EXAMPLE 7 A Continuous Distribution of Voters’ Attitudes

Suppose that voters’ attitudes are described by the continuous distribution over the left-right spectrum given in Figure 12.4a. The red and blue lines represent the policy positions of Candidates and , respectively. The voters represented by the tan-shaded area vote for Candidate ; these are voters with ideal points less than and to the left of the midpoint between the policy positions of and . The voters represented by the blue-shaded area vote for Candidate ; these are voters with ideal points greater than and to the right of the midpoint between policy positions of and . Because of the symmetry of the distribution—which means there is a line in which the distribution looks the same to the left as it does to the right—it is easy to see that more voters vote for than for because the tan region is greater in size than the blue region. Hence, would win the election.

As in the discrete model, there is a maximin position given by the median position of the distribution. For a continuous distribution, a median position covers half the area beneath the curve to the left and half the area beneath the curve to the right. Because the distribution is symmetric, the median position is the middle of the distribution. In Figure 12.4b, moves to this position—and wins the election! Candidate has no response to this move and can do no better than to join at the median.