[EM] Random Ballot Condorcet

[Ross Hyman]

Sadly, the random ballot Condorcet method I posted about earlier does not in general elect the highest ranked Smith candidate on the random ballot when there are more than three Smith candidates.

The following method will work though.
 Chose a random ballot. If it is not complete, draw others to break ties until there is a complete ranking.  Elect the highest ranked candidate for which there is a beat path from it to every other candidate.

This can be formulated in a way similar to the previous method:  Candidates are either hopeful or discarded.  All candidates are initially hopeful. All candidates, hopeful and discarded are available to be used in beat paths.  Consider the two lowest ranked hopeful candidates.  Discard the lower ranked of the two if there is a beat path from the higher ranked candidate to the lower ranked candidate. (And it doesn't matter what its strength is or if there is a stronger beat path going the other way.)  If there is no beat path from the higher to the lower candidate and there is at least one beat path from the lower to the higher candidate then discard the higher candidate.  If there is no beat path either way, then define a beat path from the higher to the lower candidate and discard the lower candidate. Repeat until one hopeful candidate remains.  Elect that candidate.

The N seat Random Ballot Condorcet STV method is similar.  Construct a ranking of every relevant set of N candidates from random ballots (you will generally need more than one ballot even if all candidates are ranked. I will give a mechanism for doing this in another post.)  Elect the highest ranked candidate set for which there is a beat path from it to every other candidate set.

This method differs from a fully deterministic method because the only elections that must be considered to create the initial set of beat paths are all elections with N+1 candidates for N seats.  For each of these elections, create beat paths from the winning N seat candidate set of that election to each of looser sets of that election that is the winner of at least one N+1 candidate set election. Other beat paths are created as needed as the method proceeds when the lowest ranked hopeful candidate sets have no beat paths between them.    















On Tuesday, May 20, 2014 5:55 PM, Ross Hyman <rahyman at sbcglobal.net> wrote:
 


A better random ballot Condorcet method is: Chose a random ballot (and if it is not complete, draw others to break ties until there is a complete ranking).  Eliminate the pair-wise loser of the two lowest ranked candidates.  Repeat until one candidate remains.  Elect that candidate.

I believe it has the following desired properties: monotonic, clone independent, only Smith candidates get a non-zero probability of being elected, independence of zero probability alternatives, and it requires the fewest number of pair comparisons and chooses the candidate that tends to be higher ranked than the previous version. In the three candidate case, if there is a cycle, it will always choose the top ranked candidate from the random ballot.  

One can form a complete social ranking by starting from the lowest ranked candidate and moving candidates down if they lose to the one below it.  The social ranking from the previous method is equivalent to starting from the highest ranked candidate and moving candidates up if they beat the one above it.



   


 







On Wednesday, May 7, 2014 6:51 PM, Ross Hyman <rahyman at sbcglobal.net> wrote:
 


Random Ballot Condorcet:  Choose a random ballot.  Elect the lowest ranked candidate that pairwise beats all higher ranked candidates.

Has this method been discussed before?  I believe that the following are true:  It will always elect a Condorcet candidate if there is one.  Otherwise it will elect a member of the Smith set with some nonzero probability for each member of the Smith set.  Non-Smith set candidates will have zero probability of being elected.  It is monotonic in that raising a candidate on some ballots cannot decrease its probability of being elected.  It is clone proof in that the probability of electing from the clone set is independent of the number of clones in the set. It is independent of irrelevant alternatives in that deleting a candidate with zero probability of winning cannot effect the probabilities for electing other candidates.  
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