Arrow's Impossibility Theorem - Statement of The Theorem

Statement of The Theorem

The need to aggregate preferences occurs in many different disciplines: in welfare economics, where one attempts to find an economic outcome which would be acceptable and stable; in decision theory, where a person has to make a rational choice based on several criteria; and most naturally in voting systems, which are mechanisms for extracting a decision from a multitude of voters' preferences.

The framework for Arrow's theorem assumes that we need to extract a preference order on a given set of options (outcomes). Each individual in the society (or equivalently, each decision criterion) gives a particular order of preferences on the set of outcomes. We are searching for a ranked voting system, called a social welfare function (preference aggregation rule), which transforms the set of preferences (profile of preferences) into a single global societal preference order. The theorem considers the following properties, assumed to be reasonable requirements of a fair voting method:

Non-dictatorship

The social welfare function should account for the wishes of multiple voters. It cannot simply mimic the preferences of a single voter.

Unrestricted domain

(or universality) For any set of individual voter preferences, the social welfare function should yield a unique and complete ranking of societal choices. Thus:

• It must do so in a manner that results in a complete ranking of preferences for society.
• It must deterministically provide the same ranking each time voters' preferences are presented the same way.

Independence of irrelevant alternatives (IIA)

The social preference between x and y should depend only on the individual preferences between x and y (Pairwise Independence). More generally, changes in individuals' rankings of irrelevant alternatives (ones outside a certain subset) should have no impact on the societal ranking of the subset. (See Remarks below.)

Positive association of social and individual values

(or monotonicity) If any individual modifies his or her preference order by promoting a certain option, then the societal preference order should respond only by promoting that same option or not changing, never by placing it lower than before. An individual should not be able to hurt an option by ranking it higher.

Non-imposition

(or citizen sovereignty) Every possible societal preference order should be achievable by some set of individual preference orders. This means that the social welfare function is surjective: It has an unrestricted target space.

Arrow's theorem says that if the decision-making body has at least two members and at least three options to decide among, then it is impossible to design a social welfare function that satisfies all these conditions at once.

A later (1963) version of Arrow's theorem can be obtained by replacing the monotonicity and non-imposition criteria with:

Pareto efficiency

(or unanimity) If every individual prefers a certain option to another, then so must the resulting societal preference order. This, again, is a demand that the social welfare function will be minimally sensitive to the preference profile.

The later version of this theorem is stronger—has weaker conditions—since monotonicity, non-imposition, and independence of irrelevant alternatives together imply Pareto efficiency, whereas Pareto efficiency and independence of irrelevant alternatives together do not imply monotonicity. (Incidentally, Pareto efficiency on its own implies non-imposition.)

Remarks on IIA

1. The IIA condition can be justified for three reasons (Mas-Colell, Whinston, and Green, 1995, page 794): (i) normative (irrelevant alternatives should not matter), (ii) practical (use of minimal information), and (iii) strategic (providing the right incentives for the truthful revelation of individual preferences). Though the strategic property is conceptually different from IIA, it is closely related.
2. Arrow's death-of-a-candidate example (1963, page 26) suggests that the agenda (the set of feasible alternatives) shrinks from, say, X = {a, b, c} to S = {a, b} because of the death of candidate c. This example is misleading since it can give the reader an impression that IIA is a condition involving two agenda and one profile. The fact is that IIA involves just one agendum ({x, y} in case of Pairwise Independence) but two profiles. If the condition is applied to this confusing example, it requires this: Suppose an aggregation rule satisfying IIA chooses b from the agenda {a, b} when the profile is given by (cab, cba), that is, individual 1 prefers c to a to b, 2 prefers c to b to a. Then, it must still choose b from {a, b} if the profile were, say, (abc, bac) or (acb, bca) or (acb, cba) or (abc, cba).