A Mathematical Look at Politics: 1st Edition (Hardback) book cover

A Mathematical Look at Politics

1st Edition

By E. Arthur Robinson, Jr., Daniel H. Ullman

CRC Press

477 pages | 14 B/W Illus.

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pub: 2010-12-09
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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.

Reviews

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

Table of Contents

Preface, for the Student

Preface, for the Instructor

Voting

Two Candidates

Scenario

Two-candidate methods

Supermajority and status quo

Weighted voting and other methods

Criteria

May's Theorem

Exercises and problems

Social Choice Functions

Scenario

Ballots

Social choice functions

Alternatives to plurality

Some methods on the edge

Exercises and problems

Criteria for Social Choice

Scenario

Weakness and strength

Some familiar criteria

Some new criteria

Exercises and problems

Which Methods are Good?

Scenario

Methods and criteria

Proofs and counterexamples

Summarizing the results

Exercises and problems

Arrow's Theorem

Scenario

The Condorcet paradox

Statement of the result

Decisiveness

Proving the theorem

Exercises and problems

Variations on the Theme

Scenario

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

Apportionment

Hamilton's Method

Scenario

The apportionment problem

Some basic notions

A sensible approach

The paradoxes

Exercises and problems

Divisor Methods

Scenario

Jefferson's method

Critical divisors

Assessing Jefferson's method

Other divisor methods

Rounding functions

Exercises and problems

Criteria and Impossibility

Scenario

Basic criteria

Quota rules and the Alabama paradox

Population monotonicity

Relative population monotonicity

The new states paradox

Impossibility

Exercises and problems

The Method of Balinski and Young

Scenario

Tracking critical divisors

Satisfying the quota rule

Computing the Balinski-Young apportionment

Exercises and problems

Deciding Among Divisor Methods

Scenario

Why Webster is best

Why Dean is best

Why Hill is best

Exercises and problems

History of Apportionment in the United States

Scenario

The fight for representation

Summary

Exercises and problems

Notes on Part II

Conflict

Strategies and Outcomes

Scenario

Zero-sum games

The naive and prudent strategies

Best response and saddle points

Dominance

Exercises and problems

Chance and Expectation

Scenario

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

Scenario

The best response

Prudent mixed strategies

An application to counterterrorism

The -by- case

Exercises and problems

Conflict and Cooperation

Scenario

Bimatrix games

Guarantees, saddle points, and all that jazz

Common interests

Some famous games

Exercises and Problems

Nash Equilibria

Scenario

Mixed strategies

The -by- case

The proof of Nash's Theorem

Exercises and Problems

The Prisoner's Dilemma

Scenario

Criteria and Impossibility

Omnipresence of the Prisoner's Dilemma

Repeated play

Irresolvability

Exercises and problems

Notes on Part III

The Electoral College

Weighted Voting

Scenario

Weighted voting methods

Non-weighted voting methods

Voting power

Power of the states

Exercises and problems

Whose Advantage?

Scenario

Violations of criteria

People power

Interpretation

Exercises and problems

Notes on Part IV

Solutions to Odd-Numbered Exercises and Problems

Bibliography

Index

Subject Categories

BISAC Subject Codes/Headings:
BUS069000
BUSINESS & ECONOMICS / Economics / General
MAT000000
MATHEMATICS / General