October 04, 2003
Condorcet Details
I've been doing a little more research on Condorcet methods, and found a couple more items of interest.First, I've written before about the Smith sets and the Schwartz sets. As I said before, the Condorcet winner is that candidate that would beat every single other candidate in a one-on-one matchup. However sometimes there isn't a Condorcet winner.
An example is college football. If Colorado beats Oklahoma, and Oklahoma beats Nebraska, then you'd expect Colorado to be better than Nebraska. But then Nebraska blows into town and stomps on Colorado. Well, that sucks since I'm a Buffalo. But, who's better? The polls deal with this by bring in all sorts of tiebreakers.
It's the same with Condorcet voting. There can be cycles where a voting population collectively prefers A>B, B>C, and then C>A.
Well, they deal with this in a few ways. First, they can isolate a group of candidates that as a group beats every candidate outside of that group. Like, A, B, and C *all* would beat D in a one-on-one matchup. That's called the Smith set. Sometimes with a really large Smith set there's a smaller group within that Smith set that would beat everyone else in that Smith set. It's kind of a silly distinction, but the *smallest* possible group of candidates that would beat every single other candidate in a race is called the Schwartz Set.
So here's the interesting part. Every single Condorcet winner is perfect* right up until the identification of the Schwartz Set. But in cases where there is a multi-member Schwartz Set, things start to break down.
There are various tie-breaking techniques to determine how to pare down the Schwartz set. They can be lazily compared, again, to football. Some methods seek to deliberately keep or lock in the candidate with the most overall votes, or the biggest win differential. Others seek to exclude the candidate with the least votes, or the biggest loss differential. Some recalculate after every lock or exclusion, some don't.
They've all got different flaws, though, and some of them are kind of strange. Like if a ballot changed the respective ranking of two candidates, it could mean that the candidate that is then ranked lower has a higher chance of winning. And in a proof I just found out about, *all* of these tiebreaker techniques suffer from a bizarre flaw where if some voters hadn't voted in the first place, their favorite candidate might have had a better chance of winning. A spinny way of putting that is that their candidate was penalized for having support expressed for them.
So this just keeps on leaving me at one conclusion. Implement Condorcet voting for elections, but only up until the Schwartz set. Then at that point, use our legislative procedures to break ties, much like how our Congress would decide an election if there is an electoral tie. Or, go for another round of education among the Schwartz set candidates, and have a runoff.
* Now for that pesky asterisk. In a group election, find the candidate that would beat all other candidates in a one-on-one matchup. Is that really what we want? It sounds great at first, but sometimes there's something that bugs me about that. Maybe it's not irrational to want a different winner out of a group than you'd want out of a series of one-on-one matchups. For one thing, the one-on-one matchups are closed votes. They frame the question. For a group of candidates, it communicates more about the overview of the race - gives parameters, and perhaps communicates to the voters about what is important to consider in the race. There's something inherently educational about it to the voters. If I saw a group of candidates to vote among, I might rank them differently as a group than I might vote for them in a series of one-on-one matchups. So in that sense only, maybe the Condorcet winner is not the appropriate winner.