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.

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, $x\succ x$ does not hold.
• 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, $x\succeq x$ always holds.

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

Incomplete preferences

Debreu's theorems were later extended to preference relations that are not complete. In particular:

• Richter
• Peleg
• Ok
• Ok and Evren