A matrix of pairwise margins is antisymmetric: its transpose has reversed signs on all entries.
Suppose we are given an antisymmetric matrix with integer entries. Can we always be sure that it is the pairwise margin matrix for some possible set of ballots? If not, what additional condition(s) on the matrix would suffice to ensure the existence of such a ballot set? If so, what is the simplest way to construct such a set of ballots? How complicated does the set of ballots have to be? For example, when the matrix is a five by five array, how many factions are needed (worst case) in a set of ballots for which this would be the pairwise margin matrix? To be more concrete, here's a "randomly" chosen five by five antisymmetric matrix: [[0,-6,2,1,-9],[6,0,5,-3,4],[-2,-5,0,8,7],[-1,3,-8,0,-2],[9,-4,-7,2,0]] Is there a set of ballots (making use of fewer than the 120 distinct permutations of the candidates A, B, C, D, and E) which gives rise to this matrix as its matrix of pairwise margins? Forest
