Here's a suggested hybrid method that completely orders the candidates while respecting the order of the various Condorcet equivalence classes:
Seed Bubble Sort with the Modified Bucklin order described below. [Bubble Sort recursively sorts the top m-1 seeded candidates, and then percolates the m_th candidate as far up the list as possible by pairwise comparisons with adjacent candidates.] Forest On Fri, 4 Jan 2002, Forest Simmons wrote: > I want to make a (hopefully) final modification to my previous versions of > Modified Bucklin. Here it is: > > The context is a single winner election with N candidates. Each ballot has > (potentially) R distinguishable levels (counting truncations as the lowest > level), some of which may go unused by some or all of the voters. > > For each candidate C let L be the highest level at which candidate C has > fewer than 1/N of the ballots showing that candidate below level L. > > [In some cases level L will be the lowest possible level, below which > every candidate has zero showings, which certainly represents fewer than > 1/N of the ballots.] > > Also for candidate C let k be the difference in the number of ballots > showing C above the level L and the number of ballots showing C below the > level L. > > So now each candidate C has an associated ordered pair of numbers (L,k). > > Order the candidates according to the lexicographical order of their > associated number pairs. > > This means that candidate C' is higher than C in the ordering > > if and only if > > L' is greater than L, > OR > L'= L AND k' > k. > > The highest candidate in this lexicographical order is the Modified > Bucklin winner. > > This method is summable. A running sum of how many ballots each candidate > receives in each level is possible via an N by R matrix, where N is the > number of candidates and R is the number of possible levels. > > Question. Is this method consistent or even monotone? > > Question. Given an N by R matrix of this type computed from some set S of > ballots, what is the smallest number of factions that a set S' (of > Cardinal Ratings style ballots) can have while yielding the same N by R > matrix of candidate level summaries? > > Forest > >
