A Winner's Mean Earnings in Lottery and Inverse Moments of the Binomial Distribution

Abstract

We study the mean earnings of a lottery winner as a function of the number 𝑛 of participants in the lottery and of the success probability 𝑝. We show, in particular, that, for fixed 𝑝, there exists an optimal value of 𝑛 where the mean earnings are maximized. We also establish a relation with the inverse moments of a binomial distribution and suggest new formulas (exact and approximate) for them.

1. Introduction

The game of lottery is both popular and simple. Focusing on the essentials, and leaving aside additional features of secondary importance, which vary across different lottery implementations, the rules of the game are as follows: each player submits to the lottery organizers a ticket consisting of 𝑀 integers (selected by the player, without repetitions, selection order being unimportant) from the range 1,…,𝑁; within the prespecified time period the game is set to last; upon the expiry of this period no more ticket submissions are accepted, and an 𝑀-tuple of distinct integers (the “winning set”) is selected uniformly at random by the lottery organizers; each submitted ticket gets compared against the winning set, and, if they match, the corresponding player “wins.” This is known as an 𝑀/𝑁 lottery system, and the winning probability is clearly 𝑝=1/(^𝑁_𝑀). The money the winners earn depends on the number 𝑛 of submitted tickets: each submitted 𝑀-tuple incurs a certain fee (which we assume, without loss of generality, to be equal to 1), and some fixed ratio of the total sum collected (which we assume, again without loss of generality, to be 100%, namely, the entire sum) is returned as prize money back to the winners and equally split among them. As long as no winner is found, earnings of earlier games accumulate until winners are found, who then split equally the entire sum. This is an important feature in the implementation of lottery systems in practice, known as rollover.

For a given success probability 𝑝, what is the effect of the total number of participants 𝑛 on a winner's mean earnings? This is the object of study of this article. Clearly, more participants lead not only to more prize money, but also to more potential winners. Intuitively, and by the elementary properties of the binomial distribution, we expect that the area of the (𝑛,𝑝)-plane where 𝑛𝑝≈1 is an important borderline: as long as 𝑛𝑝≪1, existence of winners is highly improbable, so most likely the mean earnings will trivially be 0, while, as long as 𝑛𝑝≫1, the law of large numbers applies and suggests that there will be approximately 𝑛𝑝 winners; each of when will receive an amount of money equal to 𝑛/(𝑛𝑝)=1/𝑝. We first analyze the independent draws scenario (without rollover), and then use this result to deduce the corresponding result with rollover.

An additional complication rollover presents that it is not only that the number of participants in the various draws can vary, but also that this variation can potentially exhibit strong correlation, specifically be strictly increasing: as a rule, larger money prizes motivate more extensive participation [1, 2]. A reliable model for this variation can only be obtained through detailed statistical analysis and the psychology of gambling, which both lie beyond the scope of the present article (see, e.g., [1–4] for an analysis of the playing style of lottery participants). Perhaps surprisingly, though, some analysis we performed on Greek lottery data [5] did not invariably lead to the conclusion that the number of participants in the various games during a rollover round shows a clear upward trend (a similar phenomenon is observed in [3]). On many occasions, these fluctuations appear indeed to be random and, more importantly, the order of magnitude of the number of participants does not vary. Accordingly, we will derive the general formula for an arbitrary fluctuation of the number of participants during lottery games in a single rollover round, but we will then focus on the special case where this number is constant, as it is not only interesting but also mathematically tractable.

Needless to say, actual lottery implementations are actually more complicated, incorporating a plethora of secondary features. For example, lower winning levels may be introduced, corresponding to partial matches between the tickets and the winning set, and the prize money they win may either be smaller fractions of the total sum or simply a fixed sum. Alternatively, an extra “bonus number” may be introduced, acting, in a sense, like a “second chance”: players are allowed to submit a guess for this number, in addition to their guess for the winning set, while the selection of the winning set is followed by the (also uniformly random amongst the remaining integers) bonus number. If a player's choice has an overlap of 𝑀−1 integers with the winning set, but the player also guessed correctly the bonus number, a lesser prize is offered to the player. This may, of course, be combined with all lower winning levels, whereby an overlap of 𝐾−1 integers (𝐾≤𝑀) plus correct guess of the bonus number leads to a smaller prize compared to an overlap of 𝐾 integers.

To the best of our knowledge, the issue of winners' mean earnings has not been considered in the lottery literature before. It is a problem of mathematical interest, relating lottery to the study of inverse moments of the positive binomial distribution (as will be seen below), and, for this reason, this article can be considered falling into the same group of publications by the present author [6–8], as well as other authors (e.g., [4, 9]), whose objective is to consider nontrivial probability issues related to the lottery. We should perhaps state that the purpose of this article is not to consider winning strategies or to advise a prospective player about the investive value of the lottery. For readers interested in these aspects of the lottery, we recommend the study in [1], a recent and mathematically sophisticated analysis of the lottery's viability as an investment, and also those in [10, 11].

