Elasticity of Demand and Optimal Prize Distribution for
Scratch-Off Lottery Tickets
By Michael Coon and Gwyn Whieldon∗
This study explores the optimal payout structure and prize distribution of scratch-off lottery games. Using ticket sales data for 185
scratch-off lottery games sold between 2007 and 2011 by the Maryland State Lottery and Gaming Control Agency (MLGCA), we
calculate the price elasticity of demand across a range of ticket denominations. Our findings suggest that increasing the payout rate
of scratch-off tickets will increase revenue, particularly among low
denomination tickets. We also find that reallocating the current
prize distributions within a game towards fewer but more valuable
low-tier prizes will increase the number of tickets sold.
I.
Introduction
State-run lotteries currently operate in the majority of US States and Territories, with
43 states, the District of Columbia and the US Virgin Islands running some combination
of draw and instant games. The Maryland State Lottery and Gaming Control Agency
(MLGCA), for example, reported over $1.756 billion dollars in lottery net sales for 2013
across draw games (subdivided further into numbers, monitor, and matrix games) and
instant games (Maryland Lottery and Gaming Control Agency, 2013). As lotteries represent
such a large source of state government revenue, and as these games create an ideal public
laboratory to model consumer behavior and purchase decisions, studying the structure and
microeconomic theory of lotteries provides a rich source of problems for economists.
Instant lottery games, also called scratch-offs, accounted for approximately 27.7% of all
lotto sales offered in Maryland in 2013. With instant ticket sales making up such a large part
of the portfolio of the MLGCA and that of other state lottery operators, identifying ways
to improve the structure of (and revenue from) these games is highly desirable. Unlike draw
lottery games, instant games vary notably in cost, with tickets available in denominations
of $1, $2, $3, $5, $10, and $20. Even within a fixed price point, the wide range of effective
price, prize distributions and jackpots make for a rich set of options potential players may
choose amongst, and over fifty games are available to players in any given week. We examine
here the elasticity of demand in response to effective price and payout structure for such
instant ticket games, and argue that in the instant game market, the lottery operator’s
current prize payout rates do not optimize revenue in scratch-off tickets.
Almost all current work in the economics of lotteries focuses on draw lottery games,
primarily studying features of “matrix” and “numbers” games.1 In draw games, players
select between three and seven numbers from a fixed list, with prizes awarded based on
type of game and number of matches on a player’s ticket to those selected in a random
drawing. Draw games may be held within a single state for small prize lotteries or run
∗ M. Coon: Department of Economics and Management, Hood College, Frederick, MD 21701, [email protected]. G.
Whieldon: Department of Mathematics, Hood College, [email protected].
1 For an extensive review of the current literature on the economics of lotteries, see Grote and Matheson (2011).
1
2
collaboratively between several lottery markets, typically with several draw game options
available to players simultaneously. There are mixed empirical results on whether such
games are complements or substitutes for one another, with Forrest et al. (2004) finding
some substitution effects between instant ticket purchases and draw lottery games. However,
Grote and Matheson (2006) found that US lottery games tend to complement one another
with players purchasing tickets for multiple types of games concurrently.
One style of draw lottery include daily numbers games (e.g. Pick 3 and Pick 4,) where
each drawing has a fixed prize payout based on type of bet placed (“straight” bets where
the numbers must be matched in an exact order pay out 500 to 1, “6 way” bets where
three distinct digits may be matched in any order pay out 80 to 1, etc.) Although the prize
levels for daily numbers games do not vary from draw to draw, the elasticity of demand
for tickets in response to jackpot size in other lotteries (the “halo effect” or “lottomania”)
has been studied in Chen and Chie (2008) and Matheson and Grote (2005). Some recent
work by Combs et al. (2013) has focused on estimating own-price elasticity in response to
promotions run by the lottery operators.
The most frequently studied draw games are matrix lotteries, where prizes are awarded
according to how many numbers in a drawing match those selected by the player. Most
large-jackpot draw lotteries operate as perimutuel games with roll-over if no jackpot winning
numbers have been selected by players in the pool. The effective price, or the difference
between the purchase cost of a ticket and the expected value of potential winnings, is
highly-dependent on the value of these jackpots. While the ticket puchase price is fixed,
the long odds against drawing a large-prize winning ticket (and the corresponding rollover
contribution to the next jackpot) result in a spread of effective prices for these numbers
games.
