Excerpt
WILLIAM TRACY For this part of the afternoon, we’re going to shift gears a little bit and move from the ontological perspective that really defined the first set of talks to a more epistemological perspective, getting down into some of the details. Ole Peters began this work or has been involved in this work with Murray Gell-Mann for some time. His talk is titled “Ergodicity Economics.” Ole is both a member of the London Mathematical Laboratory and the Santa Fe Institute. With that, I’ll turn it over to you, Ole.
OLE PETERS Thank you very much, Will. So, I’ll be talking about ergodicity economics. I will start with a game that many of you will know, but it’s still the best way to introduce this topic that I know of.
The game is the following: It’s a simple multiplicative game. You toss a coin. If it shows heads, you win fifty percent of your wealth. If it shows tails, you lose forty percent of your wealth. You play this game once a minute, so you’re tossing a coin once a minute. As you’re playing, your wealth follows some sort of a random trajectory. I could ask this room who would like to play this game, and I’ll actually do it. Who wants to play this game?
AUDIENCE MEMBER The people in the other room.
[Laughter]
O. PETERS That’s good. This is a sign of success. I’ve asked this question many times and by now almost no one wants to play anymore. There are good reasons why you might want to play the game, but I’m glad you don’t.
Okay. Let’s look at one sequence over sixty minutes, so we were playing this game for an hour. You get some kind of a random trajectory, as you would expect. What do you do with a random trajectory? Well, it doesn’t really tell you much about the system, for example, whether this is really a game worth playing or not. It’s just some noise.
In 1654, Fermat and Pascal came up with a recipe for treating situations like that. And they said, “Oh, just imagine everything that could possibly happen.” So, all the possible trajectories over the sixty minutes and an average over them. And if that average looks good to you—in this case, if it goes up, if you’re winning—then that’s a good game. And we’ll call this average the expectation value.
So how do you construct this thing? You construct it by just running many, many, many of these trajectories—infinitely many. These are also called an ensemble or the ensemble of possible trajectories. So you’d run ten, you’d run twenty, you get more and more colorful lines, and then you start averaging. So at each moment of time you take the average over all of these colorful lines and you end up with some sort of slightly less noisy trajectory. If you average over more trajectories, the noise goes away and you’ll find what you’d probably expected to find, which is exponential growth in the expected wealth coming out of this game.
The conclusion from this perspective is that the game is worth playing, on average, but not everyone here wanted to play it. The question is, why is that? Why didn’t you want to play this game? The classic solution to this question is called expected utility theory. It starts from that observation that not everyone is optimizing the expected wealth or changes in expected wealth. And it introduces a new concept, which is the usefulness of wealth: actually, your dollars don’t matter. It’s what they’re worth to you that matters. And it’s the expectation value of that that you will optimize. So, in technical terms, you optimize the expected changes in the utility function.