Ghanshyam B. Mehta

Representations of preferences orderings

1 Ordered Sets and Order Homomorphisms.- 2 Order Homomorphisms in Euclidean Space.- 3 The Fundamental Theorems.- 4 A Miscellany of Representations.- 5 The Urysohn-Nachbin Approach.- 6 Interval Orders.- 7 Differentiable Order Homomorphisms.- 8 Jointly Continuous Order Homomorphisms.- References.

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.

Utility and entropy

Summary. In this paper we study an astonishing similarity between the utility representation problem in economics and the entropy representation problem in thermodynamics.

Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof

Abstract In this paper, we present a new unified approach and an elementary proof of a very general theorem on the existence of a semicontinuous or continuous utility function representing a preference relation. A simple and interesting new proof of the famous Debreu Gap Lemma is given. In addition, we prove a new Gap Lemma for the rational numbers and derive some consequences. We also prove a theorem which characterizes the existence of upper semicontinuous utility functions on a preordered topological space which need not be second countable. This is a generalization of the classical theorem of Rader which only gives sufficient conditions for the existence of an upper semicontinuous utility function for second countable topological spaces.

Existence of an order-preserving function on normally preordered spaces

The object of this paper is to generalize the classic theorems of Eilenberg and Debreu on the existence of continuous order-preserving transformations on ordered topological spaces and to prove them in a different way. The proof of the theorems is based on Nachbins generalization to ordered topological spaces of Urysohns separation theorem in normal topological spaces.

Infinite-dimensional Gale-Nikaido-Debreu theorem and a fixed-point theorem of Tarafdar

This paper proves the equivalence of five fixed-point theorems in topological vector spaces and then uses them to deduce an infinite-dimensional generalization of the classic Gale-Nikaido-Debreu theorem. The proof is based on the Hahn-Banach theorem and the Tarafdar fixed point theorem. There is also a discussion of the relationship of this generalization to the existence theorems of Aliprantis and Brown, Border, Yannelis and others.

Some general theorems on the existence of order-preserving functions

This paper proves some general theorems on the existence of order-preserving functions on topological ordered spaces. The proofs are based on Nachbins generalization to topological ordered spaces of the theorems of Urysohn and Weil in general topology.

Utility functions on locally connected spaces

We investigate the role of local connectedness in utility theory and prove that any continuous total preorder on a locally connected separable space is continuously representable. This is a new simple criterion for the representability of continuous preferences, and is not a consequence of the standard theorems in utility theory that use conditions such as connectedness and separability, second countability, or path-connectedness. Finally we give applications to problems involving the existence of value functions in population ethics and to the problem of proving the existence of continuous utility functions in general equilibrium models with land as one of the commodities.

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 ) .

Maximal elements of condensing preference maps

Abstract We use the methods of nonliner analysis [1–4] to prove the existence of a maximal element for a class of preference maps defined on a closed, bounded, and convex, but not necessarily compact, subset of a Banach space.