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The problem you have described is a single-step problem, and the process you work through gives a satisfactory answer to the question you have asked. So my answ

by mrow84 6y ago

The problem you have described is a single-step problem, and the process you work through gives a satisfactory answer to the question you have asked. So my answer to > 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.

kgwgk 6y ago
Take a step back and look at his description of the bet and equation (2). 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.
- mrow84 6y ago
  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.
kgwgk 6y ago
> 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) 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).
- mrow84 6y ago
  Ah yeah, fair point - I think it would actually be 0.5 log(1.5 W) + 0.5 log(0.6 W), without the additional log(W) term, because the reward is multiplicative, but it works out the same if we let W=1. 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.
  - kgwgk 6y ago
    Note that 0.5 log(1.5 W) + 0.5 log(0.6 W) = 0.5 log(1.5) + 0.5 log(0.6) + log(W) I wrote it that way to make the comparison with the alternative log(W) more obvious.
  - kgwgk 6y ago
    You may not have to add arbitrary preference functions but you have to add the arbitrary assumption that people don’t care about anything else than the hypothetical asymptotic growth if the current “bet” was to be replayed endlessly. (I don’t think that’s what the agent experiences either.) 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!

by mrow84 6y ago

kgwgk 6y ago

Take a step back and look at his description of the bet and equation (2). 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.

mrow84 6y ago
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.

kgwgk 6y ago

> 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) 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).

mrow84 6y ago
Ah yeah, fair point - I think it would actually be 0.5 log(1.5 W) + 0.5 log(0.6 W), without the additional log(W) term, because the reward is multiplicative, but it works out the same if we let W=1. 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.
- kgwgk 6y ago
  Note that 0.5 log(1.5 W) + 0.5 log(0.6 W) = 0.5 log(1.5) + 0.5 log(0.6) + log(W) I wrote it that way to make the comparison with the alternative log(W) more obvious.
- kgwgk 6y ago
  You may not have to add arbitrary preference functions but you have to add the arbitrary assumption that people don’t care about anything else than the hypothetical asymptotic growth if the current “bet” was to be replayed endlessly. (I don’t think that’s what the agent experiences either.) 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!