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thelittledoctor comments on Terminal Bias - Less Wrong
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Risk aversion as a terminal value follows pretty naturally from decreasing marginal utility. For example imagine we have a paperclip-loving agent whose utility function is equal to sqrt(x), where x is the number of paperclips in the universe. Now imagine a lottery which either creates 9 or 25 paperclips, each with 50% probability - an expected net gain of 17 paperclips. Now give the agent a choice between 16.5 paperclips or a run of this lottery. Which choice maximizes the agent's expected utility?
In your example, given this utility function, risk aversion would correspond to consistently preferring guaranteed 16 paperclips to the bet you describe. In this case, by Savage's theorem (see postulate #4) there must exist a finite number δ > 0 such that you would also prefer a guaranteed payoff of 16 to the bet defined by {P(25) = 0.5 + δ, P(9) = 0.5 - δ}, costing you an expected utility of 2δ > 0.
I'm not sure I understand why. The lottery has an expected utility of (sqrt(9)+sqrt(25))/2=4, so shouldn't the agent express indifference between the lottery and 16 guaranteed paperclips? This behavior alone seems risk-averse to me, given that the lottery produces an expected (9+25)/2=17 paperclips.
Sidenote, is there a way to use LaTeX on here?
John Maxwell made a LaTeX editor (which gives you Markdown code you can paste into a comment).
Sorry, I made a mistake in the example, it's of course 16 not 15. Edited to correct.
Yes, the agent should - given the defined utility function and that the agent is rational. If, however, the agent is irrational and prone to risk aversion, it will consistently prefer the sure deal to the bet, and therefore be willing to pay a finite cost for replacing the bet with the sure deal, hence losing utility.
That's not risk aversion, it's just decreasing marginal utility. They look different to me.
And it's still not a terminal value, it would be instrumental.
They're really mathematically equivalent ways of expressing the same thing. If they look different to you that's a flaw in your intuition, you may want to correct it.
Ok, let's taboo "risk aversion", I'm talking about what a minimax algorithm does, where it comes up with possibilities, rates them by utility, and takes actions to avoid the worst outcomes. This is contrasted to a system that also computes probabilities to get expected utilities, and acts to maximize that. Sure you can make your utility function strongly concave to hack the traits of the minimax system into a utility maximizer, but saying that they are "mathematically equivalent" seems to be missing the point.
That's called "certainty effect" and no one is claiming that it's a terminal value.
Ok, thanks for the terminology help.