As the figure illustrates, our assumptions imply that the group which on average is ideologically neutral also has the largest number of ideologically neutral voters.9 It is natural to think of this group as consisting of ” middle class” voters.
We can also use this figure to illustrate how the parties evaluate the announcement of different policies. Suppose party A contemplates a deviation from a common policy announcement qA = qB. Such a deviation alters the number of votes party A can expect, by changing the identity of the swing voters. For example, a lower tax rate t or more public goods g benefit voters in all groups symmetrically. Taken separately, such measures thus push the identity of the swing voter in all groups to the right by the same distance, say to a’, and party A can expect to capture the voters between 0 and a’ in all groups (as the expected value of ± is equal to zero). Similarly, more transfers to group 1, financed by less transfers to group 3, shift the swing voter in group 1 to the right and the swing voter in group 3 to the left by the same distance (recall that we assume the groups to have the same size). This redistribution implies a net gain in votes, as there are more swing voters in group 1 than in group 3; that is, s1 > s3. Finally, higher rents r mean losing votes in all three groups, and a lower probability of winning. As the announced policies must respect the budget constraint, the two parties effectively trade off votes for votes, or rents for votes, when designing their platforms.
As a final preliminary, we define KA,i, the vote share of party A in group i. Given our assumptions about the group-specific distributions, KA,i can be expressed as:
where the expression within square brackets is a formal definition of the swing voter in group i. Clearly, the vote share of party B in group i is given by 1 — KA,i. Note that, from the point of view of both candidates, кA;i is a random variable, since it is a transformation of the random variable ± capturing the average popularity of party B.
We first use our model of pre-electoral politics to study policy outcomes under an electoral rule corresponding to proportional representation. Specifically, we study a very stylized case where (as in the Netherlands) there is only one voting district, comprising all the voters in the population. Winning the election thus corresponds to obtaining more than 50% of the total vote. Under this electoral rule, pA is given by: payday loans direct lender
where s = г s*/3 is the average density across groups. By symmetry, party B’s probability of winning is (1 — pA).