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__ 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 pr

• kgwgk 6y ago
  __ A simple investment decision application __ Imagine that the outcomes are different levels of your wealth at the end of the year. For example A is $500k, B is $50k, C is $200k. What is the utility function that describes your preferences? (There should be one if you're rational in the sense of the EUT axioms.) Surely we agree that you prefer A to C and C to B. But would you prefer to have $500k or $50k with 50%/50% probability or to be certain of having $200k at the end of the year? EUT doesn't give you an answer. Of course it's easy to calculate the expected value of the first alternative ($275k) but nobody says that you should prefer the "risky" $275k to the certain $200k. That's the point of introducing utility functions in the analysis, that preferences may be non-linear. What EUT says is that if you're rational (etc) there are three numbers U($500k), U($50k) and U($275k). The expected utility of the first option is 0.5 U($500k) + 0.5 U($50k) and the expected utility of the second option is U($200k). These expected utilities can be compared to see if you which option do you prefer. If you prefer one, you could also be indifferent. EUT says that the rational decision is to chose the option with higher expected utility. __ Remarks __ EUT doesn't fix the form of the utility function. Usually it's assumed that instead of linear it's concave. This means that a certain dollar is better than a dollar in expectation and you would never take fair bets. (The fact that people plays in the casino where bets are not even fair could be explained with convex utility. One could explain it as well saying that there is utility obtained from the entertainment that offsets the monetary loss.) The utility function describing someone's preferences could be very complex. Simple models are used for many reasons including tractability or theoretical properties. For example power functions (logarithmic utility is a special case). There are some reasons to like logarithmic utility (including growth-optimality arguments) but it doesn't work too well empirically. This is why, at least in some settings, it may be better to use the more general power utilities that have a parameter that can be adjusted. Using logarithmic utility, the $200k option is preferable to the $500k/$50k bet. The certainty equivalent is $158k. Logarithmic utility (like the rest of the power utilities) is scale invariant. The amounts could by 100 times larger or 100 times smaller and the answer would be the same. Does Ole Peters claim that the use of EUT here is wrong? I guess so. Parallel universes again, probably. Would you say that there is a problem with EUT up to this point? What are the problematic assumptions made by EUT? In my opinion it's EE who cannot really solve this problem without making "unphysical" assumptions. The solution corresponds to using the logarithmic utility above, EE declares any other preference the agent may have wrong. (To be continued, maybe.)
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• 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.
    • 6y ago