Strict Weak Ordering - Representing Weak Orderings By Functions

Representing Weak Orderings By Functions

If X is any set and f a real-valued function on X then f induces a strict weak order on X by setting

• a < b if and only if f(a) < f(b)

The associated total preorder is given by

• a b if and only if f(a) ≤ f(b)

and the associated equivalence by

• a b if and only if f(a) = f(b)

The relations do not change when f is replaced by g f (composite function), where g is a strictly increasing real-valued function defined on at least the range of f.

Thus e.g. a utility function defines a preference relation.

If X is finite or countable, every weak order can be represented by a function in this way (see, e.g., Roberts 1979, Theorem 3.1). However, there exist strict weak orders that have no corresponding real function. For example, there is no such function for the lexicographic order on Rn. Thus, while in most preference relation models the relation defines a utility function up to order-preserving transformations, there is no such function for lexicographic preferences.

More generally, if X is a set, and Y is a set with a strict weak ordering "<", and f a function from X to Y, then f induces a strict weak ordering on X by setting

• a < b if and only if f(a) < f(b)

The associated total preorder is given by

• a b if and only if f(a) f(b)

and the associated equivalence by

• a b if and only if f(a) f(b)

f is not necessarily an injective function, so for example a class of 2 equivalent elements on Y may induce a class of 5 equivalent elements on X. Also f is not necessarily an surjective function, so a class of 2 equivalent elements on Y may induce a class of only one element on X, or no class at all. There is a corresponding injective function g that maps the partition on X to that on Y. Thus, in the case of finite partitions, the number of classes in X is less than or equal to the number of classes on Y.