EXAMPLE 10 The Median of the Medians May Not Be an Equilibrium Strategy

Student Council is divided into groups , , and , so that and each consist of 3 students and consists of 8 students. Each student votes for a candidate based on the candidates’ announced policy positions on the left-right continuum from 1 to 9. Assume that the median positions of the distribution of the voters’ ideal points for groups , , and are 5, 6, and 8, respectively. Assume that a candidate must win a majority of the “group votes” and that groups , , and award 2, 2, and 5 group votes, respectively, to the candidate who receives a majority of the votes from the students in the group.

The median of the medians is 6 because 6 is the middle value in 5, 6, and 8. However, if a candidate announces 6 as his or her policy position, then an opponent who announces 8 would defeat the candidate. This follows because group is so large in comparison with the other groups and its group votes alone are a majority of all group votes. (In Chapter 11, group is referred to as a dictator, and groups and are referred to as dummies in the corresponding simple weighted-voting game that models this voting scenario.)

The median of each group is weighted by the number of group votes each group has to award. Groups , , and have weights 2, 2, and 5, respectively. The weighted median is the median of 5, 5, 6, 6, 8, 8, 8, 8, 8. The data string 5, 5, 6, 6, 8, 8, 8, 8, 8 includes the median of each group multiple times, according to the group’s weight. Thus, 5 is listed twice because group has weight 2, 6 is listed twice because group has weight 2, and 8 is listed five times because group has weight 5. The median of this new data is 8, which is an equilibrium strategy.