can typically observe—and that is through the effect
of variations in the size of the jackpot on sales. The size of
the jackpot is a random variable in our data because rollovers
are statistically random events. The value of each of the moments
of the prize distribution depend on the game design parameters
and on the level of sales—for example, for any given design
the mean return on a ticket is higher the higher is sales. Thus
rollovers cause there to be exogenous variation in the nature
of the prize distribution.
21. Figure 1 shows the "expected value"
of a lottery ticket for common types of design in a regular (non-rollover)
draw. Expected value is the average return on buying a £1
ticket. In the figure the take-out rate r is set at 0.55 which
is a typical value and approximately the value used in the UK
Camelot lotto game. The shape of this figure has given rise to
what has been called lotto's "peculiar economies of scale"
since it shows that the game gets cheaper to play (in the sense
that the expected loss is smaller) the higher is sales.[20]
This is because: the higher is sales the smaller is the chance
of a rollover occurring because more of the possible combinations
are sold;[21]
this makes the return higher in the current draw because rollovers
take money from the current draw and add it to the next draw;
and your ticket in this draw gives you a possible claim on prizes
in this draw but not the next. So the higher the chances that
a jackpot rolls over the less a ticket for the current draw is
worth. Note that at very large levels of sales all games have
the same mean return which simply equals the 1-r, because the
chance of a rollover is small when ticket sales are large since
most possible combinations will be sold. Notice also that at any
given level of sales easier games offer better value in regular
draws since the rollover chance is smaller.
22. Figure 1 shows the situation for regular
draws. However, when a rollover occurs, the mean, variance and
skewness all change and the way in which they change can be calculated
from a knowledge of the determinants of these moments. In Figure
2 we show, for a 6/49 design, how mean, variance and skewness
vary with sales and how these relationships are shifted when there
is a small rollover (£4m) and a large rollover (£8m).
Rollovers make a difference in kind to the relationship between
the moments and the level of sales for the following reason. In
regular draws players simply play against each other for a slice
of the overall prize pool which comes from stakes in the current
draw—since players play against each other any addition
to the prize pool is matched by additional potential winners.
But in rollover draws players are also playing for the jackpot
pool from the previous draw[22]
and the value of this extra gets spread more thinly as more tickets
are sold. Thus, the relationship between sales and the mean of
the prize distribution (also known as the expected value) is made
up of what would happen in a regular draw (as shown in Figure
1) that would have the upward sloping economies of scale characteristic
plus the value of the previous jackpot which falls as sales rise
because its value gets spread more thinly the more players are
competing for this fixed sum.
23. Thus, overall, as the top panel of Figure
2 shows, the expected value first rises (as the economies of scale
effect dominates) and then falls (as the competition for the fixed
rolled-over amount takes over and the economies of scale effect
flattens out).
24. The probability distribution implied
by the Camelot 6/49 prize structure has a large spike at zero,
since mostly players lose, and a further smaller spike at £10,
where 1 in 57 tickets match 3. For the larger prizes however,
it is not possible to describe the distribution as further spikes
because the amount won depends on the number of people who also
win a share in each prize pool. Instead of a spike, there is a
small peak with a (local) maximum in the distribution for each
prize type, which corresponds to the most probable number of winners
for that type, but around this is a distribution that arises because
there may be fewer winners each getting a large share of the pool
or more winners than expected each getting a smaller share. Successive
peaks, corresponding to the mean winnings of bigger prizes are
lower (as the chance of winning is smaller) and wider (because
the variance in the number of prize winners is higher for the
more difficult to win prizes). The overall distribution is thus
left skewed. The bottom two panels in Figure 2 show how the variance
and skewness of the prize distribution vary with sales and the
rollover size. A rollover decreases the left skew (ie increases
(right) skewness since it increases the size of the jackpot pool.
25. Figure 3 shows the effects of rollover
size on the mean, variance and skewness for two levels of sales,
typical of Wednesday and Saturday draw levels. A rollover affects
only the top prize increasing (right) skewness. Increasing ticket
sales has no impact on the two mass points corresponding to winning
nothing or ten pounds but increases the prize pool for the other
prizes and also the likely number of winners. With no rollover,
the first effect dominates[23]
and the increase in sales increases the expected value and the
peaks of the distribution corresponding to the higher value prizes
move rightward. With a rollover, however, the second effect dominates
for high sales, and, although the expected amount won for the
4, 5 and 5+ bonus prizes increases, the expected amount won in
the jackpot prize may decrease.
26. Table 1 shows the actual values of the
moments for typical examples. The message is that rollovers have
a large effect on the mean, variance and skewness of the prize
distribution, especially at low levels of sales, while the effects
of variation in sales (for a given rollover size) is relatively
small, especially at large levels of sales.
27. Few previously published papers have
looked at the modelling of lotto sales. One US example[24]
suggests that decreasing the take-out rate for the jackpot prize
could increase revenues (for the Florida state lottery).[25]
The decrease in take-out rate would have two opposing effects:
it would decrease revenue since, ceteris paribus, less
money is taken as profits, but the larger prizes made possible
would, however, increase sales and thus increase the "tax"
revenue raised. While the increase in sales would also decrease
the probability of a rollover, the increase in the probable size
of any rollover which does occur more than makes up for this.
