4 ms·

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, bu

by 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!
- mrow84 6y ago
  The assumption that one's life will be composed of a series of risky choices doesn't seem to be a very strong one. The use of a single form of bet is just illustrative - the point applies equally well to any series of bets with similar overall properties (multiplicative rather than additive changes in wealth). 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?
  - kgwgk 6y ago
    The logarithmic utility doesn’t fit observed behavior particularly well. But it does lead to growth optimization. If you think the problem call for the maximization of asymptotic growth use logarithmic utility. If you don’t, don’t. Again, it’s exactly the same assumption. Not more arbitrary. 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”.
    - mrow84 6y ago
      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?
      - kgwgk 6y ago
        > what other realistic goal would there be? 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?
        mrow84 6y ago
        I do, and no, not precisely, but I would prefer to have more wealth than less at any time step - that is a preference that is invariant. However, what may change is the structure of the problem in front of me - the nature of the bets, or my time horizon (as I age). Realistic spending vastly complicates the situation for any analysis, but some basic spending pattern does not, and is effectively just an exogenous event that doesn't affect the analysis. 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.

by mrow84 6y ago

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!

mrow84 6y ago
The assumption that one's life will be composed of a series of risky choices doesn't seem to be a very strong one. The use of a single form of bet is just illustrative - the point applies equally well to any series of bets with similar overall properties (multiplicative rather than additive changes in wealth). 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?
- kgwgk 6y ago
  The logarithmic utility doesn’t fit observed behavior particularly well. But it does lead to growth optimization. If you think the problem call for the maximization of asymptotic growth use logarithmic utility. If you don’t, don’t. Again, it’s exactly the same assumption. Not more arbitrary. 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”.
  - mrow84 6y ago
    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?
    - kgwgk 6y ago
      > what other realistic goal would there be? 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?
      - mrow84 6y ago
        I do, and no, not precisely, but I would prefer to have more wealth than less at any time step - that is a preference that is invariant. However, what may change is the structure of the problem in front of me - the nature of the bets, or my time horizon (as I age). Realistic spending vastly complicates the situation for any analysis, but some basic spending pattern does not, and is effectively just an exogenous event that doesn't affect the analysis. 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.