Arrow’s impossibility theorem

The conditions introduced by Arrow do not concern a particular class of aggregation functions. Their definitions never necessitate a condition of transitivity or other collec­tive rationality property.

Condition U (universality) A profile π may include any individual complete preorder.

This means that the individual preferences are not restricted; they are complete preor­ders, but no extra rationality conditions are postulated, as it is the case, for instance, with Black’s single-peakedness. For instance, with three social states, there are 13 complete preorders. Each of these 13 is feasible.

Condition I (independence of irrelevant alternatives) Consider two social states a and b and two profiles π¹ and π². If, for each individual, the preference regarding a and b is the same in profile π¹ and profile π², then the social preference regarding a and b must be identical for both profiles.

This means that the information used in the aggregation is myopic, and, given the definition in terms of preferences, ordinal. For instance, when there is a set of ten social states, the fact that individual j ranks a first and b tenth will have the same effect as if she had ranked a first and b second. Also, one should remark that this condition uses two profiles; it is a multi-profile condition. Since preferences could be represented by ordinal utility functions, this means that the individual utility functions are possibly different so that an Arrovian social welfare function is quite different from a Bergson-Samuelson social welfare function where the utility functions were fixed.

Condition P (Pareto principle) Let a and b be two social states and π be a profile in which every individual prefers a to b (for all i ∈ N, a >_i b); then a is socially preferred to b (a >_s b).

This is simply a unanimity principle. As a consequence, if individuals can either prefer a to b or b to a so that there is a tiny diversity among the feasible profiles (which is the case, of course, under Condition U), then it is impossible that the aggregation function f is a constant function. Are functions then excluded that would be based on a moral or religious code, independently of the individuals constituting the society, since such func­tions will always generate the same social preference whatever the profiles of individual preferences are?

A dictator would be an individual i such that for any social states x and y, the aggrega­tion function would generate x >_s y whenever x >_i y. One can see that a dictator imposes his strict preferences on the society - in Arrow’s framework, the dictator does not impose his indifferences.

Condition D (no-dictatorship) There is no dictator.

Arrow’s impossibility theorem If N is finite and includes at least two individuals, and if there are at least three social states, there is no social welfare function satisfying condi­tions U, I, P and D.

Dishonest comments were made about the impossibility of democracy. First, the theorem is about transitivity, whose violation entails that there are three social states. With only two, there is no problem. Furthermore, we can challenge the conditions. Condition U was in fact challenged in Black (1948), that is, before the publication of Arrow’s first paper. But it is condition I that has been contested most often - among others by Sen (1970b) and by Saari (1995) - mainly because it excludes intra- and inter-personal comparisons of the intensities of preferences. We can also consider a weakening of the condition of transitivity to quasi-transitivity of the social prefer­ence. In this case, that is, for QT-social decision function, Gibbard showed that if the function satisfied conditions U, I and P, we obtained an oligarchy. An oligarchy is a group of individuals who have the power of a dictator when they agree and where each member has a veto power (that is, loosely speaking, a power sufficient to pre­clude that the social preference be the inverse of his preference). If the number of oli­garchs is small, we are not so far from dictatorship. As was shown by Mas-Colell and Sonnenschein (1972), there is not much to gain from a further weakening to acyclicity of the asymmetric part of the social preference, that is, in considering social decision functions.

Independently of the beauty of Arrow’s result, it is most remarkable that it was devel­oped in an elegant framework that became the framework of the whole subject (for the origin of this framework, see Suppes 2005).