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.
George Washington University, Washington, District of Columbia, USA
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
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