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By Gaertner W.

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Dn ) = 0. 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 , .

2001). ‘Arrow’s Theorem and the Gibbard–Satterthwaite Theorem: A Unified Approach’. Economics Letters, 70: 99–105. Sen, A. K. (1970). Collective Choice and Social Welfare, chapter 3. San Francisco, Cambridge: Holden-Day. HISTORICAL SOURCE Arrow, K. J. (1951, 1963). ), Chapter 5. New York: John Wiley. MORE ADVANCED Sen, A. K. (1995). ‘Rationality and Social Choice’. American Economic Review, 85: 1–24. 1. The simple majority rule We have seen in Chapter 2 that any social welfare function satisfying unrestricted domain and the weak Pareto condition as well as Arrow’s independence condition and the requirement that the generated social relation be an ordering is doomed to be dictatorial.

Chapter 3 examines various domain restrictions of individual preferences. The purpose of this exercise is to see what can be done under the method of majority decision to avoid ‘irrational’ social choice such as preference cycles. The most prominent restriction examined is the condition of single-peaked preferences. Chapter 4 discusses the exercise of individual rights. The starting point is Sen’s very influential result of the ‘impossibility of a Paretian liberal’. We ask under which conditions the consistent exercise of personal rights becomes possible.

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