<p>Junior econ professor here. I saw Ole Peter's work and I was intrigued, so I actually looked into it to see if there was something original/insightful for me to learn. I even run a little simulation of what he calls the &quot;St. Petersburg Paradox&quot;, to make sure I understood what the guy is actually talking about.</p>
<p>Just to be clear, I didn't spend days looking into this. I &quot;only&quot; spent a few hours, until I was confident that I had an informed opinion. I eventually came to the conclusion that his point has actually little to do with the ergodicity of stochastic processes, which is a well-defined concept that (i agree) most economists are probably not familiar with. However, most time-series econometricians are definitely familiar with it</p>
<p><a href="https://en.wikipedia.org/wiki/Ergodic_process" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Ergodic_process</a></p>
<p>His point has instead a lot to do with what's called Jensen's inequality, something that most economists learn by end of year 1 of grad school</p>
<p><a href="https://en.wikipedia.org/wiki/Jensen%27s_inequality" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Jensen's_inequality</a></p>
<p>What settled it for me was going online, and finding out that a much more senior colleague of mine (who unlike me is a published expert on time series) had - completely independently - reached a similar conclusion:</p>
<p><a href="https://www.rogerfarmer.com/rogerfarmerblog/2020/5/6/the-peters-paradox" rel="nofollow noreferrer">https://www.rogerfarmer.com/rogerfarmerblog/2020/5/6/the-peters-paradox</a></p>
<p>So I initially thought &quot;well the poor guy [Peters] is just being naive: he thinks he's made a discovery and literally he's just struggling with Jensen's inequality&quot;. That is until I saw his Ted Talk and looked into his background (he works at a Climate Change/Social Justice center).</p>
<p>Now I'm not so sure he's being naive. I think more likely he's muddying the waters on purpose. He is giving this &quot;stuff&quot; (calling it a theory is too much of a stretch in my view) the pompous name of &quot;ergodicity&quot; so that he can sound smart and go around pontificating about inequality, climate change etc. and argue for extreme policies that would be rejected by traditional economists, because (so goes his argument) they don't understand how randomness and time interact.</p>
<p>That's a very old trick: you come up with some pseudo-scientific mumbo-jumbo full of technical jargon from different disciplines that most people can't understand, and then you use that to put yourself forward as an &quot;expert&quot;, and advance ideological positions as if they were based on hard science.</p>
<p>Now what to make of the fact that this &quot;ergodicity&quot; stuff got published in Nature Physics? Well we should note that in this case the editors went out of their way and published an article about economics, which is outside of their field of expertise - with predictable consequences. That is why we have specialized journals for different disciplines.</p>
<p>Now if you don't believe me and you suspect that we are biased because we are &quot;traditional economists&quot;, of course that's a possibility. Then my encouragement is: do what I did. Go ahead and figure it out for yourself. Read his paper, read the two wiki articles I linked, pick up a statistics textbook. And if you do figure out something interesting out of this, come back and do let me know. I'm happy to be proven wrong.</p>

CC BY-SA 4.0
2021-08-06T02:54:31.373Z
Thank you for a hands-on perspective.
Richard Hardy
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2021-08-06T10:25:52.303Z
The https://en.wikipedia.org/wiki/St._Petersburg_paradox was analyzed by Nicolas Bernoulli in 17-something and is thus much older than Ole Peters. Already back then he suggested utility functions and that is why these concave function $u$ that we use are called Bernoulli utility functions.
Bayesian
CC BY-SA 4.0
2021-08-09T06:07:30.883Z
I find this answer very interesting, for a couple of reasons.  To begin with, Jensen’s inequality is usually taught to first-year maths undergrads, so it’s perhaps not as closely-guarded a secret of classically-trained economists as is hinted at here.  But secondly, your Wikipedia link about ergodicity itself acknowledges that econometrics and signal processing use a different definition of ergodicity (which uses a temporal sample) than that used in physics (which uses a spatial sample at a single instant).  Are you sure you’re talking about the same thing Peters is?
JCW
CC BY-SA 4.0
2021-08-10T01:46:58.810Z
Never hinted at nor thought that Jensen's inequality was an economists' secret. With respect to the definition of ergodicity, my conclusion was not dependent a particular definition of ergodicity. My point in fact is that you don't need any notion of ergodicity (neither physical nor econometric) to explain the phenomenon that OP is concerned with. 

And even if ergodicity had anything to do with it (remains to be proven) OP's unsubstantiated claim that, as a consequence of that, we should change the way we model income inequality in economics is a non-sequitur.
bbecon
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2024-12-26T11:10:45.913Z
I somewhat doubt that @bbecon has actually read the paper, because it clearly states:

"That the geometric mean exp〈ln x〉 is less than the arithmetic mean 〈x〉 was known to Euclid (Elements, Book V, Proposition 25), and it is a special case of Jensen’s inequality of 1906. Its connection to gambling and investment problems was noted by Whitworth in 1870, is implied by Itô’s work of 1944, and is well known among gamblers as Kelly’s criterion16 of 1956. Our modest contribution is to frame these observations as a question of ergodicity, which we have found to be a fruitful perspective."
NeverMind
CC BY-SA 4.0
2021-08-05T18:21:37.717Z
2021-08-11T14:30:01.643Z
2021-08-11T14:30:01.643Z
bbecon
<p>I’am the guy who wrote the short note that was mentioned in the original question.
You can reach the note here.</p>
<p><a href="https://osf.io/preprints/socarxiv/axkfg/" rel="noreferrer">https://osf.io/preprints/socarxiv/axkfg/</a></p>
<p>I got interested in Peter's paper because of my interest in evolutionary games, where the question of whether the equilibrium selection process is or is not ergodic plays a crucial role. I agree that most economists find Peters' contribution irrelevant. However, by reading Peters’ original paper and Wakker e.a. reaction to it, a casual reader may get the impression that Peters has a point after all, but that he tends to oversell his results. The same impression one gets by reading other informal contributions written online by economists, including the present discussion.</p>
<p>In my note, I provide a pedantic presentation of Peters' running example and prove a very simple proposition, which, if correct, may put the entire matter at rest. This proposition says that in Peters’ example no subject who obeys the axioms of EU would accept the lottery unless he has an unbounded utility for money. In this respect, Peters' example is just a more involved instance of the St. Petersburg paradox and can be solved the way K. Menger solved the original one in 1934: the paradox only arises because we let utility for money be unbounded. Notice that it is easy to be fooled on this matter: risk-aversion will not do the trick, just as Menger proved in 1934.</p>
<p>The reactions to my result have been mixed. O. Peters was extremely polite, but he said that he receives way too many contributions to ergodicity economics to evaluate them all. No follower of ergodicity economics has shown any interest.</p>
<p>I had a brief mail exchange with P. Wakker, and he alerted me about the difference of opinion between K. Arrow (who believed unbounded utility to be a violation of EU) and P. Samuelson (who held the opposite view). In the end, he convinced me to side with Samuelson, although I still think Arrow's position may be defended. In any case, my result proves that Peters provided a very small contribution to a literature that in economics is half a century old. (In fact, you can find Samuelson’s view on Peters’ example in a PNAS paper published in 1971.)</p>
<p>I had some encouragement from some respected decision theorists, but most economists were not interested. From a prominent game theorist, I got a reply that rang something like: “nice work, but I have no patience for this matter”.
Finally, the note has been downloaded more than 100 times, but none has sent me an email for a comment or for any other reason. I concluded that the profession is so busy that even a moderately complex result as mine is seen as overkill for ergodicity economics.</p>
<p>Anyway, any comment is welcomed!</p>
<p>Luciano</p>
<p>PS If you have a bent for philosophical digressions, my note also discusses Peters’ claim that, when facing a compound lottery, EU assumes that a decision-maker experiences all alternative histories. In this case, Peters’ argument collides with the fact that EU is based on the independence axiom, which forces a DM to consider only the branch of the tree she happens to visit. In fact, just the opposite of what Peters claims is true.</p>

