November 10, 2002

When We Settle

Here's the scenario that is most aggravating about all voting theory:

Say you have a vote with candidate A and candidate C, and it would end up:

40 A
60 C

C wins. But now say that B enters the race. And for various reasons, the new preference makeup is like this:

40 A B C
35 B C A
25 C A B

The thinking goes, to be REALLY fair, if C would have normally beat A, and you're introducing a candidate that wouldn't win, shouldn't C still beat A?

It seems obvious, doesn't it? This is one of the main requirements that Kenneth Arrow described for "fair voting", which he then proved could never happen.

And it looks like in each of our voting systems, it flunks in one way or another:

  • Approval: (Irrelevant in this case)
  • Plurality: A now wins with 40% of the vote
  • IRV: C is eliminated first, A then wins
  • Borda: A now wins with 40% of the awarded points
  • Condorcet: This one's interesting - by listing the head-to-head matchups in order, we get:

    B/C: 75-25
    A/B: 65-35
    C/A: 60-40

    And that's a cycle. Condorcet has various methods of breaking these tie votes, some by strength of victory (B/C would be 75), others by margin of victory (B/C would be 50). But in all cases, their tie-breaking scenario would eliminate C/A as the weakest win here, which would eliminate C from the tallies, which would once again lead to A winning.

So in all cases, C would have first won, B is introduced, B never wins, and C is now always in last place. Doesn't that just seem wrong?

Well, here's the trick. Maybe it isn't wrong. Maybe that's entirely fair. I (and many others, it turns out) think the whole requirement (it's called the "Irrelevant Alternatives" criteria) is flawed to begin with.

First, it's important to recognize that just because a third candidate may change the order of two other candidates, it doesn't mean that will always happen. But second, it's just as important to consider the emotional dynamics that could create such a vote. Look at the vote results above again. What really happened? Well first, look at relative loyalty. There are the people that preferred A to C, and the people that preferred C to A. The A fans were much more loyal to A than the C fans were to C. The new candidate never bisected the difference between C and A for the C fans, and in many cases they decided they liked B better. C lost power among his base here, while A did not. There was greater strength of passion among the A voters than the C voters. And also, it suggests that the C voters didn't see a huge difference between C and A, while the A voters did. Perhaps A had a very strong regional base that wasn't compromised by B joining in. Perhaps the original C voters was made up of people that weren't strong C voters, but merely a coalition instead, and then the introduction of B busted the coalition. These aren't just maybe-perhaps, they are (IMO) the more likely explanations of what could have enabled this vote. When you start looking at the story behind the number, the results starts to sound a bit more rational. The numbers themselves suggest that of C's original supporters, many of them felt they were compromising.

The other thing is, each candidate is a vote in and of itself. If I'm trying to put a vote in for my favorite film, I'm not voting for the best ending, or the best poignant smile, or the best sunset - I'm weighing it all in my head and picking my favorite, all things considered. And since I'm summing up criterias in my head, I can have cycles amongst *my* preferences just the same as an overall vote could. If I had to consider all my films on a head-to-head basis, I doubt I'd be consistent. I'd pick Fearless over Zoolander because I like its depth. I'd pick Titanic over Fearless because of its sheer majesty. I'd pick Zoolander over Titanic, because comparatively, Titanic exhausts me. Cycles are rational, because when people rank candidates over each other, they aren't one-quality candidates. Each candidate has a collection of qualities that are more important depending upon the matchup. In other words, it's all about the matchup. (Think of college football!!)

So, there are plenty of defenses of how a new candidate can make the tallies turn out that way, and how the tallies can then make the vote turn out that way. The Irrelevant Alternatives criteria is more complicated than it sounds - just because an alternative is introduced and doesn't win does not make that alternative irrelevant. Posted by Curt at November 10, 2002 03:35 PM