Powerball Right Now and Rational Gambling

January 10, 2016 • John Vandivier

Powerball is so hot right now.

Did you know that gambling is sometimes rational? There are at least two ways it can be rational:

Expected value exceeds expected cost
Utility gains from gambling cause willingness to pay to match price

Joan Ginther had a Ph.D. from Stanford and she gambled like an addict. Except she kept winning. Some theorize that she went from state to state looking for different rational lottery gambles.

Now the Powerball is <a href="http://abcnews.go.com/US/powerball-winners-jackpot-grows-estimated-13-billion/story?id=36184696">at an all time high with a $1.3B payout, but how much might you actually take home on winning?

You can take a one-time payment or payment over time. Long story short, because of time preference and interest rates it's rational to take the immediate payment.
The immediate payment, or cash value, of winning the jackpot is $806M.
We also need to correct for the possibility of multiple winners having to split the jackpot. A possibly ratchet Wikipedia listing of recent jackpot winners shows that this happened with about 3/13 = 23% of recent Powerball payouts, or with 12/26 = 46% of jackpot lottery payouts more broadly. Let's call this loss 30%.
Now there are taxes. Dr. Tabarrok did a recent rough CBA of this sort and he pegged taxes at 40%, so I'll use that. However, he did forget two things:
1. Lottery losses can be written of against lottery winnings for tax purposes
2. There are smaller winnings which make a big difference
Let's say you aren't going to write off any gambling costs then your cash prize would be 806M*.7*.4 = $225.68M.
The chance of winning is ~1/292.2M and the cost of a ticket is $2. So on the grand prize alone this deal is solidly irrational, even if you could write off the full tax cost on winnings.

But all hope is not lost: The small winnings make a big difference in the overall expected value calculation. Plus, gambling can be fun in and of itself. Finally, there is an unappreciated value to risk in economics which I will sum up with a story at the end of this article.

Group buying of lottery tickets makes a huge difference. It's possible to buy every possible combination of lottery numbers with a cost of 2*(292.2M) = $584.4M.

You probably can't spare that change, but a collective buying agreement could crowdfund it. Then the crowdfunded entity could write of the losing tickets for tax purposes. Unfortunately it's hard to pull a group like this together in a trustworthy way.

Still, it's worth asking around your friends and family. If they are going to buy anyway you could at least coordinate your number selection to reduce duplication and thus increase possible outcome coverage.

Small Winnings and PowerPlay

You can buy a Powerball ticket for $2 or a Powerball PowerPlay ticket for $3. PowerPlay multiplies your small-dollar winnings by a random multiplier of 2, 3, 4, 5, or 10.

I think we've pretty well established that a single buyer isn't expected to win the grand prize, but they are expected to win a small-dollar prize if they purchase as little as 25 tickets.

However, the random selection is weighted so that the expected multiplier is 2.5 this week.

The 10X multiplier is only available on jackpots below $150M, so it's not available for this week's prize.

Basically, the effect of small winnings is to reduce the cost of a $2 ticket by $.25. This can be seen by adding up small dollar prizes <a href="http://www.businessinsider.com/powerball-lottery-expected-value-jan-9-draw-2016-1">on the breakeven calculation here.

That's if you buy a regular Powerball ticket. With a PowerPlay ticket, the effect of small winnings is to reduce the cost of a $3 ticket by $.25*2.5 = $62.5.

Since the expected value of an additional Powerball ticket is larger than the difference between the insurance effect of small winnings, it is probably not a good idea to do PowerPlay.

The Unappreciated Value of Risk Taking in Economics

There was a man who went to church every Sunday. After church he would pray, \"God, please let me win the lottery this week.\"

He was a very good Christian but he never seemed to win. It went on like this for years.

Finally, one day, he seemed to hear a small voice. Perhaps it was the Holy Spirit, or maybe just his own mind, or it may have been the preacher whispering from the back of the room.

"Jim, you can't win if you don't buy a lottery ticket."

In Conclusion

Participating in Powerball right now is not price-justified, but I am going to do it anyway. Maybe that's utility justification, maybe I'm a bit irrational, or both.

But I'm not going to spend, like, that much.

Also, I love how snarky the Powerball FAQ is.