Rubinstein2005-page140

Rubinstein2005-page140 - isfying the above axioms. e. Are...

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October 21, 2005 12:18 master Sheet number 138 Page number 122 Problem Set 10 Problem 1. ( Moderately difFcult. Based on May 1952. ) Assume that the set of social alternatives, X , includes only two alternatives. DeFne a social welfare function to be a function that attaches a preference to any proFle of preferences (allow indifference for the SW± and the indi- viduals’ preference relations). Consider the following axioms: Anonymity If σ is a permutation of N and if p ={ % i } i N and p 0 ={ % 0 i } i N are two proFles of preferences on X so that % 0 σ( i ) = % i , then % ( p ) = % ( p 0 ) . Neutrality ±or any preference % i deFne ( % i ) as the preference satis- fying x ( % i ) y iff y % i x . Then % ( {− % i } i N ) =− % ( { % i } i N ) . Positive Responsiveness If the proFle { % 0 i } i N is identical to { % i } i N with the exception that for one individual j either ( x j y and x  0 j y )o r ( y  j x and x 0 j y ) and if x % y then x  0 y . a. Interpret the axioms. b. Does anonymity imply non-dictatorship? c. Show that the majority rule satisFes all axioms.
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Unformatted text preview: isfying the above axioms. e. Are the above three axioms independent? Problem 2. ( Moderately difFcult ) N individuals choose a single object from among a set X . We are interested in functions that aggregate the individuals’ recommendations ( not prefer-ences , just recommendations!) into a social decision (i.e., F : X N → X ). Discuss the following axioms: • Par : If all individuals recommend x ∗ then the society chooses x ∗ . • I : If the same individuals support an alternative x ∈ X in two proFles of recommendations, then x is chosen in one proFle if and only if it chosen in the other. a. Show that if X includes at least three elements, then the only aggre-gation method that satisFes P and I is a dictatorship. b. Show the necessity of the three conditions P , I , and | X | ≥ 3 for this conclusion....
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This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.

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