New PDF release: Arrow Impossibility Theorems

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

ISBN-10: 0124033504

ISBN-13: 9780124033504

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Hence bP(u)e and by Lemma 4-1, N — {S KJ {i}) is a pairwise decisive set. , N > k + 1. Moreover, since i, S, a, b, and c are arbitrary, pairwise decisiveness of the n — (k + 1) element set, N — (S u {*}) yields x(/c + l)j; for all distinct x, y. □ Fishburn has also established, using measure-theoretic tools, the result that if |7V| is infinite, the other Arrow conditions are consistent. Fishburn's example and a new impossibility theorem by Kirman and Sondermann for the infinite |Af| case will be taken up in Chapter 8 where we can use the filter theory tools developed there.

74] We will reconsider Sen's position in the next chapter. SIMPLICITY : IMPOSSIBILITY THEOREMS 49 The condition of transitive rationality has not been the only object of criticism in Arrow's second theorem. Independence of irrelevant alternativeshas probably been the principle target of critics. Of course we already have Sen's theorem on the impossibility of a Paretian liberal which uses neither IIA nor any regularity condition. (B). Theorem 4-12 There is no collective choice rule / satisfying (i) the standard domain constraint, (ii) | £ | < o o , (iii) each Cu = f(u) has a complete, reflexive, and transitive rationaliza­ tion R(u), (iv) independence of single irrelevant alternatives, (v) the weak pairwise Pareto condition, (vi) general nondictatorship.

There­ fore xPDsy -► rPDst. All that remains is a demonstration that these local results can all be converted to global results. Still assuming xPDsy, let u be any profile with xPty for all i e S: S: xy, N-S: [xy]. Insert a z to get a u0 for which the restricted orderings are S: N-S: xzy, z[xy~]. By xPDsy we see from the first part of our proof that xPDsz. Therefore CMo({x,z}) = {x}. By the weak Pareto condition, CUo({y,z}) = {z}. Then, by base quasitransitivity, CUo({x,y}) = {x}. By independence of irrelevant alter­ natives, Cu({x, y}) = {x} as was to be shown.

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

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