CC BY-SA 4.0
2021-08-11T07:21:49.890Z
How cool it is to hear from the author himself (I mean you)! Thank you so much! I would love to be able to contribute to the debate, but I only have rudimentary knowledge of decision theory, game theory and ergodicity. I nevertheless find the topic interesting and your paper helpful and illuminating!
Richard Hardy
CC BY-SA 4.0
2022-12-02T13:49:55.813Z
Have a look at https://www.amazon.co.uk/Prediction-Learning-Games-Nicolo-Cesa-Bianchi/dp/0521841089 and I think Chapter 9. Universal portfolios etc. Worst case learning bounds. This stuff is quite fundamental and economists have no idea. I have yet to see a derivation of DCF from a control/decision/RL framework point of view. Would be interested if anyone knows.
safetyduck
CC BY-SA 4.0
2022-12-06T20:58:53.410Z
Not sure I understand this comment. That book is a very long exposition of topics developed in the 50s by obscure scientists working on the then-new field of game theory (Hannan and Blackwell mostly). The relevance of these results was discovered decades later with fundamental papers published (in top journals in Econ) by Adreeu-Mas Collell and Sergiu Hart, Fudenberg and Levine, and a few others.  Maybe economists do have a clue.
Luciano Andreozzi
CC BY-SA 4.0
2021-08-11T06:47:47.423Z
2021-08-11T06:47:47.423Z
Luciano Andreozzi
<p>Well there is no opinion poll among economists on specifically this problem, but what can be judged from reaction of economists the consensus is that the Ole Peters paper is misguided and irrelevant at best. I think the economists' consensus was already very well and succinctly summed up by Doctor, Wakker and Wang you cite (and is an example of <a href="https://www.smbc-comics.com/comics/20120321.gif" rel="noreferrer">this</a> phenomena).</p>
<p>For starters, on twitter R. Thaler (Nobel Prize) called <a href="https://twitter.com/r_thaler/status/1027592614688653315?lang=en" rel="noreferrer">it hogwash</a>, if a Nobel Prize winner for work on decision making and behavioral economics so readily dismisses your idea about how humans make decision under uncertainity its a red flag. This work was also criticized by other notable economists such as <a href="https://www.rogerfarmer.com/rogerfarmerblog/2020/5/6/the-peters-paradox" rel="noreferrer">Farmer</a>.</p>
<p>However, even more can be judged by the non-reaction of economists. The Ole Peters paper was so widely circulated that it is safe to assume that majority of economists know about (I would bet that if you will ask at your university department about &quot;that ergodicity paper&quot; most of them will know what you are talking about). It is unbelievable but this paper got so much free press that it should an case study in marketing (it was covered and <em>uncritically</em> cited by major media outlets such as <a href="https://www.bloomberg.com/news/articles/2020-12-11/everything-we-ve-learned-about-modern-economic-theory-is-wrong" rel="noreferrer">Bloomberg</a> and it even got a <a href="https://www.youtube.com/watch?v=LGqOH3sYmQA&amp;t=241s" rel="noreferrer">TED talk</a>).</p>
<p>So it is safe to assume at very least most economists are aware of the Peter's work existence. Yet, if you look at which articles cite the Peters work, you will see most are either A) criticisms, B) not even in the field of economics and C) save for the criticisms most are not published in any reputable journal.</p>
<p>The Peters work is now already 2 years old, given that it is so widely known, and given that the work basically claims that our whole expected utility framework even in its general and behavioral applications is both wrong and not useful, you would expect that people would jump at the idea and start widely applying it, or at least testing it.</p>
<p>This is because expected utility is widely used workhorse model, and even though one can criticize it on a behavioral grounds it remains useful, the same way as Newtonian Physics, remains useful in presence of general relativity. In the areas where it is not useful we have behavioral models that still build upon the idea of expected utility or alternative theories that do not require ergodicity (e.g. like prospect theory etc., see Kahneman Thinking Fast and Slow for discussion).</p>
<p>Yet we do not see people at mass abandoning either expected utility or other more generalized concepts amass. Rather the paper was met by a silence occasionally interrupted by cricket's chirp.</p>
<p>Now either there is some conspiracy going on in our profession, or simply most economists do not even consider the paper worth while to respond to or engage with. Given how large our profession is conspiracy is unlikely (given that likelihood of keeping conspiracy secret declines drastically with number of people involved in e.g. see work of <a href="https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0147905" rel="noreferrer">Grimes 2016</a> on this). So really the most straightforward explanation for the utter lack of influence of the paper on profession is that the general consensus is that it is not even worth discussing.</p>
<p>One could also argue that the idea is being suppressed by the 'old guard'. There are several historical examples of new ideas in a field being suppressed, <em>for some time</em>, by established scientists e.g. like <a href="https://en.wikipedia.org/wiki/Antoine_Lavoisier#Chemical_revolution_and_opposition" rel="noreferrer">this example</a>. While it is not impossible that is happening to the ergodicity idea as well, one has to remember that for any good idea that is too eagerly suppressed by the 'old guard' there is always a large number of ideas that were dismissed by the old guard and actually also turned out to be bad or irrelevant. A good example are ideas like <a href="https://en.wikipedia.org/wiki/EmDrive" rel="noreferrer">EM drive</a> or think of all the 'theories' like <a href="https://en.wikipedia.org/wiki/Ancient_astronauts" rel="noreferrer">ancient astronaut theory</a>. So while it could turn out that ergodicity is being suppressed, it is more likely that it is actually not.</p>
<p>Of course, an important caveat is that absence of evidence is not necessary evidence of absence. Ideally, you would want some poll among economists, or at least well known economists.   This being, said the evidence and lack of thereof that we currently have strongly points toward the conclusion that the consensus is that Peters work is irrelevant.</p>

