Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
49,612 characters · 13 sections · 29 citation commands
\abstract{ Despite its unusual payout structure, the Canadian 6/49 Lotto\textcopyright is one of the few government sponsored lotteries that has the potential for a favorable strategy we call “buying the pot.” By “buying the pot” we mean that a syndicate buys each ticket in the lottery, ensuring that it holds a jackpot winner. We assume that the other bettors independently buy small numbers of tickets. This paper presents (1) a formula for the syndicate's expected return, (2) conditions under which buying the pot produces a significant positive expected return, and (3) the implications of these findings for lottery design. }
moffitt:ziemba:2017a show that expected returns of $10\%$-$25\%$ can be achieved under certain conditions from betting all the tickets in a lottery that pays its entire jackpot in equal shares to winning ticket holders. For many large government lotteries, this strategy of “buying the pot” is not feasible because the logistical problems are insurmountable. In the California Powerball Lottery\textcopyright, for example, the number of ticket combinations is over $175,000,000$ and the rules do not allow betting large numbers of combinations on single paper tickets.
The Canadian 6/49 Lotto\textcopyright , however, has a large but manageable number of ticket combinations ($13,983,816$) and allows paper tickets that have multiple combinations. The 6/49 is played in other countries, including the UK. Here we focus on the Canadian version.
The purpose of this paper is threefold: (1) to modify the pure jackpot model in moffitt:ziemba:2017a to accomodate the irregular payout features of the Canadian 6/49 Lotto, (2) to derive conditions under which the expected return from buying the pot is positive, and (3) to discuss the implications of our findings for lottery design.
Each lottery has the following rules --- players buy tickets and the winning ticket is selected using an equiprobable drawing. Those who hold the winning ticket share equally in a jackpot that consists of a carryover pot from the previous lottery plus an after tax portion the monies wagered. If there is no winner, the jackpot pool carries over to the next drawing. There can be multiple carryovers.
moffitt:ziemba:2017a use the following assumptions and notation to analyze the pure jackpot model:
moffitt:ziemba:2017a show the following for the pure jackpot model:
Several studies of lottery strategy and design have appeared in the economic literature. chernoff1980analysis,Chernoff1981 studied the Massachusetts Numbers Game, proposing that playing unpopular numbers might be a winning strategy. However, the results from a test were disappointing because of learning (unpopular numbers became less unpopular) and gambler's ruin (betting funds were exhausted). ziemba1986dr carry this further and study various Canadian lotto games, their unpopular numbers and the uniformity of betting. 10.2307/2290349 has additional discussion of this latter point and 10.2307/3314913, 10.2307/2290073 and Ziemba2008183 further analyze unpopular numbers. citeulike:1337256 review the behavioral evidence in efficient markets for a persistence of betting at unfavorable odds. RePEc:inm:ormnsc:v:38:y:1992:i:11:p:1562-1585 investigate the use of Kelly optimal wagering on unpopular tickets and find that this strategy has positive expectation, but the waiting time to achieve reliable gains with high probability is millions of years! 10.2307/1942724 discuss behavioral bases of betting and along with Walker2008459, discuss design considerations for lotteries. None of these studies consider the strategy that in a short time achieves reliable gains --- buying the pot.
There are anecdotal accounts of successful buyings of the pot. One putative attempt involved a syndicate that tried but failed to buy all tickets. But they were lucky, having had time to bet only about $70\%$ of all tickets according to one source and $85\%$ according to another (NYTimes:US:BuyThePot1992). The syndicate ostensibly bet about \$5 million and won about \$27 million.
There are examples of when it was optimal to buy the pot or betting was advantageous. In June 1984 four western Canadian provinces jointly ran the Lotto West 6/8/56, in which players choose six numbers from a field of 56, but eight winning numbers and a bonus number are drawn. The jackpot is shared among all tickets that select six of the eight drawn, second price among all that had five of six, and other payouts to those having five of six plus the bonus, four of six or three of six. These rules make the jackpot about twelve times easier to hit than the 6/49 Lotto: 1 in $1,159,587$ (See ziemba1986dr).
In 1987, the provinces went their own ways, at which time the BC Lotto Corporation had about \$10 million in unclaimed prize money. Rather than donate it, they created a version of Lotto 6/8/56 to give it back on March 27, 1987. As before eight numbers were drawn from 56, but players could now choose 1, 2, 3, 4, 5 or 6 numbers on a ticket. A schedule of payouts was published for 1/1, 2/2, /3/3, 4/4, 5/5 and for 3/6, 4/6, 5/6 and 6/6. With these payouts, the expected payback on a \$1 ticket was $\$0.385$. To promote the game, the Corporation offered six tickets for the price of one, for an expected return of \$0.385 times 6, or \$2.31, a $131\%$ edge (drz:ColumnsOnRacing). Ziemba and colleagues at the University of British Colombia knew that individual tickets had a positive expected return, and in a makeshift effort, they bought about 13,000 of the combinations. They made a nice return, but spent hours buying and then locating the winning tickets. Some $\$3.5$ million was paid out of the unclaimed prize fund.
A game where it was optimal to buy the pot was the 5/40 Lotto played in British Colombia and Rhode Island. Ninety-one percent of the net pool went to 5/5 with a minimum shared pool of $\$150,000$ and maximum of $\$300,000$. There were small prizes for 1+, 2+, 3, 4 and 4+ where “+” means getting the sixth bonus number correct. There were $658,008$ combinations. But the jackpot that had built up slowly fell because the public viewed it as unwinnable, so it became a prime target for buying the pot; see drz:ColumnsOnRacing.
A ticket in the 6/49 Lotto is a unique choice of $6$ different numbers from integers $1$ to $49$. Thus the total number of tickets is the number of combinations of $49$ things taken $6$ at a time:
The Canadian 6/49 Lotto holds drawings twice a week and lumps together the monies wagered for purposes of awarding prizes, whose allocation is described below. On the drawing day, 6 numbers (the “winning numbers”) are selected equiprobably and without replacement from 1, 2, \ldots 49. Following that, a $7^{th}$ “bonus number” is selected.
We introduce notation to describe types of prize-wining tickets. A $x$/6- ticket is one that contains exactly $x$ of the six winning numbers but does not contain the bonus number and a $x$/6+ ticket is one that contains exactly $x$ of the 6 numbers plus the bonus number. A x/6 ticket contains x of the 6 numbers, irrespective of the status of the bonus number; it is therefore a union of types x/6- and x/6+. A 5/6-, for example, contains exactly 5 of the 6 winning numbers with the other not being the bonus number, and a 5/6+ ticket contains exactly 5 of the 6 winning numbers plus the bonus number. For example, if the six numbers drawn were 46, 13, 4, 21, 38, 25 and the bonus number was 43 then ticket 1-4-20-21-32-43 would be a 2/6+ ticket because it contains 4 and 21 from the six plus the bonus number. Similarly, ticket 4-13-21-25-43-46 would be a 5/6+ ticket.
Table (ref) has the initial 6/49 payout scheme (1982-2004) for 3/6, 4/6, 5/6-, 5/6+ and the Jackpot 6/6. The cost of a single ticket was \$1, with the lottery sponsors taking 55% of each daily betting pool and committing the remaining 45% (the “prize pool”) for player payouts. The 45% prize pool was allocated as follows: all 3/6 tickets were paid \$10, and the remainder was paid to holders of 4/6, 5/6-, 5/6+ and 6/6 using percentage allocation rules in Table (ref). That game is analyzed thoroughly in ziemba1986dr. See also 10.2307/2290073. For other analyses of such games, see citeulike:1337256 and Haigh2008481.
\scriptsize
Figure (ref) shows the leveraging effect of the fixed \$10 3/6 prize when there are average numbers, popular numbers, and unpopular numbers. The impact of popular vs. unpopular numbers selected in the drawing is significant, producing a 17% versus a 36% jackpot share. The large prizes 5/6-, 5/6+ and 6/6 for unpopular numbers in the drawing are typically seven times larger than for popular ones. See examples in ziemba1986dr.
In the 6/49 Lotto, new rules were introduced in June, 2004 and again on September 18, 2013. We discuss only the latter rules. These included (1) a single ticket cost of \$3, (2) three fixed prizes, the same 3/6 paying \$10, a 2/6+ paying \$5 and a 2/6- that earns a free play at the next drawing, and (3) altered payout percentages for 4/6, 5/6-, 5/6+ and 6/6 (Table (ref)), (4) an increase in the take from 55% to 60%, and (5) a greater allocation to 6/6 winners. The intention of these changes was to increase sales by growing jackpots faster, and creating of many small consolation prices (2/6-, 2/6+ and 3/6). This is a typical convex prize structure where most of the daily payout goes to the smallest (to make them feel that the lottery is winnable) and to the largest (to show that a huge gain can be made). Ziemba has used this in lottery consulting over the years. RePEc:eee:jfinec:v:13:y:1984:i:2:p:253-282 call this a “silver lining” for non-winners.
We call the number of tickets bet at a drawing (twice a week in the 6/49), the ticket pool, contributors to which are the crowd in amount $c$ and the syndicate in amount $t$. Thus the total number of tickets bet is $c + t$. The betting pool $d_{\!_{BP}}$ is the total number of dollars contributed by the bettors. The betting pool is divided among the lottery sponsors and the bettors as follows:
Table (ref) gives the fixed parameters of the lottery, namely those that do not involve betting strategies of the syndicate or crowd.
Table (ref) has the notation for the random variables that account for stochasticity and strategy in playing the lottery.
Using the notation in Tables (ref) and (ref), the number of dollars in each fund is
Since $f$ is non-random, the second entry of the above table is the only non-stochastic entry.
We calculate first the expected return to a syndicate that buys the pot when the crowd chooses tickets independently and equiprobably. As we discuss in Section (ref), this is the crowd's optimal strategy, although they do not employ it in practice --- and the cost of this “mistake” is considerable.
In Appendix (ref), we develop a formula for the syndicate's expected gain $G(c)$ from the wagering of $\$41,951,448 = \$3 * 13,983,816$ on $13,983,816$ tickets:
where
The term $\nu(c)$ is calculated using the recursive formula in Appendix (ref) and appears as the last column of Table (ref).
Consider the implications of the 6/49 rules and of Formula (ref). Because the lottery sponsors take such a high percentage of the betting pool (60%), a large jackpot is needed for a syndicate to have a positive expected return. When a syndicate bets one of each ticket, previous analysis showed the syndicate's numbers of winning tickets are known exactly, irrespective of the winning numbers from the drawing. There will always be exactly 1 winning ticket, exactly 6 5/6+ tickets, exactly 252 5/6- tickets, and so on. The RHS of first line of formula (ref) dominates the others when a jackpot $a$ is large.
Table (ref) shows the results of applying formula (ref) for 10 levels of total crowd betting (c) to solve for the sizes of carryover pools (a) that produce expected returns of 0%, 10% and 20% for the syndicate. Since the cost of buying the pot is $\$3 * 13,983,816 =\$41,951,448$, a return of 10% is $\$4,195,145$. When the crowd bets \$30 million, for example, any carryover larger than \$36.92 million is a potential play for the syndicate, and carryovers of \$42.80 and \$48.67 million have expected returns of 10% and 20%, respectively. The last three columns of Table (ref) provide insight into the payout structure. The sixth column shows the expected amount in the Pools Fund and the next column is its percentage in the prize pool. Thus when the crowd bets \$40 million, the expected Pools Fund is \$19.97 million, which is 49.68% of the prize pool. Thus, the charges for fixed payout tickets amount to \$50.32% of the prize pool. The final column is the expected value for the 5/6+ factor:
The expected value declines when the crowd bets more, as one expects since $X_{\text{5/6+}}$ is generally larger.
Recall from Example (ref) that the crowd bet a net \$27,000,000 on 10 million tickets and the carryover was \$30,000,000 --- yet the syndicate won over \$6 million. According to Table (ref), the syndicate should not bet under these conditions, since a minimum carryover of \$36.92 million is necessary. There is no problem here, since the numbers in the table are expected values and it is quite possible for a syndicate to win despite making an unfavorable bet. The syndicate in that example just got lucky.
The calculations in Section (ref) assumed that the crowd bets independently using $q = \frac{1}{t} 1_t$, where $1_t$ is a t-vector of all ones. What happens when the crowd bets using $q \ne \frac{1}{t} 1_t$?
In part (ref), we stated a result from moffitt:ziemba:2017a, that for pure jackpot lotteries (ones having a single prize, a non-stochastic jackpot\footnote{We are assuming that the crowd's number of tickets, $c$, is known.} $v$) the expected payoff is
where $q \ne 1/t 1_t$, $N_1$ is the random number of 6/6 tickets held by the crowd. However, formula (ref) does not apply in the present case because $v$ is stochastic, depending on the size of the Pools Fund.
