Lotteries as Investments

There has been a lot of excitement lately about the Mega Millions lottery reaching its largest-ever jackpot of $476 million. This can happen whenever a long period of time goes by without a winner, as the jackpot continues to "roll over" and accumulate. Having co-workers and friends that enjoy thinking about probabilities means that a lot of impromptu discussions have sprouted regarding the general question: "Should I play?" The answer, unless you are already a billionaire, is no.

This post is based almost solely on the information in the excellent and accessible paper by Abrams and Garibaldi entitled, "Finding Good Bets in the Lottery, and Why You Shouldn't Take Them". I will cover some of the same ground, but with a focus on working up the current Mega Millions jackpot as an example, using their paper as a starting point. I'll show, specifically for this $476 million jackpot, that Mega Millions may be a good bet this week, depending on the number tickets sold. Spoiler alert: but it's still a bad investment.

Betting Vs Investing

Abrams and Garibaldi define "good bet" to mean any wager with positive expected return. My fellow poker players are very familiar with this concept, and it's why we head straight for the poker room in a casino instead of stopping to play craps or slots (well, most of us). If you're sitting there with pocket aces, you know it's a good idea, mathematically, to get all your chips committed pre-flop. In fact, you're pretty close to an 85% favorite to win against a single opponent.

A "good investment", on the other hand takes risk of ruin into account, and is much harder to understand. Under very specific circumstances, it might be right to fold those pocket aces pre-flop! If you're on the bubble in a satellite with a hard cut-off and you see two other big stacks get all-in ahead of you, it's probably better to fold AA. For those of you that don't follow poker jargon, imagine the case of real estate auction sales: you can get properties at fractions of their worth, but there is a reasonable chance (say 25%) that one of them will be totally worthless - and you must pay cash. The auction is only a good investment if you have enough capital to absorb some bad runs of luck: 3 out of your 4 houses you bought this time around turn out to be structurally unsound and worthless, instead of the expected 1 out of 4.

That's Nice - Should I Play or Not?

OK, enough background. Let's get to the analysis! I'll save you the formulas: the probability of any randomly selected ticket in Mega Millions being a jackpot winner is 1 in 175,711,536. There are also lesser prizes with correspondingly higher probabilities, and on the downside the jackpot needs to be diluted for taxes and the probability of having to split the prize with another winner. To begin, we must calculate the simple expected rate of return (eROR), which requires knowing the ticket sales. Since we don't know that for March 30, I plugged in the numbers from the previous drawing, and then gave some projections for color on the next one. I drew heavily from Abrams to calculate these numbers:

March 27: $363M jackpot, 190,922,875 tickets sold. eROR=-3%
March 30: $476M jackpot:
    100million tickets sold: eROR=58%
    200million tickets sold: eROR=24%

    300million tickets sold: eROR=Dead Even

    400million tickets sold: eROR=-17%
(Note: all returns assume 25% taxes withheld)

Abrams converts these financialish returns into break-even curves, which I won't show here (see page 14 of his paper if you're curious). Then he uses some calculus that I also won't include here and generates an upper and lower bound for all possible break-even curves for lotteries like Mega Millions. The end result, though, is that we want the jackpot to be large relative to the number of tickets sold. But you didn't need all the math above to tell you that. What the math tells us is how much larger it needs to be.

Well it turns out that anytime the Mega Millions jackpot (lump sum, after tax) is 20% more than its ticket sales - and as I said you'll need to project this number - it's a good bet. By the way, it also has to be larger than 171million (after tax), which is the straight-up probabilistic break-even regardless of sales. Abrams predicts that this will never happen, and that the surge in ticket-buying that surrounds a big jackpot more than makes up for the improved probability-weighted return. I think he fails to see the large numbers that can happen in a roll-over jackpot like Mega Millions, which is a natural human mistake.

There is an undefined upper-bound to the public's appetite for lottery tickets regardless of mathematical edge, as well as a diminishing stimulus to ever-higher jackpots. Eventually, an extra $100million in jackpot fails to generate an extra 100million ticket sales in a few days. And that's what we're seeing right now: the jackpot has gotten so big that the habitual and some-time players have all already joined the list. Even the almost-never players are about tapped out at this point, which lets the jackpot grow without adding additional ticket sales.

Yes Yes, 300 Million Tickets - Should I Play or Not?

Will March 30 be a good bet? This really comes down to your opinion of whether 300 million tickets will be sold. If I were a betting man, and I am, I would set my over-under line at just about that figure. But I'll look one drawing farther into the future, and say this: if it rolls over one more time, the jackpot will outpace demand. At that point, it will certainly be a good bet.

But will it be a good investment? The short answer is no. Abrams goes into a detailed portfolio analysis of the variance on the Texas Lottery, using the Nobel Prize-winning techniques developed by Markowitz and Sharpe in 1990. He calculates that for that particular lottery, a reasonable investor should invest nothing in any syndicate buying lottery tickets unless that syndicate buys at least 145,000 per week. These numbers are impressive, but they don't really apply to Mega Millions. To do that, I would have to retrace the rest of Abrams' work, which I leave as an exercise for the reader. But intuitively, I think we can all see that the variance on a 1-in-176million investment is crazy-high, and should be such a tiny sliver of any investment portfolio that only the largest investors would buy any tickets at all.

I am not that investor, so I won't be buying a ticket next week, regardless of the March 30 outcome.

As a post-script, as I was chasing down all the links for this post, I stumbled upon a less technical version (Word document) of the paper I wish I had found before working through the math for this post. If you want Abrams' own words, I suggest starting there. And if you want the spreadsheet I used to do the calculations, I put it into a public Google Docs file. You'll need to download it in order to play with the numbers, but help yourself.