CC BY-SA 4.0
2021-08-03T10:13:05.450Z
You write of Peters's paper "*the work basically claims that our whole expected utility framework is wrong (and expected utility is widely used workhorse model)*", but you are also referring to [Doctor et al.'s paper](https://personal.eur.nl/wakker/pdfspubld/20.3ergodic.econ.suppl.info.pdf), where they have a section titled "**5. Falsifications of EU Are Well-Known**". So is Peters's claim of EU being wrong incorrect or is it just that his claim is not now or not more convincing than other critiques of EU?
Giskard
CC BY-SA 4.0
2021-08-03T10:17:04.020Z
"*Now either there is some conspiracy going on in our profession, or simply most economists do not even consider the paper worth while to respond to or engage with.*" This does not seem to consider that EU experts have little incentive to admit if EU is not useful in models - rendering their previous work obsolete - even without coordination. I make no claims about EU, but your argument is not very insightful. As Peters also critices behavioralism, Thaler is similarly lacking in incentives to embrace him.
Giskard
CC BY-SA 4.0
2021-08-03T10:34:38.890Z
@Giskard 1. As they explain in the chapter 5 the falsifications of EU they are referring to are falsifications of some particular specific cases of EU, in fact if you read any behavioral work they still use EU with behavioral twists. EU was not really falsified generally but specifically and the paper explains that in the section 5 you cite. So this is a moot point although I could probably rewrite it to be more nuanced. 2. I am sorry but I am unconvinced with that idea. Take simple prisoners dilemma. Sure two old guards could benefit from suppressing new ideas if they both do it but if one
1muflon1
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2021-08-03T10:35:56.587Z
1. I have read section 5 and disagree with your characterization of it.
Giskard
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2021-08-03T10:36:23.630Z
Deviates that one researcher can get all the citations from this new approach, so it’s not really stable equilibrium to have them suppress the idea and in Nash equilibrium they will both deviate. Plus there are always many young and scrappy scientists always looking for new ideas
1muflon1
CC BY-SA 4.0
2021-08-03T10:36:50.600Z
2. So if there are two kings, one would collaborate with communists? Seems strange. (Perhaps not everything should/can be described by a two-player one-shot prisoners' dilemma.)
Giskard
CC BY-SA 4.0
2021-08-03T10:37:47.323Z
2. Yes, young scientists could criticize EU. But perhaps the ones that do do not get published in good journals headed by the old guard?
Giskard
CC BY-SA 4.0
2021-08-03T10:42:36.580Z
For a case study of self interest in science I also recommend reading about [Lavoisier's oppisition](https://en.wikipedia.org/wiki/Antoine_Lavoisier#Chemical_revolution_and_opposition).
Giskard
CC BY-SA 4.0
2021-08-03T10:48:09.440Z
@Giskard 1, Kings would have no incentive to do so since they would loos even if they are the only one collaborating. 2. Journal publication is not entirely in hand of editors and also ideas like the markets from lemon that were going contrary to mainstream understanding of markets managed to get published in good journals. 3. Yes I know that sometimes these things happen but you have to have sense of proportion, for every 1 idea that is *postponed* by the old guard (and these ideas are always just postponed as at some point the evidence will simply mount too much to be ignored)
1muflon1
CC BY-SA 4.0
2021-08-03T10:49:21.800Z
So the devil is in the details, right? Then perhaps the simple examples don't explain that much, and 2 years is perhaps not too long?
Giskard
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2021-08-03T11:01:40.610Z
@Giskard you did not gave me time to finish my point...     there are dozens if not hundreds of crazy ideas  that are actually nonsense and were correctly disregarded, you basically have to make claim that this is actually one of those ideas that were suppressed because of the old guard. For example, even now there are few heterodox biologists who argue that covid-19 vaccines are not as safe as assumed by mainstream biologists e.g. see [this](https://www.youtube.com/watch?v=81mEJ2Rph1Q). These concerns are (in my opinion correctly) being rejected by mainstream
1muflon1
CC BY-SA 4.0
2021-08-03T11:01:43.033Z
biologists because even though we don't have any long term studies on effect of these covid vaccines given the evidence that we have as of now they are very likely safe and should be widely used. In the same fashion, given the evidence (or lack of thereof) we have now about this ergodic theory it is likely one of the nonsense ideas correctly rejected. However, I am open to changing my mind as new evidence is presented and revising my answer if in the future we see the idea pick up.
1muflon1
CC BY-SA 4.0
2021-08-03T11:06:55.693Z
I think your point should be finished in *your answer*. I am not defending Peters's claims, I am criticizing the arguments presented in the answer.
Giskard
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2021-08-03T11:17:29.060Z
@Giskard thanks for your critique, I tried to address the points in my last edit
1muflon1
CC BY-SA 4.0
2021-08-03T11:59:51.013Z
@1muflon1 I think you pretty much hit the nail on the head. The only thing I might recommend adding are perhaps specific examples of nonEU theories being used (rank dependent utility being used to model insurance markets, for example, or Cumulative Prospect Theory being used to model the impacts of defaults and dominated options in choice lists). You could also make the analogy to Newtonian physics--we know conclusively that classical physics models are wrong, but they're really easy to use and model most "normal" situations with incredible accuracy.
AndrewC
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2021-08-03T12:27:54.950Z
@AndrewC thanks for suggestions I edited my answer to try to incorporate them
1muflon1
CC BY-SA 4.0
2021-08-03T13:55:08.243Z
Thanks for Thaler and Farmer quotes; this is what I was looking for. Btw, TED talks tend to include a fair share of ridiculous stuff; Andrew Gelman has covered a few such instances in his blog over the years, so at least for me TED is not a stamp of quality. Not sure if you meant it that way, but mentioning this just as a heads up.
Richard Hardy
CC BY-SA 4.0
2021-08-03T22:01:41.213Z
that comic is *brilliant*
duckmayr
CC BY-SA 4.0
2021-08-03T22:30:46.307Z
Funny you point to the EM Drive. Had one been hoisted to the ISS it would have worked despite the theory being bonkers. It's solved now; a magnetic drive. In deep space it would not have worked.
Joshua
CC BY-SA 4.0
2021-08-04T08:53:37.983Z
@duckmayr Which comic...?
Giskard
CC BY-SA 4.0
2021-08-04T08:56:39.840Z
@Giskard I think he means [this one](https://www.smbc-comics.com/comics/20120321.gif) I linked at the beginning
1muflon1
CC BY-SA 4.0
2021-08-04T09:14:34.797Z
Thanks! Not realizing you had already done so, I also linked to the same comic [at the other question](https://economics.stackexchange.com/questions/47066/why-are-utility-functions-typically-assumed-to-be-concave#comment75440_47069).
Giskard
CC BY-SA 4.0
2021-08-04T09:33:13.727Z
@Giskard great minds think alike ;)
1muflon1
CC BY-SA 4.0
2022-12-02T13:45:27.850Z
Peter's work is mostly about clarification and historical work. A whole section weird economists appear to be vague or confused. I don't think real employed economists (i.e. quants) are confused about any of this. And they barely call themselves economists anyway.

So it's more about fights amongst academics.

