Additivity of Expected Value

THE LOTTERY SCAM THAT WASN’T

On July 12, 2005, the Compliance Unit of the Massachusetts State Lottery received an unusual phone call from an employee at a Star Market in Cambridge, the northern suburb of Boston that houses both Harvard and MIT. A college student had come into the supermarket to buy tickets for the state’s new Cash WinFall game. That wasn’t strange. What was unusual was the size of the order; the student had presented fourteen thousand order slips, each one filled out by hand, for a total of $28,000 in lottery tickets.

No problem, the lottery told the store; if the slips are filled out properly, anybody can play as much as they want. Stores were required to get a waiver from the lottery office if they wanted to sell more than $5,000 in tickets per day, but those waivers were easily granted.

That was a good thing, because the Star wasn’t the only Boston-area lottery agent doing a vigorous business that week. Twelve more stores contacted the lottery in advance of the July 14 drawing to ask for waivers. Three of those were concentrated in a heavily Asian-American neighborhood of Quincy, just south of Boston on the bay shore. Tens of thousands of Cash WinFall tickets were being sold to a small group of buyers at a handful of stores.

What was going on? The answer wasn’t secret; it was in plain sight, right there in the rules for Cash WinFall. The new game, launched in the fall of 2004, was a replacement for Mass Millions, which had been phased out after going an entire year without paying out a jackpot. Players were getting discouraged, and sales were down. Massachusetts needed to shake up its lottery, and state officials hit on the idea of adapting WinFall, a game from Michigan. In Cash WinFall, the jackpot didn’t pile higher and higher with each week it went unclaimed; instead, every time the pot went over $2 million, the money “rolled down” to enhance the lesser prizes that weren’t so hard to win. The jackpot reset to its minimum $500,000 value for the following drawing. The lottery commission hoped the new game, which made it possible to take in serious winnings without hitting the jackpot, would seem like a good deal.

They did their job too well. In Cash WinFall, Massachusetts had inadvertently designed a game that actually was a good deal. And by the summer of 2005, a few enterprising players had figured that out.

On a normal day, here’s how the prize distribution for Cash WinFall looked:

match all 6 numbers	1 in 9.3 million	variable jackpot
match 5 of 6	1 in 39,000	$4,000
match 4 of 6	1 in 800	$150
match 3 of 6	1 in 47	$5
match 2 of 6	1 in 6.8	free lottery ticket

If the jackpot is $1 million, the expected value of a two-dollar ticket is pretty poor:

($1 million / 9.3 million) + ($4,000 / 39,000) + ($150 / 800) + ($5 / 47) + ($2 / 6.8)= 79.8 cents.

That’s a rate of return so pathetic it makes Powerball players look like canny investors. (And we’ve generously valued a free ticket at the $2 it would cost you instead of the substantially smaller expected value it brings you.)

But on a roll-down day, things look very different. On February 7, 2005, the jackpot stood near $3 million. Nobody won that jackpot—unsurprising, considering that only about 470,000 people played Cash WinFall that day, and matching all six numbers was about a 1-in-10 million long shot.

So all that money rolled down. The state’s formula rolled $600,000 to the match-5 and match-3 prize pools and $1.4 million into the match-4s. The probability of getting 4 out of 6 WinFall numbers right is about 1 in 800, so there must have been about 600 match-4 winners that day out of the 470,000 players. That’s a lot of winners, but $1.4 million dollars is a lot of money. Dividing it into 600 pieces leaves more than $2,000 for each match-4 winner. In fact, you’d expect the payout for matching 4 out of 6 numbers that day to be around $2,385. That’s a much more attractive proposition than the measly $150 you’d win on a normal day. A 1-in-800 chance of a $2,385 payoff has an expected value of

$2385 / 800= $2.98

In other words, the match-4 prize alone makes the ticket worth its $2 price. Throw in the other prizes, and the story gets even sweeter.

