Introduction

State-sponsored lotteries are extensively used to raise revenue for good causes such as education and the arts. Worldwide the annual value of tickets bought in 1998 exceeded $51bn (see La Fleur and La Fleur¹ for references). States often contract out the running of such games to an operating company with a proportion of the cost of each ticket going to good causes or to general tax revenue and the contractor taking a further proportion as operating costs; most of the remainder is returned to participants as prizes. For example, in 1998 out of every pound spent on the UK National Lottery (UKNL), about 50 pence went to prizes, 28 pence to good causes, 13 pence to Customs and Excise as excise duties, 5 pence to the retailer as commission, 3 pence to cover the operating company (Camelot) operating costs, and Camelot kept 1 pence as profits. The tax rate and the level of participation will affect the total good-causes revenue. Participation depends on the tax rate and the design of the lottery, in particular, the probability of winning a jackpot prize. In order to study the effects of changing such parameters on the good-causes revenue it is necessary to model their effect on participation. In this paper we present such a model and use it to address the issue of whether the current tax rate and jackpot probability are the appropriate ones to maximize good-causes revenue in the UKNL.

A common feature of lotteries is that, if there are no winners in a given draw, the jackpot prize pool from that draw is added to the pool for the next draw: a rollover. The increased pool typically induces a higher level of participation. This suggests good-causes revenue may be increased by making rollovers more likely and this can be achieved by reducing the probability of winning a jackpot. However, doing so may also have a disincentive effect on participation in draws that do not feature a rollover and thus offset any increase in revenue. To capture such a trade-off effectively requires a dynamic model of individual participation. One possible approach would be to fit a 'black box' model (eg, regression-based) using standard statistical methods. However, the similarity and stability of lottery designs means that the available data offers little natural variation and poor prospects of a good fit, particularly if one controls for differences in country-specific propensities to gamble. Our approach is to work from first principles and build a model of individual participation based on standard inter-temporal models of individual behaviour under uncertainty. Lack of variation in the data forces us to adopt the simplest model parameterization. Nonetheless, this is sufficient to capture the principal features of observed behaviour. Since quantitative verification of the model is not possible, we check that the qualitative predictions of the model are plausible and consistent with casual observation. Our final step is to aggregate the behaviour of individual agents into a full model of demand for lottery tickets, which enables us to examine the effects of changing lottery design and tax rates.

There is little literature on the issues discussed here and (to our knowledge) none which bases the optimal design problem on a dynamic model of individual behaviour. Farrell et al² equate the effective price of a ticket in the UKNL to the ticket price less the expected value of the prizes and use the variation in the latter in rollover weeks to estimate the elasticity of demand. Farrell et al³ analyse the same data using use time-series methods. Although both these papers find values for the short- and long-run elasticities roughly consistent with revenue maximisation, they implicitly assume risk neutrality and neither addresses inter-temporal substitution by participants. Furthermore, they do not disentangle the effects of the tax rate and the lottery design and, using point estimates, they cannot investigate non-local changes of the parameters.

Scoggins⁴ offers rank-dependent utility as an explanation of the complex prize structure. An interesting paper by Simon⁵ attempts to construct a model that fits the first couple of years of participation in the UKNL. Although it is too tuned to UK data to be useful as a general tool for addressing design problems, perhaps its most interesting contribution is an attempt to model how observed attrition in participation could be counteracted by the high probability of winning small fixed prizes. This is usually offered as justification for the presence of small fixed prizes. We have not explicitly included this (small) effect in our model as it will not show up in the high-frequency limit we employ.

We conclude the Introduction by giving an informal description of our model. Full details of each component of the model are then set out in subsequent sections.

