By Gaertner W.
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Extra info for A primer in social choice theory
Dn ). We consider another proﬁle (d1 , . . , dn ) with di = di for i = k and dk = 0. For the latter proﬁle, we have N (1) = N (−1) + m and from induction g (d1 , . . , dn ) = +1. Using positive responsiveness, we obtain g (d1 , . . , dn ) = +1. To summarize, in this third step we have shown that if N (1) > N (−1), then D = g (d1 , . . , dn ) = +1. From this and the neutrality condition, we can infer that if N (1) < N (−1), then D = −1. All three steps together deﬁne the simple majority rule, and sufﬁciency is proved.
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.
Then, our argumentation used the ordering property of the social preference relation. What about the independence condition? We arrived at xPz without any information about the preferences of individuals other than person J on alternatives x and z. We have, of course, assumed yPi x and yPi z, but according to condition I , these preferences have no role to play in the social decision between x and z. Therefore, xPz must be the consequence of xPJ z alone, regardless of the other orderings (remember that individual preferences are assumed to be transitive).
A primer in social choice theory by Gaertner W.