Voting Methods: Making the case for the Schulze Method

If you're curious about voting methods, and/or want to make a case for a Condorcet method to your friends and co-voters, this post is for you!  

First, I think the Condorcet criterion is pretty compelling on its own.  (The Condorcet criterion is: if there's one candidate that is preferred in head-to-head matchups to every other candidate, that candidate wins.)  It's easy to come up with examples where the plurality method (this is the method that gets applied most often - whoever gets the most votes wins, even if that isn't more than half the votes -- boo plurality!) doesn't give you what seems the right solution.  If there is a Condorcet winner (there isn't always), it's the one that seems like the right answer.

So what if there isn't a Condorcet winner?  What voting method do you pick?  This is a case for the Schulze method: if there isn't a Condorcet winner, that's because there's a "rock-paper-scissors" cycle where candidate A is preferred (head-to-head) to candidate B, B is preferred to C, ... , Y is preferred to X, and X is preferred to A.  (The cycle can be size 3, like rock-paper-scissors, or can be longer - the point is it circles back on itself.)

At first glance, a cycle like that seems intractable - how do we rank any of the candidates in the cycle higher than the other?  But this is ignoring the strength of the preference:

Suppose 90% of voters prefer Rock to Scissors, 85% prefer Scissors to Paper, and 51% prefer Paper to Rock.  The cycle is there, but clearly the Paper > Rock preference is the weakest link.  The preference path Rock > Scissors > Paper has a "strength" of 85%, much higher than the 51% for Paper > Rock.  Based on path "strength", we can rank these Rock>Scissors>Paper.

Is Rock a Condorcet winner?  No.  But it's the closest thing to a Condorcet winner, in the sense that its pairwise loss is the weakest.

That's the Schulze method.  It's written in terms of "beat paths", but those are just breaking a cycle into two parts, like we did splitting "Rock>Scissors>Paper>Rock" into "Rock>Scissors>Paper" and "Paper>Rock" and comparing them.  There's some added detail to deal with multiple cycles between the same candidates, but that's really all there is to the idea of the method.

Next I hope to show another case to be made for the Schulze method, which might be more intuitive.

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