Aggregation and Representation of Preferences: Introduction to Mathematical Theory of Democracy

ISBN-13: 9783642765186 / Angielski / Miękka / 2011 / 331 str.

Andranick S. Tanguiane

Opis

Aggregation is the conjunction of information, aimed at its compact represen tation. Any time when the totality of data is described in terms of general ized indicators, conventional counts, typical representatives and characteristic dependences, one directly or indirectly deals with aggregation. It includes revealing the most significant characteristics and distinctive features, quanti tative and qualitative analysis. As a result, the information becomes adaptable for further processing and convenient for human perception. Aggregation is widely used in economics, statistics, management, planning, system analysis, and many other fields. That is why aggregation is so important in data pro cessing. Aggregation of preferences is a particular case of the general problem of ag gregation. It arises in multicriteria decision-making and collective choice, when a set of alternatives has to be ordered with respect to contradicting criteria, or various individual opinions. However, in spite of apparent similarity the problems of multicriteria decision-making and collective choice are somewhat different. Indeed, an improvement in some specifications at the cost of worsen ing others is not the same as the satisfaction of interests of some individuals to the prejudice of the rest. In the former case the reciprocal compensations are considered within a certain entirety; in the latter we infringe upon the rights of independent individuals. Moreover, in multicriteria decision-making one usu ally takes into account objective factors, whereas in collective choice one has to compare subjective opinions which cannot be measured properly."

Spis treści

1 Different Ways of Aggregation of Preferences.- 1.1 Aggregation of Preferences as a Voting Problem.- 1.2 Aggregation of Preferences by a Majority Rule.- 1.3 Aggregation of Preferences with Account of Degree of Preference.- 1.4 Ordinal and Cardinal Approaches to Aggregation of Preferences.- 1.5 Alternate View at Aggregation of Preferences.- 2 Preferences and Goal Functions.- 2.1 Introductory Remarks.- 2.2 Binary Relations and Orderings.- 2.3 Continuous Orderings.- 2.4 Numerical Representation of Orderings.- 3 Aggregation of Fixed Independent Preferences.- 3.1 Introductory Remarks.- 3.2 Construction of Approximation of Additive Goal Function.- 3.3 Accuracy of Approximation of Additive Goal Function.- 3.4 Applicability of the Construction of Additive Goal Function.- 3.5 Notes on Interpretation.- 4 Accounting Degree of Preference in Aggregation.- 4.1 Introductory Remarks.- 4.2 Method of Marks.- 4.3 Certainty of Predominance of One Preference over Another.- 4.4 Notes on Interpretation.- 5 General Model of Aggregation of Preferences.- 5.1 Introductory Remarks.- 5.2 Aggregating Operators and Deciding Hierarchies.- 5.3 Bijection Theorem.- 5.4 Theorems of Arrow and Fishburn.- 5.5 Representation of Model of Aggregation of Preferences.- 5.6 Topological Formulation of Model of Aggregation of Preferences.- 5.7 Notes on Interpretation.- 6 Aggregation of Infinite Set of Preferences.- 6.1 Introductory Remarks.- 6.2 Imbeddings of Models of Aggregation of Preferences.- 6.3 Limit Theorem.- 6.4 Sequential Aggregation of Preferences.- 6.5 Reduction of Model and Notes on Interpretation.- 7 Interpretation of Dictator as Representative.- 7.1 Introductory Remarks.- 7.2 Indicators of Representativeness.- 7.3 Representativeness of Optimal and Random Dictators.- 7.4 Overcoming Arrow’s Paradox.- 7.5 Independence of Individuals.- 7.6 Average Representativeness for Independent Individuals.- 7.7 Approximation of Majority Representativeness.- 7.8 Geometric Interpretation of Dictators.- 7.9 Numerical Estimation of Representativeness.- 7.10 Notes on Interpretation.- 8 Representation of Collective by Few Individuals.- 8.1 Introductory Remarks.- 8.2 Cabinets and Indicators of Their Representativeness.- 8.3 Representativeness of Optimal and Random Cabinets.- 8.4 Recurrent Construction of Cabinets.- 8.5 Representativeness of Cabinets for Independent Individuals.- 8.6 Consistency of Two Definitions of Optimal Cabinets.- 8.7 Numerical Estimation of Representativeness of Cabinets.- 8.8 Notes on Interpretation.- 9 Representation of Collective by a Council.- 9.1 Introductory Remarks.- 9.2 Councils and Indicators of Their Representativeness.- 9.3 Representativeness of Optimal and Random Councils.- 9.4 Representativeness of Councils for Independent Individuals.- 9.5 Consistency of Two Definitions of Optimal Councils.- 9.6 Numerical Estimation of Representativeness of Councils.- 9.7 Notes on Interpretation.- Conclusions.- Appendix 2. Tables A.2.1–A.2.6. Lower Bound of Representativeness of Optimal Cabinets.- Appendix 3. Tables A.3.1–A.3.6. Lower Bound of Representativeness of Optimal Councils.- References.- Name Index.- Designations.