I really appreciate Peter's work on Microfoundations of Discounting and consider the foundations of discounting still rather unsolved.
safetyduck
CC BY-SA 4.0
2021-08-03T09:48:24.563Z
2021-08-03T12:40:40.897Z
2021-08-03T12:40:40.897Z
1muflon1
<p>The final consensus is probably that there is no consensus. But most people have misunderstood the proposition of Ergodicity Economics (EE) anyway. I even wonder if these people have read Peter's paper at all, because it clearly states the following rather at the end of the paper (the bold formatting is mine):</p>
<blockquote>
<p>That the geometric mean exp〈ln x〉 is less than the arithmetic mean 〈x〉 was known to Euclid (Elements, Book V, Proposition 25), and it is a special case of Jensen’s inequality of 1906. Its connection to gambling and investment problems was noted by Whitworth in 1870, is implied by Itô’s work of 1944, and is well known among gamblers as Kelly’s criterion of 1956. <strong>Our modest contribution is to frame these observations as a question of ergodicity, which we have found to be a fruitful perspective</strong>. It enforces physical realism by precluding interactions among members of a statistical ensemble, it enables us to consider dynamics other than additive and multiplicative (corresponding to linear and logarithmic utility functions), and
it naturally leads to treatments of problems whose solutions are less readily visible in previous framings of the issue.</p>
</blockquote>
<p>Peter's message in his paper is nothing fundamentally new and is also not claimed to be. He is well aware that the basic concepts on which EE is based have been known for a long time. In a sense, his paper formalizes old knowledge, which is useful (also for economics) as it enables a more mathematically rigorous treatment of the matter. His emphasis is on the fact that decision-making over time typically differs for an individual and a group of independent individuals (depending on the (non)-ergodicity of the problem under consideration). While the latter on aggregate operates in the regime of large numbers, the former does not. In particular, individuals face only a single path over time, which is why probability distributions (or their derivatives in the form of expected values or higher moments) typically don't matter to them in contrast to time-average growth rates. As Peter's work reveals, this fact was fixed by economics in the view of homo oeconomicus by introducing a logarithmic utility function which transforms the maximization of the rate of change of expected utility into the maximization of a time-series growth rate in the case of multiplicative dynamics (e.g. compound growth).</p>
<p>In summary, EE has not developed anything new. It has helped us to formalize and better understand the key differences of decision-making between an individual and a group of independent individuals. Both perspectives are legitimate perspectives and neither of them is right or wrong in the first place. When to use which perspective ultimately depends on the topic or perspective in question. But this insight itself is an important gain in knowledge, formalized by the ergodicity question and rigorously linked to the underlying dynamics (instead of putting forward obscure psychological arguments for rational decision-making).</p>

CC BY-SA 4.0
2024-12-26T15:25:39.023Z
I am not intending to start another long discussion here but I was just wondering if this Peter's ergodicity discussion is at all connected to the 2 envelope problem here. In this problem, if one uses log utility, then the paradox dissappears. Thanks. https://en.wikipedia.org/wiki/Two_envelopes_problem
mark leeds
CC BY-SA 4.0
2024-12-26T22:50:31.477Z
The implication of the ergodicity question comes by many names or paradoxes. Yes, the two envelopes problem is another one to be resolved by the considerations outlined above. Only a group of independent individuals on aggregate will be able to realize the ensemble average over time, which is why from such a group perspective the gamble would be benefitial. In contrast, for an individual who plays the game repeatedly over time in a compounded way, the time-average growth rate is relevant. Log utility converts between the two views, as it renders multiplicative dynamics into additive dynamics.
NeverMind
CC BY-SA 4.0
2024-12-27T14:41:13.367Z
Hi  NeverMind. That's a nice explanation. That envelope problem always boggled my mind so atleast now I have a thread to go to if I want to understand it further. I always found it interesting that the use of the log utility removed the paradox !!!!! Thanks for your help.
mark leeds
CC BY-SA 4.0
2024-12-26T11:32:50.367Z
2024-12-29T13:48:17.253Z
2024-12-29T13:48:17.253Z
NeverMind
<p><a href="https://www.nature.com/articles/s41567-019-0732-0" rel="noreferrer">Peters (2019)</a> made a splash criticizing the theory of expected utility on the grounds that it implicitly assumes ergodicity where this is unwarranted. He stated this applies widely in economics, to the point of making the whole field suspect:</p>
<blockquote>
<p>economics is firmly stuck in the wrong conceptual space. Because the core mistake is 350 years old, the corresponding mindset is now firmly institutionalized.</p>
</blockquote>
<p>He also proposed a fix of the theory based on maximizing time-average growth rate.</p>
<p><a href="https://www.nature.com/articles/s41567-020-01106-x" rel="noreferrer">Doctor et al. (2020)</a> responded stating that Peters essentially missed the target. Economists are aware of the problem (and have been for a while) and as a rule do not apply the theory of expected utility in the naive way that Peters suggests they do (though of course there are exceptions, as everyone tends to make a mistake every now and then). Briefly, what Peters got right is not new, while what is new is not right.</p>
<p><a href="https://www.nature.com/articles/s41567-020-01108-9" rel="noreferrer">Peters (2020)</a> responded that he does not see much disagreement between what he originally said and what <a href="https://www.nature.com/articles/s41567-020-01106-x" rel="noreferrer">Doctor et al. (2020)</a> state.</p>
<p>Among some other reactions, <a href="https://osf.io/download/609277df6801ab03e42a3fee/" rel="noreferrer">Andreozzi (2021)</a> and Kim (<a href="https://files.osf.io/v1/resources/7gm8a/providers/osfstorage/5debad72b1bf4c000a405f38?action=download&amp;version=1&amp;direct" rel="noreferrer">undated, a</a>; <a href="https://files.osf.io/v1/resources/7gm8a/providers/osfstorage/5debad72b1bf4c000a405f38?action=download&amp;direct&amp;version=3" rel="noreferrer">undated, b</a>) were largely skeptical of Peters.</p>
<p>So there is the original paper and the subsequent exchange with Doctor et al., and a couple of other responses. There must also have been some reactions of economists in, say, blogs and other spaces, given the publicity the original paper has received. <strong>Do we have a consensus among economists regarding the merit (relevance, validity) of Peters' critique?</strong></p>
<p><em>This is not a question of opinion. I am trying to objectively gauge the consensus in the profession, i.e. has the matter become clear to most and has the majority opinion converged on anything concrete. References to back up an answer would be most appreciated.</em></p>
<p><sub>The question is motivated by a discussion in the comments of <a href="https://economics.stackexchange.com/questions/47066">&quot;Why are utility functions typically assumed to be concave?&quot;</a></sub></p>
<p><strong>References</strong></p>
<ul>
<li>Andreozzi, L. (2021). <a href="https://osf.io/download/609277df6801ab03e42a3fee/" rel="noreferrer">Ergodicity in Economics: a Decision Theoretic
Evaluation</a>.</li>
<li>Doctor, J. N., Wakker, P. P., &amp; Wang, T. V. (2020). <a href="https://www.nature.com/articles/s41567-020-01106-x" rel="noreferrer">Economists’ views on the ergodicity problem</a>. <em>Nature Physics, 16</em>(12), 1168-1168.</li>
<li>Kim, M. (undated, a). <a href="https://files.osf.io/v1/resources/7gm8a/providers/osfstorage/5debad72b1bf4c000a405f38?action=download&amp;version=1&amp;direct" rel="noreferrer">A comment on ergodicity economics</a>.</li>
<li>Kim, M. (undated, b). <a href="https://files.osf.io/v1/resources/7gm8a/providers/osfstorage/5debad72b1bf4c000a405f38?action=download&amp;direct&amp;version=3" rel="noreferrer">A letter to economists and physicists: on ergodicity economics</a>.</li>
<li>Peters, O. (2019). <a href="https://www.nature.com/articles/s41567-019-0732-0" rel="noreferrer">The ergodicity problem in economics</a>. <em>Nature Physics, 15</em>(12), 1216-1221.</li>
<li>Peters, O. (2020). <a href="https://www.nature.com/articles/s41567-020-01108-9" rel="noreferrer">Reply to: Economists’ views on the ergodicity problem</a>. <em>Nature Physics, 16</em>(12), 1169-1169.</li>
</ul>

CC BY-SA 4.0
2021-08-03T07:06:11.583Z
I've heard economists mention it as an example of utter trash coming from econophysics. I doubt you get more "consensus" on this, clearly the author is ignorant of even the most basic decision theory.
Michael Greinecker
CC BY-SA 4.0
2021-08-03T07:08:35.707Z
@MichaelGreinecker, thank you! Links to such examples would be appreciated.
Richard Hardy
CC BY-SA 4.0
2021-08-03T08:42:25.540Z
This is a paper in a physics journal stating nonsense about economics without even quoting any economic research. I doubt you will find much public information. Time is scarce.
Michael Greinecker
CC BY-SA 4.0
2021-08-03T08:45:41.360Z
@MichaelGreinecker, I understand that, though I think the publicity Peters (2019) received is a bit of a game changer. I think it has made commenting on the paper / the matter attractive enough for at least some respected economists. But then I might be wrong.
Richard Hardy
CC BY-SA 4.0
2021-08-03T12:10:06.363Z
I still haven't met any economit who takes Peters seriously. You can take this as evidence for economists' arrogance if you want to.
Bayesian
CC BY-SA 4.0
2021-08-03T23:11:34.277Z
Overblown claims of this kind are almost surely incorrect.
PatrickT
CC BY-SA 4.0
2021-08-09T06:27:01.313Z
Not sure it is accurate to characterise the second Kim paper as 'sceptical' of Peters - he writes: "Many people, having studied graduate-level introductory microeconomics, would have initially responded with immediate skepticism when they read [1]. I initially felt the same way, but after correspondence and some reflections, I believe I now understand where Dr. Peters and some of those following ergodicity economics come from."
JCW
CC BY-SA 4.0
2021-08-09T10:17:15.820Z
@JCW, my impression was that even tough Peters' ideas might deserve immediate criticism, Kim did not want to attack Peters personally and was trying to be kind in the paper he wrote. A very reasonable approach IMHO.
Richard Hardy
CC BY-SA 4.0
2024-12-31T04:58:12.853Z
Just stumbled upon this [relatively recent paper by Toda](https://arxiv.org/abs/2306.03275), adding to the so-called consensus.
Herr K.
<expected-utility><soft-question>
What is the consensus (if any) on Peters "The ergodicity problem in economics" (2019)?
Richard Hardy
# What is the consensus (if any) on Peters "The ergodicity problem in economics" (2019)?

(Asked by: username_0 on 2021-08-03)

[Peters (2019)](https://www.nature.com/articles/s41567-019-0732-0) made a splash criticizing the theory of expected utility on the grounds that it implicitly assumes ergodicity where this is unwarranted. He stated this applies widely in economics, to the point of making the whole field suspect:

> economics is firmly stuck in the wrong conceptual space. Because the core mistake is 350 years old, the corresponding mindset is now firmly institutionalized.

He also proposed a fix of the theory based on maximizing time-average growth rate.

[Doctor et al. (2020)](https://www.nature.com/articles/s41567-020-01106-x) responded stating that Peters essentially missed the target. Economists are aware of the problem (and have been for a while) and as a rule do not apply the theory of expected utility in the naive way that Peters suggests they do (though of course there are exceptions, as everyone tends to make a mistake every now and then). Briefly, what Peters got right is not new, while what is new is not right.

[Peters (2020)](https://www.nature.com/articles/s41567-020-01108-9) responded that he does not see much disagreement between what he originally said and what [Doctor et al. (2020)](https://www.nature.com/articles/s41567-020-01106-x) state.

Among some other reactions, [username_8 (2021)](https://osf.io/download/609277df6801ab03e42a3fee/) and Kim ([undated, a](https://files.osf.io/v1/resources/7gm8a/providers/osfstorage/5debad72b1bf4c000a405f38?action=download&version=1&direct); [undated, b](https://files.osf.io/v1/resources/7gm8a/providers/osfstorage/5debad72b1bf4c000a405f38?action=download&direct&version=3)) were largely skeptical of Peters.

So there is the original paper and the subsequent exchange with Doctor et al., and a couple of other responses. There must also have been some reactions of economists in, say, blogs and other spaces, given the publicity the original paper has received. **Do we have a consensus among economists regarding the merit (relevance, validity) of Peters' critique?**

*This is not a question of opinion. I am trying to objectively gauge the consensus in the profession, i.e. has the matter become clear to most and has the majority opinion converged on anything concrete. References to back up an answer would be most appreciated.*

The question is motivated by a discussion in the comments of ["Why are utility functions typically assumed to be concave?"](https://economics.stackexchange.com/questions/47066)

**References**

* username_8, L. (2021). [Ergodicity in Economics: a Decision Theoretic
  Evaluation](https://osf.io/download/609277df6801ab03e42a3fee/).
* Doctor, J. N., Wakker, P. P., & Wang, T. V. (2020). [Economists’ views on the ergodicity problem](https://www.nature.com/articles/s41567-020-01106-x). *Nature Physics, 16*(12), 1168-1168.
* Kim, M. (undated, a). [A comment on ergodicity economics](https://files.osf.io/v1/resources/7gm8a/providers/osfstorage/5debad72b1bf4c000a405f38?action=download&version=1&direct).
* Kim, M. (undated, b). [A letter to economists and physicists: on ergodicity economics](https://files.osf.io/v1/resources/7gm8a/providers/osfstorage/5debad72b1bf4c000a405f38?action=download&direct&version=3).
* Peters, O. (2019). [The ergodicity problem in economics](https://www.nature.com/articles/s41567-019-0732-0). *Nature Physics, 15*(12), 1216-1221.
* Peters, O. (2020). [Reply to: Economists’ views on the ergodicity problem](https://www.nature.com/articles/s41567-020-01108-9). *Nature Physics, 16*(12), 1169-1169.

**Tags:** expected-utility, soft-question

> **Comments:**
> **username_1**: I've heard economists mention it as an example of utter trash coming from econophysics. I doubt you get more "consensus" on this, clearly the author is ignorant of even the most basic decision theory.  
> **username_0**: @username_1, thank you! Links to such examples would be appreciated.  
> **username_1**: This is a paper in a physics journal stating nonsense about economics without even quoting any economic research. I doubt you will find much public information. Time is scarce.  
> **username_0**: @username_1, I understand that, though I think the publicity Peters (2019) received is a bit of a game changer. I think it has made commenting on the paper / the matter attractive enough for at least some respected economists. But then I might be wrong.  
> **username_2**: I still haven't met any economit who takes Peters seriously. You can take this as evidence for economists' arrogance if you want to.  
> **username_3**: Overblown claims of this kind are almost surely incorrect.  
> **username_4**: Not sure it is accurate to characterise the second Kim paper as 'sceptical' of Peters - he writes: "Many people, having studied graduate-level introductory microeconomics, would have initially responded with immediate skepticism when they read [1]. I initially felt the same way, but after correspondence and some reflections, I believe I now understand where Dr. Peters and some of those following ergodicity economics come from."  
> **username_0**: @username_4, my impression was that even tough Peters' ideas might deserve immediate criticism, Kim did not want to attack Peters personally and was trying to be kind in the paper he wrote. A very reasonable approach IMHO.  
> **username_5**: Just stumbled upon this [relatively recent paper by Toda](https://arxiv.org/abs/2306.03275), adding to the so-called consensus.  

