Representations of Preferences Orderings
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422
Douglas S. Bridges
Ghanshyam B. Mehta
Representations of Preferences Orderings
Springer
Authors Douglas S. Bridges University of Waikato Department of Mathematics and Statisties Hamilton, New Zealand Ghanshyam B. Mehta University of Queensland Department of Eeonomies Queensland 4072, Brisbane Australia
ISBN 978-3-540-58839-9 DOI 10.1007/978-3-642-51495-1
ISBN 978-3-642-51495-1 (eBook)
Library of Congress Cataloging-in-Publication Data. Bridges, D. S. (Douglas S.), 1945- . Representations of preferenee orderings / Douglas Bridges, Ghanshyam Mehta. p. em. - (Leeture notes in eeonomics and mathematieal systems; 422) Ineludes bibliographieal referenees and index. 1. Consumers' preferenees-Mathematieal models. I. Mehta, Ghanshyam. II. Title. III. Series. HF5415.3.B69 1995 658.S'343-dc20 9445869
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Originally published by Springer-Verlag Berlin Heidelberg New York in 1995 TypesettiDg: Camera ready by author SPIN: 10486729 42/3142-543210 - Printed on acid-free pa per
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Preface A basic assumption made by pioneers of classical microeconomics such as Edgeworth and Pareto was that the ranking of a consumer's preferences could always be measured numerically, by associating to each possible consumption bundle a real number that measured its utility: the greater the utility, the more preferred was the bundle, and conversely. It took several decades before the naivety of this assumption was seriously challenged by economists, such as Wold, attempting to find conditions under which it could be justified mathematically. Wold's work was the first in a long chain of results of that type, leading to the definitive theorems of Debreu and others in the 1960s, and subsequently to the refinements and generalisations that have appeared in the last twenty-five years. Out of this historical background there has appeared a general mathematical problem which, as well as having applications in economics, psychology, and measurement theory, arises naturally in the study of sets bearing order relations:
Given some kind of ordenng t on a set 5, fina a real-valued mapping u on 5 such that for any elements x, y of 5, x t yif and only if u(x) 2: u(y). If also 5 has a topology (respective/y, differential structure), find conditions that ensure the continuity (respectively, differentiability) of the mapping u. A mapping ·u of this kind is called a representation of the ordering C:::. In thi
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