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conchis comments on Post Your Utility Function - Less Wrong
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For the specific quote: I know that, for a small enough change in wealth, I don't need to re-evaluate all the deals I own. They all remain pretty much the same. For example, if you told me a had $100 more in my bank account, I would be happy, but it wouldn't significantly change any of my decisions involving risk. For a utility curve over money, you can prove that that implies an exponential curve. Intuitively, some range of my utility curve can be approximated by an exponential curve.
Now that I know it is exponential over some range, I needed to figure out which exponential and over what range does it apply. I assessed for myself that I am indifferent between having and not having a deal with a 50-50 chance of winning $400K and losing $200K. The way I thought about that was how I thought about decisions around job hunting and whether I should take or not take job offers that had different salaries.
If that is true, you can combine it with the above and show that the exponential curve should look like u(x) = 1 - exp(-x/400K). Testing it against my intuitions, I find it an an okay approximation between $400K and minus $200K. Outside that range, I need better approximations (e.g. if you try it out on a 50-50 shot of $10M, it gives ridiculous answers).
Does this make sense?
It makes sense however you mention that you test it against your intuitions. My first reaction would be to say that this is introducing a biased variable which is not based on a reasonable calculation.
That may not be the case as you may have done so many complicated calculations such that your unconscious "intuitions" may give your conscious the right answer. However from the millionaires biographies I have read and rich people I have talked to a better representation of money and utility according to them is logarithmic rather than exponential. This would indicate to me that the relationship between utility and money would be counter-intuitive for those who have not experienced those levels which are being compared.
I have not had the fortune to experience anything more than a 5 figure income so I cannot reasonably say how my preferences would be modeled. I can reasonably believe that I would be better off at 500K than 50K through simple comparison of lifestyle between myself and a millionaire. I cannot make an accurate enough estimation of my utility and as a result I would not be prepared to make a estimation of what model would best represent it because the probability of that being accurate is likely the same as coin flipping.
Ed: I had a much better written post but an errant click lost the whole thing - time didn't allow the repetition of the better post.
Just to be clear, you know that an exponential utility function (somewhat misleadingly ) doesn't actually imply that utility is exponential in wealth, right? Bill's claimed utility function doesn't exhibit increasing marginal utility, if that's what you're intuitively objecting to. It's 1-exp(-x), not exp(x).
Many people do find the constant absolute risk aversion implied by exponential utility functions unappealing, and prefer isoelastic utility functions that exhibit constant relative risk aversion, but it does have the advantage of tractability, and may be reasonable over some ranges.
Example of the "unappealingness" of constant absolute risk aversion. Say my u-curve were u(x) = 1-exp(-x/400K) over all ranges. What is my value for a 50-50 shot at 10M?
Answer: around $277K. (Note that it is the same for a 50-50 shot at $100M)
Given the choice, I would certainly choose a 50-50 shot at $10M over $277K. This is why over larger ranges, I don't use an exponential u-curve.
However, it is a good approximation over a range that contains almost all the decisions I have to make. Only for huge decisions to I need to drag out a more complicated u-curve, and they are rare.