The Independence of "Irrelevant" Alternatives
It will be remembered that Arrow's Condition 3, the "independence of irrelevant alternatives," is as follows:
"Let, R1,..., R'n and, R'1,..., R'n. be two sets of individual orderings and let C(S) and C′(S) be the corresponding social choice functions. If, for all individuals i and all x and y in a given environment S, xRiy if and only if xR'iy, the C(S) and C′(S) are the same." 1
Of the five conditions offered by Arrow, this one has come in for by far the most criticism. Most efforts to resolve Arrow's paradox and construct (at least in principle) an "acceptable" Social Welfare Function have consequently begun by dropping or modifying this condition.
In this chapter I shall examine what really is involved in Condition 3 and attempt to evaluate its reasonableness as an acceptability condition. In following chapters I shall describe various attempts which have been made to formulate acceptable Social Welfare Functions without imposing Condition 3.
Let us begin by considering what, in the context of Condition 3, is an "irrelevant" alternative, and why it may be reasonable to ask that social orderings be independent of such alternatives. Arrow's own explanation is:____________________