Arrow Possibility Theorem

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Arrow Possibility Theorem

The Arrow (im)possibility theorem can be viewed as a generalization of the Condorcet paradox. Assume that three girls, Ann, Beryl, and Cathy, wish to have dinner together in a restaurant. They have a choice of three restaurants: a Chinese (c ), a French (f ), and an Italian (i ). Ann, Beryl, and Cathy have different preferences and unanimously decide that they will choose the restaurant on the basis of majority rule: a restaurant x will be ranked before a restaurant y if at least two of them prefer x to y. But Ann prefers the Chinese to the French and the French to the Italian (and, of course, the Chinese to the Italian); Ann’s ranking is cfi (c is ranked first, f ranked second, and i ranked third). Beryl prefers the French to the Italian and the Italian to the Chinese (and the French to the Chinese); Beryl’s ranking is fic. Cathy prefers the Italian to the Chinese and the Chinese to the French (and the Italian to the French); Cathy’s ranking is icf. Using majority rule, the Chinese is ranked before the French, the French is ranked before the Italian, and the Italian is ranked before the Chinese (cfic …). A choice is impossible.

For Arrow’s theorem, one considers a set X of alternatives (social states, candidates to an election, etc.) and a set of individuals (in the case of an election, voters), the number of individuals being a positive integer. Individuals have preferences over alternatives. If the number of alternatives is a positive integer (i.e., if X is finite), they rank these alternatives from the most preferred to the least preferred with possible ties. For instance, with three alternatives, one can have a first ranked, a second ranked, and a third ranked (six possibilities), or two alternatives ranked first and the third alternative ranked last (three possibilities), or an alternative ranked first and the other two ranked second (three possibilities), or the three alternatives tied. This makes thirteen possibilities (thirteen complete preorders or weak orderings as they are indifferently known in mathematics). The main question of social choice is to associate a social preference or a choice—an alternative in X or a subset of X —to the individual rankings (one ranking by individual). An Arrovian social welfare function is a rule that associates a social preference that is a weak ordering (i.e., a social ranking if X is finite) to the individual weak orderings (i.e., individual rankings if X is finite). Among these rules, one can consider the rule saying that the social weak ordering is the weak ordering of some specified individual, or the rule saying that the social weak ordering is fixed whatever are the individual weak orderings. To avoid these kinds of rules, Arrow imposes four conditions.

Condition U (Universality). This condition states that the individuals can have any weak ordering (there is no extra rationality condition where some weak orderings could be excluded due, for instance, to some homogeneity in the preferences of the individuals). Consequently, for three alternatives and three individuals, there are 13³ data of individual weak orderings and 13²¹⁹⁷ social welfare functions (10²⁰⁰⁰ is 1 followed by 2,000 zeros!).

Condition I (Independence). The social preference between two alternatives, say a and b, depends only on the individual preferences between a and b. For instance, the social preference between a and b, given five alternatives a, b, c, d, and e, other things being equal, will be the same whether some individual ranks the alternatives abcde or acdeb. Majority rule satisfies this property, but scoring rules where points are attributed to alternatives according to their ranks in the individual rankings and the social preference is based on the sums obtained do not. This excludes, in particular, the standard American and British voting rules and the famous Borda’s rule. (For Borda’s rule, if individuals rank k alternatives without ties, the alternative ranked first in an individual ranking gets k -1 points, the alternative ranked second gets k -2 points, and so forth, and the alternative ranked last gets no point. The social ranking is determined by the sum of points obtained by the alternatives, with the alternative socially ranked first being the alternative whose sum of points is the greatest. Ties are possible in the social ranking; to avoid ties, one can use some tie-breaking device. For instance, if alternatives are election candidates, their age may be used to break a tie.)

Condition P (Pareto Principle). If the individuals unanimously prefer any alternative, say a, to any other alternative, say b, then in the social preference a is preferred to b. This condition, given Condition U, excludes constant rules. In particular, rules that would be imposed by a moral or religious code are excluded.

A dictator is an individual who imposes the “strict” part of his preference, that is, of his weak ordering (he does not impose indifferences—ties); a is socially preferred to b whenever he prefers a to b.

Condition D (Nondictatorship). There is no dictator.

The Theorem can now be stated: If there are at least two individuals and at least three alternatives, there is no social welfare function satisfying Conditions U, I, P, and D. The enormous number of social welfare functions has been “reduced” to none.

A common but somewhat dishonest interpretation is that democracy is impossible or possible only in a two-party system. It is clear that, from a formal point of view, one can challenge Condition U. This is done by restricting individual weak orderings—for instance, the concept of single-peaked preferences introduced by Duncan Black (1958) just does this. One can also challenge Condition I. This has been done by Donald Saari (1995) in his studies on scoring rules.

Kenneth Arrow’s book and papers on this topic are among the most brilliant scientific works of the last century. They were crucial in the introduction of the use of modern logical and mathematical concepts in economics and other social sciences, and were at the origin of a new scientific domain, social choice theory, that is now flourishing at the frontier of economics, political science, mathematics, philosophy, psychology, and sociology.

SEE ALSO Arrow, Kenneth J.; Choice in Economics; Preferences; Social Welfare Functions