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Rubinstein2005-page140

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

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October 21, 2005 12:18 master Sheet number 138 Page number 122 Problem Set 10 Problem 1. ( Moderately difficult. Based on May 1952. ) Assume that the set of social alternatives, X , includes only two alternatives. Define a social welfare function to be a function that attaches a preference to any profile of preferences (allow indifference for the SWF 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 = { i } i N are two profiles of preferences on X so that σ( i ) = i , then ( p ) = ( p ) . Neutrality For any preference i define ( i ) as the preference satis- fying x ( i ) y iff y i x . Then ( {− i } i N ) = − ( { i } i N ) . Positive Responsiveness If the profile { i } i N is identical to { i } i N with the exception that for one individual j either ( x j y and x j y ) or ( y j x and x j y ) and if x y then x y . a. Interpret the axioms. b. Does anonymity imply non-dictatorship? c. Show that the majority rule satisfies all axioms. d. Prove May’s theorem by which the majority rule is the only SWF sat-
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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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