OR, The Mathematical Case Against Representational Democracy

Spurred on by what I read over here, in regards to Vermont’s Governor recently vetoing a Bill that would have made Instant Runoff Voting (IRV) the methodology of choosing certain elected posts in our State, I’d like to bother you with the inconvenient fact that it is mathematically impossible for elections (for representational democracy) to be fair, free, consistent, equitable, and ultimately, for it to work.

Voting, we’re told, is the path to freedom; it is equality, it is our right and our duty.  Elections, referendums, budgets, we just friggin love to mark a piece of paper and forfit our power over to others.  However, what is rationally, logically true is that elections fail entirely to provide us with a fair, consistent method for making a choice.  “Democracy” then must look like something entirely different than elections.  Before we look at that, lets be sure we’re on the same page as to the how and why of electoral failings.


The Boring Thoery Stuff You Need to Know

It would seem that few- if any- would suggest that there’s anything fair or right about a system in which no more than two choices are allowed to be put to the voters at one time.  Even among those who decry third party election candidates, they do so usually by insisting that one or the other choice has the best shot of winning against the more evil other candidate.  We could probably even say that to make it law that only two candidates (or two possible budgets, or two possible stadium locations, etc) can appear on a ballot is just plain dictatorship.  Freedom and accessibility to getting on to the ballot seems just as important to “democracy” as actually being allowed to vote.

Often (except here in the U.S.) voting is undertaken using what is known as a preference ballot; meaning, rather than just putting down your first choice, you actually rank all the choices in order of preference.  This allows us to figure a winner beyond just relying on whoever receives the plurality (the most first place votes).  This is important, as we’ll see bellow, because first place votes can end up telling us very little of who is actually the preferred choice of a majority of voters.  For example, though choice A may receive 40 first place votes, and choice B & C each 30 votes, without a system of ranking (preferences) we ignore the possibility (or fact) that the voters choosing B or C could easily prefer one or the other over choice A, who we’ll suppose is despised by everyone except the 40 who voted for him.  This scenario dictates that someone (or thing) that is not supported by a majority could easily win, because in order to defeat him/it, the supporters of B or C would have to either have an opportunity to rank their preferences or vote for the other candidate that is not their favorite (when we vote for someone or something that is not our favorite in order to prevent a greater evil from winning, we can hardly be said to be voting in a free and fair way: we have fallen victim to lesser-evilism, and therefore our democratic will is being suppressed by the whims of the voting system itself).

But in the 1940’s, the mathematical economist Kenneth Arrow came across a curious fact: whenever there are more than two choices on a ballot, there is no consistent, fair, democratic method for arriving at the winner.  This became known as Arrow’s Impossibility Theorem, and in 1972 he was awarded a Nobel Prize in Economics for it.  In order to understand why and how this is, we need to know about the four major points which help us define what we mean by “fair and democratic” elections.  These are known to mathematicians as the Fairness Criteria.  We’ll look at some examples after the definitions to help illustrate what these look like in the real world.

The Fairness Criteria:

Majority Criterion.  Simply put, if there is a choice on the ballot that is the selection of a majority of people then that choice should win.

Condorcet Criterion.  If there is a choice that is preferred by the majority over the other choices, that one should win.

Monotonicity Criterion.  If choice X wins an election, and in a re-election all the changes made to the ballot are favorable to X, then X should still win after the re-election.

Independence-of-Irrelevant-Alternatives Criterion.  If choice X wins, and then one or more of the other (losing) choices drops out, choice X should still win.

Before moving on to some examples, we also need to remember one important thing about preference ballots: the transitivity of individual preferences, which is fancy math-geek talk which simply points out that if a voter prefers choice A over B, and choice B over C, than we can take this to also mean the voter prefers choice A over C.  This will be important to us, because if we’re trying to figure out if we’ve accomplished a fair, consistent, “democratic” election we’ll have to be sure that the ballots pick the choice that the voter wants or prefers. 


Some Examples of The Most Common Election Methods And How They Fail

The first is the obvious: Plurality Method.  This is how we vote here in the U.S. (with a couple of local exceptions).  Plurality Method simply states that the choice with the most first place votes wins.  Lets say we have an election and there are 4 choices on the ballot.  Lets say the results look like this:

A gets 14 first place votes

B gets 4 first place votes

C gets 11 first place votes

D gets 8 first place votes

Using the plurality method, the obvious winner is choice A, who got more first place votes than any other choice.  This is so popular and wide-spread not only because it is simple and straight-forward, but also because it seems to be an extension of the idea of majority rule, namely, whoever gets the most votes in an election between two candidates is the winner (50% + 1).  But if we look at the above results, we see that there are 37 votes cast, and so 19 first-place votes would be needed to arrive at majority.

In the above example of plurality voting, we can see that we have not violated the Majority Criterion because the choice with the most first place votes wins.  However, what we’ve failed to do is to take into account the preferences of the voters; notice that while choice A does get the most first place votes, the majority actually voted for someone else, meaning, a majority of voters preferred someone other than the winner.  If the above election had used a preference schedule, lets say the expanded results looked like this:

A gets 14 first place votes, 0 second place votes

B gets 4 first place votes, 8 second place votes

C gets 11 first place votes, 27 second place votes

D gets 8 first place votes, 2 second place votes

In this example, while choice A does win using the plurality method, we can see that choice A is in no way the overall favorite: choice C is in fact the overwhelming preference of the voters.  Yet choice C does not win using the plurality method.  Because of this, the plurality method is in violation of the Condorcet Criterion (the choice preferred by the majority of voters does not win).  So the plurality method fails to produce a fair outcome.  Perhaps there’s another way to hold this election?  A few of the other more popular voting methods:

