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Source Title: Decision Control, Management, and Support in Adaptive and Complex Systems: Quantitative Models

Copyright: © 2013
|Pages: 19
DOI: 10.4018/978-1-4666-2967-7.ch004

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TopIn the previous chapter, we presented the fundamental concepts in the theory of measurement and scaling. There we defined the term value or utility, as a measurement scale of the preferences of the decision maker. Let (** X,** (R)) is a system with relations (SR) defined over the set

This value function is also called perfect value function. If the homomorphism is defined by the inclusion (** X**⊆u

In other words, the value function is a homomorphism from the empiric system (Decision Maker (DM), preferences) to (subset of the set of the real numbers with relation greater). We assume that the set ** X** is countable, i.e. bijective to the set of the natural numbers (the set of integers). Then the following propositions are fulfilled (Fishburn, 1970).

Value function u(.) for the preference relation () of ** X** exists if and only if the relation is acyclic (i.e. there does not exist (

This proposition is obvious and follows directly from the properties of the relation (>) defined over the real numbers and the definition of the homomorphism. We define the relation “indifference, indiscernibility or equivalence” as ((*x≈y*) ⇔ *¬*((*x**y*) ∨ (*x**y*))). In Fishburn (1970), the following theorem is proved.

Let the relation () is weak order (asymmetric and negatively transitive relation) over the set of alternatives. Then it is fulfilled:

For any two alternatives (*x*, *y*) one of the following relations is fulfilled (*x**y*), (*y**x*) or (*x≈y*):

*•*The relation () is transitive;

*•*The relation (

*≈*) is equivalence;*•*((

*x**y*)∧ (*y≈z*)) ⇒ (*x**z*), ((*x≈y*)∧ (*y**z*)) ⇒ (*x**z*);*•*The relation ((

*x**y*)∨(*x≈y*)) “more preferred or equivalent” is transitive and connected;

The relation (’) over the factor set (** X**/

We investigated part of the above propositions in the previous chapter 3. We saw that asymmetricity and the negative transitivity lead to the transitivity of the preference relation () and transitivity of (*≈*). Now we will state a fundamental theorem ensuring the existence of perfect value function—Fishburn’s terminology (Fishburn, 1970).

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