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1991 | Buch

Positive Operators, Riesz Spaces, and Economics

Proceedings of a Conference at Caltech, Pasadena, California, April 16–20, 1990

verfasst von: Charalambos D. Aliprantis, Kim C. Border, Wilhelmus A. J. Luxemburg

Verlag: Springer Berlin Heidelberg

Buchreihe : Studies in Economic Theory

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Über dieses Buch

Over the last fifty years advanced mathematical tools have become an integral part in the development of modern economic theory. Economists continue to invoke sophisticated mathematical techniques and ideas in order to understand complex economic and social problems. In the last ten years the theory of Riesz spaces (vector lattices) has been successfully applied to economic theory. By now it is understood relatively well that the lattice structure of Riesz spaces can be employed to capture and interpret several economic notions. On April 16-20, 1990, a small conference on Riesz Spaces, Positive Opera­ tors, and their Applications to Economics took place at the California Institute of Technology. The purpose of the conference was to bring mathematicians special­ ized in Riesz Spaces and economists specialized in General Equilibrium together to exchange ideas and advance the interdisciplinary cooperation between math­ ematicians and economists. This volume is a collection of papers that represent the talks and discussions of the participants at the week-long conference. We take this opportunity to thank all the participants of the conference, especially those whose articles are contained in this volume. We also greatly ap­ preciate the financial support provided by the California Institute of Technology. In particular, we express our sincerest thanks to David Grether, John Ledyard, and David Wales for their support. Finally, we would like to thank Susan Davis, Victoria Mason, and Marge D'Elia who handled the delicate logistics for the smooth running of the confer­ ence.

Inhaltsverzeichnis

Frontmatter
Valuation and Optimality in Exchange Economies with a Countable Number of Agents
Abstract
We present versions of the two fundamental welfare theorems of economics for exchange economies with a countable number of agents and an infinite dimensional commodity space. These results are then specialized to the overlapping generations model.
Charalambos D. Aliprantis, Donald J. Brown, Owen Burkinshaw
Equilibrium Points of Non-Cooperative Random and Bayesian Games
Abstract
We provide random equilibrium existence theorems for non-cooperative random games with a countable number of players. Our results yield as corollaries generalized random versions of the ordinary equilibrium existence result of J. Nash [22]. Moreover, they can be used to obtain equilibrium existence results for games with incomplete information, and in particular Bayesian games. In view of recent work on applications of Bayesian games and Bayesian equilibria, the latter results seem to be quite useful since they delineate conditions under which such equilibria exist.
Nicholas C. Yannelis, Aldo Rustichini
Equilibria of Large Games with Imperfect Observability
Abstract
In this paper we present two formulations of an equilibrium notion for large games in which each player cannot observe precisely the moves of the other players in the game. In the context of large anonymous games where the moves of the other players are summarized by a probability measure on the action space, imperfect observability is formulated as a map from the space of such measures to the space of probability measures on this space. In the context of large non-anonymous games where the moves of the other players are summarized by a measurable function from the space of players to the action space, imperfect observability is formulated as a conditional expectation of such a function with respect to a σ-subalgebra of the measure space of players. We report results both on the existence and upper hemicontinuity of equilibrium.
Subir K. Chakrabarti, M. Ali Khan
Functional Analytic Tools for Expected Utility Theory
Abstract
Depending on the school of thought, expected utility theory states that choices among lotteries either should be made or actually made by maximizing the expected value of a real valued function of the outcomes—a utility function. This article provides a look at some of the functional analytic results used in expected utility theory. I concentrate on applications to the theory of stochastic dominance relations and the revealed preference approach to expected utility. Few of these results are deep, given the underlying tools, but many of them are not widely known, and their combination is novel. In particular, the revealed preference results of Border [4] are extended to higher degree stochastic dominance relations.
Kim C. Border
Remarkable Points and X (N) -spaces
Abstract
In 1974 Veksler, Koldunov, and Lozanovsky discovered that in the vicinity of some points in the maximal ideal space of the classical Banach algebra [0, 1] every measurable function is integrable. One of the purposes of this paper is to familiarize the reader with these points (referred to as remarkable points) and to indicate several directions for applications. The paper also studies and presents some applications of a new type of extension for Dedekind complete vector lattices.
Yuri A. Abramovich
Integration with Respect to Finitely Additive Measures
Abstract
This essay interprets the theory of finitely additive measures within the framework of the theory of Riesz spaces. The following topics are discussed: the extension procedures of measures, the Riemann and the Dunford integration procedures, the Radon-Nikodym Theorem and the Hahn Decomposition Theorem, the representation theory of the Radon- Nikodym derivatives as generalized functions, conditional expectation operators, the theory of L p -spaces, and the norm completeness problem.
The nature of the classical axiom of countable additivity is examined from Carathéodory’s algebraic measure-theoretic point of view.
Wilhelmus A. J. Luxemburg
Lattice-Ordered Algebras and f-Algebras: A Survey
Abstract
In this paper we present a survey of results on lattice-ordered algebras, particularly on f-algebras, almost f-algebras, and d-algebras. Example 1.2(v) and the description of nilpotents in various complex lattice- ordered algebras (Section 6) have not appeared before.
Charles B. Huijsmans
Approximating Derivative Securities in f-Algebras
Abstract
If E is a finitely generated Riesz subspace of C(X), where X is a compact Hausdorff space, containing the unit e, then Brown-Mertens- Ross have shown that the uniform closure of E is isomorphic to the uniform closure of the tensor product of the Riesz subspaces generated by s i , e (i = 1,…, m), where s 1, s 2,…, s m generate E. We extend their theorem to Archimedean f-algebras with unit and give applications to the theory of financial markets.
Donald J. Brown, Charles B. Huijsmans, Bernardus de Pagter
Some Unpleasant Objects in a Non-separable Hilbert Space
Abstract
In this paper we present examples that show the importance of the separability assumption in the theory of measurable correspondences.
M. Ali Khan, Aldo Rustichini
The Hopf Decomposition in Riesz Spaces
Abstract
Our goal in the paper is to discuss two extensions of the Hopf ergodic decomposition to Riesz spaces. The article is intended to be an outline of results which we have obtained recently on the topic, although it also contains material that has not been discussed elsewhere.
Radu Zaharopol
Frobenius Decomposition of Positive Compact Operators
Abstract
This work discusses the contributions of the authors to the analysis of nonnegative reducible operators on Banach lattices. In the first portion, we let (Ω, Σ, μ) denote a σ-finite measure space and L p (Ω, Σ,μ) (1 ≤ p < ∞) the usual Banach lattice of real-valued p th summable functions. Suppose, moreover, K is an integral operator with nonnegative kernel, mapping L p (Ω, Σ, μ) into itself, while possessing a compact iterate. We give necessary and sufficient conditions (Theorem 3.6) for the integral operator equation λf = K f + g to possess a nonnegative solution fL p (Ω, Σ, μ), where 0 ≤ gL p (Ω, Σ, μ) and λ > 0. In the second half of this work, we study the structure of the algebraic eigenspace corresponding to the spectral radius of a nonnegative reducible linear operator having a completely continuous iterate and defined on a Banach lattice E with order continuous norm. The combinatorial characterization of the Riesz index of the spectral radius and of the dimension of the algebraic eigenspace is made possible by a decomposition of the underlying operator in a form generalizing the Frobenius normal form of a nonnegative reducible matrix.
Ruey-Jen Jang, H. Dean Victory Jr.
Irreducible Positive Operators and Hyperinvariant Ideals
Abstract
This is an announcement (without proofs) of several results obtained recently by the authors. We present sufficient conditions for a positive operator to have strictly positive spectral radius. The conditions are given in terms of commutativity properties. Some of these results can be proven by formulating a lattice hyperinvariant subspace theorem è Lomonosov. The results obtained are either new or far reaching extensions of well known theorems.
Yuri A. Abramovich, Charalambos D. Aliprantis, Owen Burkinshaw
Metadaten
Titel
Positive Operators, Riesz Spaces, and Economics
verfasst von
Charalambos D. Aliprantis
Kim C. Border
Wilhelmus A. J. Luxemburg
Copyright-Jahr
1991
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-58199-1
Print ISBN
978-3-642-63502-1
DOI
https://doi.org/10.1007/978-3-642-58199-1