One final point that needs to be addressed is our choice to assume throughout this article that players choose their numbers independently of each other, namely, that the probability distribution describing the choice of an 𝑀-tuple of integers by any player is uniform over all possible 𝑀-tuples, and to actually do so despite the fact that this has been extensively investigated and found, perhaps surprisingly, not to hold in practice [1, 2, 10–13]! Indeed, humans seem to favor some classes of 𝑀-tuples more than others (for several reasons), and the net effect of this is that the probability distribution of the number of winners depends on the winning set. Lottery organizers seem to encourage this phenomenon, as it leads to more frequent rollovers, which has been documented, at least, in some lotteries (e.g., US and UK), to increase participation [1, 2]. This already suggests a playing strategy: assuming that players have information on which 𝑀-tuples are “unpopular” and that a player should choose one of those. Indeed, the probability distribution of the winning set is uniform (assuming that the lottery organizers are honest), and, if the chosen “unpopular” 𝑀-tuple wins, the likelihood of other winners existing will be minimal. In this article, however, we choose to ignore this complication, as mentioned above, on account of three reasons: (a) there appears to be no universal and conclusive statistical evidence to support the position that the skewness present in the players' choices is strong enough to unquestionably reject the validity of the uniform players' choices model, at least as an approximation; (b) as lottery players become more mature and aware, this skewness may progressively disappear (there is already evidence that, in some lotteries, more than 70% of the choices entered are picked randomly by computers [1]); and (c) as mentioned earlier, this article focuses on the principles of the lottery rather than its actual implementations.

2. Results without Rollover

Consider the lottery game as described in the Introduction: the probability distribution for the number of winners is the binomial 𝐵(𝑛,𝑝). Assuming that 𝑚>0 winners exist, each one's earnings will be 𝑛/𝑚, while, for 𝑚=0, there is no winner to earn something, so earnings are conventionally taken to be 0. It follows that the mean earnings 𝐺(𝑛,𝑝) are 𝐺(𝑛,𝑝)=^𝑛_𝑚=1𝑛𝑚𝑝^𝑚(1−𝑝)^𝑛−𝑚𝑛𝑚=𝑛(1−𝑝)^𝑛𝑛_𝑚=1𝑛𝑚1𝑚(𝑝)1−𝑝^𝑚. (2.1)

This expression is essentially the inverse first moment of the positive binomial distribution 𝐵₊(𝑛,𝑝). Assuming that 𝑋∼𝐵(𝑛,𝑝) and setting 𝑞=1−𝑝, 𝑋₊∼𝐵₊(𝑛,𝑝)⇔ℙ[𝑋₊==𝑚]=ℙ[𝑋=𝑚∣𝑋>0]ℙ[𝑋=𝑚]=1ℙ[𝑋>0]1−𝑞^𝑛𝑛𝑚𝑝^𝑚𝑞^𝑛−𝑚,𝑚=1,…,𝑛, (2.2) and the inverse 𝑘th moment is defined as 𝜇_−𝑘=𝔼[𝑋₊^−𝑘1]=1−𝑞^𝑛𝑛_𝑚=11𝑚^𝑘𝑛𝑚𝑝^𝑚𝑞^𝑛−𝑚=𝑞^𝑛1−𝑞^𝑛𝑛_𝑚=11𝑚^𝑘𝑛𝑚(𝑝𝑞)^𝑚,𝑘∈ℕ^∗. (2.3) Clearly, (2.1) and (2.3) (for 𝑘=1) consist of the same sum (with different multiplicative factors): 𝐺(𝑛,𝑝)=𝑛(1−𝑞^𝑛)𝜇₋₁. (2.4)

The problem of the inverse moments of the positive binomial distribution was first considered by Stephan in 1946 [14], and has received a fair share of attention in the literature ever since. For example, Grab and Savage [15] provided a recursive formula in 𝑛 for 𝑘=1, which was later generalized by Govindarajulu [16], while Chao and Strawderman [17] considered 𝔼[1/(𝑋+𝑐)], where 𝑐∈ℝ such that ℙ[𝑋+𝑐>0]=1. Recently, Wuyungaowa and Wang provided asymptotic expansions of 𝜇_−𝑘,𝑘∈ℕ^∗.

We now proceed to derive a more convenient formula for (2.1), hence, in view of (2.4), for 𝜇₋₁ as well. To the best of our knowledge, this formula has not appeared in the literature before.

Theorem 2.1. 𝐺(𝑛,𝑝)=𝑛∑^𝑛_𝑚=1((𝑞^𝑛−𝑚−𝑞^𝑛)/𝑚). Furthermore, setting 𝐹(𝑥)=𝑒^−𝑥∫^𝑥₀((𝑒^𝑢−1)/𝑢)𝑑𝑢, |𝐺(𝑛,𝑝)−𝑛𝐹(−𝑛ln(𝑞))|≤𝑛ln²(𝑞)24√𝑞+2ln(𝑞)√𝑞+𝑂(𝑛⁻¹). (2.5) For 𝑝≪1, in particular, −𝑛ln(𝑞)≈𝑛𝑝,ln²(𝑞)/√𝑞≈𝑝², and 2(ln(𝑞)/√𝑞)≈−2𝑝.

