Debreu theorems
In economics, the Debreu theorems are several statements about the representation of a preference ordering by a real-valued function. The theorems were proved by Gerard Debreu during the 1950s. 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.). All the responses are recorded. Then, preferences of that person are represented by a numeric "utility function", such that the utility of option A is larger than option B if and only if the agent  prefers A to B. The Debreu theorems come to answer the following basic question: what conditions on the preference relation of the agent guarantee that such representative utility function can be found? Existence of ordinal utility function. The 1954 Theorems say, roughly, that every preference relation which is complete, transitive and continuous, can be represented by a continuous ordinal utility function. Statement. The theorems are usually applied to spaces of finite commodities. However, they are applicable in a much more general setting. 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 formula_1: 1. The set of equivalence classes of the relation formula_14 (defined by: formula_15 iff formula_16 and formula_17) are a countable set. 2. There is a countable subset of X, formula_18, such that for every pair of non-equivalent elements formula_19, there is an element formula_20 that separates them (formula_21). 3. X is separable and connected. 4. X is second countable. This means that there is a countable set S of open sets, such that every open set in X is the union of sets of the class S. The proof for the  fourth result had a gap which Debreu later corrected. Examples. A. Let formula_22 with the standard topology (the Euclidean topology). Define the following preference relation: formula_23 iff formula_24. It is continuous because for every formula_25, the sets formula_26 and formula_27 are closed half-planes. 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. Hence, there exists a continuous function which represents formula_1. An example of such function is formula_29. B. Let  formula_22 with the standard topology as above. The lexicographic preferences relation is not continuous in that topology. For example, formula_31, but in every ball around (5,1) there are points with formula_32 and these points are inferior to formula_33. Indeed, this relation cannot be represented by a continuous real-valued function. Extension. Diamond applied Debreu's theorem to the space formula_34, 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. In addition to the requirement that formula_1 be total , transitive and continuous, He added a "sensitivity" requirement: Under these requirements, every stream formula_36 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. The existence result is valid even when the topology of X is changed to the topology induced by the discounted metric: formula_43 Additivity of ordinal utility function. Theorem 3 of 1960 says, roughly, that if the commodity space contains 3 or more components , and every subset of the components is preferentially-independent of the other components, then the preference relation can be represented by an "additive" value function. Statement. These are the general assumptions: The function formula_47 is called "additive" if it can be written as a sum of "n" ordinal utility functions on the "n" factors: where the formula_51 are constants. Given a set of indices formula_52, the set of commodities formula_53 is called "preferentially independent" if the preference relation formula_1 induced on formula_53, given constant quantities of the other commodities formula_56, does not depend on these constant quantities. If formula_47 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 formula_1), then formula_47 is additive. Moreover, in that case formula_47 is unique up to an increasing "linear" transformation. For an intuitive constructive proof, see Ordinal utility - Additivity with three or more goods. Theorems on Cardinal utility. Theorem 1 of 1960 deals with preferences on lotteries. It can be seen as an improvement to the von Neumann–Morgenstern utility theorem of 1947. 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?"). Formally, there is a set formula_61 of sure choices. The set of lotteries is formula_62. 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 deals with agents whose preferences are represented  by frequency-of-choice. When they can choose between "A" and "B", they choose "A" with frequency formula_72 and "B" with frequency formula_73. The value formula_72 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: