Wednesday, June 8, 2016

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.


  1. Replies
    1. I think I found an issue with Peters' non-ergodic finding in resolving the St. Petersburg paradox ...

      ... but there is also more on dealing with infinity in infinite sums.

    2. Interesting post. I might need to do think about your last point about sneaking in the logs a little more.

    3. Cheers.

      Wasn't obvious to me at first. But general point about non-ergodicity is important and should be considered.

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    2. Perhaps it's worth pointing out what will be obvious to mathy readers, that Peters' coin flipping example fails to be ergodic because the variable he has chosen to average (at last initially, I didn't watch the whole video) is inappropriate. Rather than focusing on the yield Y on investment, one should work with X=log(1+Y). This X is ergodic in the current sense, and E(X)<0.

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  4. Dumb question to ask, no doubt, about such a nice post and discussion, but does this do more than develop the point that "the results concerning fluctuations in coin tossing show that widely held beliefs about the law of large numbers are fallacious" in Feller's Introduction to Probability Theory and Its Applications, Volume 1, Third Edition, Chapter 3*, 1968?

  5. You are just betting too much, which makes the retirement analogy make little sense.

    Try betting .045*bankroll. So the first bet, you lose 4.50 or win $4.95. If you win your next bet is $4.72.

    1. Mathematically your point is correct, but in reality, we're always betting all (or almost all) of our bankroll and, while splitting it up into many smaller bets theoretically improves our odds against losing, the reality is, individual bets cast during the same period of time have a strong tendency to have the same, non-random outcome.


    The problem is the probability of loss and the size of that loss together with the size of the bet (100%).

  7. I don't think that it is a question of calculating expectations. The problem, as has been pointed out, is that the bet is too large, relative to the stake, for the expected payoffs. It violates the Kelly criterion.

    If you maximize the monetary return on investment you get the Kelly criterion. The question is not how you calculate expectations, but what you maximize.

    1. Thanks for the comment Bill (and also john and Charlie etc). But I guess for me the more interesting point is thinking about how we even know what the underlying process is when we observe only a single run of that process. What should I be thinking after playing this game for 20 rounds and having $1 in my pocket? Then trying to learn more about the process by waiting another 20 rounds and having 1c in my pocket. Obviously I would never infer that I should expect a 5% compound growth rate if I just keep waiting. And I shouldn't act as if I do. That's really a general idea that I've become interested in.

    2. Well, in my case I certainly didn't learn much from experience. As a young card player I tended to push every hand as hard as I could. Then one afternoon I was thinking about a series of hands, and it hit me that trying to maximize my expected payoff for each hand was wrong, as betting a lot on a small advantage would often leave me unable to capitalize on a large advantage later. But I couldn't figure out what was right. Later I learned about Bernoulli's work on the probability of going broke in the long run. Even later I ran across the Kelly criterion while browsing in the library. Only then did I realize the importance of focusing on the return on investment over time.

      If you look at the biographies of renowned gamblers, many of them have gone bust more than once. They probably learned from experience, but what about possibly equally talented gamblers who went bust and never made a comeback?

    3. A related question is that of prudence. Going for -- ahem -- broke all the time indeed leaves nearly everybody broke pretty quickly. But prudence does not seem to be valued much these days, certainly not as much as greed. ;) When I read about pension funds paying their investment advisors and managers bonuses for high returns in any year I just shake my head. That is only encouraging them to take on too much risk. (!) And what about prudently managed public corporations? Are they not vulnerable to corporate raiders who claim that they are being mismanaged?

    4. Bill,
      that last point about corporate raiders is one of my hobby horses. If you think about it, one of the arguments for capitalism is the evolutionary argument of trying out different strategies. But corporate raiders, make this impossible for public companies, imposing a uniform management strategy. Shouldn't hostile takeovers be banned? The arguments for seeing them as a good idea seem very weak to me. Bankruptcy also punishes unsuccessful management, why do we actually need hostile takeovers - which mostly seem to me to encourage high leverage and monopoly.

  8. Re the insurance puzzle - isn't the bigger puzzle "why is there BOTH insurance and lotteries".

    1. Yep. I actually don't gamble or get insurance. So you've raised an interesting case: the people who a) don't gamble, but b) take out insurance. Action a) corresponds to expectations, while option b) does not.

      I personally don't gamble and don't have many insurances (no private health, no home contents, no comprehensive car insurance)

    2. Not sure I understand your response here. I was thinking more about why are there people who insure themselves against big losses (reducing their risk), but also make gambles with expected loss (increasing their risk). I don't think this is such a puzzle so long as the prise is very high and the stakes are very low (since people might gain utility from having the dream of winning). But for repetitive small stakes gambling, it is much more problematic.

      I know that Nick Rowe points out this apparent conflict when talking about whether people desire a more equal distribution of income or not - if people are so egalitarian and support social insurance and progressive taxation - why are there lotteries? I think it is actually hard to explain with normal utility models.

    3. I agree this stuff is hard to explain with normal utility models.

      My answer is that I personally go with expected payoff by the ensemble mean - insurance and gambles are both negative.

      A person who insures themselves and gambles is going with expected losses on both. Though by doing so they are reducing their downside and increasing their upside in both, but the expectation (ensemble mean) of both choices is that they lose money and their average future wealth is lower.

      Maybe I should run some simulations of this game where people are able to insure themselves against unexpected losses, and also where they can enter lotteries (with both have negative expectation values).

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