Lottery draws take place at regular intervals and, at each draw, consumers decide how many tickets to purchase. The cost is measured in foregone expenditure on other goods. Each ticket displays a set of positive integers (1 to 49 for the UKNL) from which participants must choose a 'combination' ie, a subset of the available integers (of size 6 for the UKNL). After all purchases of tickets have been made, the Lottery organisers randomly draw a combination according to a uniform distribution (see Haigh^6,7 and references therein). A ticket holder whose chosen combination matches that drawn is a jackpot winner and his/her prize is determined by dividing the jackpot prize pool equally amongst all the jackpot winners. The jackpot prize pool is a fixed proportion of the prize pool augmented by any amount rolled over from previous draws. Such rollovers occur if there are no jackpot winners in a given draw, in which case the jackpot prize pool is held over and added to the jackpot prize poo1 in the next draw. A run of draws with no winners can lead to such a large jackpot prize pool that behaviour regarded as undesirable (such as attempting to buy all combinations) can occur. To avoid this, a limit on the number of consecutive draws that can be rolled over is imposed. If this limit is reached, we assume, for expositional convenience, that the money in the accumulated jackpot prize pool is simply lost from the system. The remainder of the prize pool is allocated by looking at partial matches of the subsets chosen with that drawn. In our model, we concentrate on jackpot prizes where winning is rare (1 in 14 million chance for the UKNL) with values large enough to change the life of winners and other prizes that are low in value and common (more than 1 in 60 chance for fixed-value prizes in the UKNL).

Knowing the prize structure, we model participants' decisions of how many tickets to purchase as a dynamic optimisation problem. Since the number of tickets bought is very large (over 60 million per draw in the UKNL), we assume that consumers treat rollovers as exogenous, ignoring the minuscule effect their individual actions have on their occurrence. Nevertheless, aggregate levels of participation determine (stochastically) the rollover process. This results in a circular model in which the rollover process affects participation and the latter affects the rollover process. Rather than specify the dynamics of the adjustment process, we analyse the equilibrium in which we have rollover-contingent participation levels that are optimal given the rollover process determined by those participation levels. Once the full network of terminals was installed, participation levels rapidly stabilised in the UKNL, suggesting that equilibrium had been reached.³

In the next section, we develop our model of consumer choice as a stochastic dynamic optimisation problem. In the third section and the Appendix, we analyse the limit as the time interval between draws approaches zero. By regarding this limit as a first order approximation to the dynamic problem, we are able to recast the consumer's optimisation problem as a static, deterministic, non-linear programming problem. In the fourth section we study the Kuhn–Tucker conditions for this problem and show that the unique optimal solution captures several qualitative properties of the behaviour we are seeking to model. The next section closes the model by relating rollovers to participation levels. This necessitates the construction of a simple sub-model of how combinations are chosen. Finally, we apply the model to the UKNL and draw some tentative conclusions on the current values of the parameters.

Individual behaviour

We will suppose participants live indefinitely (or participants leaving the system are replaced with others having similar tastes) and divide their expenditure in period t between y_t on lottery tickets and x_t on other goods so as to maximise the expected value of the lifetime utility:

where =e^{- t} is the discount factor, is the instantaneous rate of discount, t is the time interval between draws and u is the instantaneous, von Neumann–Morgenstern rate-of-utility function in period t.

We have included lottery expenditure as a continuous variable, which will permit the use of calculus-based methods in the sequel. As tickets are indivisible, this raises the question of the interpretation of non-integral y. If restricted to purchasing whole numbers of tickets, it is not hard to see that, even in the absence of rollovers, a participant's optimal number of tickets will typically vary from week to week. However, if purchases are made more frequently (as we shall assume), the expected utility will be nearly the same as that of a stationary policy in which the (non-integral) long-run average number of tickets is purchased in each draw.

Friedman and Savage⁸ explain how gambling can be reconciled with risk averse behaviour in a static expected utility framework by including convex sections in the utility function. Dynamic versions of this approach based on (1) are easy to write down but Farrell and Hartley⁹ show that repeated purchase of lottery tickets cannot be explained using expected utility functions unless consumers derive direct value ('fun') from their purchases of tickets (or capital markets are imperfect). We shall also assume an absence of income effects in lottery purchases since this allows us to simplify the utility function and is broadly consistent with observation. The most general form of utility function consistent with these observations is

where v captures the consumer's attitude to risk and will be assumed increasing and concave, whilst the 'fun function' f expresses the direct consumption value of lottery purchases in terms of expenditure on other goods. We also permit the consumption value to depend on external factors, a^e, such as a rollover draw, as well as past history, a^h, including the consumer's past winnings, if any. We will specify these variables more fully in the sequel. Under such a specification, participation is induced both by the direct enjoyment gained from purchasing tickets (which is independent of whether a prize is won) as well as the extra wealth in the event of winning.

