Liberal paradox

The liberal paradox is Amartya Sen’s result that unrestricted social preferences cannot always satisfy both Pareto efficiency and minimal individual liberty.

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Social welfare function: A rule that maps individual preference profiles to a social ranking of alternatives. Sen expresses the paradox as an impossibility for rules aggregating individual rankings.
Preference relation: A formal ranking of alternatives that represents which options an agent prefers or regards as equally good. The theorem compares individual rankings with the social ranking they jointly determine.
Arrow's impossibility theorem: Kenneth Arrow’s theorem shows that no social welfare function satisfies a specified set of fairness conditions for unrestricted preferences. Both are impossibility results, but Arrow’s conditions and the source of conflict differ.
Rights: Rights are normative claims or protections that constrain what others or institutions may do. Sen’s formal liberty condition models a limited domain of individual rights within collective choice.
Pareto efficiency: A condition requiring society to prefer an alternative when everyone prefers it. The paradox shows this seemingly modest efficiency condition can conflict with individual liberty.
Social preference: A collective ranking of alternatives that represents how a society compares them. Sen’s conditions constrain this ranking rather than prescribing a particular voting procedure.
Dictatorship: A social-choice rule is dictatorial when one person’s preferences always determine the collective ranking. The paradox is not a claim that minimal liberty makes society dictatorial.
Welfare economics: The branch of economics that evaluates how resource allocation and policy affect social well-being. The paradox challenges the assumption that efficiency and individual choice can always be jointly secured.
Minimal liberalism: A social-choice condition giving at least two individuals decisive control over particular pairs of alternatives. This precise liberty condition is the paradox’s other central requirement.
Decisiveness: A property of a person or group whose preference determines society’s ranking over specified alternatives. Minimal liberalism grants individuals decisiveness over limited pairs, not unrestricted sovereignty.