Lottomania and other anomalies in the market for lotto
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Journal of Economic Psychology 22 (2001) 721±744
www.elsevier.com/locate/joep
Lottomania and other anomalies in the market for lotto
Michael Beenstock *, Yoel Haitovsky
Department of Economics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel
Received 2 September 2000; received in revised form 21 December 2000; accepted 27 February 2001
Abstract
This research is concerned with the determination of the demand for ``lotto'' in Israel.
While an important focus of our research is upon the eects on the demand for lotto of ticket
pricing and jackpot announcements, we also investigate several empirical phenomena that are
apparently inconsistent with expected utility theory. These include an eect we call ``lotto-
mania'' which is induced by rollover, and ``prize fatigue'' when the jackpot does not increase.
Another aberration from expected utility theory is that the underlying odds of winning have
no measurable eect on sales. Ó 2001 Elsevier Science B.V. All rights reserved.
PsycINFO classi®cation: 2300; 2340
JEL classi®cation: D4; H0; L8
Keywords: Gambling; Statistical probability; Rational choice
That the chance of gain is naturally overvalued, we may learn from the univer-
sal 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 tick-
ets are not really worth the price which is paid by the original subscribers, and
yet commonly sell in the market for 20%, 30%, and sometimes 40% advance.
*
Corresponding author. Tel.: +972-2-5883120; fax: +972-2-5816071.
E-mail address: msbin@mscc.huji.ac.il (M. Beenstock).
0167-4870/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 4 8 7 0 ( 0 1 ) 0 0 0 5 7 - 5722 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744
The vain hope of gaining some of the great prizes is the sole cause of this de-
mand. The soberest people scarce look upon it as a folly to pay a small sum for
the chance of getting 10 000 or 20 000 pounds; though they know that even that
small sum is perhaps 20% or 30% more than the chance is worth. In a lottery in
which no prize exceeded 20 pounds, though in other respects it approached
much 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 mathematics, than that the more tickets you adventure upon,
the more likely you are to be a loser. Adventure upon all the tickets in the lot-
tery, and you lose for certain; and the greater the number of your tickets the
nearer you approach to this certainty.
Adam Smith
The Wealth of Nations
(Chapter 10)
1. Introduction
In this paper we use Israeli time series data to estimate the demand for
lotto. Our work goes beyond previous empirical eorts (e.g. Borg, Mason, &
Shapiro, 1991; Cook & Clotfelter, 1993; Walker, 1998) in that in addition to
investigating the eects of prizes on lotto demand we estimate the eects of
ticket prices and the probability of winning. Our work extends the eorts of
Shapira and Venezia (1992, 1994), who used data for Israel to investigate the
eects of ticket prices, the probability of winning, and the prize structure on
the demand for lotto. Using lotto data such as ours they show that ticket
sales vary directly with the announced jackpot and the number of winning
tickets in the previous game. The latter result, they argue, re¯ects the practice
of using winnings to buy tickets in the next game. They also ran an experi-
ment using a ®xed pay-out rate of 50% among 100 students who rank-or-
dered dierent lotteries by price, probability of winning and prize structure.
Their choices indicate that larger jackpots are preferred to larger secondary
prizes, and more frequent secondary prizes are preferred to lower ticket
prices.
We also investigate the presence of psychological phenomena, which might
aect the demand for lotto, and which are inconsistent with expected utility
theory (EU). According to EU the demand for lotto should increase withM. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 723 rollover because the expected value of winnings increases. We devise a test to see whether the demand for lotto increases by more than what EU would predict. We ®nd that rollover has a positive eect on demand, which is in- dependent of its eect via the announced jackpot. We refer to this eect as ``lottomania'', which is inconsistent with EU. The estimation of many of these eects is made possible by the length of our observation period which covers close to 600 weekly games of lotto played over 1112 years. During this period the real price of a lotto ticket varied considerably as did the odds of winning the jackpot. By contrast, Cook and Clotfelter used data for 17 months' drawings in Massachusetts (where the game is played twice weekly) in the 1980s, and Shapira and Venezia used data over a span of 26 months, in Israel, where the game was played weekly. While time series research may shed light on the determinants of lotto demand as they evolve over time, microdata are required to understand the determinants of lotto demand by individuals. Scott and Garen (1994) used microdata for Kentucky to estimate individuals' demand for lotto. They conclude that the demand for lotto has a \-shaped relationship with income and age, that the unemployed are more likely to play lotto, while married people are less likely to play. This type of evidence suggests that the demand for lotto is likely to depend on the socio-demographic structure of the pop- ulation. Eects such as these are naturally dicult to estimate using time series data since the necessary (weekly) data on income distribution, etc., are not generally available. On the other hand, time series data lend themselves to investigating ``lottomania'' and related phenomena. Quiggin (1991) has argued that when lotto players are homogeneous, ex- pected utility theory (based on Friedman & Savage, 1948) suggests that a single prize lotto is optimal, whereas rank-dependent expected utility (RDEU) theory suggests that a multiplicity of prizes may be optimal. Since multiple prize lotto is standard (in Israel there are six prizes), this appears to be inconsistent with EU. However, there are at least two objections to this line of argument. First, lotto players are unlikely to be homogeneous. Sec- ondly, the policy of oering multiple prizes may be sub-optimal. Our methodology for investigating aberrations from EU is based on the behavior of ticket sales rather than the structure of prizes. We propose an empirical model of the demand for lotto in which lotto players are heterogeneous and where their observed behavior is not necessarily consistent with EU, RDEU or any other particular nonexpected utility theory. Instead, we ®nd that the demand for lotto varies directly with the declared jackpot and inversely with the price of a lottery ticket. Lotto demand increases dramatically as rollover
724 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744
becomes more protracted, and it also depends on the number and compo-
sition of the minor prizes.