Proof. We can transform (2.1) using integration: 𝐺(𝑛,𝑝)=𝑛(1−𝑝)^𝑛₀^{𝑛𝑝/(1−𝑝)}_𝑚=1𝑛𝑚𝑢^𝑚−1𝑑𝑢=𝑛(1−𝑝)^𝑛₀^{𝑝/(1−𝑝)}(1+𝑢)^𝑛−1𝑢𝑑𝑢. (2.6) We now use the identity 𝑥^𝑛−1=(𝑥−1)(𝑥^𝑛−1+⋯+1) to get 𝐺(𝑛,𝑝)=𝑛(1−𝑝)^𝑛₀^{𝑝/(1−𝑝)𝑛−1}_𝑚=0(1+𝑢)^𝑚𝑑𝑢=𝑛(1−𝑝)^𝑛𝑛_𝑚=1(1+𝑝/(1−𝑝))^𝑚−1𝑚=𝑛^𝑛_𝑚=1(1−𝑝)^𝑛−𝑚−(1−𝑝)^𝑛𝑚=𝑛^𝑛_𝑚=1𝑞^𝑛−𝑚−𝑞^𝑛𝑚, (2.7) which proves our first claim.
We now approximate the sum in (2.7) by an integral (considering it to be a Riemann sum, or, even better, through the Euler-McLaurin formula [18]): 𝐺(𝑛,𝑝)=𝑛^𝑛_𝑚=1𝑞^𝑛−𝑚−𝑞^𝑛𝑚=𝑛𝑞^𝑛𝑛_𝑚=1𝑞^−𝑚−1𝑚=𝑛𝑞^𝑛𝑛_𝑚=1(𝑞^𝑛)^−𝑚/𝑛−11𝑚/𝑛𝑛→𝑛𝑞^𝑛1₀𝑞^−𝑛𝑥−1𝑥𝑑𝑥=𝑛𝑞^𝑛𝑛₀𝑞^−𝑥−1𝑥𝑑𝑥=𝑛𝑟^−𝑛𝑛₀𝑟^𝑥−1𝑥𝑑𝑥=𝑛𝑒^{−𝑛ln(𝑟)}₀^𝑛ln(𝑟)𝑒^𝑥−1𝑥𝑑𝑥=𝑛𝐹(𝑛ln(𝑟)), (2.8) where 1𝑟=𝑞,𝐹(𝑥)=𝑒^−𝑥𝑥₀𝑒^𝑢−1𝑢𝑑𝑢. (2.9) But this can be further simplified as ln(𝑟)=−ln(𝑞)=−ln(1−𝑝)≈𝑝for0<𝑝≪1, (2.10) implying that 𝐺(𝑛,𝑝)𝑛≈𝐹(𝑛ln(𝑟))≈𝐹(𝑝𝑛)=𝑒^−𝑝𝑛₀^𝑝𝑛𝑒^𝑥−1𝑥𝑑𝑥. (2.11)
The error of this approximation is readily given by the lowest-order Newton-Cotes numerical integration formula known as the midpoint rule [19], which, applied in the function at hand, yields |^𝑛_𝑚=1𝑞^𝑛−𝑚−𝑞^𝑛𝑚−𝑞^𝑛1₀𝑞^−𝑛𝑥−1𝑥𝑀𝑑𝑥|≤𝐸(𝑛,𝑝)∶=²24𝑛²+|𝑞^𝑛₀^1/(2𝑛)𝑞^−𝑛𝑥−1𝑥𝑑𝑥|+|𝑞^𝑛₁^1+1/(2𝑛)𝑞^−𝑛𝑥−1𝑥𝑑𝑥|, (2.12) where 𝑀₂=max_{𝑥∈[1/(2𝑛),1+1/(2𝑛)]}(𝑞^{𝑛(1−𝑥)}−𝑞^𝑛𝑥)^′′=𝑛²ln²(𝑞)(1+1/2𝑛)√𝑞+2𝑛ln(𝑞)(1+1/2𝑛)²√𝑞+2(1/√𝑞−𝑞^𝑛)(1+1/2𝑛)³, (2.13) as the second derivative is monotonically increasing, and, therefore, the maximum corresponds to the right endpoint of the interval, namely, 𝑥=1+1/(2𝑛). For large 𝑛, the contribution of the two integral error terms becomes insignificant, while, asymptotically, 𝑀₂=𝑛²ln²(𝑞)√𝑞+2𝑛ln(𝑞)√𝑞+2√𝑞+𝑂(𝑛⁻¹). (2.14) Assuming further that 𝑝≪1, we find that ln²(𝑞)/√𝑞=ln²(1−𝑝)/1−𝑝≈𝑝², and 2ln(𝑞)/√𝑞=2ln(1−𝑝)(1−𝑝)^−1/2≈−2𝑝−𝑝²; hence the total error is approximately bounded above by 𝑝𝐸(𝑛,𝑝)=²−𝑝2412𝑛+𝑂(𝑛⁻²). (2.15) This completes the proof.

The approximation (2.11) proved in the theorem above is compared in Figure 1 against the exact (2.6), or, equivalently, (2.7). Our experiments show that for 𝑝<0.01 the approximation is almost exact, while for larger values of 𝑝, where (2.15) does not yet hold, there is noticeable deviation between the two curves.

(a)

(b)

Figure 1 

Comparison of $G(n,p)/n$ as given by (2.6) versus $F$ (in blue and red, resp.) for $p=0.1$ (a) and $p=0.01$ (b). For $p\le 0.01$ the two curves are indistinguishable.

Figure 1 suggests that, for fixed 𝑝,𝐺(𝑛,𝑝) attains a global maximum for some 𝑛. We now prove this to be the case.

Theorem 2.2. For fixed 𝑝∈(0,1),𝐺(𝑛,𝑝) attains a global maximum for some 𝑛.

