Rubinstein2005-page107 - volves a procedural aspect of...

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October 21, 2005 12:18 master Sheet number 105 Page number 89 Expected Utility 89 is preferred to another if it functions better in the worst case independently of the likelihood of the worst case occurring. Comparing the most likely prize : The decision maker considers the prize in each lottery which is most likely (breaking ties in some arbitrary way) and compares two lotteries according to a basic preference relation over Z . Lexicographic preferences : The prizes are ordered z 1, ... , z K and the lottery p is preferred to q if ( p ( z 1 ) , ... , p ( z K )) L ( q ( z 1 ) , ... , q ( z K )). Expected utility : A number v ( z ) is attached to each prize and a lottery is evaluated according to its expected v , that is, accord- ing to ± z p ( z ) v ( z ) . Thus, p ± q if U ( p ) = ± z Z p ( z ) v ( z ) U ( q ) = ± z Z q ( z ) v ( z ). The richness of examples calls for the classification of preference relations over lotteries and the study of properties that these rela- tions satisfy. The methodology we follow is to formally state general principles (axioms) that may apply to preferences over the space of lotteries. Each axiom carries with it a consistency requirement or in-
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Unformatted text preview: volves a procedural aspect of decision making. The axiomatization of a family of preference relations provides justication for focusing on that specic family. Von Neumann-Morgenstern Axiomatization The version of the von Neumann-Morgenstern axiomatization pre-sented here uses two axioms, the independence and continuity ax-ioms. The Independence Axiom In order to state the rst axiom we require an additional concept, called Compound lotteries (g. 8.1): Given a K-tuple of lotteries ( p k ) and a K-tuple of nonnegative numbers ( k ) k = 1, ... , K that sum up to 1, dene K k = 1 k p k to be the lottery for which ( K k = 1 k p k )( z ) = K k = 1 k p k ( z ) . Verify that K k = 1 k p k is indeed a lottery. When only two lotteries p 1 and p 2 are involved, we use the notation 1 p 1 ( 1 1 ) p 2 ....
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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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