EXAMPLE 4 A Discrete Distribution with an Even Number of Voters

Suppose that a Home Owner Association of a condominium complex wants to build a swimming pool and clubhouse for its residents along a 1-mile stretch of road. Residents live in one of six buildings, and each building gets a single vote to indicate its ideal point for where the clubhouse and pool should be located. Let the line segment from 0 to 1 represent the 1-mile stretch of road on which to build the facility. Suppose the six buildings have ideal points, ordered from smallest to largest, of 0.1, 0.3, 0.3, 0.5, 0.8, and 0.9. A Building Company (ABC) and Barry’s Builders are to propose different locations on the road—policy positions—for the new facility. Where should they build it?

Because there are 6 locations, there is no middle item in the list: 0.1, 0.3, 0.3, 0.5, 0.8, 0.9. That is, there is no median value in the list. In the absence of such a median position, is there still an equilibrium? It is not too difficult to show that if a discrete distribution has an even number of voters, then there is at least one equilibrium position. In fact, there may be a range of policy positions that, when paired, are in equilibrium with one another.

Suppose that ABC proposes to build at 0.4 and Barry proposes to build at 0.5. Each builder will receive half of the votes: Buildings with ideal points 0.1 and 0.3 vote for ABC, while the other buildings vote for Barry. If Barry were to change the proposed building site, notice that he could do no better than receiving 3 of the 6 votes. Indeed, this is the case for ABC, too.

This example can be generalized as follows: If ABC and Barry each announce distinct positions between 0.3 and 0.5, inclusive, then each builder will receive half the votes. This is no accident, as 0.3 and 0.5 are the two middle values in the ordered list of ideal points. Furthermore, if each builder announces a position between 0.3 and 0.5, inclusive, then neither builder will have an incentive to change. They will be in equilibrium. For this example, we call the interval [0.3, 0.5] of policy positions the extended median of the discrete distribution.