• Explore the following question: If a majority tournament turns out to be transitive, does any voting method yield a different winner than the condorcet winner?
• Prove: Every tournament with more than 3 vertices has at least one transitive triple.
• Prove: A tournament has exactly one king if and only if that king is an emperor (ie, condorcet winner).
• Prove: No tournament has exactly two kings.
• Prove that the definition of a Banks set (B(T)) is identical to the defintion of a Miller set (M(T)).
• Prove B(T) is a subset of K(T) (set of kings).

Quiz 2:due Monday Jan. 21st

1. Prove: No tournament has exactly two kings.
2. Prove: B(T) is a subset of K(T).
3. Refer to the example given in class (1/3 votes for xwzy; 1/3 for wzyx; 1/3 for yxwz). Suppose that you are an agenda setter and you do not like x. Give an agenda so that x does not win by sophisticated voting method. Give an agenda so that x does not win by sincere voting method. Can you give an agenda so that x does not win by both sophisticated and sincere? If yes, give it, and if no, prove that you cannot.