Consider two parties or candidates, labeled A and B. Before elections take place, these parties commit to policy platforms, qA and qB. They act simultaneously and do not cooperate. The platform of the winning party is implemented. As we emphasize below, the precise conditions for winning depend on the electoral rule. Consider, say, party A. When announcing its policy platform, it maximizes the expected value of rents, namely:
where the term (± + aj) = 0, reflects voter j’s ideological preference for party B. This term includes two components; a is common to all voters and aj idiosyncratic itat on.
We can think of ± as the general popularity of party B. We assume that ± is a random variable with a uniform distribution on [— 2d, 2d]. Thus, the density of this distribution is given by d and the expected value of ± is zero. Furthermore, ± is realized between the announcement of the party platforms and the election. This assumption means that parties announce their platforms under uncertainty about the election outcome. Clearly, such uncertainty is very plausible. It is also technically convenient; as we shall see below, it smooths the optimization problem facing the parties.
The variable aj reflects the individual ideology of voter j. The distribution of aj differs across groups. These distributions are uniform on [— + a*, ^ + 0*] , i = 1, 2, 3. They are fully characterized by two parameters, a* and sг, and groups differ over both. In other words, groups differ in their average ideology, captured by the group-specific means a*. But they also differ in their ideological homogeneity, a higher density s* being associated with a more narrow distribution of aj. We make specific assumptions about these differences in distribution. Suppose we label the three groups according to their average ideology a*: a1 s1 ,s3. This is the substantial assumption. For convenience, we also assume that a2 = 0 and that a1s1+ a3s3 = 0.7
The meaning of these assumptions is illustrated in Figure 1, where we have drawn the distributions for aj in the three groups. Each of the three groups has an “ideologically neutral” voter with aj = 0, and the further to the right we go in the figure, the more likely we are to find a voter who will vote for party B. Assume, for concreteness, that ± = 0 and, furthermore, that the two candidates have announced the same policies qA = qB. In this event, the ideologically neutral voters with aj = 0 in each group are indifferent between the two candidates. We label these indifferent voters “swing voters”. Every voter with aj below (above) 0 finds it optimal to vote for party A (party B).