Voting is a Sham! Mathematically Speaking.

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The recent elections remind me of interesting paradoxes when you study the mathematics of voting. I first learned of this class of paradoxes as an undergraduate at Occidental College in Los Angeles (well technically Eagle Rock, emphasis always on the Rock!). As a student, I spent a couple of summers as an instructor for OPTIMO, a science and math enrichment program for kids about to enter high school. You know, that age when young men and women’s minds are keenly focused on mathematics and science. What could go wrong?!

For several weeks, these young kids would stay in dorm rooms during the week and attend classes on a variety of topics. Many of these classes were taught by full professors, while others were taught by us student instructors. Every Friday they’d go home for the weekend and leave us instructors with a little time for rest and relaxation that we mostly used to gossip about the kids. I am convinced that programs like this are the inspiration for reality television shows such as The Real World and The Jersey Shore given the amount of drama these teenage kids could pack in a few short weeks.

But as per the usual, I digress.

So how do you keep the attention of a group of hormonally charged teenagers? I still don’t know, but I gave it the best effort. I was always on the lookout for little math nuggets that defied conventional wisdom. One such problem I ran into was the voting paradox.

Voting is a method that a group of people use to pick the “best choice” out of a set of candidates. It’s pretty obvious, right? When you have two choices, the method of voting is pretty obvious. Majority wins! But when you have more than two choices, things become interesting.

extended-forehead-edition_2Suppose you have a contest for the best (not biggest) forehead between three candidates. I’ll use my former forehead endowed co-authors for this example.

You’ll notice I left out last year’s winner, Rob Conery, to keep the math simple.

Also suppose you have three voters who are asked to rank their choices in order of preference. Let’s take a look at the results. In the following table, the candidates are on the top and the voters are on the left.

Hanselman Haack Guthrie
Mariah Carey 1 3 2
Nicki Minaj 2 1 3
Keith Urban 3 2 1

Deadlock! In this particular case, there is no apparent winner. No matter which candidate you pick, two others voters prefer another candidate to that candidate.

Ok, let’s run this like a typical election where you simply get a vote or a non vote (rather than ranking candidates), but we’ll also add more voters. There will be no hanging chads in this election.

Hanselman Haack Guthrie
Mariah Carey X
Nicki Minaj X
Keith Urban X
Randy Jackson X
Simon Cowell X
Paula Abdul X
Jennifer Lopez X

In this case, Hanselman is the clear winner with three votes, whereas the other two candidates each have two votes. This is how our elections are held today. But note that Hanselman did not win with the majority (over half) of the votes. He won with a plurality. So can we really say he is the choice of the voters when a majority of people prefer someone else to him?

Both of these situations are examples of Condorcet’s Paradox. Condorcet lived in the late 1700s and was a frilly shirt wearing (but who wasn’t back then?) French mathematician philosopher who advocated crazy ideas like public education and equal rights for women and people of all ages.

I see you over there, to my right.

He also studied these interesting voting problems and noted that collective preferences are not transitive but can by cyclic.

Transitive and Nontransitive Relations

For those who failed elementary math, or simply forgot it, it might help to define what we mean by transitive. The transitive relation is a relationship between items in a set that has the following property for every item: if the first item is related to a second in this way, and that second item is related to a third in the same way. The first item is also related to the last item.

The classic example is the relation, “is larger than”. If Hanselman’s forehead is larger than Guthrie’s. And Guthrie’s is larger than mine. Then Hanselman’s must be larger than mine. One way to think of it is that this property transitions from the first element to the last.

But not every relationship is transitive. For example, if you are to my right, and you’re friend is to your right. Your friend isn’t necessarily to my right. She could be to my left if we formed an inward triangle.

Condorcet formalized the idea that group preferences are also non-transitive. If people prefer Hanselman to me. And they prefer me to Guthrie. It does not necessarily mean they will prefer Hanselman to Guthrie. It could be that Guthrie would pull a surprise upset when faced head to head with Hanselman.