Rollover technology

We allow expenditure choices to be contingent on the number of weeks that the jackpot has been rolled over, to reflect the observed increase in ticket sales in rollover weeks.² We choose this way of allowing for rollovers to simplify the model. In fact making purchases contingent on the actual amount of money rolled over makes no difference in the high frequency limit used in the sequel. Furthermore, measured levels of participation contingent on weeks rolled over is subject to rather little variation.²

Let the (non-negative) integer-valued random variable R_t denote the number of weeks the jackpot has been rolled over into week t. More precisely, R_t=k if and if only the jackpot was won in week t- k- 1 but not in weeks t- k,...,t- 1 (or, if the jackpot has never been won and t=k+1). It follows that the transition probability P(R_t+1=j|R_t=k) is zero if j{0,k+1}. Thus, if K is the maximum permitted value of R_t, the latter follows a Markov chain with Transition Probability Matrix (TPM):

where _k=P(R_t+1=k+1|R_t=k)>0 for k=0,..., K- 1, and the rows and columns are entered in the order 0,1,2,...,K- 1,K. Since Q is uni-chained and acyclic, there is a unique limiting probability vector p=(p₀,p₁,...,p_K)0 which satisfies Q^Tp=p and _j=0^Kp_j=1. It is readily verified that p₀=^{- 1} and p_k=^{- 10}..._{k- 1} for k1, where =1+_j=0^K+1₀..._j.

Our assumptions on how a^e and a^h affect f are designed to reflect certain observations. Participation is typically higher in rollover draws than in normal draws. Higher potential winnings are an obvious explanation but it is also possible that the increased publicity and general excitement of participation associated with rollover draws increase the direct consumption value. To maintain the simplicity of the model, we ignore other environmental factors so that a^e depends only on R_t and write f_k(y) for the value of expenditure y on tickets when R_t=k. We also assume that the fun associated with purchasing tickets derives predominantly from contemplation of winning a prize large enough to change the winner's lifestyle substantially. We reflect this by including only a past jackpot win in the history a^h and, in particular, setting f0 for a consumer who has won the jackpot in the past. This will mean that such a consumer no longer participates.

Since participation and other consumption will typically depend on the rollover state and past winnings, if any, they are random variables. We write Y_t[X_t] for expenditure on lottery tickets [other goods] in period t where the process {X_t,Y_t}_t=0 is adapted to the rollover process {R_t}_t=0 and the individual winning process.

Thus, consumers maximise:

where the stopping time T denotes the period in which the consumer wins the jackpot. Note that T is a random variable that depends (probabilistically) on {Y_t}_t=0, and takes the value + for a non-participant.

Constraint

A consumer's expenditure is subject to the lifetime budget constraint:

where r=exp(t)- 1 is the rate of interest, m is income per period (assumed constant) and W_t* (y) is a non-negative random variable representing winnings in period t (taking the value {0} if the consumer does not win in period t) if y is spent on lottery tickets. We interpret the constraint as holding with probability one. This prevents borrowing on the strength of future lottery wins since for any >0 and pattern of participation, there is a positive probability that the net present value at t=1 of future winnings is less than .

For t>T, lottery winnings cease so that (3a) can be written:

where G_t(y) is the random variable representing the value of a non-jackpot prize in period t when y is spent on lottery tickets and takes the value 0 if no such prize is won. Similarly, ignoring the fact that a ticket cannot win more than one type of prize, W^k is a random variable representing the value of a jackpot prize in state k.

To solve the consumer's problem, we first look at the situation faced by a jackpot winner. This involves examining the optimisation problem for a consumer conditional on T=t^ and W^{k(t^)}=w. Discounted to the first period, this yields optimal utility of t^{t^- 1} (w;t) where:

We have assumed that winners continue to receive per-period income m (or its equivalent from quitting a job in monetary value). Since v () is concave, we may use the fact that {r/(1+r)^t}_t=1 is a probability distribution, and that the constraint says that the mean value of {x_t}_t=1 is x¯=m+wr/t(1+r), to apply Jensen's inequality and deduce that the optimal solution is x_t=x¯ for every t. Hence

We shall focus on the case when t is small and note that, as t0⁺, t(w;t)(1/)v(m+w).

