Gerald Selbee arrived not long after; he told me he met with lottery lawyers at Braintree in August 2005, to let them know his Michigan corporation would be buying lottery tickets in Massachusetts. The existence of high-volume betting was no secret to the state.
But why would Massachusetts allow Harvey, Doctor Zhang, and the Selbees to cart off state money by the millions? What kind of casino lets the players beat the house, week after week, and takes no action?
To unravel this requires thinking a little more closely about how the lottery actually works. Out of every $2 lottery ticket sold, Massachusetts kept 80 cents. Some of that money was used to pay commissions to stores that sell tickets and to operate the lottery itself, and the rest was sent out to city and town governments across the state; almost $900 million in 2011, paying police officers, funding school programs, and generally spackling over the holes in municipal budgets.
The other $1.20 was plowed back into the prize pool, to be distributed among the players. But remember the computation we did at the very beginning? The expected value of a ticket, on a normal day, is just 80 cents, meaning the state is giving back, on average, 80 cents per ticket sold. What happens to the extra 40 cents? That’s where the roll-down comes in. Giving out 80 cents per ticket isn’t enough to exhaust the prize pool, so the jackpot grows bigger each week until it hits $2 million and rolls down. And that’s when the lottery changes its nature; the floodgates are opened and the accumulated money pours out, into the hands of whoever’s smart enough to be waiting.
It might look like the state’s losing money that day, but that’s taking a limited view. Those millions never belonged to Massachusetts; they were earmarked as prize money from the beginning. The state takes its 80 cents out of each ticket and gives back the rest. The more tickets sold, the more revenue comes in. The state doesn’t care who wins. The state just cares how many people play.
So when the betting cartels cashed in the fat profits on their roll-down bets, they weren’t taking money from the state. They were taking it from the other players, especially the ones who made the bad decision to play the lottery on days without a roll-down. The cartels weren’t beating the house. They were the house.
Like the operators of a Las Vegas casino, the high-volume bettors weren’t totally impervious to bad luck. Any roulette player can go on a hot streak and take the casino for a lot of money, and the same thing could have happened to the cartels if an ordinary bettor had hit all six numbers, diverting all the roll-down money to their own jackpot. But Harvey and the others had done the math carefully enough to make this outcome rare enough to tolerate. Only once in the whole course of Cash WinFall did somebody win the jackpot on a roll-down day. If you make enough bets with the odds tilted in your favor, the sheer volume of your advantage dilutes any bad luck you might experience.
That makes playing the lottery less exciting, to be sure. But for Harvey and the other high-volume bettors, excitement wasn’t the point. Their approach was governed by a simple maxim: if gambling is exciting, you’re doing it wrong.
If the betting cartels were the house, then what was the state? The state was . . . the state. Just as Nevada charges the casinos on the Strip a percentage of their profits, in exchange for maintaining the infrastructure and regulation that allows their business to thrive, Massachusetts took its steady cut from the money the cartels were raking in. When Random Strategies bought 700,000 tickets to trigger the roll-down, the towns of Massachusetts got 40 cents out of each of those tickets, a $560,000 take. States don’t like to gamble, good odds or no. States like to collect taxes. That, in essence, is what the Massachusetts State Lottery was doing. And not unsuccessfully, either. According to the inspector general’s report, the lottery took in $120 million of revenue on Cash WinFall. When you walk away with a nine-figure haul, you probably didn’t get scammed.
So who did get scammed? The obvious answer is “the other players.” It was their cash, after all, that ended up rolling into the cartels’ pockets. But Inspector General Sullivan concludes his report in a tone of voice that suggests no one got scammed at all:
As long as the Lottery announced to the public an impending $2 million jackpot that would likely trigger a roll-down, an ordinary bettor buying a single ticket or any number of tickets was not disadvantaged by high-volume betting. In short, no one’s odds of having a winning ticket were affected by high-volume betting. Small bettors enjoyed the same odds as high-volume bettors. When the jackpot hit the roll-down threshold, Cash WinFall became a good bet for everyone, not just the big-time bettors.
Sullivan is right that the presence of Harvey and the other cartels didn’t affect the chance of another player’s ticket being a winner. But he’s making the same mistake Adam Smith did—the relevant question isn’t just how likely you are to win, but how much, on average, you can expect to win or lose. The cartels’ purchases of hundreds of thousands of tickets substantially increased the number of pieces into which each roll-down prize would be sliced, which makes each winning ticket less valuable. In that sense, the cartels were hurting the average player.
Analogy: if hardly anyone shows up for the church raffle, it’s pretty likely that I’ll win the casserole pot. When a hundred new people show up and buy raffle tickets, my chance of winning the casserole pot goes way down. That might make me unhappy. But is it unfair? What if I find out that those hundred people are actually all working for one mastermind, who really, really wants a casserole pot and has calculated that the cost of a hundred raffle tickets is about 10% less than the retail price? That’s unsporting, somehow—but I can’t really say I’d feel cheated. And of course the crowded raffle is a lot better than the empty raffle at making money for the church, which is, in the end, the point of the enterprise.
