Rubinstein2005-page109

Rubinstein2005-page109 - Expected utility : Note that the...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
October 21, 2005 12:18 master Sheet number 107 Page number 91 Expected Utility 91 The Continuity Axiom Once again we will employ a continuity assumption that is basically the same as the one we employed for the consumer model. Conti- nuity means that the preferences are not overly sensitive to small changes in the probabilities. Continuity (C): If p  q , then there are neighborhoods B ( p ) of p and B ( q ) of q (when presented as vectors in R | Z | ), such that for all p 0 B ( p ) and q 0 B ( q ) , p 0  q 0 . The continuity assumption implies (verify!) the following prop- erty that is sometimes presented as an alternative deFnition of con- tinuity: C : If p  q  r , then there exists α ( 0, 1 ) such that q ∼[ α p ( 1 α) r ] . Let us check whether some of the examples we discussed earlier satisfy these two axioms.
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Expected utility : Note that the function U ( p ) is linear: U ( K k = 1 k p k ) = X z Z [ K k = 1 k p k ] ( z ) v ( z ) = X z Z [ K X k = 1 k p k ( z ) ] v ( z ) = K X k = 1 k [ X z Z p k ( z ) v ( z ) ] = K X k = 1 k U ( p k ). It follows that any such preference relation satisFes I . Since the function U ( p ) is continuous in the probability vector, it also satisFes C . Increasing the probability of a good consequence : Such a pref-erence relation satisFes the two axioms since it can be repre-sented by the expectation of v where v ( z ) = 1 for z G and v ( z ) = 0 for z B ....
View Full Document

Ask a homework question - tutors are online