The consumer's problem can therefore be written as:

An approximation to the consumer's problem

Since lottery draws occur at least weekly, the discount factor will be close to one. In a Markov reward process with this property, a good approximation to the expected discounted reward per period is the long-run average reward obtained by taking the expectation of the reward per period with respect to the long-run stationary probabilities. (See, for example, Whittle,¹⁰ p. 123). However, the objective function and constraint in CP are complicated by the presence of additional terms representing the possibility of exiting the reward process upon winning the jackpot. Since the probability of winning the jackpot is so low, for a consumer who buys the same small number of tickets each week, wins can be regarded as events in a (discrete approximation to a) Poisson process with expected inter-event times of the order of 10⁷ periods. When considering such rare events, it is quite inappropriate to approximate the expected discounted value of the jackpot by the expected average reward as the discount rate , though small, is much larger than the quit rate . To cope with these two distinct components of the reward, we adopt a hybrid approach in which consumers receive the long-run average reward per period until they win the jackpot, the value of which is discounted to the current period.

The consumer's problem CP is actually slightly more complicated than the preceding description suggests, for the probability of winning the jackpot will depend on the number of tickets purchased and this is contingent on the rollover state. Hence, quitting probabilities are state-dependent and the corresponding process is not Poisson. Nevertheless, the expected number of periods until a jackpot win will still be very large and, if we assume that the pre-quit process can be regarded as in the steady state, it is plausible that exiting can still be well-approximated by a Poisson process. In this process, we take the probability of a jackpot win leading to a quit as the expected probability of winning, taken with respect to the steady state probabilities. A formal statement of these results, with proofs, is offered in the Appendix.

Objective function

For consumers who have not yet won the jackpot we concentrate on stationary decision rules in which consumers' decisions depend on the rollover state k but not on t and write y^k[x^k] for the rate of expenditure on lottery tickets [other goods] when R_t=k. To allow for winning the jackpot, the environmenzt of any particular consumer can be summarised by the extended state space:

where, at period t, the extended state is (k,0) if the consumer has not yet won the jackpot and R_t=k is (k,1) if the consumer won the jackpot in period t- 1 and R_{t- 1}=k and is if the consumer won the jackpot in a period before t- 1. Writing D for the order-(K+1) diagonal matrix with d_kk=y^k and for the probability that a single ticket wins the jackpot, the transition matrix for the extended state space becomes:

where every component of e is 1 and the 0 s represent appropriately dimensioned zero matrices or vectors. (Note that since Q is a transition matrix, the elements in each row sum to one). We have used the fact that the probability of winning the jackpot when R_t=k is t y^k to a very good approximation. Define the (K+1)-vectors:

Then the expected value of the objective function of the consumer's problem CP is the (0,0)th (in the extended state space) component of:

where A=_t=1t^{t- 1}P^{t- 1}.

In the appendix, we demonstrate that this component is well approximated by:

This result demonstrates our earlier discussion. The first term is the net present value of receiving the long run expected reward every period for a long time. The second term is the expected utility of winning the jackpot when the rollover variable is in the steady state, discounted to the initial time.

Constraint

We shall treat the budget constraint in (CP) in a similar way. We assume that, conditional on R_t=k, G_t(y^k)>0 with probability t(y^k) where t(y^k) is the probability of winning a non-jackpot prize. Similarly, conditional on G_t(y^k)>0 and the rollover state being k, we assume that G_t(y^k) is distributed as G^k: the non-jackpot prize.

We write g_k for E[G^k]. In the Appendix, we demonstrate that the left-hand side of the constraint in CP is well approximated by:

which says that the budget constraint must be satisfied on average using the limiting probabilities.

Since v is increasing in x^k, the consumer's optimisation problem becomes:

Properties of the optimal solution

An important test of our model of individual participation is that the optimal solution of CP* exhibits plausible qualitative properties if reasonable assumptions are made about the preferences. We now turn to such a set of assumptions. These are stated in terms of derivatives so we suppose throughout that v,f₀,f₁,...,f_K and are twice differentiable for positive arguments.

Assumptions:

1. For all x>0, we have v' (x)>0 and v"(x)<0.
2. For k=0,...,K, we have f_k(0)=0 and 0<f'_k(y)<1, f"_k(y)<0 for all y>0.
3. For Kk>j0 and all y>0, we have f'_k(y)f'_j(y).
4. We have <(0)=0, and '(y)>0, "(y)0 for all y>0.
5. For Kk>j0, we have g_kg_j, and E(W^k)EW^j).

