Read e-book online A primer in social choice theory PDF

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By Gaertner W.

ISBN-10: 0199565309

ISBN-13: 9780199565306

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Extra info for A primer in social choice theory

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Since R ∗ is an ordering, three ways of ranking points in region II against u¯ are possible: the points in II could be preferred, indifferent, or worse. In our argument above, we had postulated a strict preference against u. ¯ We could also have started by assuming u¯ to be preferable to all points in II. 6. a2 32 ARROW’S IMPOSSIBILITY RESULT would have been completely analogous. However, indifference between points in II and u¯ would lead to a contradiction. We would, for example, have aI ∗ u¯ and bI ∗ u.

Positive responsiveness now requires that g (d1 , . . , dn ) = +1. The last step is by mathematical induction. Suppose that N (1) = N (−1)+m implies g (d1 , . . , dn ) = +1. We have to prove that N (1) = N (−1) + (m + 1) implies g (d1 , . . , dn ) = +1. Therefore, suppose that N (1) = N (−1) + (m + 1). Again, there is at least one di = +1 in the profile (d1 , . . , 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 , .

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. In his paper from 1952, May showed that the four conditions that define the simple majority rule are logically independent.

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A primer in social choice theory by Gaertner W.


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