The geometric theory of difference

It is nearly impossible to formulate differentiability conditions over R within the language and conceptual apparatus of the algebraic theory of difference. What is required is a certain ‘intermediate’ language.

It is possible, and it will turn out to be convenient, to express the properties that are attributed to R in terms of relations among geometric intervals within the given Euclidean straight line. What this means is that we, as theoreticians, can represent the comparison of the differences felt by the consumer, expressed by symbol ‘R’, by means of comparisons among intervals in Λ. My proposal is to build the theory of relation R by means of these comparisons, trying to express intuitive, empirical (idealized) properties of R in terms of such comparisons.

To that end, let me introduce the function s: Ω² → Λ, as an application that assigns to each satisfaction interval ab the equivalence class of all the line inter­vals whose length is intended to represent the distance that the agent associates to ab (how ‘far’ is a from b in terms of satisfaction), and whose direction is intended to represent whether the motion from a to b would be an improvement, a wors­ening, or indifferent for the agent. In particular, σ will assign to any interval aa in the diagonal the (equivalence class of the) null line interval, which of course does not exist but we can create by a convenient fiat.

If l represents (on the straight line) the motion from option a to b, and m rep­resents that from b to a, the addition of l and m must be the null interval.

7.5.1 Definition

Let Λ be the family of equivalence classes of elements of Λ modulo =, including the class of the null vector AA, which we shall denote as e. We define the binary operation ®, closed over Λ, by means of the following conditions:

7.5.2 Theorem

Structure (Λ, ®, e) is an Abelian group totally ordered by ≤, with e as identity element.

PROOF: A rigorous proof can be given using Definition 7.5.1 and the conceptual apparatus and axioms of Holder’s theory (Holder 1996, 1997), a task that is involved, as it requires lengthy arguments and the consideration of several cases. For example, the proof that ® is associative is tedious, as eight cases

Table 7.1 Representation of motions by means of σ

7.5.3 Definition

The following result is immediate, as it is based upon the existence of the already mentioned numerical representation.

7.5.4 Theorem

I will prove in what follows that the existence of a geometric representation of difference structure D implies that D is an algebraic difference structure.

7.5.5 Lemma

If abRcd then dcRba.

PROOF: Suppose that both ab and cd are in A⁺ U A⁰. This means that both σ(ab) and σ(cd) are I or null, with σ(ab) > σ(cd). Hence, σ(ba) and σ(dc) are II or null, with σ(ba) > σ(dc). It follows that dcRba.

Ifboth are II, abRcd implies that ab⅛cd and that ba and dc are I. Hence, again, dcRba.

Notice that abRcd implies that cd cannot be I or null if ab is II. Hence, the only remaining case is when ab is I or null, and cd is II. In this case, ba is II or null and dc is I. It follows that dcPba and so, finally, dcRba. ?

7.5.6 Lemma

If abRa'b' and bcRb'c' then acRa'c'.

PROOF: The proof of this lemma is easy but laborious, since there are several cases to be considered. Excluding the cases precluded by the hypothesis of the proposition, there are still nine cases to consider. They are given in Table 7.2. The proof is interesting because it yields more insight into the meaning of the geometric representation.

Table 7.2 Feasible cases in Lemma 7.5.6

feasible Casesinlemma 7.5.6

The great advantage of the language of intervals over the languages typically used to formulate preference theories is that it provides resources by means of which we can also express natural, intuitive differentiability conditions for the preference relation. We turn now to these.