Antag att ett antal kandidater tävlar om att vinna ett val. Då finns det många tänkbara röstregler, enligt vilka röster i ett val leder till ett visst vinstutfall. Man kan tänka sig att den kandidat som får flest röster vinner; man kan också tänka sig att den kandidat som får en majoritet av rösterna vinner; och det finns många andra alternativa regler. En central fråga är: Vilken röstregel är bäst?

Ett sätt att besvara den formulerades av ekonomipristagaren Kenneth Arrow, som i Arrows omöjlighetsteorem ställde upp ett antal axiom som en röstregel bör tillfredsställa. I ”The Arrow Impossibility Theorem: Where Do We Go from Here?” presenterar Arrows tidigare doktorand, ekonomipristagaren Eric Maskin, axiomen:

The first is the requirement that an election be decisive, i.e., that there always be a winner and that there shouldn’t be more than one winner. The second is what an economist would call the Pareto principle and what a political theorist might call the consensus principle: the idea that if all voters rank candidate X above candidate Y and X is on the ballot (so that X is actually available), then we oughtn’t elect Y. The third axiom is the requirement of nondictatorship—no voter should have the power to always get his way. … The final Arrow axiom is called independence of irrelevant alternatives, which in our election context could be renamed “independence of irrelevant candidates.” Suppose that, given the voting rule and voters’ rankings, candidate X ends up the winner of an election. Now look at another situation that is exactly the same except that some other candidate Y—who didn’t win—is no longer on the ballot. Well, candidate Y is, in a sense, “irrelevant;” he didn’t win the election in the first place, and so leaving him off the ballot shouldn’t make any difference. And so, the independence axiom requires that X should still win in this other situation.

Det nedslående resultat av Arrows analys var att ingen röstregel tillfredsställde samtliga axiom. Maskin ställer dock en relevant fråga:

Given that no voting rule satisfies the five axioms all the time, which rule satisfies them most often? In other words, if we can’t achieve the ideal, which voting rule gets us closest to that ideal and maximizes the chance that the properties we want are satisfied?

Svaret:

It turns out that there is a sharp answer to this problem, provided by a “domination theorem.” The theorem can be expressed as follows. Take any voting rule that differs from majority rule, and suppose that it works well for a particular class of rankings. Then, majority rule must also work well for that class. Furthermore, there must be some other class of rankings for which majority rule works well and the voting method we started with does not. In other words, majority rule dominates every other voter rule: whenever another voting rule works well, majority rule must work well too, and there will be cases where majority rule works well and the other voting rule does not.

Ett intressant resultat, och något mer uppiggande än Arrows omöjlighetsresultat.

Se även inläggen ”Teorem och konst” och ”Nationalekonomins intellektuella bidrag”.