EXAMPLE 9 Spatial Modeling of an Election with Winner-Take-All Districts

Let’s return to the mayoral election between Ann and Bob described in Example 3. Suppose that the tiny town is divided into three districts. One district consists of 5 voters with ideal points 2, 3, 3, 3, and 6; we denote this district by {2, 3, 3, 3, 6}. The other two districts have 7 and 9 voters, respectively; their ideal points are represented as {0, 2, 2, 3, 3, 6, 6} and {3, 4, 4, 4, 6, 8, 8, 10, 10}. Assume that the candidate who wins a majority of the votes from a district with voters wins district votes—akin to electoral votes. The candidate with the majority of the district votes is elected mayor.

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Bob is aware of the median of the ideal points of the 21 voters and announces as his policy position. Ann is shrewder. She realizes that she needs to win two of the three districts and therefore calculates the medians of each district. The median of districts {2, 3, 3, 3, 6}, {0, 2, 2, 3, 3, 6, 6}, and {3, 4, 4, 4, 6, 8, 8, 10, 10} are 3, 3, and 6, respectively. By announcing a policy position of 3, she wins a majority of the votes in two of the three districts, becoming mayor by a tally of 12 to 9 district votes. Indeed, 3 is the unique equilibrium policy position and cannot be defeated by any other policy position.