Self Check Answers

1. Suppose the two candidates are and and that has won the election using majority rule, with votes to ’s votes where . First, if any two voters exchange ballots, then still receives votes (although from a slightly different collection of voters) and still receives votes (again from a slightly different collection of voters). Thus, is still the winner. Second, if every voter were to change his or her ballot, then would receive the votes that previously received and vice versa. Hence, would be the new winner with votes. Third, if some voter who had voted for changed his or her vote to one for , then ’s total would become and ’s total would become . Hence, would still win.
2. Beth gets the next offer using Condorcet’s method because she defeats Adam and Dan one on one by identical scores of 3 to 2.
3. No. According to the definition, a voting system is manipulable only if there is an election in which a voter gets a more preferred outcome (rather than a less preferred outcome) by submitting a disingenuous ballot.
4. Arizona would have lost 24 points (for losing a second-place vote) and 22 points (for losing a fourth- place vote), for a total loss of 46 points. Hence, if Arizona received 17 third-place votes instead of 15, it would gain back the 46 lost points and still have a total of 1517.
5. No, Candidate is not a Condorcet winner, because loses to in a one-on-one contest.
6. No, Candidate is not a Condorcet winner, because loses to (and, incidentally, to as well) in a one-on- one contest.
7. No. An example is given in the text on page 419.