‘UTILITY WITHOUT THE COMPLETENESS AXIOM’.

                                                            -Robert Aumann.

                                                              -A Review.

                                                                                                              

                     MICROECONOMICS ASSIGNMENT

                                     SEMESTER - 1

                                                        

                                                                            Submitted by,

                                                                                  Arya.U.R.,

                                                                                 1 M A Economics,

                                                                                       Class No: 3                 

                                             Department of Economics, Kariyavattom,

                                                            University of Kerala    Submitted to,

        Siddik Rabiyath,

        Associate Professor, Department of Economics,

        University of Kerala.

     

                     CONTENTS 

·     Introduction

·     Utility Theory with Completeness Axiom

·     Views of Some Authors regarding Completeness Axiom

·     The Axioms

·     The Utility

·     Two Examples

·     Discussion of the Axioms

·     Maximization Problems ,Games and Shapley’s Theorem

·     The Structure of Partially Ordered Mixture Spaces

·     Duality

·     Proofs and Examples

·     Conclusion

·     Reference

       

                           INTRODUCTION

“Nothing can have value without being an object of utility.”

                                 -Karl Marx.

The usual ‘Utility Theory’ is based on four axioms- namely, Reflexivity, Transitivity, Completeness and Continuity.

This particular writing is based on an article titled, ‘Utility without the Completeness Axiom’ which is written by Robert Aumann. This article is published in ‘Econometrica’, the journal of the Econometric Society in its 30^th volume, Number-3 in July, 1962 (pp. 445-462).

Robert Aumann (born on 8 June, 1930) is an Israeli-American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel. He has made many contributions to the arena of Mathematical Economics and Game Theory. He has received the Nobel Memorial Prize in Economics in 2005 for his work on conflict and cooperation through game –theory analysis. He shared the prize with Thomas Schelling. He has also received the John von Neumann Theory Prize, Harvey Prize in Science and Technology and Israel Prize for Economical Research.

Through the article, Utility without the Completeness Axiom, Aumann has tried to build a utility theory that parallels the one developed by John von Neumann and Oskar Morgensetrn. The basic factor that makes Aumann’s theory different from Neumann-Morgenstern theory is that Aumann makes no use of the assumption that preferences are complete (i.e., that any two alternatives are comparable).

UTILITY THEORY WITH COMPLENESS     

                                   AXIOM  

Before going into that Utility Theory which is formulated without using the Completness Axiom, it is important to deal with the Utility Theory which uses Completeness axiom.

In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied the following four axioms has a utility function.

These axioms are:

·       Reflexivity:-

For every x in X such that, (x R X).

·       Completeness:-

For every x, y in X such that, either (x R y) is true or (y R x) is true or both the statements are true.

·       Transitivity:-

For any x, y and z in X such that, if both (x R y) and (y R z) are true then, (x R z).

·       Continuity:-

For any x in X such that, the not better than x-set and the not worse than x-set are closed in X.

The basic theorem of Utility Theory asserts that there exists a real –valued function u on the set of all lotteries, called the utility function, which enjoys the following properties:-

(a)          u represents the preference order, in the sense that a lottery x is preferred to a lottery y if and only if u(x)>u(y); and

(b)           u obeys the “expected utility hypothesis”, according to which the utility of a lottery is equal to the expected utility of all its prizes.

Furthermore, the u satisfying (a) and (b) is uniquely determined up to an additive and multiplicative constant.

Many axioms that govern the preference order of Neumann- Morgenstern have been questioned, mainly regarding their validity with respect to real life behavior. Some Authors have examined the effects or impacts by dropping or modifying one or the other axiom. For instance, Hausner’s multi-dimensional utilities dropped the so-called Continuity axiom.

Here we are concerned with the ‘Completeness’ axiom which says that ‘given any pair of lotteries, the individual either prefers one to the other or is indifferent between them’. It deliberately excludes the possibility that an individual may be willing or able to arrive at preference decision only for certain pair of lotteries, which for others he will be unwilling or unable to arrive at a decision.

