The Ergodicity Problem in Economics
nature.com
nature.com
The author seems to miss that "economics" is a vast field spanning the whole spectrum from applied economics, over theoretical economics, mathematical finance, financial mathematics, to pure mathematics (with physicists working along the whole spectrum, so this is not a consequence of the author's background per se).
He fails to engage at the right level. From a theoretical mathematician's point of view all models are wrong - they are just deductions from assumptions. From an applied economist's point of view are models are right - they explain some observed phenomena.
Ergodicity is not a niche topic, most intermediate courses on stochastic processes will cover it. Will loosening an assumption about the properties of stochastic processes yield different, potentially better models? Maybe. Will it lead to a revolution in economic theory? Unlikely.
Again, odd that this has been peer-reviewed.
See also the reply here, which is rather damning [1]
And the author's reply to the reply. [2]
They argue that Peters' model produces extreme risk aversion in lotteries with (near) zero payoffs, and such risk aversion is not backed by empirical work or even by intuition.
"Would a person ever prefer a process [A] that, after three rounds, diminishes wealth from US$10,000 to 0.5 cents over one [B] that yields a 99.9% chance of US$10,000,000 and otherwise US$0? Ergodic theory predicts [A] [... but] [v]irtually everyone will prefer B"
That is, growth models don't behave well at or near zero.
Personally, I enjoyed reading Peters solution to the St. Petersburg paradox and how the ergodicity framework is applied to economics in such a concise and intuitive manner.
But I am reluctant to think that this ergodicity framework will completely remove the "psychological" aspects of economics. The utility of agents still needs to be accounted for. The ensembles that Peters describes are heterogeneous, and agents can derive different amounts of utility even if wealth or growth rates are equal. Think about insurance.
https://en.m.wikipedia.org/wiki/Kelly_criterion
(2) Expected utility theory makes no assumptions about ergodicity. In intertemporal setting, portfolio allocations maximising expected utility will chose something close to their metric as well as giving some insight into the split between investment and consumption.
https://en.m.wikipedia.org/wiki/Intertemporal_choice
(3) with intertemporal analysis, the rate of discounting is important. This has been heavily discussed by economists and philosophers already. If you don't have heavy discounting, the future becomes much more important relative to the present than would be consistent with many people and government's actions.
Poker players adopt a utility function similar to u(x)=x and apply some very basic risk management heuristics (bankroll management) on top in order to handle ergodicity. Their whole thought process is centred on E[u(x)] (E[x]), and they have much more affinity for expected utility theory than the more "correct" Kelly.
Professional investors adopt a slightly risk-averse u(x) (not quite ln(x) in my experience) and do just fine with that. Risk of ruin is managed again using simple practical heuristics, such as capping the maximum downside on each decision, and sizing up only when confident (in Kelly-like fashion - but very subjective).
That it doesn't explicitly handle ergodicity isn't a huge flaw in practice. It's still a useful (although imperfect) mental model of what's going on in people's brains and offers some practical usefulness.
It's also not a wrong assumption technically, because it's not claiming to model such phenomena, its claim is to model the utility of a single discrete choice.
EUT + Hacks works but anything built upon of EUT is subject to sudden collapse due to unstable foundations.
Ergonomic economics is able putting down stable foundations that other theories can be built upon of.
Kelly, for example, is fatally flawed because it ignores utility:
(1) Ruin isn't an absorbing state in most contexts, due to bankrupticy laws, limited liability, relationships, and so on. Ruin is an absorbing state in other contexts. Only utility can capture this difference.
(2) Implementing Kelly requires the emotional fortitude of a rock. In practice, it becomes a losing strategy since the practitioner will collapse emotionally and make bad decisions whenever the distribution of the outcome is unknown.
(3) Implementing Kelly forgets that investors will redeem capital if the downswing is too severe for them.
(4) etc.
Only a conceptual framework built around expected utility as a bedrock principle can handle these practical realities, and moreover capture those practical realities empirically.
Kellys is fatally flawed as it's merely one of a whole class of functions that be used, where Kellys happens to be on the aggressive side of things.
Ole Peters gives the entire class of the functions that can be used. There is obviously a whole band of less to more aggressive options you can take.
Do you say this because EUT currently doesn't handle ergodicity? Why do those two things need to be mutually exclusive?
> whole band of less to more aggressive options
Right, and which option I pick will be based on utility preference considerations both over the lifecycle of the decision making process as well as over the terminal outcome's distribution.
Expected utility is still inextricably part of this.
Older people are going to choose low aggression options on their retirement portfolio because the utility consequences of ruin are much different to a 20 year old's.
Expected utility is the point. Handling ergocidity just tells us how to get there correctly.
"Why do those two things need to be mutually exclusive?" EUT is basically the formula of ergodicity with slight changes. It doesn't retrofit.
"their retirement portfolio" The ergodicity economic formulas generate slightly more money on average than their regular economics counter parts. [You can see where this is going...]
We're talking about a theory of humans' (financial) decision making.
Old people make less money than young people on purpose because of expected utility preferences.
Ignoring expected utility is therefore automatically fatal and a non-starter, as this single example demonstrates.
You need to prove this whilst keeping in mind that ergodicity economics takes account of factors that regular economics does not. You would need to show it wasn't one of those factors being responsible.
Ole Peters has already done the studies on people (outsourced to a psychology department) to show they are following his economics model.
"You need to prove this "
"You would need to show it wasn't one of those factors"
Secondly, you're misunderstanding my objection. What I'm objecting to are the very conceptual foundations of the theory. Old people demonstrably accept a lower EV than young people because of a difference in expected utility over the distribution of near-term outcomes. To throw the concept of expected utility in the bin is therefore a departure from reality and as such the theory automatically fails on conceptual grounds.
You can look all through the comment section, you won't find anyone claiming the maths is wrong.
Ergodicty economics disproves EUT. EUT is rejected on solid mathematical grounds.
Saying Expected Utility is correct is no different than claiming that 2 + 2 = 5. Though the maths involved is a bit more complicated.
The math of EE is correct given the axioms from which it is deduced. Nobody anywhere is disputing that.
What I'm disputing is whether it is a theory that explains human financial decision making, in the same way that some physicists dispute that string theory explains physical reality (despite acknowledging that the math behind string theory is deductively correct).
"Saying Expected Utility is correct"
Once again - the math IS deductively correct, but that same math fails as a theory of financial decision making since it doesn't explain the observed reality.
It's the other way around EUT fails as a theory of financial decision making since it doesn't explain the observed reality.
Without this concept, tell me how EE is supposed to grapple with:
(1) gamblers who take on negative EV bets
(2) old people who shift into fixed income
(3) low risk-tolerance young people who keep only cash
(4) high risk-tolerance young people who put everything into crypto
(5) why some people buy insurance and some don't, despite earning the same income
The fact is, you can't explain this heterogeneous behaviour without the concept of expected utility of outcomes.
Our brains are emotional, irrational vehicles designed by evo psych, and you can't grapple with that reality without some notion of subjective preference pertaining to expected outcome.
EE shows that EUT only gives correct results when U = log (Wealth), that means as soon as you set U to anything other than log (Wealth), it is no longer giving correct results.
So it would be fair to say EUT also has no concept of utility.
Talking about portfolio selection, for example, the EE - a.k.a. U=log(wealth) - solution may be the "correct solution" to the "we never spend a dollar problem and we have an infinite horizon" problem.
But EE cannot get any results, correct or otherwise, for many other problems that are much more interesting where EUT can be applied.
Like investment decisions when your horizon is not infinite and you intend to use the money at some point.
There's no such thing as a "correct result" because people's preferences (utility) varies by individual.
What's "correct" for a risk-seeking gambler is very different to what's "correct" for an investor who's trying to build generational wealth.
That's why we have a U(x) to begin with. Without addressing this concept, you're no longer attempting to describe reality, you're making a prescriptive normative assertion that everyone should follow a specific strategy of your choosing.
You seem to think that this invalidates EUT.
On the contrary, it's a vindication of EUT.
In that particular case, EUT works and the preferences of the agent would be correctly described by that particular utility function. Otherwise you wouldn't say that EE and U=log(w) give "correct results".
It can also happen in other cases that EE cannot be used to explain the preferences of the agent while EUT is still applicable because a utility function (maybe logarithmic, maybe not) can be found which describes them adequately.
But there are incorrect results and setting U to anything other than log (Wealth) results in an incorrect result.
"It can also happen in other cases that EE cannot be used to explain the preferences of the agent while EUT is still applicable because a utility function (maybe logarithmic, maybe not) can be found which describes them adequately."
