Rubinstein2005-page138

Rubinstein2005-page138 - i , . . . , n ) , is a...

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October 21, 2005 12:18 master Sheet number 136 Page number 120 120 Lecture Ten Related Issues Arrow’s theorem was the starting point for a huge literature. We mention three other impossibility results. 1. Monotonicity is another axiom that has been widely discussed in the literature. Consider a “change” in a proFle so that an alterna- tive a , which individual i ranked below b , is now ranked by i above b . Monotonicity requires that there is no alternative c such that this change deteriorates the ranking of a vs. c . Muller and Satterth- waite (1977)’s theorem shows that the only SW±’s satisfying Par and monotonicity are dictatorships. 2. An SW± speciFes a preference relation for every proFle. A social choice function attaches an alternative to every proFle. The most striking theorem proved in this framework is the Gibbard- Satterthwaite theorem. It states that any social choice function C satisfying the condition that it is never worthwhile for an individual to mispresent his preferences, namely, it is never that C ( Â 1 , ... , Â 0 i , ... , Â n ) Â i C ( Â 1 , ... ,
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Unformatted text preview: i , . . . , n ) , is a dictatorship. 3. Another related concept is the following. Let Ch ( 1 , . . . , n ) be a function that assigns a choice function to every proFle of orderings on X . We say that Ch satisFes unanimity if for every ( 1 , . . . , n ) and for any x , y A , if y i x for all i then, x / Ch ( 1 , . . . , n )( A ) . We say that Ch is invariant to the procedure if, for every proFle ( 1 , . . . , n ) and for every choice set A , the following two approaches lead to the same outcome: a. Partition A into two sets A and A 00 . Choose an element from A and an element from A 00 and then choose one element from the two choices. b. Choose an element from the unpartitioned set A . Dutta, Jackson, and Le Breton (2001) show that only dictatorships satisfy both unanimity and invariance to the procedure. Bibliographic Notes Recommended readings : Kreps 1990, chapter 5; Mas-Colell et al. 1995, chapter 21....
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