Assumption 1 says that 'other goods' are desirable and participants are risk averse. The sign constraints on the derivative in Assumption 2 state that lottery tickets are desirable but that the additional fun derived from the purchase of an extra ticket falls as the number of tickets purchased rises. The requirement f'_k(y)<1 implies that the 'fun' generated by purchasing tickets does not exceed the foregone consumption of other goods. Hence, both fun and the prospect of winning prizes are required to induce lottery participation; neither alone is sufficient. Assumption 3 specifies that marginal fun is non-decreasing in the size of the rollover, and Assumption 4 says that is concave. This will hold if participants choose the numbers on their tickets optimally. When the number of tickets purchased is small, we expect to be (nearly) linear. Assumption 5 requires the expected size of prizes to be non-decreasing in the rollover state variable. The total prize pool always increases but the number of participants and therefore, on average, the number of winners sharing this pool will also increase. Our assumption, which is supported by empirical evidence, is that this effect is not large enough to fully offset the increase in the prize pool.

Assumptions 1 and 2 mean that v(x+f_k(y)) is a strictly concave function of (x,y) for k=0,...,K. Hence, this is also true for the objective function in CP*. Assumption 4 means that the left-hand side of the equality constraint is a convex function of (x,y). Since the objective function is increasing in x^k the constraint may be replaced by a weak inequality. Furthermore, this inequality can be satisfied strictly (by setting x and y small and assuming m>0). It follows that the first-order Kuhn–Tucker conditions are necessary and sufficient and have a unique solution.¹¹ For k=0,...,K, these conditions can be written in the form:

where is a Kuhn–Tucker multiplier for the first constraint. We have cancelled p_k from both sides of these inequalities, using the fact that p_k>0 for all k.

We shall consider only solutions that are interior for other goods. Thus, all x_k>0 and, since v' is strictly decreasing by Assumption 1, we derive:

Optimality condition 1

It follows from Assumption I that >0 and, hence, that the second part of the optimality condition can be written:

where

Assumptions 2, 3 and 5 imply that:

1. _k(y) is strictly decreasing in y for a given k,
2. _k(y) is strictly increasing in k for a given y.

From (a) and (8), we deduce that, if _k(0)>1, then y^k>0 and _k(y^k)=1; otherwise, y^k=0. When _k(0)>1 for some k=0,...,K, we can define J to be the smallest k such that _k(0)>1. Then, (a) and (b) lead to:

Optimality condition 2

If _K(0)1, then y⁰==y^K=0, (Case NP) otherwise, y^k=0 for k<J and y^k>0 with _k(y^k)=1 for kJ. (Case P)

Case P corresponds to lottery participation (when the rollover state variable is at least J) and NP to non-participation for any rollover state.

Properties (a) and (b) further imply that y^k>y^j when k>jJ. Now observe that, by integrating the inequality in Assumption 3 from 0 to y, and using Assumption 2, we obtain f_k(y)f_j(y) for all y0 when k>j. Hence,

using Assumption 2. In view of Optimality condition 1, we can conclude that x^k<x^j.

If k>j, then it follows from Assumption 2 that

Suppose, further, that

ie, additional participation following an increase in the rollover state is induced solely by the enlarged prize pool rather than by any additional fun. Then, using Optimality condition 1,

Theorem 1 The optimal solution of CP* has one of the following forms:

Case NP: For all k=0,..., K, we have y^k=0 and x^k=m.

Case P: There exits a J{0,..., K} such that

For a potential participant the theorem says that positive participation will take place if the number of rollovers is at least J, as the rollover state increases, more is spent on lottery tickets, less on other goods. Nevertheless total expenditure increases if the fun function is independent of the rollover state.

Equilibrium/consistency

In this section we recognise that the collective decisions of all consumers will determine aggregate lottery expenditure and this influences both the transition matrix Q and the (random) jackpot prize W=(W⁰,...,W^K). Current lack of suitable data on heterogeneous purchases forces us to consider a homogenous population of identical individuals who engage in the gamble whatever the rollover state. Nonetheless, the arguments and formulae we develop below are readily extended to a population of heterogeneous agents.