Still, even if the high-volume bettors aren’t scammers, there’s something discomfiting about the Cash WinFall story. By virtue of the game’s quirky rules, the state ended up doing the equivalent of licensing James Harvey as the proprietor of a virtual casino, taking money month after month from less sophisticated players. But doesn’t that mean the rules were bad? As William Galvin, the Massachusetts secretary of state, told the Globe: “It’s a private lottery for skilled people. The question is why?”
If you go back to the numbers, a possible answer suggests itself. Remember, the point of switching to WinFall was to increase the lottery’s popularity. And they succeeded—but maybe not as well as they’d planned. What if the buzz around Cash WinFall had gotten so strong that the lottery started selling 3.5 million tickets to ordinary Bay Staters each time roll-down day arrived? Remember, the more people who play, the bigger the state’s 40% cut. As we computed before, if the state sells 3.5 million tickets, it comes out ahead even on the roll-down days. Under those circumstances, high-volume betting isn’t profitable anymore: the loophole closes, the cartels dissolve, and everybody, except maybe the high-volume players themselves, winds up happy.
Selling that many tickets would have been a long shot, but lottery officials in Massachusetts might have thought that if they got lucky they could pull it off. In a way, the state liked to gamble after all.
TWELVE
MISS MORE PLANES!
George Stigler, the 1982 Nobelist in economics, used to say, “If you never miss the plane, you’re spending too much time in airports.” That’s a counterintuitive slogan, especially if you’ve actually missed a flight recently. When I’m stuck in O’Hare, eating a cruddy $12 chicken Caesar wrap, I seldom find myself applauding my economic good sense. But as weird as Stigler’s slogan sounds, an expected value computation shows it’s completely correct—at least for people who fly a lot. To simplify matters, we can just consider three choices:
Option 1: arrive 2 hours before flight, miss flight 2% of the time
Option 2: arrive 1.5 hours before flight, miss flight 5% of the time
Option 3: arrive 1 hour before flight, miss flight 15% of the time
; How much it costs you to miss a flight depends very strongly on context, of course; it’s one thing to miss the shuttle to DC and hop on the next one, quite another to miss the last flight out when you’re trying to get to a family wedding at ten the next morning. In the lottery, both the cost of the ticket and the size of the prize are denominated in dollars. It’s much less clear how to weigh the cost of the time we might waste sitting in the terminal against the cost of missing the flight. Both are annoying, but there’s no universally recognized currency of annoyingness.
Or at least there’s no such currency on paper. But decisions must be made, and economists aspire to tell us how to make them, and so some version of the annnoyingness dollar must be constructed. The standard economic story is that human beings, when they’re acting rationally, make decisions that maximize their utility. Everything in life has utility; good things, like dollars and cake, have positive utility, while bad things, like stubbed toes and missed planes, have negative utility. Some people even like to measure utility in standard units, called utils.* Let’s say an hour of your time at home is worth one util; then arriving two hours before your flight costs you two utils, while arriving one hour before costs you only one. Missing a plane is clearly worse than wasting an hour of your time. If you think it’s worth about six hours of your time, you can think of a missed plane as costing you six utils.
Having translated everything into utils, we can now compare the expected values of the three strategies.
Option 1
−2 + 2% × (−6) = −2.12 utils
Option 2
−1.5 + 5% × (−6) = −1.8 utils
Option 3
−1 + 15% × (−6) = −1.9 utils
Option 2 is the one that costs you the least utility on average, even though it comes with a nontrivial chance of missing your flight. Yes, getting stuck in the airport is painful and unpleasant—but is it so painful and unpleasant that it’s worth spending an extra half hour at the terminal, time after time, in order to reduce the already small chance of missing your plane?
Maybe you say yes. Maybe you hate missing your plane, and missing a plane costs you twenty utils, not six. Then the computation above changes, and the conservative option 1 becomes the preferred choice, with an expected value of
−2 + 2% × (−20) = −2.4 utils.
But that doesn’t mean Stigler is wrong; it just moves the tradeoff to a different place. You could reduce your chance of missing the plane even further by arriving three hours earlier; but doing so, even if it reduced your chance of missing the plane essentially to zero, would come with a guaranteed cost of 3 utils for the flight, making it a worse choice than option 1. If you graph the number of hours you leave yourself at the airport against your expected utility, you get a picture that looks like this:
It’s the Laffer curve again! Showing up fifteen minutes before the plane leaves is going to slam you with a very high probability of missing the plane, with all the negative utility that implies. On the other hand, arriving many hours before also costs you many utils. The optimal course of action falls somewhere in between. Exactly where it falls depends on how you personally feel about the relative merits of missing planes and wasting time. But that optimal strategy always assigns you some positive probability of missing the flight—it might be small, but it’s not zero. If you literally never miss a flight, you may be off to the left of the best strategy. Just as Stigler says, you should save your utils and miss more planes.
