# [EM] Pairwise Margins

Forest Simmons fsimmons at pcc.edu
Sat Jan 5 08:37:04 PST 2002

```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

```