Since I’m working on non-ergodic behaviour in economic models, I was intrigued by recent claims from Ole Peters.[1] In a series of articles, blogs and notably in a TED talk (here), Ole has made some rather strong assertions about the way economists model choice under uncertainty. According to Peters, economists do not understand the concept of ergodicity. As a consequence, we have apparently made some rather bad blunders.
What, you may ask, is ergodicity and why does it matter? Imagine you are repeatedly confronted with an uncertain world. A good example is the one that Ole gives us. You start with $100 and a casino offers you a gamble in which the house flips a fair coin. If it comes up heads you win $50. If it comes up tails, you lose $40. The first question you might reasonably ask is; should you trust the casino? Is the coin really fair? Does it really have a 50% chance of heads and a 50% chance of tails or is the house shading the odds? To answer that question, you consult a friend who has a Ph.D. in statistics, and she advises you to observe someone else playing the game for a while.
If each flip takes 30 seconds then after watching for roughly eight and a half hours you will have acquired a list of 1,000 observations and, if the coin is fair, roughly 500 of the times you should have seen a head and the other 500 observations should be tails. I say roughly because the chance of 500 heads in 1,000 flips is itself a random variable so you might for example, see 503 heads and 497 tails. But you are very unlikely to see 200 heads and 800 tails unless the casino has been cheating. The exact statement is that if the process is ergodic then the proportion of heads in n tosses will converge to p where p=0.5 for a fair coin.
Notice I slipped in the word ergodic to this definition. That’s a very important idea in problems like the one I just described, where we are trying to estimate an unknown quantity. In this case, we are estimating the probability, p, that a single toss of a coin will come up heads. If the coin is fair, p is equal to 0.5. If the casino is cheating by weighting the coin, p might be different from 0.5. That’s what we hope to find out by observing repeated flips of the same coin.
So far so good. But how do we know the casino doesn’t cheat occasionally. Suppose that the house has a whole boatload of different coins. Some of them are fair and some of them are not. Now your statistician friend advises you that the experiment she proposed of counting the frequency of heads in a series of flips will tell you nothing about the next flip. Averaging a sequence of flips only works if each flip has the same value of p. For the estimation of p using sample averages to make sense, the process must be ergodic.
Now we know a little bit about ergodicity, let’s look at Ole’s experiment. You watch Ole’s TED talk video and explain it to your friend. The one with the Ph.D. She listens carefully but seems a bit puzzled. The first thing she points out is that Ole is certainly not assuming non-ergodicity of the coin flip since he explicitly assumes that the coin that is flipped is fair. Peters is not asking about the distribution of wins or losses in repetitions of a game: no, he is instead asking about the distribution of your wealth if you play the game n times. Let’s call this random variable W(n). The assumption that you start with $100 means that W(0)=100. What Peters studies are sequences {W(i)}_(i=1)^N where you play this game N times and you reinvest all of your wealth at every stage.
You explain this to your friend and she now understands a little better. The random variable W(i) is not ergodic. In fact for i≠j, W(i) and W(j) do not even have the same probability distribution. If you play the game once you will have $150 with probability 0.5, or $60 with probability 0.5. There are only two possible outcomes. If, on the other hand, you play the game twice by reinvesting your wealth after stage 1 you will have $36 with probability 0.25, $90 with probability 0.5 and $225 with probability 0.25. If you win twice, you win a lot, but the most likely outcome (statisticians call this the mode) is that you will be $10 poorer if you play the game twice.
