Chapter 13 - Voting and Apportionment - 13.2 Flaws of Voting Methods - Exercise Set 13.2 - Page 860: 3

(a) Since Amsterdam is favored over Vancouver and Rio de Janeiro, Amsterdam is favored over all others using a head-to-head comparison. (b) Vancouver has the highest number of first-place votes, so Vancouver wins the vote using the plurality method. (c) The head-to-head criterion is not satisfied.

(a) With a head-to-head comparison, each candidate is compared with every other candidate. For each pair of candidates, a candidate is favored over the other candidate if the candidate is ranked higher than the other candidate on a majority of ballots. We can compare Amsterdam and Vancouver. Amsterdam: 12 + 4 = 16 Vancouver: 9 + 4 = 13 Since Amsterdam is ranked higher than Vancouver on more ballots, Amsterdam is favored over Vancouver. We can compare Amsterdam and Rio de Janeiro. Amsterdam: 12 + 4 = 16 Rio de Janeiro: 9 + 4 = 13 Since Amsterdam is ranked higher than Rio de Janeiro on more ballots, Amsterdam is favored over Rio de Janeiro. We can compare Rio de Janeiro and Vancouver. Rio de Janeiro: 12 + 4 = 16 Vancouver: 9 + 4 = 13 Since Rio de Janeiro is ranked higher than Vancouver on more ballots, Rio de Janeiro is favored over Vancouver. Since Amsterdam is favored over Vancouver and Rio de Janeiro, Amsterdam is favored over all others using a head-to-head comparison. (b) Using the plurality method, the candidate which has the most first-place votes is declared the winner. We can find the number of first-place votes for each city. The number of first-place votes for Amsterdam is 12 votes. The number of first-place votes for Rio de Janeiro is 4 votes. The number of first-place votes for Vancouver is 9 + 4 which is 13 votes. Vancouver has the highest number of first-place votes, so Vancouver wins the vote using the plurality method. (c) The head-to-head criterion is not satisfied. Amsterdam is favored over all others using a head-to-head comparison, but Amsterdam did not win the vote using the plurality method.