Of all the axioms of utility, the completeness axiom is the most questionable. Like the other axioms, it is inaccurate in describing real life. Unlike others, we find it hard to accept even from the normative viewpoint.

VIEWS OF SOME AUTHORS REGARDING COMPLETENESS AXIOM

Some Authors expressed their views regarding completeness axiom as:

·       Describing the intransitivities in experimental work on utility theory, Luce and Raffia mentioned the possibility of intransitivity occurring when choice is made between inherently comparable alternatives.

·       For Thrall, ‘from practical point of view, if the number of judgments needed is finite, but large, there is a time difficulty. E.g.) By the time the Judge has reached the 1000000^th choice, his standards of comparisons are not same as initially. The theory calls for instantaneous and simultaneous judgments between all pairs.

·       Shapely says the outcome of a game sometimes take the form of a vector having numerical components whose relative values are ascertained. According to him, the partial difference ordering are useful for describing preferences of groups because they enable us to distinguish between indecision and indifference. Following up his idea, we can set up the theory of games, in which mixed outcomes to an individual player is partially ordered.

·       Neumann-Morgenstern themselves say that it is very doubtful, whether reality treats this property as a valid one is appropriate.

Historically, the first mention of the possibility of a utility theory without the ‘Completeness’ axiom was by Neumann-Morgenstern. Since the details were not published, Aumann made a prediction by saying that what they probably had in mind was some kind of mapping from the space of lotteries to a canonical partially ordered Euclidian space, rather than real – valued mappings used here.

1.   AXIOMS

The space on which utility will be defined is called mixture space. It is assumed that on our mixture space X, there is defined a transitive and reflexive relation called preference-or-indifference and is denoted by’ ≥’ If x ≥ y and y ≥ x, it could be said that x is indifferent to y and write x~y; if x≥y but not x~y, it could be said that x is preferred to y. and write x~ y.

(1.1) if 0< γ<1 and z is arbitrary, then, x ≥ y if and only if γx+(1-γ)z≥γy+(1-γ)z;

(1.2) archimidean or continuity axiom.

The relation ≥will be called partial order; the space together with the partial order ≥ will be called a partially ordered mixture space.

                     2.  THE UTILITY

A utility on a partially ordered mixture space X is a function from X to the reals for which

(2.1) u(γx+(1-γ)y)=γu(x)+(1-γ)u(y),

(2.2)x>y implies u(x)>u(y),

(2.3)x~y implies u(x)=u(y).

Condition (2.1) shows “expected utility hypothesis”, whereas (2.2) and (2.3) state that u represents the preference order.

THEOREM A: There is at least one utility on X.

       3.  EXAMPLES

An example of a mixture space is Euclidean n-space Rⁿ, considered as a vector space over the real numbers. Two of the partial orders most frequently  encountered  in the literature  are the weak and the strong partial orders on  Rⁿ, which are denoted by >w, and>s respectively. 

 4. DISCUSSION OF THE AXIOMS

This is done by taking into account various matters like;

·       The lotteries (objects of study in utility theory) may other lotteries as well as basic alternatives as their prizes. Lotteries that have other lotteries for their prizes are called compound lotteries and those that have only basic alternatives as prizes are called simple lotteries. Every compound lottery is equivalent to a simple one; for if one plays out the component lotteries one must eventually get to basic alternatives.

·       There are situations in which it is convenient to work with a model in which certain parameters (such as money) take a continuum of values. In such situations we it might be able to obtain a finite dimensional mixture space, by dividing out the indifference relation.

·       Axiom1.1 is taken from Hausner’s set of axioms. It asserts that a preference is not changed by” dilution”, and conversely that if we have a diluted preference, then the corresponding undiluted preference also holds.