Numerous articles currently in the literature seek to model elasticity of demand in response to the effective price of numbers game lotteries. For example, Forrest et al. (2000)
suggest that the UK National Lottery (a parimutuel numbers game) has an optimal current
take-out rate of 50% based on an estimated price elasticity of demand close to -1. Similarly,
Papachristou (2006) evaluates a pair of numbers games offered in Greece with different prize
structures and odds, examining if lottery demand elasticity in response to effective price
is a reliable marketing tool. Modeling has also been done on the elasticity of demand in
response to the size of the jackpot itself. Forrest et al. (2002) argue that the demand for
the lottery is more a function of jackpot size than of effective price.
These empirical studies have found that state and national lotteries have close to an
optimal takeout rate, but the scope of these articles has been limited to large-jackpot
numbers games, which make up only a part of most state lottery portfolios. Little appears
in the literature on the elasticity of demand for scratch-offs or other quick-play games in
response to either jackpot size or design of payout structure. Similarly, there are no studies
currently available that assess if the government payout rate for prizes is optimal for instant
tickets.
Beyond the proportion of ticket purchase-price paid out to players in the form of prizes,
we also examine whether the distribution of prizes in instant games have optimal structure.
In previous studies of numbers lotteries, De Boer (1990) and Thiel (1991) respectively
conclude that lengthening the odds of winning a jackpot (and increasing roll-over jackpot
sizes) attracts more players and increases revenue. In a more recent paper, Forrest et al.
(2010) also find that sales of some numbers games increase as the jackpot odds decline. In
the same work, they suggest that demand for numbers lottery will increase as the number
SCRATCH-OFF TICKET ELASTICITY
3
of lower tier prizes rises.
We find similar patterns hold in instant games, with more ticket sales expected in response
to reallocating prizes. Our results suggest that increasing prize amounts at the low-tier and
jackpot levels, while decreasing the overall number of available prizes, will increase both
ticket sales and revenue within a given denomination.
Some of the differences in structure between previously studied draw games and our
instant ticket sales lead to results which are slightly more nuanced than those currently
appearing in the literature. In particular, while an increase in the jackpot of various numbers
games has been shown to increase ticket sales and overall participation in the lottery, our
results suggest that for instant games jackpot size is only significant when interacted with
the price-point of the ticket. This would suggest that increasing the jackpot of an instant
game may not induce players to purchase higher denomination tickets (for example, to buy
a $20 ticket over a $10 ticket,) but it may increase the number of tickets purchased within
the player’s desired range.
II.
Data and Empirical Model
Our analysis utilizes weekly sales data for 185 scratch-off lottery games offered by the
Maryland Lottery between July 2007 and June 2011. Following standard practice in the
lottery literature we measure price elasticity as the percentage change in quantity of tickets
sold with respect to a change in the effective price of the ticket, where the effective price
of the ticket is measured as the expected loss per dollar gambled. Variation in effective
price for our dataset comes from two sources. The first is that scratch-off tickets vary in
the denomination of the ticket. That is, consumers have the option of purchasing tickets
ranging in price from $1 to $20. As an enticement to get consumers to gamble in bulk,
higher denomination tickets have a higher expected payout rate. For example, $1 tickets
are designed to pay 59% of the revenue they generate in prizes, whereas a $10 ticket pays,
on average, 75% of the revenue it generates in prize money. Thus, consumers can expect
to lose 41 cents for each $1 ticket they purchase, but if they purchase a $10 ticket they
will only expect to lose 25 cents per dollar gambled. The second source of variation occurs
within each ticket denomination, and is largely a result of the production process. When the
MLGCA orders tickets from their supplier they set specific parameters on quantity, payout,
and jackpot sizes, but allow for a margin of error on some of the parameters. For example,
they may place an order of 2 million tickets with a denomination of $1, a payout rate of
59%, and twenty-five tickets which pay jackpots of $1000 each. During printing, however,
a certain number of tickets may be destroyed because of misprints, or may be torn or
improperly cut, so that the producer only delivers 1.987 million tickets. Such an occurrence
is typically deemed acceptable as long as the number of jackpot tickets is correct and the
actual payout rate falls within some acceptable range around the desired 59%. Therefore,
while each price point has an expected payout rate, the realized payout rates for each game
within that price point will vary around that expected rate.