Proof. To begin with, note that 𝐺(1,𝑝)=𝑝 by (2.1). Furthermore, we obtain from (2.7) that 𝐺(𝑛,𝑝)=𝑛^𝑛−1_𝑚=0𝑞^𝑚𝑛−𝑚−𝑛𝑞^𝑛=𝐻(𝑛)_{𝑛−1𝑚=0}𝑞^𝑚+^𝑛−1_𝑚=0𝑚𝑞𝑛−𝑚^𝑚−𝑛𝑞^𝑛=𝐻(𝑛)1−𝑞^𝑛+1−𝑞^𝑛−1_𝑚=0𝑚𝑞𝑛−𝑚^𝑚−𝑛𝑞^𝑛=1𝐻(𝑛)𝑝−𝑞^𝑛(1𝑝+𝑛𝐻(𝑛))+^𝑛−1_𝑚=0𝑚𝑞𝑛−𝑚^𝑚=1𝑝−𝑞^𝑛(1𝑝1+𝑛𝐻(𝑛))+𝑛^𝑛−1_𝑚=1𝑚𝑞^𝑚+1𝑛^𝑛−1_𝑚=1𝑚²𝑞𝑛−𝑚^𝑚, (2.16) where 𝐻(𝑛)=∑^𝑛_𝑘=1(1/𝑘) denotes the 𝑛th harmonic number. We further break the last sum into two sums: 1𝑛^𝑛−1_𝑚=1𝑚²𝑞𝑛−𝑚^𝑚=1𝑛^𝑁1_𝑚=1𝑚²𝑞𝑛−𝑚^𝑚+1𝑛^𝑛−1_𝑚=𝑁1+1𝑚²𝑞𝑛−𝑚^𝑚≤1𝑛^𝑁1_𝑚=1𝑚²𝑞𝑛−𝑚^𝑚+1𝑛^𝑛−1_𝑚=𝑁1+1𝑚²𝑞^𝑚, (2.17) where 𝑁₁<𝑛 is a positive integer to be specified later. Since ∑^∞_𝑚=0𝑚²𝑞^𝑚∶=𝐶<∞ for 𝑞<1, for any 𝜖₁>0 there exists a suitable 𝑁₁ such that ^𝑛−1_𝑚=𝑁1+1𝑚²𝑞^𝑚<^∞_𝑚=𝑁1+1𝑚²𝑞^𝑚<𝜖₁. (2.18) On the other hand, for this specific 𝑁₁, and for any 𝜖₂>0, there exists 𝑁₂>𝑁₁ such that, for any 𝑛>𝑁₂, ^𝑁1_𝑚=1𝑚²𝑞𝑛−𝑚^𝑚<1𝑛−𝑁₁^𝑁1_𝑚=1𝑚²𝑞^𝑚<𝐶𝑛−𝑁₁<𝜖₂. (2.19)
Putting (2.17), (2.18), and (2.19) together, we obtain that, for any 𝜖₁,𝜖₂>0, there exist 𝑁₁>0 and 𝑁₂>𝑁₁ such that, for any 𝑛>𝑁₂, 1𝑛^𝑛−1_𝑚=1𝑚²𝑞𝑛−𝑚^𝑚=1𝑛^𝑁1_𝑚=1𝑚²𝑞𝑛−𝑚^𝑚+1𝑛^𝑛−1_𝑚=𝑁1+1𝑚²𝑞𝑛−𝑚^𝑚<𝜖₁+𝜖₂𝑛. (2.20) Furthermore, ^𝑛−1_𝑚=1𝑚𝑞^𝑚→^∞_𝑚=1𝑚𝑞^𝑚=𝑞𝑝², (2.21) so that, for any 𝜖₃>0,∃𝑁₃>0 such that, for any 𝑛>𝑁₃, 𝑞0<𝑝²−^𝑛−1_𝑚=1𝑚𝑞^𝑚<𝜖₃. (2.22) Finally, for any 𝑛>1/𝑝, 𝑛𝑞^𝑛(1𝑝+𝑛𝐻(𝑛))<𝑛𝑞^𝑛(𝑛+𝑛²)<2𝑛³𝑞^𝑛, (2.23) and we know that this decays to 0 much faster than any power: in particular, for any 𝜖₄>0, there exists 𝑁₄>0 such that, for any 𝑛>𝑁₄, 𝑛𝑞^𝑛(1𝑝𝜖+𝑛𝐻(𝑛))<⁴𝑛². (2.24) From (2.16), then, using (2.22) and (2.24), we obtain 1𝐺(𝑛,𝑝)−𝑝=−𝑞^𝑛(1𝑝1+𝑛𝐻(𝑛))+𝑛^𝑛−1_𝑚=1𝑚𝑞^𝑚+1𝑛^𝑛−1_𝑚=1𝑚²𝑞𝑛−𝑚^𝑚>𝑞/𝑝²−𝜖₃𝑛−𝜖₄𝑛², (2.25) for 𝑛>𝑁=max{𝑁₂,𝑁₃,𝑁₄}, whence, if we ensure that 𝑞/𝑝²>𝜖₃+𝜖₄, it follows that 1𝐺(𝑛,𝑝)−𝑝>0for𝑛>𝑁. (2.26) On the other hand, 1|𝐺(𝑛,𝑝)−𝑝|<𝑞/𝑝²+𝜖₁+𝜖₂𝑛+𝜖₄𝑛²,𝑛>max{𝑁₂,𝑁₄1}⇒lim𝐺(𝑛,𝑝)=𝑝. (2.27) To recapitulate, we have shown that 𝐺(1,𝑝)=𝑝<1/𝑝, whereas, as 𝑛→∞, 𝐺(𝑛,𝑝) converges to 1/𝑝  from above: more specifically, we have shown that, for all 𝑛 sufficiently large, 10<𝐺(𝑛,𝑝)=𝑝+𝑞𝑝²𝑛+𝑜(𝑛⁻¹). (2.28) Hence, 𝐺(𝑛,𝑝) exhibits a global maximum for some value of 𝑛, for fixed 𝑝, and this concludes the proof.

