Utility representation theorem

In economics, a preference representation theorem is a theorem asserting that, under certain conditions, a preference ordering can be represented by a real-valued utility function, such that option A is preferred to option B if and only if the utility of A is larger than that of B.

Background

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.). If the agent prefers A to B, we write $A\succ B$ . 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 $u(A)>u(B)$ if and only if $A\succ B$ .

Not every preference-relation has a utility-function representation. For example, if the relation is not transitive (the agent prefers A to B, B to C, and C to A), then it has no utility representation, since any such utility function would have to satisfy $u(A)>u(B)>u(C)>u(A)$ , which is impossible.

A utility representation theorem gives conditions on a preference relation, that are sufficient for the existence of a utility representation.

Often, one would like the representing function u to satisfy additional conditions, such as continuity. This requires additional conditions on the preference relation.

Definitions

The set of options is a topological space denoted by X. In some cases we assume that X is also a metric space; in particular, X can be a subset an Euclidean space R^m, such that each coordinate in {1,...,m} represents a commodity, and each m-vector in X represents a possible consumption bundle.

Preference relations

A preference relation is a subset of $X\times X$ . It is denoted by either $\succ$ or $\succeq$ :

The notation $\succ$ is used when the relation is strict, that is, $A\succ B$ means that option A is strictly better than option B. In this case, the relation should be irreflexive, that is, $A\succ A$ does not hold. It should also be asymmetric, that is, $A\succ B$ implies that not $B\succ A$ .
The notation $\succeq$ is used when the relation is weak, that is, $A\succeq B$ means that option A is at least as good as option B (A may be equivalent to B, or better than B). In this case, the relation should be reflexive, that is, $A\succeq A$ always holds.

Given a weak preference relation $\succeq$ , one can define its "strict part" $\succ$ and "indifference part" $\simeq$ as follows:

$A\succ B$ if and only if $A\succeq B$ and not $B\succeq A$ .
$A\simeq B$ if and only if $A\succeq B$ and $B\succeq A$ .

Given a strict preference relation $\succ$ , one can define its "weak part" $\succeq$ and "indifference part" $\simeq$ as follows:

$A\succeq B$ if and only if not $B\succ A$ ;
$A\simeq B$ if and only if not $B\succ A$ and not $A\succ B$ .

For every option $A\in X$ , we define the contour sets at A:

Given a weak preference relation $\succeq$ $\succeq$ , the weak upper contour set at A is the set of all options that are at least as good as A: $\{B\in X:B\succeq A\}$ $\{B\in X:B\succeq A\}$ . The weak lower contour set at A is is the set of all options that are at most as good as A: $\{B\in X:A\succeq B\}$ $\{B\in X:A\succeq B\}$ .
- A weak preference relation is called continuous if its contour sets are topologically closed.
Similarly, given a strict preference relation $\succ$ $\succ$ , the strict upper contour set at A is the set of all options better than A: $\{B\in X:B\succ A\}$ $\{B\in X:B\succ A\}$ , and the strict lower contour set at A is is the set of all options worse than A: $\{B\in X:A\succ B\}$ $\{B\in X:A\succ B\}$ .
- A strict preference relation is called continuous if its contour sets are topologically open.

A preference-relation is called:

Countable - if the set of equivalence classes of the indiffference relation $\simeq$ is countable.
Separable - if there is a countable subset of X, $Z=\{z_{0},z_{1},...\}$ , such that for every pair $A\succ B$ , there is an element $z_{i}\in Z$ that separates them, that is, $A\succ z_{i}\succ B$ (an analogous definition holds for weak relations).

As an example, the strict order ">" on real numbers is separable, but not countable.

Utility functions

A utility function is a function $u:X\to \mathbb {R}$ .

A utility function u is said to represent a strict preference relation $\succ$ , if $u(A)>u(B)\iff A\succ B$ .
A utility function u is said to represent a weak preference relation $\succeq$ , if $u(A)\geq u(B)\iff A\succeq B$ .

