Bravo, Eliezer. Anyone who says the answer to this is obvious is either WAY smarter than I am, or isn’t thinking through the implications.
Suppose we want to define Utility as a function of pain/discomfort on the continuum of [dust speck, torture] and including the number of people afflicted. We can choose whatever desiderata we want (e.g. positive real valued, monotonic, commutative under addition).
But what if we choose as one desideratum, “There is no number n large enough such that Utility(n dust specks) > Utility(50 yrs torture).” What does that imply about the function? It can’t be analytic in n (even if n were continuous). That rules out multaplicative functions trivially.
Would it have singularities? If so, how would we combine utility functions at singular values? Take limits? How, exactly?
Or must dust specks and torture live in different spaces, and is there no basis that can be used to map one to the other?
The bottom line: is it possible to consistently define utility using the above desideratum? It seems like it must be so, since the answer is obvious. It seems like it must not be so, because of the implications for the utility function as the arguments change.
Edit:
After discussing with my local meetup, this is somewhat resolved. The above desiderata require the utility to be bounded in the number of people, n. For example, it could be a staurating exponential function. This is self-consistent, but inconsistent with the notion that because experience is independent, utilities should add.
Interestingly, it puts strict mathematical rules on how utility can scale with n.
Bravo, Eliezer. Anyone who says the answer to this is obvious is either WAY smarter than I am, or isn’t thinking through the implications.
Suppose we want to define Utility as a function of pain/discomfort on the continuum of [dust speck, torture] and including the number of people afflicted. We can choose whatever desiderata we want (e.g. positive real valued, monotonic, commutative under addition).
But what if we choose as one desideratum, “There is no number n large enough such that Utility(n dust specks) > Utility(50 yrs torture).” What does that imply about the function? It can’t be analytic in n (even if n were continuous). That rules out multaplicative functions trivially.
Would it have singularities? If so, how would we combine utility functions at singular values? Take limits? How, exactly?
Or must dust specks and torture live in different spaces, and is there no basis that can be used to map one to the other?
The bottom line: is it possible to consistently define utility using the above desideratum? It seems like it must be so, since the answer is obvious. It seems like it must not be so, because of the implications for the utility function as the arguments change.
Edit: After discussing with my local meetup, this is somewhat resolved. The above desiderata require the utility to be bounded in the number of people, n. For example, it could be a staurating exponential function. This is self-consistent, but inconsistent with the notion that because experience is independent, utilities should add.
Interestingly, it puts strict mathematical rules on how utility can scale with n.