Prize	Chance of winning	Expected number of winners	Roll-down allocation	Roll-down per prize
match 5 of 6	1 in 39,000	12	$600,000	$50,000
match 4 of 6	1 in 800	587	$1.4m	$2,385
match 3 of 6	1 in 47	10,000	$600,000	$60

So the average ticket could be expected to bring home cash winnings of

$50,000 / 39,000 + $2385 / 800 + $60 / 47= $5.53.

An opportunity to make three and a half bucks of profit on a $2 investment is not one to pass up.[*](text00116.html#footnote-2505-80)

Of course, if one lucky person hits the jackpot, the game turns back into a pumpkin for everybody else. But Cash WinFall was never popular enough to make that outcome likely. Out of forty-five roll-down days during the lifetime of the game, only once did a player match all six numbers and stop the roll-down in its tracks.[*](text00117.html#footnote-2505-81)

Let’s be clear—this computation doesn’t mean that a $2 bet is sure to win you money. On the contrary, when you buy a Cash WinFall ticket on a roll-down day, your ticket is most likely a loser, just as it is on any other day. The expected value is not the value you expect! But on roll-down day, the prizes, in the unlikely event that you do win, are bigger—a lot bigger. The expected payout per ticket in the roll-down calculation is $5.53. Buying a hundred, a thousand, or ten thousand tickets increases the total expected return, but additivity alone does not show how close the outcome will be to that expectation. Any given ticket is probably worthless, and a large purchase can still lose money; the result depends on the winning numbers, the tickets held, and the prize distribution.

Who buys a thousand lottery tickets at a time?

Kids at MIT, that’s who.

The reason I can tell you the WinFall payoffs on February 7, 2005, down to the last dollar is because this figure is recorded in the exhaustive and, frankly, kind of thrilling account of the WinFall affair submitted to the state in July 2012 by Gregory W. Sullivan, the inspector general of the Commonwealth of Massachusetts. I think I’m safe in saying this is history’s only state fiscal oversight document which inspires the reader to wonder: Does someone have the movie rights to this?

And the reason it’s this particular day for which this data is recorded is that February 7 was the first roll-down day after James Harvey, an MIT senior working on an independent study project comparing the merits of various state lottery games, realized that Massachusetts had accidentally created an insanely profitable investment vehicle for anyone quantitatively savvy enough to notice it. Harvey got a group of friends together (at MIT, it’s not hard to get a group of friends together who can all compute expected value) and bought a thousand tickets. Just as you might expect, one of those 1-in-800 shots came through, and Harvey’s group took home one of those $2,000 prizes. They won a bunch of match-3s too; in all, they just about tripled their initial investment.

It won’t surprise you to hear that Harvey and his co-investors didn’t stop playing Cash WinFall. Or that he never did get around to finishing that independent study—at least not for course credit. In fact, his research project quickly developed into a thriving business. By summer, Harvey’s confederates were buying tens of thousands of tickets at a time—it was a member of his group who placed the mammoth order at the Cambridge Star Market. They called their team Random Strategies, though their approach was anything but scattershot; the name referred to Random Hall, the MIT dorm where Harvey had originally cooked up his plan to make money on WinFall.

And the MIT students weren’t alone. At least two more betting clubs formed up to take advantage of the WinFall windfall. Ying Zhang, a medical researcher in Boston with a PhD from Northeastern, formed the Doctor Zhang Lottery Club. It was the DZLC that accounted for the spike in sales in Quincy. Before long, the group was buying $300,000 worth of tickets for each roll-down. In 2006, Doctor Zhang quit doctoring to devote himself full-time to Cash WinFall.

