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 $\succeq$, the weak upper contour set at A is the set of all options that are at least as good as A: $\{B\in X:B\succeq A\}$. The weak lower contour set at A is is the set of all options that are at most as good as A: $\{B\in X:A\succeq B\}$.
• Similarly, given a strict preference relation $\succ$, the strict upper contour set at A is the set of all options better than 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: $\{B\in X:A\succ B\}$.

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

Main article: Debreu's representation theorems

Debreu proved the existence of a contiuous representation of a weak preference relation $\succeq$ satisfying the following conditions:

1. Transitive;
2. Complete, that is, for every two options A, B in X, either $A\succeq B$ or $B\succeq A$ or both;
3. For all $A\in X$, both the upper and the lower contour sets are topologically closed;
4. 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. Second-countability is implied by the following properties (from weaker to stronger):
   • The space X is separable and connected.
   • There is a countable subset of X, $Z=\{z_{0},z_{1},...\}$, such that for every pair of non-equivalent elements $x\prec y$, there is an element $z_{i}\in Z$ that separates them ($x\preceq z_{i}\preceq y$).
   • The set of equivalence classes of the indiffference relation $\simeq$ 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.

• Richter
• Peleg
• Ok
• Ok and Evren

See also

• Von Neumann-Morgenstern utility theorem