In economics, the Debreu's theorems are preference representation theorems—statements about the representation of a preference ordering by a real-valued utility function.
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.).
All the responses are recorded and form 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 that maps a real number to each option, such that the utility of option A is larger than that of option B if and only if the agent prefers A to B. Debreu's theorems address the following question: what conditions on the preference relation guarantee the existence of a representing utility function?
The 1954 Theorems[1][2] say, roughly, that every preference relation which is complete, transitive and continuous, can be represented by a continuous ordinal utility function.
The theorems are usually applied to spaces of finite commodities.
These are the general assumptions: Each one of the following conditions guarantees the existence of a real-valued continuous function that represents the preference relation
Condition 1 is violated because the set of equivalence classes is uncountable.
However, condition 2 is satisfied with Z as the set of pairs with rational coordinates.
Condition 3 is also satisfied since X is separable and connected.
The lexicographic preferences relation is not continuous in that topology.
For 1, use the proposition that any countable linear ordering is isomorphic to a subset of
, construct its upper and lower Dedekind cuts
have the same ordering iff their Dedekind cuts are equal.
to squeeze the extended real line to a finite interval.
Thus the number of gap pair representatives is at most countable.
Diamond[4] applied Debreu's theorem to the space
, the set of all bounded real-valued sequences with the topology induced by the supremum metric (see L-infinity).
X represents the set of all utility streams with infinite horizon.
is equivalent to a constant-utility stream, and every two constant-utility streams are separable by a constant-utility stream with a rational utility, so condition #2 of Debreu is satisfied, and the preference relation can be represented by a real-valued function.
is called additive if it can be written as a sum of n ordinal utility functions on the n factors: where the
is called preferentially independent if the preference relation
is additive, then obviously all subsets of commodities are preferentially-independent.
If all subsets of commodities are preferentially-independent AND at least three commodities are essential (meaning that their quantities have an influence on the preference relation
For an intuitive constructive proof, see Ordinal utility - Additivity with three or more goods.
The earlier theorem assumes that agents have preferences on lotteries with arbitrary probabilities.
Debreu's theorem weakens this assumption and assumes only that agents have preferences on equal-chance lotteries (i.e., they can only answer questions of the form: "Do you prefer A over an equal-chance lottery between B and C?").
Debreu's theorem states that if: Then there exists a cardinal utility function u that represents the preference relation on the set of lotteries, i.e.: Theorem 2 of 1960[5] deals with agents whose preferences are represented by frequency-of-choice.
can be interpreted as measuring how much the agent prefers A over B. Debreu's theorem states that if the agent's function p satisfies the following conditions: Then there exists a cardinal utility function u that represents p, i.e: