Mathematics and Politics: Strategy, Voting, Power, and Proof

 Alan D. Taylor e Allison M. Pacelli 

Springer | 2010 -2ª edição | 382 páginas | pdf | 1,2 Mb

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As a text for an undergraduate mathematics course for nonmajors, Mathematics and Politics requires no prerequisites in either area while the underlying philosophy involves minimizing algebraic computations and focusing instead on some conceptual aspects of mathematics in the context of important real-world questions in political science.

Five major topics are covered including a model of escalation, game theoretic models of international conflict, yes-no voting systems, political power, and social choice. Each topic is discussed in an introductory chapter and revisited in more depth in a later chapter. This new edition has added co-author, Allison Pacelli, and two new chapters on "Fairness" and "More Fairness." The examples and the exercises have been updated and enhanced throughout.

Contents
Preface vii
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1 SOCIAL CHOICE
1.1. Introduction 1
1.2. May’s Theorem for Two Alternatives 4
1.3. Six Examples of Social Choice Procedures 5
1.4. Five Desirable Properties of Social Choice Procedures 10
1.5. Positive Results—Proofs 13
1.6. Negative Results—Proofs 20
1.7. A Glimpse of Impossibility 28
1.8. Approval Voting 31
1.9. Conclusions 34
Exercises 35
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2 YES–NO VOTING
2.1. Introduction 49
2.2. Four Examples of Yes–No Voting Systems 50
2.3. Weighted Voting and the U.N. Security Council 53
2.4. Swap Robustness and the Nonweightedness of the Federal System 56
2.5. Trade Robustness and the Nonweightedness of the Canadian System 59
2.6. Statement of the Characterization Theorem 62
2.7. Conclusion 63
Exercises 64
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3 POLITICAL POWER
3.1. Introduction 71
3.2. The Shapley-Shubik Index of Power 73
3.3. Calculations for the European Economic Community 78
3.4. The Banzhaf Index of Power 83
3.5. Two Methods of Computing Banzhaf Power 85
3.6. The Power of the President 90
3.7. The Chair’s Paradox 98
3.8. Conclusions 104
Exercises 105
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4 CONFLICT
4.1. Introduction 112
4.2. Two-By-Two Games 113
4.3. Dominant Strategies and Nash Equilibria 116
4.4. Prisoner’s Dilemma and the Arms Race 117
4.5. Chicken and the Cuban Missile Crisis 122
4.6. The Yom Kippur War 126
4.7. The Theory of Moves 128
4.8. Conclusions 137
Exercises 137
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5 FAIRNESS
5.1. Introduction 152
5.2. The Problem of Apportionment 153
5.3. Divisor Methods of Apportionment 155
5.4. A Glimpse of Impossibility 157
5.5. Dispute Resolution and Fair Division 159
5.6. An Alternative to Divide-and-Choose 163
5.7. Adjusted Winner 165
5.8. Adjusted Winner and the Middle East 171
5.9. Conclusions 173
Exercises 174
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6 ESCALATION
6.1. Introduction 179
6.2. The Dollar Auction 180
6.3. Game-Tree Analyses 181
6.4. Limitations and Back-of-the-Envelope Calculations 188
6.5. Statement of O’Neill’s Theorem 192
6.6. Vickrey Auctions 195
6.7. Conclusions 199
Exercises 200
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7 MORE SOCIAL CHOICE
7.1. Introduction 205
7.2. Social Welfare Functions 206
7.3. A Generalization of May’s Theorem 209
7.4. Arrow’s Impossibility Theorem 211
7.5. The Gibbard-Satterthwaite Theorem 222
7.6. Single Peakedness—Theorems of Black and Sen 231
7.7. Conclusions 240
Exercises 240
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8 MORE YES–NO VOTING
8.1. Introduction 247
8.2. A Magic Square Voting System 248
8.3. Dimension Theory and the U.S. Federal System 251
8.4. Vector-Weighted Voting Systems 255
8.5. Conclusions 260
Exercises 261
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9 MORE POLITICAL POWER
9.1. Introduction 264
9.2. The Johnston Index of Power 265
9.3. The Deegan–Packel Index of Power 271
9.4. Ordinal Power: Incomparability 275
9.5. Ordinal Power: Comparability 279
9.6. A Theorem On Voting Blocs 285
9.7. Conclusions 288
Exercises 289
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10 MORE CONFLICT
10.1. Introduction 293
10.2. Models of Deterrence 293
10.3. A Probabilistic Model of Deterrence 298
10.4. Two-Person Zero-Sum Games 303
10.5. Conclusions 309
Exercises 309
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11 MORE FAIRNESS
11.1. Introduction 314
11.2. Efficiency in Adjusted Winner 315
11.3. Adjusted Winner and Manipulability 319
11.4. Fair Division Procedures for Three or More Parties 322
11.5. Envy-Free Procedures 325
11.6. Envy-Free Procedures for Four or More Parties 328
11.7. Another Impossibility Result 330
11.8. Conclusions 331
Exercises 332
Contents xv
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12 MORE ESCALATION
12.1. Introduction 337
12.2. Statement of the Strong Version of O’Neill’s Theorem 337
12.3. Proof (By Mathematical Induction) of the Strong Version of O’Neill’s Theorem 344
12.4. Vickrey Auctions as a Generalized Prisoner’s Dilemma 346
12.5. Conclusions 348
Exercises 349
Attributions 350