We assume that there are n individuals sharing the same preferences, reflected in the utility function v and the fun functions f_k, and the same income m. In rollover state k, the number of tickets sold is n^k=ny^k and this affects both Q and W through the probability distribution of the number of prizewinners; if this number is 0 the jackpot prize pool is rolled over. To obtain this distribution, the simplest assumption to make would be that each combination of numbers (6 out of 49 for the UKNL) is chosen independently and with equal probability (ie, =). In this case, the number of winners would follow the binomial distribution B(,n^k) or, to a very good approximation, the Poisson distribution P(n^k). However, the equal probability assumption conflicts with observation in that it significantly under-predicts the probability that there are no winners^2,7 and a consequent rollover.

We will therefore assume that any ticket design induces a probability distribution over the set I of permitted combinations, such that the probability of the combination iI being chosen on a randomly selected ticket is (i), independently of all other choices. Since all combinations are drawn with probability |I|^{- 1}, the distribution Z_k() of the number of winners when the rollover state variable is k is an equally weighted combination of Poisson distributions. In particular,

The uniform case corresponds to (i)==|I|^{- 1} for all iI; a simple, two-parameter generalisation is discussed in the next section. The probability of a rollover is given by

and the expression for the stationary probabilities implies that

We now turn to the distribution of W^k, the jackpot prize from the perspective of a participant who will use () to choose their combination. The total jackpot prize pool in rollover state k is a proportion of the expenditure on tickets net of taxes and operating costs augmented, if k1 by the jackpots rolled over from the previous k- 1 draws. Hence the total jackpot prize in this state is (1- - )n¯^k, where n¯^k=_l=0^{k- 1}n^l, and is the proportion of revenue attributed to operating cost and profit. For any ticket, conditional on the combination iI being drawn, the number of winning tickets excluding that ticket is a random variable Z_i^k where Z_i^k P{(i)(n^k- 1)}. It follows that a winning ticket will receive a jackpot prize of (1- - )n¯^k/(Z_i^k+1). Since the probability that i is chosen, given that a ticket wins, is (i), we deduce that the probability distribution of the size of jackpot prize is:

(We have ignored the highly unlikely possibility that any participant wins two or more jackpot prizes in a given draw.) Using (12), we can write, by rearranging the series, for k=0,...,K:

Since non-jackpot prizes are not subject to rollovers, a proportion (1- )(1- - ) of the total expenditure on tickets is allocated to non-jackpot prizes and the average prize is worth g_k in state k. As all tickets are equally likely to win, the probability of winning must be (1- )(1- - )/g_k. We shall further assume that is linear (since y is small):

in which case the term (y^k)g_k in the constraint of CP* can be replaced with (1- )(1- - )y^k. We can therefore write the (necessary and sufficient) first-order conditions for households in state k as:

Note that our assumption of homogeneity forces y^k>0 for all k so that the inequality in (15) is satisfied as an equation. The equilibrium solutions (x,y) are the solutions of equations (10), (11) and (13) to (16).

Simulations

In this section, we apply the model to the UKNL and so we need to select specific functional forms for the utility function, fun functions, and the distribution . The limited variability in the data dictates the use of parsimoniously parameterized functions and, for the utility and fun functions, we choose functions with a single parameter. Individual preferences are assumed to exhibit constant relative risk aversion (CRRA):

The parameter characterizes relative risk aversion (defined as - xv"(x)/v' (x)) and is assumed constant. The literature tends to favor CRRA as a parsimonious and accurate description of individual behaviour.^12,13

A convenient model for the fun function is a power function.¹⁴ However, in order to satisfy Assumptions 2 and 3, we need to transform it into the form

where 0<_jk<1 for Kk>j0. For simplicity, in the simulations we assume that _k=, k. These are the conditions in the last part of Theorem 1, which allow us to deduce that expenditure increases with the rollover.

