Monday, February 20, 2012

Just What the Heck is Non-Monotonicity?

I am still working on the remaining analysis of Duluth municipal elections - hold your horses! - but I thought that as a breather I would talk about something that comes up a lot when people are talking about switching over to instant-runoff voting. Normally I would use the term ranked-choice voting, which it the prefered term that's used by FairVoteMN, but non-monotonicity isn't a feature (bug) of all ranked-choice voting systems.

So what the heck is it?
A voting system is non-monotonic if, by switching your vote from a losing candidate to a winning candidate, you can make the winner into a loser (the actual definition is a bit more complex, but that's the gist of it for our purposes). This doesn't have to happen in all of the cases; it just has to be possible in some of them.

But that sounds horrible!
I agree, it does sound horrible. When we make a vote, we want to know that our vote is actually helping a candidate and our interests wouldn't have been better served by voting to someone else. (There's a different voting system which is called the non-participation criterion, which says that it should always be better for you if you go vote instead of staying home, which is pretty closely related - and which instant-runoff voting also violates.)

So instant-runoff voting is non-monotonic?
Unfortunately, yes. Check out the simplest case of an instant-runoff where this is true:
  • A>B>C = 10 votes
  • B>A>C = 8 votes
  • C>B>A = 9 votes
Under vanilla instant-runoff voting, B gets eliminated, his votes go to A, and A has more than a simple majority so he wins. But check this out; if two of the people who would otherwise have voted for C change their vote from C to A:
  • A>B>C = 12 votes
  • B>A>C = 8 votes
  • C>B>A = 7 votes
Then B wins! If your preference is C>B>A, it would have been better for you to have voted for the guy you like least! That's horrible! This is also how the non-participation criterion is violated; it would be better for the people who voted for C to stay home, because then if

So why the heck would you advocate for a system like that?
Here's a fun exercise; let's pretend that the voters have the same preferences as in the first example, but instead of using instant-runoff voting, we use two-stage first-past-the-post, the current system used in Duluth municipal elections. Here's the primary:
  • A = 10 votes
  • B = 8 votes
  • C = 9 votes
A and C go on to the general:
  • A = 18 votes
  • C = 9 votes
And A wins the general election. Seems a little familiar, right? Well, that's because it's exactly the same thing that happens in instant-runoff voting. If those two people who voted for C in the primary had stayed home or voted for A, then A would be the loser and B would be the winner. You can't argue against implementing instant-runoff voting on because it's non-monotonic and violates the non-participation criterion when the current system is non-monotonic and violates the non-participation criterion. That's like saying you'd rather keep eating your rotten apple because the banana I'm offering you is a fruit.

So how can we make a monotonic voting system?
There are a couple of ways. The trick is in making a monotonic system that doesn't suck. For example, we could take our current system and lop off the primary to make it monotonic, but you would have a lot of problems with split votes, favorite betrayal, strategy gaps, etc. There's also the problem that many of these solutions are more complicated to set up, because many of them require making pairwise calculations. Complex solutions cost more to implement and have a greater chance of failure. There's also another problem; some of the proposed solutions would require voting machines that aren't available for purchase. Mostly my position is that it's just something that we should stop worrying about for now - what's most important is replacing the current broken system.

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