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Rubinstein2005-page28

Rubinstein2005-page28 - tions for example the relation xSy...

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October 21, 2005 12:18 master Sheet number 26 Page number 10 Problem Set 1 Problem 1. ( Easy ) Let % be a preference relation on a set X. DeFne I ( x ) to be the set of all y X for which y x . Show that the set (of sets!) { I ( x ) | x X } is a partition of X , i.e., ±or all x and y , either I ( x ) = I ( y ) or I ( x ) I ( y ) =∅ . ±or every x X , there is y X such that x I ( y ) . Problem 2. ( Standard ) Kreps (1990) introduces another formal deFnition for preferences. His prim- itive is a binary relation P interpreted as “strictly preferred.” He requires P to satisfy: Asymmetry : ±or no x and y do we have both xPy and yPx . Negative-Transitivity : ±or all x , y , and z X ,if xPy , then for any z either xPz or zPy (or both). Explain the sense in which Kreps’ formalization is equivalent to the tra- ditional deFnition. Problem 3. ( Standard ) In economic theory we are often interested in other types of binary rela-
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Unformatted text preview: tions, for example, the relation xSy : “ x and y are almost the same.” Suggest properties that would correspond to your intuition about such a concept. Problem 4. ( DifFcult. Based on Kannai and Peleg 1984. ) Let Z be a Fnite set and let X be the set of all nonempty subsets of Z . Let % be a preference relation on X (not Z ). Consider the following two properties of preference relations on X : a. If A % B and C is a set disjoint to both A and B , then A ∪ C % B ∪ C , and if A Â B and C is a set disjoint to both A and B , then A ∪ C Â B ∪ C . b. If x ∈ Z and { x } Â { y } for all y ∈ A , then A ∪ { x } Â A , and if x ∈ Z and { y } Â { x } for all y ∈ A , then A Â A ∪ { x } ....
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