Thoughts on "Bernoulli's error" and Samuelson v. J L Kelly

EUT is the foundation of modern economics. Despite this, I have yet to find a practitioner who uses it.  --ergodicityeconomics

I was reading an interesting post at ergodicityeconomics.com yesterday (The trouble with Bernoulli 1738). If I may be permitted to oversimplify, the poster's point is that there is a possible error in the way Bernoulli deals with fees in the context of the utility curve for a lottery. That means for him that to some extent Bernoulli is not "the same" as modern utility economics which he then uses as a platform to advocate a non-utility approach (time-average growth rates) that is "better."  I can't comment on Bernoilli's error (or lack) but it is definitely worth the read to see the poster pick apart the original paper.

On the other hand the thing that my amateur brain did notice was that in the remainder of his post, where he starts to tease out some of the differences between evaluating utility (disfavored by poster) and evaluating expected exponential rates of growth in wealth (favored) it was starting to smell and sound really, really familiar to me, like someone speaking in some difficult dialect but who also starts to slowly become more and more understandable (usually in the presence of beer).  Here are some excerpts. These may be tendentiously selected in a way that is in my post's favor but are representative I think:

• The main thrust of our work is, of course, to replace expected utility theory and instead work with time-average growth rates of wealth.  
• This basis of economics contains an error that invalidates commonly held beliefs and puts tens of thousands of studies into a different light.  
• What’s this? In any field other than economics, this object is called the expected exponential growth rate of wealth. No utility required, no psychology required.  
• Wealth dynamics are fundamentally multiplicative — we can invest wealth to generate more wealth. For such dynamics the exponential growth rate is an ergodic observable, which means its expectation value tells us what happens in the long run. Mystery solved! People just optimize what happens to their wealth over time 
• Economics textbooks and papers miss this point. The problem is treated in an a-temporal space, as a so-called “one-shot game”. The mathematics is the mathematics of things happening in parallel universes, not of things happening over time. Where time is mentioned, it is usually wrongly assumed that [delta x] indicates what happens over time. Endless arguments ensue over whether ln(x) is the correct psychological re-weighting people apply to wealth — in reality this is a question about dynamics, with psychology as a second-order effect  
• Depending on which part of Bernoulli we read, and how carefully we read it, we will come away with a different impression of what utility means and what EUT is. Not surprisingly, the economics literature is littered with arguments and disagreements and invalid studies that seem to arise from a confused use of EUT.  
• Finally: a plea for treating the problem with the modern mathematical concepts we now have. By this, I mean: start worrying about time, and compute time-average growth rates. The “expected change of logarithmic utility” is nothing but the time-average growth rate of wealth, under multiplicative dynamics.  
• Once this concept has sunk in — that gambles are evaluated according to the growth rates they generate for those who engage in them — the type of confusion that surrounds EUT becomes almost impossible.  
• Economics is devoted to the quantitative evaluation of risky prospects, but the people who quantitatively evaluate risky prospects for a living make no use of the techniques it has developed.  
• The fundamental and fatal flaw is conceptual — parallel universes are used where there should be time and dynamics. Because of this flaw, there’s nothing to check against, the theory is not falsifiable because it depends on unobservable states of happiness or discomfort.

As best as I can tell, these ideas are presented as ergodicityeconomics.com's critical business insights and/or positions. In fact a post like this, given the first bullet, may even be a form of marketing. I'll assume yes.  If not, though, I was a little surprised then that there was no reference to some of the history of the "debate" between utility and analytically determined exponential growth rates because this should all start to sound awfully familiar.  The ergodicityeconomics argument, in different words but related notation, is (to me...and if I'm wrong I'll post the errors) when "unpacked" nothing more than a discussion of the well known Samuelson-Kelly face-off as it has been fought by others over a number of decades.  While the blog post is a great read, an even better read, I think, is chapter 9 of Risk-Return Analysis Vol 2 pp 141-166, Markowitz 2016 titled The Mossin-Samuelson Model."  This is a much deeper and more general and surprisingly accessible discussion of what I consider to be the same topic.  Markowitz in that chapter provides some context and history but then squares himself and his own theory with Kelly and becomes parti pris on the side of what he calls Max E log(1+Rt) [MEL] (what I call geometric mean maximization [GMM] and which I take to be the same as what ergodicity calls "expected exponential growth rates" or "time-average growth rates of wealth") vs. Samuelson's arguments from Utility.  In a multiplicative context over multiple time periods/games with Samuelson's game converted to games and the utility "scoring" adjusted from terminal wealth to growth rates (a subtle but important shift) Markowitz makes the case that at some time Tn MEL always will start to win big which I believe to be the linked poster's main point. Markowitz tweaks a Samuelson unable to respond thus:

Theorem: If Harry repeatedly invests in a portfolio whose E log(1+Rt) is greater than that of Paul, then -- with probability 1.0 -- there will come a time (T0) when Harry's wealth exceeds Paul's and remains so forever thereafter. [emphasis in original pp150] 

I don't have deep knowledge of the history of the debate and the various critiques of MEL or GMM but I recall something about Samuelson's objection to using criteria that arise only from statistical artifacts rather than rational human narratives. Also, some have correctly pointed out that Tn can be very very far in the future, well past human life spans and that (we aren't in consumption mode here by the way) the intervening years where one might be waiting for MEL to eventually "win" with probability 1.0 are also the same years that might be soul crushing periods of very low wealth...so what is defined as long run matters (RiversHedge reminds us here to remember that the presence of consumption can crash a MEL process to and through zero in timeframes that are of great interest to the living). Pros and cons as always. Best, though, to at least be aware of the debate. Here's Markowitz again:

If in a given context there is any doubt as to which meaning of "long run" is intended, one could refer to one as investing for the long run in the Kelly sense, or in the Samuelson sense. In fact the Kelly sense is used much more often than the Samuelson sense [there's an implied response to the epigraph at the beginning of my post]...This does not prove that one is right and one is wrong; only which is more likely to be meant if it is not explicitly specified. [emphasis in original; pp 155-6]

Given some of the heat that has come from this debate over the years I was surprised that the name Kelly (or others) did not specifically come up in the Ergodicity post.  Again, Markowitz is a good general source for some of these missing antecedents:

Kelly    '56

Latané  '57 '59

Breiman '61

Markowitz '59 '76 2006

Hakansson '71

McLean Thorp Ziemba 2011

Here, by the way, are some other related resources I've used in the past here at RH:

• Growth Optimal Portfolios, Corey Hoffstein.
• "A Practical Framework for Portfolio Choice" by Richard Michaud
• Diversification, Rebalancing, And The Geometric Mean Frontier, William J. Bernstein and David Wilkinson 1997.
• On the Relationship between Arithmetic and Geometric Returns, Dimitry Mindlin, CDI Advisors 2011. 
• Quant Nugget 2: Linear vs. Compounded Returns – Common Pitfalls in Portfolio Management, A. Meucci 2010.

Final point

If I were to have a point it would be this. I am not skilled enough to know whether Bernoulli did or didn't make an error nor do I have an opinion on whether an error would or would not in some way subvert modern utility-based economics.  What I do know is that one does not really need the platform of an error in 1738 in order to have, in 2018, a robust discussion about "growth optimal portfolios" (to use Hoffstein's phrasing) which has a rich literature of its own going back at least sixty years if not longer.