Analysis of the Model of Campaigns
in Mass Elections
P is an array of two dimensions. It represents public opinion — the themes that constitute the political environment in which the electorate and the candidates operate. The two dimensions are bounded by negative one and one, and they are indexed by j = 1, 2. The model can be generalized to more than two dimensions, but in the interest of simplicity only the case with two dimensions is considered. There is one voter who has an ideal point Pj on each dimension. These ideal points are distributed uniformly on the interval negative one and one, so they have an expected value of zero. Voter ideal points are independent across dimensions. The model can also be generalized to multiple voters.
There are two candidates indexed by k, k = D, R. Each candidate has an ideal point on each dimension Pjk. These ideal points are constrained to the “left” and “right”: PjD ∈ [−1,0] and PjR ∈ [0,1]. Each candidate has a budget B and makes allocations Bjk from that budget to the dimensions each chooses to discuss. These allocations are constrained: B1k + B2k = B and Bjk ≥ 0.
The voter chooses between the two candidates according to a voting strategy V. The candidate who receives the vote wins and sets policies equal to their ideal points Pj*. Thus, if D wins Pj* = PjD and if R wins Pj* = PjR.
The voter's overall utility from electing a particular candidate Uvoter, k depends on the utility derived from each dimension. The voter also weights each dimension according to a weight ωjk: