Debreu's representation theorems

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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.

Background

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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?

Existence of ordinal utility function

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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.

Statement

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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:

  • X is a topological space.
  •   is a relation on X which is total (all items are comparable) and transitive.
  •   is continuous. This means that the following equivalent conditions are satisfied:
    1. For every  , the sets   and   are topologically closed in  .
    2. For every sequence   such that  , if for all i   then  , and if for all i   then  

Each one of the following conditions guarantees the existence of a real-valued continuous function that represents the preference relation  . The conditions are increasingly general, so for example, condition 1 implies 2, which implies 3, which implies 4.

1. The set of equivalence classes of the relation   (defined by:   iff   and  ) are a countable set.

2. There is a countable subset of X,  , such that for every pair of non-equivalent elements  , there is an element   that separates them ( ).

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]

Examples

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A. Let   with the standard topology (the Euclidean topology). Define the following preference relation:   iff  . It is continuous because for every  , the sets   and   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  . An example of such function is  .

B. Let   with the standard topology as above. The lexicographic preferences relation is not continuous in that topology. For example,  , but in every ball around (5,1) there are points with   and these points are inferior to  . Indeed, this relation cannot be represented by a continuous real-valued function (in fact, it cannot be represented even by non-continuous functions).

Proofs

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Proofs from.[2]

Notation: for any  , define  , and similarly define other intervals.

Proof of 1, 2

For 1, use the proposition that any countable linear ordering is isomorphic to a subset of  .

For 2, first use the proposition to construct a utility   that preserves the ordering. Then for each   not equivalent to one of the  , construct its upper and lower Dedekind cuts  . By density of the set  , two such   have the same ordering iff their Dedekind cuts are equal.

Then, define  . This defines a utility function  .

Finally, use the hyperbolic tangent function   to squeeze the extended real line to a finite interval.

Proof of 3

If   is trivial on   then define  . So assume it's not trivial.

If   is dense in  , then if   in  , there exists   such that  

The intervals   are nonempty since  .
By continuity of  , both intervals are open subsets of  . By totality of  , their union is all of  . Since   is connected, their intersection is nonempty. Thus exists some   such that  .
Since   is dense in  , and   is continuous, there exists a close enough   such that  .

Since   is separable, we apply part 2.

Proof of 4

Enumerate the countable set of basis sets  . For each  , pick one representative  , and gather them into one set  . This means that any   if   and   is nonempty, then there exists some  , so that  . It remains to deal with the exceptions.

Define a "gap pair" to be   such that   and   is empty. Pick a set of representatives  , such that for any gap pair   there exists exactly one pair of representatives   such that  .

For each pair  , choose some   such that  , and  . It's easy to check that if   then we must have  . Thus the number of gap pair representatives is at most countable.

Now the set   is countable, and we use part 2.

Applications

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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.

In addition to the requirement that   be total, transitive and continuous, he added a sensitivity requirement:

  • If a stream   is smaller than a stream   in every time period, then  .
  • If a stream   is smaller-than-or-equal-to a stream   in every time period, then  .

Under these requirements, every stream   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:  

Additivity of ordinal utility function

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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.

Statement

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These are the general assumptions:

  • X, the space of all bundles, is a cartesian product of n commodity spaces:   (i.e., the space of bundles is a set of n-tuples of commodities).
  •   is a relation on X which is total (all items are comparable) and transitive.
  •   is continuous (see above).
  • There exists an ordinal utility function,  , representing  .

The function   is called additive if it can be written as a sum of n ordinal utility functions on the n factors:

 

where the   are constants.

Given a set of indices  , the set of commodities   is called preferentially independent if the preference relation   induced on  , given constant quantities of the other commodities  , does not depend on these constant quantities.

If   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  ), then   is additive.

Moreover, in that case   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

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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   of sure choices. The set of lotteries is  . Debreu's theorem states that if:

  1. The set of all sure choices   is a connected and separable space;
  2. The preference relation on the set of lotteries   is continuous - the sets   and   are topologically closed for all  ;
  3.   and   implies  

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. When they can choose between A and B, they choose A with frequency   and B with frequency  . The value   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:

  1. Completeness:  
  2. Quadruple Condition:  
  3. Continuity: if  , then there exists C such that:  .

Then there exists a cardinal utility function u that represents p, i.e:

 

See also

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References

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  1. ^ Debreu, Gerard (1954). Representation of a preference ordering by a numerical function.
  2. ^ a b Debreu, Gerard (1986). "6. Representation of a preference ordering by a numerical function". Mathematical economics: Twenty papers of Gerard Debreu; introduction by Werner Hildenbrand (1st pbk. ed.). Cambridge [Cambridgeshire]: Cambridge University Press. ISBN 0-521-23736-X. OCLC 25466669.
  3. ^ Debreu, Gerard (1964). "Continuity properties of Paretian utility". International Economic Review. 5 (3): 285–293. doi:10.2307/2525513.
  4. ^ Diamond, Peter A. (1965). "The Evaluation of Infinite Utility Streams". Econometrica. 33: 170. doi:10.2307/1911893. JSTOR 1911893.
  5. ^ a b c Debreu, Gerard. Topological Methods in Cardinal Utility Theory (PDF).