NOTE:  See my important update in the comments section…. there’s a lot going on here!

Almost a year ago I wrote a post called Elections Theory 101 which has proven to be one of the more popular things to come from this blog.  If you haven’t read it, you should, and at least a cursory familiarity with the material from it is required to read this post.  While I extrapolate a big picture conclusion in 101 (that because it is mathematically impossible, therefore by definition illogical, to have a free and fair election whenever there are more than two choices on the ballot- and because making it illegal for more than two choices to be on a ballot would be even more unfair and freedom-hindering- electoralism and representative democracy should be recognized as unjust and inherently unfree and un-democratic) in this post I want to look briefly at the recent mayoral election in Burlington, VT which used Instant Runoff Voting (IRV).  The reason I say that the ideas from 101 are “required” is twofold: First, the average person is generally unaware that there are a number of different methods for voting or holding elections, simple majority or plurality being what a lot of people consider to be “how you vote”.  As the issue of IRV has come-up increasingly throughout the U.S., that is of course a positive but the fact remains that IRV or majority or plurality voting are only a few of the options on the table and no one is going to be able to understand the issue and make an intelligent, informed decision if they don’t know all of the options available.  Second, but most importantly, Arrow’s Impossibility Theorem must be reiterated over and over so that we can all understand: whenever there are more than two choices on the ballot there is no way to hold an entirely consistent, free, and fair vote.  The very best methods (IRV, range voting) have their strong points over the others, but their own faults too.

For instance, IRV (also known as preference voting or STV, Single Transferable Voting) suffers from the following paradoxes:

1. Thwarted-Majorities Paradox, in which a candidate who can defeat every other candidate in head-to-head races may lose the election

2. More-Is-Less Paradox, when if the winner of the election were ranked higher on the ballot by some voters this can cause a different candidate to actually win (this is also known as the Monotonicity Criterion from the Fairness Criterion, which is the standard we are trying to meet to know that we have a consistent, free, and fair election)- this is by far the biggest problem with IRV, I’ll speak more to this bellow.

3. No-Show Paradox, whereas if x’s supporters don’t show up to vote for her/him/it this may actually cause x to win and if x’s supporters do show up to vote she/he/it may lose.

4. Multiple-District Paradox, in which a candidate can win in each district separately yet lose the general election of the combined districts.

For the recently concluded mayoral election in Burlington (which I hashed-through here) some clever math students up at UVM decided to put the results through the ringer and see what really happened, beyond the headlines and the popular analysis.  Hopefully, this was done for honest purposes of applying mathematical concepts to the real-world, rather than any disingenuous political motives (I honestly just don’t know).  And the results aren’t shocking at all, if you are educated about these things: the 2009 mayoral election in Burlington, VT (which was held using the system of voting known as IRV) resulted in a Thwarted-Majorities Paradox.  According to the analysis of voter’s ballots done by UVM, Democrat Andy Montroll was favored over Republican Kurt Wright 56% to 44% and over Progressive Bob Kiss 54% to 46%, despite the fact that in the actual election Montroll came in third and Kiss won in the third round of voting with 51.5%.  According to the preferences indicated by voters’ on their ballots, if Montroll went head-to-head with either Kiss or Wright in a two-person race, he would be mayor.

Now, before IRV detractors get on their high-horses and their soapboxes and throw their “I told you so’s” and their cries for reform or return to the old system it’s important that we understand what exactly this means and what else we know of the election.  The most basic fact is that while it does seem that Montroll would have won head-to-head against any other candidates (we can only assume this extends to the other two candidates who were on the five-person ballot though the UVM work appears to only look at the top three candidates) the fact is that the race was not a two person race and to suggest in any way that democracy, fairness and freedom would be at all served to attempt to enact laws or rules which would prevent more than two people from being on the ballot in the future would be simply Fascist.  So while the method of voting Burlington currently uses (IRV) did in this instance produce this odd little paradox, lets be clear that that is what it is: a paradox.  I can’t imagine Montroll getting very far if he were to start crying that somehow the system had failed him.

One of the nice things about having a preference schedule (a ballot on which voters rank the candidates in order of their preference) is that we know more than if we simply had a check in a box next to one person’s name.  For instance, if this election were carried out using the plurality method (i.e., the way we elect the president) then Kurt Wright would have won because he got the most (first place) votes (35.2%).  But the problem here, of course is that Wright wasn’t preferred by the majority of voters (wasn’t preferred by 64.8% if we look just at that first round of IRV voting with 5 candidates, wasn’t preferred by 60.7% of the voters when the race was down to him, Kiss, and Montroll, as well as wasn’t preferred by 51.5% of the voters when it came down to just him and Kiss and now thanks to UVM we see he wasn’t even preferred by 56% of the voters head-to-head with Montroll).  No matter how the ballots are divided up, Wright was not the preferred choice for a majority of voters and so if Burlington chose its mayor the same we the U.S. chooses its president we would have had an election that violates the Condorcet Criterion, which is a serious affront against our task of holding a consistent, free, and fair election.

Now, before IRV Burlington actually required that a candidate receives over 40% to be declared winner, so if this were the rule the top two vote getters (Wright and Kiss) would have had a run-off election (which would have cost a whole shit load of money for the electorate).  Assuming that everyone that voted the first time came back to vote in the run-off (something that almost never happens- which means, oops, a run-off election disenfranchises voters; again, weren’t we looking for consistent, free, and fair?) we know that Kiss would have won.  However, because we can also assume that voter turn-out in a run-off drops (often heavily) it’s impossible to predict who would actually have benefited or suffered from this.  The best we can do is extrapolate the numbers we have from IRV and assume the results would have been similar, and vow never to use a system that has the horrible effect of disenfranchising voters and thus skewing the results (and leaving our ideas of consistent, free and fair far behind).

Back to that inconvenient little fact- that IRV can, and in this case did, violate monotonicity.  The fact of the matter is that, as a simple by-product that there were more than 2 candidates on the ballot, no matter what method Burlington used to elect its mayor we were bound to find some sort of inconsistency or violation of the Fairness Criterion or Paradox.  The question then becomes (for Burlington, as well as the rest of us) was there a better way? do we feel that the person who was declared ‘winner’ is someone preferred by most voters (either as their choice, or as a “good enough” choice over one or more of the alternatives).  In my opinion, the answer is that this was the best, most consistent, most fair, and most free way to hold the election and the person who won deserved to (not because I may like him- I’m speaking strictly to what voters marked on their ballots).

Of course, we could do away with this whole problem by doing away with government entirely and leading free, liberated lives in which we cooperate socially for the health, safety, happiness and em-betterment of all- but that’s for the voters to decide.