Thursday, October 13, 2011

A Crack in the Foundation: An error that has wended its way through economics for 77 years

This post is an introduction to a detective story about an error. An error that passed undetected by some of the greatest minds of the twentieth century, and led economics down a path that now must be cast into question. It is an error from 1934 Vienna that has lain hidden for the better part of a century, uncovered in 2011 in the academic halls of Imperial College of London. It is an error that is both obvious and startling after the fact, and is the result of a calculation that literally is off by an order of infinity.

A few weeks ago I attended a conference sponsored by the Santa Fe Institute, where I participated on a panel with Henry Kaufman, Bill Miller and Marty Leibowitz. The conference topic was Forecasting in the Face of Risk and Uncertainty. One of the presentations was by Ole Peters, from the Department of Mathematics at the Imperial College of London. His presentation compared time series analysis with ensemble analysis. Time series analysis takes one realization of a process and runs it over a very long time period and then looks at the distribution over the course of that run, whereas ensemble analysis creates many copies of the process and runs these over a shorter period, and then looks at the distribution of those results. Time series analysis is what you see over many years in one universe, ensemble analysis is what you see when you take many universes and integrate across them to look at the distributional properties.

Even if we use the same process for generating the paths as we do for the time series, these two approaches can lead to surprisingly different results for the ultimate distribution. This will always be true if a process is not ergodic, that is if it doesn't have the properties of creating a defined and unique distribution and leading to that distribution without regard to the starting point. Another way to think of an ergodic process is that over time it visits every possible state in proportion to its probability, and does so with the same proportions no matter where you start the process off. And one of the keystone processes analyzed in economics, the process of the inter-temporal compounding of wealth, is an example of a non-ergodic process. Peters presents a disarmingly simple example to show the difference between the time series and ensemble approaches for this process. Using a progression of simulated coin flips, he shows a case where the ensemble approach has a positive mean while the time series approach has one that is negative. On average people will make money while for the individual wealth will follow a straight line (on a log scale, at least) toward zero.

As Peters recounts in his presentation, economics has been almost unwavering in applying the ensemble approach. The reason is that in 1934 the Austrian mathematician Karl Menger wrote a paper that rejected unbounded utility functions. These unbounded functions include, for example, logarithmic utility, a particularly useful one because it corresponds to exponential growth, and thus is a natural for many time series processes (like compounded returns). Because his result is wrong, the motivation for focusing on the ensemble approach is ill founded. And, to make matters worse, in many important cases it is a time series approach that makes the most sense. After all, we only live one life, and we care about what is dealt to us in that life. If we enjoyed (and recognized that we enjoyed) reincarnation so that we could experience many alternative worlds – and better yet, if we experienced them all simultaneously -- perhaps it would be a different matter.

What is fascinating is that Menger’s paper has been cited widely by notable economists, including Samuelson, Arrow and Markowitz, Nobel laureates all. Peters recounts a number of these: In 1951, Arrow wrote a clearer version of Menger's argument, but failed to uncover the error while doing so. Ironically, by performing this service he helped propagate the development of economic theory along the wrong track. Arrow more recently wrote that "...a deeper understanding [of Bernoulli's St. Petersberg paradox] was achieved only with Karl Menger’s paper”. Markowitz accepted Menger's argument, stating that “we would have to assume that U[tility] was bounded to avoid paradoxes such as those of Bernoulli and Menger”. Samuelson waxed effusive regarding Menger's 1934 paper: “After 1738 nothing earthshaking was added to the findings of Daniel Bernoulli and his contemporaries until the quantum jump in analysis provided by Karl Menger”, and further that “Menger 1934 is a modern classic that stands above all criticism”. (And up until Peters’ paper, it seems it did indeed stand above all criticism, if there was any at all). That the paper was such a focus for so stellar a group of economists gives you a hint of its importance to the path economics has taken.

