I think you are not allowed to refer explicitly to utility in the options. That is an option of “I do not choose this option” is selfdefeating and illformed. In another post I posited a risk-averse utility function that references amount of paperclips. Maximising the utility function doesn’t maximise expected amount of paperclips. Even if the physical objects of interest are paperclips and we value them linearly a paperclip is not synonymous with utilon. It’s not a thing you can give out in an option.
I think you are not allowed to refer explicitly to utility in the options.
I was going to answer that I can easily reword my example to not explicitly mention any utility values, but when I tried to that it very quickly led to something where it is obvious that u(A) = u(C). I guess my rewording was basically going through the steps of the proof of VNM theorem.
I am still not sure I am convinced by your objection, as I don’t think there’s anything self-referential in my example, but that did give me some pause.
In a case where you are going to pick less variance less expected value over more variance more expected value it will mean that option needs to have a bigger “utility number”. In order to get that you need to mess with how utility is calculated. Then it becomes ambigious whether the “utility-fruits” are redefined in the same go as we redefine how we compare options. If we name them “paperclips” it’s clear that they are not touched by such redefining.
It triggerred a “type-unsafety” trigger but the operation overall might be safe as it doesn’t actualise the danger. For example having an option of “plum + 2 utility” could give one agent “plum + apple” if it valued apples and “plum + pear” if it valued pears. I guess if you consistenly replace all physical items for their utility values it doesn’t happen.
In the case of “gain 1 utility with probability 1” if your agent is risk-seeking it might give this option “actual” utility less than 1. In general if we lose the distribution independence we might need to retain the information of our suboutcomes rather than collapsing it to he a single number. For if an agent is risk-seeking it’s clear that it would prefer A=( 5% 0,90% 1, 5% 2) to B=(100%, 1). But same risk-seeking in combined lotteries would make it prefer C=(5% , 90% A, 5% A+A) over A. When comparing C and A it’s not sufficent to know that their expected utilities are 1.
I think you are not allowed to refer explicitly to utility in the options. That is an option of “I do not choose this option” is selfdefeating and illformed. In another post I posited a risk-averse utility function that references amount of paperclips. Maximising the utility function doesn’t maximise expected amount of paperclips. Even if the physical objects of interest are paperclips and we value them linearly a paperclip is not synonymous with utilon. It’s not a thing you can give out in an option.
I was going to answer that I can easily reword my example to not explicitly mention any utility values, but when I tried to that it very quickly led to something where it is obvious that u(A) = u(C). I guess my rewording was basically going through the steps of the proof of VNM theorem.
I am still not sure I am convinced by your objection, as I don’t think there’s anything self-referential in my example, but that did give me some pause.
In a case where you are going to pick less variance less expected value over more variance more expected value it will mean that option needs to have a bigger “utility number”. In order to get that you need to mess with how utility is calculated. Then it becomes ambigious whether the “utility-fruits” are redefined in the same go as we redefine how we compare options. If we name them “paperclips” it’s clear that they are not touched by such redefining.
It triggerred a “type-unsafety” trigger but the operation overall might be safe as it doesn’t actualise the danger. For example having an option of “plum + 2 utility” could give one agent “plum + apple” if it valued apples and “plum + pear” if it valued pears. I guess if you consistenly replace all physical items for their utility values it doesn’t happen.
In the case of “gain 1 utility with probability 1” if your agent is risk-seeking it might give this option “actual” utility less than 1. In general if we lose the distribution independence we might need to retain the information of our suboutcomes rather than collapsing it to he a single number. For if an agent is risk-seeking it’s clear that it would prefer A=( 5% 0,90% 1, 5% 2) to B=(100%, 1). But same risk-seeking in combined lotteries would make it prefer C=(5% , 90% A, 5% A+A) over A. When comparing C and A it’s not sufficent to know that their expected utilities are 1.