Arrow's impossibility theorem

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Arrow's impossibility theorem, Arrow's theorem, or Arrow's paradox is a statement from social choice theory, named after economist Kenneth Arrow, who first described it in 1950: Suppose there is a vote, and voters have at least three different options to choose from. Each voter will then rank the options according to his or her preference. Arrow said, that in such a case, there is no way to convert these rankings into a community-wide complete and transitive ranking while also meeting certain criteria.

Arrow demonstrated the theorem in his doctoral thesis. He popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare".

In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

  • If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
  • If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
  • There is no "dictator": no single voter possesses the power to always determine the group's preference.

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