I just told you that you can not use an utility function other than log (Wealth). Any other utility function you use will give you a mathematically incorrect result. The log (Wealth) term covers up the maths error, so it can't be changed to another term.
If you want to do something like that then you need to use the maths from EE.
"Any other utility function you use will give you a mathematically incorrect result."
The role of the utility function in EUT is to represent the agents preferences.
Preferences can be rational (i.e. consistent) and not be represented by the logarithm of wealth.
If Mr. X has some amount to invest now to pay for his child’s college in five years it’s not “irrational” to opt for something less risky than taking a loan to get a leveraged equity investment.
Mr. X may not care that his portfolio wouldn’t growth at the optimal highest possible rate if left untouched forever, if that’s what you mean by “mathematically incorrect result”.
Mr. X doesn’t care about your idea of “correct result”, he cares about being reasonably certain to have enough money available in five years.
Now, you tell me to use the maths from EE to find the “correct” utility functions.
How can I use the maths from EE to select a portfolio if I want to take out a certain amount of money in five years?
1) It seems to me that it is only utility, rather than its expectation, that is the concept you are treating as necessary. There are an infinity of ways to reduce a distribution of utility-weighted outcomes to a single summary, albeit not with the same simplicity (and perhaps value) as the expectation.
2) All of the phenomena you list could be described by some mechanism other than the agents involved computing expected utilities - whether or not this is a useful or effective description is beside the point, it is possible. (Expected) Utility is not required to describe these phenomena.
3) The basic EE claim is that the ergodic hypothesis, roughly that the temporal and ensemble distributions are the same, is false in the context of these economic systems. This has nothing to do with whether or not you can associate utility values with states, nor whether it is possible to compute expected utilities, but instead is a claim about how, and from where, those utilities should be measured, in particular when considering problems like optimising long-term returns.
3) Ok. Some people seem to think that EE disproves EUT somehow, though. That's the context of the comment your reply to.
More generally, there are two aspects of the value of EUT being discussed:
1) Does expected utility theory describe observed economic behaviour. There is evidence that it does not, and my previous points concern that fact. 2) Can expected utility theory be used to design a system that will produce optimal outcomes. EE confronts this question, and claims that, in the usual formulation of EUT, it cannot (because the ergodic hypothesis does not apply).
As others have mentioned, in practice some people do account for non-ergodic behaviour. Others, however, do not, and being explicit about the limitations of any given model rarely hurts anything except people's egos.
I guess it's true that one can also have economic theories which are not compatible with rational decision making as understood in EUT so they could be completely different.
> claims that, in the usual formulation of EUT, it cannot
EE claims that, but it's a baseless claim. The growth optimization arguments used by EE can be perfectly used (and have been used) in the usual formulation of EUT. If the agent has a preference for growth that can be described with a utility function.
(I agree that EUT doesn't explain all behaviour. EE even less, being even more restrictive.)
The disagreement seems to boil down to what is considered the "usual" formulation of EUT. My understanding of the basic formulation is in accordance with that in the paper, namely that one typically assumes single-period uncertainties either explicitly or in effect (e.g. assume they are IID), and time is treated by discounting - but I admit I am no expert.
That it is possible to extend that formulation is, I think, not in doubt, but we should be able to agree that what I have described above does, implicitly, make an ergodic assumption, and thus the EE critique would apply.
You may contest my description of what the basic formulation is, and, as I suggested, lack of clarity about that does seem to be driving a lot of the discussion.
These discussions are typically made more problematic by the fact that practitioners often use more advanced methods than the basic theory, to overcome such problems whilst remaining within the same broad intellectual frame. It seems to me that the claim of EE is that the basic theory itself should be replaced because it fails to account for many important real-world phenomena, so that even the "what is the basic formulation" question would become moot.
EUT is a theory of decision making under uncertainty. If an agent preferences are rational (as in they verify a number of properties) his preferences can be described assigning a number to each outcome. If the outcome is uncertain, the utility is the weighted average of the outcomes utilities.
For example, for outcomes A, B, C and D there will be four numbers U(A), U(B), U(C), U(D) such that
if U(A)>U(B) the agent prefers A to B
if the agent is indifferent between C and D
Say that A is "in the beach, it's sunny", B is "in the beach, it's raining", C is "at home, it's sunny" and D is "at home, it's raining".
With the equations above, if I'm in the beach I prefer that it's sunny. If I'm at home, I'm indifferent to rain.
Let's make a couple of additional assumptions U(A)>U(C) and U(B)<U(D). If it's sunny, I prefer to be at the beach. If it's raining, I prefer to be at home.
Let's say that my preferences are described by the following values: U(A)=10, U(B)=-20 and U(C)=U(D)=0.
If the probability of rain tomorrow is 50% do I prefer to go to the beach or to stay at home?
EUT allows me to calculate U(beach)=0.5 U(A)+0.5 U(B)=-5 and U(home)=0 (it doesn't depend on the weather). I prefer to stay at home.
What is the probability of rain that makes me I'm indifferent between going to the beach or staying at home?
U(beach)=(1-x) U(A)+x U(B)=10-30 x = U(home)=0 => x=1/3
__ Remarks __
The probability doesn't have to be "right" for the theory to work. It only has to be a faithful description of the expectations of the agent. If I believe that the chance of rain is higher than 1 in 3 is rational that I stay at home.
Would Ole Peters say that this use of probabilities to make decisions is incorrect because it assumes that I'm interacting with a copy of myself in parallel universe? I suspect so.
Would you say that there is a problem with EUT up to this point?
Imagine that the outcomes are different levels of your wealth at the end of the year. For example A is $500k, B is $50k, C is $200k. What is the utility function that describes your preferences? (There should be one if you're rational in the sense of the EUT axioms.)
Surely we agree that you prefer A to C and C to B.
But would you prefer to have $500k or $50k with 50%/50% probability or to be certain of having $200k at the end of the year?
EUT doesn't give you an answer. Of course it's easy to calculate the expected value of the first alternative ($275k) but nobody says that you should prefer the "risky" $275k to the certain $200k. That's the point of introducing utility functions in the analysis, that preferences may be non-linear.
What EUT says is that if you're rational (etc) there are three numbers U($500k), U($50k) and U($275k). The expected utility of the first option is 0.5 U($500k) + 0.5 U($50k) and the expected utility of the second option is U($200k).
These expected utilities can be compared to see if you which option do you prefer. If you prefer one, you could also be indifferent.
EUT says that the rational decision is to chose the option with higher expected utility.
__ Remarks __
EUT doesn't fix the form of the utility function. Usually it's assumed that instead of linear it's concave. This means that a certain dollar is better than a dollar in expectation and you would never take fair bets.
(The fact that people plays in the casino where bets are not even fair could be explained with convex utility. One could explain it as well saying that there is utility obtained from the entertainment that offsets the monetary loss.)
The utility function describing someone's preferences could be very complex. Simple models are used for many reasons including tractability or theoretical properties. For example power functions (logarithmic utility is a special case).
There are some reasons to like logarithmic utility (including growth-optimality arguments) but it doesn't work too well empirically. This is why, at least in some settings, it may be better to use the more general power utilities that have a parameter that can be adjusted.
Using logarithmic utility, the $200k option is preferable to the $500k/$50k bet. The certainty equivalent is $158k.
Logarithmic utility (like the rest of the power utilities) is scale invariant. The amounts could by 100 times larger or 100 times smaller and the answer would be the same.
Does Ole Peters claim that the use of EUT here is wrong? I guess so. Parallel universes again, probably.
Would you say that there is a problem with EUT up to this point?
What are the problematic assumptions made by EUT? In my opinion it's EE who cannot really solve this problem without making "unphysical" assumptions. The solution corresponds to using the logarithmic utility above, EE declares any other preference the agent may have wrong.
(To be continued, maybe.)
> Would you say that there is a problem with EUT up to this point?
Is no, not for this problem, and question.
However, consider figure 2 (and correspondingly equation 2) from the paper. It considers a problem very similar to the one you pose, but with a couple of important differences:
1. We maintain a stock of "utility", and are aiming to maximise this stock over multiple iterations of the bet.
2. The utility response is multiplicative rather than additive.
If we consider the expected utility of taking the bet at any point in time given some wealth W we obtain a positive utility (0.5 * 1.5W + 0.5 * 0.6W = 1.05W). However, multiplicative wealth is not an ergodic process, and so the expected change over time does not reflect that expected utility, and after T timesteps is something like (1.5^(0.5T) * 0.6^(0.5T)).
Of course, as mentioned in the paper, some processes are ergodic - in particular changing the utility response to be additive rather than multiplicative. Overall, the point is that one must consider the nature of the problem carefully.
