Sunday, January 30, 2011

The Virginia Lottery, Scalability and Expected Payout

The following excerpt is from The Drunkard's Walk by Leonard Mladinow.

"In 1992 some investors in Melbourne, Australia, noticed that [only 7,058,052 number choices existed in the Virginia Lottery.] The lottery jackpot was $27 million, and with second, third, and fourth prizes included, the pot grew to $27,918,561. The clever investors reasoned..." that since each ticket cost $1.00, $7,059,052 would buy a ticket for every combination.
   There were some risks in their plan. For one, since they weren't the only ones buying tickets, it was possible that another player or even more than one other player would also choose the winning ticket... In the 170 times the lottery had been held, there was no winner 120 times, a single winner 40 times, and two winners 10 times. If those frequencies reflected their odds, then there was a 120 in 170 chance  they would get the the pot all to themselves, a 40 in 170 chance they would end up with half the pot, and a 10 in 170 chance they would win just a third of it. Recalculating their expected winnings... they found them to be

(120/170 x $27.9 million) + (40/170 x $13.95 million) + (10/170 x $6.975 million) = $23.4 million.

That is $3.31 per ticket... a 231% return over a very short time frame.
   But there was another danger: the logistic nightmare of completing the purchase of all the tickets by the lottery deadline. That could lead to the expenditure of a significant portion of their funds with no significant prize to show for it.
   The members of the investment group made careful preparations. They filled out 1.4 million slips by hand, as required by the rules, each slip good for five games. They placed groups of buyers at 125 retail outlets and obtained cooperation from grocery stores, which profited from each ticket they sold. The scheme got going just seventy-two hours before the deadline. Grocery-store employees worked in shifts to sell as many tickets as possible. One store sold 75,000 in the last forty-eight hours. A chain store accepted bank checks for 2.4 million tickets, assigned the work of printing the tickets among its stores, and hired couriers to gather them. Still, in the end, the group ran out of time: they had purchased just 5 million of the 7,059,052 tickets.
   ...The consortium had won but... A month of legal wrangling ensued before the officials concluded they had no valid reason to deny the group. Finally, they paid out the prize."

3 Billion Percent Annualized Return

Since the group won but only purchased 5 million tickets, their return was actually 377%. I would conservatively translate this to a compounded annual return of 3 billion percent ((100%+377%)^(52/3) assuming that preparations took 3 weeks), while the Virginia Lottery lost a great deal of credibility.

The circumstance of profits exceeding the cost of all tickets was owed to two factors:
      1) A low ticket price.
      2) The cumulative wagers of many drawings without a winner.
I therefore wonder if arbitrageurs monitor state lotteries for other occasions where the cost of buying all tickets is less than the pot. I am curious as to the tax implications though, whether the tax on proceeds was factored into the expected winnings calculations, or whether it is even applicable since the investors lived in a different country.

Is It Scalable?

More importantly though, I think this story emphasizes the value of a certain perspective. For example, in The Google Story, it's said that Google asks "Is it scalable?" about every product, service, algorithm, and feature that they develop... a question that would have saved the Virginia Lottery some heartache. Furthermore, this story demonstrates the value of a quantitative perspective. After all, would you rather be the party profiting from a quantitative perspective or be exploited due to its absence?
From The Drunkard's Walk by Leonard Mladinow.