Economists normally scorn the lottery as a “tax on people who can’t do math.” But sometimes — very, very rarely — a prize is so big that even with an extremely low probability, it can be “rational” to play. That is, the expected value exceeds the cost.
Last week, when the Powerball was a mere $800 million, economist Alex Tabarrok asked, “Is the prize high enough to make it worth playing for an economist? In other words, is the prize high enough to be a net gain in expected value terms?”
The odds of winning are 1 in 292.2 million. So the expected value of a ticket is $800*1/292.2=$2.73. A ticket only costs $2 so that’s a positive expected value purchase!
We do have to make a few adjustments, however.
The $800 million is paid out over 30 years while the $2 is paid out today. The instant payout is about $496 million so that makes the expected value $496*1/292.2=$1.70.
We also have to adjust for the possibility that more than one person wins the prize. If you play variants of your birthday or “lucky” numbers that’s a strong possibility. If you let the computer choose your chances are better, but with so many people playing it wouldn’t be surprising if two people had the same number — I give it at least 25%.
So that knocks your winnings down to $372 million in expectation.
Finally the government will take at least 40% of your winnings, leaving you with $223 million in expectation. At a net $223 million the expected value of a $2 ticket is about 75 cents. Thus, a Powerball ticket doesn’t have positive net expected value, but the net price of $1.25 is significantly less than the sticker price of $2.
Given that the jackpot has nearly doubled since then, I wondered how much that changes the calculations.
The odds of winning are still terrible: 1 in 292.2 million. But the jackpot is now expected to be $1.5 billion.
$1,500*(1/292.2) = $5.13
So far, so good! But the lump-sum payout is only $930 million.
$930*(1/292.2) = $3.18
Tabarrok's 25% probability of a split pot is probably about right, but his calculation (multiplying the jackpot by .75) is a mistake. If you win, you'd have a 75% chance of keeping all of it, and a 25% chance of keeping half. So:
$930*.75 + $465*.25 = $813.75
$813.75*(1/292.2) = $2.79
At this point, expected return for a $2 ticket is still positive. But we have to account for taxes. Virtually all of the winnings will hit the top federal tax rate of 39.6% — let's assume you live in a state that doesn't tax lottery winnings, like California or Pennsylvania — so you'll owe about $322 million to Uncle Sam.
$813.75*.604 = $491.505
$491.505*(1/292.2) = $1.68
Currently, the net expected value of a $2 Powerball ticket is about -$0.32. So close!
It doesn’t yet make sense as a purely financial decision, but as a consumption good, if you think you’ll get more than 32 cents worth of enjoyment out of it, the math says go for it.
But because hope springs eternal, let’s see if we can find a way to close that gap.
Some simple math shows that to make a ticket’s expected value equal $2, the net expected payout has to be at least $585 million, so we need to find a way to get you another $93 million — about 19 percent more than the current expected after-tax payout.
One way would be to reduce your tax liability by giving away a bunch of money to charity. A back of the envelope calculation shows you'd need to donate about $235 million of your expected payout to reduce your tax bill by $93 million.
If you’re a charitable type, and if (for instance) you prefer what Doctors Without Borders would do with your money to what the government would do with it, you might not mind having $142 million less for yourself in order to do $235 million worth of good (and not do $93 million worth of bad).
This way, your expected winnings (discounted for the possibility of a split jackpot) are $814 million, with $235 million going to charity, $229 million to the government, and $350 million left for you. Not too shabby.
But the most rational thing to do would be to not take the money right away. Take the $1.5 billion in installments over 30 years. Josh Barro explains:
There are big tax advantages to the annuity. The main one is that taking the annuity is basically like letting the government hold onto part of your prize for a while and invest it for you — and the government does not pay tax on investment income.
Of course, once you get the annuity checks, you’ll have to pay income tax on them. But if you take the lump-sum cash prize, you’ll pay tax twice: on the prize when you win it, and on the income you get by investing it.
This adds up. If you invested all your prize money in the same way Powerball does (essentially by putting it in government bonds), you’d end up with 20 percent more cash in 2045 if you took the annuity option rather than the cash option, thanks to the tax savings.
There you go: we needed another 19 percent to make the value of the Powerball ticket positive, and we got 20 percent (over three decades).
Now, depending on your risk tolerance and time preferences, this might not be for you. You might be getting on in years and don’t have thirty to waste. You might be a maverick investor who wants to let it all ride on the stock market for bigger potential returns than the ultra-safe bonds Powerball will invest it in.
But it’s interesting that, for once, the math says the expected financial value of a lottery ticket isn’t necessarily negative for everyone. And if we go another drawing without a winner, things will probably look even better.
Of course, now that I’ve shown how it could be economically rational to play the Powerball, I’m obliged to point out that it’s only as “rational” as worrying about anything with a 1 in 292.2 million probability.