Complete preference relations

Debreu^[1]^[2] proved the existence of a contiuous representation of a weak preference relation $\succeq$ satisfying the following conditions:

Transitive;
Complete, that is, for every two options A, B in X, either $A\succeq B$ or $B\succeq A$ or both;
For all $A\in X$ , both the upper and the lower weak contour sets are topologically closed;
The space X is second-countable. This means that there is a countable set S of open sets, such that every open set in X is the union of sets of the class S.^[3] Second-countability is implied by the following properties (from weaker to stronger):
- The space X is separable and connected.
- The relation $\succeq$ is separable.
- The relation $\succeq$ is countable.

Incomplete preference relations

Incomplete preferences are partial order relations. This means that some options may be incomparable: it is possible that neither $A\succeq B$ nor $B\succeq A$ holds. Since real numbers are always comparable, it is impossible to have a representing function u with $u(A)\geq u(B)\iff A\succeq B$ . There are several ways to cope with this issue.

Peleg^[4] defined a one-directional representation: u represents $\succ$ if $A\succ B\implies u(A)>u(B)$ . He proved the existence of a one-dimensional continuous utility representation of a strict preference relation $\succ$ satisfying the following conditions:

Transitive;
For all $A\in X$ , the lower strict contour set is topologically open;
Separable: there is a countable subset of X, $Z=\{z_{0},z_{1},...\}$ , such that for every pair $A\succ B$ , there is an element $z_{i}\in Z$ that separates them ( $A\succ z_{i}\succ B$ ).
Spacious:
$\succ$

Richter^[5]
Ok^[6]
Ok and Evren^[7]

References

^ Debreu, Gerard (1954). Representation of a preference ordering by a numerical function.
^ Debreu, Gerard (1986). "6. Representation of a preference ordering by a numerical function". Mathematical economics : twenty papers of Gerard Debreu ; introduction by Werner Hildenbrand (1st pbk. ed.). Cambridge [Cambridgeshire]: Cambridge University Press. ISBN 0-521-23736-X. OCLC 25466669.
^ Debreu, Gerard (1964). "Continuity properties of Paretian utility". International Economic Review. 5 (3): 285–293. doi:10.2307/2525513.
^ Peleg, Bezalel (1970). "Utility Functions for Partially Ordered Topological Spaces". Econometrica. 38 (1): 93–96. doi:10.2307/1909243. ISSN 0012-9682.
^ Richter, Marcel K. (1966). "Revealed Preference Theory". Econometrica. 34 (3): 635–645. doi:10.2307/1909773. ISSN 0012-9682.
^ Ok, Efe (2002). "Utility Representation of an Incomplete Preference Relation". Journal of Economic Theory. 104 (2): 429–449. ISSN 0022-0531.
^ Evren, Özgür; Ok, Efe A. (2011-08-01). "On the multi-utility representation of preference relations". Journal of Mathematical Economics. 47 (4): 554–563. doi:10.1016/j.jmateco.2011.07.003. ISSN 0304-4068.

[Debreu54-1] Debreu, Gerard (1954). Representation of a preference ordering by a numerical function.

[:0-2] Debreu, Gerard (1986). "6. Representation of a preference ordering by a numerical function". Mathematical economics : twenty papers of Gerard Debreu ; introduction by Werner Hildenbrand (1st pbk. ed.). Cambridge [Cambridgeshire]: Cambridge University Press. ISBN 0-521-23736-X. OCLC 25466669.

[Debreu64-3] Debreu, Gerard (1964). "Continuity properties of Paretian utility". International Economic Review. 5 (3): 285–293. doi:10.2307/2525513.

[4] Peleg, Bezalel (1970). "Utility Functions for Partially Ordered Topological Spaces". Econometrica. 38 (1): 93–96. doi:10.2307/1909243. ISSN 0012-9682.

[5] Richter, Marcel K. (1966). "Revealed Preference Theory". Econometrica. 34 (3): 635–645. doi:10.2307/1909773. ISSN 0012-9682.

[6] Ok, Efe (2002). "Utility Representation of an Incomplete Preference Relation". Journal of Economic Theory. 104 (2): 429–449. ISSN 0022-0531.

[7] Evren, Özgür; Ok, Efe A. (2011-08-01). "On the multi-utility representation of preference relations". Journal of Mathematical Economics. 47 (4): 554–563. doi:10.1016/j.jmateco.2011.07.003. ISSN 0304-4068.

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