1. then no matter what experiment the expert performs, he can always force the principal to pay more than U ? U (p 0 ) in expectation (by just always sending this message), so the principal's expected payoff must be less than U (p 0 ), and the principal is worse off than by not hiring the expert. Since W is convex, this restriction implies W (p) ? W for all p. Now say that a function W : ?(?) ? R + is a reduced-form contract if it is the reduced form of some contract. We make two claims: ? Claim 1: The set of reduced;U ? U;/p 0 (?) in some state ?
2. Note that each W k must be a Lipschitz function with constant W (relative to the L 1 norm on ?(?)). By passing to a subsequence, we may assume that W k converges pointwise at each rational point p ? ?(?);Together, these claims imply that V P attains a maximum over the reduced-form contracts whose values never exceed W , which is then a global maximum, as needed
3. Using Dummy Bridging Faults to Define a Reduced Set of Target Faults
4. Now we claim that W k ? W in sup norm. If not, there exists some ? > 0 and a subsequence of k's and points p k along which |W k (p k ) ? W (p k )| > ?. Again by taking a subsequence, we may assume the p k converge to some point p. Now, we can find a rational point q such that the L 1 distance between p and q is less than ?/4W . Then, for k high enough;There is some subsequence along which the m k converge to some m ? . Then, m ? (p) = W ? (p), and m ? (p ? ) ? W ? (p ? ) for each other rational p ? , so that m ? ? M . Therefore W (p) ? W ? (p), and equality follows