From aggregative value comparisons to hyperreal values

Suppose that we have n objects α₁, ..., α_n, 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 + ... + m_nα_n, where the m_i 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 + ... + m_nα_n ≤ p₁α₁ + ... + p_nα_n as saying that the “aggregative value” of having m_i copies of α_i for all i is less than or equal to the “aggregative value” of having p_i copies of α_i 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 + ... + m_nα_n where the m_i are allowed to be positive or negative by stipulating that:

• m₁α₁ + ... + m_nα_n ≤ p₁α₁ + ... + p_nα_n iff (k+m₁)α₁ + ... + (k+m_n)α_n ≤ (k+p₁)α₁ + ... + (k+p_n)α_n for some natural k such that k + m_i and k + p_i 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 + ... + r_nα_n where the r_i are rational numbers by stipulating that:

• r₁α₁ + ... + r_nα_n ≤ s₁α₁ + ... + s_nα_n iff ur₁α₁ + ... + ur_nα_n ≤ us₁α₁ + ... + us_nα_n for some positive integer u such that ur_i and us_i 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, ..., α_n.

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, ..., α_n. 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,...,x_n) with x₁ + x₂ϵ + ... + x_nϵ^n − 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.