Alternatively, the eventual positivity of 𝐺(𝑛,𝑝)−1/𝑝 also follows immediately from the more general asymptotic formulas presented in [20]. Having now demonstrated that 𝐺(𝑛,𝑝) has a global maximum, the next step is to locate and calculate its value. To do this, we do not work on 𝐺(𝑛,𝑝) directly, but rather on the approximation 𝐺_𝑎(𝑛,𝑝)=𝑛𝐹(𝑛𝑝), derived in Theorem 2.1.

Theorem 2.3. For fixed 𝑝, 𝐺_𝑎(𝑛,𝑝) is maximized for 𝑛_opt≈4.168485/𝑝, and 𝐺_𝑎(𝑛_opt,𝑝)≈0.310724𝑛_opt≈1.295248/𝑝. Furthermore, 𝐺_𝑎(𝑛,𝑝)/𝑛 is maximized for 𝑛_opt,r≈1.502861/𝑝, and 𝐺_𝑎(𝑛_opt,r,𝑝)/𝑛_opt,r≈0.517351.

In particular, the maximal mean earnings possible are about 29.5% above the asymptotic value, and about 31% of the total prize money.

Proof. Considering 𝑛 to be a continuous quantity, we determine the global maximum of 𝐺_𝑎 from the condition 𝜕𝐺_𝑎(𝑛,𝑝)/𝜕𝑛=0: 𝜕𝐺_𝑎(𝑛,𝑝)𝜕𝑛=(1−𝑛𝑝)𝑒^−𝑛𝑝₀^𝑛𝑝𝑒^𝑥−1𝑥𝑑𝑥+𝑛𝑒^−𝑛𝑝𝑝𝑒^𝑛𝑝−1⇔𝑛𝑝=0_𝑠0𝑒^𝑥−1𝑥𝑒𝑑𝑥=^𝑠−1,𝑠−1 (2.29) where 𝑠=𝑛𝑝. This equation can be solved numerically to yield 𝑠≈4.168485. It follows that the optimal 𝑛 is asymptotically given by the formula 𝑛_opt≈𝑠𝑝≈4.168485𝑝. (2.30) The value of the maximum, consequently, is 𝐺_𝑎(𝑛_opt𝑠,𝑝)≈𝑝𝑒^−𝑠𝑠₀𝑒^𝑥−1𝑥𝑠𝑑𝑥=𝑝1−𝑒^−𝑠≈𝑠−11.295248𝑝⇔𝐺(𝑛_opt,𝑝)𝑛_opt≈0.310724. (2.31)
What about maximizing the relative mean earnings 𝐺_𝑎(𝑛,𝑝)/𝑛=𝐹(𝑛𝑝)? Going back to (2.11), we need to determine the roots of the derivative of 𝐹: 0=𝐹^′(𝑠)=−𝑒^−𝑠𝑠₀𝑒^𝑥−1𝑥𝑑𝑥+𝑒^−𝑠𝑒^𝑠−1𝑠⇔^𝑠₀𝑒^𝑥−1𝑥𝑒𝑑𝑥=^𝑠−1𝑠. (2.32) Solving numerically we obtain 𝑠=𝑛_opt,r𝑝≈1.502861, whence 𝐹(𝑠)=𝑒^−𝑠𝑒^𝑠−1𝑠≈0.517351. (2.33) This completes the proof.

As a brief numerical example, consider the Greek lottery, a 6/49 lottery system, for which 𝑝=1/(₆⁴⁹)=1/13983816. Theorem 2.3 shows that 𝐺_𝑎(𝑛,𝑝) is maximized for 𝑛_opt≈4.168485/𝑝≈58.3 million, while 𝐺_𝑎(𝑛,𝑝)/𝑛 is maximized for 𝑛_opt≈1.502861/𝑝≈21.0 million. The actual number of tickets normally played, however, is significantly lower, ranging from 1 to 8 million [5], so, for the given level of participation, a 6/49 system is not optimal.

