# Question

Many states supplement their tax revenues with state-sponsored lotteries. Most of them do so with a game called lotto. Although there are various versions of this game, they are all basically as follows. People purchase tickets that contain r distinct numbers from 1 to m, where r is generally 5 or 6 and m is generally around 50. For example, in Virginia, the state discussed in this case, r = 6 and m = 44. Each ticket costs $1, about 39 cents of which is allocated to the total jackpot10. There is eventually a drawing of r = 6 distinct numbers from the m = 44 possible numbers. Any ticket that matches these 6 numbers wins the jackpot.

There are two interesting aspects of this game. First, the current jackpot includes not only the revenue from this round of ticket purchases but also any jackpots carried over from previous drawings because of no winning tickets. Therefore, the jackpot can build from one drawing to the next, and in celebrated cases it has become huge. Second, if there is more than one winning ticket—a distinct possibility—the winners share the jackpot equally. (This is called pari-mutuel betting.) So, for example, if the current jackpot is $9 million and there are three winning tickets, then each winner receives $3 million.

It can be shown that for Virginia’s choice of r and m, there are approximately 7 million possible tickets (7,059,052 to be exact). Therefore any ticket has about one chance out of 7 million of being a winner. That is, the probability of winning with a single ticket is p _ 1/7,059,052—not very good odds. If n people purchase tickets, then the number of winners is binomially distributed with parameters n and p. Because n is typically very large and p is small, the number of winners has approximately a Poisson distribution with rate λ = np. For example, if 1 million tickets are purchased, then the number of winning tickets is approximately Poisson distributed with λ = 1/7.

In 1992, an Australian syndicate purchased a huge number of tickets in the Virginia lottery in an attempt to assure itself of purchasing a winner. It worked! Although the syndicate wasn’t able to purchase all 7 million possible tickets (it was about 1.5 million shy of this), it did purchase a winning ticket, and there were no other winners. Therefore, the syndicate won a 20-year income stream worth approximately $27 million, with a net present value of approximately $14 million. This made the syndicate a big profit over the cost of the tickets it purchased. Two questions come to mind:

(1) Is this hogging of tickets unfair to the rest of the public?

(2) Is it a wise strategy on the part of the syndicate (or did it just get lucky)?

To answer the first question, consider how the lottery changes for the general public with the addition of the syndicate. To be specific, suppose the syndicate can invest $7 million and obtain all of the possible tickets, making it a sure winner. Also, suppose n people from the general public purchase tickets, each of which has 1 chance out of 7 million of being a winner. Finally, let R be the jackpot carried over from any previous lotteries. Then the total jackpot on this round will be [R + 0.39(7,000,000 + n)] because 39 cents from every ticket goes toward the jackpot. The number of winning tickets for the public will be Poisson distributed with λ = n/7,000,000.

However, any member of the public who wins will necessarily have to share the jackpot with the syndicate, which is a sure winner. Use this information to calculate the expected amount the public will win. Then do the same calculation when the syndicate does not play. For values of n and R that you can select, is the public better off with or without the syndicate? Would you, as a general member of the public, support a move to outlaw syndicates from hogging the tickets?

The second question is whether the syndicate is wise to buy so many tickets. Again assume that the syndicate can spend $7 million and purchase each possible ticket. Also, assume that n members of the general public purchase tickets, and that the carryover from the previous jackpot is R. The syndicate is thus assured of having a winning ticket, but is it assured of covering its costs? Calculate the expected net benefit (in terms of net present value) to the syndicate, using any reasonable values of n and R, to see whether the syndicate can expect to come out ahead. Actually, the analysis suggested in the previous paragraph is not complete. There are at least two complications to consider. The first is the effect of taxes. Fortunately for the Australian syndicate, it did not have to pay federal or state taxes on its winnings, but a U.S. syndicate wouldn’t be so lucky. Second, the jackpot from a $20 million jackpot, say, is actually paid in 20 annual $1 million payments. The Lottery Commission pays the winner $1 million immediately and then purchases 19 “strips” (bonds with the interest not included) maturing at 1-year intervals with face value of $1 million each. Unfortunately, the lottery prize does not offer the liquidity of the Treasury issues that back up the payments. This lack of liquidity could make the lottery less attractive to the syndicate.

