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.