To control for these two sources of variation we estimate the following model
(1)
ln(y) = α + β1 ln(ep) + β2 ln(d) + β3 ln(ep) × ln(d) + ΓX + ,
where y is the number of tickets sold, ep is the effective price measured as expected loss
per dollar gambled, d is the denomination, or face value, of the ticket, and X is a vector of
4
controls specific to each game. Thus, the elasticity can be estimated as
(2)
ED = b1 + b2 ln(d)
for each ticket denomination.
Additional controls include the probability of winning any prize, the share of revenue
allocated to prizes under $25, the share of revenue allocated to prizes between $26 and
$600, the size of the jackpot and its interaction with the ticket denomination, and the
number of other games available at the time the game was launched. The probability of
winning any prize ranges from 22.6% to 40.8% for all tickets, but the mean value increases
as the ticket denomination increases (see Table 1). Our controls for distribution of prize
values follows MLGCA’s classification of prize tiers. Tier 1 prizes are any prize of $25 or
less. The share of prizes that fall in Tier 1 range from 26% to 89% with a mean value
of 71.7%. The mean value declines with respect to ticket denomination, however there is
substantial variation within each price category. Tier 2 prizes are any prize valued between
$26 and $600. The share of prizes in Tier 2 ranges between 4.4% and 56% with a mean
of 23.4%. This mean value tends to rise with ticket price, but again with wide variation
within each denomination. Tier 3 prizes are any prize valued at more than $600. The share
of prizes that fall into Tier 3 range from 0 (some games have a jackpot as low as $333) to
23.9% with a mean of 4.9%. These means also tend to rise with ticket denomination. Our
final control with respect to prizes is (the log of) jackpot size. As mentioned above, the
minimum jackpot for any game in our study is $333. The largest jackpot is $1 million (see
Table 2). Jackpot size is directly related to ticket denomination. In almost all cases the
maximum jackpot for a given ticket denomination is less than the minimum jackpot for the
next highest ticket denomination. The only exception to this rule is that approximately
10% of the $5 games have jackpots exceeding the minimum jackpot of $100000 for the $10
tickets. Because of this direct correlation between jackpot size and ticket denomination, we
also include a term to account for their interaction. Our final control variable is the number
of other games that were on sale at the same time, which allows us to control for the degree
of competition each game faces while on sale.
A new game’s ticket sales follow a typical pattern in which sales start off very strong,
then drop rapidly over the first ten weeks. This is followed by a slow steady decline over the
duration of the game, which usually ends when 97% of the tickets are sold (see Figure 1).
Because of the dynamic nature of weekly sales and the static nature of our independent
variables, we estimate our model using cumulative tickets sold in the first 13 weeks and 26
weeks after a game is launched. Estimating our model in this manner allows us to treat the
data as cross-sectional.
III.
A.
Results
Elasticity
Table 3 presents OLS estimates of Equation 1. The dependent variable in Column 1
is cumulative sales after 13 weeks, and the dependent variable in Column 2 is cumulative
26 week sales. As expected, in both columns the coefficients on both the effective price
of the tickets and their denomination are negative and significant. The coefficient on the
interaction term is positive and significant. This indicates that while consumers respond
SCRATCH-OFF TICKET ELASTICITY
5
0
Mean # of tickets sold in week
50000
100000
150000
Mean Weekly Ticket Sales Across All Games
0
10
20
30
40
50
60
70
# of weeks since launch
80
90
100
Figure 1. Average Game Sales by Duration
positively to increases in the payout rate, i.e. a decrease in effective price, the response
is smaller for individuals purchasing larger denomination tickets. Thus, ticket sales can
be increased by reducing the effective price of the tickets, i.e. increasing the payout rate.
Whether this increases total revenue, however, will depend on the price elasticity of demand
for scratch-off tickets.
