or a non-EUM with legible goals (like a risk-averse money-maximizer)
Our prototypical examples of risk-averse money-maximizers are EUMs. In particular, the Kelly bettor is probably the most central example: it maximizes expected log wealth (i.e. log future money). The concavity of the logarithm makes it risk averse: a Kelly bettor will always take a sure payoff over an uncertain outcome with the same expected payoff.
I bring this up mainly because the wording makes it sound like you’re under the impression that being an EUM is inconsistent with being a risk-averse money-maximizer, in which case you probably have an incorrect understanding of the level of generality of (nontrivial) expected utility maximization, and should probably update toward EUMs being a better model of real agents more often than you previously thought.
I think my thought process when I typed “risk-averse money-maximizer” was that an agent could be risk-averse (in which case it wouldn’t be an EUM) and then separately be a money-maximizer.
But I didn’t explicitly think “the risk-aversion would be with regard to utility not money, and risk-aversion with regard to money could still be risk-neutral with regard to utility”, so I appreciate the clarification.
Note that the same mistake, but with convexity in the other direction, also shows up in the OP:
Alice and Bob could toss a coin to decide between options #1 and #2, but then they wouldn’t be acting as an EUM (since EUMs can’t prefer a probabilistic mixture of two options to either option individually).
An EUM can totally prefer a probabilistic mixture of two options to either option individually; this happens whenever utility is convex with respect to resources (e.g. money). For instance, suppose an agent’s utility is u(money) = money^2. I offer this agent a $1 bet on a fair coinflip at even odds, i.e. it gets $0 if it loses and $2 if it wins. The agent takes the bet: the bet offers u = 0.5*0^2 + 0.5*2^2 = 2, while the baseline offers a certainty of $1 which has u = 1.0*1^2 = 1.
Your example bet is a probabilistic mixture of two options: $0 and $2. The agent prefers one of the options individually (getting $2) over any probabilistic mixture of getting $0 and $2.
In other words, your example rebuts the claim that an EUM can’t prefer a probabilistic mixture of two options to the expectation of those two options. But that’s not the claim I made.
This is a tangent, but might be important.
Our prototypical examples of risk-averse money-maximizers are EUMs. In particular, the Kelly bettor is probably the most central example: it maximizes expected log wealth (i.e. log future money). The concavity of the logarithm makes it risk averse: a Kelly bettor will always take a sure payoff over an uncertain outcome with the same expected payoff.
I bring this up mainly because the wording makes it sound like you’re under the impression that being an EUM is inconsistent with being a risk-averse money-maximizer, in which case you probably have an incorrect understanding of the level of generality of (nontrivial) expected utility maximization, and should probably update toward EUMs being a better model of real agents more often than you previously thought.
I think my thought process when I typed “risk-averse money-maximizer” was that an agent could be risk-averse (in which case it wouldn’t be an EUM) and then separately be a money-maximizer.
But I didn’t explicitly think “the risk-aversion would be with regard to utility not money, and risk-aversion with regard to money could still be risk-neutral with regard to utility”, so I appreciate the clarification.
Note that the same mistake, but with convexity in the other direction, also shows up in the OP:
An EUM can totally prefer a probabilistic mixture of two options to either option individually; this happens whenever utility is convex with respect to resources (e.g. money). For instance, suppose an agent’s utility is u(money) = money^2. I offer this agent a $1 bet on a fair coinflip at even odds, i.e. it gets $0 if it loses and $2 if it wins. The agent takes the bet: the bet offers u = 0.5*0^2 + 0.5*2^2 = 2, while the baseline offers a certainty of $1 which has u = 1.0*1^2 = 1.
Your example bet is a probabilistic mixture of two options: $0 and $2. The agent prefers one of the options individually (getting $2) over any probabilistic mixture of getting $0 and $2.
In other words, your example rebuts the claim that an EUM can’t prefer a probabilistic mixture of two options to the expectation of those two options. But that’s not the claim I made.