Not that the error is all that obscure, at least with the benefit of hindsight and a clear exposition. It boils down to Menger saying that the sum of a quantity A and a quantity B tends to infinity in the limit. Menger shows that A tends to infinity, and then argues that because of this, it doesn't matter what is going on with B, because infinity plus anything else is still infinity. Unless, of course, it happens that B is tending toward negative infinity even faster. Which, it turns out, is the case. So the sum, rather than having infinity as its limit, has negative infinity as its limit!


  1. What is the significance of this, in terms of modern economic theory? Are reevaluations taking place or is this something professors may bring up as just a curiosity? Followed over from Thanks!

  2. It has led to a focus on ensemble rather than time series approaches, to not using log utility and other unbounded utility functions. A log function can allow for a preference-free approach for looking at some problems. It basically has cut off one path that economics could have taken. Because we didn't take it, it is difficult to know what has been lost as a result.

  3. If we had only ensemble averages rather than time-series averages to understand chemical systems, then we'd have no understanding of non-equilibrium processes, or indeed of any system in which the distribution of observable statistical variables does not change over time. We'd have thermodynamics, but not kinetics or non-equilibrium statistical mechanics. We'd understand that both diamond and graphite were possible equilibria for carbon, but not why one was more common.

    Actually, not all economists seem to have had their eyes closed. Mandelbrot, in particular, questioned the assumption of stationarity, which I believe is implicit in the assumption of ergodicity.

    Douglass North has been talking about non-ergodicity for a while...

  4. Seems theres more going on here than non-ergodicity. Very neat post! Thanks!! ... Please expand.

  5. I have been doing some thinking on this subject, particularly regarding the "so whats". I have written a short paper - the front end summarises Ole Peters's work (so doesn't add much to Rick's post above), but the back end contains what I believe to be the practical implications, primarily from an investment point of view.

  6. Peters could have benefited from a better understanding on the extensive literature on the Petersburg game.

    The game was the subject of significant debate in the eighteenth century, as discussed by Jorland (in The Probabilistic Revolution: Volume 1: Ideas in History). The debate was essentially ended when Laplace observed there was not an issue in cases when the game could be repeated an infinitely number of times. Or, Laplace makes the equivalent observation that Peters makes, only 200 years ago.

    The model Peters develops appears to be remarkably similar to the one Durand proposed in 1957 (The Journal of Finance, 12, 348–363) and is discussed by Szezkely and Richards (The American Statistician, 2004, Vol. 58, No. 3).

    I do not disagree with your assessment that there has been an error in economics for the past 77 years (just one?) but mathematicians working in finance have generally ignored Samuelson's attacks on logarithmic utility. Poundstone's book on the Kelly Criterion is a good description of the battle in the 1960s and I there is a rich contemporary literature that develops Kelly's ideas, for example Platen and Heath (of HJM)"A Benchmark Approach to Quantitative Finance" or Fernholtz and Karatzas "Stochastic Portfolio Theory: an Overview", a working paper available

    My advice to economics: be careful of physicists bearing gifts. Peters recognises the essential problem of non-ergodicity in finance and economics, but the assumption of ergodicity is so central to the physical sciences that is never going to be easy for a trained physicists to come to terms with it. I have discused these issues in more detail

  7. Bernoulli's solution to the SPP is simply ill conceived. Concave utility functions are rational for wealth consumed on tangible items. However, wealth has a distinct intangible quality to it as well that enables a convex utility function, enjoying an economies of scale (philanthropy is but one instance). Much research has been done on reverse-S shaped utility functions, namely the Markowitz Stochastic Dominance curve. We expand such analysis here

    Furthermore on the SPP, there exists a very large premium to the expected value as soon as one selects an amount to play. If I offer $25 to play, I am expecting at least 5 heads in a row (through whatever analysis I perform). The expected value for that specific outcome is the same as all other outcomes, $0.5. What differentiates this gamble from a lottery is the certainty I am the sole participant, thus sole winner. The lottery does not enjoy such a consideration, supporting the considerable premium to expected value for the SPP when compared.

    By stating that more wealth must mean less (as Bernoulli stipulates) thus is governed by an increasing risk-averse concave utility function is shortsighted, and is not empirically supported.