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A Mathematical Look at Politics

E. Arthur Robinson, Jr. and Daniel H. Ullman
Chapman & Hall/CRC
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Bill Wood
, on

Social choice has emerged as a popular topic in application-oriented liberal arts mathematics texts. The typical unit on voting theory in such a course might introduce preference balloting, run through some voting methods and fairness criteria, and conclude with a statement of Arrow’s Theorem, which says roughly that no voting system can satisfy all fairness criteria all the time. Game theory, apportionment, and power are other standard topics.

It is a fun unit to teach because we can demonstrate the utility and practice of mathematical reasoning with little mathematical background. Students enjoy it because they understand the importance of elections and there are plenty of ways to connect the material to their own studies and experiences (however one may feel about the 2000 U.S. presidential election, it was undeniably a boon for this course). So why not spend the whole term on it? Robinson and Ullman’s book is designed as a textbook for such a course.

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. For example, the book is very explicit about what the goals are and that mathematicians are not casual about words like “impossible.” This is a real strength of the book — the patient reader will discover not just the content, but how and why mathematics is written the way it is. This is, therefore, a book that must be read and not skimmed. There are no colorful text boxes with vocabulary words, biographies, and anecdotes. Students who demand approaching exercises by finding similar examples in the chapter and skipping the narrative will find great frustration here.

There are four units: Voting, Apportionment, Conflict (probability and game theory), and The Electoral College (weighted voting, Banzhaf power). The exercises that close each chapter are interesting and often quite challenging — there are few straightforward computational practice problems. Solutions to the odd-numbered exercises are in the back. No college-level mathematics is required, but there is a fair bit of algebra and notation that, while generally introduced and explained with care, will intimidate some students. Though appropriate for an introductory course, the text is 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 examples and discussion are grounded in the American system of government and a general familiarity with American politics and history are useful. The downside to this approach is that other possible applications of the broad ideas in the text get little attention (and, of course, international students might need some background and persistent reminders).

There is a deliberate effort throughout to expose students to topics in mathematical reasoning that flow naturally from the content. For example, binomial coefficients come up in calculating the probabilities of ties and other election scenarios, and the value of approximation is covered when Stirling’s formula is employed to understand these quantities. By grabbing a specific application, students have the opportunity to explore when an exact value is needed, when an approximation is acceptable, and why it is often more important to understand the asymptotics of a function rather than its particular values. These distinctions are notoriously tough to sell in isolation and the authors do well including them here. The general goal, however, is the precise articulation of problems, the process of solving them, and the creation and recognition of ironclad solutions (often a proof).

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.

Bill Wood is an Assistant Professor of Mathematics at the University of Northern Iowa.

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