Thursday, November 8, 2012

Ergodicity -- the Biggest Mistake in Economics?


I'm increasingly convinced that Ole Peters has identified the nub of an utterly essential problem in the framework of contemporary (i.e. last 50 years) economics. In a series of recent papers (here, here, here), he has argued with impressive clarity that the usual ensemble averages used to compute "expected" returns in finance are, in many cases, simply inappropriate to making decisions in the real world. Take a risky gamble, and the usual average over different outcomes mixes potential worlds in which we go broke with others in we get rich, and, importantly, takes the often irreversible consequences of these outcomes (bankruptcy, for example) out of the picture. If you make hugely risky investments, this average gives you full credit for all the wonderful possible outcomes, weighted appropriately for their likelihood, which of course seems sensible. What it doesn't do is account for the very real fact that the bad outcomes may effectively wipe you out entirely and take you out of the game, making it impossible to play again -- in which case you will never get to experience those eventual big payoffs.

Maybe the best thing to read about this is this wonderful paper by people from the financial firm Towers Watson (credit: I learned of this from Rick Bookstaber's blog). The potential implications of this are really huge, as Peters' perspective suggests that the standard way of assessing risk versus reward in financial economics is wrong and systematically underestimates risks (and not merely because it ignores fat tails). The paper above, the first paper of Peters I mentioned above, and this interview with Peters are among the most interesting things I've read this year.

I'm going to do an in depth post on this stuff soon, but I must admit that I need to study it in detail a little more. I'm convinced that Peters insight -- which brilliantly resolves the centuries old "St Petersburg paradox" of probability theory proposed originally by Bernoulli -- also has a lot to do with the work of Doyne Farmer and John Geanakoplos on economic discounting, which I've written about before. Both suggest that our basic thinking about probability in time series suffers from some terrible misconceptions, and generally makes us underestimate risks. More coming on this soon.

290 comments:

  1. Love your posts! Paul Davidson, a prominent Post-Keynesian economist, has been stressing this theme for a long time, a theme also stressed by Keynes, himself, though he didn't use the term, ergodicity.

    ReplyDelete
  2. I've skimmed the paper on St. Petersburg paradox and, by chance, I've also been thinking about this topic recently, so here are few quick thoughts:

    1) Basing ones decisions on time average makes sense only when facing a long sequence of bets. Thus this doesn't seem to resolve the original paradox, which was about single bet.

    2) Peters considers utility "arbitrary" - I'd instead say it allows us to incorporate people's preferences, which seems like a reasonable thing to do, with economics being about people. The time-average approach in the end boils down to choice of one particular utility function (log utility) - and _that_ is arbitrary.

    Also, in contemporary economics, expected utility is not really justified by reference to "parallel universes". Instead it can be derived from properties of preferences over "bets", even when probabilities of particular states are unknown (some keywords: "subjective expected utility" and "Savage axioms").

    3) Of course, if we were dealing with truly nonergodic processes where ensemble and time averages differ, one could make a case that basing decisions on expected utility is unreasonable. But I don't think Peters' examples show this. Basically, his core argument is that we have a sequence of random gross returns, and we compute two averages: _arithmetic_ ensemble average and _geometric_ time average. The two numbers differ (in Petersburg paradox the first one doesn't even exist), which is why we should focus on time average.

    But wait, one average is geometric and the other arithmetic, so naturally they will differ! A proper comparison would be to use log returns and compute arithmetic averages in both settings, in which case the difference would disappear. AFAIK, formal mathematical theorems dealing with ergodicity and equality of ensemble/time averages are also formulated in this way (i.e. using additive averaging in both cases).

    Maybe there is some deeper point about difference between geometric and arithmetic averaging that I've missed. But anyway, this seems to have little to do with what people usually mean when speaking about nonergodicity (i.e. the world is supposed to be nonstationary and path-dependent, so we can't estimate probabilities from past data).

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  4. Economists don't consider data dispersion
    in the Petersburg paradox, the Allais paradox and others.
    But the data dispersion can create restrictions
    and can bias the results of experiments. See, e.g.,
    Theorem of existence of ruptures in probability scale. ...
    http://econpapers.repec.org/paper/pramprapa/23319.htm
    and
    Data Dispersion in Economics (I) --- Possibility of Restrictions
    http://www.bapress.ca/refissue-v2-n3.php
    Data Dispersion in Economics(II)---Inevitability and Consequences of Restrictions
    http://www.bapress.ca/refissue-v2-n4.php

    Alexander Harin
    http://econpapers.repec.org/RAS/pha243.htm

    ReplyDelete
  5. I have read only the first paper by Peters (on St.Petersburg's paradox, at http://rsta.royalsocietypublishing.org/content/369/1956/4913.full.pdf+html). I have some substantive comments, or at least I like to think so.

    First, even on its own account, the paper is unnecessarily complex. It would have sufficed to say that the process is non-ergodic, which takes two lines to show, and to give the time and ensemble averages, which is a trivial exercise.

    Second, non-ergodicity is the norm in non-recurring processes. But the criticism that economists use ensemble averages is just unfounded. In fact, in many papers in finance, only the existence of long-term averages is posited, which is weaker than ergodicity. When they do, they make sure to assume that the states of the economy live in a compact space. I would refrain from calling this a "problem" actually (an "utterly essential problem in the framework of contemporary (i.e. last 50 years) economics."), and rather spend my time reading actual original articles.