Some of our empirical ®ndings are consistent with EU and its derivatives.
These include the negative eect on lotto sales of higher ticket prices and the
positive eect of higher announced jackpots. However, there are several
®ndings, which are apparently inconsistent with EU. One is the ``lottomania''
eect mentioned above. Another aberration is ``prize fatigue''; if the same
jackpot is announced (so that expected value is unchanged) the demand for
tickets tends to fall. A further aberration is that the demand for lotto tickets
is apparently independent of the underlying odds of winning.
We stress that these results are obtained from real-life data as distinct from
experimental data. For example, the results of Shapira and Venezia con-
cerning the eects of ticket prices and prize structure on the demand for lotto
were based on experimental data. There is an obvious advantage to using real
data as opposed to experimental data, since the former re¯ect genuine, real-
life behavior whereas the latter may re¯ect arti®cial behavior. Our data on
lotto games provide a rare empirical opportunity to investigate the deter-
minants of risky behavior in general, and gambling in particular. Therefore,
the aberrations mentioned in the previous paragraph cannot be dismissed as
resulting from experimental design. In Section 2 we present our data. The
theoretical framework is developed in Section 3. In Section 4 we estimate the
demand for lotto in Israel. Section 5 concludes.
2. Lotto in Israel
In this section we describe lotto as played in Israel during the sample
period, 1 and we present the key variables that are used in the econometric
exercises reported in Section 4. Lotto in Israel is managed by a government-
owned company, Mifal Hapayis, which also operates a number of other
lotteries. Lotto players currently choose 6 out of 49 numbers on the ticket.
The weekly draw normally took place on Tuesday evening when six numbers
were drawn from an urn. An ``additional number'' was also drawn which
plays a role in the lesser prizes. To win the ®rst prize it is necessary to guess
correctly all six numbers. If there is more than one winner the prize is shared
equally. If there are no winners the prize money is rolled over. If the jackpot
1
Since 2000 the game is played two times per week instead of only once.M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 725
is not won for four consecutive weeks the game normally recommences and
the unwon prizes are distributed as lesser prizes. Exceptionally, there may be
more than three rollovers. For example, during the Gulf War in early 1991
there were ®ve rollovers due to the paucity of ticket sales.
Fig. 1 presents the distribution of rollover over time. In the late 1980s there
was a ®rst-week winner in about 70% of draws. However, this subsequently
fell to about 50%. The incidence of second-week winning has grown from
around 15% in 1985 to around 35% ten years later. The incidence of third-
and fourth-week winning has grown to 20% from 14%. These trends re¯ect
the lower odds of winning as discussed below.
To win the second prize it is necessary to guess 5 out of 6 and the ``ad-
ditional number''. To win the third prize it is necessary to guess 5 out of 6, the
fourth prize 4 out of 6, and the ®fth 3 out of 6. A sixth prize was introduced
in May 1993 (2 out of 6 and the ``additional number'').
A lower bound for the ®rst prize is announced on Wednesday for the fol-
lowing game. It is based on the lotto managers' assessment of ticket sales. If
sales exceed their expectations the prize will be greater than the announcement
because there is a predetermined pay-out rate set by the lotto management.
But if sales are less than expected, the prize is equal to the announcement, and
the loss is absorbed by Mifal Hapayis. A comparison between the announced
prize and its actual value indicates an average dierence (the announced prize
Fig. 1. Percentage of time that the jackpot was won by week.726 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744
being less than the actual prize) of about 4% between 1985 and 1994 and only
0.5% in 1995±1996. The correlation between the announced prize and its ac-
tual value (in the event of a win) is 0.976. The announced prize is a policy
variable. The more generous the announcement, the more likely it is that
Mifal Hapayis will suer a loss, but ticket sales may be boosted. Below we
exploit this institutional detail to identify ``lottomania''.
Lotto policy consists of deciding on pay-out rates, ticket prices and the
number of items on the card. Since 1993 the latter has been 49. However, it has
been increased from 38 in 1985, to 42 in 1989 and to 45 in 1991, thereby re-
ducing the odds of winning the jackpot from one in 2,760,681 to one in
13,983,816. At the same time the real price of a ticket has more than doubled
(Fig. 2) with most of the increase incurring in the second half of the 1980s. The
curious pattern in Fig. 2 is due to the fact that in¯ation has ranged between
10% and 20% during the observation period, while ticket prices are adjusted
periodically. Hence in¯ation erodes the real price until the next adjustment.
The pay-out rate on all prizes was 50% until May 1993 when it was raised
to 52% (see Table 1). It was increased further to at least 55% in September
1996. Therefore the eective price of lotto has slightly fallen to the consumer
since the game has become fairer. On the other hand, the combined eect of
higher ticket prices and lower odds has made lotto eectively more expensive
to consumers solely interested in the jackpot. However, the game has been
made more attractive by increasing the declared jackpot (Fig. 3). Although
Fig. 2. Price of lottery ticket.M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 727
Table 1
Pay-out rates (%)
Prize 1987 September 1989 August 1993 March 1996 September
1 15 20 21.4 20.9
2 2.5 1.5 1.1 1
3 5 4 2.7 2.6
4 7.5 6 4.8 4.7
5 20 18.5 18.2 17.6
6 0 0 5.4 8.3
Total 50 50 52±55 55
rollover naturally imparts a high degree of variance to Fig. 3, the underlying
jackpot may be discerned from the trend in the ®rst-week prize which rose 10-
fold in constant 1993 prices from 0.3 million shekels in 1985 to 3 million
shekels in 1996.
