By Gaertner W.

ISBN-10: 0199565309

ISBN-13: 9780199565306

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**Example text**

Rn ) implies xP y under (R1 , . . , Rn ). Positive responsiveness considers the effect on the social preference relation when some person k expresses a change in favour of x. Either a strict preference for y over x turns into at least an indifference between the two, or an indifference turns into a strict preference in favour of x, with everyone else’s preference between x and y remaining the same. Positive responsiveness requires for such a case that social preference move in the direction of x.

Let us turn to the sufﬁciency part of the proof. The ﬁrst thing to notice is that since g satisﬁes condition A , the value of g (d1 , . . , dn ) only depends on the number of +1’s, −1’s, and the number of 0’s in the list and not on the positions of the +1’s, −1’s, and 0’s in the list. The number of 0’s, however, is determined by |N | − N (1) − N (−1). Therefore, condition A implies that the value of g (d1 , . . , dn ) entirely depends on N (1) and N (−1). THE SIMPLE MAJORITY RULE 41 Secondly, if N (1) = N (−1), then D = 0.

All this is rather debatable and we shall return to the welfarism issue later on in this book. For the present analysis, however, the welfaristic set-up has a great advantage. Instead of considering the orderings RU generated by F , we can focus on an ordering R ∗ of IR n , the space of utility n-tuples, that orders vectors of individual utilities which correspond to the social alternatives from the given set X . The formal result that Blackorby et al. state in this context (it is due to 30 ARROW’S IMPOSSIBILITY RESULT D’Aspremont and Gevers (1977)) says that if the social evaluation functional F satisﬁes the three axioms of welfarism (viz.

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

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