NotesECN741-page33

NotesECN741-page33 - exist = *-1 n ( ) A n such that X -n (...

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ECN 741: Public Economics Fall 2008 Proposition 9 (Revelation Principle) A allocation ( c n ,y n ) N n =1 is implementable if and only if it is truthfully implementable in a direct mechanism. Proof. Suppose allocation ( c n ,y n ) N n =1 is implementable as outcome of some mechanism ( A,g c ,g y ) . We construct a truth-full mechanism , ˜ g c , ˜ g y ) as the following ˜ g c ( θ 1 ,...,θ N ) = g c ( α * 1 ( θ 1 ) ,...,α * N ( θ N )) , ˜ g y ( θ 1 ,...,θ N ) = g y ( α * 1 ( θ 1 ) ,...,α * N ( θ N )) In which { α * n } N n =1 is the BNE of the mechanism ( A,g c ,g y ) . We only need to show that truth-telling is a BNE of , ˜ g c , ˜ g y ) . Suppose not, i.e., suppose there is a type θ n and a report ρ Θ such that X θ - n π ( θ - n ) ± u g c n ( ρ,θ - n )) - v ± ˜ g y n ( ρ,θ - n )) θ n ²² > X θ - n π ( θ - n ) ± u g c n ( θ n - n )) - v ± ˜ g y n ( θ n - n )) θ n ²² = X θ - n π ( θ - n ) ± u ( g c n ( α * n ( θ n ) * - n ( θ - n ))) - v ± g y n ( α * n ( θ n ) * - n ( θ - n )) θ n ²² in which the last inequality follows from definition of g c , ˜ g y ). This implies that there must
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Unformatted text preview: exist = *-1 n ( ) A n such that X -n ( -n ) u ( g c n ( , *-n ( -n )))-v g y n ( , *-n ( -n )) n > X -n ( -n ) u ( g c n ( * n ( n ) , *-n ( -n )))-v g y n ( * n ( n ) , *-n ( -n )) n this is a contradiction. Therefore, ( c n ,y n ) N n =1 can be implemented by c n = g c ( 1 ,..., N ) , y n = g y ( 1 ,..., N ) Using Revelation Principle we can restrict attention to direct mechanism and allocations that are truthfully revealing. This means that the set of implementable allocations are the 33...
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