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E-raamat: Mathematics of Politics

(George Washington University, Washington, D.C., USA), (George Washington University)
  • Formaat: 477 pages
  • Ilmumisaeg: 18-Nov-2016
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781498798884
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  • Formaat: 477 pages
  • Ilmumisaeg: 18-Nov-2016
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781498798884

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It is because mathematics is often misunderstood, it is commonly

believed it has nothing to say about politics. The high school

experience with mathematics, for so many the lasting impression

of the subject, suggests that mathematics is the study of numbers,

operations, formulas, and manipulations of symbols. Those

believing this is the extent of mathematics might conclude

mathematics has no relevance to politics. This book counters this impression.



The second edition of this popular book focuses on mathematical reasoning

about politics. In the search for ideal ways to make certain kinds

of decisions, a lot of wasted effort can be averted if mathematics can determine that

finding such an ideal is actually impossible in the first place.

In the first three parts of this book, we address the following three

political questions:

(1) Is there a good way to choose winners of elections?

(2) Is there a good way to apportion congressional seats?

(3) Is there a good way to make decisions in situations of conflict and

uncertainty?

In the fourth and final part of this book, we examine the Electoral

College system that is used in the United States to select a president.

There we bring together ideas that are introduced in each of the three

earlier parts of the book.
Preface for the Reader xi
Preface for the Instructor xv
I Voting
1(126)
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(8)
2.4 Some Methods on the Edge
38(1)
2.5 Exercises and Problems
39(6)
3 Criteria for Social Choice
45(18)
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(6)
4 Which Methods Are Good?
63(18)
4.0 Scenario
63(1)
4.1 Methods and Criteria
64(1)
4.2 Proofs and Counterexamples
65(11)
4.3 Summarizing the Results
76(1)
4.4 Exercises and Problems
77(4)
5 Arrow's Theorem
81(14)
5.0 Scenario
81(1)
5.1 The Condorcet Paradox
81(3)
5.2 Statement of the Result
84(2)
5.3 Decisiveness
86(2)
5.4 Proving the Theorem
88(4)
5.5 Exercises and Problems
92(3)
6 Variations on the Theme
95(32)
6.0 Scenario
95(1)
6.1 Inputs and Outputs
96(1)
6.2 Vote-for-One Ballots
97(3)
6.3 Approval Ballots
100(6)
6.4 Mixed Approval/Preference Ballots
106(2)
6.5 Cumulative Voting
108(2)
6.6 Condorcet Methods
110(5)
6.7 Social Ranking Functions
115(3)
6.8 Preference Ballots with Ties
118(1)
6.9 Exercises and Problems
119(8)
Notes on Part I
123(4)
II Apportionment
127(104)
Introduction to Part II
129(2)
7 Hamilton's Method
131(16)
7.0 Scenario
131(1)
7.1 The Apportionment Problem
132(2)
7.2 Some Basic Notions
134(3)
7.3 A Sensible Approach
137(2)
7.4 The Paradoxes
139(5)
7.5 Exercises and Problems
144(3)
8 Divisor Methods
147(22)
8.0 Scenario
147(1)
8.1 Jefferson's Method
148(3)
8.2 Critical Divisors
151(3)
8.3 Assessing Jefferson's Method
154(3)
8.4 Other Divisor Methods
157(2)
8.5 Rounding Functions
159(6)
8.6 Exercises and Problems
165(4)
9 Criteria and Impossibility
169(18)
9.0 Scenario
169(1)
9.1 Basic Criteria
169(3)
9.2 Quota Rules and the Alabama Paradox
172(2)
9.3 Population Monotonicity
174(2)
9.4 Relative Population Monotonicity
176(2)
9.5 The New States Paradox
178(1)
9.6 Impossibility
179(3)
9.7 Exercises and Problems
182(5)
