It’s well known that the voting methods we use are highly defective, as they fail to meet fundamental criteria like positive responsiveness, the Pareto principle, and stability. Positive responsiveness (monotonicity) means that if a candidate improves on some voters’ ballots, this should not reduce the candidate’s chances of winning. Yet, many voting methods, including runoffs and ranked-choice voting, fail positive responsiveness. In other words, candidates who became more preferred by voters can end up losing when they would have won when they were less preferred! It’s even more shocking that some voting systems can fail the Pareto principle, which simply says that if every voter prefers x to y then the voting system should not rank y above x. Everyone knows that in a democracy a candidate may be elected that the minority ranks below another possible candidate but how many know that there are democratic voting procedures where a candidate may be elected that the majority ranks below another possible candidate or even that democratic voting procedures may elect a candidate that everyone ranks below another possible candidate! That is the failure of the Pareto principle and the chaos results of McKelvey–Schofield show that this should be expected.
Almost all researchers in social choice understand the defects of common voting systems and indeed tend to agree that the most common system, first past the post voting, is probably the most defective! But, as no system is perfect, there has been less consensus on which methods are best. Ranked choice voting, approval voting and the Borda Count all have their proponents. In recent years, however, there has been a swing towards the Borda Count.
Don Saari, for example, whose work on voting has been a revelation, has made strong arguments in favor of the Borda Count. The Borda Count has voter rank the n candidates from most to least preferred and assigns (n-1) points to the candidates with the candidate with the most points winning. For example if there are 3 candidates a voter’s top-ranked candidate gets 2 points, the second ranked candidate gets 1 point and the last ranked candidate 0 points.
The Borda Count satisfies positive responsiveness, the Pareto principle and stability. In addition, Saari points out the Borda Count is the only positional voting system to always rank a Condorcet winner (a candidate who beats every other candidate in pairwise voting) above a Condorcet loser (a candidate who loses to every other candidate in pairwise voting.) In addition, all voting systems are to some extent gameable, but Saari shows that the Borda Count is by some reasonable measures the least or among the least gameable system.
My own early work in voting theory shows, with a somewhat tongue in cheek but practical example, that the Borda Count would have avoided the civil war! I also show that other systems such as cumulative voting or approval voting are highly open to chaos, as illustrated by the fact that under approval voting almost anything could have happened in the Presidential election of 1992, including Ross Perot as President.
One reason the Borda Count performs well is that it uses more information than other systems. If you just use a voter’s first place voters, you are throwing out a lot of information about how a voter ranks second and third candidates. If you just use pairwise votes you are throwing out a lot of information about the entire distribution of voter rankings. When you throw out information the voting system can’t distinguish rational from irrational voters which is one reason why the outcomes of a voting system can look irrational.
Eric Maskin has a very important new contribution to this literature. Arrow’s Independence of Irrelevant Alternatives (IIA) says that if no voters change their rankings of x and y then the social ranking of x and y shouldn’t change. In other words, if no voter changes their ranking of Bush and Gore then the outcome of the election shouldn’t change regardless of how Nader is ranked (for the pedantic I exclude the case where Nader wins.) The motivation for IIA seems reasonable, we don’t want spoilers who split a candidate’s vote allowing a less preferred candidate, even a Condorcet loser to won. But IIA also excludes information about preference intensity from the voting system and throwing out information is rarely a good idea.
What Maskin shows is that it’s possible to keep the desirable properties of IIA while still measuring preference intensity with what he calls modified IIA, although in my view a better name would be middle IIA. Modified or middle IIA says that an alternative z should be irrelevant unless it is in the middle of x and y, e.g. x>z>y. In other words, we allow the voting system to change the ranking of x and y if the ranking of z moves in or out of the middle of x and y but not otherwise (recall IIA would forbid the social ranking of x and y to change if no voter changes their ranking of x and y).
Maskin shows that the Borda Count is the the only voting system which satisfies MIIA and a handful of other desirable and unobjectionable properties. It follows that the Borda Count is the only voting system to both measure preference intensity and to avoid defects such as a spoilers.
The vote wars will probably never end. Indeed, voting theory itself tells us that multi-dimensional choice is always subject to some infirmities. Nevertheless, we can conclude that plurality rule is a very undesirable voting system and the case for the Borda Count is strong.
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