Answer
(a) Since Boston is favored over Atlanta and Chicago, Boston is favored over all others using a head-to-head comparison.
(b) Atlanta has the highest number of first-place votes, so Atlanta wins the vote using the plurality method.
(c) The head-to-head criterion is not satisfied.
Work Step by Step
(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 Boston and Atlanta.
Boston: 19 + 5 = 24
Atlanta: 20
Since Boston is ranked higher than Atlanta on more ballots, Boston is favored over Atlanta.
We can compare Boston and Chicago.
Boston: 20 + 19 = 39
Chicago: 5
Since Boston is ranked higher than Chicago on more ballots, Boston is favored over Chicago.
We can compare Chicago and Atlanta.
Chicago: 19 + 5 = 24
Atlanta: 20
Since Chicago is ranked higher than Atlanta on more ballots, Chicago is favored over Atlanta.
Since Boston is favored over Atlanta and Chicago, Boston 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 Boston is 19 votes.
The number of first-place votes for Chicago is 5 votes.
The number of first-place votes for Atlanta is 20 votes.
Atlanta has the highest number of first-place votes, so Atlanta wins the vote using the plurality method.
(c) The head-to-head criterion is not satisfied. Boston is favored over all others using a head-to-head comparison, but Boston did not win the vote using the plurality method.