## Wednesday, April 4, 2012

### I still wouldn't play the lottery

Having a lot of math friends, I heard a few of them bought lottery tickets last week, when the expected value of the ticket was greater than the price.  Shit math people say, amirite?

The positive expected value is due to the record-breaking size of the jackpot.  It was \$640 million if chosen to be received over 26 years, or \$462 million if received all at once.  It's less than that when you subtract tax, but it's still enough to overcome the \$1 ticket price and 1/175M probability of winning the jackpot.  (There are smaller prizes, but they have negligible expected value.)

My understanding is that a small percentage of ticket sales goes to the jackpot.  So the record-breaking jackpot is the result of a lot of people spending money on tickets when there was a negative expected outcome, and winning even less money than is statistically expected.

The positive expected value of lottery tickets is greatly mitigated by the fact that if multiple people win the jackpot, they split it amongst themselves.  I am not sure how many people bought tickets, but according to CBS news, ticket sales that week were projected to be \$1.46 billion.  With those numbers, you'd expect a lot of simultaneous winners.  There were only three.

But even ignoring the probability of splitting the jackpot, I still don't think it is a good idea to play the lottery.  Why?  \$462 million is not 462 million times more useful than \$1.

Put another way, imagine that 462 million people all gave a dollar to one person.  That person is rich, rich, rich, on account of having received money from a lot of people.  Wait, why are we all paying this rich person? If we're going to give away money, isn't it better to give it to the poor, and equalize wealth?

Put another way, if I start increasing the amount of money I have, starting from zero, what will I spend that money on?  At first I'd buy food.  Then I'd rent an apartment.  Eventually I'd have enough to get a car.  Much later, I could afford a house.  Later I could afford to retire early, or hire servants (or whatever rich people do with their money).  The point is that the money I get early on is used to buy essential goods, while later money is used to buy luxuries.  I could choose to buy luxuries even when I'm poor, but I'd much rather buy the essentials, because that's more for my money's worth.  The result is that a dollar is more useful to a poor person than a rich person.

One way to capture this intuitive idea is with a utility function.  (Insert standard disclaimer about not having any economics education.)  A utility function is just a way to express how preferable something is.  So if we're considering money, U(\$x) tells you the utility of having x dollars. The greater the utility, the more preferable it is.

If we assign a numerical value to utility, we can speak of the expected utility of buying a lottery ticket.  Suppose we start with \$10000.  Then the expected utility of buying a lottery ticket is approximately as follows (ignoring tax):
U(\$9999) * (1 - 1/175M) + U(\$462M) * (1/175M)
And of course, this needs to be compared with the utility of not buying a lottery ticket, which is U(\$10000).

Common sense says that U(\$10000) is greater than U(\$9999).  But common sense also says that U(\$x) increases more slowly as x becomes greater (ie U is concave), because the more money you have, the less useful each dollar is.  One possible function which fits these criteria is U(\$x) = Log(x).  This is not the only possible utility function, though in some sense it is the "correct" one, for reasons I don't at all understand.  Using this utility function, here's the expected utility:
Expected utility of not buying a ticket: U(\$10000) = 4
Expected utility of buying a ticket: U(\$9999) * (1 - 1/175M) + U(\$462M) * (1/175M) = 3.9999566
Expected utility of dropping a dollar in a hole: U(\$9999) = 3.9999566
So yeah...

Buying more tickets is actually worse, because the difference between U(\$9999) and U(\$9998) is greater than the difference between U(\$10000) and U(\$9999).  I estimate you'd need about \$100M before it becomes worthwhile to buy a single ticket.

Of course, as long as we're talking about utility rather than hard money, you could argue that there is utility in being entertained by the lottery.

I leave analysis of Pascal's Wager as an exercise to the reader.

(via Uncertain Principles)