Playing the lottery is not a Sharpe choice

The Powerball Jackpot is approaching $500 million dollars tonight.  The chances of winning are 1 in 175 million. Which gives the expected value of $2.86 per ticket, and at a cost of $2 per ticket, that’s a great value!

Of course, the 500 million is an annuity, the “lump sum” or the real value is 337, which gives a value of $1.92 per ticket. This is increased by all the other prizes, as one can win $4, $7, $100, $10,000 and $1,000,000 for various combinations. Doing the math (which I won’t spell out in this post) gives you about $0.36 per ticket, so for tonight’s jackpot we can estimate the expected value of a single ticket at $2.28, or roughly a 10% average return.

If, as we’ve calculated, the expected value of the ticket is $2.28, is buying a ticket the economically intelligent thing to do? What, for that matter, does expected value actually mean?

One way to think of it is to imagine buying all possible tickets. There are 175 million possible number combinations (thus the 1 in 175 million chances), if you bought every possible ticket, you would be guaranteed a winner, (and multiples of all the lesser prizes) which means that buying all the tickets will make you money, about $400 million to be approximate. If buying all the tickets will get you x amount of money, then buying 1 ticket will get you x/175 million in money; doing this math gets us to the $2.28 figure.

But there’s something else to consider, that is the possibility of another winner. If two people win the prize, then the prize amount drops in half. Lets say, for the sake of simplicity, that there is a 40% chance of having two winners (which, if anything seems too low, it seems like when lotteries get this high). That reduces the expected value of the ticket from 2.28 to 1.90, which makes it fall below our magical $2 level.

But what is this telling us? Well, it tells us that if we buy all the tickets, now we will expect to lose money. But why does this have any relevance for anyone who buys a single ticket? Lets imagine a scenario. First, lets assume that, for whatever reason, the jackpot is rarely split in two so you barely figure that into your calculations (maybe a 1% chance of two winners instead of a 40% chance). Now, imagine that you buy a Powerball ticket, and to your joy while watching the drawing you see each of your 6 numbers match those being drawn. As you’ve just one the jackpot, you are ecstatic. The next day, as you get the morning newspaper, you see the headline: “Record Powerball Jackpot has 2 winners” Instead of winning the full jackpot, you only win half. Now, do you regret buying the ticket? Of course not, ex ante, the math worked in your favor, ex post, the math worked in your favor.  So why, if when event x happens it does not cause you to change any rewards scheme, should the possibility of event x enter into your calculations?   No realistic number of winners will ever push the jackpot down to a level where you wouldn’t be happy winning it.

Lets change gears a moment and talk about taxes. Assume that taxes takes away half the winnings.  Business Insider did the math here: basically the taxes push the expected value to $1.32 (for some insane reason they are reporting the expected return on buying a ticket, instead of the expected value of a ticket.) They figure that you should apply the after tax return to the cost of buying a ticket to determine the expected value.

But what does our analysis actually mean? If we return to the scenario of buying every ticket, all in all we’d spend about $350 million, get the jackpot and all the other prizes, for a value of about $400 million. Now, if you assume that you would pay half that $400 million in taxes, you’d be left with $200 million, or a loss of $150 million, so you figure don’t buy the tickets. But in reality you wouldn’t be charged $200 million in taxes, because of the ability to write off gambling losses against gambling wins. So instead of paying $200 million, you would only pay $25 million in taxes (on 50 million net winnings), which pushes the net value to positive territory.  So which value is correct? One the one hand, you’re not actually buying all the tickets, and the value of a single ticket as a tax write off is negligible. On the other hand, it’s not like paying the taxes on the winnings is going to make a big difference, you’ll still be horribly rich. So what is the answer?

The answer is that there is no answer. Using something like expected value is a tool which helps us understand our world, but it is not the world. For any Powerball lottery, there is always a very small chance that you will become very rich. Exactly how rich is hardly an interesting question. Imagine yourself with $100 million. Now imagine yourself with $200 million. There isn’t much of a difference. Or to put it differently, think of all the things that you could do with $200 million that you couldn’t do with $100 million: not a gigantic difference. So if the human difference between 100 and 200 million dollars isn’t big, what is the point in taking the jackpot, multiplying it by the probability of winning, and comparing that number to the ticket price? If you win, you’ll be happy. If you lose, you wasted money. In other games with lower variance in returns (such as roulette) can help you understand whether you will win or lose over the long run, in the case of roulette the long run is over an hour or a few hours. But in Powerball, the long run would last millions of years (potentially billions if you only play when the Powerball odds are in your favor), so how is that useful?

In finance, there is a concept called the Sharpe ratio. It measures the marginal change in return to marginal change in risk. Basically, it’s a way of adjusting returns for risk. If I did my math right, in our situation we an expected return of about .14 (winning 2.28 on a 2 dollar ticket is a 14% return, we can assume the risk free rate of return is zero overnight (close enough anyway), and a standard deviation on the return of 162 million. Which gives us a Sharpe ratio of 0.00001123293428; well below the advised value of 1 for a decent risk adjusted investment. It’s not perfect resolution of the problem, but it does illuminate the basic problem with Powerball, if it’s worth $2 for a 1 in a 175 million chance to win $300 million, it’s probably still worth $2 for a 1 in a 175 million chance to win $40 million, big changes in very low probability events just aren’t significant.

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