Friday, November 30, 2012

What Democracy Looks Like

A while ago I wrote a tongue-in-cheek post about what democracy looks like amid the Occupy Protests.

It was pretty silly. I promise from here on out to only be serious. I swear.

I also posted a while ago about different electoral systems that are in existence, including First Past the Post, Borda Count, and Instant Runoff Voting. The second two electoral systems rely on ranking candidates, while First Past the Post only depends on one candidate chosen per ballot. Because some of the lower rankings can take effect when counting votes, an election with first rank results that would cause a winner in a First Past the Post system may not necessarily cause the same winner in a Borda or Instant Runoff election.

With that being said, this is what Democracy looks like:

This is a ternary plot based on a three-candidate First Past the Post system. The bottom axis ranks the percentage of votes received by the reference candidate, and each other axis ranks the other two candidates, respectively. The colour of each triangle is the chance of that candidate winning, where green means 100% and red means 0%. The First Past the Post plot makes intuitive sense - as long as you have more votes than anyone else, you win, otherwise you lose. This graph is pretty straightforward, so let's move on.

This plot is based on Instant Runoff Voting. What's interesting here is that the middle has opened up a little bit - a candidate could have more votes than anyone else but lose as long as their opponents were close to each other. The outsides of the plot are still similar to the First Past the Post plot, though. An example of this could be a race where the vote is split 49%-46%-5% - First Past the Post would give the win to the first candidate, and almost every time the first candidate would win in an Instant Runoff Vote too, as the second candidate would need all of the third candidate's votes to win (which is unlikely). Once we get to the middle though, the fraction of vote needed by the second place candidate from the losing candidate is less, and so there's a range of voting combinations that give a candidate a chance of winning. This gets even more noticeable in...

The Borda Count! This was the count where you get a certain number of points for each first-place vote, fewer points for a second place vote, etc. Here the boundary is significantly blurred between each candidate's corner - in fact, it is statistically possible for a candidate to win the election even if they only have 16% of the first preference votes, provided their opponents split the difference and hand them a lot of second-round votes. Funky.

Two last fun graphs:

(You may have to click on it to expand it...)

This is the same Instant Runoff graph based on the vote distribution for each race in the 2012 SU elections. In each case the winner is circled with the appropriate vote distribution, at the point in the contest when only three candidate remained in the count. In each case, the winner was the candidate with more votes than any other candidate, but both Andy Cheema and Brent Kelly were located at a point where they had more than a 10% chance of losing still.

Last graph: This is a more visual interpretation of the 2012 Board of Governors race. After NotA gets eliminated, its vote share goes to 0%, which is shown by the arrow pointing to the second circle. In this case it looks like the NotA vote share was mostly split evenly, with a bit of favoritism towards Brent Kelly. As the NotA vote was a sizeable 18%, it could have potentially swung the election towards Rebecca Taylor. Cool visualization, eh?

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