A Mathematical Look at Politics

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