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E-raamat: Mathematical Look at Politics

(George Washington University, Washington, D.C., USA), (George Washington University, Washington, D.C., USA)
  • Formaat: 477 pages
  • Ilmumisaeg: 09-Dec-2010
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781439891179
  • Formaat - PDF+DRM
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  • Formaat: 477 pages
  • Ilmumisaeg: 09-Dec-2010
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781439891179

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What Ralph Nader's spoiler role in the 2000 presidential election tells us about the American political system. Why Montana went to court to switch the 1990 apportionment to Deans method. How the US tried to use game theory to win the Cold War, and why it didnt work. When students realize that mathematical thinking can address these sorts of pressing concerns of the political world it naturally sparks their interest in the underlying mathematics.

A Mathematical Look at Politics is designed as an alternative to the usual mathematics texts for students in quantitative reasoning courses. It applies the power of mathematical thinking to problems in politics and public policy. Concepts are precisely defined. Hypotheses are laid out. Propositions, lemmas, theorems, and corollaries are stated and proved. Counterexamples are offered to refute conjectures. Students are expected not only to make computations but also to state results, prove them, and draw conclusions about specific examples.

Tying the liberal arts classroom to real-world mathematical applications, this text is more deeply engaging than a traditional general education book that surveys the mathematical landscape. It aims to instill a fondness for mathematics in a population not always convinced that mathematics is relevant to them.

Arvustused

The book finds a nice compromise between formality and accessibility. The authors take care to build from examples, isolate what is important, and generalize into theorems. It is expected that the reader has only limited mathematical experience, so much effort is put toward making very clear what is and is not being said. The exercises that close each chapter are interesting and often quite challenging Topics are introduced and motivated thoughtfully. Definitions are clear, and the authors take the time to explain why they need to be with well-chosen examples. When the proofs come (and they do come), they are set up properly. The book has plenty of uses other than as a textbook. Instructors teaching a broader liberal arts mathematics course could use it to add depth to these topics or craft supplemental readings and projects. Students of mathematics or politics will find independent study opportunities here, and mathematicians from other areas will find this an enjoyable introduction. This is a very thoughtfully written text that should be made available to anyone with an interest in learning or teaching this topic. MAA Reviews, July 2011





Tying the liberal arts classroom to real-world mathematical applications, this text is more deeply engaging than a traditional general education book that surveys the mathematical landscape. It aims to instill a fondness for mathematics in a population not always convinced that mathematics is relevant. BULLETIN BIBLIOGRAPHIQUE, 2011

