Proper preferences and quasi-concave utility functions

Notre Dame Journal of Formal Logic, 1985

Some years ago J. H. Silver proved that a co-analytic equivalence relation on a Polish space has either countably many or continuum many equivalence classes. Later L. Harrington greatly simplified the complicated original proof. The present paper is a sort of footnote to Harrington's lectures on these matters. It will be shown that information developed in his proof settles a problem of (hyper-)theoretical mathematical economics first investigated by Wesley [13] and Mauldin [8]. Namely, it will be shown that any family of closed preference orders that is parametrized in a Borel fashion can be represented by a family of continuous utility functions parametrized in an absolutely measurable fashion. Though the author is greatly indebted to Mauldin's work [8], the treatment of the problem here will be self-contained. Background and motivation for problems of this kind can be found in [6], Section 2.1. Terminology and notation pertaining to descriptive set theory will be as in [9]. 2 Definitions Throughout let ψ be a topological space. A preference order on 'ψ is any transitive, connected binary relation <*. Associated are the strict preference and indifference relations given by: x <* y <-> x <* y & ~y <* x X =* y +-> x <* y & y <* χ m Note that Ξ* i s an equivalence relation, and that <* induces a linear order on its equivalence classes, [x]* will denote the equivalence class of x. <* will be *Research in part supported by USA National Science Foundation Grant MCS 8003254.

Economic Theory Bulletin, 2018

We construct a complete space of smooth strictly convex preference relations defined over physical commodities and monetary transfers. This construction extends the classical one by assuming that preferences are monotone in transfers, but not necessarily in all commodities. We thereby provide a natural framework to perform genericity analyses in situations involving inventory costs or decisions under risk. The space of preferences we construct is contractible, which allows for a natural aggregation procedure in collective decision situations.

Journal of Mathematical Economics, 1986

This paper primarily demonstrates the existence of Arrow-Debreu equilibria in a general class of topological vector spaces of commodity bundles. Two conditions based on production possibilities, preferences, and the topological nature of bounded sets are shown to substitute, in any locally convex space, for the advantages of the Euclidean topology. Examples fulfilling these conditions are supplied. The approach is that of Bewley, demonstrating equilibria on finite-dimensional sub-economies and establishing a net of these equilibria that converges to an equilibrium on the whole commodity space. An example of equilibrium with a storage technology is given. An auxiliary result concerns the price support of efficient allocations. This paper was distributed in draft in July 1983. A version appeared in a dissertation completed in the Engineering-Economic Systems Department at Stanford University. I am grateful for the comments of David Luenberger, David Kreps, Kenneth Arrow, Donald Brown, Andreu Mas-Colell, William Zame, and a referee.

Proceedings of the 2019 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (EUSFLAT 2019), 2019

Convexity of preferences is a canonical assumption in economic theory. In this paper we consider a generalized definition of convex preferences that relies on the abstract notion of convex space.

Springer eBooks, 2006

A direct construction of concave utility functions representing convex preferences on finite sets is presented. An alternative construction in which at first directions of supergradients ("prices") are found, and then utility levels and lengths of those supergradients are computed, is exhibited as well. The concept of a least concave utility function is problematic in this context.

In this paper we are concerned with numerical representation of preferences over bundles. Specifically, we provide an axiomatic characterization of preferences that have a numerical representation that look similar to a concave function defined on a convex set. We call such preferences concave. We also show that the concept of a super-gradient is inherent to rational choice. We discuss additively separable preferences and their axiomatization due to Fishburn (1970) since they are an important example of concave preferences. Using the same methods that we use to prove earlier results we show that the first fundamental theorem of welfare economics holds for combinatorial assignment problems.

Annals of Operations Research, 1998

Journal of Mathematical Economics, 2000

We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder. 

Mathematical Social Sciences, 1995

We consider a class of relations which includes irreflexive preference relations and interdependent preferences. For this class, we obtain necessary and sufficient conditions for representation of the relation by two numerical functions in the sense of a ~ x if and only if u(a) < vex).

Journal of Mathematical Economics, 1987

It is shown that if a consumer's preference ordering is strictly convex and is representable by means of a concave, twice continuously differentiable utility function, then the partial derivative of a demanded commodity with respect to its price is bounded from above in a neighborhood of a price vector at which the demand fails to be differentiable.