[EM] Why Condorcet

[Dave Ketchum]

Real-life voters need a way to express their most serious desires, and  
be heard, with reasonable effort and expense:
      Condorcet claims to offer such.  Better than Plurality, for  
which voters can vote for only one.  Better than Approval, which  
assumes equal desires for all voted for.  Better than Score for, while  
rating can be more fine-tuned than ranking, ranking is simpler to  
decide.
      Condorcet also means less election effort and expense than some  
of the competition.  It has less need than Plurality for primaries  
(tolerable for voters to vote for all of a set of such as clones) or  
runoffs (since voters can more completely express their desires).   
Does not have IRV's way of counting.
      To honor what the voters say.  IRV uses the same ranking (except  
normally forbidding equal ranking) - but, by its way of selectively  
reading only part of what voters say, IRV too often awards the win to  
other than who the voters actually want.
      Bucklin and Borda also use ranking, but each has a more complex  
way for voters to express their desires.
      Score demands more.  A voter thinking of A>B>C>D has no trouble  
offering max and min ranks or ratings to A and D.   With Score the  
voter is expected to diligently assign the available rating space  
among A>B, B>C, and C>D.

When one candidate is clearly preferred, Condorcet finds it as the CW  
for being preferred among every pair of candidates that include it.   
If there is no CW there is a cycle of 3 or more candidates, of which  
each would be CW if all other cycle members were suppressed, but is  
not preferred over every other cycle member.
      There are many methods offered for resolving Condorcet cycles  
when there is no CW - the chosen method should be both reasonable and  
have details explainable to most voters.  Methods deserving  
consideration share ballot format, voting rules, and the matrix  
produced when ballots are counted.
      Note that cycles mean different desires of moderate liking - a  
strong candidate gets to be CW, while a very weak one is weaker than  
any cycle member.

I claim Condorcet is not difficult, and describe details here:
      For the voter, who simply ranks higher those liked better.
      For the counter, whose life is more tolerable if some
optimization gets applied:
     Agreeing that having several winners in a single race, such as of  
legislature members,  requires a different method.

Now some thought about keeping it simple, yet doable.

I will lean toward Ranked Pairs with margins, but amending toward  
other Condorcet methods should be doable.

Ballots:  Must support write-ins and, perhaps, 3 ranks (do not need to  
rank rejects and can do equal ranking).

Voting:  Voter can rank one or more candidates, much as would be done  
in various methods:
     Bullet voting, ala Plurality - simply rank one.  This is a  
suitable vote for many voters in many races.
     Approval - just give them the same rank.
     Condorcet - Equal ranking permitted.  Counters care only which of  
any pair of candidates ranks higher, not how voter decides on ranking.
     Rank below unranked candidates?  Can't, but can rank all others  
above those most hated.
     Rank, but number not clear - rules could have counters treat such  
as a rank below the lowest real rank.
     Write-ins permitted (if few write-ins expected, counters may lump  
all such as if a single candidate - if assumption correct the count  
verifies it; if incorrect, must recount - if many expected for one  
person, that name could be added in for counting).

Counting:  Have an N array with one entry per candidate plus an N*N  
matrix with one row and column per candidate.  From each ballot simply  
step each ranked candidate's entry in the array.  When this is later  
copied into the N*N matrix it will supply exactly what is needed for  
pairs with no ranking, for pairs with one ranked, and for winner if  
both ranked.
     For pairs with a winner and loser the above will count both as  
winner, so give loser a negative count now in N*N to adjust; for ties  
you can leave both winning; or mark both losing via negative counts.
     (For example, a ballot with 3 ranks gets 3 counts in N, and  
adjustments for 3 pairs in N*N - even if there are a dozen pairs on  
the ballot)

Note that the above means minimum labor for counting:
     Q elements in the N array if the voter ranks Q candidates.
     P elements in N*N if the Q elements were composed of P pairs (0  
for a bullet vote; 6 for Q=4).

Completing matrix N*N:
      If such as write-ins mean a matrix with one less candidate than  
others, simply add an empty element to its N and an empty row and  
column to its N*N.
      Sum all the matrices and all the arrays.  Then add each array  
element into its matrix row as wins by its candidate in each of its  
pairs.
      The diagonals (A,A thru N,N) should be zero - make them thus.

Finding the winner.  What I suggest here is less labor than gets  
described for many methods, especially if there are many candidates -  
so, for candidates A-N:
     A single loss disqualifies a candidate from being CW, so start  
with A vs B.  If A loses, B continues; if B loses A continues; for a  
tie try a different pair among not-yet-losers (if any; else punt).
     Check final row when all but one have lost.  If there are no  
losers in this row we have the CW; else we have a cycle member and  
each loss is to another cycle member.
     By checking each cycle member found for such losers, we make a  
list of all such (for the simplest cycles each loses to one other).
     There are many methods for resolving cycles.  For RP I see  
deleting the smallest margins from the list until what remains is not  
a cycle, but does identify a winner.

Reporting results:  Besides the winner, contents of the final N*N  
matrix can be useful.  N*N matrices for districts such as a town or  
county can also be done and published.