Lotteries

I tend to think about the lotto probably more than I should, especially for someone who very rarely or almost never plays it. I'm often fascinated by things we do as a human mob, as the aggregate. Often enough, through sheer randomness, a mass of people begins to organize and group and do really smart things without even meaning to. There's a radiolab episode about emergence which discusses this very thing. But sometimes, plenty of times, really, we do exactly the opposite. We do really dumb things. The lottery is one of those times (health insurance is another*).

In the lottery, a very large group of people get together and each of them buys a ticket with the same face value in the chances that one of them will win a very large sum. Here's the problem: that very large sum never equals the total amount that was collected for tickets. In the aggregate it's like having a person say "I want to spend a billion dollars so that I can win $350 million." Only a fool would do this. But break this process into the mob and we are millions of fools, who do this over and over and over again, and have practically since the beginning of time.

Let's talk about the value of a ticket for a second. Let's pretend that nobody else is taking a cut. There is no lottery commission taking a fee to run the game, and there are no taxes to be paid on winnings. Let's also pretend that every ticket costs exactly one dollar and has exactly the same chance of winning. In this case, in this perfect game, the actual expected value of each ticket really is $1. Let's say it's the mega millions game, where your chances of winning the jackpot are 1 in 258,890,850. For simplicity's sake, I'm going to forget about all of the minor prizes and how they affect overall odds as well as multiple winners and how they affect the jackpot. That $1 has a 0.00000038626316843566% chance of winning the jackpot. If the jackpot were say, only $100 million, then you could say that the value of the ticket was 0.0000000038626316843566 * $100,000,000 or roughly 39 cents. In order for the expected value of the ticket to rise to the full $1, the jackpot has to rise to $258,890,850.

The way that lotteries actually run is that the ticket is never worth the full $1. Not even close. In fact, it's usually closer to the 39 cents, meaning that every time we buy a ticket, we are instantly throwing away nearly two thirds of our money (or, in my experience, all of it). So why do we do this? It comes down to utility.

If you are completely new to economics utility is essentially the value that we perceive an item to have based on its usefulness to us. The more of the item we have, typically the less utility each one has. Imagine the value of a banana if you were starving to death. It has a lot of utility.

Now imagine that you are a billionaire. I have for sale two lottery tickets. Each one costs $10,000. One ticket, the billion dollar lottery, has a 1% chance of winning $1,000,000,000. The other ticket, the one hundred thousand dollar ticket, has a 99% chance of winning $100,000. Which one would you buy? I think most billionaires who could afford the $10,000, and who might scoff at $100,000 would look at the expected value of each ticket. The expected value of the billion dollar ticket is one percent of one billion, or $10 million. The expected value of the hundred thousand dollar ticket, likewise, is 99% of $100,000, or $99,000. In either case, the ticket is a smart bet, but the billion dollar ticket is way smarter. But try to remember you have a 99% chance of losing ten grand.

Now imagine that you're you. You don't have $10,000 to lose. Which one do you buy? Do you even skip out altogether and not even play, choosing not to risk it on the 1% chance that you lose everything? This is the difference that utility makes.

People play the lottery because it gives them something that they value. It gives them hope, a thrill that they just might overcome the staggering odds and win huge. It just blows my mind that this utility is worth 2/3 of their money. It blows my mind that so many people do it.

I tend to believe that around 90% of the time that money changes hands between two parties, one of those parties is getting ripped off. The only time they aren't is when the value equals the price, which is rare. If you're not sure which party is getting ripped off, it's almost certainly you. Why we do it en masse is beyond me.

*I say this about health insurance because on the aggregate level the insurance industry really just exists to pay the bill. Imagine if the total number of patients were collectively pooled into Group A. Next, the insurance industry can be pooled into Group B. Finally, the medical community is thrown together in Group C. What we do in this country is akin to saying "Group A owes Group C $10. Let's pay Group B $15 to pay Group C." It boggles my mind.