A historical summary of pay-out rates is given in Table 1. In the second
half of the 1980s the pay-out rate on the jackpot was 15%, i.e., 30% of the
overall pay-out rate of 50%. In 1989 it was raised to 20% and in 1995 to 22%
of revenue. Pay-out rates on other prizes (excluding the sixth prize) have
fallen. For example, in the case of the second prize it fell from 2.5% to 1%,
while the ®fth prize fell from 20% to 17.5%.
Fig. 4 plots ticket sales (by volume) over time. Here too, although rollover
imparts a high degree of volatility, the underlying trend may be discerned by
Fig. 3. Announced jackpot.728 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744
Fig. 4. Ticket sales.
Fig. 5. Relationship between ticket sales and announced jackpot.
®rst-week sales. Sales were static until late 1990 whence they have grown by
about 50%.
Fig. 5 plots ticket sales against the announced jackpot. It clearly suggests a
positive relationship between them. Since the overall pay-out rate has beenM. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 729
more or less constant, Fig. 5 suggests that the policy to give a greater weight
to the jackpot has boosted ticket sales.
3. Theoretical framework
3.1. Decision calculus
We assume that there are K heterogeneous potential lotto customers with
archetypal utility function of the form
Uk F Gi ; pi ; P ; Zk ek ; 1
where Gi denotes the money received if the individual wins the ith prize
i 1; 2; . . . ; N , pi denotes the probability of winning the ith prize by
purchasing a single ticket (e.g. in the case of the ®rst prize it is currently
1:13,983,186), P denotes the price of a lottery ticket, and ek captures
unobserved heterogeneity. Note, for example, that Gi is not the prize
money itself (which we denote by Ji ), but its shared value after other
winners have been taken into consideration, i.e., Gi Ji =wi , where wi
denotes the number of winners of the ith prize. We denote by /i the
density function for wi as de®ned in Eq. (5) below. N, the number of
prizes, is currently 6. Zk denotes a vector of socio-economic controls in-
cluding income and other variables that might in¯uence the individual's
demand for lottery tickets. We assume that partial derivatives of k's utility
function with respect to prizes and their probabilities are positive and that
Uk varies inversely with P.
Although Gi , pi and P are the same for all individuals, individual k can
increase his/her chance of winning by buying more tickets. Denoting the
number of tickets by Tk , the probability that he/she will win the ith prize will be
pi Tk
pki 1 e : 2
According to expected utility theory the form of Eq. (1) from playing lotto
would be
XN X1
Ji
E Uk pki /i wi F Wk Tk P
i1 w1
wi
!
XN
1 pki F Wk Tk P ek ; 3
i1730 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744
where Wk denotes the initial level of wealth. Since lotto is an unfair game, i.e.,
X
N
pi Ji aP ; 4
i1
where a, the pay-out rate, is less than 1, a positive demand for lotto implies,
as demonstrated by Friedman and Savage (1948), that the marginal utility of
wealth eventually increase. A formulation based on RDEU would replace p
in Eq. (3) by f p such that f 0 > 0, f 0 0, f 1 1, and f > p when p is
close to 0 and f < p as it approaches unity. It can be shown that in this case a
positive demand for lotto does not require that the marginal utility of wealth
eventually increase. Since we do not know whether EU, RDEU, or any other
speci®cation is appropriate we prefer not to restrict Eq. (1).
Given Ji , pi and P the individual will play lotto only if Uk > 0, in which
case Tk P 1, otherwise Tk 0, and the number of tickets purchased will vary
directly with Uk .
3.2. Prize dilution
We now turn to the relationship between G J =w and J. Since more than
one person may win, the rational individual, before deciding to play the
game, has to allow for the fact that the prize may be shared and therefore
diluted. To simplify matters we assume that each player buys one ticket. If
there are T players, each choosing a combination of n numbers randomly, the
probability that there will be w winners is
pT w e pT
/ w : 5
w!
In the simple case in which there is only one prize (N 1) and each player
buys only one ticket the expected value of G will be equal to pJ paPT
divided by the expected number of winners, which from Eq. (5) is
X
T
1 pT w
pT
E G p R aP 1 T e ; 6
w0
1w w!
where R denotes the amount rolled over. It may be shown that when R 0
the ®rst derivative of E G with respect to T is positive, while the second is
negative, implying that E G increases monotonically with T, but at a de-
creasing rate. Furthermore, it tends to aP as T tends to in®nity, even in the
case when R > 0. This implies that expected winnings are increasing in theM. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 731
number of players, and that the scale effect due to larger T unambiguously
dominates the dilution effect. However, when there is a particularly large
rollover E G may cease to increase monotonically with T; it may converge
on aP from above instead of from below as T tends to in®nity. In this case the
dilution effect dominates the scale effect.
Cook and Clotfelter (1993) have shown that Eq. (6) may be approximated
as
R aP 1 T pT
E G 1 e : 7
T
It turns out that the approximation is very close for plausible values of T and
p.
Obviously, G in Eq. (1) is a random variable because the number of players
and the number of winners are both random variables. It also suggests that
the rational individual must have formed expectations about T when deciding
to play lotto. Eq. (6) implies that Uk varies directly with T in which case the
more players there are the more likely it is that the the faint-hearted will join
in. As the prize increases Uk rises and increases the number of players for
given assumptions about the unobserved heterogeneity, e. Whether or not
they join in according to expected utility theory remains to be seen in Section
4.
3.3. Equilibrium
In principle, we need to determine in equilibrium both the number of
players and the number of tickets sold. This is because the latter determines
the prize money while the former is necessary because dilution refers to in-
dividuals rather than tickets. A complete solution would require solving Eqs.