Still another betting group was led by Gerald Selbee, a retiree in his seventies with a bachelor’s degree in math. Selbee lived in Michigan, the original home of WinFall; his group of thirty-two bettors, mostly made up of his relatives, played WinFall there for about two years until the game shut down in 2005. When Selbee found out the gravy train was getting back on the tracks out East, his course was clear; in August 2005, he and his wife Marjorie drove to Deerfield, in the western part of Massachusetts, and placed their first bet—sixty thousand tickets. They took home a little over $50,000 in pure profit. Selbee, with the benefit of his experience playing the game in Michigan, added an extra profit-making venture to his Cash WinFall tickets. Stores in Massachusetts got a 5% commission on lottery ticket sales. Selbee cut deals directly with one store, offering hundreds of thousands of dollars’ worth of business at a time in exchange for going halfsies on the 5% commission. That move alone made Selbee’s team thousands of dollars in extra profit every roll-down.

You don’t need an MIT degree to see how the influx of high-volume players affected the game. Remember: the reason the roll-down payoffs were so swollen was that a lot of money was being split among just a few winners. By 2007, a million or more tickets were being sold for each roll-down drawing, most of them to the three high-volume syndicates. The days of the $2,300 prize for matching four out of six numbers were long gone; if a million and a half people bought tickets, and one person in eight hundred matched 4, then you’d typically see almost two thousand match-4 winners. So each share of the $1.4 million kitty was now more like $800.

It’s pretty easy to figure out how much a big player stood to gain from Cash WinFall—the trick is to look at it from the point of view of the lottery itself. If it’s roll-down day, the state has (at least!) $2 million of accumulated jackpot money it’s got to get rid of. Let’s say a million and a half people buy tickets for the roll-down. That’s $3 million more in revenue, of which 40%, or $1.2 million, goes into the state’s coffers, and the other $1.8 million gets plowed into the jackpot fund, all of which is to be disbursed to bettors before the day is through. So the state takes in $3 million that day and hands out $3.8 million:[*](text00118.html#footnote-2505-82) $2 million from the money already in the jackpot fund and $1.8 million from that day’s ticket receipts. On any given day, whatever the state makes, the players, on average, lose, and vice versa. So this day is a good day to play; ticket buyers, in the aggregate, took $800,000 from the state.

If players buy 3.5 million tickets, it’s a different story; now the lottery takes $2.8 million as its share and pays out the remaining $4.2 million. On top of the $2 million already in the kitty, that amounts to $6.2 million, less than the $7 million of revenue the state took in. In other words, despite the generosity of the roll-down, the lottery has gotten so popular that the state still ends up making money at the expense of the players.

This makes the state very, very happy.

The break-even point comes when the 40% share of the roll-down day revenue exactly matches the $2 million already in the pot (that is, the money contributed by the players who were unsophisticated or risk-loving enough to play WinFall without a roll-down). That’s $5 million, or 2.5 million tickets. More sales than that, and WinFall is a bad bet. But any fewer—and over the life span of the WinFall game, it always was fewer—and WinFall offers players a way to make some money.

What we’re really using here is a wonderful, while at the same time commonsensical, fact called additivity of expected value. Suppose I own a McDonald’s franchise and a coffee shop, and the McDonald’s has an expected annual profit of $100,000, while the coffee shop’s expected net is $50,000. The money might go up and down from year to year, of course; the expected value means that, in the long run, the average amount of money the McDonald’s makes will be about $100,000 a year, and the average amount from the coffee shop $50,000.

Additivity says that, on average, my total take from Big Macs and mochaccinos together is going to average out to $150,000, the sum of the expected profits from each of my two businesses.

In other words:

ADDITIVITY: The expected value of the sum of two things is the sum of the expected value of the first thing with the expected value of the second thing.

Mathematicians like to sum up that reasoning in a formula, just as we summed up the commutativity of multiplication (“this many rows of that many holes is the same thing as that many columns of this many holes) by the formula a × b= b × a. In this case, if X and Y are two numbers whose values we’re uncertain about, and E(X) is short for “the expected value of X,” then additivity just says

E(X+Y)= E(X) + E(Y).

This equality does not require X and Y to be independent, provided their expected values are finite. Dependence does matter when assessing the variability of their combined outcome.