Ergodicity Economics and von Neumann-Morgenstern utility
sivark.me
sivark.me
Really?
I've seen the opposite claims.
> EE is not in opposition with the vNM framework!
I agree.
Ole Peters doesn't agree, though. That Nature Physics articule is full of nonsense:
> Expected utility theory predicts that people are insensitive to changes in the dynamics. (...) Ergodicity economics predicts something quite different.
> A positive result (...) would corroborate ergodicity economics and falsify expected utility theory (insofar as experiments falsify models).
vNM does not (technically) assume a probability distribution, but EUT as commonly applied add one extra piece on top of vNM -- implicitly ignoring dynamics and assuming a distribution which leads to "ensemble averaging".
vNM doesn’t assume any particular probability distribution (or any particular utility function).
But it does assume the existence of a probability distribution (and a utility function).
How could they define an expected value otherwise?
But how does an application of EUT decide what distribution to use? (That's where the subtlety/assumption lies)
Example situation: 1 million tosses of a fair coin, with multiplicative payoff -- wealth doubles for heads and halves for tails. Say you start with a dollar -- what payoff should you expect? Is "expected payoff" (ensemble averaging) the right calculation to model that?
The distribution is not something that you choose arbitrarily. It's given. It's a part of the problem statement. Starting with $1 the distribution of outcomes after 4 doubles-or-halves flips is P($0.0625) = 1/16, P($0.25) = 4/16, P($1) = 6/16, P($4) = 4/16, P($16) = 1/16. After one million flips it will be more complex (and more skewed to the upside, it will go from close to $0 to $10^301030) but also well defined.
When you say "what distribution to use" you may mean "what utility function to use".
You're right that one may not one want to use the final payoff to represent the utility of each outcome. That's the whole point of introducing a utility function.
The logarithm has been a popular choice of utility function for centuries. Among other reasons, because it maximizes the asymptotic growth rate.
The distribution of utility values becomes in the 4 flips case (for simplicity, the million flips case would be similar):
P(u = -log(16)) = 1/16, P(u = -log(4)) = 4/16, P(u = 0) = 6/16, P(u = log(4)) = 4/16, P(u = log(16)) = 1/16
The expected utility (the ensemble average of the utilities of the potential outcomes, if you will) is zero. The same that for $1 without any uncertainty.
Assuming logarithmic utility, you would be indifferent between keeping your dollar or getting the outcome of the million-flips game.
BUT that would be assuming that your whole fortune, present and future, depended on that dollar. Otherwise you need to include the rest of your wealth when calculating the logarithmic utility. Or your preferences may not be well represented by that utility function: you'll be more willing to risk your only dollar today when you know you're getting your paycheck tomorrow.
If you're a normal person and you're offered to play the game you will prefer to do so than to keep $1 with certainty. You have 49% probability of getting at least $50m while in the worse case your lose $1.
The fundamental claim of EE is that the distribution cannot be fixed that way to be 1/N for each of the N trajectories in the limit that the number of trajectories far exceeds the number of agents. (Ref: concentration of measure)
Whether you agree or disagree, at least we are now at the crux of the matter. I would encourage you re-read my post with this update.
Now, it’s of course true that the mean of a finite sample taken from that distribution (which has a HUGE variance) will also have a huge variance.
Why would that give a meaning to your talking about “deciding what distribution to use” in EUT, I don’t know.
And what do I care that the compounded growth over one million flips was close to 0% per flip if I have 49% probability of walking away with more that 50 million dollars?
> I’m sorry. That is the probability distribution. EE doesn’t change that mathematical fact. The distribution is not “fixed that way”, it is what it is.
That is where the mistaken assumption lies, and repeating it is simply refusing to engage with the point.
> Now, it’s of course true that the mean of a finite sample taken from that distribution (which has a HUGE variance) will also have a huge variance.
You're halfway there! The underlying distribution on trajectories has a huge variance like you pointed out, but the mean of a finite sample is likely to have only a (vanishingly) small variance, because of the central limit theorem! (aka concentration of measure for "typical" trajectories)
So, averaging using the low variance distribution on the mean of (typical) finite samples is more representative model (corresponds to real world measurements) than averaging using the high-variance distribution on the underlying ensemble (what you assumed the de facto probability distribution). The former is called a "time average" and the latter is called an "ensemble average".
Don’t you agree that there is a probability distribution that describes the outcomes of the game you proposed?
If you don’t, that’s where you’re mistaken.
Expected utility theory uses that probability distribution of outcomes. In this case it was directly derived from your description of the game. The probability distribution could also represent the probability assigned to the different potential outcomes by the agent, and another agent could have a different probability distribution.
In any case, there is no notion of ensemble average over N realizations or time average over N realizations or anything like that in EUT.
There is a set of possible outcomes, which have a value called utility attached (representing preferences).
For each possible decision, there is a probability distribution over the outcomes. An expected value can be calculated.
The optimal decision is the one with highest expectation of the utility.
Saying that utility theory has an issue because of typical trajectories and time averages and ensemble averages doesn’t make much sense. All those things are not part of EUT. You could just as well say that it has a problem with the spark plugs.
If you say that you may solve some problems differently, that’s ok.
That doesn’t make EUT wrong or disallows the use of probability distributions to calculate expected values.
And as far as I know, all the decision problems solved by EE have a straightforward correspondence to a EUT solution with some utility function (and with a perfectly standard probability distribution that can be used to calculate expected utilities).
The mean of a finite sample is a random number. This random number has a distribution and has a variance. The variance of the sample mean is the variance of the original distribution divided by N. Huge.
Saying that "the mean of a finite sample is likely to have only a small variance" doesn't mean anything. Maybe you meant that the variance of a finite sample, which is also a random number and has a distribution, is much more likely to be low (all the elements in the sample are in the high-density lower part of the original distribution) than high (a few elements in the very skewed upper tail).
And, as I said, in your example there is no concentration of measure in the original distribution. Typical trajectories will move around $1 but the probability of having exactly $1 at the end goes down as the horizon grows.
As an aside, every trajectory will eventually reach any (positive) level that you may specify. And bounce between any two levels infinite times. Well, those things may not happen for some trajectories but they have probability zero. Infinite stochastic processes are fun.
To see that concentration you have to transform the original dollar outcome distribution into a growth rate per flip distribution (which converges to 0 in your example).