(2)±(6) simultaneously for given assumptions about the distribution of cus-
tomer heterogeneity. We do not intend to engage in such an exercise. Little is
lost, however, by assuming that each customer, if he/she participates, buys
only one ticket. This avoids the problem of distinguishing between individ-
uals and tickets. In this case we can use the analysis presented in Section 3.2.
The nature of the equilibrium is illustrated in Fig. 6 where schedule OA plots
the relationship between T and E G implied by Eq. (1); the greater is E G
the more people will have positive U, in which case the number of players will
increase with E G. We have drawn OA on the assumption that participation
in lotto is convex and it has a natural upper limit of 100%. The precise
shape of schedule OA depends on the distribution of heterogeneity among732 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744
Fig. 6. Equilibrium in the lotto market.
customers. While OA is monotonically increasing in E G, in some cases it
may do so in an S-shaped fashion rather than as indicated in Fig. 6.
Schedule OB plots the relationship between E G and T that is implied by
Eq. (6) when R 0. It is drawn convex toward the horizontal axis, in the light
of the discussion of Eq. (5), and is asymptotic to aP . The jackpot varies
linearly with T and the slope of schedule OC varies directly with aP . Note
that Fig. 6 should be drawn such that J > E G since there is a positive
probability that the prize will be shared. The equilibrium will be at point e.
Since expectations are ful®lled at e, the equilibrium is consistent with rational
expectations (see Scott & Gulley, 1995).
In principle, there may be multiple equilibria since schedule OA may in-
tersect OB more than once. A necessary condition for multiple equilibria is
that OA be locally convex. In the event of multiple equilibria, the highest
stable one would generate most pro®ts for the lotto company.
Clearly this analysis is at best an approximation since we have assumed in
Fig. 6 that each player buys only one ticket. The purchase of multiple tickets
increases the probability of at least one winner, but decreases the probability
of sharing, when compared to the same number of tickets bought individu-
ally. As T increases existing players are likely to buy more tickets.
An autonomous increase in the demand for lotto due, for example, to
demographic change would shift the OA schedule upwards. The increase in
ticket sales implied by the new equilibrium will incorporate an induced eect
since E G will be greater at the new location of e. The same applies to au-M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 733
tonomous shifts in the OB schedule due, for example, to an increase in the
pay-out rate. At the new equilibrium ticket sales will be higher. An increase in
the price of a ticket lowers OA via Eq. (3) and shifts OB to the right via Eq.
(6), hence the net effect of price on the direction of change of equilibrium
sales is indeterminate, depending upon the elative slopes of the two schedules
in the vicinity of their intersection.
If e happens to occur on the upper reaches of schedule OA, changes in the
pay-out rate a (which shift OB) will have a relatively small effect upon sales.
This is likely to occur in large, saturated lotto markets. If e happens to occur
on the lower reaches of schedule OB, autonomous changes in the demand for
lotto (shifts in OA) are likely to have a relatively large effect upon sales. This
is likely to occur in small lotto markets where economies of scale have yet to
be exploited. If schedule OA happens to be S-shaped the analysis would be
considerably more complicated.
Finally, rollover shifts OB to O0 B0 as indicated, where O0 ±O aT1 P is the
amount rolled over from the previous week. As already noted, if R is large
enough O0 B0 may cross aP and converge upon it from the right. If schedule
OA remains unchanged, ticket sales in the second week of the game will rise
to T2 . If, additionally, schedule OA happens to shift upwards due to ``lot-
tomania'', the increase will be even greater. We de®ne ``lottomania'' as ap-
parently irrational shifts in the demand for lotto induced by rollover. That is,
movements along schedule OA are rational in the sense that they are con-
sistent with expected utility theory, whereas shifts in schedule OA are irra-
tional in the sense that they are inconsistent with expected utility theory. In
Section 4.1 we suggest an empirical test for distinguishing between ``lotto-
mania'' and nonconvexities in schedule OA, i.e., we can identify separately
movements along the schedule from shifts in it.
3.4. Jackpot announcement
As mentioned in Section 2 the minimum jackpot (JA) is announced a week
in advance. Because it is a minimum, Jt max JAt ; aPTt Rt . Suppose the
demand for tickets in week t is hypothesized to be
Tt c bEt 1 Jt hRt ut ; 8
where u N 0; r2 and h > 0 captures lottomania eects. The expected value
of Jt is the truncated mean:
Et 1 Jt lt r/ xt = 1 U xt ; 9734 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744
where xt JAt lt =r and lt is the untruncated mean. The truncated mean
naturally increases with JA. If expectations are rational the untruncated
mean may be solved as
lt aPEt 1 Tt =JAt 0 Rt ;
which from Eq. (8) is
lt P ac 1 akRt = 1 ab Rt : 10
Substituting Eqs. (9) and (10) into Eq. (8) implies that the demand for tickets
is of the form
Tt c bH JAt ; Rt hRt ut ; 11
where H is a nonlinear function increasing in both of its arguments. The
lotto authority will set JA at time t 1 according to the sort of criteria
discussed by Beenstock, Goldin, and Haitovsky (2000). Note that because JA
is determined at time t 1; E JAt ut 0. Hence the eect of jackpot an-
nouncements on the demand for tickets is identi®ed; JA is a natural instru-
ment for J. Moreover, JA has a high public pro®le; it is widely advertised in
all the media, and it no doubt has a major in¯uence on individuals' decision
to purchase lotto tickets. We therefore use the ex ante jackpot JA rather than
the ex post jackpot (J) in our regressions.
