Arrow's impossibility theorem

In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a pre-specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem.

Quotes

• To put it another way, if we assume everybody should count equally, then there is no method of voting which will avoid some possibility of violating the rules.
  • Kenneth Arrow, in "Interview with Dr. Kenneth Arrow". The Center for Election Science. October 6, 2012

• Contrary to the rationalist followers of the American economist Kenneth Arrow, for whom the instability of majority rule was a problem, Dahl’s Madisonian insight was that instability is actually an advantage. It keeps majorities fluid in ways that stop politics from becoming winner-take-all contests in which losers might as well reach for their guns.
  • Ian Shapiro, "Democracy Man: The Life and Work of Robert A. Dahl", Foreign Affairs (February 12, 2014)

• Arrow’s Impossibility Theorem is quite surprising. It shows that three very plausible and desirable features of a social decision mechanism are inconsistent with democracy: there is no “perfect” way to make social decisions. There is no perfect way to “aggregate” individual preferences to make one social preference. If we want to find a way to aggregate individual preferences to form social preferences, we will have to give up one of the properties of a social decision mechanism described in Arrow’s theorem.
  • Hal Varian, Microeconomics: A Modern Approach, Chapter 33. Welfare