What you described is compatible with EU maximization, except the part where you claim your utility to be linear in X. That seems like a wrong claim. The ultimate source of truth when determining an agent’s utility function is the agent’s preferences among actions. (The vNM theorem takes preferences among actions as given, and hacks together a utility function describing them.) And your preferences among actions imply a utility function that’s nonlinear in X.
How does non-linearity lead to me choosing different options in single vs iterated problems?
I’m fine with saying I maximise expected utility (I interpret that as it is possible to construct an expected utility maximising agent with some preference who would always choose the same strategy I do), but I’m not sure this is the case.
To offer insight into my utility function:
In singleton problems:
If probability of a set of states is below epsilon, I ignore that set of states.
In iterated problems, I consider it iff the probability of the set of states is high enough that I expect it to occur at least one during the number of iterations.
Only one state of the world would manifest. If I not expect to not see that state of the world, I ignore it, irrespective of the payoff of that state. You could interpret this as a bounded utility function. However, in iterated problems I might consider that state, so my utility function isn’t bounded.
I’m trying to maximise utility, and not expected utility. In problems with pathological (very unequal) probability distributions, I may completely ignore a certain set of states. This is because in a given singleton problem, I expect that state to not occur. I don’t care about other Everett branches, so some of the EU arguments also don’t move me.
DagonGod, you are clearly not getting the point here, which is that the vN-M theorem that defines utility is not compatible with you declaring values of your utility function. If you do that, you are no longer talking about the same concept of “utility”.
The concept of a utility function is only relevant insomuch as you can model rational decision makers as possessing a utility function that they try to maximise in some way. I do possess a utility function (not necessarily in the VnM sense as I don’t maximise expected utility, and maximising expected utility is implicit in the definition of VnM utility (this is a point of contention for me)). If I make choices that don’t maximise expected utility, then you must be able to demonstrate that I am irrational on some way (without special pleading to my failure to maximise EU). Either that, or maximising expected utility is not the perfect performance measure for rational choice.
What you described is compatible with EU maximization, except the part where you claim your utility to be linear in X. That seems like a wrong claim. The ultimate source of truth when determining an agent’s utility function is the agent’s preferences among actions. (The vNM theorem takes preferences among actions as given, and hacks together a utility function describing them.) And your preferences among actions imply a utility function that’s nonlinear in X.
How does non-linearity lead to me choosing different options in single vs iterated problems?
I’m fine with saying I maximise expected utility (I interpret that as it is possible to construct an expected utility maximising agent with some preference who would always choose the same strategy I do), but I’m not sure this is the case.
To offer insight into my utility function:
Only one state of the world would manifest. If I not expect to not see that state of the world, I ignore it, irrespective of the payoff of that state. You could interpret this as a bounded utility function. However, in iterated problems I might consider that state, so my utility function isn’t bounded.
I’m trying to maximise utility, and not expected utility. In problems with pathological (very unequal) probability distributions, I may completely ignore a certain set of states. This is because in a given singleton problem, I expect that state to not occur. I don’t care about other Everett branches, so some of the EU arguments also don’t move me.
DagonGod, you are clearly not getting the point here, which is that the vN-M theorem that defines utility is not compatible with you declaring values of your utility function. If you do that, you are no longer talking about the same concept of “utility”.
The concept of a utility function is only relevant insomuch as you can model rational decision makers as possessing a utility function that they try to maximise in some way. I do possess a utility function (not necessarily in the VnM sense as I don’t maximise expected utility, and maximising expected utility is implicit in the definition of VnM utility (this is a point of contention for me)). If I make choices that don’t maximise expected utility, then you must be able to demonstrate that I am irrational on some way (without special pleading to my failure to maximise EU). Either that, or maximising expected utility is not the perfect performance measure for rational choice.