5 ms·

> 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)

by 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!
  - 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”.
      - 6y ago

by kgwgk 6y ago

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”.
    - 6y ago