The distribution needs to be parsimoniously parameterised and not tied too closely to a particular ticket design (since changing entails changing the design). A simple way to do this is to assume that a proportion 1- , 0<1, of the |I| possible combinations are never chosen and that the remaining combinations are equally likely to be chosen, with probability /. A slightly more sophisticated version assumes that some participants use this distribution and others choose uniformly. Indeed, in many lotto games, computerized pseudo-random generation of combinations is offered to participants (called 'Lucky Dip' in the UKNL). Let (0,1) be the proportion of tickets on which combinations are chosen in this way. Then (i)=(=₁) if i is a forbidden combination, and (i)=(1- +)(/)(=₂), otherwise. It follows from (10) that the probability of a rollover in state k is:

and (13) can be written

where, for i=1,2,

We take and , the jackpot proportion and operating costs, from the current design and consider values of , the tax rate, and in a range of 4% either side of the current values, while we put income equal to the average UK weekly household expenditure and N to the number of UK households. We assume an annual interest rate of 10% and set the maximum number of rollovers K to 2 reflecting the maximum observed value. This leaves setting the values of the parameters: , the proportion of combinations available for selection, , the coefficient of relative risk aversion, and , the fun function parameter, to complete the calibration. The values are chosen such that (i) sales in normal (non-rollover) weeks (ii) sales in single rollover weeks, and (iii) the long run proportion of normal weeks agree with observed values. We note that the fitted value of falls within the commonly observed range of values (ie, between 0.5 and 3).

In Figure 1 we present contour plots in the (+,) plane for the parameter values in Table 1. The panels illustrate how local changes in the design affect the expected tax revenue (top left panel), the expenditure per household on the lottery during a non-rollover week (top right panel) and during a single rollover week (middle right panel), the expected number of jackpot winners during a non-rollover week (middle left panel), the expected size of a jackpot in a non-rollover week (bottom right panel), and the long run proportion of non-rollover weeks (bottom left panel).

Figure 1.

Simulation of design changes, contour plots.

Full figure and legend (61K)

Table 1 - Simulation parameters.

Full table (6K)

With the parameters set at the calibrated values, we see that increasing , and more surprisingly, decreasing , leads to an increase in expected tax revenue. In the former case, although the number of rollovers and therefore of potential additional revenue associated with more rollovers falls, this is more than offset by the increased participation induced by the increased chance of a jackpot win; the number of winners also increases, indeed sufficiently so that expected (normal week) jackpots fall in spite of the enlarged jackpot prize pool. Decreasing also increases participation sufficiently to offset the reduced tax take per ticket. Normal and rollover week expenditure rise (in contrast to increasing ), the number of rollovers falls and the expected number of winners and the expected jackpot both increase. Changing the parameters from the current values in a direction of steepest ascent of the expected tax revenue function has the same effect on all the graphed functions as increasing . (The expected jackpot decreases slightly).

Before drawing policy conclusions from these results, it is appropriate to examine the sensitivity of the results to the parameter values and functional forms. Indeed, it is precisely because they will be particularly sensitive to such model details that we have refrained from suggesting optimal values. To investigate these issues we have experimented by varying the parameter values (re-calibrating where appropriate) and with different functional forms such as a constant absolute risk aversion utility function. The results of these experiments are too numerous to include in the paper but most of the qualitative features of the functions plotted in Figure 1 remain robust to these changes (details are available from the authors). In particular, the expected tax revenue function always appears to be upward sloping at current parameter values in a west to northwest direction. We suggest that this evidence is sufficient to prompt further investigation into whether the tax rate in the UKNL may be too high and whether the probability of winning the jackpot may be too low.

References

1. La Fleur T, La Fleur B (1999). La Fleur's 1999 World Lottery Almanac. TLF Publications Incorporated: Maryland, USA.
2. Farrell L, Lanot G, Hartley R, Walker I (1999). The demand for lotto: the role of conscious selection. J Bus Econ Statist 18: 228–241.
3. Farrell L, Morgenroth E, Walker I (1999). A time series analysis of UK lottery sales: the long run price elasticity. Oxf Bull Econ Statist 61: 513–526. | Article |
4. Scoggins JF (1995). The lotto and expected net revenue. Nat Tax J 5: 61–70.
5. Simon J (1998). Dreams and disillusionment: A dynamic model of lottery demand. Four essays and a note on the demand for lottety tickets and how lotto players choose their numbers, unpublished Ph.D., Department of Economics, European University Institute (Florence, Italy).
6. Haigh J (1996). Lottery—the first 57 draws. Roy Statist Soc News 23: 1–2.
7. Haigh J (1997). The statistics of the national lottery. J Roy Statist Soc (Series A) 160: 187–206. | Article |
8. Friedman M, Savage LJ (1948). Utility analysis of choices involving risk. J Polit Econ 56: 279–304. | Article |
9. Farrell L, Hartley R (2002). Can Friedman-Savage utility functions explain gambling? American Economic Review 92: 613–624. | Article |
10. Whittle P (1982). Optimization Over Time. John Wiley and Sons: Chichester.
11. Rockafellar RT (1970). Convex Analysis. Princeton University Press: Princeton.
12. Deaton A, Muellbauer J (1980). Economics and Consumer Behavior. Cambridge University Press: Cambridge.
13. Hirshleifer J, Riley JG (1992). The Analytics of Uncertainty and Information. Cambridge University Press: Cambridge.
14. Johnson JEV, Shin HS (1995). A violation of dominance and the consumption value of gambling. Department of Economics. University of Southampton, WP 9525.
15. Cox DR, Miller HD (1965). The Theory of Stochastic Processes. Chapman and Hall: London.