> Would Ole Peters say that this use of probabilities to make decisions is incorrect because it assumes that I'm interacting with a copy of myself in parallel universe?
I think the answer to this is no - not in the problem as you have framed it. However, if we extend it in the way I have outline above then we clearly need to be more careful.
Is says “a simple gamble”. It doesn’t say anything about an infinite series of iterations of the bet. It’s a single step problem. Do you play once or do you pass?
(If the question is “do you want to play twice (or N times)” it’s also effectively a single-period problem. One just has to consider the distribution of outcomes after two (or N) rounds.)
The usual EUT resolution is what I just described, which you don’t find problematic. He does find it problematic, because for him calculating an expectation is interacting with a copy of yourself in a parallel universe or something.
The reason why he talks about infinite sequences of games is not because the problem is about an infinite sequence of games. To solve the simple problem he has to hypothesize that there is an infinite sequence of them.
original comment:
Yes, you are correct that in the initial framing it is just a single gamble, but of course the point is that an individual's life is made up of many gambles. Expected utility assures us that this bet has a positive expectation, and so naively we might think that iterating it also produces a positive expectation sequence, but it does not, as we saw.
We can recover a correct answer by considering the expected utility of the whole sequence, but there is nothing in the problem to suggest that we should do so, unless we acknowledge the cause of the issue, which is that the ergodic hypothesis does not hold.
The point of the whole "parallel universe" thing is that even though the expectation in a single step may be positive, an individual never realises that ensemble average - they only ever realise a time average. Thus, the time average is the more useful object of study.
No. That is not the expected utility in the usual EUT solution.
In the usual EUT solution (with the usual logarithmic utility that would be used in a textbook problem like this one) the expected utility is 0.5 log(1.5) + 0.5 log(0.6) + log(W) which is lower than the expected utility of not playing log(W).
I read this paper when it was on here previously, and only skimmed choice parts again this time - rereading some more of it I can see that part of his point is that the introduction of utility, and its loose association with psychology and preference, was required to explain why you need to put a logarithm (or something similar) into the expectation in order to generate correct results, and not, as I was wrongly suggesting, that EUT generates incorrect results for problems like these. His claim is that you can recover identical behaviour, without needing to add arbitrary preference functions, by considering the time-average behaviour. Furthermore, as I suggested in another comment, this actually makes sense, because this is what an individual agent experiences.
I wrote it that way to make the comparison with the alternative log(W) more obvious.
Using EUT with a logarithmic utility function is not more arbitrary than that. Assume that people look to maximize asymptotic growth under repeated bets and you know that the logarithmic utility function is what you need.
And as a bonus EUT can be applied in many cases where people don’t care just about the asymptotic growth rate!
I think that the choice of a logarithmic utility function is more arbitrary - do you know what the explanation for it is, other than it fits with observed behaviour.
Another point is that the "(non-)ergodic theory" neatly explains why behaviour would change given different expectations about repetition. If you only had one bet left in your life then using logarithmic utility would produce what I would argue would be "incorrect" results for the equation 2 bet - I think that the rational choice would be to bet, because you don't really have anything to lose. It is only with iteration that losing starts to factor in. In EUT this would be explained with a change in preferences - but the point is that we don't need these additional mechanisms, it all falls out of the dynamics of the problem.
To your latter point, about the broader applicability of EUT - whilst I agree it is very convenient, fundamentally it doesn't seem that insightful that by choosing different scoring functions, and taking expectations over distributions of scores, we can recover all kinds of behaviours that might be of interest.
More explicitly, EUT doesn't really seem to tell us much - we can find a function that gives us any desired behaviour, sure, but it doesn't tell us why that behaviour is expected. As far as I know this is indeed where the theory stops, and the behaviours get effectively "written off" as preferences. A theory that explains the same behaviours without requiring these additional choices is surely preferable?
EE tells you that using the logarithmic utility is equivalent. The article mentions “the correspondences between linear utility and additive dynamics; and between logarithmic utility and multiplicative dynamics”.
Assuming that we want to maximise growth over whatever our horizon is (be it one period, multiple periods, or an infinity of periods) is not much of an assumption - what other realistic goal would there be?
Your earlier point that EUT can be applied in other situations still holds, but I think that is a consequence of the fact that it is so flexible that it can be fit to all manner of situations.
With EE you recover different behaviours depending on the structure of the problem, both the reward structure and the temporal structure (e.g. number of periods), whereas with EUT you have to inject different utility functions to recover the desired behaviours for any given problem - they don't just fall out of the structure.
What is the EUT answer to the problem I posed in the previous comment - consider the same equation 2 bet we have been discussing, but in both the iterated case and the single-period case. Is there a utility function that correctly solves both cases?
Don’t you ever spend any money? Is your only goal really to maximize the growth rate of your wealth in the infinite-time limit?
In EUT some of the response to changes in situation is modelled as a change of preferences, and encoded in the utility function - for example, older people may have different preferences to younger people, and those who have a lot of "good fortune" (perhaps through a good network) may have different preferences to those with few good opportunities.
This is fine, and it produces good results, but a more satisfying theory would be able to derive the behaviours from the problem itself. One claim is that this is not possible - that preferences are a fundamental primitive, but it isn't obvious to me that this is necessarily true.
To be clear, I would distinguish between irrational behaviours caused by lack of information, poor estimation or analysis, etc. and differences in rational behaviour caused by problem structure - here I am only considering the latter.
See my other comment about prescriptive vs. descriptive. You can postulate some utility and derive the theoretical behaviour. You can observe some behaviour and try to infer the utility that would be consistent with it (assuming that the behaviour is rational, for some definition of rationality).
EE can only do the former. Facing the same problem, people is not allowed to have different preferences. EE prescribes what rational behaviour is.
EUT and EE are not incompatible. When solving an optimization problem EUT can use the same arguments that EE does to find the utility function to use. I don't think these results are new, but even if they were they wouldn't invalidate EUT.
The point is that in the "prescriptive" setting the EUT framework can be used with whatever utility function makes sense for the problem. The basic theory doesn't tell you either what's the utility of inundating a valley to build a dam, or the utility of a floods if you don't. When you use EUT to find a solution to a problem you have to think about the problem.
If the problems are different a different utility function may be appropriate. (Nobody says “don’t look at the structure of the problem”.)
> With EE you recover different behaviours depending on the structure of the problem, both the reward structure and the temporal structure (e.g. number of periods)
How so? EE works only with an infinite number of periods.
The claim of the paper is that you can derive an appropriate rational solution for each problem using a single technique.
> How so? EE works only with an infinite number of periods.
Afaik this is not correct - EE does not require the problem to have an infinite number of periods, rather it is saying that you cannot assume that the temporal integral is equal to the ensemble integral, and that you must act accordingly. You have to (in principle) integrate the whole time series of interest - this can be done over your uncertainty, but requires no utility function, a simple expectation will do.
Can you derive a rational solution for this problem using that technique?
"An investor must choose how much to consume and must allocate his wealth between stocks and a risk-free asset"
https://en.wikipedia.org/wiki/Merton%27s_portfolio_problem
In the problems that EE has actually solved the solution is known and is derived using similar arguments.
Note that EUT has two main uses, descriptive (how do people behave?) and normative (what should you do in this situation?). Descriptive: you try to find empirically a utility function that describes people's preferences when facing decisions.
Normative: you look at the problem (whether it's portfolio selection or deciding where to construct a dam), make assumptions about the probabilities of outcomes and their desirability and calculate what is the prefer solution that you should take.
Optimizing long-term growth because it's the defining property of a multiplicative process is not a new idea. The logarithmic utility function is derived for this problem using those arguments even in undergraduate textbooks. It's discussed for example in pages 232-234 of http://dl.rasabourse.com/Books/Finance%20and%20Financial%20M... (I don't necessarily agree with everything said there, it's just an example.)
Empricically, though, people don't quite behave as growth-optimizers. They would be leveraged to be invested 200% in equities to have the optimal portfolio.
> EE does not require the problem to have an infinite number of periods
Ok. But then the infinite repetion where the time-average is justified by the ability of the agent to experience every possible outcome infinite times is just a mental construct without any relation to physical reality. Which is fine for me, mind you.
Still, that doesn't explain how this is correct:
"With EE you recover different behaviours depending on the structure of the problem, both the reward structure and the temporal structure (e.g. number of periods)"
In the question of whether you take that bet, you get the same answer whether you play once, or ten times, or one million. (Or course the same is true when you analyse the problem using logarithmic utility, because it's mathematically equivalent to growth optimization in a multiplicative process).
> In the question of whether you take that bet, you get the same answer whether you play once, or ten times, or one million.