·       Axiom1.2 is an extremely weak version of “archimidean” or continuity principle. According to Aumann, it is weaker than any variant he has seen. It serves only to exclude the case in which the direction of strict preference between a point z and a closed line segment [xy] goes in one direction for one of the end points y and in opposite direction for the entire remainder of the segment. Two cases which are not excluded are illustrated by weak and strong orders respectively (figure 1).In the figure, y and z are on the same horizontal line. For both orders all points in the half-open segment [xy) are preferred to z; the difference between the two orders is expressed in relation between z and the end point y. For the weak order y and z are incomparable, and for the strong order y is preferred to z; in neither case, though, is z actually preferred to y.

 

                                                                                                       x

                                                                                                      

                                                                          .z                         y

 

It is suggested that (1.2) could be slightly weakened.

(4.1) if γx+(1-γ)z>y for all γ>0, then z≥y.

The adoption of (4.1) would simplify the proof of the existence of utility (Theorem A).

(4.2)γ₀x+(1-γ₀)z>y implies that for all γ sufficiently close to γ_0,γx+(1-γ)z>y.

(4.1) excludes weak order and (4.2) excludes strong order; and when there is completeness, the two are equivalent. Even if we drop (1.2), we can still build a utility theory. It will still be possible to solve maximization problems and games under exactly the same conditions as before.

5. MAXIMISATION PROBLEMS, GAMES AND

                     SHAPLEY’S THEOREM

The convex hull of a finite set of points in X is called a convex polyhedron.

THEOREM B:   Let E be a convex polyhedron in X, and let x ᵋE.  Then x is maximal in E under the partial order ≥, if and only if there is a utility u on X such that x maximizes u over E.                        

What this theorem does for maximization problems defined on the partially ordered space X can also be done for two-person zero-sum game played over X.

THEOREM C: (c⁰,d⁰) is an equilibrium point in the matrix game A if and only if there is a pair (u,v) of utilities on X such that     ( c⁰,d⁰) is an equilibrium point (in the sense of Nash in the bimatrix game (u(A), -v(A)).

This theorem generalizes the result of Shapely. Shapely defined weak and strong equilibrium points accordingly and by exhibiting the utilities explicitly, proved what amounts to Theorem C. This theorem could be extended to n-persons games.

6. THE STRUCTURE OF PARTIALLY

                            ORDERED MIXTURE SPACES

Assumptions (1.1) and (1.2) can be restated as follows:

(6.1.1) x≥y implies x +z ≥ y +z;

(6.1.2) x≥y and α>0 implies αx≥αy;

(6.2) x>kz for all positive integers k implies not z>0.

(6.3)S is a convex cone,

(6.4)T ᵔ (-T)= ᵩwhere the bar denotes closure. T is also a convex cone but does not contain the origin.

7. DUALITY

The dual of a cone C is Rⁿ is defined to be the cone C ^* consisting of all u ᵋ C For example, the open positive orthant and the closed positive orthant without the origin are mutually dual, as are Rⁿ and ᵩ, an open half space and the ray normal to its bounding hyperplane, and concentric open and closed right circular cones (the latter without the origin) whose half- angles add to 90⁰. The cone {x e R²: x₁ > 0, X₂ > 0} is self-dual. In all the above examples C** = C.  If C** is calculated, it could be found that it is precisely the intersection of the open supports of C.

THEOREM D: A necessary and sufficient condition that C^**=C is that C be the intersection of its open supports.

The importance of Theorem D lies in the fact that the condition given is of wide applicability. A regular cone must be convex, and unless it is all of Rⁿ , it may not contain the origin; but aside from these restrictions, almost any cone “likely to come up in practice” is regular. Any open cone is regular. If C  is a convex cone obtained from a closed cone by removing the origin, then C is regular. The set of all x satisfying a given set of homogenous linear inequalities, which may contain both weak and south inequalities, is regular if it contains at least one strong inequality. If the positive half of one of the axes is added to the open positive octant in R³ , the result is a cone which is not regular, though it is convex  and does not contain  the origin.

A concept  closely  related  to regularity  is Fenchel's  "even  convexity" ;a set  is said to  be evenly convex if it is the intersection of open half  spaces. Clearly a regular cone is the same thing as an evenly convex cone without the origin.

