Example 8

EXAMPLE 8 Entry of a Third Candidate in a Two-Candidate Race

Look at Figure 12.6, in which and are both at and therefore split the vote. If a third candidate, , enters and takes a position on either side of (say, to the right), then voters with ideal positions to the right of the midpoint of and ’s policy position vote for Candidate . Even though the area under the distribution to ’s right may encompass less than of the total area, may still win a plurality of votes, that is, more votes than either or .

To show why this is so, consider the portion of the electorate’s vote that will receive and the portion that will receive. If ’s area (tan) is greater than half of ’s area (blue), will win more votes than or , because ’s area includes not only the votes to the right of his or her position but also some votes to the left. More precisely, will attract voters up to the point midway between his or her position on the horizontal axis and that of ; and will split the votes to the left of this midway point. Because picks up some votes to the left of his or her position, less than of the electorate may lie to the right and still enable to win a plurality of more than of the total vote.

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Figure 12.6: Figure 12.6 Unimodal distribution with three candidates. Candidate can take a position with less than of the voters to his or her right and still win if candidates and are at the median and split the remainder of the vote.

By similar reasoning, it is possible to show that a fourth candidate, , could take a position to the left of and further chip away at the total of the two centrists. Indeed, could beat Candidate , as well as and , by moving closer to from the left than moves from the right.

Clearly, has little appeal, and is in fact quite vulnerable, to a third or fourth candidate contemplating a run against two centrists. Indeed, it is not difficult to show that whatever positions two candidates adopt—the same or different—at least one of these candidates will be vulnerable to a third candidate.