Arrow Impossibility Theorems - download pdf or read online

By Jerry S. Kelly and Karl Shell (Auth.)

ISBN-10: 0124033504

ISBN-13: 9780124033504

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Extra info for Arrow Impossibility Theorems

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11) 52 CHAPTER 4 Again we will settle the Bx° case; the B2° case follows analogously. N — b ç c;if N — b ^ c, then we have {b,N-b)B^c\ (12) Since (b, c) e ^ 0 , then for every distinct pair x, y, (b, c) e 5 1 °(x,y), and (12) then tells us (b, N — b) $ F(x, y). Completeness of the social relation then tells us (N — b,b) e F(y,x) and this holds for all (y,x). , b = 0 and, since b u c = N,c = N = N — b, contrary to our assumption that n — b # c. Thus our assumption is wrong and c = N — b. This disjointness of the two components of any member of B^ or B2° can be combined with a voting paradox profile to get very deep information about those two sets: (b9 N - b) G B^ -> |b| < 1, (N - c, c) G £ 2 ° -> |c| < 1.

Theorem 5-5 (Blau and Deb [51], Sen [318]).

35 36 CHAPTER 4 CU0{{x,y}) = {x} by xPDsy. CUo({\\z}) = {y} by the weak Pareto condition. Therefore, by base quasitransitivity, CUo({x, z}) = {x}. By independence of irrelevant alternatives, CM({x, z}) = {x}, so xPDsz. xPDsy^xPDsz. (1) Start again, letting u satisfy the condition that zPty for all i e S while yPtz for all ie N — S. Consider a profile ux for which the restrictions of the orderings to {x, v, z} are S: N - S: zxy, yzx. CMl({-M'}) = W by xPDsy. CMi((x,z}) = {z} by the weak Pareto condi­ tion.

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Arrow Impossibility Theorems by Jerry S. Kelly and Karl Shell (Auth.)

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