Preface for the Reader xi
Preface for the Instructor xv
I Voting
1(124)
Introduction to Part I
3(2)
1 Two Candidates
5(20)
1.0 Scenario
5(1)
1.1 Two-Candidate Methods
6(3)
1.2 Supermajority and Status Quo
9(1)
1.3 Weighted Voting and Other Methods
10(3)
1.4 Criteria
13(5)
1.5 May's Theorem
18(3)
1.6 Exercises and Problems
21(4)
2 Social Choice Functions
25(20)
2.0 Scenario
25(1)
2.1 Ballots
25(3)
2.2 Social Choice Functions
28(2)
2.3 Alternatives to Plurality
30(7)
2.4 Some Methods on the Edge
37(2)
2.5 Exercises and Problems
39(6)
3 Criteria for Social Choice
45(16)
3.0 Scenario
45(1)
3.1 Weakness and Strength
46(2)
3.2 Some Familiar Criteria
48(2)
3.3 Some New Criteria
50(7)
3.4 Exercises and Problems
57(4)
4 Which Methods Are Good?
61(18)
4.0 Scenario
61(1)
4.1 Methods and Criteria
62(1)
4.2 Proofs and Counterexamples
63(11)
4.3 Summarizing the Results
74(1)
4.4 Exercises and Problems
75(4)
5 Arrow's Theorem
79(14)
5.0 Scenario
79(1)
5.1 The Condorcet Paradox
79(3)
5.2 Statement of the Result
82(2)
5.3 Decisiveness
84(2)
5.4 Proving the Theorem
86(4)
5.5 Exercises and Problems
90(3)
6 Variations on the Theme
93(32)
6.0 Scenario
93(1)
6.1 Inputs and Outputs
94(1)
6.2 Vote-for-One Ballots
95(3)
6.3 Approval Ballots
98(6)
6.4 Mixed Approval/Preference Ballots
104(2)
6.5 Cumulative Voting
106(2)
6.6 Condorcet Methods
108(5)
6.7 Social Ranking Functions
113(3)
6.8 Preference Ballots with Ties
116(1)
6.9 Exercises and Problems
117(4)
Notes on Part I
121(4)
II Apportionment
125(108)
Introduction to Part II
127(2)
7 Hamilton's Method
129(16)
7.0 Scenario
129(1)
7.1 The Apportionment Problem
130(2)
7.2 Some Basic Notions
132(3)
7.3 A Sensible Approach
135(2)
7.4 The Paradoxes
137(5)
7.5 Exercises and Problems
142(3)
8 Divisor Methods
145(22)
8.0 Scenario
145(1)
8.1 Jefferson's Method
146(3)
8.2 Critical Divisors
149(3)
8.3 Assessing Jefferson's Method
152(3)
8.4 Other Divisor Methods
155(2)
8.5 Rounding Functions
157(6)
8.6 Exercises and Problems
163(4)
9 Criteria and Impossibility
167(18)
9.0 Scenario
167(1)
9.1 Basic Criteria
167(3)
9.2 Quota Rules and the Alabama Paradox
170(2)
9.3 Population Monotonicity
172(2)
9.4 Relative Population Monotonicity
174(2)
9.5 The New States Paradox
176(1)
9.6 Impossibility
177(3)
9.7 Exercises and Problems
180(5)
10 The Method of Balinski and Young
185(14)
10.0 Scenario
185(1)
10.1 Tracking Critical Divisors
186(5)
10.2 Satisfying the Quota Rule
191(2)
10.3 Computing the Balinski-Young Apportionment
193(4)
10.4 Exercises and Problems
197(2)
11 Deciding among Divisor Methods
199(14)
11.0 Scenario
199(1)
11.1 Why Webster Is Best
199(4)
11.2 Why Dean Is Best
203(3)
11.3 Why Hill Is Best
206(4)
11.4 Exercises and Problems
210(3)
12 History of Apportionment in the United States
213(20)
12.0 Scenario
213(1)
12.1 The Fight for Representation
214(10)
12.2 Summary
224(1)
12.3 Exercises and Problems
225(2)
Notes on Part II
227(6)
III Conflict
233(122)
Introduction to Part III
235(2)
13 Strategies and Outcomes
237(18)
13.0 Scenario
237(1)
13.1 Zero-Sum Games
238(2)
13.2 The Naive and Prudent Strategies
240(4)
13.3 Best Response and Saddle Points
244(4)
13.4 Dominance
248(4)
13.5 Exercises and Problems
252(3)
14 Chance and Expectation
255(18)
14.0 Scenario
255(1)
14.1 Probability Theory
255(2)
14.2 All Outcomes Are Not Created Equal
257(3)
14.3 Random Variables and Expected Value
260(3)
14.4 Mixed Strategies and Their Payoffs
263(3)
14.5 Independent Processes
266(1)
14.6 Expected Payoffs for Mixed Strategies
267(2)
14.7 Exercises and Problems
269(4)
15 Solving Zero-Sum Games
273(18)
15.0 Scenario
273(1)
15.1 The Best Response
273(4)
15.2 Prudent Mixed Strategies
277(2)
15.3 An Application to Counterterrorism
279(3)
15.4 The 2-by-2 Case
282(4)
15.5 Exercises and Problems
286(5)
16 Conflict and Cooperation
291(22)
16.0 Scenario
291(1)
16.1 Bimatrix Games
292(1)
16.2 Guarantees, Saddle Points, and All That Jazz
293(2)
16.3 Common Interests
295(4)
16.4 Some Famous Games
299(9)
16.5 Exercises and Problems
308(5)
17 Nash Equilibria
313(18)
17.0 Scenario
313(1)
17.1 Mixed Strategies
314(2)
17.2 The 2-by-2 Case
316(7)
17.3 The Proof of Nash's Theorem
323(5)
17.4 Exercises and Problems
328(3)
18 The Prisoner's Dilemma
331(24)
18.0 Scenario
331(1)
18.1 Criteria and Impossibility
332(7)
18.2 Omnipresence of the Prisoner's Dilemma
339(4)
18.3 Repeated Play
343(2)
18.4 Irresolvability
345(2)
18.5 Exercises and Problems
347(4)
Notes on Part III
351(4)
IV The Electoral College
355(42)
Introduction to Part IV
357(2)
19 Weighted Voting
359(20)
19.0 Scenario
359(1)
19.1 Weighted Voting Methods
360(3)
19.2 Non-Weighted Voting Methods
363(4)
19.3 Voting Power
367(4)
19.4 Power of the States
371(4)
19.5 Exercises and Problems
375(4)
20 Whose Advantage?
379(18)
20.0 Scenario
379(1)
20.1 Violations of Criteria
379(2)
20.2 People Power
381(5)
20.3 Interpretation
386(3)
20.4 Exercises and Problems
389(4)
Notes on Part IV
393(4)
Solutions to Odd-Numbered Exercises and Problems 397(52)
Bibliography 449(4)
Index 453
E. Arthur Robinson Jr., Daniel H. Ullman