---

## 4 answers

### Answer by username_10 on 2021-08-03. Score: 12

Well there is no opinion poll among economists on specifically this problem, but what can be judged from reaction of economists the consensus is that the Ole Peters paper is misguided and irrelevant at best. I think the economists' consensus was already very well and succinctly summed up by Doctor, Wakker and Wang you cite (and is an example of [this](https://www.smbc-comics.com/comics/20120321.gif) phenomena).

For starters, on twitter R. Thaler (Nobel Prize) called [it hogwash](https://twitter.com/r_thaler/status/1027592614688653315?lang=en), if a Nobel Prize winner for work on decision making and behavioral economics so readily dismisses your idea about how humans make decision under uncertainity its a red flag. This work was also criticized by other notable economists such as [Farmer](https://www.rogerfarmer.com/rogerfarmerblog/2020/5/6/the-peters-paradox).

However, even more can be judged by the non-reaction of economists. The Ole Peters paper was so widely circulated that it is safe to assume that majority of economists know about (I would bet that if you will ask at your university department about "that ergodicity paper" most of them will know what you are talking about). It is unbelievable but this paper got so much free press that it should an case study in marketing (it was covered and *uncritically* cited by major media outlets such as [Bloomberg](https://www.bloomberg.com/news/articles/2020-12-11/everything-we-ve-learned-about-modern-economic-theory-is-wrong) and it even got a [TED talk](https://www.youtube.com/watch?v=LGqOH3sYmQA&t=241s)).

So it is safe to assume at very least most economists are aware of the Peter's work existence. Yet, if you look at which articles cite the Peters work, you will see most are either A) criticisms, B) not even in the field of economics and C) save for the criticisms most are not published in any reputable journal.

The Peters work is now already 2 years old, given that it is so widely known, and given that the work basically claims that our whole expected utility framework even in its general and behavioral applications is both wrong and not useful, you would expect that people would jump at the idea and start widely applying it, or at least testing it.

This is because expected utility is widely used workhorse model, and even though one can criticize it on a behavioral grounds it remains useful, the same way as Newtonian Physics, remains useful in presence of general relativity. In the areas where it is not useful we have behavioral models that still build upon the idea of expected utility or alternative theories that do not require ergodicity (e.g. like prospect theory etc., see Kahneman Thinking Fast and Slow for discussion).

Yet we do not see people at mass abandoning either expected utility or other more generalized concepts amass. Rather the paper was met by a silence occasionally interrupted by cricket's chirp.

Now either there is some conspiracy going on in our profession, or simply most economists do not even consider the paper worth while to respond to or engage with. Given how large our profession is conspiracy is unlikely (given that likelihood of keeping conspiracy secret declines drastically with number of people involved in e.g. see work of [Grimes 2016](https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0147905) on this). So really the most straightforward explanation for the utter lack of influence of the paper on profession is that the general consensus is that it is not even worth discussing.

One could also argue that the idea is being suppressed by the 'old guard'. There are several historical examples of new ideas in a field being suppressed, *for some time*, by established scientists e.g. like [this example](https://en.wikipedia.org/wiki/Antoine_Lavoisier#Chemical_revolution_and_opposition). While it is not impossible that is happening to the ergodicity idea as well, one has to remember that for any good idea that is too eagerly suppressed by the 'old guard' there is always a large number of ideas that were dismissed by the old guard and actually also turned out to be bad or irrelevant. A good example are ideas like [EM drive](https://en.wikipedia.org/wiki/EmDrive) or think of all the 'theories' like [ancient astronaut theory](https://en.wikipedia.org/wiki/Ancient_astronauts). So while it could turn out that ergodicity is being suppressed, it is more likely that it is actually not.

Of course, an important caveat is that absence of evidence is not necessary evidence of absence. Ideally, you would want some poll among economists, or at least well known economists. This being, said the evidence and lack of thereof that we currently have strongly points toward the conclusion that the consensus is that Peters work is irrelevant.

