Gerhard Herden

On the existence of utility functions ii

Abstract This is the first of two papers on (continuous) utility functions. While the second paper is mainly dedicated to the extension problem on (continuous) utility functions this paper presents general existence theorems on (continuous) utility functions on arbitrary preordered (topological) spaces. At first generalizations of the classical Birkhoff representation theorem are proved. Then as the main result of this paper a very general theorem which presents necessary and sufficient conditions for an arbitrary preordered topological space to have a continuous utility function is presented. This theorem allows one to reobtain all well known classical results on the existence of (continuous) utility functions due to Eilenberg, Debreu, Fleischer, Peleg, etc. and their generalizations due to Mehta as corollaries. Its proof follows the spirit of the work of Nachbin and its application to existence problems on (continuous) utility functions due to Mehta. Hence it proves a conjecture of Mehta.

The non-existence of a utility function and the structure of non-representable preference relations

In this paper we investigate the structure of non-representable preference relations. While there is a vast literature on different kinds of preference relations that can be represented by a real-valued utility function, very little is known or understood about preference relations that cannot be represented by a real-valued utility function. There has been no systematic analysis of the non-representation problem. In this paper we give a complete description of non-representable preference relations which are total preorders or chains. We introduce and study the properties of four classes of non-representable chains: long chains, planar chains, Aronszajn-like chains and Souslin chains. In the main theorem of the paper we prove that a chain is non-representable if and only it is a long chain, a planar chain, an Aronszajn-like chain or a Souslin chain.

On the continuous analogue of the Szpilrajn Theorem I

Abstract One of the best known theorems in order theory, mathematical logic, computer sciences and mathematical social sciences is the Szpilrajn Theorem which states that every partial order can be refined to a linear order. Since in mathematical social sciences one frequently is interested in continuous linear orders or preorders, in this paper the continuous analogue of the Szpilrajn Theorem will be discussed.

On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

The Szpilrajn theorem and its strengthening by Dushnik and Miller belong to the most quoted theorems in many fields of pure and applied mathematics as, for instance, order theory, mathematical logic, computer sciences, mathematical social sciences, mathematical economics, computability theory and fuzzy mathematics. The Szpilrajn theorem states that every partial order can be refined or extended to a total (linear) order. The theorem by Dushnik and Miller states, moreover, that every partial order is the intersection of its total (linear) refinements or extensions. Since in mathematical social sciences or, more general, in any theory that combines the concepts of topology and order one is mainly interested in continuous total orders or preorders in this paper some aspects of a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller will be discussed. In particular, necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Dushnik-Miller theorem will be presented. In addition, it will be proved that a continuous analogue of the Szpilrajn theorem does not hold in general. Further, necessary and in some cases necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Szpilrajn theorem will be presented.

Lexicographic decomposition of chains and the concept of a planar chain

Abstract In an earlier paper [Journal of Mathematical Economics, 37 (2002) 17–38], we proved that if a preference relation on a commodity space is non-representable by a real-valued function then that chain is necessarily a long chain, a planar chain, an Aronszajn-like chain or a Souslin chain. In this paper, we study the class of planar chains, the simplest example of which is the Debreu chain ( R 2 , l ) .

On some equivalent approaches to Mathematical Utility Theory

Abstract In this paper a fundamental result is proved which shows, in particular, that the continuous representation theorems of Eilenberg, Debreu, Peleg, Herden, the Debreu Open Gap Theorem, the Beardon Weak Open Gap Theorem, Nachbins Separation Theorem, the Cantor Characterization Theorem of the linear continuum and, in addition, Urysohns Separation Theorem and the Alexandroff-Urysohn Metrization Theorem can be considered to be equivalent to one another. This result, therefore, finishes a development initiated by Mehta which combines the basic approaches to mathematical utility theory with some of the most important results of elementary topology.

SOME LIFTING THEOREMS FOR CONTINUOUS UTILITY FUNCTIONS

Abstract In this paper the extension problem on continuous utility functions, which are defined on (closed) subsets of arbitrary preordered topological spaces, is solved. This solution of the general extension problem on continuous utility functions is of particular importance because of the following reasons. It generalizes the classical extension theorem due to Tietze-Urysohn for normal spaces. Furthermore, it generalizes all theorems on the existence of continuous utility functions on arbitrary preordered topological spaces (cf., for example, the Arrow-Hahn-theorem), and finally it can be used to prove general characterization theorems for goodness criteria in numerical taxonomy. This last mentioned application will be carried out in forthcoming paper.

Some Aspects of Clustering Functions

Janowitz’s concept of hierarchical clustering which includes the concepts of hierarchical clustering due to Jardine and Sibson and Matula is extended—following the main stream of the theory of partially ordered sets—to describe all connections between hierarchies and isotone functions which measure the homogeneity or compactness of sets of data. In particular a very general description of those hierarchies which correspond bijectively to Hubert’s k-clustering functions is presented. As a consequence an exact characterization of the discrepancies between the original concept of Jardine and Sibson—its generalization due to Janowitz—and Hubert’s concept of hierarchical clustering is possible.

Arbitrary torsion classes and almost free abelian groups

Using the set theoretical principle ∇ for arbitrary large cardinals κ, arbitrary large strongly κ-free abelian groupsA are constructed such that Hom(A, G)={0} for all cotorsion-free groupsG with |G|<κ. This result will be applied to the theory of arbitrary torsion classes for Mod-Z. It allows one, in particular, to prove that the classF of cotorsion-free abelian groups is not cogenerated by aset of abelian groups. This answers a conjecture of Göbel and Wald positively. Furthermore, arbitrary many torsion classes for Mod-Z can be constructed which are not generated or not cogenerated by single abelian groups.

Open Gaps, Metrization and Utility

SummaryIt is shown that each of the Debreu Open Gap Theorem and the Debreu Continuous Utility Representation Theorem can be used in order to prove the other. Furthermore, it is proved that the classical Alexandroff-Urysohn Metrization Theorem implies Debreus Continuous Utility Representation Theorem and, thus, all known results on the existence of continuous utility functions.