Dr. Steven Brams of New York University explains: "The example below… shows how nonmonotonicity can occur when all except the top two candidates are eliminated.." 12/09/07
Here's the nonmonotonicity example from Approval Voting, pp. 142-143, with four classes of voters (total: 17 voters) ranking three candidates:
Candidate “A” wins . Because nobody has a majority of 9 votes on the 1st round, C, the candidate with the fewest first-place votes (5), is eliminated. His 5 votes go to A, who wins with 11 (6 + 5) votes.
Here are the same classes of voters, with class # 4’s rankings changed:
Candidate “C” wins. Now assume the 2 class (4) voters change their preference ranking to the following: 4'. 2: ABC (as above)
Again, nobody has a majority of 9 votes on the 1st round. Because B gets the fewest first-place votes (4), he is eliminated, and his 4 votes go to C, who wins with 9 (5 + 4) votes.
In summary, candidate “A” wins when he is ranked second by the class (4) voters, but he loses (to C) when he is ranked first by these voters. Thus, “A” wins when fewer voters favor him, and he loses when more voters favor him.
This example is not as dramatic as my earlier example, when raising a candidate from last to first place--instead of second to first--causes him to lose, but it is still an example of nonmonotonicity. It shows that when one drops all except the top two candidates in the 1st round (in this example, only one candidate since there are a total of three), nonmonotonicity can still occur. This example can be easily extended--by adding voters and candidates--so that when one drops more than one candidate on the 1st round to eliminate all except the top two, nonmonotonicity can still occur.
Steven J. Brams
Dept. of Politics
19 West 4th St., 2d Fl.
E-mail: steven.brams at nyu.edu
New York University, New York, NY 10012
