Thanks for posting this! If I understand your “risky” assumptions correctly, it seems to be targeted at people who believe (as a simple example):
Apples are better than oranges, and furthermore no amount of oranges can equate to one Apple
Nonetheless, it’s better to have a high probability of receiving an orange than a small probability of getting one Apple
Is that correct?
If so, what is the argument for believing both of these? My assumption is that someone who thinks that apples are lexically better than oranges would disagree with (2) and believe that any probability of an Apple is better than any probability of an orange.
Side question: the “risky” axioms seem quite similar to the Archimedean axiom in some variants of the VNM utility theorem. I think you also assume completeness and transitivity – are they enough to recover the entire VNM theorem? (I.e. do your axioms imply that there is a real-valued utility function whose expectation we must be trying to maximize?)
Side question: the “risky” axioms seem quite similar to the Archimedean axiom in some variants of the VNM utility theorem. I think you also assume completeness and transitivity – are they enough to recover the entire VNM theorem? (I.e. do your axioms imply that there is a real-valued utility function whose expectation we must be trying to maximize?)
This is interesting. It looks like the risky versions would follow from the Archidemean axiom + their non-risky vesions.
I don’t think you could get the independence axiom from the other axioms, though. Well, technically anything satisfying all of the axioms would satisfy independence, since nothing satisfies all of the axioms, since it’s an impossibility theorem, but if you consider only the risky axioms (or the Archimedean axiom), completeness and transitivity, I don’t see how you could get the independence axiom. Maybe maximizing the median value of some standard population axiology like total utilitarianism is a counterexample?
Thanks for your comment! I think the following is a closer analogy to what I say in the paper:
Suppose apples are better than oranges, which are in turn better than bananas. And suppose your choices are:
An apple and m bananas for sure.
An apple with probability 1−p and an orange with probability p, along with m oranges for sure.
Then even if you believe:
One apple is better than any amount of oranges
It still seems as if, for some large m and small p, 2 is better than 1. 2 slightly increases the risk you miss out on an apple, but it compensates you for that increased risk by giving you many oranges rather than many bananas.
On your side question, I don’t assume completeness! But maybe if I did, then you could recover the VNM theorem. I’d have to give it more thought.
Thanks for posting this! If I understand your “risky” assumptions correctly, it seems to be targeted at people who believe (as a simple example):
Apples are better than oranges, and furthermore no amount of oranges can equate to one Apple
Nonetheless, it’s better to have a high probability of receiving an orange than a small probability of getting one Apple
Is that correct?
If so, what is the argument for believing both of these? My assumption is that someone who thinks that apples are lexically better than oranges would disagree with (2) and believe that any probability of an Apple is better than any probability of an orange.
Side question: the “risky” axioms seem quite similar to the Archimedean axiom in some variants of the VNM utility theorem. I think you also assume completeness and transitivity – are they enough to recover the entire VNM theorem? (I.e. do your axioms imply that there is a real-valued utility function whose expectation we must be trying to maximize?)
This is interesting. It looks like the risky versions would follow from the Archidemean axiom + their non-risky vesions.
I don’t think you could get the independence axiom from the other axioms, though. Well, technically anything satisfying all of the axioms would satisfy independence, since nothing satisfies all of the axioms, since it’s an impossibility theorem, but if you consider only the risky axioms (or the Archimedean axiom), completeness and transitivity, I don’t see how you could get the independence axiom. Maybe maximizing the median value of some standard population axiology like total utilitarianism is a counterexample?
Thanks! Your points about independence sound right to me.
Thanks for your comment! I think the following is a closer analogy to what I say in the paper:
On your side question, I don’t assume completeness! But maybe if I did, then you could recover the VNM theorem. I’d have to give it more thought.