The Borda Count was developed by a French military man in the late 1700’s.  Apparently, this guy had little more to do with his time than devise complicated mathematical solutions to everyday problems.  His idea for how to have fair elections:  assign points to each ranking on a preference schedule: 1 point for first place votes, 2 points for second place votes, 3 points for third place votes (etc).  The choice that receives the least amount of points in this system wins, as it will clearly be the choice preferred by the voters.  By figuring the winner in this way, Borda reaches the best compromise choice on the ballot by figuring out who (or what) is the most preferred choice on the ballot- he completely satisfies the Condorcet Criterion.  But suppose we hold a vote in which the preferance ballots return looking like this:

# of voters-  6       2       3

1st choice      A      B       C     

2nd choice     B      C       D   

3rd choice     C       D      B  

4th choice     D       A      A    

If we used the Borda Count here, we find that choice A gets 26 points (6 for first place, and 20 for forth place), choice B gets 23 points, choice C gets 25 points, and choice D gets 36 points.  So, choice B is the winner! with 23 points it is clearly the choice preferred by a majority of voters.  However, what Mr. Borda and his method fail to consider is that with 11 voters (6+2+3) choice A actually received a majority (6 votes, which is obviously more than 50% of 11).  So the Borda Count fails at a crucial point: it does not necessarily select the choice that actually wins, i.e., it is a violation of the Majority Criterion).  Also, the Borda Count violates the Condorcet Criterion, since it is true that a plurality does not imply a majority, but a majority does imply a plurality (i.e., if one has a plurality of votes that doesn’t mean one has a majority; however, if one does have a majority than one also must have a plurality).  More fancy talk, simply meaning if you violate the Majority Criterion you automatically violate the Condorcet Criterion at the same time.

A bit more complicated is Copeland’s Method, also known as the method of pairwise comparisons.  In this headache, each choice is independently compared to each other choice, head-to-head.  A point is awarded to each time in which one choice is favored over another, and you go through this until every possible head-to-head choice is accounted for.  Whoever (or whatever) has the most points, wins.  Quickly, if I mark my ballot, in order of preference, like this:

Choice A

Choice B

Choice C

Choice D

It would be tallied like so: A vs B- 1 point for A; A vs C- 1 point for A; A vs D- 1 point for A; B vs C- 1 point for B; B vs D- 1 point for B; C vs D- 1 point for C…. so in the end A gets 3 points, B gets 2, C gets 1, and D none.  You go through this with every friggin’ ballot and eventually one of the choices is a winner.  It’s highly involved and if practiced on a mass scale inevitably expensive (in terms of money and time).  Without wasting a bunch of my time by drawing up an intricate example, I’ll just say that it’s quite easy for one of the choices to drop out (either during or immediately after the vote).  This situation throws the whole count off and it would have to be re-calculated.  When this happens, it is not at all uncommon for the choice that was the winner to suddenly find that it is now the loser, which is a violation of the Independence of Irrelevant Alternatives Criterion (a fair election will be a consistent election).

Finally, there’s the Plurality With Elimination Method (popularly known as Instant Runoff Voting, or IRV).  Using this method, the voter ranks their choice in a preference ballot.  If no choice receives a majority then the last place vote-getter is eliminated; their second, third (etc) choices are then re-assigned to the other choices (so the voter’s ballot isn’t thrown out, just their vote for that losing choice- the voter’s second choice then becomes their first choice, etc).  This process repeats until one of the choices has a majority.  IRV ends up violating, very subtly, the Monotonicity Criterion.  Imagine we take a straw pole right before the election and the results look like this:

Number of Voters     7    8    10    4

First Choice              A    B      C     A

Second Choice         B    C      A     C

Third Choice            C    A      B     B

When we go through the steps of IRV, we find that choice C eventually wins.  Now, though this straw pole was supposed to be a secret, the results get leaked.  Because everyone loves to be on the winning team, several voters switch their vote when the election comes and instead of those 4 people who voted for choice A as their first choice, they vote for choice C (bandwagon jumpers- we can only assume they’re Yankee fans!).  Anyway, surely this is good news for choice C, since all that has happened is that the results of the election now give it more first place votes.  However, choice C still didn’t win a majority, and so IRV does its thing.  However, if you go through the steps (now with the 4 voters in that far right column joining the 10 voters immediately to its left) you find that suddenly choice B has won!  Because choice C loses although the only changes were in it’s favor, we’ve discovered that we’ve still failed to find a consistent, fair method for choosing the winner.

In the end, IRV is most certainly the least egregious violator of the Fairness Criteria.  For this reason, it is (and should be) preferred as the most fair, consistent way to hold an election.  However, as my friend Dave is keen on saying: being one pace ahead of a slow runner does not make one a fast runner; being the least unfair method does not make IRV a fair method unto itself.


Participation, Not Representation

The logical conclusion, that we must make if we are rational (or post-rational, to use some Integral parlance) people, is that representational “democracy”, electoralism, is not a consistent, fair, “democratic” in the popular sense, method for organizing society.  This is consistent with people (like myself) who follow that arbitrary, disproportionate concentrations of power (socially, economically, etc) are likewise irrational (pre-rational) and therefore undesirable.  As the above demonstrates, the very act of forfeiting one’s inherent self-authority over to a another body is illogical, in that the math insists it is not possible to do so in a manner that is “fair” and “free of outside restraint”.

Now, to be entirely sure, I am not arguing or even meaning to intend that these facts preclude the occasional necessity or desirability of elections.  But such acts should be confined to those rare instances when it is more rational to take such a measure; surely, the governance of our whole society cannot be thought of as occasional.  It will only be when each agent is free to participate directly in the decisions and issues which effect them that we have a manner of governance that is consistent with a rational, and eventually post-rational, world.