Probabilities of dominant candidates based on first-place votes

Suppose each of an odd number n of voters has a strict preference order on the three ‘candidates’ in {1,2,3} and votes for his most preferred candidate on a plurality ballot. Assume that a voter who votes for i is equally likely to have ijk and ikj as his preference order when {i,j,k} = {1,2,3}.Fix an integer m between $\text{1}\text{2}$(n + 1) and n inclusive. Then, given that n_i of the n voters vote for i, let f_m(n₁,n₂,n₃) be the probability that one of the three candidates is preferred by m or more voters to each of the other two.This paper examines the behavior of f_m over the lattice points in L_n, the set of triples of non-negative integers that sum to n. It identifies the regions in L_n where f_m is 1 and where f_m is 0, then shows that f_m(a,b + 1, c)>f_m(a + 1,b,c) whenever a + b + c + 1 = n, a≤c≤b, a<c<m and c≤n ? m. These results are used to partially identify the points in L_n where f_m is minimized subject to f_m>0. It is shown that at least two of the n_i are equal at minimizing points.