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Arrovian Aggregation Models : THEORY AND DECISION LIBRARY SERIES B, MATHEMATICAL AND STATISTICAL METHODS - F. Aleskerov

Arrovian Aggregation Models

THEORY AND DECISION LIBRARY SERIES B, MATHEMATICAL AND STATISTICAL METHODS

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Published: 31st March 1999
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Aggregation of individual opinions into a social decision is a problem widely observed in everyday life. For centuries people tried to invent the best' aggregation rule. In 1951 young American scientist and future Nobel Prize winner Kenneth Arrow formulated the problem in an axiomatic way, i.e., he specified a set of axioms which every reasonable aggregation rule has to satisfy, and obtained that these axioms are inconsistent. This result, often called Arrow's Paradox or General Impossibility Theorem, had become a cornerstone of social choice theory. The main condition used by Arrow was his famous Independence of Irrelevant Alternatives. This very condition pre-defines the local' treatment of the alternatives (or pairs of alternatives, or sets of alternatives, etc.) in aggregation procedures. Remaining within the framework of the axiomatic approach and based on the consideration of local rules, Arrovian Aggregation Models investigates three formulations of the aggregation problem according to the form in which the individual opinions about the alternatives are defined, as well as to the form of desired social decision. In other words, we study three aggregation models. What is common between them is that in all models some analogue of the Independence of Irrelevant Alternatives condition is used, which is why we call these models Arrovian aggregation models. Chapter 1 presents a general description of the problem of axiomatic synthesis of local rules, and introduces problem formulations for various versions of formalization of individual opinions and collective decision. Chapter 2 formalizes precisely the notion of rationality' of individual opinions and social decision. Chapter 3 deals with the aggregation model for the case of individual opinions and social decisions formalized as binary relations. Chapter 4 deals with Functional Aggregation Rules which transform into a social choice function individual opinions defined as choice functions. Chapter 5 considers another model &endash; Social Choice Correspondences when the individual opinions are formalized as binary relations, and the collective decision is looked for as a choice function. Several new classes of rules are introduced and analyzed.

`This monograph is excellent and should belong to every social choice theorist's library. it is also highly recommended to mathematicians working in discrete mathematics since it offers many applications of this mathematical domain.' Mathematical Reviews, 2001c

Forewordp. ix
Acknowledgementsp. xi
Aggregation: A General Descriptionp. 1
Introductionp. 1
Analysis of examplesp. 2
Arrow's General Impossibility Theoremp. 3
Individual opinion: a formalizationp. 8
Aggregation: the synthesis problemp. 11
Concluding remarksp. 16
Rationality of Individual Opinions and Social Decisionsp. 17
Introductionp. 17
Binary relationsp. 17
Criterial model of choicep. 22
Expansion-Contraction Axiomsp. 26
Relations between the classes of choice functionsp. 35
Concluding remarksp. 41
Social Decision Functionsp. 45
Introductionp. 45
Strong localityp. 46
Normative conditionsp. 49
Rules from Central Classp. 53
Rationality constraintsp. 56
Comparing classes in [Lambda][superscript C]p. 62
Arrow's General Impossibility Theoremp. 65
Rationality constraints: further resultsp. 67
Aggregation of equivalencesp. 72
Non-monotonic strongly local SDFsp. 74
Localityp. 78
Normative conditionsp. 80
Rules from Central Classp. 85
Rationality constraintsp. 97
Comparing classes in [Lambda][superscript C]p. 110
Concluding remarksp. 119
Functional Aggregation Rulesp. 123
Introductionp. 123
Localityp. 124
Normative conditionsp. 127
Rules from Central Classp. 133
Rationality constraints: non-emptinessp. 136
Rationality constraints: domains H, C, and Op. 142
Comparing classes in [Lambda][superscript C]p. 150
Rules from Basic Classp. 152
Non-monotonic rulesp. 158
Non-monotonic rules: dual domainsp. 165
Concluding remarksp. 174
Social Choice Correspondencesp. 177
Introductionp. 177
Localityp. 177
Normative conditionsp. 183
Boolean representation of Social Choice Correspondencesp. 190
Rules from Central Class, Ip. 192
Rules from Central Class, IIp. 200
Rules from Symmetrically Central Classp. 204
Rationality constraints: single-valuednessp. 212
Coalitional q-federation rules under rationality constraintsp. 216
Rationality constraints: domains H, C, Op. 219
Comparing classes in [Lambda][superscript C]p. 222
Concluding remarksp. 223
Bibliographyp. 227
Indexp. 239
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780792384519
ISBN-10: 0792384512
Series: THEORY AND DECISION LIBRARY SERIES B, MATHEMATICAL AND STATISTICAL METHODS
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 244
Published: 31st March 1999
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 25.4 x 17.15  x 1.91
Weight (kg): 0.54