Wednesday, December 10, 2008

Marching Two By Two

Voting is easy, at least if there are only two options. When congress is deciding on what form a bill should take, they don't ask the congresmen to vote for their favorite of 3 (or 4 or more) options; they break it into pieces, and vote yes/no on incorporating each one as an ammendment. How would this procedure work if we tried to apply it to elections?

Presume, as before, that there are three candidates, and that the 9 congressmembers opinions break down as in our previous examples:

4 prefer A > B > C
3 prefer C > B > A
2 prefer B > C > A

Three "bills" are introduced, one naming each of the candidates, A B and C, as the ultimate winner of the election. At first, each member only votes "yes" for the bill naming their first-choice candidate. Since no candidate has a majority of first-preference, none of them pass. Not picking a winner is clearly sub-optimal, so we'll have to try something else.

Let's suppose B's supporters get a clever idea: they resubmit the "C wins" bill... but vote in favor of it, along with all of C's supporters. By a one-vote margin the bill passes. This is almost analogous to what would happen in a plurality election or in a run-off; B's supporters compromised and got their second-choice candidate.

Now suppose that B's supporters, feeling a bit cheeky, get even more clever: they submit a motion to change the winner from C to B. Obviously, their allies in the "C wins" vote are against them this time, but A's supporters more than make up for it, and the motion passes. A and C's supporters can try similar tricks, but neither such bill will pass; each opposes the other's motion, and B's supporters oppose both. A temporary compromise (like the B voters voting for C) can shift the field, but if the voters always eventually fall back to their honest opinions, B always comes out on top. I contend that this means that B is, overall, the more prefered candidate, the best compromise among all the options, and that the best vote-counting method would choose them as the winner.

Multiple votes are probably out of the question as a method to replace our current election process; this is why the "instant" in instant runoff is appealing. But IRV would give C the win, and that's only half-way to the solution. But there is a voting method that, like IRV, can determine the winner in one round of ballots, and those ballots are not any more complicated than the ones used in IRV, and it's called Condorcet's Method.

Condorcet's method works by breaking the election down into a group of smaller, simpler, one-on-one elections. With only two choices, voting is easy! So, in a three way race, rather than look at it as a free-for-all of A vs. B vs. C, Condorcet looks at it as three seperate contests: A vs. B, B vs. C, and A vs. C. In our example, we see that B wins against A (since both B and C's supporters prefer B in that case), that B also wins against C (since both B and A's supporters prefer B in that case), and that C wins against A (since both B and C's supporters prefer C in that case). So who wins? Put simply (and somewhat tautologically-sounding), the winner is the candidate who doesn't ever lose. This means that there is no single opponent who a majority of voters would prefer over them. In our case, this is candidate B.

Condorcet's method is a powerfull tool for teasing a compromise out of a vote. It's not perfect (no voting method is, and I'll cover so-called "circular ambiguities" soon), but I think it gives a better result in a greater number of situations than plurality or IRV.