Appendix to: A fungibility theorem
Suppose that is a set and we have functions . Recall that for , we say that is a Pareto improvement over if for all , we have . And we say that it is a strong Pareto improvement if in addition there is some for which . We call a Pareto optimum if there is no strong Pareto improvement over it.
Theorem. Let be a set and suppose for are functions satisfying the following property: For any and any , there exists an such that for all , we have .
Proof. Let . By hypothesis, the image is convex.
For , let the Pareto volume of be the set
This is a closed convex set. Note that is a Pareto optimum precisely when . Let's assume that this is the case; we just have to prove that maximizes for some choice of .
It suffices to find a hyperplane that contains and that supports . Then the desired function can be constructed by ensuring that is a level set.
If lies in a proper affine subspace of , let be the smallest such subspace. Let be the interior of in and let be the interior of . The case where is a point is trivial; suppose it's not, so is nonempty. By convexity, is the closure of and is the closure of .
Since is convex, is convex, and we can exhaust with a nested sequence of nonempty compact convex sets . And is convex, so we can exhaust with a nested sequence of nonempty compact convex sets . By the hyperplane separation theorem, for each , there is a hyperplane separating and . I claim that has a convergent subsequence. Indeed, each must intersect the convex hull of , and the space of hyperplanes intersecting that convex hull is compact. So has a subsequence that converges to a hyperplane .
It's easy to see that separates from for each , and so separates from . So must contain and support .
Note that the theorem does not guarantee the existence of a Pareto optimum. But if is closed and bounded, then a Pareto optimum exists.
A limitation of the theorem is that it assumes a finite list of values , not an infinite one.