“Risk Aversion,” as a technical term, means that the utility function is concave with respect to its input
Risk aversion is separate from the properties of utility function. Being risk-averse rather means preferring a guaranteed payoff to a bet with the same expected utility. See here for a numerical example. It is possible to be risk averse even with a convex utility function.
That is a non-standard definition. (Standard definition.) Agents should always be indifferent between bets with identical expected utilities. (They do not always have to be indifferent between bets with identical expected payoffs.)
Preferring a guarantee to a bet is the certainty effect, like I claimed in the grandparent.
Rational agents should be. Irrational agents—in this case, prone to risk aversion—would instead be willing to pay a finite cost for the bet to be replaced with the sure deal, thus losing utility. You can fix this by explicitly incorporating risk in the utility function, making the agent rational and not risk-averse any more.
This strikes me as, though I’m unsure as to which technical term applies here, ‘liking your theory too much’. ’Tis necessary to calculate the probability of payoff for each of two exclusive options of identical utility in order to rationally process the choice. If Option A of Utilon 100 occurs with 80% probability and Option B also of Utilon 100 occurs with 79.9% probability, Option A is the more rational choice. To recognise the soundness of Vaniver’s following statement, one must acknowledge the necessity of calculating risk. [Additionally, if two options are of unequal utility, differences in payoff probabilities become even more salient as the possible disutility of no payoff should lower one’s utility estimate of whichever choice has the lower payoff probability (assuming there is one).]*
(They do not always have to be indifferent between bets with identical expected payoffs.)
Honestly the above seems so simple that I very much think I’ve misunderstood something, in which case please view this as a request for clarification.
[...]* This also seems obvious, but on an intuitive mathematical level, thus I don’t have much confidence in it; it fit better up there than down here.
Again, what you are saying is a non-standard definition. The commonly used term for the bias you’re describing is certainty effect, and risk aversion is used to refer to concave utility functions.
First, concave utility function is just a model for risk aversion which is “the reluctance of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain, but possibly lower, expected payoff.” (wiki)
Second, the certainty effect is indeed one of the effects that is captured by my preferred model, but of course it’s not limited to it, because it’s possible to behave in a risk-averse manner even if none of the offered bets are certain.
This! If you’re risk averse, then you want to avoid risk, and so in the real utility calculation upon which you base your decisions the risk-averse option gets a little extra positive term for being, well, risk-averse. And then the two options no longer have the same expected utility.
Unfortunately, under your new “fixed” utility function there will again be a point of indifference at some slightly different probability/payoff combination, where you, being risk-averse, have to go for the sure deal, so you will end up stuck in an infinite recursion trying to adjust your utility function further and further. I tried to explain this more clearly here.
I don’t think that follows. The risk-aversion utility attaches to the choice, not the outcome: I get extra utility for having made a choice with lower expected variance, not for one of the outcomes. If you then offer me a choice between choices, then sure, there will be more risk aversion, but I don’t think it’s viciously recursive.
Risk aversion is separate from the properties of utility function. Being risk-averse rather means preferring a guaranteed payoff to a bet with the same expected utility. See here for a numerical example. It is possible to be risk averse even with a convex utility function.
That is a non-standard definition. (Standard definition.) Agents should always be indifferent between bets with identical expected utilities. (They do not always have to be indifferent between bets with identical expected payoffs.)
Preferring a guarantee to a bet is the certainty effect, like I claimed in the grandparent.
Rational agents should be. Irrational agents—in this case, prone to risk aversion—would instead be willing to pay a finite cost for the bet to be replaced with the sure deal, thus losing utility. You can fix this by explicitly incorporating risk in the utility function, making the agent rational and not risk-averse any more.
That sounds like a …drumroll… terminal bias.
Enshrining biases as values in your utility function seems like the wrong thing to do.
This strikes me as, though I’m unsure as to which technical term applies here, ‘liking your theory too much’. ’Tis necessary to calculate the probability of payoff for each of two exclusive options of identical utility in order to rationally process the choice. If Option A of Utilon 100 occurs with 80% probability and Option B also of Utilon 100 occurs with 79.9% probability, Option A is the more rational choice. To recognise the soundness of Vaniver’s following statement, one must acknowledge the necessity of calculating risk. [Additionally, if two options are of unequal utility, differences in payoff probabilities become even more salient as the possible disutility of no payoff should lower one’s utility estimate of whichever choice has the lower payoff probability (assuming there is one).]*
Honestly the above seems so simple that I very much think I’ve misunderstood something, in which case please view this as a request for clarification.
[...]* This also seems obvious, but on an intuitive mathematical level, thus I don’t have much confidence in it; it fit better up there than down here.
Again, what you are saying is a non-standard definition. The commonly used term for the bias you’re describing is certainty effect, and risk aversion is used to refer to concave utility functions.
First, concave utility function is just a model for risk aversion which is “the reluctance of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain, but possibly lower, expected payoff.” (wiki)
Second, the certainty effect is indeed one of the effects that is captured by my preferred model, but of course it’s not limited to it, because it’s possible to behave in a risk-averse manner even if none of the offered bets are certain.
Another way to interpret this situation is that the “utility function” being used to calculate the expected value is a fake utility function.
This! If you’re risk averse, then you want to avoid risk, and so in the real utility calculation upon which you base your decisions the risk-averse option gets a little extra positive term for being, well, risk-averse. And then the two options no longer have the same expected utility.
Unfortunately, under your new “fixed” utility function there will again be a point of indifference at some slightly different probability/payoff combination, where you, being risk-averse, have to go for the sure deal, so you will end up stuck in an infinite recursion trying to adjust your utility function further and further. I tried to explain this more clearly here.
I don’t think that follows. The risk-aversion utility attaches to the choice, not the outcome: I get extra utility for having made a choice with lower expected variance, not for one of the outcomes. If you then offer me a choice between choices, then sure, there will be more risk aversion, but I don’t think it’s viciously recursive.