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EUT should be adjusted to handle ergodicity properly. But the concept of expected utility needs to be the centrepiece of the theory.
Kelly, for example, is fat
EUT should be adjusted to handle ergodicity properly. But the concept of expected utility needs to be the centrepiece of the theory.
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.
"expected utility needs to be the centrepiece of the theory" Fundamentally expected utility is wrong.
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.
> Fundamentally expected utility is wrong.
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.
"There's no such thing as a "correct result" because people's preferences (utility) varies by individual."
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.
I think that the errors you are making are that EE does not do away with expected utility, and expected utility is not a strict requirement for the development of a theory to describe economic outcomes.
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.
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1) If the expectation is there it's because the expected value of the utilities for a probability distribution over outcomes allows for an ordering of the available choices which is consistent with the preferences over outcomes. Is that true for any other among that infinity of ways of producing a summary?
3) Ok. Some people seem to think that EE disproves EUT somehow, though. That's the context of the comment your reply to.
The comment I replied to was discussing the necessity of the concept of expected utility for the success of any economic theory. My points were with reference to that - I am only claiming that it is not a requirement for a reasonable economic theory, for the reason I stated. There are other syntheses that might better describe actual economic behaviour.
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 think he pointed to the need of having something similar to expected utility maximization in the context of a theory of rational decision making.
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.)
> 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.
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.
__ The basic formulation of EUT __
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?
"
You acknowledge then that you were wrong when you said that "One includes the starting wealth of the gambler and the other does not."
Good. We're progressing. You've learned something today. "
No, one does include the starting wealth of the gambler and the other does not.
edit: As noted in your other comment, the expectation with a standard utility function is actually negative, thus the premise for part of the below is incorrect, and therefore so are its conclusions. A more faithful reproduction of Peters' argument is that EE recovers the correct solution without requiring the addition of an arbitrary utility function, and corresponding appeals to irrationality.
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.
I realise I am not explaining my self particularly well, but I don't think it is fair to call it the same assumption.
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?