4. Results
4.1. Empirical framework
Our objective is to estimate equations based on Eq. (11) which explain the
number of tickets sold (T) in each draw (see Fig. 4). By contrast, Cook and
Clotfelter (1993) and Walker (1998) instrument the current prize by rollover,
since it is predetermined as of time t. No doubt Mifal Hapayis uses infor-
mation on rollover when it announces JA. However, JA is not perfectly
collinear with rollover data in Israel, implying that Mifal Hapayis does not
apply a simple rule to project JA on the basis of rollover data. For example,
the R2 between JA and R (the sum rolled-over) is 0.71 and the R2 that results
from regressing JA upon the cumulative number of rollover is 0.66, both of
which statistically signi®cantly different from unity. Since, as discussed in
Section 2, the announced jackpot is a crucial marketing parameter, Mifal
Hapayis most probably determines JA strategically, and does not thereforeM. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 735 apply a simple projection rule based on rollover data alone. We argue below that because JA is suf®ciently independent of rollover data, it has identifying power. It enables the separate identi®cation of ``lottomania'' effects, as well as the more standard effect of the jackpot on ticket sales. We have already remarked in Section 2 that JA is an imperfect predictor of the jackpot itself. Since, if there is a winner, J max JA; R aPT , Mifal Hapayis loses money when JA > R aPT . Hence JA incorporates a risk premium which is part of the reason that JA cannot be simply projected from rollover data. Although Eq. (11) is presented in its static form, the estimation is carried out dynamically in terms of error correction models (ECMs), in which we nest down from unrestricted dynamic models containing lags of the explan- atory variables to restricted dynamic models. Indeed, we ®nd that the de- mand for lotto is highly dynamic both in terms of lagged dependent and explanatory variables. We use the RESET test to determine the degree of nonlinearity in the speci®cation of the model. We started out by assuming linearity, which clearly failed the RESET test. A loglinear speci®cation improves matters, but it too clearly fails the RESET test. We experimented with dierent speci®- cations until the RESET criterion was satis®ed. It turns out that the demand for lotto is not loglinear in both prices and prizes. The Z variables in Eq. (11) consist of scale effects as well as psychological effects. In the absence of weekly data on scale variables such as consumer spending, we experiment with time trends and with expenditure on other lotteries. Psychological effects are largely expressed in terms of dummy variables in the number of rollovers. For example, WEEK3 assumes a value of unity if there have been two rollovers (i.e., the game has rolled over into the third week). Rollover may induce a sort of ``lottomania'' in which ticket sales increase by considerably more than implied by the unusually large jackpot. In terms of Fig. 6 this is represented by an upward shift in schedule OA. An alternative explanation for increases in sales after rollover is that there are local nonconvexities in schedule OA. To distinguish between shifts in OA induced by ``lottomania'' from movements along nonconvex segments of OA, we specify dummy variables such as WEEK3 to capture shifts in OA in week 3 and a polynomial in JA to capture possible nonconvexities. Since, as we have already shown, the an- nounced jackpot is only imperfectly collinear with WEEKr r 1; . . . ; 4, the ``lottomania'' and the jackpot eects are separately identi®ed. Moreover, we check for possible arbitrary parametric identi®cation by specifying the level
736 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 of JA and WEEKr rather than various nonlinear transformations of JA. If JA were highly correlated with WEEKr , so that there was no identifying information in JA, nonlinear transformations of JA might induce spurious identi®cation. As reported below, it turns out that the identi®cation is not spurious; the joint signi®cance of JA and WEEKr is independent of the way JA is transformed. This is because JA is suciently independent of WEEKr , as already noted. We also test for ``prize fatigue'' by investigating whether the public ap- petite for lotto wanes if prizes are not increased. We do so by including in Z a series of dummy variables that assume a value of unity if JA is greater than some previous value and 0 otherwise. Clearly, phenomena such as ``lotto- mania'' and ``prize fatigue'' are aberrations from expected utility theory. Other components of Z include the underlying odds of winning the jackpot and changes in the structure of secondary prizes. In principle, the latter at- taches importance to all the prizes and not just the jackpot. In practice, as explained below, there has been insuf®cient independent variation in the secondary prizes to be able to obtain ®rm estimates of their effect. Cook and Clotfelter (1993) have shown for Massachusetts that specifying the jackpot ®ts the data somewhat better than models which specify rollover and ex- pected values (of all prizes). Walker (1998) speci®es expected value rather than the jackpot. Our approach, which attaches particular, but not exclusive, importance to the jackpot implies that it is the skewness of the prize distri- bution that is important and not just its expected value. Indeed, this is the intuition expressed by Adam Smith above (and in the ``Discussion'', espe- cially by Schweizer, of Walker's paper). The data presented in Section 2 and which we use for estimating Eq. (11) are evidently nonstationary. Recent developments in unit root econometrics (see e.g. Enders, 1995) suggest that the residuals from a static version of Eq. (11) will have to be stationary if the demand for tickets in the long run is to be explained by the covariates that feature in Eq. (10). It is for this reason that we present both static and dynamic speci®cations. 4.2. Static speci®cation The simplest speci®cation, model 1 in Table 2 is loglinear, ignores prize fatigue, but attaches importance to two and three rollovers. Since the test statistics reported at the foot of Table 2 indicate that the residuals of the model are autocorrelated and heteroscedastic we present Newey±West HAC- consistent standard errors (i.e., adjusted for heteroscedasticity and autocor-