Historical Examples

In fact, there are historical examples of this occurring in U.S. presidential elections. This is known as the Spoiler Effect. For example, in the 2000 U.S. election, many contend that Ralph Nader siphoned enough votes from Al Gore to deny him a clear victory. Had Nader not been in the race, Al Gore most likely would have won Florida outright. Of course, Nader is only considered a spoiler if enough voters who who voted for him would have voted for Gore had Nader not been in the race to put Gore above Bush in Florida. Multiple polls indicate that this is the case.

In the interest of bipartisanship, Scientific American has another example that negatively affected Republicans in 1976.

Mathematician and political essayist Piergiorgio Odifreddi of the University of Turin in Italy gives an example: In the 1976 U.S. presidential election, Gerald Ford secured the Republican nomination after a close race with Ronald Reagan, and Jimmy Carter beat Ford in the general election, but polls suggested Reagan would have beaten Carter (as indeed he did in 1980).

Reagan had to wait another four years to become President due to that Ford spoiler.

No party is immune from the implications of mathematics.

Condorcet Method

As part of his writings on the voting paradox, Condorcet came up with the Condorcet criterion.

Aside: I have to assume Condorcet had a different name for the criterion when he formulated it and it was named after him by later mathematicians. After all, what kind of vainglorious person applieshis own name to theorems.

A Condorcet winner is a candidate who would win every election if paired one on one against every other candidate. Going back to the prior example, if Hanselman would beat me in a one-on-one election. And he would beat Guthrie in a one-on-one election, then Hanselman would be the Condorcet winner.

It’s important to note that not every election has a Condorcet winner. This is the paradox that Condorcet noted. But if there is a Condorcet winner, one would hope that the method of voting would choose that winner. Not every voting method makes this guarantee. For example, the voting method that declares that the candidate with the most votes wins fails to meet this criterion if there are more than two candidates.

A voting method that always elect the Condorcet winner, if such a winner exists in the election, satisfies the Condorcet criterion and is a Condorcet method. Wouldn’t it be nice if our elections at least satisfied this criteria?

Arrow’s Impossibility Theorem

It might feel comforting to know methods exist that can choose a Condorcet method. But that feeling is fleeting when you add Arrow’s Impossibility Theorem to the mix.

In an attempt to devise a voting system that would be consistent, fair (according to a set of fairness rules he came up with), and always choose a clear winner, Arrow instead proved it was impossible to do so when there are more than two candidates.

In short, the theorem states that no rank-order voting system can be designed that satisfies these three “fairness” criteria:

  • If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
  • If every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y will also remain unchanged (even if voters’ preferences between other pairs like X and Z, Y and Z, or Z and W change).
  • There is no “dictator”: no single voter possesses the power to always determine the group’s preference.

On one hand, this seems to be an endorsement of the two-party political system we have in the United States. Given only two candidates, the “majority rules” criterion is sufficient to choose the preferred candidate that meets the fairness criteria Arrow proposed.

But of course, politics in real life is so much messier than the nice clean divisions of a math theorem. A voting system can only, at times, choose the most preferred of the options given. But it doesn’t necessarily present us with the best candidates to choose from in the first place.

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23 responses

  1. Avatar for Josh
    Josh November 27th, 2012

    I really like this clip showing the unfairness of the first past the post system:

  2. Avatar for Robert Slaney
    Robert Slaney November 27th, 2012

    Don't know where this fits into your scenarios, but Australia uses the "Two party preferred" solution. It requires that all candidates are ranked from 1 to x.
    If there is no clear majority ( over 50% ), then the candidate with the lowest votes is eliminated. Each vote for that candidate is individually re-assigned to next ranked candidate.
    The process repeats until 1 candidate gains over 50% of the votes.
    It means that it is possible for a candidate to get most most number of votes in the first "round" but lose the final count.
    In your second example with 7 voters/3 candidates, if the 4 votes not for Hanselman both ranked Hanselman 3rd, he would not be elected.
    Yes I realise that it is a tie for the first round loser but expand the example with 9 votes -> 4 hanselman, 3 Haack - 2 Guthrie. Using the Aussie system, Haack would be the successful candidate if both Guthrie's voters ranked Haack second.
    If gets much more complicated when more than 1 candidate can be elected.
    See, if you have a pathological need to find more information!!

