I wonder if we can “extend” utility maximization representation theorems to drop Completeness. There’s already an extension to drop Continuity by using an ordinal-indexed vector (sequence) of real numbers, with entries sorted lexicographically (“lexicographically ordered ordinal sequences of bounded real utilities”, Russell and Isaacs, 2020). If we drop Completeness, maybe we can still represent the order with a vector of independent but incomparable dimensions across which it must respect ex ante Pareto efficiency (and each of those dimensions could also be split into an ordinal-indexed vector of real numbers with entries sorted lexicographically, if we’re also dropping Continuity)?
These also give us examples of somewhat natural/non-crazy orders that are consistent with dropping Completeness. I’ve seen people (including some economists) claim interpersonal utility comparisons are impossible and that we should only seek Pareto efficiency across people and not worry about tradeoffs between people. (Said Achmiz already pointed this and other examples out.)
Intuitively, the dimensions don’t actually need to be totally independent. For example, the order could be symmetric/anonymous/impartial between some dimensions, i.e. swapping values between these dimensions gives indifference. You could also have some strict preferences over some large tradeoffs between dimensions, but not small tradeoffs. Or even, maybe you want more apples and more oranges without tradeoffs between them, but also prefer more bananas to more apples and more bananas to more oranges. Or, a parent, having to give a gift to one of their children, may strictly prefer randomly choosing over picking one child to give it to, and find each nonrandom option incomparable to one another (although this may have problems when they find out which one they will give to, and then give them the option to rerandomize again; they might never actually choose).
Maybe you could still represent all of this with a large number of, possibly infinitely many, real-valued utility functions (or utility functions representable by “lexicographically ordered ordinal sequences of bounded real utilities”) instead. So, the correct representation could still be something like a (possibly infinite) set of utility functions (each possibly a “lexicographically ordered ordinal sequences of bounded real” utility functions), across which you must respect ex ante Pareto efficiency. This would be similar to the maximality rule over your representor/credal set/credal committee for imprecise credences (Mogensen, 2019).
Then, just combine this with your policy “if I previously turned down some option X, I will not choose any option that I strictly disprefer to X”, where strictly disprefer is understood to mean ex ante Pareto dominated.
But now this seems like a coherence theorem, just with a broader interpretation of “expected utility”.
To be clear, I don’t know if this “theorem” is true at all.
Possibly also related: McCarthy et al., 2020 have a utilitarian representation theorem that’s consistent with “the rejection of all of the expected utility axioms, completeness, continuity, and independence, at both the individual and social levels”. However, it’s not a real-valued representation. It reduces lotteries over a group of people to a lottery over outcomes for one person, as the probabilistic mixture of each separate person’s lottery into one lottery.
I think you are right about the representation claim since any quasi-ordering (reflexive and transitive relation) can be represented as the intersection of complete quasi-orderings.
EDIT: Looks like a similar point made here.
I wonder if we can “extend” utility maximization representation theorems to drop Completeness. There’s already an extension to drop Continuity by using an ordinal-indexed vector (sequence) of real numbers, with entries sorted lexicographically (“lexicographically ordered ordinal sequences of bounded real utilities”, Russell and Isaacs, 2020). If we drop Completeness, maybe we can still represent the order with a vector of independent but incomparable dimensions across which it must respect ex ante Pareto efficiency (and each of those dimensions could also be split into an ordinal-indexed vector of real numbers with entries sorted lexicographically, if we’re also dropping Continuity)?
These also give us examples of somewhat natural/non-crazy orders that are consistent with dropping Completeness. I’ve seen people (including some economists) claim interpersonal utility comparisons are impossible and that we should only seek Pareto efficiency across people and not worry about tradeoffs between people. (Said Achmiz already pointed this and other examples out.)
Intuitively, the dimensions don’t actually need to be totally independent. For example, the order could be symmetric/anonymous/impartial between some dimensions, i.e. swapping values between these dimensions gives indifference. You could also have some strict preferences over some large tradeoffs between dimensions, but not small tradeoffs. Or even, maybe you want more apples and more oranges without tradeoffs between them, but also prefer more bananas to more apples and more bananas to more oranges. Or, a parent, having to give a gift to one of their children, may strictly prefer randomly choosing over picking one child to give it to, and find each nonrandom option incomparable to one another (although this may have problems when they find out which one they will give to, and then give them the option to rerandomize again; they might never actually choose).
Maybe you could still represent all of this with a large number of, possibly infinitely many, real-valued utility functions (or utility functions representable by “lexicographically ordered ordinal sequences of bounded real utilities”) instead. So, the correct representation could still be something like a (possibly infinite) set of utility functions (each possibly a “lexicographically ordered ordinal sequences of bounded real” utility functions), across which you must respect ex ante Pareto efficiency. This would be similar to the maximality rule over your representor/credal set/credal committee for imprecise credences (Mogensen, 2019).
Then, just combine this with your policy “if I previously turned down some option X, I will not choose any option that I strictly disprefer to X”, where strictly disprefer is understood to mean ex ante Pareto dominated.
But now this seems like a coherence theorem, just with a broader interpretation of “expected utility”.
To be clear, I don’t know if this “theorem” is true at all.
Possibly also related: McCarthy et al., 2020 have a utilitarian representation theorem that’s consistent with “the rejection of all of the expected utility axioms, completeness, continuity, and independence, at both the individual and social levels”. However, it’s not a real-valued representation. It reduces lotteries over a group of people to a lottery over outcomes for one person, as the probabilistic mixture of each separate person’s lottery into one lottery.
I think you are right about the representation claim since any quasi-ordering (reflexive and transitive relation) can be represented as the intersection of complete quasi-orderings.