M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 737
Table 2
Static models (dependent variable: ln T ) (observation period: 8 June 1985 ± 17 December 1996 N 594)
Model 1 2 3 4
g^ SD g^ g^ SD g^ g^ SD(^
g g^ SD g^
Intercept 9.856 0.304 10.188 0.306 10.892 0.290 24.998 3.406
ln(JA) 0.415 0.021 0.392 0.021 0.356 0.020 )1.640 0.463
ln P )0.663 0.085 )0.661 0.009 )0.191 0.122 )0.437 0.165
Week3 0.075 0.030 )3.770 1.038 )3.179 0.984
Week4 0.478 0.071 )2.330 2.395 )1.857 2.300 0.5 62 0.077
Week3ln(JA) 0.249 0.067 0.213 0.063
Week4ln(JA) 0.179 0.149 0.154 0.143
Change1 )0.139 0.059 )0.127 0.05
Change2 )0.181 0.033 )0.142 0.029
Change3 0.152 0.032 0.144 0.032
Change4 )0.005 0.031
Ln JA2 0.070 0.016
Ln P 2 )0.697 0.231
Less4 )0.311 0.085
Days 0.036 0.008
Summer )0.060 0.017
R2 (adjusted) 0.772 0.785 0.840 0.857
Standard error 0.1671 0.162 0.140 0.132
LM 146.3 152.2 91.0 62.7
ARCH 130.7 163.6 75.2 60.0
White test 9.59 7.94 9.5 4.47
Chow test 0.27 0.37 0.55
Reset test 45.52 24.74 0.36 0.045
Notes: (1) LM test for fourth-order serial correlationv20:05
9:49. (2) ARCH test for sixth-order ARCH
v20:05 14:44. (3) White test for heteroscedasticity
(v20:05
11:07 for model 1 )21.03 for model 4). (4)
Chow test for predictive stability 1995±1996 F0:05 1:11. (5) RESET test for linearity t 1:96. Pa-
rameter standard errors are corrected for heteroscedasticity and serial correlation using a ®fth-order
Newey±West lag truncation.
relation using a lag truncation of 5 periods) of the parameter estimates.
Model 1 implies that the long-run elasticity of demand for lotto with respect
to the jackpot is 0.415, while the price elasticity is )0.663. There is no sta-
tistically signi®cant rollover eect in the second week, however, the eects in
weeks 3 and 4 are 8% (the antilog of 0.075) and 61% (the antilog of 0.478),
respectively.
Model 2 interacts the rollover dummies with the jackpot to test the hy-
pothesis that the rollover eect is not independent of the jackpot. The model
implies that the rollover eect varies directly with the jackpot and that the
prize elasticities are 0.392 in weeks 1 and 2, 0.641 ( 0.392 + 0.249) in week 3
and 0.571 ( 0.392 + 0.179) in week 4. Although we do not apply formal tests738 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 for cointegration it turns out that the model residuals in Table 2 are auto- correlated but stationary. For example, in the case of model 2, the ®fth-order augmented Dickey±Fuller and Phillips±Perron statistics are )4.9 and )11.6, respectively, implying that the estimated residuals are stationary. This in turn implies that the static regressions reported in Table 2 are not spurious and are reasonable speci®cations of the long-run demand for lotto. To test for spurious identi®cation we reran model 2 specifying JA instead of its logarithm. All the parameters which were statistically signi®cant in model 2 remain so in the modi®ed model. For example, the ``t''-value for ln JA in model 2 of 18.67 becomes 11.96. Had JA been highly collinear with WEEKr the ``t'' value for JA would have been low. Here and elsewhere we conclude that the ``lottomania'' effect is separate from the jackpot effect, and that this conclusion is not the arbitrary consequence of nonlinear transfor- mations of JA. We have already noted, in relation to the discussion of Table 1, that it is dicult to investigate the eects of the minor prizes on the demand for lotto. This is because the prize distribution was changed only three times during the observation period, and more than one parameter was varied in each change. In model 3 we specify cumulative dummy variables to capture these discrete changes. Change1 takes a value of 0 before September 1987 and 1 subse- quently. Change2 is 0 prior to August 1989 and 1 subsequently. Change3 is 0 prior to March 1993 and 1 subsequently. Change4 is 0 prior to September 1996 and 1 subsequently. Model 3 indicates that 3 out of the 4 dummies are statistically signi®cant. For example, the coecient of change2 implies that the decision to lower the share of the secondary prizes in August 1989 low- ered the demand for lotto by almost 20%, while the decision to introduce a sixth prize in March 1993 raised the demand for lotto by 15%. However, the most pro®table prize regime prevailed prior to September 1987. In September 1996 the share of the sixth prize was raised further in the belief that the demand for lotto depends upon the probability of winning per se, even for small sums. However, the fact that Change4 is not statistically signi®cant may be due to its proximity to the end of the observation period. Finally, model 4 shows that the prize and price elasticities are not constant. The estimates imply that the absolute elasticities are increasing in J and P, respectively. For example, the prize elasticity (de®ned as 0:072 ln JA 1:64) rises from 0.448 to 0.617 when the prize increases from 3 million shekels to 10 millions, and the price elasticity falls from )0.126 to )0.691 when the ticket price increases from 0.8 shekels to 1.2 shekels. Note that in model 4 WEEK3 ceases to be statistically signi®cant, its role apparently has been intermediated
