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

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.