EXAMPLE 3 Determining the Winner of an Election in a Spatial Model

Suppose that there are 21 eligible voters in a tiny town who must decide between Ann and Bob for mayor. Each voter has an ideal position that describes how much of the town’s taxes should be used to subsidize child care. A voter with an ideal position closer to 0 desires a higher subsidy. As the voter’s ideal point increases, then the voter prefers a smaller subsidy. The 21 voters have 7 distinct ideal positions, as described by the discrete distribution given in Table 12.5. For example, there are 4 voters who have 6 as their ideal position.

As part of their election platforms, Ann and Bob announce their positions on the child-care subsidy. Bob is more supportive of the child-care subsidy, having announced a position of 2 on the 0 to 10 scale. Ann is less supportive, having announced a position of 7. Each voter votes for the candidate whose announced position is closest to his or her ideal position. Figure 12.1 shows a graphical representation of the voters’ ideal points, as well as the policy positions announced by Ann (red line) and Bob (blue line).

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Voters at positions 0, 2, 3, and 4 all vote for Bob because each of these positions is to the left of the midpoint and therefore closer to Bob’s position of 2 (the blue line in Figure 12.1) than Ann’s position of 7 (the red line in Figure 12.1). For example, the distance between a voter’s ideal position 4 and Bob’s position 2 is , whereas the distance to Ann’s position is . Because 2 is less than 3, a voter with ideal position 4 votes for Bob over Ann. The other voters with ideal positions at 6, 8, and 10 vote for Ann over Bob because their ideal positions are greater than the midpoint and are closer to Ann’s position of 7 than Bob’s position of 2. Consequently, 13 of the 21 voters vote for Bob and 8 of the 21 voters vote for Ann. Bob wins the election.

What if Ann now thinks about how she could have changed her policy position? If she had announced a position to the right of Bob’s 2, then the best she could have hoped for is that all voters with ideal positions greater than 2 voted for her. There are 17 such voters. If Ann wanted to attract these 17 voters and win the election, then she could have announced a policy position of 3 or any other policy position greater than 2 and less than 4. However, if she had announced a policy position of 4, she would have still won the election, receiving 11 of the 21 votes. In fact, the policy position 4 has a special property: it is the median of the distribution.