Rubinstein2005-page136

# Rubinstein2005-page136 - is decisive it is almost decisive...

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October 21, 2005 12:18 master Sheet number 134 Page number 118 118 Lecture Ten w ( k ) if x appears in the k ’th place in Â i . Attach to x the sum of the weights assigned to x by the n individuals and rank the alternatives by those sums. The Borda rule is an SWF satisfying Par but not I . Transitivity of the Social Order: The majority rule satis±es all assumptions but can induce a relation which is not transitive. | X |≥ 3: For | X |= 2 the majority rule satis±es Par and I and induces (a trivial) transitive relation. Proof of Arrow’s Impossibility Theorem Let F be an SWF that satis±es Par and I . Hereinafter, we denote the relation F ( Â 1 , ... , Â n ) by % . Given the SWF we say that a coalition G is decisive if for all x , y , [for all i G , x Â i y ] implies [ x Â y ], and a coalition G is almost decisive if for all x , y , [for all i G , x Â i y and for all j / Gy Â j x ] implies [ x Â y ]. Note that if G
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Unformatted text preview: is decisive it is almost decisive since the “almost decisiveness” refers only to the subset of pro±les where all members of G prefer x to y and all members of N − G prefer y to x . Field Expansion Lemma: If G is almost decisive, then G is decisive. Proof: We have to show that for any x , y and for any pro±le ( Â i ) i ∈ N for which x Â i y for all i ∈ G , the preference F ( Â 1 , . . . , Â n ) determines x to be superior to y . By I ∗ it is suf±cient to show that for one pair of social alternatives a and b , and for one pro±le ( Â i ) i ∈ N that agrees with the pro±le ( Â i ) i ∈ N on the pair { a , b } , the preference F ( Â 1 , . . . , Â n ) determines a to be preferred to b ....
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