Why you should play the lottery

If you know anything at all about probability theory, you know that playing the lottery is a bad idea -- your chances of winning are very low. Or more precisely, if the lottery ticket costs $10, then your expected winnings are less than $10. Even so, I'm going to convince you that it is mathematically optimal for you to play the lottery.

Imagine that all you have is $1000 in cash. That's it. This is your whole livelihood. Would you bet it all on a coin flip? If you win, you get $2000; if you lose, you lose everything! Of course you wouldn't. You would rather spend that money on food for the next few days, while you are looking for a job.

Imagine instead that you are a millionaire, and you enjoy playing poker. Would you agree to play poker with your friends for $1000? Let's say they are of a similar skill level as you are, and they are fun people to hang around. Will you play? Sure. Maybe. Why not? It's only $1000. It's only 0.1% of your net worth. Your expected winnings are about $0 (breaking even), and the fun you will have playing is definitely worth more than that.

The point I am trying to make is that the first $1000 in your bank account is worth much more to you than your most recently added $1000. The value of money, to you, decreases the more money you have. Mathematicians call this "utility". If you buy something for $20, then you want that thing more than you want the $20. That thing has higher utility for you. The person selling you the thing wants the $20 more than he wants the thing. The couch has lower utility for him. Both sides win (increase their total utility).

Coming back to the money, it seems that the more money you have, the less is the utility of $1000 to you. In fact, things are a bit more non-linear. Imagine if you had $X in your bank account, for various values of X from $0 up to $1 million. What would your quality of life be? With $0, it would suck. Let's call that zero quality of life. With $1 million, life would be pretty sweet. Let's call that 100 units of quality. There is some non-linear, increasing function that maps money in the bank to quality of life. If you had nothing, and somebody gave you $X, the blue curve shows what your quality of life would become.

There are some bumps around $10,000 (the price of a car) and $100,000 (the price of a down payment on a house). This function is not concave.

Now imagine you already have $10,000 in your bank account. How much would your life improve if somebody gave you $10? Very little. $100? Still very little. $1,000? A bit. $1 million? It would go all the way up to 100 units. The red curve shows your new quality of life after receiving $X once you already have $10,000.

So what does this have to do with playing the lottery? Let's say a lottery ticket costs $10, and it gives you a small chance to win $1 million. That chance is almost certainly smaller than 1/100,000; otherwise, the lottery organizers have made a terrible mistake. Let's say the chance is half that -- 1/200,000. Should you play the lottery?

In terms of maximizing your wealth, no, you shouldn't. Your expected winnings are -$5 (negative $5). But what about quality of life? That's a lot more important than money! It depends on how much money you have. Let's say it is $10,000. What is your expected gain in quality of life? If you lose, you lose $10, which is worth almost nothing. If you win, you can buy a house. Your quality of life jumps all the way up to above 100. It sounds like you should play! In fact, it makes sense to play whenever you find yourself in one of the convex parts of the utility function that is curved enough to compensate for the negative expected value of the lottery.

In simple terms, if you don't mind risking $10 to win an amount of money that is larger than you would be able to earn at work, then sometimes it makes financial sense to play the lottery... :)