Suppose that P is a set and we have functions v1,…,vn:P→R. Recall that for p,q∈P, we say that p is a Pareto improvement over q if for all i, we have vi(p)≥vi(q). And we say that it is a strong Pareto improvement if in addition there is some i for which vi(p)>vi(q). We call p a Pareto optimum if there is no strong Pareto improvement over it.
Theorem. Let P be a set and suppose vi:P→R for i=1,…,n are functions satisfying the following property: For any p,q∈P and any α∈[0,1], there exists an r∈P such that for all i, we have vi(r)=αvi(p)(1−α)vi(q).
Then if an element p of P is a Pareto optimum, then there exist nonnegative constants c1,…,cn such that the function ∑civi achieves a maximum at p.
Proof. Let V=v1×⋯×vn:P→Rn. By hypothesis, the image V(P) is convex.
For p∈P, let the Pareto volume of p be the set
pv(p)={(x1,…,xn)∈Rn∣xi≥vi(p) for all i}
This is a closed convex set. Note that p∈P is a Pareto optimum precisely when pv(p)∩V(P)={V(p)}. Let’s assume that this is the case; we just have to prove that p maximizes ∑civi for some choice of ci≥0.
It suffices to find a hyperplane H⊂Rn that contains V(p) and that supportsV(P). Then the desired function ∑civi can be constructed by ensuring that H is a level set.
If V(P) lies in a proper affine subspace of Rn, let Z be the smallest such subspace. Let A be the interior of V(P) in Z and let B be the interior of pv(p). The case where V(P) is a point is trivial; suppose it’s not, so A is nonempty. By convexity, V(P) is the closure of A and pv(p) is the closure of B.
Since V(P) is convex, A is convex, and we can exhaust A with a nested sequence of nonempty compact convex sets Aj. And pv(p) is convex, so we can exhaust B with a nested sequence of nonempty compact convex sets Bj. By the hyperplane separation theorem, for each j, there is a hyperplane Hj separating Aj and Bj. I claim that Hj has a convergent subsequence. Indeed, each Hj must intersect the convex hull of A1∪B1, and the space of hyperplanes intersecting that convex hull is compact. So Hj has a subsequence that converges to a hyperplane H.
It’s easy to see that H separates Aj from Bj for each j, and so H separates A from B. So H must contain V(p) and support V(P).
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Note that the theorem does not guarantee the existence of a Pareto optimum. But if V(P) is closed and bounded, then a Pareto optimum exists.
A limitation of the theorem is that it assumes a finite list of values v1,…,vn, not an infinite one.
Proof of fungibility theorem
Appendix to: A fungibility theorem
Suppose that P is a set and we have functions v1,…,vn:P→R. Recall that for p,q∈P, we say that p is a Pareto improvement over q if for all i, we have vi(p)≥vi(q). And we say that it is a strong Pareto improvement if in addition there is some i for which vi(p)>vi(q). We call p a Pareto optimum if there is no strong Pareto improvement over it.
Theorem. Let P be a set and suppose vi:P→R for i=1,…,n are functions satisfying the following property: For any p,q∈P and any α∈[0,1], there exists an r∈P such that for all i, we have vi(r)=αvi(p)(1−α)vi(q).
Then if an element p of P is a Pareto optimum, then there exist nonnegative constants c1,…,cn such that the function ∑civi achieves a maximum at p.
Proof. Let V=v1×⋯×vn:P→Rn. By hypothesis, the image V(P) is convex.
For p∈P, let the Pareto volume of p be the set
pv(p)={(x1,…,xn)∈Rn∣xi≥vi(p) for all i}
This is a closed convex set. Note that p∈P is a Pareto optimum precisely when pv(p)∩V(P)={V(p)}. Let’s assume that this is the case; we just have to prove that p maximizes ∑civi for some choice of ci≥0.
It suffices to find a hyperplane H⊂Rn that contains V(p) and that supports V(P). Then the desired function ∑civi can be constructed by ensuring that H is a level set.
If V(P) lies in a proper affine subspace of Rn, let Z be the smallest such subspace. Let A be the interior of V(P) in Z and let B be the interior of pv(p). The case where V(P) is a point is trivial; suppose it’s not, so A is nonempty. By convexity, V(P) is the closure of A and pv(p) is the closure of B.
Since V(P) is convex, A is convex, and we can exhaust A with a nested sequence of nonempty compact convex sets Aj. And pv(p) is convex, so we can exhaust B with a nested sequence of nonempty compact convex sets Bj. By the hyperplane separation theorem, for each j, there is a hyperplane Hj separating Aj and Bj. I claim that Hj has a convergent subsequence. Indeed, each Hj must intersect the convex hull of A1∪B1, and the space of hyperplanes intersecting that convex hull is compact. So Hj has a subsequence that converges to a hyperplane H.
It’s easy to see that H separates Aj from Bj for each j, and so H separates A from B. So H must contain V(p) and support V(P).
□
Note that the theorem does not guarantee the existence of a Pareto optimum. But if V(P) is closed and bounded, then a Pareto optimum exists.
A limitation of the theorem is that it assumes a finite list of values v1,…,vn, not an infinite one.