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

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

ISBN-10: 0124033504

ISBN-13: 9780124033504

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The weak pairwise Pareto condition ensures that such a W exists and is nonempty. Pick one individual keW and so partition W as {k} u W*. If W* = 0 , k is a pairwise dictator and {k} is a pairwise oligarchy; therefore assume W* # 0 . We want to show that k is a weak pairwise dictator. So pick any pair of alternatives x and y and let u be any profile in which xPkuy: k: W* yj(N -S): xy, [xy]. Picking any third alternative z, examine a profile u0 in which the orderings restricted to {x, y, z} are k : xzy, W*: [xy]z, N -W: z[xy].

Since x G I;, x ^ t;' — v, so x ^ C(v' — v). Therefore x e C(t;). Hence i; n C(v') ç C(»). Next we show that path independence implies Property e. Let v ç v' and suppose, contrary to Property 8, that C(t/) p C(v). ')) = C{C(v)). By the first part of this proof, Property a holds, so that from C(v) ^ vwc can derive C(v) n C(v) Ç C(C(Ü)), so C(i;) = C(C(v)). ') = C{v), a contradiction. Therefore Property s is satisfied. 4 Property s has been called the superset property [45]. 30 CHAPTER 3 Finally, from Properties a and s we will obtain path independence.

Thus F(x,z) contains (ft n ft', (c n c') u [(c - ft) n (ft' - c')] u [(ft - c) n (c' - ft')]). Since (ft, c j e ß / , we cannot have ft n ft' 5 ft. Thus ft ç ft' and ft n ft' = ft. Also ft ç ft' implies [(ft — c) n (c' — ft')] = 0 , so F(x, z) contains (ft, (c n c') u [(c - ft) n (ft' - c')]). Since (ft, c) G 5 ^ , we cannot have (c n c') u [(c —ft)n (ft' — c')] g; c. 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).

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


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