Appendices

Appendix

We will use the following ergodic results for which we offer an elementary proof.

Lemma Suppose {a_t}_t=1 is a sequence of independent and identically distributed random (K+1)-vectors and Ea₁ exists. Let Q* be the transition matrix in which every row is p (the stationary distribution of Q). Then, as t0⁺

with probability one.

Proof. It is well known¹⁵ that Q^tQ* as t. Hence, QQ*=Q*Q=Q*Q*=Q*. It follows inductively that, if S=Q- Q*, then Q^t=S^t+Q* for t1. Consequently, S^t0 as t, which means that I- S is non-singular. Thus, using =e^{- t},

Write

The latter inverse exists since (I- t D)Q is a sub-stochastic matrix. Note that

from which it follows that

For n1, write s_n=(1/n)_t=1ⁿa_t. By the strong law of large numbers, for almost all sample paths , s_n()Ea₁. For such an , we shall write

and prove that S(t)AEa₁ as t0⁺. Note that by rearranging the sums,

A similar argument gives

Taking the difference of these results gives

and the proof is completed by showing that the term in braces approaches 0 as t0⁺.

To achieve this, we use the vector and matrix norms x=max_k|x_k|, A=max_k'k|a_k'k|, and the fact that AxAx together with the standard results on matrix norms. We have Q=1 and I- t D=1- ty, where y=min_ky^k. Since

there exists a >0 and ₁>0 such that, if 0<t<₁, then 1- (1- t y)>t.

Let >0. Since s_t()Ea₁ as t there is an N such that s_t()- Ea₁<²(/2) for all t>N. Define

If t<min{₁,₂}, then

which concludes the proof.

For i1,

where the missing term is chosen to make the rows sum to 1. It follows that the (0,0)^th component (in the extended state space) of (5) is the 0^th (in the original state space) component of

using the lemma. (We can drop the 'almost surely' qualification in this case.) For the UKNL ^{- 1} is close to 1410⁶ and if interest rates are less than 10% then does not exceed 1/500. Hence for purchases of a few tickets a week, the inverse matrix in the expression above is close to I. Furthermore, since v is concave and increasing, it is bounded above by an affine function and below by a constant, and this allows us to apply the dominated convergence theorem when taking the limit of tE(W^k;t) as t0⁺. Expression (6) is a rewriting of the limit in (A2) and in using it in our model, we are assuming that draws are sufficiently frequent for us to apply this limit. One observation that supports this assumption is that winning small prizes has little effect on expenditure.

We can use the same approach for the constraint. In time period t, if the rollover state is k, the probability of winning a non-jackpot prize is t (y^k) and if won, the prize is g_t^k, an independent copy of g₁^k. Thus the left-hand side of the constraint of CP* is the (0,0)th component of

,

where _t=(x⁰+y⁰- m- (y⁰)g_t⁰,...,x^K+y^K- m- (y^K)g_t^K).

We can use the expression (A1) for P^t above to rewrite this as the 0th component of

where D~ is the diagonal matrix with d~_kk=(y^k). By the same argument as above, we can replace the inverse matrix in this expression with I, thus justifying (7).

Acknowledgements

Financial support from the ESRC under research grant R000236821 is acknowledged. ESRC Research Fellowship Award H53627501695 supported the second author. We would like to thank Lisa Farrell, Richard Cornes and Ian Walker for discussions and comments, as well as seminar participants at Edinburgh, Keele, Oxford and Toulouse.