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Rubinstein2005-page134

# Rubinstein2005-page134 - among Â 1 Â n(with ties broken...

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October 21, 2005 12:18 master Sheet number 132 Page number 116 116 Lecture Ten 1. F ( Â 1 , ... , Â n ) = % for some ordering % . (This is a degenerate SWF that does not account for the individuals’ preferences.) 2. De±ne x z if a majority of individuals prefer x to z . Order the alternatives by the number of “victories” they score, that is, x % y if |{ z | x z }|≥|{ z | y z }| . 3. For X ={ a , b } , a % b unless 2 / 3 of the individuals prefer b to a . 4. “ The anti-dictator ”: There is an individual i so that x is preferred to y if and only if y Â i x . 5. De±ne d ( Â ; Â 1 , ... , Â n ) as the number of ( x , y , i ) for which x Â i y and y Â x . The function d can be interpreted as the sum of the distances between the preference relation Â and the n preference relations of the individuals. Choose F ( Â 1 , ... , Â n ) to be an ordering that minimizes d ( Â ; Â 1 , ... , Â n ) (ties are broken arbitrarily). 6. Let F ( Â 1 , ... , Â n ) be the ordering that is the most common
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Unformatted text preview: among ( Â 1 , . . . , Â n ) (with ties broken in some predetermined way). Axioms Once again we use the axiomatization methodology. We suggest a set of axioms on social welfare functions and study their implica-tions. Let F be an SWF. Condition Par (Pareto): For all x , y ∈ X and for every pro±le ( Â i ) i ∈ N , if x Â i y for all i then x Â y . The Pareto axiom requires that if all individuals prefer one alter-native over the other, then the social preferences agree with the individuals’. Condition IIA (Independence of Irrelevant Alternatives): For any pair x , y ∈ X and any two pro±les ( Â i ) i ∈ N and ( Â i ) i ∈ N if for all i, x Â i y iff x Â i y , then x % y iff x % y ....
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