Rubinstein2005-page109

# Rubinstein2005-page109 - • Expected utility Note that the...

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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 Â 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 Â 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 Â q Â 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.
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Unformatted text preview: • Expected utility : Note that the function U ( p ) is linear: U ( ⊕ K k = 1 α k p k ) = X z ∈ Z [⊕ K k = 1 α k p k ] ( z ) v ( z ) = X z ∈ Z [ K X k = 1 α k p k ( z ) ] v ( z ) = K X k = 1 α k [ X z ∈ Z p k ( z ) v ( z ) ] = K X k = 1 α 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 ....
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