 If T*X is regular, the order could be recovered from the set of all utilities. Thus the set of all utilities on a given X "almost” determines the order, and determines it completely if the set of orders under consideration is suitably restricted.

Here C ^* is defined to be the set of all u such that ux < 0 for all x ᵋ C. Under that definition the necessary  and sufficient  condition  that  C** = C is that  C be the intersection  of its closed supports,  or equivalently  that  it be convex  and closed.

8. PROOFS AND EXAMPLES

These could be dealt with under:

·       Characterization of Partially Ordered Euclidean Spaces

The objective here to prove that when X =Rn. Assumptions (1.1) and (1.2) are equivalent to (6.1.1), (6.1.2), and (6.2). The proof is a straightforward computation.

·       Proof of Theorem A

It is assumed that the underlying mixture space of X is

Rⁿ;   this involves no loss of generality because any finite-dimensional mixture space can be imbedded in such a mixture space.  The proof is by induction on n. If n =1 the order must either be complete or all elements are incomparable; in either case the theorem is trivial.  Suppose the theorem has been proved for all dimensions up to but not including n. If there is an element  of X  other than 0  that  is indifferent  to 0, then the indifference relation could be  "divided out" ,  i.e., consider equivalence  classes under indifference; this  yields  a space  of lower  dimension,  to  which  the  induction  hypothesis applies. So it could therefore assume without loss of generality that the order on X is pure, so that S = T u {0}.

·       Proof of Theorem B The "if" statement follows from the definition of utility (2.2). To prove the "only if" half, it is assumed again that the underlying space is Rn and that the x Shapley's proof precisely; it is included only for the sake of completeness.

·       Theorem C follows at once from Theorem B

An Infinite-Dimensional Partially Ordered Mixture Space without a Utility.

  Let X be the set of all infinite sequences x ={x_1, x₂ ...} of real numbers.  Define a pure order on X by stipulating that x ≥y if and only if xɩ≥ yɩ for all ɩ, but x =# y (this is the strong order). It is easily verified that all the axioms except finite dimensionality are satisfied.  Suppose there were a utility; set

        u₁₌ u{1, .0,.. }

        u₂₌ u{0,   1, 0,  }

From (2.2) it follows that all the uɩ are positive. Set xɩ = 1/uɩ for all ɩ. Then for all positive integers k we have

                    u(x) = u{x_1, x_2,…}

=x₁u_2+…= x_ku_{k +}u{0,…0, x_k+1…}>k, again by  (2.2). Hence u(x) is larger than any positive integer, as per Aumann, an absurdity. In this example, the dimensionality of X is the cardinality c of the continuum.                          

CONCLUSION

  This writing is based on an article, ‘Utility without the Completeness Axiom’ by Robert Aumann. The article starts with the description of the classical Utility Theory which was developed by John von Neumann and Oskar Morgenstern. Then Aumann moves to develop his own Utility Theory.

The fundamental point that makes Aumann’s theory different from Neumann-Morgenstern-theory is that Aumann has not made use of ‘Completeness’ axiom.

It is found that much of the utility theory stayed intact even when the completeness axiom is dropped. Aumann has used the ideas expressed many economists and has also used many proofs and examples to prove his point. Thus, as per Aumann, though the completeness axiom is neglected, we still get a utility function that satisfies the expected utility hypothesis. Also this utility function represents a preference order, but now in a weaker sense.  This makes Aumann’s theory similar to that of Neumann-Morgenstern. But this is no longer true in a partially-ordered situation – as long as we restrict ourselves to a single utility function.

                     REFERENCE

·       Aumann.J.Robert,

Utility without Completeness Axiom,

Econometrica ,Vol.30.  No.3 (Jul., 1962), pp.445-462.

·       Cowell Frank (2009),

Microeconomics: Principles and Analysis,

Oxford University Press,

New Delhi.

·       Dwivedi.N.D.(2013),

Microeconomics: Theory and Applications,

Dorling Kindersley Pvt. Ltd. ,

New Delhi.