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Arrow’s impossibility theorem
 

 

The theorem provides a proof that no perfect process exists for aggregating individual rankings of alternatives into collective (or social) ranking. An example of an aggregation process is majority voting but the Condorcet paradox shows how this can fail to produce a useful outcome. A perfect process is defined as one that satisfies a set of desirable axioms. The basis of Arrow’s theorem is a set of axioms that a collective ranking must satisfy. One of several equivalent ways of expressing these axioms is the following. Independence of irrelevant alternatives: adding a new option should not affect the initial ranking of the old options, so the collective ranking over the old options should remain unchanged. Non-dictatorship: the collective ranking should not be determined by the preferences of a single individual. Pareto criterion: if every individual agrees on the ranking of the options, so should society. Hence, the collective ranking should coincide with the common individual ranking. Unrestricted domain: the collective choice method should accommodate any possible individual ranking of options. Transitivity: if option A is preferred to option B and B to C in the social ranking then C cannot be preferred to A. The impossibility theorem proves that there is no aggregation process that simultaneously satisfies these five axioms.

 

Reference: Oxford Press Dictonary of Economics, 5th edt.