Arrow’s impossibility theorem

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In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives (options), no rank order voting 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. These pre-specified criteria are called unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of election theory as it is further interpreted by the Gibbard–Satterthwaite theorem.

The theorem is named after economist Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled “A Difficulty in the Concept of Social Welfare”.[1]

In short, the theorem states that no rank-order voting system can be designed that 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.

Voting systems that use cardinal utility (which conveys more information than rank orders; see the subsection discussing the cardinal utility approach to overcoming the negative conclusion) are not covered by the theorem.[2] The theorem can also be sidestepped by weakening the notion of independence. Arrow rejected cardinal utility as a meaningful tool for expressing social welfare,[3] and so focused his theorem on preference rankings.

The axiomatic approach Arrow adopted can treat all conceivable rules (that are based on preferences) within one unified framework. In that sense, the approach is qualitatively different from the earlier one in voting theory, in which rules were investigated one by one. One can therefore say that the contemporary paradigm of social choice theory started from this theorem

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