Mathematics of Decision Theory PDF
Preview Mathematics of Decision Theory
mathematics of decision theory Unauthenticated Download Date | 11/18/19 9:24 PM methods and models in the social sciences 3 MOUTON • THE HAGUE • PARIS Unauthenticated Download Date | 11/18/19 9:24 PM mathematics of decision theory by PETER C. FISHBURN MOUTON . THE HAGUE • PARIS Unauthenticated Download Date | 11/18/19 9:24 PM Library of Congress Catalog Card Number: 75-171099 © 1972, UNESCO Jacket-design: Jurriaan Schrofer Printed in Hungary Unauthenticated Download Date | 11/18/19 9:24 PM To Jan and our daughters Susan, Kathy, and Sally Unauthenticated Download Date | 11/18/19 9:24 PM Unauthenticated Download Date | 11/18/19 9:24 PM contents 1 THE MATHEMATICS OF DECISION THEORY 11 What are the mathematics of decision theory? On the structure of axiomatic theories. Methods of proof. 2 SETS AND FUNCTIONS 18 Unions, intersections, partitions. Complements and differences. Products of sets. Real numbers. Functions. Finite and infinite sets. 3 BINARY RELATIONS 25 Basic definitions. Some relational properties. Equivalence relations. Order relations. Asymmetric orders. Reflexive orders. 4 CHOICE FUNCTIONS 35 Revealed preference. Subset choice functions. Systems of choice func- tions. 5 FINITE LINEAR SYSTEMS 43 Convex sets and solution vectors. Expected utilities. Semiorders. Additive utilities. Special majority. 6 ZORN'S LEMMA 53 Szpilrajn's theorem. Additive utility. An unusual probability measure. 7 REAL-VALUED ORDER-PRESERVING FUNCTIONS 58 An example. Order denseness. Applications of Cantor's theorem. A composite application. 8 TOPOLOGY 63 Topologies. Connected and separable spaces. Orders on topological spaces. Real-valued functions. Continuity. Product topologies. 7 Unauthenticated Download Date | 11/18/19 5:11 AM 8 Mathematics of decision theory 9 PROBABILITY 70 Algebras of subsets. Probability measures. Axioms for probability. Mathematical expectation. Probability from preference. 10 MIXTURE SETS 79 Mixture sets. Expected utility. Strict partial orders. Lexicographic NM proposition. Products of mixture sets. 11 ORDERED GROUPOIDS 87 Ordered groups. Bisymmetric linearly ordered groupoids. REFERENCES 95 INDEX 101 Unauthenticated Download Date | 11/18/19 5:11 AM preface This little book describes the mathematics used in axiomatic investigations in decision theory. Its purpose is to introduce the student to these areas of mathematics and to show how they can apply to the analysis of specific types of decision situations, including multiple-factor situations and situa- tions that involve uncertainty. Social choice or group decision is considered along with individual decision. The book is organized by specific mathematical topics. Following an introduction that includes some elementary theory of sets and functions there are two chapters on binary relations and choice functions, upon which almost all decision theories are built. These are followed by a chapter that shows how the theory of the solution of finite systems of linear inequalities applies to axiomatic investigations in virtually every area of decision theory. Our attention then turns to mathematics that are mainly concerned with the treatment of infinite sets. The first of these chapters shows how Zorn's lemma applies to several specific questions in decision theory: the second presents Cantor's Theorem on the existence of order-preserving real-valued functions. The final four chapters are based on structures of sets, or sets of sets, that are closed under one or more operations. Included are topolo- gies, algebras of sets (used in probability theory), mixture sets, and ordered groups and groupoids. In each of these areas we shall note some of the special characteristics of decision theories that arise from the special struc- tures. 9 Brought to you by | provisional account Authenticated Download Date | 1/5/20 3:39 PM 10 Mathematics of decision theory I am indebted to many friends for their help over the years. My greatest debt is to Professor Leonard J. Savage. I extend my sincerest thanks to Mrs. Virginia M. Johnson for her superb typing of the manuscript. Potomac, Maryland Peter C. Fishburn January 1970 Brought to you by | provisional account Authenticated Download Date | 1/5/20 3:39 PM
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