M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 739 by the variable elasticities. However, WEEK4 continues to be statistically signi®cant, suggesting that the increase in sales in the fourth week is due to ``lottomania'' rather than nonconvexities induced by heterogeneity. Model 4 indicates additionally that the demand for lotto is seasonal; sales tend to be 6% lower in the summer. Because of religious festivals some lotto weeks have fewer selling days. Model 4 implies that lotto sales fall by about 3.6% for each lost selling day. Less4 is a dummy variable which assumes a value of unity if the prize in week 4 is less than what it was on the previous occasion that there were three rollovers. The negative coecient implies that the public loses interest on a massive scale (36%) unless the ``big one'' is at least as big as its predecessor. All the models in Table 2 are structurally stable as suggested by the Chow- tests. However, models 1 and 2 fail the RESET test. It turns out that to pass this test it is necessary either to specify nonloglinearity in prize, or to include Change3 in the model. Curiously, we found no eect (and so do not report the estimated model) of the probability of winning (as determined by the number of items on the card) on ticket sales, implying that the public was apparently indierent to the lengthening of the odds over the observation period from 1:2,760,681 to 1:13,983,816. This result is robust under dierent nonlinear transformations, e.g. specifying the objective probability, or specifying the number of items on the card. Perhaps, following Kahneman and Tversky (1979), the public cannot distinguish between dierent very long odds when they are minute. Another variable that was not statistically signi®cant, and therefore is absent from models 1±4, is a time trend designed to pick up scale eects due to economic and demographic growth, which were substantial during the observation period. This (non)result is surprising too, implying that lotto does not share in the secular growth of the economy. Following Scott and Garen (1994), who showed that the ``lotto tax'' is regressive, it may be the case that the secular trend in income distribution in Israel in favor of the rich has oset the trend in the economy as a whole insofar as it aects the demand for lotto. Another possibility is that there is a secular trend in ``prize fatigue''; public interest in lotto declines over time suciently strongly to oset the increase in demand expected from secular economic growth. We found that lotto sales were substantially depressed during the Gulf War (from the second half of January 1991 to the beginning of March) when Israel was attacked by Iraqi scud missiles, and buying tickets was dicult. However, we do not report these ®ndings in the interest of space. In the case of model 1, for example, the addition of a Gulf War dummy left the other
740 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 slope coecients unchanged while the dummy coecient was estimated to be )0.39. Nor do we report the ®ndings of an alternative model that suggested that sales are particularly large in week 1 and depressed in week 2. 4.3. Dynamic speci®cation The dynamic regressions reported in Table 3 have been obtained by ap- plying the so-called ``general-to-speci®c'' methodology in which an unre- stricted dynamic model is nested down to the restricted dynamic models reported in the table. In model 1 a third-order lag structure on the dependent variable and ln JA turns out to be appropriate, which naturally improves the goodness-of-®t and greatly reduces the severity of serial correlation. The coecients of the lagged dependent variables imply that there is substantial inertia in lotto demand. If demand has been unusually high during the last three weeks it is likely to remain high, albeit with some regression to the mean. The coecients on the lagged jackpots are negative implying that demand becomes depressed in the wake of high prizes. This implies that to maintain demand the public has to be oered by ever-increasing prizes. The implied long-run prize elasticity of demand is 0.156/0.469 0.35 (i.e., 0:321 0:054 0:058 0:045 1 0:257 0:17 0:104), while the price elasticity is )0.43. These fall within the range of estimates suggested by the static models reported in Table 2. In this context care should be taken in calculating the long-term eects of rollover. For example, the long-run eect of rollover in the fourth week is not 0.042/0.469 since this calculation assumes that such rollovers occur weekly instead of only 3±4 times a year. The same applies to all the other variables (such as less4) which are occasional rather than weekly phenomena. The models in Table 3 include additional psychological variables which capture ``prize fatigue''. High7 and High8 are dummy variables which as- sume a value of unity if the declared jackpot exceeds 7 and 8 million shekels, respectively, while Less34 is a dummy variable which assumes a value of unity if the prize in either or both of weeks 3 and 4 is smaller than what it was on the last occasion of a third or fourth week. Model 2 in Table 3, which combines both dynamic and nonlinear speci- ®cations, has the smallest standard error of estimate (0.088) of all ®ve models reported. Several variables that were statistically signi®cant in model 1 cease to be so in model 2, and vice versa. For example, change3 is no longer sta- tistically signi®cant while time appears as a polynomial with positive coe- cients. The latter implies a positive and strengthening time trend in the
M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 741
Table 3
Dynamic models (dependent variable: ln T )
Model 1 2
g^ SD g^ g^ SD g^
Intercept 4.997 0.499 3.202 2.614
ln T 1 0.257 0.029 0.319 0.043
ln T 2 0.170 0.028 0.0038* 0.00081
ln T 3 0.104 0.027 0.00085* 0.0004
Change 2 )0.098 0.020 )0.140 0.028
ln(JA) 0.321 0.018
ln JA 1 )0.054 0.016 1.032 0.2 78
ln JA 2 )0.058 0.015 )0.045 0.014
ln JA 3 )0.045 0.015
ln JA 5 )0.392 0.194
ln JA2 0.0145 0.0008
ln JA 1 2 )0.0385 0.01
ln JA 5 2 0.0134 0.006
Change3 0.051 0.015
ln P )0.202 0.063 )0.664 0.091
ln P 2 )0.672 0.124
Week2 )0.119 0.018
Week3 )1.618 0.833
Week4 0.447 0.065
Week3 ln JA 0.119 0.054
Week4 ln JA 0.042 0.006
High7 0.177 0.055 0.217 0.051
High8 )0.163 0.050 )0.154 0.050
Less4 )0.238 0.077 )0.236 0.072
Less34 )0.087 0.037
Days 0.022 0.007 km0:10 0.034
Time3 10b 4.02 1.58
Time4 10b )6.13 2.48
Summer )0.027 0.009 )0.021 0.009
Last8 )1.590 0.601 )0.0001 0.00008
Gulf War )0.261 0.018
Standard error 0.097 0.088
LM test 6.252 4.70
Chow test 0.58 0.73
Whitetest 4.59 4.98
ARCH test 6.00 1.52
RESET test 2.51 1.47
See notes to Table 2; denotes square of covariate.