> **Comments:**
> **username_11**: You write of Peters's paper "*the work basically claims that our whole expected utility framework is wrong (and expected utility is widely used workhorse model)*", but you are also referring to [Doctor et al.'s paper](https://personal.eur.nl/wakker/pdfspubld/20.3ergodic.econ.suppl.info.pdf), where they have a section titled "**5. Falsifications of EU Are Well-Known**". So is Peters's claim of EU being wrong incorrect or is it just that his claim is not now or not more convincing than other critiques of EU?  
> **username_11**: "*Now either there is some conspiracy going on in our profession, or simply most economists do not even consider the paper worth while to respond to or engage with.*" This does not seem to consider that EU experts have little incentive to admit if EU is not useful in models - rendering their previous work obsolete - even without coordination. I make no claims about EU, but your argument is not very insightful. As Peters also critices behavioralism, Thaler is similarly lacking in incentives to embrace him.  
> **username_10**: @username_11 1. As they explain in the chapter 5 the falsifications of EU they are referring to are falsifications of some particular specific cases of EU, in fact if you read any behavioral work they still use EU with behavioral twists. EU was not really falsified generally but specifically and the paper explains that in the section 5 you cite. So this is a moot point although I could probably rewrite it to be more nuanced. 2. I am sorry but I am unconvinced with that idea. Take simple prisoners dilemma. Sure two old guards could benefit from suppressing new ideas if they both do it but if one  
> **username_11**: 1. I have read section 5 and disagree with your characterization of it.  
> **username_10**: Deviates that one researcher can get all the citations from this new approach, so it’s not really stable equilibrium to have them suppress the idea and in Nash equilibrium they will both deviate. Plus there are always many young and scrappy scientists always looking for new ideas  
> **username_11**: 2. So if there are two kings, one would collaborate with communists? Seems strange. (Perhaps not everything should/can be described by a two-player one-shot prisoners' dilemma.)  
> **username_11**: 2. Yes, young scientists could criticize EU. But perhaps the ones that do do not get published in good journals headed by the old guard?  
> **username_11**: For a case study of self interest in science I also recommend reading about [Lavoisier's oppisition](https://en.wikipedia.org/wiki/Antoine_Lavoisier#Chemical_revolution_and_opposition).  
> **username_10**: @username_11 1, Kings would have no incentive to do so since they would loos even if they are the only one collaborating. 2. Journal publication is not entirely in hand of editors and also ideas like the markets from lemon that were going contrary to mainstream understanding of markets managed to get published in good journals. 3. Yes I know that sometimes these things happen but you have to have sense of proportion, for every 1 idea that is *postponed* by the old guard (and these ideas are always just postponed as at some point the evidence will simply mount too much to be ignored)  
> **username_11**: So the devil is in the details, right? Then perhaps the simple examples don't explain that much, and 2 years is perhaps not too long?  
> **username_10**: @username_11 you did not gave me time to finish my point...     there are dozens if not hundreds of crazy ideas  that are actually nonsense and were correctly disregarded, you basically have to make claim that this is actually one of those ideas that were suppressed because of the old guard. For example, even now there are few heterodox biologists who argue that covid-19 vaccines are not as safe as assumed by mainstream biologists e.g. see [this](https://www.youtube.com/watch?v=81mEJ2Rph1Q). These concerns are (in my opinion correctly) being rejected by mainstream  
> **username_10**: biologists because even though we don't have any long term studies on effect of these covid vaccines given the evidence that we have as of now they are very likely safe and should be widely used. In the same fashion, given the evidence (or lack of thereof) we have now about this ergodic theory it is likely one of the nonsense ideas correctly rejected. However, I am open to changing my mind as new evidence is presented and revising my answer if in the future we see the idea pick up.  
> **username_11**: I think your point should be finished in *your answer*. I am not defending Peters's claims, I am criticizing the arguments presented in the answer.  
> **username_10**: @username_11 thanks for your critique, I tried to address the points in my last edit  
> **username_12**: @username_10 I think you pretty much hit the nail on the head. The only thing I might recommend adding are perhaps specific examples of nonEU theories being used (rank dependent utility being used to model insurance markets, for example, or Cumulative Prospect Theory being used to model the impacts of defaults and dominated options in choice lists). You could also make the analogy to Newtonian physics--we know conclusively that classical physics models are wrong, but they're really easy to use and model most "normal" situations with incredible accuracy.  
> **username_10**: @username_12 thanks for suggestions I edited my answer to try to incorporate them  
> **username_0**: Thanks for Thaler and Farmer quotes; this is what I was looking for. Btw, TED talks tend to include a fair share of ridiculous stuff; Andrew Gelman has covered a few such instances in his blog over the years, so at least for me TED is not a stamp of quality. Not sure if you meant it that way, but mentioning this just as a heads up.  
> **username_13**: that comic is *brilliant*  
> **username_14**: Funny you point to the EM Drive. Had one been hoisted to the ISS it would have worked despite the theory being bonkers. It's solved now; a magnetic drive. In deep space it would not have worked.  
> **username_11**: @username_13 Which comic...?  
> **username_10**: @username_11 I think he means [this one](https://www.smbc-comics.com/comics/20120321.gif) I linked at the beginning  
> **username_11**: Thanks! Not realizing you had already done so, I also linked to the same comic [at the other question](https://economics.stackexchange.com/questions/47066/why-are-utility-functions-typically-assumed-to-be-concave#comment75440_47069).  
> **username_10**: @username_11 great minds think alike ;)  
> **username_9**: Peter's work is mostly about clarification and historical work. A whole section weird economists appear to be vague or confused. I don't think real employed economists (i.e. quants) are confused about any of this. And they barely call themselves economists anyway.  So it's more about fights amongst academics.  I really appreciate Peter's work on Microfoundations of Discounting and consider the foundations of discounting still rather unsolved.  

---

### Answer by username_6 on 2021-08-05. Score: 9

Junior econ professor here. I saw Ole Peter's work and I was intrigued, so I actually looked into it to see if there was something original/insightful for me to learn. I even run a little simulation of what he calls the "St. Petersburg Paradox", to make sure I understood what the guy is actually talking about.

Just to be clear, I didn't spend days looking into this. I "only" spent a few hours, until I was confident that I had an informed opinion. I eventually came to the conclusion that his point has actually little to do with the ergodicity of stochastic processes, which is a well-defined concept that (i agree) most economists are probably not familiar with. However, most time-series econometricians are definitely familiar with it

<https://en.wikipedia.org/wiki/Ergodic_process>

His point has instead a lot to do with what's called Jensen's inequality, something that most economists learn by end of year 1 of grad school

[https://en.wikipedia.org/wiki/Jensen's\_inequality](https://en.wikipedia.org/wiki/Jensen%27s_inequality)

What settled it for me was going online, and finding out that a much more senior colleague of mine (who unlike me is a published expert on time series) had - completely independently - reached a similar conclusion:

<https://www.rogerfarmer.com/rogerfarmerblog/2020/5/6/the-peters-paradox>

So I initially thought "well the poor guy [Peters] is just being naive: he thinks he's made a discovery and literally he's just struggling with Jensen's inequality". That is until I saw his Ted Talk and looked into his background (he works at a Climate Change/Social Justice center).

Now I'm not so sure he's being naive. I think more likely he's muddying the waters on purpose. He is giving this "stuff" (calling it a theory is too much of a stretch in my view) the pompous name of "ergodicity" so that he can sound smart and go around pontificating about inequality, climate change etc. and argue for extreme policies that would be rejected by traditional economists, because (so goes his argument) they don't understand how randomness and time interact.

That's a very old trick: you come up with some pseudo-scientific mumbo-jumbo full of technical jargon from different disciplines that most people can't understand, and then you use that to put yourself forward as an "expert", and advance ideological positions as if they were based on hard science.

Now what to make of the fact that this "ergodicity" stuff got published in Nature Physics? Well we should note that in this case the editors went out of their way and published an article about economics, which is outside of their field of expertise - with predictable consequences. That is why we have specialized journals for different disciplines.

Now if you don't believe me and you suspect that we are biased because we are "traditional economists", of course that's a possibility. Then my encouragement is: do what I did. Go ahead and figure it out for yourself. Read his paper, read the two wiki articles I linked, pick up a statistics textbook. And if you do figure out something interesting out of this, come back and do let me know. I'm happy to be proven wrong.

> **Comments:**
> **username_0**: Thank you for a hands-on perspective.  
> **username_2**: The https://en.wikipedia.org/wiki/St._Petersburg_paradox was analyzed by Nicolas Bernoulli in 17-something and is thus much older than Ole Peters. Already back then he suggested utility functions and that is why these concave function $u$ that we use are called Bernoulli utility functions.  
> **username_4**: I find this answer very interesting, for a couple of reasons.  To begin with, Jensen’s inequality is usually taught to first-year maths undergrads, so it’s perhaps not as closely-guarded a secret of classically-trained economists as is hinted at here.  But secondly, your Wikipedia link about ergodicity itself acknowledges that econometrics and signal processing use a different definition of ergodicity (which uses a temporal sample) than that used in physics (which uses a spatial sample at a single instant).  Are you sure you’re talking about the same thing Peters is?  
> **username_6**: Never hinted at nor thought that Jensen's inequality was an economists' secret. With respect to the definition of ergodicity, my conclusion was not dependent a particular definition of ergodicity. My point in fact is that you don't need any notion of ergodicity (neither physical nor econometric) to explain the phenomenon that OP is concerned with.   And even if ergodicity had anything to do with it (remains to be proven) OP's unsubstantiated claim that, as a consequence of that, we should change the way we model income inequality in economics is a non-sequitur.  
> **username_7**: I somewhat doubt that @username_6 has actually read the paper, because it clearly states:  "That the geometric mean exp〈ln x〉 is less than the arithmetic mean 〈x〉 was known to Euclid (Elements, Book V, Proposition 25), and it is a special case of Jensen’s inequality of 1906. Its connection to gambling and investment problems was noted by Whitworth in 1870, is implied by Itô’s work of 1944, and is well known among gamblers as Kelly’s criterion16 of 1956. Our modest contribution is to frame these observations as a question of ergodicity, which we have found to be a fruitful perspective."  