If you are only playing one round, you should take the bet, if you are playing more than one round, you shouldn't. EUT requires you to change utility function to recover those two different answers, EE gives you the correct answer however many rounds you choose to play (one, or more) without requiring any additional modification.
Well, using EUT descriptively is useful when you're objective is to create a model from empirical observations. I agree that this is irrelevant in this discussion.
> Again, the EE claim, and one which is demonstrated for at least some simple cases, is that you can recover a range of behaviours without adding in parameters by considering time explicitly.
Sure. You can recover the same using EUT in prescriptive mode. Which is not about fitting observations to a general parametric utility function or anything like that. It's about using a model of the problem (p.ex. multiplicative dynamics) to derive the right utility function to use (p.ex. logarithmic) when possible (just like EE does). And using an approximate model that is hopefully not too wrong when the problem is not so easy.
> In the question of whether you take that bet, you get the same answer whether you play once, or ten times, or one million. If you are only playing one round, you should take the bet, if you are playing more than one round, you shouldn't.
The question in equation (2) is of the form "you are only playing one round". But the answer is not "you should take the bet", because we're told that to answer you need to consider an hypothetical infinite sequence of bets.
If I offer you the possibility to win $10000 or lose $9999 on a coin flip, only once, do you play or not?
> EE gives you the correct answer however many rounds you choose to play (one, or more) without requiring any additional modification.
So there is one answer for "one" and one answer for "more"? When you said that it could depend on "the number of periods" I thought you meant that the answer could be different for two periods than for seven, for example. (Of course EUT with logarithmic utility is equivalent to EE and won't make a difference between playing twice or seven times either.)
Well, expected utility theory has been a great success and it’s used all the time to produce useful results in many fields. And the idea of optimizing growth is not new, it has been used within the expected utility framework since the fifties.
If you find the EE derivations interesting it’s fine. But don’t think it proves that EUT is wrong. And be aware that people use the same kind of analysis of the details of problems to select utility functions within the EUT framework.
EUT can be used for many things. It can be used to study empirically the behavior of people (who are not really “rational”).It can also be used to optimize industrial processes (according to a “rational” model). It can be used anywhere that there are decisions about uncertain outcomes which are more or less desirable.
If EE just puts the U in EUT it will also fail as a theory of financial decision making in all the cases where EUT fails.
[1] Did Ergodicity Economics and the Copenhagen Experiment Really Falsify Expected Utility Theory? https://researchers.one/articles/20.02.00002
EE stats that in EUT the U term must equal log (Wealth) otherwise EUT produces wrong results.
EE uses a completely different formula. It has an open space for a (slightly restricted) function (similar to U). An example of EE with this open space filled is Kelly's criterion, which of course looks nothing like EUT.
I’m not sure if I’m reading this correctly:
Kelly’s criterion looks nothing like EUT?
Maybe I completely misunderstood what you were trying to say.
https://en.wikipedia.org/wiki/Kelly_criterion
“The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate.”
The reply button is missing below so I'll reply here.
The issue is that the major results of economics in this sort of area are been effectively destroyed by EE. Prospect theory for example is completely gone.
There may be places where people's behaviour isn't rational but research into that needs to be effectively carried out again from scratch. The main "evo psych" results in economists have been effectively disproven.
So you might be right that people don't act rationally but all existing research into that area is currently dead in the water. It's start from square 1 again.
"research ... dead in the water"
I can go down to the casino and see this. Or I can see that family member A has insurance while family member B doesn't.
You talked earlier about wanting a theory to have stable foundations. A theory that doesn't even recognize the existence of subjective expected utility is not that.
Do you claim to understand them or are you just relaying the claims from others?
We don't even know what do you mean by EUT.
Maybe they think that EUT is too limited to explain the world but then they won’t be impressed with the more restrictive EE.
If Expected Utility is not correct all the solutions proposed by Ergodicity Economics cannot be correct either.
False.
The ergodicity economic formulas generate exactly the same portfolio as the "regular economics" approach when you make the same assumptions that are implicit in the asymptotic growth maximization (no spending, infinite horizon, logarithmic utility).
Don't you think that when people choose investments their objective may be to spend at least some of their money before the end of time?
The rest is just you arguing against a strawman in your head.
Maybe there was something wrong with your code? Because optimizing asymptotic growth and maximizing logarithmic utilty are mathematically identical.
EE doesn't suffer from the same problems as Growth-optimal portfolios either (Kelly's is a hyper aggressive form of EE).
"Maybe there was something wrong with your code? Because optimizing asymptotic growth and maximizing logarithmic utilty are mathematically identical."
So naive and so wrong. That's only true if you structure the problem to make it true. In the general case, not at all.
You said it was not so. That you coded and ran the simulations.
Did you compare the asymptotic growth maximization of a multiplicative process with the logarithmic utility solution? Yes or no?
Either you simulated something else or you did something wrong trying to simulate two problems that everyone agrees that are identical.
They aren't quite identical and it does make a difference.
"that everyone agrees that are identical."
Except of course the theory of thermodynamics. But now you are going to try to explain to me that the theory of thermodynamics is wrong.
Hint: I won't be impressed.
"we had worked out in detail the correspondences between linear utility and additive dynamics; and between logarithmic utility and multiplicative dynamics"
From "The time resolution of the St Petersburg paradox":
"the time-average performance of the lottery is computed. The final result can be phrased mathematically identically to Daniel Bernoulli's resolution, which uses logarithmic utility"
"Equation (6.10) is mathematically equivalent to Bernoulli's use of logarithmic utility."
It's not a matter of opinion. There is no reason to disagree. You don't have to take my word for it. You don't even have to believe Ole Peters when he says that asymptotic growth maximization in a multiplicate process is equivalent to logarithmic utility maximization.
The maths are not that complex. You can derive it yourself of find a proof and go through it until you're satisfied.
You insists a simulation that you coded and ran shows that this mathematical identity is false. I can think of multiple reasons. For example:
- you are thinking of something other than a multiplicative process and the maximization of growth and logarithmic utility
- you coded something that doesn't does what you think it does
- the output you produced doesn't says what you think it says
- you have fun playing dumb on the internet
I'm afraid not. They are infact different. One includes the starting wealth of the gambler and the other does not.
Will you believe it from the mouth of Ole Peters himself?
This is from the Nature Physics article:
"we had worked out in detail the correspondences between linear utility and additive dynamics; and between logarithmic utility and multiplicative dynamics"
From "The time resolution of the St Petersburg paradox":
"the time-average performance of the lottery is computed. The final result can be phrased mathematically identically to Daniel Bernoulli's resolution, which uses logarithmic utility"
"Equation (6.10) is mathematically equivalent to Bernoulli's use of logarithmic utility."
On that specific problem there are other problems where they are not equal.
Good. We're progressing. You've learned something today.
Good. We're progressing. You've learned something today. "
No, one does include the starting wealth of the gambler and the other does not.
This is from the Nature Physics article:
"we had worked out in detail the correspondences between linear utility and additive dynamics; and between logarithmic utility and multiplicative dynamics"
From "The time resolution of the St Petersburg paradox":
"the time-average performance of the lottery is computed. The final result can be phrased mathematically identically to Daniel Bernoulli's resolution, which uses logarithmic utility"
"Equation (6.10) is mathematically equivalent to Bernoulli's use of logarithmic utility."
Which solution does include the starting wealth of the gambler, according to you, and which one doesn't?
They care only about the limit of the rate of growth when the horizon grows. It's effectively a single (infinite) period model.
At best, they just repeat a well-known argument supporting logarithmic utility. It's just a special case within the expected utility framework.
I'm not sure it's fair to say "independently reinvented". They know the previous work, at least some of it. Their contribution is to add the word "ergodic" here and there and tell the reader that "Growth rate optimization is now sometimes called ‘ergodicity economics’".
It complains that when people fail to act according to the theories, that the people are considered to be irrational, rather than amending the theories. While it is of course true that theories need to be made to account for how people actually behave, it seems to me that people failing to satisfy the vNM axioms, really is a way in which people fail to live up to the correct ideal of a rational agent. Well, maybe for it to make sense, instead there should be a formalization of like, a compute-limited approximation of following the vNM axioms, maybe throw in a dash of sub-agent stuff, and a few things along those lines.
Hmm.
Ok, well, if this makes better predictions of how people behave, that is valuable. Though, it would raise the question of "why?".
In his article, Peters immediately rushes into the time domain, whereas the time domain doesn't exist at all in EUT.
I feel like Peters could have made a valid point about the application of dynamic EUT, but he severely over-claims the problem he sees.
