Approval Voting in Box Societies

Under approval voting, every voter may vote for any number of candidates. To model approval voting, we let a political spectrum be the set of all possible political positions, and let each voter have a subset of the spectrum that they approve, called an approval region. The fraction of all voters who approve the most popular position is the agreement proportion for the society. We consider voting in societies whose political spectrum is modeled by d-dimensional space (Rd) with approval regions defined by axis-parallel boxes. For such societies, we first consider a Turán-type problem, attempting to find the maximum agreement between pairs of voters for a society with a given level of overall agreement. We prove a lower bound on this maximum agreement and find in the literature a proof that the lower bound is optimal. By this result we find that for sufficiently large n, any n-voter box society in Rd where at least α(2) pairs of voters agree on some position must have a position contained in βn approval regions, where α = 1 − (1 − β)2/d. We also consider an extension of this problem involving projections of approval regions to axes. Finally we consider the question of (k, m)-agreeable box societies, where a society is said to be (k, m)-agreeable if among every m voters, some k approve a common position. In the m = 2k− 1 case, we use methods from graph theory to prove that the agreement proportion is at least 1/(2d) for any integer k ≥ 2.