By Gaertner W.
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Additional info for A primer in social choice theory
Condition P (Weak Pareto principle). For any x, y in X , if everyone in society strictly prefers x to y, then xPy. Condition I (Independence of irrelevant alternatives). If for two proﬁles of individual orderings (R1 , . . , Rn ) and (R1 , . . , Rn ), every individual in society has exactly the same preference with respect to any two alternatives x and y, then the social preference with respect to x and y must be the same for the two proﬁles. In other words, if for any pair x, y and for all i, xRi y iff xRi y, and yRi x iff yRi x, then f (R1 , .
Remember that each of the two persons is totally free to map his or her utility scale into another one by a strictly increasing transformation. It is easy to ﬁnd a transformation (there are inﬁnitely many) that maps a1 into b1 and u¯ 1 into u¯ 1 . Similarly, one can ﬁnd another transformation that maps a2 into b2 and u¯ 2 into itself. 6(a) and (b) depict two such transformations. We know that since we are in the framework of ordinal and non-comparable utilities, these transformations do not change the rankings of the two persons.
The present condition is closely related to the axiom of strong neutrality that was discussed in the context of the third proof of Arrow’s impossibility result in the preceding chapter. Remember that individual utility functions and utility proﬁles formed the basis of analysis in the third proof. As a matter of fact, neutrality, the equal treatment of issues or alternatives, is satisﬁed by quite a few decision rules, for instance by the simple majority rule and the absolute majority rule which will be deﬁned in this chapter.