Table 4 presents estimates of the price elasticity of demand for the different ticket denominations based on equation 2. Elasticity estimates of 13 week sales range from ED = −6.5 for
$1 tickets to ED = −1.5 for $20 tickets. Given that revenue is maximized where ED = −1,
these results indicate that scratch-off tickets are all over-priced, particularly for low denomination tickets. Based on our estimates, we can expect that raising payout rates by 20% will
increase 13 week sales of $20 tickets by approximately 30%. Raising payout rates of $1 tickets by the same amount will cause 13 week sales to more than double. However, given the
large disparity in the elasticities across ticket prices, payout rates for lower denomination
tickets should be increased significantly more than higher denomination tickets.
It should be noted that previous studies of price elasticity of lottery games indicate an
optimal payout rate of approximately 50% (see Walker (1998) and Forrest et al. (2000).)
Given that scratch-off games already payout much more than 50%, it is worth exploring why
the evidence presented above suggests increasing, rather than decreasing, the payout rate.
The stark difference in optimal payout rates between scratch-off games and draw games is
likely due to the nature of play. That is, the instant nature of scratch-off lottery games may
lead to more dramatic changes in consumption in response to increases in the payout rate.
For example, consider a person playing a weekly drawing lottery game. Suppose this
person buys five $1 tickets on Wednesday. They must then wait three days until Saturday
to find out if they won. If they lose, they will likely wait until the following Wednesday and
purchase five more $1 tickets. If they win, say $10, they may also wait until the following
Wednesday to cash in their ticket, at which time they will purchase five more tickets and
pocket the remaining $5. Now, consider a person playing scratch-off lottery tickets. They
purchase five $1 tickets on Wednesday. After scratching off all five tickets and not winning,
they are convinced that the winning ticket is one of the next ones, so they buy five more.
Or, suppose they win $10. Feeling the intense rush of a win, they cash in their ticket and
buy ten more, in the hopes of replicating that winning feeling. Thus, the instant nature of
scratch-off games can potentially increase the likelihood that players will chase losses and
6
recycle wins, both of which will increase sales. Increasing the payout rate will enhance both
of these behaviors. Players going too long without a win may become disenchanted and
potentially stop playing. Likewise, players winning more may be more likely to continue
playing with their winnings. Thus, given the instant nature of scratch-off games, the payout
structure should more closely mirror those of casino slot machines, rather than draw games.
B.
Prize Distribution
In addition to finding the optimal payout rate, prize distribution will also influence ticket
sales. For example, as shown in Table 3, the coefficient on the overall probability of winning
any prize is negative and significant. Specifically, increasing the share of winning tickets in
a particular game by one percent, other things equal, is associated with a 3.35% decrease
in sales over the first 13 weeks, and a 3.58% decrease over the first 26 weeks. While this
may seem counter-intuitive given that the goal of consumers playing the lottery is to win,
previous research by Haisley et al. (2008) has found that players prefer long odds, large win
games to fairer games with smaller payouts. Therefore, increasing the total payout simply
by increasing the number of winning tickets may not be the most efficient way of increasing
revenue.
Perhaps the most obvious method of increasing payout while preserving (or reducing) the
overall probability of winning would be to increase the size of the jackpot. However, this
may not necessarily lead to an increase in sales. As shown in Table 3, in both columns,
the coefficient on jackpot size is not significantly different from zero. The coefficient on the
interaction term between jackpot size and denomination, however, is. Thus, the impact of
jackpot size depends entirely on the denomination of the ticket (See Table 5). For $1 tickets
changing the jackpot size has no effect. For a $2 ticket, doubling the jackpot size will lead
to 12.6% increase in the number of tickets sold in the first 13 weeks, and an 11.5% increase
over the first 26 weeks. Doubling a $20 ticket’s jackpot will increase the number of tickets
sold by 54.3% and 49.7% over 13 and 26 weeks, respectively.
A more consistent method of increasing ticket sales by increasing payout is to apply the
payout to lesser prizes. As seen in Column 2 of Table 3, increasing the share of prize money
dedicated to Tier 1 and Tier 2 prizes by one percentage point increases the number of tickets
sold in the first 26 weeks by 2.94% and 2.75%, respectively. This suggests that increases
in the payout rate, particularly for low denomination tickets, should be directed to lower
tier prizes. However, since the coefficient on overall probability is negative, it should not
be directed to increasing the number of lower tier prizes, but their value. That is, it would
be more effective to use $20 in additional prize money to convert 2 $5 prizes into 2 $15
prizes than to create 20 new $1 prizes. This fits in with the story outlined above regarding
recycling winnings. It is conceivable that a ticket winning $1 might go uncashed, if the cost
of taking it back to the store is considered to be greater than $1. But a $20 ticket is more
likely to be cashed, and recycled. Plus, if there are fewer winning tickets, the consumer is
more likely to chase losses after recycling the ticket.