demand for lotto. In model 2 ``lottomania'' ceases to be endogenous (i.e.,
interactive with the jackpot, which is endogenous) but occurs strongly in
week 4. Week2 now has a negative coecient, implying that the demand for
lotto tends to weaken with the ®rst rollover.742 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744
In summary, in both the static and dynamic models ``lottomania'' occurs
consistently in week 4, i.e., after three rollovers. Whether this eect is en-
dogenous or autonomous depends upon the speci®cation of the model. In
most of the speci®cations there is evidence of ``lottomania'' after two roll-
overs. In nearly all the models there is evidence of ``prize fatigue''.
5. Conclusion
We have used Israeli data to show that the demand for lotto varies directly
with the announced jackpot and inversely with the price of a ticket. The
respective long-run elasticities are about 0.4 and )0.65. We also conclude
that ``lottomania'' rather than customer heterogeneity explains the demand
for lotto especially following the third rollover. This implies that the eect of
jackpot announcements on the demand for lotto is path dependent. 2 A given
jackpot announcement in the fourth week will have a much greater eect
than in the ®rst. Our results suggest that rollover induces an independent
source of excitement and interest in the game. This eect con¯icts with ex-
pected utility theory (as well as RDEU), as do several other eects, such as
prize fatigue, and the absence of an eect of the underlying odds of winning
on lotto sales.
It is easier to say which theories are rejected by our results than to say
which theories they support. The ``lottomania'' eect is arguably consistent
with the ¯ip-side of Loomes and Sugden's (1982) ``regret'' theory. They argue
that the experience of regret that arises from adverse outcomes of avoidable
risks induces individuals to be more conservative than EU. The ¯ip-side of
regret is jubilation. The jubilation experienced from very favorable outcomes
of avoidable risks induces individuals to be less conservative than EU.
Winning the jackpot arguably ®ts into this category. The thrill of winning
from this particular avoidable risk exceeds the conventional utility calculus of
EU.
The ``prize fatigue'' eect is arguably consistent with rational addiction
theory of Becker and Murphy (1988), who argue that acquired tolerance to
drugs induces the need to increase the dose. In the case of gambling in general
and lotto in particular, our results regarding ``prize fatigue'' suggest that
consumers acquire tolerance to prizes, implying that the dose has to be in-
2
The policy implications of this path dependency are discussed in Beenstock et al. (2000).M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 743 creased if the public is to remain turned-on. However, this tolerance is short- lived; after a few months its eects appear to have worn o. Our results shed some light on Quiggin (1991), who asked whether it is better to have a single prize than a multiplicity of prizes. Unfortunately, the prize structure did not change with sucient frequency during the sample period to enable serious testing of this issue. Nevertheless, our ®nding that the introduction of a sixth prize in 1993 induced an increase in sales and that the decision to lower the share of the second prize in August 1989 induced a decrease in sales can be interpreted that the public prefers multiplicity. Finally, there is negative conditional autocorrelation in the demand for lotto, implying that bad weeks (in terms of sales) follow good weeks, and vice versa. This information may be used to improve short-term sales forecasts that are undertaken prior to announcing the next jackpot. Our results imply that by sharpening its forecasts the lotto authority can raise the demand for tickets by announcing more generous jackpots. The policy implications of our results for determining pay-out rates, and other policy parameters are discussed by Beenstock et al. (2000). Acknowledgements We wish to thank Mifal Hapayis for providing data and Natasha Kipnitz for her able research assistance. We are grateful to the Eshkol Institute for ®nancial assistance in completing the research. References Becker, G. S., & Murphy, K. M. (1988). A theory of rational addiction. Journal of Political Economy, 75, 675±700. Beenstock, M., Goldin, E., & Haitovsky, Y. (2000). What jackpot? The optimal lottery tax. European Journal of Political Economy, 16, 655±671. Borg, M. O., Mason, P. M., & Shapiro, S. L. (1991). The economic consequences of state lotteries. New York: Praeger. Cook, P. J., & Clotfelter, C. T. (1993). The peculiar scale economies of lotto. American Economic Review, 83, 634±643. Enders, W. (1995). Applied time series analysis. New York: Wiley. Friedman, M., & Savage, L. J. (1948). The utility analysis of choices involving risk. Journal of Political Economy, 56, 279±304. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47, 263±291.
744 M. Beenstock, Y. Haitovsky / Journal of Economic Psychology 22 (2001) 721±744 Loomes, G., & Sugden, R. (1982). Regret theory: An alternative theory of rational choice under uncertainty. Economic Journal, 92, 805±824. Quiggin, J. (1991). On the optimal design of lotteries. Economica, 58, 1±16. Scott, F. A., & Garen, J. (1994). Probability of purchase, amount of purchase, and the demographic incidence of the lottery tax. Journal of Public Economics, 54, 121±143. Scott, F. A., & Gulley, O. D. (1995). Testing for eciency in lotto markets. Economic Inquiry, 33, 175±188. Shapira, Z., & Venezia, I. (1992). Size and frequency of prizes as determinants of the demand for lotteries. Organizational Behavior and Human Decision Making, 52, 303±318. Shapira, Z., & Venezia, I. (1994). The eects of lottery prize structure on demand: Theory and ®ndings from Israel. Economic Quarterly, 215±227 (Hebrew). Walker, I. (1998). The economic analysis of lotteries. Economic Policy, 27, 357±402.
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