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.

Complete preference relations

Incomplete preferences

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

Richter^[1]
Peleg^[2]
Ok^[3]
Ok and Evren^[4]

References

^ Richter, Marcel K. (1966). "Revealed Preference Theory". Econometrica. 34 (3): 635–645. doi:10.2307/1909773. ISSN 0012-9682.
^ Peleg, Bezalel (1970). "Utility Functions for Partially Ordered Topological Spaces". Econometrica. 38 (1): 93–96. doi:10.2307/1909243. 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.

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

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

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

[4] 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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