916 CHAPTER 14 Voting and Apportionment The head-to-head criterion is also known as the Condorcet criterion . This criterion is named after the Marquis de Condorcet; see the Profile in Mathematics . Profile in Mathematics Marquis de Condorcet The Marquis de Condorcet (1743–1794) was one of the most influential mathematicians, economists, political scientists, and sociologists during the American and French revolutions. Condorcet believed that principles of fair government could be discovered mathematically. By analyzing voting methods, he discovered, to his disappointment, that occasionally there can be no clear and fair way to declare the winner of an election. The pairwise comparison method is also referred to as the Condorcet method. One of Condorcet’s discoveries was that in an election with three candidates A, B, and C, it was possible for the voters to prefer A to B and B to C but also C to A. Example 3 Applying the Head-to-Head Criterion Four candidates are running for mayor of Dunedin: Alvarez (A), Buchannon (B), Czechanski (C), and Davis (D). The election involves voters ranking the candidates with the results shown in Table 14.17. Table 14.17 Dunedin Mayor Preference Table Number of Votes 129 90 87 78 42 First D A B C C Second A C C B B Third B B A D A Fourth C D D A D a) Is there one candidate who is favored over all others using a head-to-head comparison? b) Who wins this election if the plurality method is used? Does that result violate the head-to-head criterion? c) Who wins this election if the Borda count method is used? Does that result violate the head-to-head criterion? d) Who wins this election if the plurality with elimination method is used? Does that result violate the head-to-head criterion? e) Who wins this election if the pairwise comparison method is used? Does that result violate the head-to-head criterion? Solution a) Using Table 14.17, we can determine that Alvarez is favored, using head-to-head comparison, over Buchannon. Alvarez is favored over Buchannon in columns 1 and 2, giving Alvarez + 129 90 or 219 votes. Buchannon is favored over Alvarez in columns 3, 4, and 5, giving Buchannon 87 78 42 + + or 207 votes. Alvarez is favored over Buchannon by a margin of 219 votes to 207 votes. Using this procedure, we can determine that Alvarez is also favored over Czechanski by a margin of 219 votes to 207 votes and Alvarez is also favored over Davis by a margin of 219 votes to 207 votes (you should verify these numbers). Therefore, Alvarez is favored over all the candidates, and according to the head-to-head criterion, Alvarez should be declared the winner. b) Using the plurality method, Davis is elected by virtue of the 129 first-place votes. Therefore, this result violates the head-to-head criterion. This example demonstrates that the plurality method has the potential to violate the head-to-head criterion. c) Using the Borda count method, Buchannon receives 1146 points, Czechanski receives 1140 points, Alvarez receives 1083 points, and Davis receives 891 points (you should verify these tallies). Therefore, Buchannon is elected using the Borda count method. Therefore, this result violates the head-to-head criterion. This example also demonstrates that the Borda count method has the potential to violate the head-to-head criterion. d) Using plurality with elimination, we see that Buchannon is eliminated in the first round, Alvarez is eliminated in the second round, and in the third round Head-to-Head Criterion If a candidate is favored when compared head-to-head with every other candidate, that candidate should be declared the winner.
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