In Burlington, Vermont there was a direct election. The election did not use electors: candidates were directly elected. Who wins? Why, the person who got less votes than the plurality candidate and was less popular head to head than other candidates. How does this happen?

Welcome to Arrow’s Impossibility Theorm. It is mathematical proof that democratic elections will always have flaws, or, more snarkily, mathematical proof that democracy doesn’t work.

In Burlington, Vermont in 2009, they held an election by instant runoff voting. Republican Kurt Wright won a plurality of votes. Democrat Montroll won the Condorcet vote. Bob Kiss came in second place by both measures and won.

You probably know what a plurality is, but are less likely to know what Condorcet means. Condorcet is a comparison between every potential pairing. In a race between A, B and C, if A is preferred to B in a head to head race and A is preferred to C is a head to head race, A wins a Condorcet.

What happened: Montroll was preferred to Kiss in a majority of ballots and preferred to Wright in a majority of ballots. Wright got more votes as first choice than Kiss or Montroll. There were 2 other minor candidates, too. The people who liked the minor candidates liked Kiss as their second choice, so when they were eliminated in instant runoff voting rounds, Kiss ended up with more votes than Montroll and Monstroll was eliminated. Montroll’s supporters preferred Kiss to Wright so Kiss ended up with a majority in the last round.

So here is your new hero, Bob Kiss. He was second place by every measure and ended up winning.

Now, maybe you think I am saying instant runoff voting doesn’t work. Au contraire! No form of democracy works and there is mathematical proof called Arrow’s Impossibility Theorm. The theorm posits that perfect democratic elections would have 3 properties:

1. In every election where more voters prefer candidate X to candidate Y, X would win over Y.

2. Introducing candidate Z where X is preferred to Y never causes Y to be preferred to X as a mathematical artifact.

3. No single voter can force a particular outcome. Only a group of 2 or more voters of like preferences can cause a candidate to win.

This has been proven to be impossible by economist Kenneth Arrow. So now you can say to anyone who extolls democracy that Kenneth Arrow mathematically proved that democracy doesn’t work and point to Burlington, Vermont as a real world example.

[BTW: I am being tongue in cheek. This case is unusual and all voting systems will have problems. IRV normally works smoothly. I prefer democratic elections to all other ways of selecting government officials and I prefer instant runoff voting to what we have now.]