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

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

ISBN-10: 0124033504

ISBN-13: 9780124033504

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Extra info for Arrow Impossibility Theorems

Example text

In fact, as we shall soon see, neutrality is often also combined with a nonnegative responsiveness condition as well. Thus we follow Blau and Deb [51] in defining a collective choice rule / : u\-> Cu to satisfy NIM (weak neutrality, independence, and monotonicity) if whenever u = (Rl9R2,. . , Rn) and u' = (R^R^,. ,Rn') are related by (Vf e N)[(xPiy -» zP/w) & (wP/z -+ yPtx)l then C„({x,>'})={x}-*C„,({w,z})={z}. Lemma 5-4 If / satisfies (i) neutrality, (ii) independence of irrelevant alternatives, and (iii) nonnegative pairwise responsiveness, then / satisfies NIM.

IMPOSSIBILITY THEOREMS 41 dealing with the condition of transitive rationality, in the next few theorems. Our first result stems from Sen's observation [307] that if the condition of transitive rationality is weakened to only quasitransitive rationality, choice rules do exist satisfying that and all of the remaining Arrow condi­ tions. Sen's example is the function that selects the Pareto optimals from v: Cup(v) = {x/x e v and there is no y e v such that yRiux for all i and yPiux for at least one i}.

Then z is K-best in {x, y, z}, z G C({X, y, z}). Applying Property )85 {y,z} = C\{y,z}) c C({x,y,z}). Rx, y E C({x,y}). Applying Property ß again, {x,y} = C({x,y}) ^ C({x, y, z}). But that means x is ^-maximal contrary to zPx. CASE 2: C({}', z}) = {y}. Then yPz, so neither z nor x are i^-maximal in {x,y,z}. Since C({x,y,z}) ^ 0 , we need ye C({x,y,z}). By Property a, 28 CHAPTER 3 y G C({x, y}). Applying Property ß, {x,y} = C({x,y}) c C({x,y,z}). But x G C({x,y, z}) would imply x is i^-best, contrary to zPx.

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

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