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NotesECN741-page33 - exist α = α-1 n ρ ∈ A n such that...

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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 > X θ - n π ( θ - n ) u g c n ( θ n , θ - n )) - v ˜ g y n ( θ n , θ - n )) θ n = X θ - n π ( θ - n ) u ( g c n ( α * n ( θ n ) , α * - n ( θ - n ))) - v g y n ( α * n ( θ n ) , α * - n ( θ - n )) θ n in which the last inequality follows from definition of
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Unformatted text preview: exist α = α *-1 n ( ρ ) ∈ A n such that X θ-n π ( θ-n ) ± u ( g c n ( α,α *-n ( θ-n )))-v ± g y n ( α,α *-n ( θ-n )) θ n ²² > X θ-n π ( θ-n ) ± u ( g c n ( α * n ( θ n ) ,α *-n ( θ-n )))-v ± g y n ( α * n ( θ n ) ,α *-n ( θ-n )) θ n ²² this is a contradiction. Therefore, ( c n ,y n ) N n =1 can be implemented by c n = ˜ g c ( θ 1 ,...,θ N ) , y n = ˜ g y ( θ 1 ,...,θ 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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