Before we conclude this section, let us investigate the variance of a winner's earnings. Our goal is to obtain a similar formula as the one presented in Theorem 2.1, and then to generalize it into a formula for any 𝜇_−𝑘,𝑘∈ℕ^∗. We begin by calculating the mean square earnings 𝐺⁽²⁾(𝑛,𝑝)=^𝑛_𝑚=1𝑛𝑚𝑝^𝑚𝑞^𝑛−𝑚(𝑛𝑚)²=𝑛²𝑞^𝑛𝑛_𝑚=1𝑛𝑚1𝑚²(𝑝𝑞)^𝑚=𝑛²𝑞^𝑛𝑛_𝑚=1𝑛𝑚1𝑚₀^𝑝/𝑞𝑢^𝑚−1𝑑𝑢=𝑛²𝑞^𝑛₀^𝑝/𝑞𝑑𝑢𝑢^𝑛_𝑚=1𝑛𝑚^𝑢0𝑑𝑣𝑣^𝑚−1=𝑛²𝑞^𝑛₀^𝑝/𝑞𝑑𝑢𝑢^𝑢₀𝑑𝑣𝑣^𝑛_𝑚=1𝑛𝑚𝑣^𝑚=𝑛²𝑞^𝑛₀^𝑝/𝑞𝑑𝑢𝑢^𝑢₀𝑑𝑣𝑣[(1+𝑣)^𝑚−1]=𝑛²𝑞^𝑛₀^𝑝/𝑞𝑑𝑢𝑢^𝑢₀𝑑𝑣^𝑛−1_𝑚=0(1+𝑣)^𝑚=𝑛²𝑞^𝑛₀^𝑝/𝑞𝑑𝑢𝑢^𝑛_𝑚=1(1+𝑢)^𝑚−1𝑚=𝑛²𝑞^𝑛₀^𝑝/𝑞𝑑𝑢^𝑛_𝑚=11𝑚^𝑚−1_𝑙=0(1+𝑢)^𝑙=𝑛²𝑞^𝑛𝑛_𝑚=11𝑚^𝑚_𝑙=11𝑙𝑝[(1+𝑞)^𝑙−1]=𝑛^2𝑛_𝑚=11𝑚^𝑚_𝑙=11𝑙[𝑞^𝑛−𝑙−𝑞^𝑛]. (2.34)

It follows that the variance is 𝑉(𝑛,𝑝)=𝐺⁽²⁾(𝑛,𝑝)−𝐺²(𝑛,𝑝)=𝑛²[^𝑛_𝑚=11𝑚^𝑚_𝑙=11𝑙[𝑞^𝑛−𝑙−𝑞^𝑛]−(^𝑛_𝑚=11𝑚[𝑞^𝑛−𝑚−𝑞^𝑛])²]. (2.35)

Asymptotics can be found as in Theorem 2.1: 𝐺⁽²⁾(𝑛,𝑝)=𝑛²𝑞^𝑛𝑛_𝑚=11𝑛1𝑚/𝑛^𝑚_𝑙=11𝑛1𝑙/𝑛[𝑒^{−(𝑙/𝑛)𝑛ln(𝑞)}−1]→𝑛²𝑒^{𝑛ln(𝑞)1}₀𝑑𝑢𝑢^𝑢₀𝑑𝑣𝑣[𝑒^{−𝑛𝑣ln(𝑞)}−1]=𝑛²𝑒^𝑛ln(𝑞)₀^{−𝑛ln(𝑞)}𝑑𝑢𝑢^𝑢₀𝑑𝑣𝑣[𝑒^𝑣−1]. (2.36)

The generalization is clear. For any 𝑘∈ℕ^∗, 𝐺^(𝑘)(𝑛,𝑝)=^𝑛_𝑚=1𝑛𝑚𝑝^𝑚𝑞^𝑛−𝑚(𝑛𝑚)^𝑘=𝑛^𝑘𝑛_𝑚1=11𝑚₁^𝑚1_𝑚2=11𝑚₂⋯^𝑚𝑘−1_𝑚𝑘=11𝑚_𝑘(𝑞^{𝑛−𝑚𝑘}−𝑞^𝑛), (2.37) while the asymptotic expression as 𝑛→∞ becomes 𝐺_𝑎^(𝑘)(𝑛,𝑝)=𝑛^𝑘𝐹^(𝑘)(−𝑛ln(𝑞)),where𝑒^𝑥𝐹^(𝑘)(𝑥)=^𝑥₀𝑒^𝑢𝐹^(𝑘−1)(𝑢)𝑑𝑢,𝐹⁽⁰⁾(𝑥)=1−𝑒^−𝑥𝑥. (2.38)

The relation to the inverse moments follows from the generalization of (2.4) 𝐺^(𝑘)(𝑛,𝑝)=𝑛^𝑘(1−𝑞^𝑛)𝜇_−𝑘,𝑘∈ℕ^∗. (2.39)

3. Results with Rollover

When rollover is introduced, a sequence of lottery draws is played till a winner is found, at which point the total prize money (consisting of the prize money of the current draw plus the jackpot, namely, the accumulated prize money over the previous draws) is equally split among the winners. We assume that the (infinite) sequence 𝑛∈(ℕ^∗)^∞ of the number of participants in the various draws is known and fixed, and we denote by 𝐺(𝑘,𝑛,𝑝) the mean earnings in the 𝑘th lottery draw, assuming that no winner was found in the first 𝑘−1 draws and that a winner was found in the 𝑘th draw. It is clear that this quantity depends not on the full 𝑛 but only on its starting subsequence (𝑛₁,…,𝑛_𝑘); hence, for example, 𝐺(𝑛,𝑝) of the previous section equals 𝐺(1,𝑛^′,𝑝) for any 𝑛^′∈(ℕ^∗)^∞ with 𝑛^′₁=𝑛.