IV.
Conclusion
Our findings suggest that the optimal payout rate for instant lottery games is significantly higher than that of draw lottery games previously studied in the literature. The
results presented above indicate that increasing the payout rate of instant lottery games
will increase revenues. This is particularly true for low denomination tickets, which have
SCRATCH-OFF TICKET ELASTICITY
7
a lower payout rate than higher denomination tickets. While we find some evidence that
increasing jackpot size can increase sales, we find that it is more effective to increase the
share of prize money dedicated to lower tier prizes. However, increasing the overall number
of prizes is expected to reduce sales. Thus, the most advantageous strategy would be to
increase total prize money by offering fewer, but larger low tier prizes and increase jackpot
sizes on higher denomination tickets.
V.
Tables
Table 1—Payout Rates by Ticket Denomination
Ticket Denomination
1
2
3
5
10
20
Mean
0.5918
0.6390
0.6724
0.7173
0.7495
0.7763
Standard Deviation
.0060654
.0051262
.0056522
.0062313
.0017385
.0064507
Minimum
0.5821
0.6208
0.6556
0.6905
0.7466
0.7650
Maximum
0.6040
0.6427
0.6790
0.7230
0.7539
0.7853
Table 2—Jackpot Size by Ticket Denomination
Ticket Denomination
1
2
3
5
10
20
Mean
1130.02
11460.56
31521.74
106662.59
128571.43
783500.00
Standard Deviation
674.36
5291.31
5097.86
174791.90
60356.09
155528.70
Minimum
333
7777
25000
50000
100000
650000
Maximum
5000
30000
50000
714850
250000
1000000
8
Table 3—OLS Estimates: Dep. Var. = ln(Cumulative Tickets Sold)
(1)
13 Weeks
-6.5206∗∗∗
(1.3097)
(2)
26 Weeks
-5.8393∗∗∗
(1.0310)
ln(Denomination)
-1.3255∗∗
(0.6437)
-1.4917∗∗∗
(0.4917)
ln(Effective Price) x ln(Denomination)
1.6689∗∗
(0.8118)
1.3963∗
(0.7116)
Overall Probability
-0.0335∗∗
(0.0161)
-0.0358∗∗
(0.0144)
Share of Tier 1 Prizes
0.0236
(0.0144)
0.0294∗∗
(0.0131)
Share of Tier 2 Prizes
0.0213
(0.0143)
0.0275∗∗
(0.0131)
ln(Jackpot)
-0.0708
(0.1134)
-0.0029
(0.0892)
ln(Jackpot) x ln(Denomination)
0.1812∗∗
(0.0709)
0.1659∗∗
(0.0665)
# of Games at launch
-0.0032
(0.0020)
-0.0006
(0.0017)
Constant
7.7291∗∗∗
(2.3153)
171
0.7419
64.1097
7.5688∗∗∗
(1.7796)
154
0.7886
75.2046
ln(Effective Price)
N
r2
F
Robust standard errors in parentheses
∗
p < 0.10,
∗∗
p < 0.05,
∗∗∗
p < 0.01
SCRATCH-OFF TICKET ELASTICITY
Table 4—Estimated Price Elasticity of Demand by Ticket Denomination
Ticket Denomination
1
2
3
5
10
20
Elasticity (13 Weeks)
-6.521
-5.364
-4.687
-3.835
-2.678
-1.521
Elasticity (26 Weeks)
-5.839
-4.871
-4.305
-3.592
-2.624
-1.656
Table 5—Estimated Elasticity of Jackpot Size by Ticket Denomination
Ticket Denomination
1
2
3
5
10
20
Elasticity (13 Weeks)
0
0.126
0.199
0.292
0.417
0.543
Elasticity (26 Weeks)
0
0.115
0.182
0.267
0.382
0.497
9
10
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