To American voters, it’s an all-too familiar dilemma: do you cast your lot with the candidate most likely to win, or risk spoiling the election by supporting the third-party candidate in whom you actually believe? What if, instead of choosing one candidate, voters were instead given the opportunity to rate each potential office-holder, in the same way that Olympic judges score athletes? Brian Dunning at Skeptoid takes an interesting look at the mathematics of voting systems:
In the 1969 film Putney Swope, members of the board of executives were prohibited from voting for themselves, so they all voted for the one board member they were sure nobody else would vote for. Ergo, this free, democratic election produced a chairman that no voter wanted.
In a perfect democracy, everyone gets an equal opportunity to vote, and equal representation. Therefore, we hold elections to let everyone have their say, to either vote representatives into office, or to enact certain laws. It’s a fine idea, and most countries do their level best to implement such systems. Some voters take advantage of it, and some choose apathy and don’t vote. Some try to anticipate what other voters might do, and cast a vote in an unexpected direction not to vote for a candidate, but to affect another candidate’s chances. This is what went wrong in Putney Swope: each voter cast a throwaway vote hoping to improve his own chances. In most elections, everyone has the right to do any of these things; the election is theirs, and theirs to decide. But what many of them might not know is that virtually any electoral process is flawed. Some outcomes are surprising. There are a number of different circumstances in which the candidate most desired does not win.
Democratic voting is only simple if there are just two candidates, or if it’s a Yes or No vote. In those cases, any attempt to vote tactically or to create a voting block — casting votes that don’t represent your preference — work against you. What we’re talking about today are elections where there are three or more candidates. And the idea that all the various systems for running such elections are flawed (subject to results that do not represent the group’s preference) is not just a whim or a crazy opinion of mine. It’s proven by Arrow’s Impossibility Theorem, named for the economist Kenneth Arrow, winner of the 1972 Nobel Prize in economics and the 2004 National Medal of Science. He proved it in 1951 with his Ph.D. thesis at Columbia University.
Arrow’s theorem can be simplified into one clear statement: that no fair voting system exists when there are three or more candidates. To this I ask: What do you mean by fair? That’s the key to Arrow’s theorem. It holds true, depending on a rigid definition of fair that must satisfy three criteria:
1. If every individual prefers X to Y, then the group prefers X to Y.
2. If every voter’s preference of X over Y stays the same, then the group’s preference of X to Y stays the same, even if other preferences change: such as Y to Z, or Z to X.
3. There can be no dictator, as Arrow called him; a single voter with the power to dictate the group’s preference.
Arrow’s theorem applies to election systems that require voters to rank the candidates. This is the case with most voting systems worldwide. Typically, when you vote, you mark an X in the box for one candidate. That’s a ranking; you’ve ranked that candidate first. Arrow’s theorem applies to these simple ranking systems, but its richest mathematical complexities come from systems with three or more candidates and the voters rank all candidates in order of preference. This isn’t used in many real-world elections, but it’s the theoretical basis for social choice theory.
[Read the rest at Skeptoid.com]