Self Check Answers

1. Suppose the two candidates are $A$ and $B$ and that $A$ has won the election using majority rule, with $n$ votes to $B$’s $m$ votes where $n>m$. First, if any two voters exchange ballots, then $A$ still receives $n$ votes (although from a slightly different collection of voters) and $B$ still receives $m$ votes (again from a slightly different collection of voters). Thus, $A$ is still the winner. Second, if every voter were to change his or her ballot, then $B$ would receive the $n$ votes that $A$ previously received and vice versa. Hence, $B$ would be the new winner with $n$ votes. Third, if some voter who had voted for $B$ changed his or her vote to one for $A$, then $A$’s total would become $n+1$ and $B$’s total would become $m-1$. Hence, $A$ 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 $D$ is not a Condorcet winner, because $D$ loses to $B$ in a one-on-one contest.
6. No, Candidate $A$ is not a Condorcet winner, because $A$ loses to $C$ (and, incidentally, to $D$ as well) in a one-on- one contest.
7. No. An example is given in the text on page 419.