Of course, this kind of computation is necessarily subjective; your extra hour in the airport might not cost you as many utils as mine does. (I really hate those airport chicken Caesar wraps.) So you can’t ask the theory to spit out an exact optimal time to arrive at the airport or an optimal number of planes to miss. The output is qualitative, not quantitative. I don’t know what your ideal probability of missing a plane is; I just know it’s not zero.
One warning: in practice, a probability that’s close to zero can be hard to distinguish from a probability that actually is zero. If you’re a global jet-setting economist, accepting a 1% risk of missing a plane might really mean missing a flight every year. For most people, such a low risk might well mean going your whole life without missing a plane—so if 1% is the right level of risk for you, always catching the plane doesn’t mean you’re doing anything wrong. Similarly, one doesn’t take Stigler’s argument to make a good case for “If you’ve never totaled your car, you drive too slow.” What Stigler would say is that if you have no risk at all of totaling your car, you’re driving too slow, which is trivially true: the only way to have no risk is to not drive at all!
Stigler-style argument is a handy tool for all sorts of optimization problems. Take government waste: you don’t go a month without reading about a state worker who gamed the system to get an outsized pension, or a defense contractor who got away with absurdly inflated prices, or a city agency that has long outlived its function but persists at the public expense thanks to inertia and powerful patrons. Typical of the form is an item from the Wall Street Journal’s Washington Wire blog of June 24, 2013:
The Social Security Administration’s inspector general on Monday said the agency improperly paid $31 million in benefits to 1,546 Americans believed to be deceased.
And potentially making matters worse for the agency, the inspector general said the Social Security Administration had death certificate information on each person filed in the government database, suggesting it should have known the Americans had died and halted payments.
Why do we allow this kind of thing to persist? The answer is simple—eliminating waste has a cost, just as getting to the airport early has a cost. Enforcement and vigilance are worthy goals, but eliminating all the waste, just like eliminating even the slightest chance of missing a plane, carries a cost that outweighs the benefit. As blogger (and former mathlete) Nicholas Beaudrot observed, that $31 million represents .004% of the benefits disbursed annually by the SSA. In other words, the agency is already extremely good at knowing who’s alive and who’s no more. Getting even better at that distinction, in order to eliminate those last few mistakes, might be expensive. If we’re going to count utils, we shouldn’t be asking, “Why are we wasting the taxpayer’s money?,” but “What’s the right amount of the taxpayer’s money to be wasting?” To paraphrase Stigler: if your government isn’t wasteful, you’re spending too much time fighting government waste.
ONE MORE THING ABOUT GOD, THEN I PROMISE WE’RE DONE
One of the first people to think clearly about expected value was Blaise Pascal; puzzled by some questions posed to him by the gambler Antoine Gombaud (self-styled the Chevalier de Méré), Pascal spent half of 1654 exchanging letters with Pierre de Fermat, trying to understand which bets, repeated over and over, would tend to be profitable in the long run, and which would lead to ruin. In modern terminology, he wished to understand which kinds of bets had positive expected value and which kinds were negative. The Pascal-Fermat correspondence is generally thought of as marking the beginning of probability theory.
On the evening of November 23, 1654, Pascal, already a pious man, experienced an intense mystical experience, which he documented in words as best he could:
FIRE.
God of Abraham, God of Isaac, God of Jacob
Not of the philosophers and the scholars . . .
I have cut myself off from him, shunned him, denied him, crucified him.
Let me never be cut off from him!
He can only be kept by the ways taught in the Gospel.
Sweet and total renunciation.
Total submission to Jesus Christ and my director.
Everlasting joy in return for one day’s effort on earth.
Pascal sewed this page of notes into the lining of his coat and kept it the
re the rest of his life. After his “night of fire,” Pascal largely withdrew from mathematics, devoting his intellectual effort to religious topics. By 1660, when his old friend Fermat wrote to propose a meeting, he replied:
For, to talk frankly with you about Geometry, is to me the very best intellectual exercise: but at the same time I recognize it to be so useless that I can find little difference between a man who is nothing else but a geometrician and a clever craftsman . . . my studies have taken me so far from this way of thinking, that I can scarcely remember that there is such a thing as geometry.
The Mémorial, Parchment Copy. Photograph © Bibliothèque Nationale de France, Paris.
Pascal died two years later, at thirty-nine, leaving behind a collection of notes and short essays meant for a book defending Christianity. These were later collected as the Pensées (“Thoughts”) which appeared eight years after his death. It’s a remarkable work, aphoristic, endlessly quotable, in many ways despairing, in many ways inscrutable. Much of it comes in short, numbered bursts:
199. Let us imagine a number of men in chains, and all condemned to death, where some are killed each day in the sight of the others, and those who remain see their own fate in that of their fellows, and wait their turn, looking at each other sorrowfully and without hope. It is an image of the condition of men.
209. Art thou less a slave by being loved and favored by thy master? Thou art indeed well off, slave. Thy master favors thee; he will soon beat thee.
How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843) Page 23
