A Mathematical Look at Politics  book cover
1st Edition

A Mathematical Look at Politics

ISBN 9781439819838
Published December 9, 2010 by CRC Press
480 Pages 14 B/W Illustrations

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Book Description

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 Dean’s method. How the US tried to use game theory to win the Cold War, and why it didn’t 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.

Table of Contents

Preface, for the Student
Preface, for the Instructor

Two Candidates
Two-candidate methods
Supermajority and status quo
Weighted voting and other methods
May's Theorem
Exercises and problems
Social Choice Functions
Social choice functions
Alternatives to plurality
Some methods on the edge
Exercises and problems
Criteria for Social Choice
Weakness and strength
Some familiar criteria
Some new criteria
Exercises and problems
Which Methods are Good?
Methods and criteria
Proofs and counterexamples
Summarizing the results
Exercises and problems
Arrow's Theorem
The Condorcet paradox
Statement of the result
Proving the theorem
Exercises and problems
Variations on the Theme
Inputs and outputs
Vote-for-one ballots
Approval ballots
Mixed approval/preference ballots
Cumulative voting .
Condorcet methods
Social ranking functions
Preference ballots with ties
Exercises and problems
Notes on Part I

Hamilton's Method
The apportionment problem
Some basic notions
A sensible approach
The paradoxes
Exercises and problems
Divisor Methods
Jefferson's method
Critical divisors
Assessing Jefferson's method
Other divisor methods
Rounding functions
Exercises and problems
Criteria and Impossibility
Basic criteria
Quota rules and the Alabama paradox
Population monotonicity
Relative population monotonicity
The new states paradox
Exercises and problems
The Method of Balinski and Young
Tracking critical divisors
Satisfying the quota rule
Computing the Balinski-Young apportionment
Exercises and problems
Deciding Among Divisor Methods
Why Webster is best
Why Dean is best
Why Hill is best
Exercises and problems
History of Apportionment in the United States
The fight for representation
Exercises and problems
Notes on Part II

Strategies and Outcomes
Zero-sum games
The naive and prudent strategies
Best response and saddle points
Exercises and problems
Chance and Expectation
Probability theory
All outcomes are not created equal
Random variables and expected value
Mixed strategies and their payouts
Independent processes
Expected payouts for mixed strategies
Exercises and Problems
Solving Zero-Sum Games
The best response
Prudent mixed strategies
An application to counterterrorism
The -by- case
Exercises and problems
Conflict and Cooperation
Bimatrix games
Guarantees, saddle points, and all that jazz
Common interests
Some famous games
Exercises and Problems
Nash Equilibria
Mixed strategies
The -by- case
The proof of Nash's Theorem
Exercises and Problems
The Prisoner's Dilemma
Criteria and Impossibility
Omnipresence of the Prisoner's Dilemma
Repeated play
Exercises and problems
Notes on Part III

The Electoral College
Weighted Voting
Weighted voting methods
Non-weighted voting methods
Voting power
Power of the states
Exercises and problems
Whose Advantage?
Violations of criteria
People power
Exercises and problems
Notes on Part IV
Solutions to Odd-Numbered Exercises and Problems

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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.