8 This paper draws on Ian Walker's research with colleagues
at Keele University that was funded by ESRC under research grant
R000236821, and on Juliet Young's MSc dissertation at the University
of Warwick. We exploit data that has been provided by the Lotto-shop
website, the National Lottery Commissioners, the Consumers' Association,
and the Family Expenditure Survey and the British Social Attitudes
Survey which were provided by ESRC Data Archive at the University
of Essex with the permission of the Office of National Statistics.
We are grateful for the co-operation that we have received from
these agencies. The views expressed here are those of the authors
alone and do not represent the views of our employers. Back
9
See R Munting, An economic and social history of gambling in Britain
and the USA, Manchester UP, 1994 for the history of UK lotteries. Back
10
There are three areas where we have little to say: First, technology
affects both how games can be presented to players and the kind
of game that it is possible to organise. Parimutuel games that
allow players to choose their numbers require sophisticated computer
systems. But new technology also offers the prospect of Internet-based
games and games operated via mobile phones using SMS or WAP. The
technological possibility of international competition also imposes
constraints on the domestic market as well as offering further
market possibilities. Secondly, gambling can have adverse social
consequences and intelligent game design can be used to minimise
these. However, imposing constraints on game design because of
a concern over adverse social consequences will generally have
adverse consequences for sales so a trade-off may be involved.
For example, it might be regarded as better to have a large number
of small players than a small number of large ones. Finally, scratchcards
are a part of the portfolio of the UK game and we have little
to say about this since we do not have good data for them. Back
11
In the lottery industry "online" means games where
ticket sales are recorded electronically at a dedicated terminal. Back
12
That is, it is 60m divided by 14m. Back
13
See L Farrell, E Morgenroth and I Walker, "A Time Series
Analysis of UK Lottery Sales: Long and Short Run Price Elasticities",
Oxford Bulletin of Economics and Statistics, 61, 1999. Back
14
The problem is made more complex where there are other substitution
possibilities-for example, in the US it is possible that cross-state
substitution takes place. This gives rise to incentives for neighbouring
states to collude and share the proceeds of a single large game
rather than have two competing games. Back
15
In fact, the Israeli online lotto game has just been redesigned
from 6/49 (1 in 14m) to 6/45 (1 in 8.1m) precisely because the
operators felt that it was too difficult to win and rollovers
were too frequent-it is being promoted as "Less numbers,
bigger chances"). In contrast the game in Ireland (population
3.8m) has twice been redesigned to make it harder to win to induce
more rollovers. Indeed, the redesigns followed organised attempts
to "buy the pot" because large jackpots had accrued.
Under the new design, a 6/42 game so that the odds of winning
are 1 in 5.25m, there are more frequent but smaller jackpots.
In California (population 34m) the game began as 6/49, went to
5/53 and then to 6/51 (1 in 18 million) but, since June, has a
complex 5/47+1/27 design that gives extremely long jackpot odds
of 1 in 41.4m. In Florida (population 15m) the game has also recently
become more difficult, going from 6/49 to 6/53. Back
16
Care must be taken when comparing across games to recognise that
some games pay prizes as a lump sum (in the UK, for example) while
others (most US states) pay an annuity (or some heavily discounted
lump sum). Moreover, in some countries prizes (the USA) are liable
for income tax while in other countries (UK) they are not. Back
17
That is si = pi [(1-r)S-10.N3], where i = 4,5,5+b,6, N3, is the
number of players that match 3 of the numbers drawn, and pi is
a fraction. For example p6 = 0.52. Back
18
Gerry Quinn in Ireland provides a helpful website, http://indigo.ie/-gerryq/Lotodds/lotodds.htm,
that allows probabilistically-challenged readers to compute the
odds for many common game designs. Back
19
Games that are hard to win often feature large jackpot shares.
For example, in the Florida online twice weekly lotto draw the
odds of matching 5 of the 6/53 has a (relatively) high chance
but it has such a small share of the overall prize pool that it
is only, on average, worth approximately $5,000. That is, the
Florida lotto game is both hard to win and highly skewed. It is
the large jackpot that entices people to play in regular draws
even though there is a high chance that it will be rolled over
and won by someone in subsequent weeks. Back
20
See Cook, P J and C T Clotfelter, "The Peculiar Scale Economies
of Lotto" American Economic Review 83, 1993. Back
21
The rollover probability is (1-p6)s where n6 the jackpot odds
(1/14m in the 6/49 case) and S is the level of sales. Back
22
In principle, lower prize pools could also roll over but we have
no evidence that this has ever occurred in practice. Back
23
See Clotfelter, C T and P J Cook, Selling Hope: State Lotteries
in America, Harvard University Press, 1991. Back
24
J F Scoggins, "The lotto and expected net revenue"
National Tax Journal, 48, 1995. Back
25
See D Forrest, D Gulley and R Simmons, "Elasticity of demand
for UK National Lottery tickets" forthcoming in National
Tax Journal, (2000) for UK work that follows this line but
does not support the proposition. Back
| 
