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

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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