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Stuart_Armstrong comments on Expected utility without the independence axiom - Less Wrong
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It's a good result, but I wonder if the standard deviation is the best parameter. Loss-averse agents react differently to asymmetrical distributions allowing large losses than those allowing large gains.
Edit: For example, the mean of an exponential distribution f(x;t) = L * e^(-L*x) has mean and standard deviation 1/L, but a loss-averse agent is likely to prefer it to the normal distribution N(1/L, 1/L^2), which has the same mean and standard deviation.
Once you abanndon independence, the possibilities are litteraly infinite - and not just easily controllable infinities, either. I worked with SD as that's the simplest model I could use; but skewness, kurtosis or, Bayes help us, the higher moments, are also valid choices.
You just have to be careful that your choice of units is consistent; the SD and the mean are in the same unit, the variance is in units squared, the skewness and kurtosis are unitless, the k-th moment is in units to the power k, etc...
That's true - and it occurred to me after I posted the comment that your criteria don't define the decision system anyway, so even using some other method you might still be able to prove that it meets your conditions.