25/02/2015
A straightforward introduction to the impossibility theorem of Arrow
The basic engagement of social choice with which Arrow was concerned involved evaluating and choosing from the set of available social states (x, y, …), with each x, y, etc., describing what is happening to the individuals and the society in the respective states of affairs. Arrow was concerned with arriving at an aggregate “social ranking” R defined over the set of potentially available social states x, y, etc. With his democratic commitment, the basis of the social ranking R is taken to be the collection of individual rankings {Ri}, with any Ri standing for person I’s preference ranking over the alternative social states open for social choice. It is this functional relation that Kenneth Arrow calls the “social welfare function”. Given any set of individual preferences, the social welfare function determines a particular aggregate social ranking R.
That there could be problems of consistency in voting rules was demonstrated by the Marquis de Condercet in eighteenth century. It is useful to recollect how the problem comes about, for example, for the method of majority of decision.
In majority decisions, x defeats y, which defeats z, which in turn defeats x. The R generated by majority rule violates transitivity and ever weaker conditions of consistency than that. And since each alternative is defeated by another available alternative in the available set, there is no majority winner for the available set {x, y, z}.
Majority rule is of course a very special rule, though highly appealing. Arrow’s theorem, among other things, generalizes the problem for any voting rule, and indeed it does much more than that.
Consider now the following set of axioms, which are motivated by Arrow’s original axioms but are in fact somewhat simpler which, taken together, are nevertheless adequate for the impossibility theorem.
• U (unrestricted domain): For any logically possible set of individual preferences, there is a social ordering R.
• I (independence of irrelevant alternatives): The social ranking of any pair {x, y} will depend only on the individual ranking of x and y.
• P (Pareto Principle): If everyone prefers any x to any y, then x is socially preferred to y.
• D (Nondictatorship): There is no person i such that whenever this person prefers any x to any y, no matter what others prefer.
The General Possibility Theorem: If there are at least three distinct social states and a finite number of individuals, then no social welfare function can satisfy U, I, D, and P.
One common way of putting this result is that a social welfare function that satisfies unrestricted domain, independence, and Pareto Principle has to be dictatorial. This is a repugnant conclusion meaning from a collection of reasonable looking axioms.
