Transylvanian Lottery

Transylvanian lottery

In this toy lottery, a drawing selects three numbers from seven, and each ticket lists three numbers. Matching all three wins the jackpot; matching two wins a smaller prize. I could find no evidence that such a game has ever been played in Transylvania, or by vampires.

Two out of three is a pretty easy win. So I don’t have to keep typing “two out of three,” let’s call a ticket that wins this lesser prize a deuce. If the jackpot drawing is 1, 4, and 7, for example, the four tickets with a 1, a 4, and some number other than 7 are all deuces. And besides those four, there are the four tickets that hit 1–7 and the four that hit 4–7. So twelve out of thirty-five, just over a third of the possible tickets, are deuces. Which suggests there are probably at least a couple of deuces among Gerald Selbee’s seven tickets. To be precise, you can compute that Selbee has

5.3% chance of no deuces

19.3% chance of exactly one deuce

30.3% chance of two deuces

26.3% chance of three deuces

13.7% chance of four deuces

4.3% chance of five deuces

0.7% chance of six deuces

0.1% chance of all seven tickets being deuces.

The expected number of deuces is thus

5.3% × 0 + 19.3% × 1 + 30.3% × 2 + 26.3% × 3 + 13.7% × 4 + 4.3% × 5 + 0.7% × 6 + 0.1% × 7= 2.4

The Transylvanian version of James Harvey, on the other hand, doesn’t use the Quic Pic; he fills out his seven tickets by hand, and here they are:

124

135

167

257

347

236

456

Suppose the lottery draws 1, 3, and 7. Then Harvey’s holding three deuces: 135, 167, and 347. What if the lottery draws 3, 5, 6? Then Harvey once again has three deuces among his tickets, with 135, 236, and 456. Keep trying possible combinations and you’ll quickly see that Harvey’s choices have a very special property: either he wins the jackpot, or he wins exactly three deuces. The chance that the jackpot is one of Harvey’s seven tickets is 7 out of 35, or 20%. So he has a

20% chance of no deuces

80% chance of three deuces.

His expected number of deuces is

20% × 0 + 80% × 3= 2.4

the same as Selbee’s, as it must be. The variance in the number of deuces is lower, but the important payoff feature is that a draw yielding no deuces instead gives Harvey the jackpot. That makes Harvey’s portfolio a lot more attractive to potential cartel members. Note especially: whenever Harvey doesn’t get three deuces, he wins the jackpot. That means that Harvey’s strategy guarantees a substantial minimum payoff, something the Quic-Pickers like Selbee can never do. In this toy game, the design puts a floor under the payout while preserving the expected number of deuces. Whether that payout covers the ticket cost depends on the prizes and prices.

And how do you do that? That is—literally, for once!—the million-dollar question.

First try: just ask your computer to do it. Harvey and his team were MIT students, presumably able to knock off a few dozen lines of code before their morning coffee. Why not just write a program to run through all combinations of 300,000 WinFall tickets to see which one provided the lowest-variance strategy?

That wouldn’t be a hard program to write. The one small problem would be the way all matter and energy in the universe decayed into heat death by the time your program had handled the first tiny fragment of a microsliver of the data it was trying to analyze. From the point of view of a modern computer, 300,000 is not a very large number. But the objects that the proposed program has to pick through are not the 300,000 tickets—they are the possible sets of 300,000 tickets to be purchased from the 10 million possible Cash WinFall tickets. How many of those sets are there? More than 300,000. More than the number of subatomic particles that exist or have ever existed. A lot more. You’ve probably never even heard of a number as big as the number of ways to select your 300,000 tickets.*

What we’re up against here is the dreaded phenomenon known by computer-science types as “the combinatorial explosion.” Put simply: very simple operations can change manageably large numbers into absolutely impossible ones. If you want to know which of the fifty states is the most advantageous place to site your business, that’s easy; you just have to compare fifty different things. But if you want to know which route through the fifty states is the most efficient—the so-called traveling salesman problem—the combinatorial explosion goes off, and you face difficulty on a totally different scale. There are about 30 vigintillion routes to choose from. In more familiar terms, that’s 30 thousand trillion trillion trillion trillion trillion.

Boom!

So there’d better be another way to choose our lottery tickets to tamp down variance.