We denote the mean earnings under rollover by 𝐺_𝑟(𝑛,𝑝). Letting 𝐾(𝑛) be the random variable denoting, for a fixed 𝑛, the number of draws played till a winner is found, then 𝐺_𝑟(𝑛,𝑝)=^∞_𝑘=1ℙ[𝐾(𝑛)=𝑘]𝐺(𝑘,𝑛,𝑝). (3.1)

It is, in fact, possible to determine the probability distribution of 𝐾(𝑛): 𝐾(𝑛)=𝑘 if and only if no winner is found in the first 𝑘−1 draws and a winner is found in the 𝑘th draw. Taking into account that the probability that no winner is found in the 𝑖th draw is (1−𝑝)^𝑛𝑖≈𝑒^{−𝑛𝑖𝑝} for 𝑝≪1, we immediately find ℙ[𝐾(𝑛)=𝑘]=𝑒^{−𝑝∑𝑘−1𝑖=1𝑛𝑖}(1−𝑒^{−𝑛𝑘𝑝}). (3.2) Furthermore, revisiting (2.11) in Theorem 2.1, we see that the first factor of 𝑛 in the equation denotes the prize money; whereas every other occurrence of 𝑛 is within the expression 𝑛𝑝 and denotes the number of tickets played; whence, in the present case, the asymptotic expression 𝐺_𝑎(𝑘,𝑛,𝑝) for 𝐺(𝑘,𝑛,𝑝) is given by 𝐺_𝑎(𝑘,𝑛,𝑝)∑^𝑘_𝑖=1𝑛_𝑖=𝑒^{−𝑝𝑛𝑘𝑝𝑛𝑘}₀𝑒^𝑥−1𝑥𝑑𝑥=𝐹(𝑝𝑛_𝑘). (3.3) To sum up, the asymptotic expression 𝐺_𝑟,𝑎(𝑛,𝑝) for 𝐺_𝑟(𝑛,𝑝) is given by 𝐺_𝑟,𝑎(𝑛,𝑝)=^∞_𝑘=1(^𝑘_𝑖=1𝑛_𝑖)𝑒^{−𝑝∑𝑘−1𝑖=1𝑛𝑖}(1−𝑒^{−𝑛𝑘𝑝})𝑒^{−𝑝𝑛𝑘𝑝𝑛𝑘}₀𝑒^𝑥−1𝑥𝑑𝑥. (3.4)

It would be, of course, far more interesting to enrich the model by allowing 𝑛 to be random, following some infinite-dimensional discrete probability distribution ℙ[𝑁=𝑛], in which case the averaged 𝐺_𝑟,𝑎 would be 𝐺_{𝑟,𝑎,avg}(𝑝)=_{𝑛∈(ℕ∗)∞}ℙ[𝑁=𝑛]𝐺_𝑟,𝑎(𝑛,𝑝). (3.5)

In general, (3.4) cannot be simplified further in a meaningful way. This, however, is possible in the special case, where, for some 𝑛∈ℕ^∗,𝑛_𝑖=𝑛 for all 𝑖∈ℕ^∗, in which case we may substitute the vector 𝑛∈(ℕ^∗)^∞ appearing in the preceding expressions by the value 𝑛∈ℕ of its first coordinate (since all coordinates are equal) and prove the following theorem, corresponding to Theorem 2.3.

Theorem 3.1. Assuming that the number of tickets submitted remains constant throughout rollover draws, the asymptotic expression for 𝐺_𝑟(𝑛,𝑝) as 𝑛→∞ and 𝑝≪1 is 𝐺_𝑟,𝑎(𝑛,𝑝)=𝑛𝑒^−𝑛𝑝1−𝑒^−𝑛𝑝₀^𝑛𝑝𝑒^𝑥−1𝑥1𝑑𝑥=1−𝑒^−𝑛𝑝𝐺_𝑎(𝑛,𝑝). (3.6) This function is maximized at 𝑛^′_opt=3.750147/𝑝 and 𝐺_𝑟,𝑎(𝑛^′_opt,𝑝)=1.320264/𝑝=0.352057𝑛^′_opt.

In particular, the maximal mean earnings possible are about 32% above the asymptotic value, and about 35% of the (first draw) total prize money.

