Why can no ranked voting system be perfectly fair?

No ranked voting system can be perfectly fair because, as the economist Kenneth Arrow proved in 1950, every method that turns voters' rankings into a group decision breaks at least one common-sense rule. The rule that always gives way is this one: the choice between A and B should not depend on a third, unrelated option C. In practice, that means every ranked voting system is open to the spoiler effect, where adding or removing a candidate who doesn't win can change who does.

The philosopher Sidney Morgenbesser is said to have captured the absurdity over dessert. Offered apple or blueberry pie, he picks apple. The waitress returns: there's also cherry. "In that case," he says, "I'll have blueberry." From one person it sounds silly. Arrow showed that, for a whole society voting by rankings, this kind of flip can never be ruled out.

The root of the problem was spotted long before Arrow. In the late 18th century the Marquis de Condorcet noticed that majorities can go in circles, like rock-paper-scissors: most voters prefer A to B, most prefer B to C, and yet most prefer C to A, even though each voter on their own is perfectly consistent. Whatever you pick, a majority prefers something else.

That does not make all voting systems equally bad. Spoilers are far more common in simple choose-one voting than in methods that compare candidates head to head, and cycles turn out to be rare in big elections. Systems where voters grade each candidate, instead of ranking them, escape Arrow's theorem altogether, as Arrow himself later admitted.