  3. Avatar for Rob Conery
    Rob Conery November 27th, 2012

    GREAT article! It reminded me of one of my favorite tomes on cognitive sublimation:
    I think we need more wonkery. I really do :). If only to hit Wikipedia more often so I could try to prove you wrong :):):).
    Silliness aside, thanks for the mental exercisee. Reminds me of "the dialog" in Anathem :)

  4. Avatar for Jon Galloway
    Jon Galloway November 27th, 2012

    Great post.
    Also, wow, is that some fine artwork on the foreheads?!? It's obvious the artist spent at least 15 minutes in GIMP, mostly because he kept cracking himself up while applying the clone brush.
    Also, slightly related - an interesting article in Wired earlier this year suggested that a small, randomly selected group of voters could do a better job than an open popular vote. Sort of like a jury trial is (hopefully) more accurate than a random poll of uninformed people on the street.

  5. Avatar for Bertrand Le Roy
    Bertrand Le Roy November 27th, 2012

    Aha, excellent. Voting systems are so much fun! Our current system is abysmally wrongerest of course, but one should also ask the question of why we are voting. Usually, it's to achieve some sort of representation. Once in office, the candidate is supposed to represent the people who voted. What the article is neglecting to consider in this perspective is whether the set of candidates being voted for does actually achieve representation. They usually don't, as the reason for becoming a candidate already selects you for certain unrepresentative qualities (in the US, that's wealth apparently).
    Here's the thing: the only system (that I know of) that does achieve representation was the one used in ancient Greek democracy: chance. Just do away with elections and pick people at random instead. I wrote this about the idea:

  6. Avatar for John Ludlow
    John Ludlow November 27th, 2012

    Interestingly, this is actually the situation we had in the last election in the UK. Conservatives got the most votes, but didn't get the majority (>50%), which meant we had a hung parliament (I was disappointed to find that the term wasn't literal).
    They had the option of forming a minority government, but the general consensus amongst the press at the time was that this wouldn't be viable. What they actually did was form a coalition government with the 3rd-running party, the Liberal Democrats, which is hilarious because they both come from the opposite ends of the political spectrum (Conservatives are moderate right, while LibDems are far left). The two parties combined had the majority.
    The tragic thing is that, as much of a mess as this coalition is, it's still better than Labour.

  7. Avatar for WilliamBZA
    WilliamBZA November 27th, 2012

    I'd like to see the workings of Arrow's theorem as the great Charles Dodgson spent a great deal of time working on Elections - particularly on voting in situations with more than two candidates and more than two outcomes. He was able to work around cyclical voting conditions in almost all cases (perhaps even all? Can't remember)
    He never got around to publishing his work, but a great deal of it is still available. I mourn when I think of what the modern political landscape would be now if he had completed and published his work.

  8. Avatar for James Hancock
    James Hancock November 27th, 2012

    You're assuming that public education is good for us. It's not, no more kids have access to education than they did immediately before public education became law and no matter how much money you throw into public education the result is diminishing. Not diminishing returns on that investment, because there has been absolutely no return on increased investment in public education. Diminishing as in every year that goes by, public education gets worse compared to private and privately funded education.
    You're also assuming that democracy is desirable. It's not. It results in the rule of the mob. In the US it results in the candidate that gets 30% of the eligible voter's vote. That's right, only ~30% voted for Obama, even less for Romney. There there is no way that you can call that an endorsement by the majority, and even if you do, then all you're doing is enslaving the minority to the will of the majority for 4 years.
    Further democracy's greatest flaw is that as soon as the people recognize that they can vote themselves free stuff, the democracy will collapse. That's exactly what is happening in Europe and indeed in the US.
    As soon as any government system accepts that government is not subject to the same laws of theft and slavery as individuals, all governments become evil, and all governments seal their own ending.
    The desirable governmental system is one of local governance based on base principles of criminal law and NOTHING else. It is only a loose affiliation for protection against foreign aggression and the enforcement of non-aggression (against person or property) by everyone including government that results in a positive system that is sustainable. To do so requires that the government must stay small and strictly limited because only the small and limited government is directly controllable by the people either through force or by walking down the road to a different community.
    Thus I say that yes, the math is bogus and results in an insular government that doesn't actually answer to the people AND the entire concept of voting is a sham that makes is no more free than no true government.

