Monday, April 29, 2024

From aggregative value comparisons to hyperreal values

Suppose that we have n objects α1, ..., αn, and we want to define something like numerical values (at least hyperreal ones, if we can’t have real ones) on the basis of comparisons of value. Here is one interesting way to proceed. Consider the space of formal sums m1α1 + ... + mnαn, where the mi are natural numbers, and suppose there is a total preorder (total transitive reflexive relation) on this space satisfying the axioms:

  1. x + z ≤ y + z iff x ≤ y

  2. mx ≤ my iff x ≤ y for all positive m.

We can think of m1α1 + ... + mnαn ≤ p1α1 + ... + pnαn as saying that the “aggregative value” of having mi copies of αi for all i is less than or equal to the “aggregative value” of having pi copies of αi for all i. The aggregative value of a number of objects is the “sum value”, where we don’t take into account things like the diversity or lack thereof or other “arrangement values”.

Now extend ≤ to formal sums m1α1 + ... + mnαn where the mi are allowed to be positive or negative by stipulating that:

  • m1α1 + ... + mnαn ≤ p1α1 + ... + pnαn iff (k+m1)α1 + ... + (k+mn)αn ≤ (k+p1)α1 + ... + (k+pn)αn for some natural k such that k + mi and k + pi are non-negative for all i.

Axiom (1) implies that the choice of k is irrelevant. It is easy to see that ≤ still satisfies both (1) and (2). Moreover, ≤ is still total, transitive and reflexive.

Next extend ≤ to formal sums r1α1 + ... + rnαn where the ri are rational numbers by stipulating that:

  • r1α1 + ... + rnαn ≤ s1α1 + ... + snαn iff ur1α1 + ... + urnαn ≤ us1α1 + ... + usnαn for some positive integer u such that uri and usi is an integer for all i.

Axiom (2) implies that the choice of u is irrelevant. Again, it is easy to see that ≤ continues to satisfy (1) and (2), and that it remains total, transitive and reflexive.

Thus, ≤ is a total vector space preorder on an n-dimensional vector space V over the rationals with basis α1, ..., αn.

Let C be the positive cone of ≤: C = {x ∈ V : 0 ≤ x}. This is closed under addition and positive rational-valued scalar multiplication. Let K be the kernel of the preorder, i.e., {x ∈ V : 0 ≤ x ≤ 0} = C ∩  − C.

Now, let W be the n-dimensional vector space over the reals with basis α1, ..., αn. Let D be the smallest subset of W containing C and closed under addition and multiplication by positive real scalars: this is the set of real-linear combinations of elements of C with positive coefficients. It is easy to check that D ∩ V = C. Let L = D ∩  − D. Then L ∩ V = K.

Let E be a maximal subset of W that contains D, is closed under addition and multiplication by positive real scalars, and is such that E ∩  − E = L. This exists by Zorn’s Lemma. I claim that for any v in W, either v or  − v is in E. For suppose neither v nor  − v is not in E. Then let E′ = {e + tv : t > 0, e ∈ E}. This contains C, and is closed under addition and multiplication by positive reals. If we can show that E′ ∩  − E′ = L, then since E is a proper subset of E′, we will contradict the maximality of E. Suppose z ∈ E′ ∩  − E but not z ∈ L. Since E ∩  − E = L, we must have either z or  − z in E′ ∖ E. Without loss of generality suppose z ∈ E′ ∖ E. Then z = e + tv for e ∈ E and t > 0. Thus, e + tv ∈  − E. Hence tv ∈ (−e) + (−E) ⊆  − E, since e ∈ E and E is closed under addition. Since E is closed under positive scalar multiplication, we have v ∈  − E, which contradicts our assumption that  − v is not in E.

Define ≤* on W by letting v*w iff w − v ∈ E. Note that ≤* agrees with on V. If v ≤ w are in V, then w − v ∈ C ⊆ E and so v*w. Conversely, if v*w, then w − v ∈ E. Now, since w − v is in V, and is total, if we don’t have v ≤ w, we must have w ≤ v and hence v − w ∈ C, so w − v ∈  − C. Since E ∩  − E = L, we have w − v ∈ L. But v, w ∈ V, so w − v ∈ L ∩ V = K. Thus, v ≤ w, a contradiction.

It’s also easy to see that * is total, transitive and reflexive. It is therefore representable by lexicographically-ordered vector-valued utilities by the work of Hausner in the middle of the last century. And vector-valued utilities are representable by hyperreals (just represent (x1,...,xn) with x1 + x2ϵ + ... + xnϵn − 1 for a positive infinitesimal ϵ).

Remark 1: Here is a plausible condition on the extension ≤* that we can enforce if we like: if Q and U are neighborhoods of v and w respectively, and for all q ∈ Q ∩ V and all u ∈ U ∩ V we have q ≤ v, then v*w. For this condition will hold provided we can show that if Q is a neighborhood of v such that Q ∩ V ⊆ C, then v ∈ E. Note that any positive-real-linear combination of points v satisfying this neighborhood condition also satisfies this condition, and any sum of a point v satisfying this condition and a point in D will also satisfy it. Thus we can add to D all such points v, and carry on with the rest of the proof.

Remark 2: If we start off with being a partial preorder, * still becomes a total order. Then instead of proving it agrees with the partial preordering on V (or the initial ordering), we use the basically the same proof to show that it extends both the non-strict and strict orders: (a) if w ≤ v, then w*v and if w < v, then w<*v.

Question 1: Can we make sure that the values are real numbers?

Response: No. Suppose you are comparing a sheep and a goat, and suppose that they are valued positively and equally—the one exception is ties are broken in favor of the sheep. Thus, n+1 copies of the goat are better than n copies of the sheep and both are better than nothing, but n copies of the sheep are better than n copies of the goat. To represent this with hyperreals we need to take the value of the sheep to be ϵ + g where g > 0 is the value of the goat, and where ϵ/g is a positive infinitesimal.

Question 2: Is the representation is “practically unique”, i.e., does it generate the same decisions in probabilistic situations, or at least ones with real-valued probabilities?

Response: No. Supose you have a sheep and a goat. Now consider two hypotheses: on the first, the sheep is worth  − ϵ + π goats, and on the second, the sheep is worth ϵ + π goats, for a positive infinitesimal ϵ. Both hypotheses generate the same aggregative value comparisons between aggregates consisting of n1 copies of the goat and n2 copies of the sheep for natural numbers n1 and n2, since π is irrational. But the two hypotheses generate opposite probabilistic decisions if we are choosing between a 1/π chance of the sheep and certainty of the goat.

3 comments:

  1. This comment has been removed by the author.

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  2. I have fixed some errors in the original proof, which had assumed in a few places that the preorder on V was an order (i.e., its kernel was {0}). One probably could also have used the original proof after replacing V with its quotient over the kernel of the preorder.

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  3. It isn't hard to extend the main result to infinitely many objects. We suppose that we have a comparison satisfying (1) and (2) on finite formal sums. Then we restrict to a finite set of objects and generate a hyperreal value for them. And then we take a limit of these hyperreal values along an ultrafilter on the partially ordered set of finite sets of objects ordered by inclusion.

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