Consider a non-stochastic configuration of single ticket bets $n_j = (n_{j1}, n_{j2}, \ldots, n_{jt})'$ for individuals $j = 1, \ldots, c$, each having zeroes except for a single $1$ in some position. Define $z_k = \sum_{j=1}^{j=c} n_{jk}$ and t-vector $z = (z_1, \ldots, z_t)'$. Clearly, $\sum_{k=1}^{k=t} z_k = c$. To compute the expected values of ticket types 1, 2, \ldots 7 with respect to an equiprobable drawing, observe that as $i$ ranges over all ticket drawings $i = 1, \ldots t$, for any $n_j$, the number of 6/6 is 1, the number of 5/6+ is 6, the number of 5/6 is 252, and so on as indicated in Table (ref). Since the drawing is equiprobable, dividing each of these by $t$ gives the probability that any non-stochastic ticket will be of the indicated type under an equiprobable drawing. Define indicator functions on single ticket t-vectors $n$ as: \[ I_{x/6}(n) =
\] Applying this to fixed payout types 3/6, 2/6+ and 2/6, we obtain for $d_{\!_{AB}}$ in formula (ref)
where the notation $E_e$ emphasizes that the expectation is taken over equiprobable drawings and $\$5,696,520$ is the fixed payout/deduction for the syndicate.\footnote{$\$5,696,520 = \$10*246,820 + \$5*172,200 + \$1.41*1,678,950$.} The (stochastic) jackpot is $v = a \, + \, 0.795 d_{\!_{PF}}$ and the random 6/6 payout to the syndicate is
In (ref), the factor multiplying $1/(1 + N_1)$ is fixed. Its expectation using (ref) is
where $\lambda = c/t$. Thus for this term at least, the syndicate gets more than a fair split of the jackpot since \[ \frac{1}{\lambda} ( 1 - \exp(-\lambda) ) > \frac{t}{t + c}. \] The second term (ref) depends on $N_1$, $N_5$, $N_6$ and $N_7$, which respectively, are the numbers of 6/6, 3/6, 2/6+ and 2/6 tickets held by the crowd, and these are dependent on the crowd betting probabilities $q = (q_1, \ldots, q_t)'$. But we do not have the data to model the joint distribution of $(N_1, N_5, N_6, N_7)$ which is needed to evaluate (ref).
However, we have circumstantial evidence that $N_5$, $N_6$ and $N_7$ are positively correlated with $N_1$. Therefore we make a crude assumption that the joint crowd payouts for 3/6, 2/6+ and 2/6 tickets are increased linearly with the winning ticket, that is, the payout for ticket $i$ is proportional to $q_i$: \[ \frac{\$10 N_5 \, + \, \$5 N_6 \, + \, \$1.41 N_7}{1 + N_1} \cdot q_i / (1/t). \] Thus if the winning ticket $i$ is bet with twice the frequency of an equiprobable bet, so that $t q_i = 2$, then the fixed payouts/deductions will be twice that expected in the equiprobable case (see the discussion leading to equation (ref)).
Using $H = \$10 p_5 \, + \, \$5 p_6 \, + \, \$1.41 p_7$, we calculate
where $\lambda = c/t$ and the step (ref) follows from Jensen's inequality since $1 - e^{-cq}$ is a concave function of $q$. Jensen's inequality can be stated as follows. A function $f: [a,b] \rightarrow \mathbb{R}$ that satisfies $f(ta + (1-t)b) \le t f(a) + (1-t) f(b)$ for all $t \in (0,1)$ is called convex, and if the inequality is strict, strictly convex. For a random variable $X$ and convex function $f$, Jensen's inequality asserts that $f(E[X]) \le E[f(X)]$. Further, if $X$ is not degenerate and $f$ is strictly convex, then $f(E[X]) < E[f(X)]$. A function $f$ is (strictly) concave if $-f$ is (strictly) \emph{convex}i, so Jensen's inequality is reversed for \emph{concave} functions.
Putting (ref) together with (ref) yields
where $\lambda = c/t$ and $H = \$10 p_5 \, + \, \$5 p_6 \, + \, \$1.41 p_7$.
This calculation shows that the syndicate obtains a better result than when the crowd bets proportionally, as in the corresponding result for pure jackpot lotteries.
Lottery design includes the goal to maximize the the sponsors' earnings. Assuming fairly constant fixed costs of running the lottery, sponsors should strive to make the lottery popular, thereby increasing profitability. The most recent changes to payouts were made with that goal in mind --- these changes increased the “convexity” of payouts, meaning many little prizes and greater jackpot growth. Ziemba recommended these designs in his work in the 1980's and Walker2008459 later also recommended them. Convex designs encourage players because more “get something back,” while at the same time growing large jackpots quickly.