    Third, and perhaps more importantly, the difference between ensemble and time averages is at the core of the theory of growth optimal portfolios (or universal portfolios), as studied by Cover, Ziemba, Luenberger, Christensen etc. A good review is McLean and Ziemba, that shows that log utility maximizes time-average growth; also, universal portfolios have already been identified as having the best ones (on paper) for risk management (e.g., the only ones that have zero prob of default); the papers of Ziemba and Luenberger as well as the recent paper of Dean Foster and Sergiu Hart on operational measure of riskinness show this is the case (check out also Christensen's review on universal portfolios). Lastly, it is simply untrue that the reason St. Petersburg's wager is undesirable because a reasoning about differences between growth rates under the two averaging mechanism. Let's assume that each betting cycle is instantaneous, so we get to the outcome instantaneously. Would a bettor accept the wager? No. The reason is that there is no infinite leverage, which in turn changes the expected payoff of the wager. This is unintuitive, because the optional stopping theorem for Martingales would suggest otherwise; but it doesn't if you check the assumptions carefully. This is discussed clearly in William's "Probability with Martingales"

    So, I am puzzled by the level of attention this paper has received. But it just goes to show that Santayana's dictum is true. And that physicists
    mindlessly applying concepts from their field to other areas are usually right or original, but never at the same time.

    For the record: I am a former physicist working in a finance.

    ReplyDelete
    Replies
    1. Gappy,

      Thanks very much for this thoughtful comment. I'm going to look into some of the references you mention on growth optimal portfolios.

      I'm not quite sure I understand your explanation of why the St Petersburg wager is undesirable. As you say: " Let's assume that each betting cycle is instantaneous, so we get to the outcome instantaneously. Would a bettor accept the wager? No. The reason is that there is no infinite leverage, which in turn changes the expected payoff of the wager. This is unintuitive, because the optional stopping theorem for Martingales would suggest otherwise; but it doesn't if you check the assumptions carefully. This is discussed clearly in William's "Probability with Martingales""

      Can you outline this argument in a little more detail (if you have a moment!)? I'm tantalized by Peters' argument but I still feel I don't understand things well enough to judge it properly.

      Delete
    2. Mark, thank for the kind reply. Now, consider a setting in which the casino is running the bet sequentially on a reasonably fast computer. It is easily possible to run 1e9 rounds of the bet within a minute. The probability that the bet will last that long is very, very, small. The time element is now removed from the bet. If the losses exceed the casino's current net worth, the lottery stops. Would the casino sell the lottery at a finite price? I believe so, and I believe this price would be determined by capital structure, risk aversion, client risk aversion, etc. I'd rather not simplify the decision process underlying the pricing. What is relevant here is that it that a) the price is not infinity; b) time has nothing to do with it.

      Now, the connection to martingale theory. I play against the casino. At stage t, a fair coin is tossed. If tails comes up, I win 2^n dollars, otherwise I lose 2^n dollars. Let X(t) be the cumulative payoff at time t and let T the first time heads comes up. The St.Petersburg wager consists of X(T). Now, X(t) is a martingale: E(X(t+1)|X(t))=X(t), because the bet at stage t has zero expectation. Iterating, E(X(t))=X(0)=0. Doob's optional stopping time says that this result applies if we replace t with a random time T whose realization at time t can be determined by the history up to that time (this is very informal--see any graduate text on probability). St.Petersburg Paradox is exactly based on such a random time. We know if T <= t at time t, based on the realization of a head. So, according to the theorem E(X(T))=X(0)=0, the fair value of the bet for a risk-neutral bettor. But we know that X(T)=2^(n-1)! What gives? This is another version of the paradox. A closer inspection on the condition of stopping times helps understand why it doesn't hold. And the presence of finite leverage helps "fix" the concept of stopping time and to give a non-trivial expected value. Details in Williams and any graduate-level probability textbook (except Billingsley). Again, no need to introduce time considerations or ensemble vs time averages. But regarding the latter (and the relationship with risk management), there is a ton of literature on optimal growth portfolios. That's for another post.

      Delete
  6. My 2c here is just to highlight that the geometric method will give a junk answer of "zero" in any evaluation of a Probability density function that contains even a vanishingly small element at (or below)zero payoff. E.g. let's compute the expected geometric value of me leaving my house today. Oh wait, there is a vanishingly small chance I may die in a road accident, or kill multiple others due to some freakish negligent action. Clearly the answer is to stay at home and starve, everyday!

    FWIW I use geometric and ensemble averages all the time, where appropriate. But there is no black or white interpretation or endorsement of absolute authority. Context is everything.

    ReplyDelete
  7. Outside of fantasy dice rolling games, I posit that in the vast majority of realworld scenarios it is possible to assign an extremely small chance of complete derailment due to some obscure and rarely realized risk factor. Thus confounding the pure use of the geometric average method.

    In models, perfect is the enemy of good enough, wouldn't you say?

    On a related note, I have found some arithmetically biased risk metrics that are horrendously skewed unless recast in a geometric fashion(e.g. the MAR, or Calmar ratio is very distorted by varying leverage outside of a low band). All my attempts to persuade others generally are accepted as valid, but have never been seriously acted upon as far as I know. This may day something bad about human willingness to incumbent frames of reference, I don't know.

    Interestingly, the arithmetic versions often don't diverge that dramatically from the geometric in terms of the comparative rankings they give to a set of competing datasets at low stepsizes relative to initial "wealth" (i.e.low leverage) and of limited runsteps(timesteps). I.e. a bad stock portfolio is a bad portfolio relative to its superior peers and whether you use naieve or complex measures the answer often remains the same. I can only really conclude that accessibility, computational simplicity and an acceptable avoidance of overprecision are more useful in reality than the "correct method", when the realworld probabilities are never truly fully known ex ante anyway.

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