478 Pages 14 B/W Illustrations
    by CRC Press

    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.

    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


    E. Arthur Robinson Jr., Daniel H. Ullman

    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.