Risk, utility, and money

Via MKB, a terrific little essay about risk, utility, and money from Mark Dominus, whose blog has a really cool name.

Consider the following game: You bet one dollar on the throw of a die. If the die comes up 6, you get your dollar back plus 25 more dollars. Otherwise, you lose your dollar. You can play as much as you want to. This is a great moneymaking proposition, because your expected winnings are four dollars on each game. Play a hundred times, you can expect to be about four hundred dollars ahead. Even if you're only allowed to play once, you would probably choose to play this game.

I pulled some sleight-of-hand in the previous paragraph. I said the game was a good deal "because" the expected winnings were positive. But that's not sufficient. If it were, the following game would also be a good deal: You bet one million dollars on the throw of a die. If the die comes up 6, you get your million back plus 25 million more. Otherwise, you lose your million.

For some people, the second game is a good deal. For most people, including me, it's obviously a very bad idea. To get a million dollars, I'd at least have to mortgage everything I owned. Then I'd be under a crushing debt for the rest of my life, with 83% likelihood. But the expected return of the two games is the same; this shows that a good expected return is not a sufficient condition for a good investment.

The difference, of course, is that the second game is much riskier than the first.

He goes on to describe how this plugs into the idea of utility, in a really crafty way.