When is ambiguity–attitude constant

Authors: Jürgen Eichberger Simon Grant David Kelsey

Publish Date: 2012/12/04

Volume: 45, Issue: 3, Pages: 239-263

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Abstract

This paper studies how updating affects ambiguity attitude In particular we focus on generalized Bayesian updating of the Jaffray–Philippe subclass of Choquet Expected Utility preferences We find conditions for ambiguity attitude to be the same before and after updating A necessary and sufficient condition for ambiguity attitude to be unchanged when updated on an arbitrary event is for the capacity to be neoadditive We find a condition for updating on a given partition to preserve ambiguity attitude We relate this to necessary and sufficient conditions for dynamic consistency Finally we study whether ambiguity increases or decreases after updatingIf nu Ealpha A =alpha sigma A +left 1alpha right bar sigma A where 0leqslant alpha leqslant 1 and sigma is a convex capacity on E then frac alpha mu A + 1alpha left 1mu left Acright right alpha left mu A mu left Acup Ecright +mu Bcup Ec mu B right +1mu Bcup Ec +mu B =alpha sigma left Aright + 1alpha bar sigma A This equation has the form frac aalpha +bcalpha +d=ealpha +f where c= mu A +mu Ac mu left Acup Ecright mu left Acup Eccright etc Cross multiplying aalpha +b=alpha 2ce+ fc+de alpha +fd Equating coefficients we obtain ce=0a= fc+de b=fdUnless sigma is the complete uncertainty capacity there exists A such that sigma A =eneq 0 which implies c=0 Note one can easily show that the result holds if sigma is the complete uncertainty capacity Hence mu A mu Acup Ec +mu Bcup Ec mu B holds □

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