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

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Extra resources for Arrow Impossibility Theorems

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Hence A n Bis locally pairwise decisive for x against y and so, by Lemma 4-1, is decisive. Now let W be the smallest decisive set (known to exist by the weak pairwise Pareto condition and quasitransitive rationality). To show that W is a general oligarchy, it remains to show that if individual A: e W, then A: is a weak pairwise dictator. Suppose not. Y,y\) = {y}. We partition N according to preferences on [x,y] in u: B{\ AT, B2: (AT), B,: yx, with k E B{. Y,y,z) given by Bx\ B2: B3 : xzy, (xy)z, yxz.

If zR(u)x, then, by transitivity, zP(u)y. By Theorem 3-17 zPDw_{i)y and so, by Lemma 4-1, W — {/} is pairwise decisive, contrary to our choice of W. Therefore xP(u)z. But then, by Theorem 3-17, xPD{i]z and so, by Lemma 4-1, {i} is pairwise decisive. With Property a, this says i is a general dictator. □ This is not quite the theorem that Arrow stated. For he dropped (i), allowing |£| > 3. But with this relaxation, it is not possible to establish the standard domain constraint and so it is not possible to establish Theorem 3-18 and Lemma 4-1 which Arrow had dealt with as if |£| = 3.

M This \^i- 47 SIMPLICITY: IMPOSSIBILITY THEOREMS By xPDwy, {x} = CU5({x,y}). Our assumption gives CU5({y,z}) = {y}. By triple acyclicity, x e C U5 ({X, Z}); but this contradicts our assumption. □ Proof of Theorem 4-10 Let 1, x and y be an individual and a pair of alternatives with the properties described in the lemma. We will show 1 is a weak pairwise dictator. t and Çïj)(jeN-{i}->tPjs) imply seCu({s,t}). (8) imply x e Cu{{s, t}). (9) We will prove only that for all t e E xPj and (Y/)(7 e N - {i} -+ tPpc) The steps from (9) to (8) are sufficiently similar to the ones we use to establish (9) that they are left to the reader to complete.

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