There are two interesting aspects of this game. First, the current jackpot includes not only the revenue from this round of ticket purchases but also any jackpots carried over from previous drawings because of no winning tickets. Therefore, the jackpot can build from one drawing to the next, and in celebrated cases it has become huge. Second, if there is more than one winning ticket—a distinct possibility—the winners share the jackpot equally. (This is called pari-mutuel betting.) So, for example, if the current jackpot is $9 million and there are three winning tickets, then each winner receives $3 million.

It can be shown that for Virginia’s choice of r and m, there are approximately 7 million possible tickets (7,059,052 to be exact). Therefore any ticket has about one chance out of 7 million of being a winner. That is, the probability of winning with a single ticket is p _ 1/7,059,052—not very good odds. If n people purchase tickets, then the number of winners is binomially distributed with parameters n and p. Because n is typically very large and p is small, the number of winners has approximately a Poisson distribution with rate λ = np. For example, if 1 million tickets are purchased, then the number of winning tickets is approximately Poisson distributed with λ = 1/7.

In 1992, an Australian syndicate purchased a huge number of tickets in the Virginia lottery in an attempt to assure itself of purchasing a winner. It worked! Although the syndicate wasn’t able to purchase all 7 million possible tickets (it was about 1.5 million shy of this), it did purchase a winning ticket, and there were no other winners. Therefore, the syndicate won a 20-year income stream worth approximately $27 million, with a net present value of approximately $14 million. This made the syndicate a big profit over the cost of the tickets it purchased. Two questions come to mind:

(1) Is this hogging of tickets unfair to the rest of the public?

(2) Is it a wise strategy on the part of the syndicate (or did it just get lucky)?

To answer the first question, consider how the lottery changes for the general public with the addition of the syndicate. To be specific, suppose the syndicate can invest $7 million and obtain all of the possible tickets, making it a sure winner. Also, suppose n people from the general public purchase tickets, each of which has 1 chance out of 7 million of being a winner. Finally, let R be the jackpot carried over from any previous lotteries. Then the total jackpot on this round will be [R + 0.39(7,000,000 + n)] because 39 cents from every ticket goes toward the jackpot. The number of winning tickets for the public will be Poisson distributed with λ = n/7,000,000.

However, any member of the public who wins will necessarily have to share the jackpot with the syndicate, which is a sure winner. Use this information to calculate the expected amount the public will win. Then do the same calculation when the syndicate does not play. For values of n and R that you can select, is the public better off with or without the syndicate? Would you, as a general member of the public, support a move to outlaw syndicates from hogging the tickets?

The second question is whether the syndicate is wise to buy so many tickets. Again assume that the syndicate can spend $7 million and purchase each possible ticket. Also, assume that n members of the general public purchase tickets, and that the carryover from the previous jackpot is R. The syndicate is thus assured of having a winning ticket, but is it assured of covering its costs? Calculate the expected net benefit (in terms of net present value) to the syndicate, using any reasonable values of n and R, to see whether the syndicate can expect to come out ahead. Actually, the analysis suggested in the previous paragraph is not complete. There are at least two complications to consider. The first is the effect of taxes. Fortunately for the Australian syndicate, it did not have to pay federal or state taxes on its winnings, but a U.S. syndicate wouldn’t be so lucky. Second, the jackpot from a $20 million jackpot, say, is actually paid in 20 annual $1 million payments. The Lottery Commission pays the winner $1 million immediately and then purchases 19 “strips” (bonds with the interest not included) maturing at 1-year intervals with face value of $1 million each. Unfortunately, the lottery prize does not offer the liquidity of the Treasury issues that back up the payments. This lack of liquidity could make the lottery less attractive to the syndicate.

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