Time to revisit how we calculate expectations?

The below presentation by Dr Ole Peters opened my mind. If there was one thing I believed was a reasonable implicit assumption of economics, it was determining the expectation value upon which agents base their decisions as the “ensemble mean” of a large number of draws from a distribution. Surely there is nothing about this simple method that could undermine the main conclusions about rational expectations, whether humans act that way or not? Surely this is a logical benchmark, regardless of whether actual human behaviour deviates from it.

But now I’m not so sure. Below is a video of Dr Peters making the case that non-ergodicity (according to the physics interpretation of the word) of many economic processes means that taking the ensemble mean as an expectation for an individual is probably not a good, or rational, expectation upon which to base your decisions.

I encourage you to watch it all.


Let me first be very clear about the terminology he is using. He uses the term ergodic to describe a process where the average across the time dimension is the same as the average across another dimension.

Rolling a dice is a good example. The expected distribution of outcomes from rolling a single dice in a 10,000 roll sequence is the same as the expected distribution of rolling 10,000 dice once each. That process is ergodic [1].

But many processes are not like this. You cannot just keep playing over time and expect to converge to the mean.

Peter’s example is this. You start with a $100 balance. You flip a coin. Heads means you win 50% of your current balance. Tails means you lose 40%. Then repeat.

Taking the ensemble mean entails reasoning by way of imagining a large number coin flips at each time period and taking the mean of these fictitious flips. That means the expectation value based on the ensemble mean of the first coin toss is (0.5x$50 + 0.5*$-40) = $5, or a 5% gain. Using this reasoning, the expectation for the second sequential coin toss is (0.5*52.5 + 0.5 * $-42) = $5.25 (another 5% gain).

The ensemble expectation is that this process will generate a 5% compound growth rate over time.

But if I start this process and keep playing long enough over time, I will never converge to that 5% expectation. The process is non-ergodic.

In the left graph above I show in blue the ensemble mean at each period of a simulation of 20,000 runs of this process for 100 time periods (on a log scale). It looks just like our 5% compound growth rate (as it should).

The dashed orange lines are a sample of runs of the simulation. Notably the distribution of those runs is heavily biased towards final balances of around $1 (remembering the starting balance was $100).

In fact, out of the 20,000 runs in my simulation, 17,000 lost money over the 100 time periods, having a final balance less than their $100 starting balance. Even more starkly, more than half the runs had less than $1 after 100 time periods. The right hand graph shows the final round balances of the 20,000 simulations on a log scale. You can read more about the mathematics here.

So if almost everybody losses from this process, how can the ensemble mean of 5% compound growth be a reasonable expectation value? It cannot. For someone who is only going to experience a single path through a non-ergodic process, basing your behaviour on an expectation using the ensemble mean probably won’t be an effective way to navigate economic variations.

I see two areas of economics where we may have been mislead by thinking of the ensemble mean as reasonable expectation.

First is a very micro level concern: behavioural biases. The whole idea of endowment effects and loses aversion make sense in a world dominated by non-ergodic processes. We hate losing what we have because it very often decreases our ability to make future gains. And we should certainly avoid being on one of the losing trajectories of a non-ergodic process.

The second is a macro level concern: insurance and retirement. Insurance pools resources at a given point in time across individuals in the insurance scheme in order that those who are lucky enough to be winners at that point in time, make a transfer to those who are losers. By doing this, risk is shared amongst the pool of insurance scheme participants [2].

Retirement and disability support schemes are social insurance schemes. They pool the resources of those lucky enough to be able to earn money at each point in time, and transfer it to those that are unable to.

But there has been a big trend towards self-insurance for retirement. In the US they are 401k plans, and in Australia superannuation schemes. Here the idea is that rather than pooling with others at each point in time (as in a public pensions systems), why not pool with your past and future self to smooth out your income?

You can immediately see the problem here. If the process of earning and saving non-ergodic and similar in character to the example here, such a system won’t be able to replace public pensions, as many individuals earning and saving paths will never recover during their working life to support their retirement. Unless you want the poor elderly living on the street, some public retirement insurance will be necessary.

Undoubtedly there are many more areas of economics where this subtle shift in thinning can help improve out understanding of the world (I’m thinking especially about Gigerenzer’s ideas of heuristics approach as being ways humans have evolved to navigate non-ergodic processes).

I will leave the last word to Robert Solow, who has had similar misgivings (for over 30 years!) about our assumptions of ergodicity (a stationary stochastic process) which undermine rational expectations.

I ask myself what I could legitimately assume a person to have rational expectations about, the technical answer would be, I think, about the realization of a stationary stochastic process, such as the outcome of the toss of a coin or anything that can be modeled as the outcome of a random process that is stationary. If I don’t think that the economic implications of the outbreak of World war II were regarded by most people as the realization of a stationary stochastic process. In that case, the concept of rational expectations does not make any sense. Similarly, the major innovations cannot be thought of as the outcome of a random process. In that case the probability calculus does not apply.

fn[1]. He does not use the term as it is often used in economics as describing what often falls under the term Lucas critique, or in sociology is called performativity. Basically, it is the idea that the introducing a model of the world creates a reaction to that modal. Take a sports example. As a basketball coach I look at the past data and see that three point shots should be take more because they aren’t defended well. I then create plays (models) that capitalise on this. But because my opponents respond to the model, the success of the model is fleeting.

fn[2]. Peters himself has a paper on The Insurance Puzzle. The puzzle is that if it is profitable to offer insurance, it is not profitable to get insurance. The typical solution invokes non-linear utility to solve it. Peters offers an alternative. My take is on the economic implications of this - if people can smooth through time for retirement than there is not logic to social insurance.