Sunday, February 22, 2015

Valuation Theory

 

Valuation Systems

A valuation system (VS) is any system by which value is assigned to things. That is, the way in which terms like "better", and "worse", "good", and "bad", are given meaning or are understood. For example, in choosing a hinge for a door, one system of saying "hinge X is better than hinge Y" is to consider price (cheaper being better, for instance), or resistance to rust, or weight, or color, or size, etc. After all, there is no unqualified way to say "hinge A is better than hinge B", and any statement that does not explicitly state the way in which hinge A is deemed better than hinge B will have some implicit VS.

All VSs have a domain, which is the set of all things which can be valued by that VS. The VS used to compare hinges won't be able to compare the value of microprocessors, or political parties, or cake recipes. It is important to keep in mind the domain of a VS when discussing it. We will denote the domain of VS X as DX.


Types of Valuation Systems


There are two general sorts of valuation systems:

  • Comparative Valuation Systems (CVSs): Determines only the ranking of value for the elements of a given, countable set. If X is a CVS and X values A above B, we will write that as \( (A>B)_X\), which we can read as "A is better than B, according to X". Note that CVSs don't have any notion of "good" or "bad", but only "better" and "worse", and possibly "best", if there is some element better than the rest.

    • A subset of CVSs are Bi-comparative VSs (bCVSs, or C2VSs), which only rank sets with exactly two elements, either with one better and one worse, or with both equal. If the bCVS has the additional property of being transitive, then the system can be used to impose a partial ordering on the elements of its domain.

  • Evaluative Valuation Systems (EVSs): Determines the plain value of every element in its domain, like a function. Namely, we can symbolize "the value of A, according to EVS X" as \(V_X(A)\). Without loss of generality, we can take the values assigned to be real numbers. If only order is important, we can take the range to be the numbers in the interval \([-1,1]\). Note that EVSs can have a notion of "good" and "bad", in that we can define "A is bad, according to EVS X" as \(V_X(A)< c \), for some number c, which we can take to be 0. Similar statements can be similarly defined. To keep notation consistent, we will write \((A>B)_X\) iff \(V_X(A)>V_X(B)\), for some EVS X.



Indifferent Extensions


We can also define the indifferent extension of a valuation system X with domain DX as the valuation system that is identical to X for any elements in DX, and is indifferent to all other things. More exactly, we can define it for the cases of CVSs and EVSs as follows:
  • CVSs:
    Let \(X\) be a CVS with domain \(D_X\). The CVS \(X'\) is the indifferent extension of \(X\), such that, for any \( a,b \notin D_X\) and \(c \in D_X\), \((a< c )_{X'} \), \((a=b)_{X'}\).

  • EVSs:
    Let \(X\) be an EVS with domain \(D_X\). The EVS \(X'\) is the indifferent extension of \(X\), such that, for any \( a\notin D_X\), \(V_{X'}(a)=0\).



Optimal Elements


We can also give meaning to statements like "t is the best element in set S, according to X", in two senses. We can say that t is the optimal element of S according to VS X if, for every element s of S such that \(s \neq t\), then \( (t > s)_X \). We can say that t is an equi-optimal element of S according to VS X if, for every element s of S, \( (t \geq s)_X \). We can also say that "t is the best element in set S, according to set A", for some set A of VSs, if, for each VS X in A, s is the optimal element in X. We might also stipulate that for every VS in A there is an optimal element in S. Similarly for equi-optimal.

If we want to say something like "t is the best element in S" without qualifying it by a VS, it must be the case that all valuation systems agree (or perhaps there is some "best VS" which would deem s optimal, but we will get to that later). Namely, we say that s is the universo-optimal (UO) element of S if, for every VS X for which there is an optimal element in S, s is the optimal element of X. We also can say that s is a universo-equi-optimal(UEO) element of S if, for every VS X for which there is an equi-optimal element in S, s is an equi-optimal element of X. Note that for there to be a universo-optimal element, all relevant VSs must agree: if there is even one VS for which there is a different optimal element than another, then there is no universo-optimal element in S.


Meta-Valuation Systems, Optimal Valuation Systems, and Recommendation


We can also have VSs whose domain includes some subset of the set of all VSs. We can call these meta-valuation systems (MVS). We can also define the set of totally meta-VSs (TMVS), which is the set of all VSs whose domain includes the set of all VSs.
Now, if there is to be some VS that can be called "the best VS", it must be the case that it is UO (or at least UEO) in the set of all VSs. Thus we define:
a VS X is the objectively best VS iff, for ever VS Y in the set TMVSs for which there is an optimal element, X is the optimal element of Y in the set of all VSs.
However, it seems not hard to very strongly suggest if not prove that there is no such VS, for all it takes are two TMVSs with optimal elements that disagree as to this optimal element, and this seems very easy to construct. Thus there simply is no such objectively best VS. We can call this the Universo-Optimality Absence Theorem.

Also, we can say that VS A recommends VS B if \((B>A)_A\). We denote this by \(A \rightarrow B\). Clearly A must be a MVS, as it includes the VS B in its domain. The relevance is that, if we hold to VS A, and A recommends B, then we should discard A and take up B instead. We may have some issues if A recommends multiple VSs, but then the solution would then be to follow the recommendation that is outranks the rest. For example, if \(A \rightarrow B\) and \(A \rightarrow C\), and \((B>C)_A\), then we should choose B, rather than C. However, we will say that a VS A is a consistent recommender if it is the case that if \(A \rightarrow B\), and \(A \rightarrow C\), and \((B>C)_A\), then \(C \rightarrow B\), and it is not the case that \(B \rightarrow C\).