Arrow’s Impossibility Theorem:

1) Do social welfare functions exist?

• Social welfare functions depart beyond the Pareto principle to complete social preferences.

• Applicant for a social welfare function is majority-rule voting. For example, each of us might vote for every pair of alternative allocations, and in each case the allocation with more votes would be deemed preferred. As well as the implied social preferences would be complete.

• There are other mechanisms for evaluating all alternative allocations- would they deliver the same social preferences?

• Arrow’s Impossibility Theorem set up those social alternatives cannot be ranked. Therefore, social welfare functions don’t exist. For instance with majority-voting, social preferences would be complete but intransitive.

2) Conditions and Proof:

• Do social preferences exist? – In special cases, yes. However generally no.

• Arrow’s conditions for social liking are:

(a) Universal Domain - Social preferences are absolute and transitive for all individual preferences that are complete and transitive.
(b) Pareto Principle - If allocation x is favoured to allocation x′ for all individuals, then x is preferred to x′ .
(c) IIA (Independence of Irrelevant Alternatives) - For any two allocations x and x′ , whether x is preferred or vice versa does not depend on the rankings of other alternatives x′′ .
(d) No Dictatorship - No entity i is a dictator in the sense that social preferences are i’s preferences no matter what the preferences of the other individuals.

• Arrow’s Impossibility Theorem - If there are at least three alternative allocations (or social outcomes), then there does not exist social preferences satisfying Universal Domain, the Pareto principle, independence of Irrelevant Alternatives, and No dictatorship.

• This is a powerful rage to social engineers looking to implement ‘what’s best’ for society. It must also frustrate economists working with social welfare functions. So, the only hope for those requiring a social welfare function is to relax or to replace one of Arrow’s assumptions (Is Arrow’s Impossibility Theorem sensitive to its assumptions?).

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