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The Mathematics of Politics Second 2nd Edition by Arthur Robinson, Daniel Ullman 0367239035 9780367239039

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ISBN 10: 0367239035

ISBN 13: 9780367239039

Author: E. Arthur Robinson; Daniel H. Ullman

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.

The Mathematics of Politics Second 2nd Table of contents:

1 Two Candidates

1.0 Scenario

1.1 Two-Candidate Methods

1.2 Supermajority and Status Quo

1.3 Weighted Voting and Other Methods

1.4 Criteria

1.5 May’s Theorem

1.6 Exercises and Problems

2 Social Choice Functions

2.0 Scenario

2.1 Ballots

2.2 Social Choice Functions

2.3 Alternatives to Plurality

2.4 Some Methods on the Edge

2.5 Exercises and Problems

3 Criteria for Social Choice

3.0 Scenario

3.1 Weakness and Strength

3.2 Some Familiar Criteria

3.3 Some New Criteria

3.4 Exercises and Problems

4 Which Methods Are Good?

4.0 Scenario

4.1 Methods and Criteria

4.2 Proofs and Counterexamples

4.3 Summarizing the Results

4.4 Exercises and Problems

5 Arrow’s Theorem

5.0 Scenario

5.1 The Condorcet Paradox

5.2 Statement of the Result

5.3 Decisiveness

5.4 Proving the Theorem

5.5 Exercises and Problems

6 Variations on the Theme

6.0 Scenario

6.1 Inputs and Outputs

6.2 Vote-for-One Ballots

6.3 Approval Ballots

6.4 Mixed Approval/Preference Ballots

6.5 Cumulative Voting

6.6 Condorcet Methods

6.7 Social Ranking Functions

6.8 Preference Ballots with Ties

6.9 Exercises and Problems

7 Hamilton’s Method

7.0 Scenario

7.1 The Apportionment Problem

7.2 Some Basic Notions

7.3 A Sensible Approach

7.4 The Paradoxes

7.5 Exercises and Problems

8 Divisor Methods

8.0 Scenario

8.1 Jefferson’s Method

8.2 Critical Divisors

8.3 Assessing Jefferson’s Method

8.4 Other Divisor Methods

8.5 Rounding Functions

8.6 Exercises and Problems

9 Criteria and Impossibility

9.0 Scenario

9.1 Basic Criteria

9.2 Quota Rules and the Alabama Paradox

9.3 Population Monotonicity

9.4 Relative Population Monotonicity

9.5 The New States Paradox

9.6 Impossibility

9.7 Exercises and Problems

10 The Method of Balinski and Young

10.0 Scenario

10.1 Tracking Critical Divisors

10.2 Satisfying the Quota Rule

10.3 Exercises and Problems

11 Deciding among Divisor Methods

11.0 Scenario

11.1 Why Webster Is Best

11.2 Why Dean Is Best

11.3 Why Hill Is Best

11.4 Exercises and Problems

12 History of Apportionment in the United States

12.0 Scenario

12.1 The Fight for Representation

12.2 Summary

12.3 Exercises and Problems

Notes on Part II

III: Conflict

Introduction to Part III

13 Strategies and Outcomes

13.0 Scenario

13.1 Zero-Sum Games

13.2 The Naive and Prudent Strategies

13.3 Best Response and Saddle Points

13.4 Dominance

13.5 Exercises and Problems

14 Chance and Expectation

14.0 Scenario

14.1 Probability Theory

14.2 All Outcomes Are Not Created Equal

14.3 Random Variables and Expected Value

14.4 Mixed Strategies and Their Payoffs

14.5 Independent Processes

14.6 Expected Payoffs for Mixed Strategies

14.7 Exercises and Problems

15 Solving Zero-Sum Games

15.0 Scenario

15.1 The Best Response

15.2 Prudent Mixed Strategies

15.3 An Application to Counterterrorism

15.4 The 2-by-2 Case

15.5 Exercises and Problems

16 Conflict and Cooperation

16.0 Scenario

16.1 Bimatrix Games

16.2 Guarantees, Saddle Points, and All That Jazz

16.3 Common Interests

16.4 Some Famous Games

16.5 Exercises and Problems

17 Nash Equilibria

17.0 Scenario

17.1 Mixed Strategies

17.2 The 2-by-2 Case

17.3 The Proof of Nash’s Theorem

17.4 Exercises and Problems

18 The Prisoner’s Dilemma

18.0 Scenario

18.1 Criteria and Impossibility

18.2 Omnipresence of the Prisoner’s Dilemma

18.3 Repeated Play

18.4 Irresolvability

18.5 Exercises and Problems

Notes on Part III

IV: The Electoral College

Introduction to Part IV

19 Weighted Voting

19.0 Scenario

19.1 Weighted Voting Methods

19.2 Non-Weighted Voting Methods

19.3 Voting Power

19.4 Power of the States

19.5 Exercises and Problems

20 Whose Advantage?

20.0 Scenario

20.1 Violations of Criteria

20.2 People Power

20.3 Interpretation

20.4 Exercises and Problems

 

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