In economics, the Debreu's theorems are preference representation theorems—statements about the representation of a preference ordering by a real-valued utility function. The theorems were proved by Gerard Debreu during the 1950s.
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. However, they are applicable in a much more general setting. These are the general assumptions:
\preceq
\preceq
x\inX
\{y|y\preceqx\}
\{y|y\succeqx\}
X
(xi)
xi\toxinfty
xi\preceqy
xinfty\preceqy
xi\succeqy
xinfty\succeqy
Each one of the following conditions guarantees the existence of a real-valued continuous function that represents the preference relation
\preceq
1. The set of equivalence classes of the relation
\sim
x\simy
x\preceqy
x\succeqy
2. There is a countable subset of X,
Z=\{z0,z1,...\}
x\precy
zi\inZ
x\preceqzi\preceqy
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.[3]
A. Let
X=R2
(x,y)\preceq(x',y')
x+y\leqx'+y'
(x,y)
\{(x',y')|x'+y'\leqx+y\}
\{(x',y')|x'+y'\geqx+y\}
\preceq
u(x,y)=x+y
B. Let
X=R2
(5,1)\succ(5,0)
x<5
(5,0)
Proofs from.
Notation: for any
x,y\inX
(x,y)=\{z\inX:x\precz\precy\}
Diamond[4] applied Debreu's theorem to the space
X=\ellinfty
In addition to the requirement that
\preceq
x
y
x\precy
x
y
x\preceqy
Under these requirements, every stream
x
The existence result is valid even when the topology of X is changed to the topology induced by the discounted metric:
infty{2 | |
d(x,y)=\sum | |
t=1 |
-t|xt-yt|}
Theorem 3 of 1960[5] 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.
These are the general assumptions:
X=
n{X | |
x | |
i} |
\preceq
\preceq
v
\preceq
The function
v
v(x1,...,xn)=\sum
n{k | |
i |
vi(xi)}
ki
Given a set of indices
I
(Xi)i\in
\preceq
(Xi)i\in
(Xi)i\notin
If
v
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
\preceq
v
Moreover, in that case
v
For an intuitive constructive proof, see Ordinal utility - Additivity with three or more goods.
Theorem 1 of 1960[5] 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
S
S x S
S
S x S
\{(A,B)\inS x S|(A,B)\preceq(A',B')\}
\{(A,B)\inS x S|(A,B)\succeq(A',B')\}
(A,B)\inS
(A1,B2)\preceq(A2,B1)
(A2,B3)\preceq(A3,B2)
(A1,B3)\preceq(A3,B1)
u(A,B)=
u(A,A)+u(B,B) | |
2 |
Theorem 2 of 1960[5] deals with agents whose preferences are represented by frequency-of-choice. When they can choose between A and B, they choose A with frequency
p(A,B)
p(B,A)=1-p(A,B)
p(A,B)
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)\leqp(C,D)\iffp(A,C)\leqp(B,D)
p(A,B)\leqq\leqp(A,D)
p(A,C)=q
p(A,B)\leqp(C,D)\iffu(A)-u(B)\lequ(C)-u(D)