To refute EUT, Peters had to start with one set of EUT axioms. He clearly doesn't want to do that. We can guess why.
The behaviour of real people matches Ole Peters maths but not EUT. This means that EUT does not apply to the real world, which makes EUT a pointless theory.
If Peters wants to propose an alternate theory, he needs to explain all of this existing evidence. For example, he needs to explain why historically the return on stocks is so high. If people only cared about the logarithm of wealth, they would hold almost entirely stocks and very little bonds. They would take their weekly paycheck and deposit it directly into their Robinhood account. (And everyone would have a brokerage account.)
"people only cared about the logarithm of wealth" But that isn't what Ole Peters is claiming.
"Multi-period Expected Utility Theory Predicts Zero Risk Aversion in Copenhagen Experiment, Same as Ergodicity Economics."
https://medium.com/@aa_goldstein/multi-period-expected-utili...
#!/usr/bin/python3
import random
#100 people start with $100
balances=[100]*100
#They take 10000 bets with an _expected_ return of 1.25x.
# ... so they should end up with ~1.2e1000 at the end...
for i in range(10000):
balances=[0 if x<=0 else random.choice([x*2,x//2]) for x in balances]
#Yet all or almost all are bankrupt:
print(len([x for x in balances if x<=0]))Who has ever claimed this? Also, people end up with zero because you are doing integer division in your code (ie the simplified expected return is wrong). Here are my results with 100 bets, and 10000 bet takers. The average return is also on the upward trend, but not really close since CLT doesn't really apply to this distribution:
>>> sum(balances)/1000000
24546192.840913434
>>> print(len([x for x in balances if x<=100]))
5426
>>> print(len([x for x in balances if x>=100]))
5381
>>> print(len([x for x in balances if x>100]))
4574You can show ruin without any flooring division:
#!/usr/bin/sage
import random
balances=[Rational(100)]*100
for i in range(10000):
balances=[0 if x<=0 else random.choice([x*2+2,x/2-1]) for x in balances]
#Yet all or almost all are bankrupt:
print(len([x for x in balances if x<=0]))
One that is the case 100 bets with 10000 betters and 10000 bets with 100 betters stop being the same thing.
And this is a very realistic and physical assumption because real investments have integerization, fees, overheads, etc. You can't invest a femto-cent in the market, and certainly not get the same relative returns as someone investing 100k.
But economics is deeply tied to politics. What type of economy you have is entirely dependent on current leadership. It's the government that sets the basic foundation and rules upon which the economy is built. Some governments decided to go to either extreme. Communism or capitalism. There is a price equilibrium, assuming one is allowed to exist, which is dependent on your government, but where exactly it lies is also entirely dependent on the laws the government set.
If the government bans garbage disposal in rivers then it would be completely unreasonable to insist that the equilibrium price should stay the same.
By this same logic if there is an inherent injustice or imbalance it exists in the laws the government created.
One prominent example would be the central bank flooding the market in a way that benefits existing asset holders, one could have done the exact opposite as well or maybe even done a little of both. It's not a question of which is the right action, the question is "What kind of economy do you want?" and the central bank has spoken.
It is absolutely silly, bordering shitposting, to respond to a claim about subtle/hidden assumptions by claiming:
> Expected utility theory makes 4 assumptions, which are stated precisely and concisely in every graduate textbook. Ergodicity is not among them.
when the whole point is that the conventional economics literature might have ignored a subtlety.
"Although it is true that our consumption of economic goods develops over time, time is not the most central aspect of all our decisions. For many of our decisions, other equally ubiquitous aspects such as risks, strategy and the balancing of pros and cons are more central. Just because something is ubiquitous, it should not be confused with being explanatory; for example, we can argue everything consists of molecules, but it is not a reason to think that all questions in economics, geography and throughout life should be answered by molecular dynamics.... Economists use static EU for static decisions, when dynamics are not central. Otherwise, a dynamic model is used."
Also, the field has gone far beyond von Neumann-Morganstern, so I'm not sure where you're getting that, either. I'm simply making the point that expected utility does not require ergodicity. If you read the proofs, it's clear. If you don't want to read the proofs, the best that I can do is provide indirect historical evidence.
Okay well that point is wrong, as in very wrong.
Ergodicity can be directly defined as it is valid to take the Expected Value. That's just what Ergodicity means.
Expected Utility is defined on top of Expected Value.
So Expected Utility assumes Ergodicity.
Peters (and you, I assume) are taking a strong philosophical stance on the meaning of probability. Lots of people would disagree with you. Essentially all Bayesians would disagree with you, for example.
"strong philosophical stance on the meaning of probability" ???
It's taken directly from thermodynamics: https://en.wikipedia.org/wiki/Ergodic_hypothesis
You would be arguing directly with the laws of thermodynamics.
There are more things in heaven and earth than dreamt of in your philosophy, Horatio: https://plato.stanford.edu/entries/probability-interpret/
https://bayes.wustl.edu/etj/articles/theory.1.pdf
“It is concluded that statistical mechanics need not be regarded as a physical theory dependent for its validity on the truth of additional assumptions not contained in the laws of mechanics (such as ergodicity, metric transitivity, equal a priori probabilities, etc.).”
Peters' claims the contrary, and then claims to "destroy" a whole academic field with one fell swoop. That's academic shitposting.
That EUT is based on assumptions that have nothing to do with time is quite crucial, and one major misunderstanding. However, if you read on, you see that Peters makes other claims that don't hold up.
Look at it from this side: Here comes someone trying to carve out a heterodox niche (he ain't the first), but he does it without actually discovering something new and without the proper care for or understanding of economic theory.
This ruffles people's feathers.
(I'd also like to hold on to a copy rather than losing it buried in a comments thread :-)
Since the larger discussion is getting side-tracked and might expire before we get a chance to discuss, I made a new HN submission to focus on this, if you prefer: https://news.ycombinator.com/item?id=26378454
I looked online and found this post that explained it well: https://taylorpearson.me/ergodicity/
This is a widely studied area in dynamical systems.
I am squeamish about such popular accounts. An old account which is readable even today, is by Birkhoff [1].
About Kelly-Bernoulli criterion, the readable account is the book [2].
[1] https://doi.org/10.1080/00029890.1942.11991212
[2] "Fortune's Formula" by William Poundstone. https://archive.org/details/fortunesformulau00poun
> A central chapter that crystallizes all my work. Time to explain ergodicity, ruin and (again) rationality
https://medium.com/incerto/the-logic-of-risk-taking-107bf410...
Basically, Expected Utility Theory (EUT) is wrong. It happens to give the right result by coincidence when you set the fudge factor U to log(wealth), which all economists do without any justification.
Well EUT is corrected using the fudge factor, other parts of economics built using it are not.
The nobel prize winning Prospect theory, when you correct the maths, dissolves away into nothing. The theory says people deviate away from the expected rational answer due to psychological reasons. It turns out people use the rational answer, the maths in Prospect theory is just wrong.
Additionally, the famous St. Petersburg paradox isn't a paradox and has an exact answer.
This is a pretty big blow for economics and the economists on social media aren't actually being mature about it.
https://www.nature.com/articles/s41567-020-01106-x
Read both sides!
EUT was developed by von Neumann, someone slightly familiar with ergodicity. It simply is not based on dynamics of out-of-equilibrium systems. Any such thing is an application that adds assumptions to the construct.
Ergodicity is an obvious addition in the time domain, it is so obvious that the insight is not even new. However, Peters restricts the problem to a very simple dynamic gamble and then claims that all of economics must be wrong.
The very first sentences of Peter's article already get this completely wrong.
If you insist on his results, despite the quoted article, the most one can say is that the application of EUT to these problems is questionable.
To put it in terms you may be more familiar with. It's like saying that Object Oriented Programming is "wrong", because Python doesn't work for my problem at hand.
"Ergodicity is an obvious addition in the time domain" Great, except Ole Peters if you understood his work is adding non-ergodicity.
Ole Peters is effectively claiming that static gambles don't exist (outside of utterly bizarre circumstances like parallel universes or co-operatives).
The set of problems that you can use EUT on and be correct is almost zero.
To simplify:
You must always use the Kelly's criteria even for single one-off gambles otherwise you have got the wrong answer.
(There are other valid criteria then Kelly's that you could also use but you need to read Ole Peters' paper for them)
But his theory is applicable in a strict subset of those...
"People behave as rational actors"
Rather than the EUT's result of:
"Faced with a risky choice leading to gains, individuals are risk-averse (concave value function).
Faced with a risky choice leading to losses, individuals are risk-seeking(convex value function)."
Basically ergonomics economics shows that Prospect theory is a math's error. This is one of the reasons why Ole Peters work is important.
What's the solution to this problem using his methods, for example?
https://en.wikipedia.org/wiki/Merton%27s_portfolio_problem
How do his methods apply to the uncountable situations where expected utility theory is used?
A random example: https://www.researchgate.net/publication/240488954_Risk_of_d...
Is your point here that Ole Peters should convert all problems in economics to ergonomics economics all at once? Cause that's a bit of a silly point to make.
There is also a fundamental problem with expected utility theory in that U can set to any formula. It's pretty close to just making stuff up on the fly.
Is expected utility theory wrong and ergodicity economics is an alternative?
Or is EUT too broad and ergodicity economics cannot blamed for being nothing more than a well-known way to pick a specific utility function within the EUT framework in a very constrained subset of problems?
I think it's the latter.
My point is that the claims of having subverted centuries of flawed economic thinking based on parallel universes are delusional.
My opinion may of course improve if any of those great research papers is ever written. It would also improve if they changed their attitude and stopped saying idiotic things. I'm not sure what of those is more likely...
EUT is claiming there are parallel universes not EE. And your right EUT is delusional to make that claim.
About your opinion:
"It is difficult to get a man to understand something when his salary depends upon his not understanding it." - Upton Sinclair
Can you back this up with a reference to anything not coming from Ole Peters (and known associates)?
> "It is difficult to get a man to understand something when his salary depends upon his not understanding it."
Helping to clear the confusion he creates in his full-time occupation is a just a hobby for me.
Let's say there is a lottery L, it has a million tickets $1 each, it pays out 1 million and 1 dollars on the winning ticket.
Should you buy a ticket?
EE says no. EV says yes since it has a positive EV.
EE says no because in 99.999999% of cases you lose $1.
EV says yes because you share the 1 million and 1 dollars out in equal portions with versions of you living in a million parallel universes, one of which won.
I hope that clears this up for you.
---
@drdeca:
Due to missing reply button I'll reply here.
The claim being made is that U is a hack.
When you set U = log (Wealth), EUT gives the same result as EE. When you set U to any other function EUT gives bad answers.
EE has it's own set of functions that take place of U in EUT and all of those functions give the right result whereas EUT only gives the right result when you set U = log (Wealth)
That would only hold if you assume that utility is either linear or superlinear in money (or, more precisely, that the marginal utility of losing a dollar, times 1-(1/1 million), plus the marginal utility of gaining a million dollars, divided by 1 million, is greater than 0)
If this is the sort of argument that people advocating for EE are making, it makes EE seem less worthy of attention.
Oh, you said EV not EU? Ok, but no one claims that people maximize expected money. That would be stupid.
Using EU rather than E$ isn’t some hack to add a fudge factor, it is capturing that people have preferences about things in general, not merely how much money they have. Placing money centrally is silly; money isn’t some universal terminal goal. It is a convergent instrumental goal. Depending on what I care about, the use of money to me will scale differently, just like how the scale of other things to each-other will. The idea of utility is to pick the quantity which I do value linearly in probability. However I take probability into account, provided I do so in a coherent way, there is a unique-up-to-positive-affine-transformation utility function which corresponds to that.
If it does give the right result why do you say elsewhere that "they aren't quite identical and it does make a difference" and you coded and ran a simulation that proves it?
If EE is just a way to choose the U in EUT how does that mean that EUT is incorrect?
EUT is incorrect only when there is no U that can describe the preferences of the agent.
"If EE is just a way to choose the U in EUT how does that mean that EUT is incorrect?"
EE doesn't choose the U in the EUT. EE has something similar to U in it's math where you can select any function of a certain class and plug it in.
One of those function when plugged in gives out the Kelly criteria.
"If it does give the right result why do you say elsewhere that "they aren't quite identical and it does make a difference" and you coded and ran a simulation that proves it?"
The simulation ran the game purposed by Ole Peters where they give different results. Only in special games like the ones typically purposed by economists, do they give the same result.
EUT doesn't say what the U is. It only says that one can be found if the agent's preferences are rational (for some definition of rationality).
Can you point to an example where EE provides a solution that cannot be represented by some utility function?
I don't know if that game is related to "retirement portfolios", probably not.
I understand then that you don't object to my claim that ergodicity economic formulas generate exactly the same portfolio as the "regular economics" approach when you make the same assumptions that are implicit in the asymptotic growth maximization (no spending, infinite horizon, logarithmic utility).
And that you agree with Ole Peters and yours truly that asymptotic growth maximization of a multiplicative process gives the same solution as the maximization of logarithmic utility.
Nope. Unless you claim that all discussion of probability does the same.
If an agent satisfies the vNM axioms , that is sufficient to conclude that the agent’s actions are equivalent to maximizing the expectation of some function, which we call the utility. These axioms do not require any assumptions about “parallel universes” beyond simply the concept that “there is such a thing as probability”. Presumably, the same idea would work even if the “probabilities” in question were all from logical uncertainty due to limited computation time, taking place in an entirely deterministic universe. In such a case, there explicitly cannot be the “alternate universes”, because they would be logically inconsistent, but due to computational limits, the agent is still uncertain, and so it still makes sense to deal with expected utility.
It's not going to be an easy solution, because for a sufficiently small x, the optimal strategy for a small number of bets is going to be to bet $0 each time (since the x-th percentile of the terminal outcome will be a loss on capital, or ruin), but the optimal strategy for a large number of bets is going to be to bet > $0 (since the x-th percentile of the terminal outcome will be positive).
So we can already see that this is going to be a function of the number of bets, unlike with Kelly, and therefore is going to be a much harder problem.
“This paper provides an alternative behavioral foundation for an investor's use of power utility in the objective function and its particular risk aversion parameter. The foundation is grounded in an investor's desire to minimize the objective probability that the growth rate of invested wealth will not exceed an investor-selected target growth rate. Large deviations theory is used to show that this is equivalent to using power utility, with an argument that depends on the investor's target, and a risk aversion parameter determined by maximization. As a result, an investor's risk aversion parameter is not independent of the investment opportunity set, contrary to the standard model assumption.”
I think that's basically the key point. Ergodicity economics and the real world are both based on dynamic out of equilibrium systems. Where regular economics is not.
Real people behave according to ergodicity economics according to Ole Peters study and also the data set used in Prospect's theory.
What does being a Comp Sci PhD have to do with anything?
This is fundamentally a modelling error. They are using valid maths that models the wrong scenario.
Consider a group of 1,000,000 people making gambles. In traditional economics a rational actor will make decisions that maximize the SUM of the 1,000,000 rational actors money. In ergodicity economics a rational actor will make decisions that maximize it's money.
A subtle distinction, but as a Comp Sci PhD I can tell you that small modelling error has utterly fatal consequences for large parts of economics.
Prospect theory for example is written off.
Physics PhDs also do modelling. You can see this as a knowledge gap in understanding of a typical economist or stats person, which is why they are having such difficultly in understanding Ole Peters work.
You have to be aware of how this comes off, right? This is all tongue in cheek?!? I mean your comment might as well have said “So, why does <your field> need a whole journal, anyway?”
If you want to determine who is right that is all you need to know.
To simplify this for you, if you saw someone who in their maths equations swapped mean and median averages and told you that their maths usually gives the right answer. What would you think?
"how can you possibly so sure of yourself because the maths looks good to you?" Because I know for certain.
"Especially when experts are saying" A nobel prize winning physicist has signed their name on it (Previous paper in the line).
Now do you really think the nobel prize winning physicist is wrong or some annoyed economists on twitter are wrong?
"Would a person ever prefer a process [A] that, after three rounds, diminishes wealth from US$10,000 to 0.5 cents over one [B] that yields a 99.9% chance of US$10,000,000 and otherwise US$0? Ergodic theory predicts [A] [... but] [v]irtually everyone will prefer B"
This suggests to me like the maths may not be all you need to know about who is right and wrong.
Secondly, you've appealed to the authority of one advocate's Nobel prize IN PHYSICS to assert that this paper in economics is correct. I believe the bunch of salty critics on Twitter (and Reddit, this thread, also in official rebuttals published in Nature), from people who actually work in the domain - should at least make you entertain the possibility that you are wrong about the correctness/importance/applicability of this paper.
The answer is that the person prefers A not B. The physical interpretation of that is that you have a 99.9% chance of getting $10,000,000 and 0.01% of being killed where you stand. People naturally avoid sudden losses of resources as it usually means death. [If you want me to explain to you, why it's totally irrational to choose B, I'm happy to do so]
You are using maths without understanding what that maths actually means.
You could argue that you should be calculating the effect of making a decision a very large number of times rather than infinity times. But that would be a minor tiny issue rather than a reasonable refute.
I have read the paper of refutes and they are no good.
@sweezykeezy At the end of the day, Ergodicity Economics is a revolutionary idea for economics and like with most revolutionary ideas it has a lot of people who don't want the status quo to be upset. In this case an outside's unbias opinion is more valid than an insider's heavy bias opinion.
The other incredibility telling thing is that all the detractors can't agree on what actually their disagreement is with Ole Peters work.
They can't attack the maths, they can't attack that it applies to the real world, so they are all over the place.
"Nobel prize IN PHYSICS" I think there is a big hint here that Economics is in the scientific dark ages where people are more interested in advancing their pet theories then are actually concerned about the real world or what is correct. If economics didn't want people from other fields coming in, then they shouldn't of let their field fall into a state of disrepair.
In your next reply to this, explain to me what an ensemble average and an time average are and why they are different. At least show me you understand at least a little of what you are talking about.
There are certainly no comments on this article that successfully refute Ole Peters, I have read them all.
The economists here are behaving actually how researchers from problem search fields typically behave.
One typically problem behaviour is argument by example, two different research groups have their own theory and then they use research papers and more recently social media to send each other ridicious examples of where their theory works and the other research group's does not.
An example of argument by example is: "Would a person ever prefer a process [A] that, after three rounds, diminishes wealth from US$10,000 to 0.5 cents over one [B] that yields a 99.9% chance of US$10,000,000 and otherwise US$0? Ergodic theory predicts [A] [... but] [v]irtually everyone will prefer B" from the rebuttal paper. Along with all the other ones in the rebuttal paper.
That's a telltale sign of poor quality researchers.
They didn't even bother spend any time to work out what Ole Peters was even saying and since they didn't know what Ole Peters was saying, they failed to refute it.
Good researchers, who's opinions are worth listening to, don't behave in that manner.
??? Have you not heard of competitive equilibria at all? Is game theory outside of the realm of traditional economics?
> Prospect theory for example is written off.
Yeah, it's so written off that the authors received the biggest award in economics for their works in behavioral economics.
Ole Peters work is at least a step in the right direction.
Funny stuff. They predict that in aggregate 100% of the wealth is invested in stocks and investing more than that would be suboptimal: the optimal leverage is 1.
They "submit the hypothesis to a rigorous test" and find that the optimal leverage for the S&P 500 is around 1, given its volatility.
There is a small problem with their rigorous test. They forgot to include dividends: "a real investment in the S&P500 outgrew federal deposits at only 2.0% p.a".
Including dividends would bring the optimal leverage to 2 or 3. But that's ok. It's close enough to 1 for them, apparently.
Even better: ignoring dividends was not so wrong anyway because the S&P 500 overestimates the return of a real equity investment due to survivorship bias.
"The S&P500 is an index of five hundred large companies, listed publicly in the United States. We use it as a proxy for a generic diversified investment in US stocks, but we note some caveats. Firstly, the index does not account for dividends paid to stockholders. This means it will tend to underestimate the performance of a real investment. Secondly, the index suffers from survivorship bias, representing a portfolio of the largest and most successful companies in the US, in which less successful companies are routinely replaced. This acts in the opposite direction to the first caveat."
Below are some of my thoughts after a glance through the paper:
1. The critique of the article by Doctor et al. (linked from Ben Golub's Twitter) summarizes my thoughts succinctly. I am left wondering: if everyone used "ergodicity" as the basis of their decisions, doesn't this imply that, given some scenario with risk, everyone would make the same decision? But this is certainly not true. The author mentions that different people might care about "additive growth" vs. "geometric growth", but this is equivalent to using different functional forms of utility (e.g. CARA vs CRRA). Perhaps the author is saying that his ergodicity-derived decisions are "optimal"? But in what sense? It seems to me that his maximization of the growth rate is equivalent to using log utility. This result is well-known (see the Kelly criterion).
2. Ergodicity is covered in first-year graduate econometrics (at least where I was taught). See Hayashi Chapter 2. Perhaps not everyone reads it this way, but the tone of the article seems to suggest that this topic is completely foreign to economics.
3. There is a rich economics literature in decision theory. Savage's work on subjective expected utility shows that under a certain set of axioms of rationality, the decisions that a person makes can be completely captured by a subjective probability distribution and a personal utility function. The sentence "expected utility theory implicitly assumes that individuals can interact with copies of themselves, effectively in parallel universes" is completely misguided.
4. There are a number of well-known documented violations of expected utility (e.g. the Allais paradox). These represent challenges to the axioms of rationality. Nevertheless, the point of using expected utilities is to model human behavior, even if it is only approximate. From Rubinstein's great book "Economic Fables": "I remember the moment as a student when I realized that the models in economic theory do not assume that the decision maker consciously tries to maximize his preferences, but only assume that the behavior of a decision maker can be described as if he had maximized some objective function" (emphasis mine).
5. It is true that time-separable utility (usually with constant relative risk aversion, or CRRA) is often assumed for many mainstream models for tractability reasons. However, in recent years substantial progress has been made in extending utility to recursive preferences, e.g. Epstein-Zin. In fact, Epstein-Zin is essentially as tractable as the conventional time-separable CRRA utility, and allows for the separation of elasticities between risk and time. I expect that these more sophisticated preferences will soon be the norm. There are even stranger preferences in use, for example hyperbolic discounting. A whole field (behavioral economics) is founded on the notion that humans are not rational. The point is, lots of work has been done on generalizing utility functions.
6. I am not exactly sure what point the author is trying to make with Figure 2. This appears to be nothing more than a demonstration of Ito's Lemma (the convexity adjustment necessary for solving stochastic differential equations). In fact, the author mentions as much in the last paragraph of the third section.
7. My (ungenerous) interpretation of the experiment section: "Look, experiments calibrate the coefficient of relative risk aversion to be about 1 (log utility). And there doesn't seem to be much heterogeneity in risk aversions across people. This is consistent with my ergodicity-derived decisions, which is essentially log utility. Hence my model is supported!" (For reference, the usual calibrations I encounter for CRRA range from 1 to 5.)
8. Finally, the "Outlook" section. This section definitely rubbed me the wrong way. The author patronizes the entire field of economics, boasts of the ingenuity of his invention, and claims to somehow link together everything from the equity premium of the stock market to optimal monetary policy. It sounds too overconfident of itself and dismissive of others. References to works I expect to see are absent, and self-citations are abundant.
After reading this paper, I was gently reminded of Baez's "Crackpot Index". I imagine that what I am feeling is similar to what a physics professor might feel after reading an idea for a perpetual motion machine. There is a mixture of correctness and sophistry, with a dash of illusions of grandeur. It is comprehensible enough to understand and reasonable-sounding enough to require some effort to criticize. Looking back, writing such a lengthy critique might not have been the best use of my time.
I am rather surprised this paper was published. I am not familiar with Nature Physics, but based on this article alone, my regard for it is not high. In my opinion, this paper would not be sent out for refereeing, let only pass the referees, at any reputable economics or finance journal.
2. You clearly don't understand it. Ole Peters is claiming that the ergodicity assumption is wrong and purposing a non-ergodicity version of economics to fix it.
3. "expected utility theory implicitly assumes that individuals can interact with copies of themselves, effectively in parallel universes" is completely correct. EUT implies that is true as one of it's implicit assumptions.
4. Which raises the question of why has no one but Ole Peters has come up with a new theory to fix the violations? It's poor show by the field of economics. Our theories don't work, let's just sit on our broken theories.
Given Ole Peters work is mathematically correct and shows EUT to be mathematically incorrect. It's pretty pointless calling him a "crackpot". There is an objective right answer here and Ole Peters has that right answer.
A reference to the paper or textbook that you learnt it from, for example, would suffice.
Even better if you can point to the specific place where there is something mathematically incorrect and/or an implicit assumption of individuals interacting with copies of themselves in parallel universes.
Some thoughts on this: Peters has published with Gell-Mann, so physicists might be unreasonably impressed. My feeling based on interacting with theoretical physicists at university is that there is a good portion of them who have little knowledge about and interest in other quantitative disciplines. They sometimes have a tendency to like to feel superior to "less rigorous" disciplines, so it's not unlikely that the particular reviewers were attracted to this?
Peters probably submits his papers to all kinds of journals, and probably very similar papers many times over. His more recent papers are kind of mono-thematic. The maths is not actually incorrect and easy to check, so once in a while they might get through the review process. Maybe that's the "ergodicity approach to publishing"?
He talks about a paper that has been "rejected by 14 economics journals, if I've counted correctly, for questioning the ergodic hypothesis."
I've not looked at that paper. I don't know how bad it is. But I suspect that "questioning the ergodic hypothesis" may not be the only reason.
https://old.reddit.com/r/badeconomics/comments/kcmtce/guy_wi...
Which seems a great simplifying assumption where applicable.
https://sci-hub.se/https://www.nature.com/articles/s41567-02...
Personally I think almost all of it is outdated and have no respect for economists. To back systems that perpetuate such gross inequity it seems most must have a very dated and reductionist Social Darwinist mindset.
The first thing we need is to start to incorporate technology into society, such as by comprehensively tracking resources. Things like distributed technologies are the best starting point for this. Then when we use point systems for distributing control over resources (money) that should be a high tech system that is integrated with resource tracking and regulation.
The way the current system works is kind of like if someone built an MMORPG and only included one stat --- money. Then, instead of server code for managing things, the powerful Economic Cult characters estimated what was happening and tried to regulate everything by controlling the spawn of gold coins.
Yes.
It's called "economics".
Empiricism is absolutely central to modern economics. Here's the latest AER, the top journal.[1] 7 out of 8 articles are empirical. 4 out of 8 actually use the word "Evidence" in the title; by now this is practically a meme.
Incorporates high tech advances: interested in machine learning? Here's an overview for you: https://www.nber.org/system/files/chapters/c14009/c14009.pdf
Accounting for sustainability? First, environmental economics is an entire subfield. William Nordhaus won the Nobel prize[2] for estimating the costs of global warming, oh, and have you heard of the Stern Review? More broadly, the field of economics was famously defined as "Economics is the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses" (my italics). Sounds like it might be relevant to finite resource tracking.
Lastly, I'm not sure why you think economists support the current level of inequality. I'd wager most don't. Maybe you've heard of Capital in the 21st Century, which put inequality squarely back on the political agenda. It's by Thomas Piketty, an economist.
[1] https://www.aeaweb.org/issues/625
[2] I beseech you not to tell me how It's Not Really The Nobel Prize. Everyone knows. Nobody cares.
The paper itself is pretty interesting if you know some maths and haven't thought about non-linear utility functions. You can also read that "original" paper by Bernoulli, he has some simple examples there (for example why does it make sense to buy insurance despite it having negative expected payoff, he brings the example of ship cargo insurance but it applies to any kind of insurance really).
EDIT: Daniel Bernoulli's paper: https://www.semanticscholar.org/paper/Exposition-of-a-New-Th...
I am not qualified to judge whether those feelings are correct however.
Using the wrong type of average mildly screws up everything in very subtle ways as you can imagine.
A lot of the important results in the field including one that got a nobel prize turn out to be wrong.
The economists are responding to this in a very mature way and have told Ole Peters "he can go kick a rock".
I found this to be an excellent overview of ergodicity and its implications.
The published version of Peter Wakker et al's critique [1] is remarkably polite. You have to read their supplementary information [2] for the full, hilarious horror. Here are some juicy quotes to tempt you:
"The EU [expected utility] value does not actually have to be realized or consumed in any sense.... However, Peters erroneously thinks that the EU value must actually be realized in some sense.... EU involves imagining, a priori, some outcomes that later may not have actually been received. This procedure involves imagining consequences that will never happen. But we do this every day, and such is the nature of every probabilistic decision. We do not need to believe in “parallel universes” or the existence of “multiverse clones"...."
"A more fundamental problem in dynamic decisions is that we do not just maximize our entire wealth at the end of our life, but intermediate consumption patterns virtually always play a role. For nonquantitative outcomes, growth rates cannot even be defined. Dynamic questions as discussed here are central, for instance, in economic growth theory and in life-cycle consumption theory."
"Peters suggests that economists should primarily study intertemporal processes, the topic of ergodic theory. For example, he suggests that risk attitudes and risky variance are not important and that interpersonal variations are not important, and then, in one blow, that neither is any economic theory."
"Peters’ claims that, because of the ubiquity of time, we should always study intertemporal growth. Similarly, a risk theorist can claim that we always face uncertainties and, therefore, we should always study risk theories.... In the annotated bibliography Wakker (2020), the keyword “own small expertise = meaning of life” gives references to other authors falling victim to this ubiquity fallacy.
... and then there's the experiment. Oh boy, I'd forgotten that part!
"Meder et al. applied expected utility and prospect theory in a way that we call static: they applied EU and PT to each choice in each round separately, as if it was the only choice made and as if intermediate outcomes were actually received. This static analysis is incorrect. The intermediate outcomes are not outcomes received and consumed by subjects."
Translation: Sorry, kid, you failed your midterm.
tl:dr; This is nonsense, an embarrassment to the authors, and an embarrassment to physics. Yet it pops up on Hacker News from time to time as a Deep Mathematical Takedown of Economists by Physicists!!!!... No.[3]
[1] https://www.nature.com/articles/s41567-020-01106-x
[2] https://static-content.springer.com/esm/art%3A10.1038%2Fs415...
[3] https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu...
It was during the reading of O. Peters and A. Adamou "Ergodicity Economics" [3] that I better understood the idea.
Imagine a basic gamble that repeats indefinitely. In each iteration and with equal probability, the player can win 60% more or lose 40% of the initial capital. For this gamble, the ensemble average (aka expectation) of the player's wealth one step ahead is simply 1/2 x 160% + 1/2 x 60% = 110%. A good gamble, right? However, the time average of the same step (i.e. the average gain of a single individual playing) is sqrt((1.6)x(0.6)) x 100 = 98%. So, the individual looses money with time. This was quite surprising to me although obvious a posteriori given that the multiplicative stochastic process is not ergodic. In other words, this simple gamble shows that the expectation does not have the intuitive meaning we sometimes assign to it specially for some repetitive gambles.
In other words. the time and ensemble average differ in general for non-ergodic stochastic processes and in particular for multiplicative stochastic processes (note that for additive processes the expectation of the wealth increment can be used).
And here comes the important implication... Given that several economic processes can be modeled as a first approximation as multiplicative random processes (e.g. stock markets, real investments, GDP growth, etc.), it is not a rational strategy to use the ensemble average (aka expectation of wealth increment) to take some economic decisions.
There are several implications of the above simple fact including the optimality of the Kelly criterion; the optimal leverage being below 1 in all cases involving multiplicative processes; the incorrect measurement of inequality; or the known inadequacy of the average income, instead of the median, to measure the average well-being to name a few.
A possible controversial corollary of the above is that the concept of utility is unnecessary and incorrect as a first approximation to the micro-economic behaviour. Instead an ergodic measurable should be used. In the specific cases of multiplicative stochastic processes the difference of the walth logarithm is ergodic and a rational decision maker should use it to optimize his wealth. This will require further debate within the scientific community because it is not clear that what is an optimal decision is a good model for the people's behavior. In any case, if the expected utility is not optimal, it also does not make much sense as a model for the Homo Economicus.
In any case, I really recommend reading instead of rushing to conclusions [3].
[1] J. L. Kelly, A new interpretation of information rate. Bell System Technical Journal, 35 (1956), 917-926. [2] O. Peters and M. Gell-Mann. Evaluating gambles using dynamics. Chaos, 26:23103, February 2016. [3] Peters, Ole, and Alexander Adamou. "Ergodicity economics." London Mathematical Laboratory (2018).
This is wrong.
Instead I should have written: "there is an optimal leverage point, most likely close to 1. The optimal leverage point does not depend on the individual risk preferences of the investor".
Honestly, the whole approach is a step back. People discussed these ideas back when Kelly first published the Kelly rule, and the conclusion is that most people are not that aggressive in their investment decisions. People still study it, though, (it's called the growth optimal portfolio), and its properties are well-known.
Peters also sneaks in a second assumption, which is that the growth rate of wealth is ergodic. Is it? That's not clear to me at all.
That's because Kelly's criterion is a special case of Ole Peters, where Kelly's is a very aggressive option. There are other criterion that can be used, which are not so aggressive. Ole Peters lists the class of functions that can be used in his paper.
So basically economics has rejected it for the wrong reason.
If you try to match a power utility model with stock market data, to match prices on the stock market, you would reject both the additive model (eta = 0) and the multiplicative model (eta = 1), in favor of an eta greater than 10. This is known as the equity premium puzzle.
The thrust of research over the past 30 years in explaining stock prices has been to reject both Peters' models, and expected utility in general. People build their entire careers on investigating alternatives. There are hundreds of experiments to explain, plus all of the real-world data. It turns out that explaining all of this with one unified model is hard.
The maths you are looking for is here: https://ergodicityeconomics.files.wordpress.com/2018/06/ergo...