Suppose a person is asked questions of the form "Do you prefer A or B?"
(when A and B can be options, actions to take, states of the world, consumption bundles, etc.).
The set of all such preference-pairs forms the person's preference relation.
Instead of recording the person's preferences between every pair of options, it would be much more convenient to have a single utility function - a function u that assigns a real number to each option, such that
Often, one would like the representing function u to satisfy additional conditions, such as continuity.
This requires additional conditions on the preference relation.
In some cases we assume that X is also a metric space; in particular, X can be a subset of a Euclidean space Rm, such that each coordinate in {1,..., m} represents a commodity, and each m-vector in X represents a possible consumption bundle.
, we define the contour sets at A: Sometimes, the above continuity notions are called semicontinuous, and a
[1] A preference-relation is called: As an example, the strict order ">" on real numbers is separable, but not countable.
Debreu[2][3] proved the existence of a continuous representation of a weak preference relation
satisfying the following conditions: Jaffray gives an elementary proof to the existence of a continuous utility function.
[5] Preferences are called incomplete when some options are incomparable, that is, neither
Since real numbers are always comparable, it is impossible to have a representing function u with
Peleg defined a utility function representation of a strict partial order
[6] Peleg proved the existence of a one-dimensional continuous utility representation of a strict preference relation
, we can apply Peleg's theorem by defining a strict preference relation:
is separable) is implied by the following three conditions: A similar approach was taken by Richter.
[8] Jaffray defines a utility function representation of a strict partial order
Sondermann defines a utility function representation similarly to Jaffray.
He gives conditions for existence of a utility function representation on a probability space, that is upper semicontinuous or lower semicontinuous in the order topology.
[10] Herdendefines a utility function representation of a weak preorder
on X has a continuous utility function, if and only if there exists a countable family E of separable systems on X such that, for all pairs
, there is a separable system F in E, such that B is contained in all sets in F, and A is not contained in any set in F. He shows that this theorem implies Peleg's representation theorem.
The concept was introduced by Efe Ok.[13] Every preorder (reflexive and transitive relation) has a trivial MUR.
[1]: Prop.1 Moreover, every preorder with closed upper contour sets has an upper-semicontinuous MUR, and every preorder with closed lower contour sets has a lower-semicontinuous MUR.
[1]: Prop.2 However, not every preorder with closed upper and lower contour sets has a continuous MUR.
[1]: Exm.1 Ok and Evren present several conditions on the existence of a continuous MUR: All the representations guaranteed by the above theorems might contain infinitely many utilities, and even uncountably many utilities.
Evren and Ok prove there exists a finite MUR where all utilities are upper[lower] semicontinuous for any weak preference relation
satisfying the following conditions:[1]: Thm 3 Note that the guaranteed functions are semicontinuous, but not necessarily continuous, even if all upper and lower contour sets are closed.
[13]: Exm.2 Evren and Ok say that "there does not seem to be a natural way of deriving a continuous finite multi-utility representation theorem, at least, not by using the methods adopted in this paper".