By Gaertner W.

ISBN-10: 0199565309

ISBN-13: 9780199565306

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**Sample text**

The conditions of unrestricted domain, anonymity, neutrality, and positive responsiveness are necessary and sufﬁcient for a social aggregation rule to be the simple majority rule. In the sequel, we wish to give a sketch of May’s original proof. In order to do this, we can introduce a somewhat different yet simpler notation which is made to ﬁt the simple majority rule. For each individual i, we associate a variable di that takes the values +1, −1, 0, respectively, when xPi y, yPi x, xIi y. Similarly, for society, we write D = g (d1 , .

If for any two social alternatives x, y ∈ X and two utility proﬁles U and U , both x and y obtain the same n-tuple of utilities in U and U , then RU and RU must coincide on {x, y}. 3). Nor do we want to redeﬁne the weak Pareto condition which, of course, also has the same meaning as before. However, we now introduce a condition called Pareto indifference, which requires that if all members of the society are indifferent between a pair of alternatives, the same should hold for society’s preference over this pair.

Dn ). Since g is a function, the hypothesis that D = +1 therefore leads to a contradiction. The same argumentation can be used to show that D = −1 is not possible either. Thus, the only possibility in the case of N (1) = N (−1) is D = 0. Thirdly, assume that N (1) > N (−1). We want to show that D = g (d1 , . . , dn ) = +1. Let us suppose that N (1) = N (−1) + m, where m is a positive integer with m ≤ |N | − N (−1). Let us ﬁrst assume that m = 1 so that N (1) = N (−1)+1. Because of the latter, there exists at least one dk = +1.

### A primer in social choice theory by Gaertner W.

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