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.
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.
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).
frac{sqrt{9} sqrt{25}}{2}=4
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.