---

### Answer by username_8 on 2021-08-11. Score: 8

I’am the guy who wrote the short note that was mentioned in the original question.
You can reach the note here.

<https://osf.io/preprints/socarxiv/axkfg/>

I got interested in Peter's paper because of my interest in evolutionary games, where the question of whether the equilibrium selection process is or is not ergodic plays a crucial role. I agree that most economists find Peters' contribution irrelevant. However, by reading Peters’ original paper and Wakker e.a. reaction to it, a casual reader may get the impression that Peters has a point after all, but that he tends to oversell his results. The same impression one gets by reading other informal contributions written online by economists, including the present discussion.

In my note, I provide a pedantic presentation of Peters' running example and prove a very simple proposition, which, if correct, may put the entire matter at rest. This proposition says that in Peters’ example no subject who obeys the axioms of EU would accept the lottery unless he has an unbounded utility for money. In this respect, Peters' example is just a more involved instance of the St. Petersburg paradox and can be solved the way K. Menger solved the original one in 1934: the paradox only arises because we let utility for money be unbounded. Notice that it is easy to be fooled on this matter: risk-aversion will not do the trick, just as Menger proved in 1934.

The reactions to my result have been mixed. O. Peters was extremely polite, but he said that he receives way too many contributions to ergodicity economics to evaluate them all. No follower of ergodicity economics has shown any interest.

I had a brief mail exchange with P. Wakker, and he alerted me about the difference of opinion between K. Arrow (who believed unbounded utility to be a violation of EU) and P. Samuelson (who held the opposite view). In the end, he convinced me to side with Samuelson, although I still think Arrow's position may be defended. In any case, my result proves that Peters provided a very small contribution to a literature that in economics is half a century old. (In fact, you can find Samuelson’s view on Peters’ example in a PNAS paper published in 1971.)

I had some encouragement from some respected decision theorists, but most economists were not interested. From a prominent game theorist, I got a reply that rang something like: “nice work, but I have no patience for this matter”.
Finally, the note has been downloaded more than 100 times, but none has sent me an email for a comment or for any other reason. I concluded that the profession is so busy that even a moderately complex result as mine is seen as overkill for ergodicity economics.

Anyway, any comment is welcomed!

username_8

PS If you have a bent for philosophical digressions, my note also discusses Peters’ claim that, when facing a compound lottery, EU assumes that a decision-maker experiences all alternative histories. In this case, Peters’ argument collides with the fact that EU is based on the independence axiom, which forces a DM to consider only the branch of the tree she happens to visit. In fact, just the opposite of what Peters claims is true.

> **Comments:**
> **username_0**: How cool it is to hear from the author himself (I mean you)! Thank you so much! I would love to be able to contribute to the debate, but I only have rudimentary knowledge of decision theory, game theory and ergodicity. I nevertheless find the topic interesting and your paper helpful and illuminating!  
> **username_9**: Have a look at https://www.amazon.co.uk/Prediction-Learning-Games-Nicolo-Cesa-Bianchi/dp/0521841089 and I think Chapter 9. Universal portfolios etc. Worst case learning bounds. This stuff is quite fundamental and economists have no idea. I have yet to see a derivation of DCF from a control/decision/RL framework point of view. Would be interested if anyone knows.  
> **username_8**: Not sure I understand this comment. That book is a very long exposition of topics developed in the 50s by obscure scientists working on the then-new field of game theory (Hannan and Blackwell mostly). The relevance of these results was discovered decades later with fundamental papers published (in top journals in Econ) by Adreeu-Mas Collell and Sergiu Hart, Fudenberg and Levine, and a few others.  Maybe economists do have a clue.  

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### Answer by username_7 on 2024-12-26. Score: 0

The final consensus is probably that there is no consensus. But most people have misunderstood the proposition of Ergodicity Economics (EE) anyway. I even wonder if these people have read Peter's paper at all, because it clearly states the following rather at the end of the paper (the bold formatting is mine):

> That the geometric mean exp〈ln x〉 is less than the arithmetic mean 〈x〉 was known to Euclid (Elements, Book V, Proposition 25), and it is a special case of Jensen’s inequality of 1906. Its connection to gambling and investment problems was noted by Whitworth in 1870, is implied by Itô’s work of 1944, and is well known among gamblers as Kelly’s criterion of 1956. **Our modest contribution is to frame these observations as a question of ergodicity, which we have found to be a fruitful perspective**. It enforces physical realism by precluding interactions among members of a statistical ensemble, it enables us to consider dynamics other than additive and multiplicative (corresponding to linear and logarithmic utility functions), and
> it naturally leads to treatments of problems whose solutions are less readily visible in previous framings of the issue.

Peter's message in his paper is nothing fundamentally new and is also not claimed to be. He is well aware that the basic concepts on which EE is based have been known for a long time. In a sense, his paper formalizes old knowledge, which is useful (also for economics) as it enables a more mathematically rigorous treatment of the matter. His emphasis is on the fact that decision-making over time typically differs for an individual and a group of independent individuals (depending on the (non)-ergodicity of the problem under consideration). While the latter on aggregate operates in the regime of large numbers, the former does not. In particular, individuals face only a single path over time, which is why probability distributions (or their derivatives in the form of expected values or higher moments) typically don't matter to them in contrast to time-average growth rates. As Peter's work reveals, this fact was fixed by economics in the view of homo oeconomicus by introducing a logarithmic utility function which transforms the maximization of the rate of change of expected utility into the maximization of a time-series growth rate in the case of multiplicative dynamics (e.g. compound growth).

In summary, EE has not developed anything new. It has helped us to formalize and better understand the key differences of decision-making between an individual and a group of independent individuals. Both perspectives are legitimate perspectives and neither of them is right or wrong in the first place. When to use which perspective ultimately depends on the topic or perspective in question. But this insight itself is an important gain in knowledge, formalized by the ergodicity question and rigorously linked to the underlying dynamics (instead of putting forward obscure psychological arguments for rational decision-making).

> **Comments:**
> **username_15**: I am not intending to start another long discussion here but I was just wondering if this Peter's ergodicity discussion is at all connected to the 2 envelope problem here. In this problem, if one uses log utility, then the paradox dissappears. Thanks. https://en.wikipedia.org/wiki/Two_envelopes_problem  
> **username_7**: The implication of the ergodicity question comes by many names or paradoxes. Yes, the two envelopes problem is another one to be resolved by the considerations outlined above. Only a group of independent individuals on aggregate will be able to realize the ensemble average over time, which is why from such a group perspective the gamble would be benefitial. In contrast, for an individual who plays the game repeatedly over time in a compounded way, the time-average growth rate is relevant. Log utility converts between the two views, as it renders multiplicative dynamics into additive dynamics.  
> **username_15**: Hi  username_7. That's a nice explanation. That envelope problem always boggled my mind so atleast now I have a thread to go to if I want to understand it further. I always found it interesting that the use of the log utility removed the paradox !!!!! Thanks for your help.  

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