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

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

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Rn) and u' = (Rx\ . . , Rn) satisfy xRty -► xP/y, then with Cu = j\u) and Cu. (v) ). 33 SIMPLICITY: CONDITIONS Proof R'*) as Define two new profiles u* =(R1*9.. } n {x,y} = 0 or a e {x, y}. #,* is obtained from # £ by taking x and y to the top of Rt but otherwise preserving the Rt relations between x and y and among the members of E — {x,y}. R'j* is obtained from Rt by taking x and y to the top, preserving the Rt relations among the members of E — {x,y}, but the R/ relations between x and y.

Hence c ç c' and (c n c') u [(c — ft) n (ft' — c')] = c. Thus (ft, c) G F(x, z) and by its global maximality, (ft, c) e B1°(x, z). An analo­ gous argument allows us to derive (ft, c) e B^iw^y) for w <£ {x,y}. Linking these two proves (ft, c) e Bl°(w, z) for all distinct w, z as was to be shown. Now suppose (ft, c) e B^. Then by the above, (ft, c) G ß^vv,*). The pre­ vious paragraph then shows that if (ft', c') e F(x, y), then ft ç ft'. Similarly, if (ft, c) e ß 2 °, then c ç c' whenever (ft', c') G F(X, y).

Theorem 4-3 (Arrow's second impossibility theorem) There is no collective choice rule / satisfying (i) the standard domain constraint (2), (ii) each Cu = f(u) has a complete, reflexive, and transitive rationaliza­ tion R(u), (iii) independence of irrelevant alternatives, (iv) the weak pairwise Pareto condition, (v) general nondictatorship. Proof As before, Lemma 4-1 allows us to talk about pairwise decisive sets. If W is the smallest decisive set, nondictatorship and Property a tell us \W\ > 2.

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