Before we start, I'd like to thank listener "Cvithlani" for posting another nice review on iTunes. This probably helped Math Mutation get back in iTunes's Science & Medicine "What's Hot" podcast list in the past few weeks. So keep those reviews coming!
Anyway, another election day has come and gone, and this reminded me that I haven't yet discussed one of the election-related topics on my to-do list, Arrow's Theorem. This theorem was first proven by economist Kenneth Arrow in 1951, as part of his Ph.D. work-- just the start of a long career that later won him a Nobel Prize. This is a theorem that basically says that according to some common criteria that we should use to define a fair voting system, no rank-order voting system can ever meet those criteria. In some ways it can be considered an extension of Condorcet's voting paradox, which I described back in podcast 183. I think the easiest way to introduce Arrow's Theorem is through an anecdote I read in the Cafe Hayek blog by Don Boudreaux.
"You walk into an ice-cream store and ask what flavors are available today. The clerk says 'We’ve got vanilla and strawberry.' You ponder for a moment and tell the clerk 'I’ll have strawberry.' Just before the clerk starts to scoop out your strawberry ice cream, he turns to you and says, 'Oh, I almost forgot. We also have pistachio.' In response, you ponder for another second and then tell the clerk, 'Well, in that case, I’ll have vanilla.'
Seems pretty absurd, right? Somehow the availability of a flavor you don't like changes your first-choice selection? But when we're talking about voting, our society really does choose like this. In the most recent election, people point to the Virginia governor's race, where many believe that Republican Cuccinelli only lost to Democrat McAuliffe because of the presence of 'spoiler' Libertarian candidate Robert Sarvis. McAuliffe won the election, but if Sarvis had not been running, then Cuccinelli would have won. Even if you disagree in this instance, articles on the spoiler effect appear during pretty much every election season in our country.
This spoiler effect is related to one of the criteria for fair voting in Arrow's Theorem, the "Independence of Irrelevant Alternatives"-- the idea that if you prefer X over Y, your feelings about some third alterative Z should not change that. There are three other criteria in the theorem. First, "non-dictatorship": no single voter should be able to decide the outcome. Second, "universality": for any set of votes, the system must provide a complete, deterministic ranking of the society's preferences as a whole. Third, "unanimity": if every individual prefers one choice over another, then so must the society as a whole. The theorem then states that if you have at least two voters and at least three options to decide among, it will never be able to meet all of these four fairness criteria. Proving this is a little messy for an audio podcast, but you can see an outline of a proof at the Wikipedia page in the show notes.
As you would expect, Arrow's theorem has led to lots of discussions about how to improve democratic voting. One simple way is to cheat and just relax one of the fairness criteria-- in fact, this has largely been done in practice, as we do have voting systems in many countries, including the United States, that do allow the spoiler effect & violate the independence of irrelevant alternatives. Another method is to always limit votes to two alternatives, since then we can have a 'fair' system according to the Arrow criteria-- but unfortunately if we divide a larger group of alternatives into pairs to try to use this method, then we find the collective choice using these multiple pairwise votes is in effect a larger tournament that is subject to Arrow-like problems. The order of pairing can have a big influence on the ultimate winner. There are also systems not based on rank order, for example 'Range Voting', giving a score to each candidate instead of a simple rank order and adding the populations's total ratings: this has its own problems though, sometimes giving a result close to society's average judgement but disagreeing with the true majority choice.
Some mathematicians have also pointed out that if you drop the assumption of finitely many voters, Arrow's theorem can be fixed, but I think there might be a few other problems if we increased our birth rate to infinity just to fix our voting system. Plus, until they reached voting age, we wouldn't be able to fairly elect a school board to oversee the education of our infinite number of children anyway, so they would not grow up to be informed voters.
All these improved voting systems, or at least the set of them that are actually possible in real life, suffer the disadvantage that they make voting more complicated in general. Given the contentiousness and error rates we have now regarding the simple problem of counting direct votes, I think that is likely a fatal flaw. At some level, we just have to recognize that any system of governance will have its inherent flaws, and that we just need to learn to cope with the fact that life is never 100% fair. And the fact that after every election, advocates for the losing side will come out and declare society fatally flawed, is just one of the prices we pay for the public having some kind of influence on its government-- every fix they advocate will result in some other form of unfairness.
Personally, I still say the best system would be the one I proposed in podcast 172, where we simply admit at some point that the election system is uncertain, and roll a die to help randomly determine the final result, with probabilities determined by the votes counted so far. Or at least, that would be the best stand-in until society gains the wisdom to make me the supreme dictator and king.
And this has been your math mutation for today.