Expected Value: Utility vs. Dollars
- $540M the amount the Mega Millions jackpot is going for this weekend, which is a record by the way
- $176M the amount it would cost to theoretically buy every single ticket combination possible in the lottery
- $117M the amount, after federal (not state) taxes, you’d gain if you bought…
Still, it’s one of few lotteries that actually offers a really good Expected Value (EV) on a risk-adjusted basis. You should buy a few tickets if you believe in gambling at all.
Fun aside - this expected value thing used to create teams of slot machine players. With the old mechanical tumbler-based slot machines with displayed group jackpots, it was possible to mathematically generate a positive EV. At that point, a team would occupy all the slot machines and play until the jackpot ran out. This is why modern machine odds are computer-determined - to keep flocks from descending when it’s profitable (on an infinite horizon) to play until you hit the big one.
When gambling for odds that long and a payout that high, it is important to consider the potential payout in terms of how useful the money is rather than how much money is expected. For most of us, the utility of the first million dollars is a lot higher than the utility of the second million.
Here’s an example:
Suppose I offer you a choice between $10 million or a fifty-fifty shot at $100 million.1 With ten million, you’re comfortably set for life. And it’s a sure bet. With the $30 million, there’s a fifty-fifty chance you get nothing.
Your expected payout for the fifty-fifty chance is $15 million. Nevertheless, most people will go with the sure bet at the $10 million. This is not because they are risk averse. The same people would happily take a 50-50 shot at $30 over $10. It is because at very large (and very small) numbers the utility of a dollar goes down.
With $10 million, you’re comfortably set for life. The calculation most of us make would be “comfortably set for life” vs. “comfortably set for life with some really expensive toys.” The second one is not worth twice as much as the first.
Use your five bucks to buy a nice sandwich instead of that lottery ticket2. You’ll enjoy the sandwich. You would probably enjoy the money a million times more than the sandwich. Maybe even ten million times more. But not a hundred million times more.
Or you enjoy the feeling that there’s a small chance that you might have a $500 million dollar ticket in your pocket, maybe that’s worth more than the sandwich. Maybe buying a ticket will facilitate a dream about what you could do with that kind of money. Perhaps that’s worth a few bucks.
But unless you already have a few hundred million to your name, it’s not a good decision on purely economic grounds.
Some restrictions apply. Specifically, your 50-50 shot is not transferable, so you can’t just go sell it to a bank for $14 million. Also, you have to spend it all yourself. This should prevent some altruistic jerk from screwing up my hypo by pointing our that you should give away the money to the Red Cross or some organization that would benefit in a reasonably linear manner from the bigger payout. ↩
I don’t even know how much that ticket costs. Is it a $1 ticket? Whatever. ↩