Proof. Under the stated assumptions, (3.4) becomes 𝐺_𝑟,𝑎(𝑛,𝑝)=^∞_𝑘=1ℙ(𝐾=𝑘)𝐺_𝑎=(𝑘,𝑛,𝑝)_∞𝑘=1𝑒^{−𝑛𝑝(𝑘−1)}(1−𝑒^−𝑛𝑝)𝑘𝑛𝑒^−𝑛𝑝₀^𝑛𝑝𝑒^𝑥−1𝑥𝑑𝑥=𝑛(1−𝑒^−𝑛𝑝)₀^𝑛𝑝𝑒^𝑥−1𝑥𝑑𝑥^∞_𝑘=1𝑘𝑒^{−𝑛𝑝𝑘}=𝑛𝑒^−𝑛𝑝1−𝑒^−𝑛𝑝₀^𝑛𝑝𝑒^𝑥−1𝑥⇔𝐺𝑑𝑥_𝑟,𝑎(𝑛,𝑝)𝑛=𝐹(𝑛𝑝)1−𝑒^−𝑛𝑝=𝐹_𝑟𝑒(𝑛𝑝)=^−𝑛𝑝1−𝑒^−𝑛𝑝₀^𝑛𝑝𝑒^𝑥−1𝑥𝑑𝑥, (3.7) and this proves our first claim.
We now note that the asymptotic behavior of 𝐺_𝑎 and 𝐺_𝑟,𝑎 is identical as 𝑛→∞, while lim_𝑛→0𝐺_𝑟,𝑎(𝑛,𝑝)=0 (𝑛 is treated here as a continuous quantity). As in Theorem 2.2, it follows that 𝐺_𝑟,𝑎 has a global maximum, which, as before, we locate through the condition 𝜕𝐺_𝑟,𝑎(𝑛,𝑝)/𝜕𝑛=0: 0=(1−𝑛𝑝)𝑒^−𝑛𝑝(1−𝑒^−𝑛𝑝)−𝑛𝑝𝑒^−2𝑛𝑝(1−𝑒^−𝑛𝑝)^20𝑛𝑝𝑒^𝑥−1𝑥𝑑𝑥+𝑛𝑒^−𝑛𝑝1−𝑒^−𝑛𝑝𝑝𝑒^𝑛𝑝−1𝑒𝑛𝑝=1+^−𝑛𝑝(1−𝑛𝑝−𝑒^−𝑛𝑝)(1−𝑒^−𝑛𝑝)^20𝑛𝑝𝑒^𝑥−1𝑥⇔𝑑𝑥_𝑠0𝑒^𝑥−1𝑥𝑑𝑥=(1−𝑒^−𝑠)²𝑒^−𝑠(𝑠+𝑒^−𝑠.−1) (3.8) This equation can be solved numerically to yield 𝑠≈3.750147, whence 𝑛^′_opt≈𝑠𝑝≈3.750147𝑝. (3.9) The value of the maximum, consequently, is 𝐺_𝑟,𝑎(𝑛^′_opt𝑠,𝑝)≈𝑝𝑒^−𝑠1−𝑒^−𝑠𝑠₀𝑒^𝑥−1𝑥=𝑠𝑑𝑥𝑝𝑒^−𝑠1−𝑒^−𝑠(1−𝑒^−𝑠)²𝑒^−𝑠(𝑠+𝑒^−𝑠=𝑠−1)𝑝(1−𝑒^−𝑠)𝑠+𝑒^−𝑠≈−11.320264𝑝⇔𝐺_𝑟,𝑎(𝑛^′_opt,𝑝)𝑛^′_opt≈0.352057. (3.10) This completes the proof.

Continuing the numerical example for the Greek lottery given right below Theorem 2.3, 𝐺_𝑟,𝑎(𝑛,𝑝) is maximized for 𝑛^′_opt≈3.750147/𝑝≈52.4 million, which is also significantly higher than the actual number of tickets normally played.

We may attempt, as in Theorem 2.3, to maximize the relative (to the prize money accumulated in the first draw) mean earnings. To do this, we refer back to (3.7) and ask for the roots of the derivative of 𝐹_𝑟: 0=𝐹^′_𝑟𝑒(𝑠)=−^𝑠(𝑒^𝑠−1)^2𝑠0𝑒^𝑥−1𝑥1𝑑𝑥+𝑒^𝑠𝑒−1^𝑠−1𝑠⇔^𝑠₀𝑒^𝑥−1𝑥𝑑𝑥=(𝑒^𝑠−1)²𝑠𝑒^𝑠. (3.11) This equation, however, has no root! It turns out that 𝐹_𝑟 is strictly decreasing. A more appropriate quantity to consider is the mean earnings relative to the mean accumulated prize money 𝑔_𝑟=^∞_𝑘=1𝑘𝑛𝑒^{−𝑛𝑝(𝑘−1)}(1−𝑒^−𝑝𝑛𝑛)=1−𝑒^−𝑝𝑛, (3.12) whence it follows that 𝐺_𝑟,𝑎/𝑔_𝑟=𝐹.

Figure 2 plots 𝐺_𝑎,𝐺_𝑟,𝑎,𝐹_𝑟, and 𝐹 for 𝑝=0.01.

(a)

(b)

Figure 2 

(a) $G_a$ (in red) and $G_{r,a}$ (in blue) for $p=0.01$, as given by (2.11) and (3.7), respectively. (b) $F_r$ (in blue) and $F$ (in red) for $p=0.01$.

4. Conclusion

We investigated the mean earnings of a lottery winner as a function of the number 𝑛 of players participating in a lottery with success probability 𝑝, under the assumption that the total amount of money offered as a prize to the winners is equal (or, in general, linearly proportional) to 𝑛. We considered two versions of the lottery game, namely, with and without rollover. We concluded that both with and without rollover, both the absolute mean earnings and the fraction of the mean earnings over the total accumulated sum for an individual winner are maximized when 𝑛𝑝 equals a certain constant (different, in general, for the four different stated cases), which we determined numerically. In the course of our investigation, we linked the mean earnings formula to the classical problem of determining the inverse moments of a binomial distribution, offering for them novel formulas, both exact and approximate.

Acknowledgments

The author would like to thank the two anonymous referees for their detailed and insightful comments, which greatly improved this paper; in particular, for bringing to the author's attention that the problem studied here is related to the classical problem of determining the inverse moments of the binomial distribution, and for providing an extensive list of additional references.