Mathematical Theory of Democracy

Transcription

1 Andranik Tangian Mathematical Theory of Democracy 4y Springer

2 Parti History 1 Athenian Democracy Introduction Path to Democracy Establishing Democracy Culmination of Democracy Philosophy of Democracy Aristotle's Mathematical Model of Two Types of Governance Summary 31 References 33 2 Echoes of Democracy in Ancient Rome Introduction Typology of States and the Theory of Mixed Government Rotation of Government Types The Roman Republic Pliny's Logical Analysis of a Senate Hearing Summary 53 References 54 3 Revival of Democracy in Italian Medieval City-Republics Introduction 58

3 xvi 3.2 The Venetian Republic The Florentine Republic Theory of Mixed Government Revitalized First Mathematical Approaches to Elections Summary 89 References 90 4 Enlightenment and the End of Traditional Democracy Introduction Montesquieu and the Separation of Powers Rousseau and the General Will The BordaMethod of Election Laplace's Justification of the Borda Method Problems Posed by the Borda Method Condorcet's Social Choice Theory The Condorcet Method Versus the Borda Method Summary Proofs 149 Proof of Theorem Proof of Jury Theorem Proof of Lemma References Modernity and Schism in Understanding Democracy Introduction American and French Revolutionary Constitutions Principles of Representation Tocqueville's Understanding of Democracy Evolution of Representative Government Universal Suffrage and Democracy, General Commitment to Democracy Democratic Perspectives Voting Theory in the Age of Representative Government From Egalitarianism to Hierarchies 206

4 xvii 5.11 Summary Proofs 217 Proof of Theorem Proof of Theorem References 220 Part II Theory 6 Direct Democracy Introduction Example: Evaluating Athenian Politicians in 462 BC Indices of Representativeness : Geometric Interpretation of the Indices Decisive Bodies Selected from the Society Ambiguous Effects of the Enlargement of Decisive Bodies Magistrate Versus Parliament with Regard to Their Functions Inefficiency of Democracy in an Unstable Society Summary Proofs 264 Proof of Theorems 6.1 and Proof of Theorem Proof of Theorem Proof of Theorem References Dictatorship and Democracy Introduction Measuring Representativeness for one Preference Profile Indices of Representativeness for the Simplest Model Indices of Representativeness for the Combinatorial Model Leaders Versus Dictators Finding Leaders in a Large Society Discussion of Definitions Summary 306

5 xviii 7.9 Proofs 307 Proof of Theorem Proof of Theorem Proof of Theorem References Representative Democracy Introduction, Data Structure for Analysis Evaluation of Parties and Bundestag Evaluation of Party Coalitions Universality Versus Popularity Summary Proof of Theorem Internet Links to Public Opinion Polls 347 References Statistically Testing the Representative Capacity Introduction Geometric Method Algebraic Method Probabilistic Method Interpolation Methods Example: Significance of Representativeness of Parties and Coalitions Summary 422 References Concluding Discussion: Bridging Representative and Direct Democracies Democracy and Voting Shortcomings of Election Reconsideration of Election by Name Architecture of Election Procedure 431

6 xix 10.5 Example: Enhancing the Representativeness of the Bundestag Election Calculus Versus Axiomatic Studies Toward Truly Democratic Elections 442 References 444 Part III Applications 11 Simple Applications Planning a Public Opinion Poll with no Cyclic Majorities Planning a Representative Public Opinion Poll Invitation to a Round Table Sufficiency of Limited Information for Decision Making References Application to Collective Multicriteria Decisions Introduction Specifying Travel Alternatives for an Outing Selecting One Travel Alternative Selecting Several Travel Alternatives Measuring the Individual Satisfaction Summary 476 References Application to Stock Exchange Predictions Introduction Dow Jones Stocks as Representatives of DAX Stocks Testing Stand-Alone Predictors Testing the Best Predictor out of Many Testing a Predictor Selected by Lot Groups of Dow Jones Stocks as Collective Predictors Totality of Dow Jones Stocks as an Aggregate Predictor Summary Proof of Theorem References 502

7 xx 14 Application to Traffic Control Introduction Traffic Forecasts Using the Model of Representatives Testing a Stand-Alone Forecaster Testing the Best Forecaster out of Many Testing a Forecaster Selected by Lot Groups of Intersections as Collective Forecasters Totality of Intersections as an Aggregate Forecaster Summary 518 References 519 Part IV Appendix A Computational Formulas 523 A. 1 Chebyshev's Inequality and the Central Limit Theorem 523 A.2 Beta Distribution and Incomplete Beta Function 524 A.3 Multinomial Sums and Falling Factorials 525 A.4 Proofs 527 References 535 B Probabilities of Unequal Choices by Vote and by Candidate Scores 537 How to Use B-Tables 537 Probability Tables 539 C Statistical Significance of Representative Capacity 547 How to Use C-Tables 547 Significance of Representativeness of a Single Candidate (k = 1) Significance of Representativeness of a Two-Member Coalition (Jfc = 2) 558 Significance of Representativeness of a Three-Member Coalition (Jfc = 3) 562 Index 571