(Hemi)continuity of additive preference preorders

It is shown that the two common notions of topological continuity for preference preorders, which require closed contour sets and a closed graph respectively, are equivalent even when completeness is not assumed, provided that the domain is a normed linear space or a topological group and the preorder is additive.     

On financial equilibrium with intermediation costs

This paper studies the set of competitive equilibria in financial economies with intermediation costs. We consider an arbitrary dividend structure, which includes options and equity with limited liabilities. We show a general existence result and upper-hemi continuity of the equilibrium correspondence. Finally, we prove that when intermediation costs approach zero, unbounded volume of asset trades is a necessary and sufficient condition, provided that, there is no financial equilibrium without intermediation costs.     

Implementation with partial verification

This paper examines the implementability of social choice functions when only partial verification of private information is possible. Green and Laffont (1986) used this framework to derive a necessary and sufficient condition for the revelation principle to continue to hold with partial verification. We provide economically interesting characterizations of this condition, which suggest that it may be too restrictive. This leads us to consider implementation (not necessarily truthful) in general, when there is partial verification. We consider the case where compensatory transfers are allowed, giving the mechanism designer further leeway. We show how partial verification may allow efficient implementation of bilateral trade, where it would otherwise not be possible. Received: 1 August 1998 / Accepted: 5 September 2000     

Afriat’s theorem for indivisible goods

We identify a natural counterpart of the standard GARP for demand data in which goods are all indivisible. We show that the new axiom (DARP, for “discrete axiom of revealed preference”) is necessary and sufficient for the rationalization of the data by a well-behaved utility function. Our results complement the main finding of Polisson and Quah (2013), who rather minimally modify the original consumer problem with indivisible goods so that the standard GARP still applies.     

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

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.     

Utility functions on ordered convex sets

The work of Cantor in set theory near the end of the last century showed that a linearly ordered set need not be order isomorphic to a subset of the real numbers. To obtain order isomorphism, it is necessary and sufficient that some countable subset of the linearly ordered set be order- dense in the entire set. The present paper proves a negative result that is much stronger than Cantor's. It shows that a weakly ordered convex set whose order relation is ‘continuous’ and ‘totally convex’ need not be order homomorphic to a subset of the reals. To obtain order homomorphism, it is necessary and sufficient that the weakly ordered set be countably bounded, i.e., have countable coinitial and cofinal subsets. Countable boundedness is a significantly weaker condition than countable order-denseness. Connections between these results and economic utility theory are discussed.     

Continuous extension of preferences

We consider the problem of extending preferences from a subset of a commodity space to the entire space. It is a simple consequence of the Tietze extension theorem that continuous preferences can be extended if they are defined on closed subsets of a normal space and are representable by utility functions. We show the following: If the space is a non-separable metric space, then extension of preferences is not always possible. In fact for (path-connected) metric spaces, extension property, utility representation property, and separability are equivalent to each other.     

Large games with countable characteristics

We consider anonymous games with an atomless probability space of players in which players’ characteristics are countable. Our main result shows that the set of equilibrium distributions coincides with the set of distributions induced by equilibrium strategies together with the function assigning characteristics to players. This result implies the existence of Nash equilibria in continuous large games with countable characteristics.     

Consistent choice under uncertainty

Suppose ex post preferences are defined upon prizes and ex ante preferences are defined upon lotteries. Then the consistent choice of decision rules reigns whenever ex post optimality is equivalent to ex ante optimality. This essay provides a necessary and sufficient condition for consistent choice in terms of revealed preferences. Indeed, ex ante revealed preferences must be induced from ex post revealed preferences in a manner which requires them to satisfy the independence axiom from expected utility theory.     

Efficiency and monotonicity of bargaining solutions

A continuity axiom for bargaining solutions is introduced, which is satisfied by all Pareto optimal and continuous (in the Hausdorff metric) solutions. It is shown by two examples how this axiom can be used to characterize solutions having certain kind of monotonicity properties. One of the solutions is the lexicographic maximin solution. The other is the lexicographic extension of the Kalai-Smorodinsky solution. The former is an efficient (Pareto optimal) extension of the symmetric proportional solution. The latter is an efficient extension of the Kalai-Smorodinsky solution.     

The Computational Complexity of Rationalizing Behavior

We study the computational complexity of rationalizing choice behavior. We do so by analyzing two polar cases, and a number of intermediate ones. In our most structured case, that is where choice behavior is defined in universal choice domains and satisfies the “weak axiom of revealed preference,” finding the complete preorder rationalizing choice behavior is a simple matter. In the polar case, where no restriction whatsoever is imposed, either on choice behavior or on choice domain, finding a collection of complete preorders that rationalizes behavior turns out to be intractable. We also show that the task of finding the rationalizing complete preorders is equivalent to a graph problem. This allows the search for existing algorithms in the graph theory literature, for the rationalization of choice.     

A utility representation theorem with weaker continuity condition

We prove that a mixture continuous preference relation has a utility representation if its domain is a convex subset of a finite dimensional vector space. Our condition on the domain of a preference relation is stronger than Eilenberg (1941) and Debreu (1959, 1964), but our condition on the continuity of a preference relation is strictly weaker than the usual continuity assumed by them.     

Preference aggregation and atoms in measures

This paper examines the aggregation of preferences with a finitely additive measure space of agents. We consider three types of non-dictatorship axioms: non-dictatorship, coalitional non-dictatorship, and atomic non-dictatorship. First, we show that the existence of an atom is a necessary and sufficient condition for the existence of a social welfare function that satisfies weak Pareto, independence of irrelevant alternatives, and coalitional non-dictatorship. Second, we simultaneously impose non-dictatorship and coalitional non-dictatorship, and specify a necessary and sufficient condition for the finitely additive measure that guarantees the compatibility among the axioms. Third, we impose all non-dictatorship axioms and show that the corresponding measure is extremely restricted.     

Bayesian Analysis in the <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Norm of the Mixing Proportion Using Discriminant Analysis

We consider the mixing proportion π in a mixture of two independent distributions, and establish the expression of its posterior density, in closed form and in terms of L ¹-norms of various related functions, using a prior beta and the optimal classification rule for the two populations provided by Discriminant analysis. A numerical example fully illustrates the concepts presented.Research partially supported by CRSNG 9249 (Canada). The authors wish to thank the Faculty of Science and the Department of Statistics of UNISA for their generous support that has led to this joint work. Also, thanks to Ms. Jeannette LeBlanc for her excellent technical support, and to an anonymous referee for very helpful comments that have helped to improve the presentation of the paper.     

Pareto optimality in continuous-time OLG economies

The aim of this paper is the achievement of a complete characterization of the Pareto optimality of competitive equilibria for deterministic, pure exchange, continuous-time economies with a countable number of overlapping generations, where each consumer’s life-span consists in a bounded interval of time. For such an environment, we obtain separate sufficient and necessary conditions for Pareto optimality in the form of the Cass’ criterion, that is, in terms of the equilibrium prices. However, these conditions are not equivalent in general. Therefore, in order to get that equivalence we are compelled to impose certain restrictions, either on consumers’ lifetimes, assuming that all of them have the same longevity, or on the dynamic behaviour of relative intertemporal equilibrium prices. In both cases, we are able to derive a single condition that is sufficient and necessary for efficiency, thus achieving full characterizations.     

Stochastic comparisons of multivariate reversed frailty model

Multivariate frailty approaches are most commonly used to define distributions of random vectors, which represent lifetimes of individuals or components and stochastically compare them in terms of various multivariate orders. In this paper, we study a multivariate shared reversed frailty model and a general multivariate reversed frailty mixture model, and derive sufficient conditions for some of the stochastic orderings to hold among the random vectors. We also consider a particular case of a general multivariate mixture model in which the baseline distribution function is represented in terms of a copula and study stochastic comparisons (stochastic and lower orthant order) among the two random vectors.     

Dual Representations of Non-Parametric Technologies and Measurement of Technical Efficiency

This paper extends the recent work by Frei and Harker on projections onto efficient frontiers (1999) in three ways. First, we provide a formal definition of the production set as the intersection of a finite number of closed halfspaces. We emphasize the necessity of a complete enumeration of the supporting hyperplanes to define the production set properly. We focus on the problem of exhaustive enumeration of the supporting hyperplanes to characterize the production set. Second, we consider the problem of an arbitrary-norm projection on the boundary of the production set. We use the concept of the Hölder distance function and we derive the necessary mathematics to calculate distances and projections of inefficient DMUs onto the efficient frontier. Third, we introduce a relevant weighting scheme for inputs and outputs so that the Hölder distance function respects the commensurability axiom defined by Russell (1988). Finally, we present an illustration using the same data set as Frei and Harker (1999) to highlight some of the extensions proposed in the paper.     

Menu-dependent self-control

The literature on self-control problems has typically put forth models that imply behavior that is consistent with the weak axiom of revealed preference (WARP). We argue that when choice is the outcome of some underlying internal conflict, the resulting choices may not be perfectly consistent across choice problems: an agent’s ability to resist temptation may well depend on what alternatives are available to him. We generalize Gul and Pesendorfer (2001) so that self-control weakens in the presence of temptation. To model choices from menus explicitly, we consider a choice correspondence as well as a preference over menus and relax both the Independence axiom for the preference and the WARP condition for the choice correspondence. The model is shown to unify a range of well-known findings in the experimental literature on choice under risk and over time within a single specification.     

CONTINUOUS PREFERENCE ORDERINGS REPRESENTABLE BY UTILITY FUNCTIONS

This paper surveys the conditions under which it is possible to represent a continuous preference ordering using utility functions. We start with a historical perspective on the notions of utility and preferences, continue by defining the mathematical concepts employed in this literature, and then list several key contributions to the topic of representability. These contributions concern both the preference orderings and the spaces where they are defined. For any continuous preference ordering, we show the need for separability and the sufficiency of connectedness and separability, or second countability, of the space where it is defined. We emphasize the need for separability by showing that in any nonseparable metric space, there are continuous preference orderings without utility representation. However, by reinforcing connectedness, we show that countably boundedness of the preference ordering is a necessary and sufficient condition for the existence of a (continuous) utility representation. Finally, we discuss the special case of strictly monotonic preferences.     

Concepts of similarity for utility functions

A generalization of the compact-open topology is defined for a space of utility functions with different choice sets. The space is a complete separable metric space. A continuous representation theorem of Levin (1983) gives a homeomorphism between this space and the space of preference relations, topologized by closed convergence. A map into the space is measurable with regard to the Borel algebra iff the choice set correspondence is measurable and the utility function is measurable on its graph. A coarser topology (that of subgraph convergence) is also studied. This topology is coarser than the compact-open topology when the utility functions are defined on the same choice set. However the demand correspondence is still upper-hemicontinuous. A homeomorphism is given between this space and the space of preference relations, with the latter given a certain topology coarser than closed convergence.