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Much of our daily life is spent taking part in various types of what we might call “political”procedures. Examples range from voting in a national election to deliberating with othersin small committees. Many interesting philosophical and mathematical issues arise whenwe carefully examine our group decision-making processes.
There are two types of groupdecision making problems that we will discuss in this course. A voting problem: Supposethat a group of friends are deciding where to go for dinner. If everyone agrees on whichrestaurant is best, then it is obvious where to go. But, how should the friends decide whereto go if they have different opinions about which restaurant is best? Can we always find achoice that is “fair” taking into account everyone’s opinions or must we choose one personfrom the group to act as a “dictator”? A fair division problem: Suppose that there is a cake anda group of hungry children. Naturally, you want to cut the cake and distribute the piecesto the children as fairly as possible. If the cake is homogeneous (e.g., a chocolate cake withvanilla icing evenly distributed), then it is easy to find a fair division: give each child a piecethat is the same size. But, how do we find a “fair” division of the cake if it is heterogeneous(e.g., icing that is 1/3 chocolate, 1/3 vanilla and 1/3 strawberry) and the children each wantdifferent parts of the cake?
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Week 1: Voting Methods The Voting Problem A Quick Introduction to Voting Methods (e.g., Plurality Rule, Borda Count,
Plurality with Runoff, The Hare System, Approval Voting) Preferences The Condorcet Parado How Likely is the Condorcet Paradox? Condorcet Consistent Voting Methods Approval Voting Combining Approval and Preference Voting by Grading
Week 2: Voting Paradoxes Choosing How to Choose Condorcet's Other Parado Should the Condorcet Winner be Elected? Failures of Monotonicity Multiple-Districts Parado Spoiler Candidates and Failures of Independence Failures of Unanimity Optimal Decisions or Finding Compromise? Finding a Social Ranking vs. Finding a Winne
Week 3: Characterizing Voting Methods Classifying Voting Methods The Social Choice Model Anonymity, Neutrality and Unanimity Characterizing Majority Rule Characterizing Voting Methods Five Characterization Results Distance-Based Characterizations of Voting Methods Arrow's Theorem Proof of Arrow's Theorem Variants of Arrow's Theorem
Week 4: Topics in Social Choice Theory Introductory Remarks Domain Restrictions: Single-Peakedness Sen’s Value Restrictio Strategic Voting Manipulating Voting Methods Lifting Preferences The Gibbard-Satterthwaite Theorem Sen's Liberal Parado
Week 5: Aggregating Judgements Voting in Combinatorial Domains Anscombe's Parado Multiple Elections Parado The Condorcet Jury Theorem Paradoxes of Judgement Aggregatio The Judgement Aggregation Model Properties of Aggregation Methods Impossibility Results in Judgement Aggregatio Proof of the Impossibility Theorem(s)
Week 6: Fair Division Introduction to Fair Divisio Fairness Criteria Efficient and Envy-Free Divisions Finding an Efficient and Envy Free Divisio Help the Worst Off or Avoid Envy? The Adjusted Winner Procedure Manipulating the Adjusted Winner Outcome
Week 7: Cake-Cutting Algorithms The Cake Cutting Problem Cut and Choose Equitable and Envy-Free Proocedures Proportional Procedures The Stromquist Procedure The Selfridge-Conway Procedure Concluding Remarks
