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.
Contents
Background
Suppose we interrogate a person and ask him questions of the form "Do you prefer A or B?" (when A,B can be options, actions to take, states of the world, consumption bundles, etc.). We write down all the answers. Then, we want to represent the preferences of that person 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 we can find such representative utility function?
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:
- For every
x ∈ X , the sets{ y | y ⪯ x } and{ y | y ⪰ x } are topologically closed inX . - For every sequence
( x i ) such thatx i → x ∞ x i ⪯ y thenx ∞ ⪯ y , and if for all ix i ⪰ y thenx ∞ ⪰ y
Each one of the following conditions guarantees the existence of a real-valued continuous function that represents the preference relation
1. The set of equivalence classes of the relation
2. There is a countable subset of X,
3. X is separable and connected.
4. X is perfectly separable. 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.
Examples
A. Let
B. Let
Extension
Diamond applied Debreu's theorem to the space
In addition to the requirement that
Under these requirements, every stream
The existence result is valid even when the topology of X is changed to the topology induced by the discounted metric:
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
where the
Given a set of indices
If
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
Moreover, in that case
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
- The set of all sure choices
S is a connected and separable space; - The preferece relation on the set of lotteries
S × S is continuous - the sets{ ( A , B ) ∈ S × S | ( A , B ) ⪯ ( A ′ , B ′ ) } and{ ( A , B ) ∈ S × S | ( A , B ) ⪰ ( A ′ , B ′ ) } are topologically closed for all( A , B ) ∈ S ; -
( A 1 , B 2 ) ⪯ ( A 2 , B 1 ) and( A 2 , B 3 ) ⪯ ( A 3 , B 2 ) implies( A 1 , B 3 ) ⪯ ( A 3 , B 1 )
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
Debreu's theorem states that if the agent's function p satisfies the following conditions:
-
p ( A , B ) + p ( B , A ) = 1 -
p ( A , B ) ≤ p ( C , D ) ⟺ p ( A , C ) ≤ p ( B , D ) - Continuity: if
p ( A , B ) ≤ q ≤ p ( A , D ) , then there exists C such that:p ( A , C ) = q .
Then there exists a cardinal utility function u that represents p, i.e: