Liberal paradox

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The liberal paradox is a logical paradox advanced by Amartya Sen, building on the work of Kenneth Arrow and his impossibility theorem, which showed that within a system of menu-independent social choice, it is impossible to have both a commitment to "Minimal Liberty", which was defined as the ability to order tuples of choices, and Pareto optimality.

Since this theorem was advanced in 1970, it has attracted a wide body of commentary from philosophers such as James M. Buchanan and Robert Nozick.

The most contentious aspect is, on one hand, to contradict the libertarian notion that the market mechanism is sufficient to produce a Pareto-optimal society—and on the other hand, argue that degrees of choice and freedom, rather than welfare economics, should be the defining trait of that market mechanism. As a result it attracts commentary from both the left and the right of the political spectrum.

The theorem

The formal statement of the theorem is as follows.

Suppose there is a set of social outcomes <math>X</math> with at least two alternatives and there is a group of at least two individuals each with preferences <math>succsim_i</math> over <math>X</math>.

A benign social planner has to choose a single outcome from the set using the information about the individuals' preferences. The planner uses a social choice function. For every possible set of preferences, a social choice function selects a choice <math>x in...
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