Learning Module 6 - Concepts
Before we start that, I'd like to suggest you look at another discussion of voting methods from (I think!) Washington State University. This site covers Plurality, Run-off (not quite the same as Instant Run-Off; in the more common - here in the US - Run-Off method only two candidates survive to the second and final round; in Instant Run-off we eliminate one at a time until one candidate has a majority), and Borda points, as well as looking at the fairness criteria we'll cover this time and giving you some practice in being insincere (sometimes you get closer to what you want by voting for something other than what you most want).
I have also found two articles about voting theory from just before the 2000 presidential election. They cover the various voting systems and make interesting predictions. Have a look, if you like; they also cover some of the material from this section.
Chronicle of Higher Education: "When Votes Don't Add Up"
SIAM News (It's a math magazine - don't be frightened): "Making Sense of Consensus"
The San Francisco Measure A people (see link from Module 5, if you haven't already) are arguing that Instant Run-off is better than Run-off (where only two candidates survive to the second election). Other people might argue that either of these is better than Plurality Winner.
A problem with Plurality Winner is the "splitting the vote" phenomenon. If B and C have roughly the same positions, and 60% of the voters agree with both of them, while A has 40% of the voters, A might win 40-35-25 - since B and C split the majority vote for their joint position. Situations like this work well with either kind of run-off (whichever of B and C gets more votes wins the second round) or with Borda (since B's voters and C's voters both place A last - one of them will defeat A: try some numbers and see). This is a common situation!
The other systems have problems, too. An example in Module 5 Activity shows that changing votes in a candidate's favour might actually LOSE that candidate an election he/she would have WON. Borda's point method is easy to "finesse" with insincere voting - to get your candidate elected, don't rank the candidates in their true order of preference - but put the biggest rival LAST. (Borda himself, when this was pointed out to him, shrugged and said "My method is for honest men" ). And it seems fair that a Condorcet winner - preferred head-to-head over everyone else - should always win the election. However NONE of the methods (except Condorcet's which doesn't always produce a winner) guarantees this.
In the 1950's, American mathematician Kenneth Arrow made a list of requirements for a system to be, in his opinion, democratic. His "fairness criteria" were:
We say a voting system passes one of these criteria if: no matter how the votes are cast, the result according to the voting system in use is the result demanded by the fairness criterion. We say a sytem fails if SOMETIMES the result doesn't match what the fairness criterion demands.
Let's look at these one by one.
A voting system is a dictatorship if there's one voter who always gets his/her way - only that one vote gets counted. These aren't democratic!
None of the systems we're discussing are dictatorships, so all get a passing grade on this one.
The majority criterion: If a majority (remember, a majority is more than half; a plurality is just the largest number, even if not a majority) of voters rank a candidate first, that candidate should always win the election.
Plurality Winner and all kinds of Run-off pass this test - someone with a majority wins right away. However, the Borda count does NOT pass this test: Candidate A gets 10 first place votes and 9 fourth place votes (out of 4). Candidate B gets 10 second place votes and 9 first place votes. A gets 30 points (3 points each for 10 first place votes, nothing for the last place votes); B gets 47 ( 2 points x 10 for the seconds; 3 points x 9 for the firsts). It's not even close!
Remember, a Condorcet winner is a candidate who would win a mini-election against EVERY other single candidate. There is no guarantee that a Condorcet winner exists.
A voting system passes the Condorcet Criterion if whenever there is a Condorcet winner, that candidate always wins the election.
Sadly, most of our systems fail the Condorcet criterion - which means we can make up examples where there is a Condorcet winner but that candidate doesn't win the election under Plurality, or under Borda, or under instant run-off.
This might be the most obvious one of all. It says: if voters change their mind and rank a candidate higher than they used to, that shouldn't HURT the candidate. Or, put another way: if we run an election and candidate X wins, and then re-run the election with the same votes except that one or more voters move X (the winner) UP their preference list - then X should still win.
Instant Run-Off fails the monotonicity criterion, because changing the votes can lead to different candidates being eliminated - and the final (2 candidate) showdown might have a different pair of candidates and a different result. We saw this in Module 5, Activity. Borda and Plurality satisfy the monotonicity criterion.
Independence of Irrelevant Alternatives (IIE)
This one's a little odd. Suppose we run the election by our system (whichever one we're using at the time), and candidate A wins. Then we realise that candidate B was not a legal candidate - it doesn't seem to matter, B didn't win. But we re-run the election with the SAME preferences except that B (the "irrelevant alternative") is removed. But wait - candidate C wins this election! If our system chose A over B and C and... before, why should it choose C over A, when no votes are changed? That seems wrong!
That scenario is a violation of IIE. Adding, or removing, a non-winning candidate shouldn't change the results. A real-life example was the figure skating one from the Chronicle of Higher Education article (linked above and here too) where Michelle Kwan skated so well she came 4th - and switched the positions of the previously 2nd and 3rd skaters! This is of course ridiculous - and it's an example of a violation of IIE.
Borda, plurality, and instant run-off can all sometimes fail IIE!
All of Arrow's requirements seem reasonable. When he wrote them down in 1952 he spent some time playing around trying to create a voting system that NEVER broke any of these rules. He had no luck. After thinking about why, he ended up proving his theorem (and winning the Nobel Prize in Economics. There is no Nobel Prize in Math). Arrow's theorem says No voting system that satisfies Arrow's conditions will ever be discovered! So we have to put up with some of the rules above being broken, no matter what system is chosen. It also means - when we have elections - we need to agree on a system and that system will have flaws!