Andranik Tangian Mathematical Theory of Democracy 4y Springer
Parti History 1 Athenian Democracy 3 1.1 Introduction 4 1.2 Path to Democracy 6 1.3 Establishing Democracy 15 1.4 Culmination of Democracy 19 1.5 Philosophy of Democracy 21 1.6 Aristotle's Mathematical Model of Two Types of Governance 28 1.7 Summary 31 References 33 2 Echoes of Democracy in Ancient Rome 37 2.1 Introduction 38 2.2 Typology of States and the Theory of Mixed Government... 41 2.3 Rotation of Government Types 44 2.4 The Roman Republic 46 2.5 Pliny's Logical Analysis of a Senate Hearing 51 2.6 Summary 53 References 54 3 Revival of Democracy in Italian Medieval City-Republics 57 3.1 Introduction 58
xvi 3.2 The Venetian Republic 60 3.3 The Florentine Republic 64 3.4 Theory of Mixed Government Revitalized 74 3.5 First Mathematical Approaches to Elections 82 3.6 Summary 89 References 90 4 Enlightenment and the End of Traditional Democracy 95 4.1 Introduction 96 4.2 Montesquieu and the Separation of Powers 102 4.3 Rousseau and the General Will 105 4.4 The BordaMethod of Election 118 4.5 Laplace's Justification of the Borda Method 123 4.6 Problems Posed by the Borda Method 128 4.7 Condorcet's Social Choice Theory 133 4.8 The Condorcet Method Versus the Borda Method 143 4.9 Summary 148 4.10 Proofs 149 Proof of Theorem 4.2 149 Proof of Jury Theorem 4.6 151 Proof of Lemma 4.12 151 References 154 5 Modernity and Schism in Understanding Democracy 159 5.1 Introduction 160 5.2 American and French Revolutionary Constitutions 162 5.3 Principles of Representation 167 5.4 Tocqueville's Understanding of Democracy 171 5.5 Evolution of Representative Government 177 5.6 Universal Suffrage and Democracy, 183 5.7 General Commitment to Democracy 187 5.8 Democratic Perspectives 195 5.9 Voting Theory in the Age of Representative Government... 201 5.10 From Egalitarianism to Hierarchies 206
xvii 5.11 Summary 216 5.12 Proofs 217 Proof of Theorem 5.6 217 Proof of Theorem 5.7 218 References 220 Part II Theory 6 Direct Democracy.77 233 6.1 Introduction 234 6.2 Example: Evaluating Athenian Politicians in 462 BC 236 6.3 Indices of Representativeness : 238 6.4 Geometric Interpretation of the Indices 244 6.5 Decisive Bodies Selected from the Society 251 6.6 Ambiguous Effects of the Enlargement of Decisive Bodies... 255 6.7 Magistrate Versus Parliament with Regard to Their Functions 257 6.8 Inefficiency of Democracy in an Unstable Society 262 6.9 Summary 263 6.10 Proofs 264 Proof of Theorems 6.1 and 6.4 264 Proof of Theorem 6.6 270 Proof of Theorem 6.8 279 Proof of Theorem 6.11 280 References 282 7 Dictatorship and Democracy 285 7.1 Introduction 286 7.2 Measuring Representativeness for one Preference Profile... 289 7.3 Indices of Representativeness for the Simplest Model 292 7.4 Indices of Representativeness for the Combinatorial Model.. 295 7.5 Leaders Versus Dictators 297 7.6 Finding Leaders in a Large Society 300 7.7 Discussion of Definitions 304 7.8 Summary 306
xviii 7.9 Proofs 307 Proof of Theorem 7.1 307 Proof of Theorem 7.3 311 Proof of Theorem 7.6 312 References 317 8 Representative Democracy 319 8.1 Introduction,320 8.2 Data Structure for Analysis 324 8.3 Evaluation of Parties and Bundestag 331 8.4 Evaluation of Party Coalitions 336 8.5 Universality Versus Popularity 342 8.6 Summary 343 8.7 Proof of Theorem 8.6 344 Internet Links to Public Opinion Polls 347 References 350 9 Statistically Testing the Representative Capacity 353 9.1 Introduction 354 9.2 Geometric Method 358 9.3 Algebraic Method 379 9.4 Probabilistic Method 396 9.5 Interpolation Methods 408 9.6 Example: Significance of Representativeness of Parties and Coalitions 420 9.7 Summary 422 References 422 10 Concluding Discussion: Bridging Representative and Direct Democracies 425 10.1 Democracy and Voting 426 10.2 Shortcomings of Election 428 10.3 Reconsideration of Election by Name 430 10.4 Architecture of Election Procedure 431
xix 10.5 Example: Enhancing the Representativeness of the Bundestag 435 10.6 Election Calculus Versus Axiomatic Studies 440 10.7 Toward Truly Democratic Elections 442 References 444 Part III Applications 11 Simple Applications 449 11.1 Planning a Public Opinion Poll with no Cyclic Majorities... 450 11.2 Planning a Representative Public Opinion Poll 455 11.3 Invitation to a Round Table 458 11.4 Sufficiency of Limited Information for Decision Making... 460 References 463 12 Application to Collective Multicriteria Decisions 465 12.1 Introduction 466 12.2 Specifying Travel Alternatives for an Outing 467 12.3 Selecting One Travel Alternative 470 12.4 Selecting Several Travel Alternatives 472 12.5 Measuring the Individual Satisfaction 475 12.6 Summary 476 References 477 13 Application to Stock Exchange Predictions 479 13.1 Introduction 480 13.2 Dow Jones Stocks as Representatives of DAX Stocks 486 13.3 Testing Stand-Alone Predictors 488 13.4 Testing the Best Predictor out of Many 490 13.5 Testing a Predictor Selected by Lot 492 13.6 Groups of Dow Jones Stocks as Collective Predictors 493 13.7 Totality of Dow Jones Stocks as an Aggregate Predictor 496 13.8 Summary 497 13.9 Proof of Theorem 13.1 497 References 502
xx 14 Application to Traffic Control 505 14.1 Introduction 506 14.2 Traffic Forecasts Using the Model of Representatives 508 14.3 Testing a Stand-Alone Forecaster 512 14.4 Testing the Best Forecaster out of Many 514 14.5 Testing a Forecaster Selected by Lot 515 14.6 Groups of Intersections as Collective Forecasters 516 14.7 Totality of Intersections as an Aggregate Forecaster 518 14.8 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).. 549 Significance of Representativeness of a Two-Member Coalition (Jfc = 2) 558 Significance of Representativeness of a Three-Member Coalition (Jfc = 3) 562 Index 571