  9. Avatar for Peter Beckley
    Peter Beckley November 27th, 2012

    Dang it, Phil, you made me (re)learn math by tricking me with a well writing, timely article that also disarmed my suspicions by the use of humor! You're a tricky, clever bastard...good job!

  10. Avatar for Matt
    Matt November 27th, 2012

    Brian Dunning did an episode of his (excellent) Skeptoid podcast on the science of voting last year. He covers a lot of the same ground, but it is a good listen in any case.

  11. Avatar for Terje
    Terje November 27th, 2012

    To quote Douglas Adams (from memory): A person capable of making him self president should on no account be allowed to do the job.
    Also: [The job of the (Galactic) President] is not to wield power but to draw attention away from it.

  12. Avatar for Bertrand Le Roy
    Bertrand Le Roy November 27th, 2012

    And... you've attracted the crazy libertarians. Well done ;)

  13. Avatar for November 27th, 2012

    We are trying to address these problems with voting and collective community ignorance with a computational engine that organizes political communities like a brain. And will integrate git to implement line item voting on complex legislation.

  14. Avatar for Dave
    Dave November 27th, 2012

    If only we had a perfect electoral system that would keep all the "crazy libertarians" out of politics--you know, one that makes it impossible for an independent third party to win! Oh wait, that's the system we have.
    Why don't you work out the math for voting your conscience instead of voting for Tweedledum or Tweedledee? The conscience loses every time.

  15. Avatar for Jbone
    Jbone November 27th, 2012

    William Poundstone wrote a book (Gaming the Vote) about this several years back. It presented a history of voting mechanisms and provided arguments in favor of range voting. It's not as though computer scientists and economists aren't actively studying this. Be careful not to interpret Arrow's theorem as an argument against democracy relative to markets.

  16. Avatar for JS
    JS November 27th, 2012

    It's a little late, but just two quick comments:
    1. Dictatorship criterion in Arrow's theorem really is a misnomer, because a person is not really a dictator if there is no intent of his to be. What happens here is that it's not known in advance who will be a "dictator" in a particular vote (it is determined by relative preferences); calling that a dictatorship is just silly. It really seems like a more subtle version of (also a ridiculous argument) calling democracy "dictatorship of the majority".
    2. There are voting systems, such as my favorite Range voting, to which the Arrow theorem does not apply, since they don't just use ordering of preferences, but also their weights. And it is a very simple, obvious system, which has also many other good properties (mostly stemming from the fact that it doesn't throw away any information provided by voters).

  17. Avatar for Radu
    Radu November 28th, 2012

    I love this type of math conundrums that come up every now and then. That being said though,

  18. Avatar for haacked
    haacked November 28th, 2012

    It’s not "name dropping" if these people are your friends and co-authors. :P

  19. Avatar for Mark
    Mark November 30th, 2012

    Awesome post, I guess the U.S. has the edge on Canada where it's more a 3 or 4 party system. We can still beat you in hockey though.
    PS - Put my vote down for Guthrie.

  20. Avatar for Craig Johnson
    Craig Johnson December 7th, 2012

    Range voting rocks. Bayesian regret minimized. Tough to game.

  21. Avatar for Tim Long
    Tim Long December 11th, 2012

    I'm not sure you do in fact have a 'two party system'. If your system is anything like ours in the UK, anyone is free to stand in elections and in fact we regularly have 6 or more parties standing. Nevertheless, two 'main parties' dominate.
    I wonder if two-party domination is itself a side-effect of first-past-the-post voting?

  22. Avatar for sarkari naukri
    sarkari naukri December 15th, 2012

    I feel that nerds should not allowed to stand for voting...Only People above IQ 125 should be allowed

  23. Avatar for Guest
    Guest April 29th, 2013

    testing something... will delete.