## Ergodicity – probabilistic thinking gone awry

5 Apr, 2013 at 10:58 | Posted in Economics, Statistics & Econometrics | 1 Comment

In 1956 … John Larry Kelly published a … profound paper looking at how a gambler — facing a series of risky bets — could optimize his winnings in the long run and avoid ruin along the way. Kelly offered a concise answer: the gambler should at each stage wager a specific fraction of his or her current wealth, the fraction determined by the odds and potential winnings …

Consider how much you might be willing to pay to play a lottery based on a coin flip. If the first flip is heads, you win \$1. If tails, you flip again. Heads on the second toss and you win \$2, otherwise you flip again, with heads on the third toss giving \$4 and so on. The lottery pays out 2n dollars if the first head comes up on the nth roll. An easy calculation shows that the expected payout of the lottery is actually infinite, as the size of the payout grows just as fast as its likelihood decreases …

No sensible person would pay much to play this game, despite the infinite expected pay-off. Real people do not find this lottery appealing and generally offer less than \$10 or so to play …

After all, the familiar calculation based on expected value actually entails supposing that the gamble plays out simultaneously in several parallel worlds, one for each possible outcome. The result of the calculation is influenced by every one of these no matter how unlikely. This is the essence of probability, of course, yet it clearly introduces an artificial element into the situation. An alternative way to treat the problem — Kelly’s way — is instead to calculate the expected pay-off from a string of such wagers actually playing out in real time, as a real person would experience if trying to learn how to play the game by trial and error …

There is one other aspect … of Kelly’s way of thinking that holds special interest for physicists. We’re familiar with the notion of ergodicity — the property of the dynamics of a system that makes a time average equal to an ensemble average. This is a neat trick, hugely useful, as ensemble averages are typically much easier to calculate. The assumption of ergodicity lies at the basis of statistical mechanics.

And it is the failure of this assumption that distinguishes time and ensemble averages in gambling problems also. Economists have long relied on the equality of ensemble and time averages, assuming that the probabilities they deal with often have this feature. But the multiplicative growth process involved in any situation of repeated gambles is necessarily not ergodic. Go broke at one time step, and you are permanently out of the game, stuck at wealth = 0, a situation never captured by the ensemble average, which assumes continued exploration of the space of outcomes.

Mark Buchanan

[Yours truly has argued that it is untenable to assume that ergodicity is essential to economics here here here here and here.]

## 1 Comment

1. Since I am working with and developing non-ergodic chemistry, I find the notion of the ergodic assumption in economics laughable. I guess the argument for it must have been: The physicists do it.

Considering how easy it is to make a chemical system that deviates from ergodic behavior, it must be hard to find ergodic behavior in economics at all.

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