Arrow’s impossibility theorem

Arrow’s impossibility theorem is a theorem in the context of social choice theory in which is an Impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked 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 It is also kwnon as the General Possibility Theorem or Arrow’s paradox. The theorem is often cited in discussions of voting 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”.

In short, the theorem states that no rank-order voting 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.

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. The theorem can also be sidestepped by weakening the notion of independence. Arrow rejected cardinal utility as a meaningful tool for expressing social welfare, 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.

See also

Game Theory, Social Choice Theory

Papers

• Arrow, Kenneth J. (1950). A Difficulty in the Concept of Social Welfare. Journal of Political Economy 58 (4): 328-346.
• Geanakoplos, John (2005). Three Brief Proofs of Arrow’s Impossibility Theorem. Economic Theory 26 (1): 211-215.
• Tao, Terence (2012). Arrow’s Theorem. cit, 05-24.
• Hansen, Paul (2002). Another Graphical Proof of Arrow’s Impossibility Theorem. The Journal of Economic Education 33 (3): 217-235.

Books

• Arrow, Kenneth J.; Sen, Amartya K.; Suzumura, Kōtarō, eds. (2002). Handbook of social choice and welfare Volume 1. Amsterdam, Netherlands: Elsevier.