This design is supported by research in behavioral finance. Lopes' SP/A (Security-Potential/Aspiration) model (Lopes1987255), an improved version of the classic Friedman/Savage (1948) utility curves, argues that many unsophisticated gamblers prefer strategies of buying safe prospects with a few longshots (the “Cautiously Hopeful” pattern of SP/A). Regarding large jackpots, Daniel Kahneman has written
There is another aspect of lottery design, namely, discouraging syndicates from buying the pot. There are two ways to accomplish this: (1) creating a large number of tickets making it logistically difficult to buy the pot, and (2) using convex designs, which reduces the likelihood that pot buying situations will occur. Method (1) is not feasible except for large lotteries like the California Powerball lottery. The reason is that if the number of tickets sold are too small relative to the total number of tickets, the jackpot may build slowly and seldom be won. On the other hand, method (2) can be effective regardless of the size of the lottery. To illustrate, consider a pure jackpot lottery with the same carryover, take and crowd betting as in Table (ref). The results are shown in Table (ref). The first column has the number of tickets, which after a 10% deduction for free plays, equals the crowd contribution to the betting pool shown in the second column. Then assuming a take of 60%, breakeven thresholds of 0%, 10% and 20% for the pure jackpot lottery are shown in columns 3-5 and for the 6/49 in columns 6-8. The results show that buying the pot thresholds are lower in the pure lottery, but not as much as might be expected.
But one can see the reason by a simple argument. When the sponsors takes 60%, only 40 cents is returned as prizes for each dollar wagered. Therefore, a syndicate needs to recover 60% of the covering bet, or $0.6*\$3*13,983,816 = \$25,170,869$, regardless of the lottery's rules. As we have shown, the syndicate earns its fair share of consolation prizes, but the free plays it earns are not worth too much since after the lottery is hit the next lottery when those tickets will be used will have a small purse. Assuming the the crowd bets $\$1,000,000$ on the next lottery the expected value of these $1,678,950$ tickets will under optimal wagering be worth about $\$150,000$.
We conclude the discussion by examining the impacts of design choices in the 6/49 Lotto. The 6/49 Lotto's convex design according to Table (ref) raised the bar for buying-the-pot strategies, making carryover thresholds roughly 12% to 20% higher. We now compare the impacts of the 6/49's design features toward increasing the threshold for buying the pot. We identify four factors: (1) the take, (2) the payouts for small prizes, (3) the payouts for large, non 6/6 prizes, and (4) free plays. Then we compare by
Table (ref) shows breakeven carryover thresholds for these design factors. The factor is indicated in the first column and the other 5 columns are carryover thresholds when the crowd bets the indicated millions of dollars, 20, 40, etc. In the second column (corresponding to a crowd bet of \$20 million), the numbers in parenthesis are differences of threshold carryovers from the current 6/49 values (second row, second column). Since the relative impacts of these factors are the same for the five crowd betting amounts, their impacts on the buying the pot strategy can be assessed using this column. The greatest factor impact is due to free plays; removing them drops the threshold by $\$3.39$ million ($\sim 10\%$). The largest inhibitor is clearly the take --- increasing it by $0.05\%$ from to $65\%$ has a large impact on breakeven carryovers.
Based on these statistics, we make recommendations for state lotteries using ratings of the form $(+=-\pm\mp, +=-\pm\mp)$. The first sign is for popularity, the second for inhibiting buyers of the pot. For example, $(+,-)$ indicates that a factor increases the lottery's popularity, but encourages buying the pot.
Increasing the allocation to 6/6 allows quicker build-up of jackpots, which encourages greater crowd participation. However, we did not address the question of build-up speed of the jackpot, nor the acceleration of betting on larger jackpots. These need to be studied in order to design prizes and allocations to optimize betting flows.
In this paper, we have shown conditions under which buying the pot in the 6/49 Lotto has positive expected return when the crowd bets equiprobably. We also indicated that equiprobable betting is optimal for the crowd, that is, expected return is lower when it does not bet equiprobably. We illustrated the advantages of lotteries with convex designs by calculating 6/49 carryover thresholds and comparable pure jackpot carryover thresholds. We then rated various design features for their likelihood of increasing a lottery's popular, and decreasing the likelihood of buyers of the pot.