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Douglas_Knight comments on Winning the Unwinnable - Less Wrong
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Comments (51)
or is it just a standard assumption? I've never heard anything more precise than declining marginal utility.
Pretty sure it's the standard result that people don't consistently assign utilities to levels of wealth.
Logarithmic u-functions have an uncomfortable requirement that you must be indifferent to your current wealth and a 50-50 shot at doubling or halving it (e.g. doubling or halving every paycheck/payment you get for the rest of your life). Most people I know don't like that deal.
That's only a requirement for risk-neutral people. Most people you know are not risk-neutral.
Logarithmic utility functions are already risk-averse by virtue of their concavity. The expected value of a 50% chance of doubling or halving is a 25% gain.
Nitpick: you put the values in utiles, which should include risk-aversion. If you put the values in dollars or something, I would agree.
I would say that such a person doesn't have preferences representable by a utility function.
I don't think opportunities to make choices are usually considered to be in the domain of a utility function. (If I'm wrong, educate me. I'd appreciate it.)
Ok, I looked it up and it looks like you and thomblake (ETA: and Technologos. Thanks for correcting me!) are right: the usual way of doing it is to include risk aversion in the utility function. Sorry about that.
Wikipedia on risk-neutral measures does discuss the possibility of adjusting the probabilities, rather than the utility, when calculating the expected value of a choice, but it looks like that's usually done for ease of financial calculation.
So, one explanation for why people don't take the "half or double" gamble is that they do have the log(x) utility function, but don't behave accordingly because of loss aversion (as opposed to risk aversion).
The post is technical, but Stuart_Armstrong analyzed some special cases of not-quite-utility-function agents.
That's just plain false. Risk-aversion is a valid preference, and can be included as a term in a utility function (at slight risk of circularity, but that's not really a problem).
ETA: well, the stated units were utils, so risk-aversion should be included, so I think you're correct.
I'm confused about what is uncomfortable about this, or what function of wealth you would measure utility by.
Naively it seems that logarithmic functions would be more risk averse than nth root functions which I have seen Robin Hanson use. How would a u-function be more sensitive to current wealth?
I think the uncomfortable part is that bill's (and my) experience suggests that people are even more risk-averse than logarithmic functions would indicate.
I'd suggest that any consistent function (prospect theory notwithstanding) for human utility functions is somewhere between log(x) and log(log(x))... If I were given the option of a 50-50 chance of squaring my wealth and taking the square root, I would opt for the gamble.
Hmm, good question. Quick Google search doesn't turn up anything...