impossibility theorem

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Proof that something cannot be done or cannot be had. The most famous such result in politics, due to K. J. Arrow, proves that if a choice or ordering system (such as an electoral procedure) produces results that are transitive and consistent (see economic man), satisfies ‘universal domain’ (that is, works for all possible combinations of individual preference), satisfies the weak Pareto condition, and is independent of irrelevant alternatives, then it is dictatorial. ‘Dictatorial’ here has a technical meaning, namely, that the preferences of one individual may determine the social choice, irrespective of the preferences of any other individuals in the society. A non‐technical interpretation of Arrow's theorem is as follows. In a society, group choices, or group rankings, often have to be made between courses of action or candidates for a post. We would like a good procedure to satisfy some criteria of fairness as well as of logicality. Arrow's startling proof shows that a set of extremely weak such criteria is inconsistent. We would like a good procedure to satisfy not only these but much more besides. But that is logically impossible.

Impossibility theorems save time. For instance, much work by electoral reformers amounts to trying to evade Arrow's theorem. As we know it cannot be done, this removes the need to scrutinize many such schemes in detail. This is not to say that all electoral systems are equally bad, however; there remains an important job for electoral reformers within the limit set by Arrow's and other impossibility theorems.

Subjects: Politics.

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