October 21, 2005 12:18masterSheet number 110 Page number 9494Lecture EightFurthermore, assume thatW(p)=zp(z)w(z)represents the pref-erencesas well.We will show thatwmust be a positive affinetransformation ofv. To see this, letα >0 andβsatisfyw(M)=αv(M)+βandw(m)=αv(m)+β(the existence ofα >0 andβis guaranteed byv(M) >v(m)andw(M) >w(m)). For anyz∈Zwe have[z] ∼v(z)M⊕(1−v(z))m, soit must be thatw(z)=v(z)w(M)+(1−v(z))w(m)=v(z)[αv(M)+β] +(1−v(z))[αv(m)+β] =αv(z)+β.The Dutch Book ArgumentThere are those who consider expected utility maximization to bea normative principle. One of the arguments made to support thisview is the following Dutch book argument. Assume thatL1L2but thatαL⊕(1−α)L2αL⊕(1−α)L1.We can perform the fol-lowing trick on the decision maker:1. TakeαL⊕(1−α)L1(we can describe this as a contingency withrandom eventE, which we both agree has probability 1−α).2. Take instead
This is the end of the preview.
access the rest of the document.