10 The Method of Balinski and Young
187(10)
10.0 Scenario
187(1)
10.1 Tracking Critical Divisors
188(5)
10.2 Satisfying the Quota Rule
193(2)
10.3 Exercises and Problems
195(2)
11 Deciding among Divisor Methods
197(14)
11.0 Scenario
197(1)
11.1 Why Webster Is Best
197(4)
11.2 Why Dean Is Best
201(3)
11.3 Why Hill Is Best
204(4)
11.4 Exercises and Problems
208(3)
12 History of Apportionment in the United States
211(20)
12.0 Scenario
211(1)
12.1 The Fight for Representation
212(10)
12.2 Summary
222(2)
12.3 Exercises and Problems
224(7)
Notes on Part II
225(6)
III Conflict
231(122)
Introduction to Part III
233(2)
13 Strategies and Outcomes
235(18)
13.0 Scenario
235(1)
13.1 Zero-Sum Games
236(3)
13.2 The Naive and Prudent Strategies
239(3)
13.3 Best Response and Saddle Points
242(4)
13.4 Dominance
246(3)
13.5 Exercises and Problems
249(4)
14 Chance and Expectation
253(18)
14.0 Scenario
253(1)
14.1 Probability Theory
253(2)
14.2 All Outcomes Are Not Created Equal
255(3)
14.3 Random Variables and Expected Value
258(3)
14.4 Mixed Strategies and Their Payoffs
261(3)
14.5 Independent Processes
264(1)
14.6 Expected Payoffs for Mixed Strategies
265(2)
14.7 Exercises and Problems
267(4)
15 Solving Zero-Sum Games
271(18)
15.0 Scenario
271(1)
15.1 The Best Response
271(4)
15.2 Prudent Mixed Strategies
275(2)
15.3 An Application to Counterterrorism
277(3)
15.4 The 2-by-2 Case
280(4)
15.5 Exercises and Problems
284(5)
16 Conflict and Cooperation
289(22)
16.0 Scenario
289(1)
16.1 Bimatrix Games
290(1)
16.2 Guarantees, Saddle Points, and All That Jazz
291(2)
16.3 Common Interests
293(4)
16.4 Some Famous Games
297(9)
16.5 Exercises and Problems
306(5)
17 Nash Equilibria
311(18)
17.0 Scenario
311(1)
17.1 Mixed Strategies
312(2)
17.2 The 2-by-2 Case
314(7)
17.3 The Proof of Nash's Theorem
321(5)
17.4 Exercises and Problems
326(3)
18 The Prisoner's Dilemma
329(24)
18.0 Scenario
329(1)
18.1 Criteria and Impossibility
330(7)
18.2 Omnipresence of the Prisoner's Dilemma
337(5)
18.3 Repeated Play
342(1)
18.4 Irresolvability
343(2)
18.5 Exercises and Problems
345(8)
Notes on Part III
349(4)
IV The Electoral College
353(38)
Introduction to Part IV
355(2)
19 Weighted Voting
357(20)
19.0 Scenario
357(1)
19.1 Weighted Voting Methods
358(3)
19.2 Non-Weighted Voting Methods
361(4)
19.3 Voting Power
365(4)
19.4 Power of the States
369(3)
19.5 Exercises and Problems
372(5)
20 Whose Advantage?
377(14)
20.0 Scenario
377(1)
20.1 Violations of Criteria
377(2)
20.2 People Power
379(5)
20.3 Interpretation
384(3)
20.4 Exercises and Problems
387(4)
Notes on Part IV 391(4)
Solutions to Odd-Numbered Exercises and Problems 395(54)
Bibliography 449(4)
Index 453
E. Arthur Robinson, Jr. is a Professor of Mathematics a Professor of mathematics at the George Washington University, where he has been since 1987. Like his coauthor, he was once the department chair. His current research is primarily in the area of dynamical systems theory and discrete geometry. Besides teaching the Mathematics and Politics course, he is teaching a course on Math and Art for the students of the Corcoran School the Arts and Design.



Daniel H. Ullman is a Professor of Mathematics at the George Washington University, where he has been since 1985. He holds a Ph.D. from Berkeley and an A.B. from Harvard. He served as chair of the department of mathematics at GW from 2001 to 2006, as the American Mathematical Society Congressional Fellow from 2006 to 2007, and as Associate Dean for Undergraduate Studies in the arts and sciences at GW from 2011 to 2015. He has been an Associate Editor of the American Mathematical Monthly since 1997. He enjoys playing piano, soccer, and Scrabble.