NotesECN741-page32

NotesECN741-page32 - Good news is that there is a very...

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ECN 741: Public Economics Fall 2008 The outcome is determined according to outcome function Definition 5 Let ( A,g c ,g y ) be a Mechanism. A Bayesian Nash Equilibrium (BNE) is a collection of strategies { α * n } N n =1 , α * n : Θ -→ A n such that α * n ( θ n ) arg max ρ A n X θ - n π ( θ - n ) ± u ( g c n ( ρ,α * - n ( θ - n ))) - v ± g y n ( ρ,α * - n ( θ - n )) θ n ²² We call g c n ( α * 1 ( θ 1 ) ,...,α * N ( θ N )) and g y n ( α * 1 ( θ 1 ) ,...,α * N ( θ N )) ( θ 1 ,...,θ N ) equilibrium out- come. Definition 6 A feasible allocation ( c n ,y n ) N n =1 is implementable if there is a mechanism ( A,g c ,g y ) and a BNE { α * n } N n =1 of that mechanism such that c n = g c n ( α * 1 ( θ 1 ) ,...,α * N ( θ N )) , y n = g y n ( α * 1 ( θ 1 ) ,...,α * N ( θ N )) So far we have made it clear what exactly do we mean by implementability. But is it helpful? Notice that our setup so far does not impose any restriction on the type of games (mecha- nisms) considered. Any equilibrium outcome of some game is implementable. Think for a moment about the following problem: we want to find the best implementable allocation. That means we need to search in the space of games, find a game that has a BNE that implements that best allocation as its outcome. This is a very complicated problem.
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Unformatted text preview: Good news is that there is a very powerful result that allows us to restrict attention to a very particular game without lose of generality. We need couple of more definitions. Definition 7 A Direct Mechanism is a a game such that A n = Θ for all n . Definition 8 A truth-telling BNE of a direct mechanism (Θ ,g c ,g y ) is α * n ( θ n ) = θ n for all n such that θ n ∈ arg max ρ ∈ Θ X θ-n π ( θ-n ) ± u ( g c n ( ρ,θ-n ))-v ± g y n ( ρ,θ-n )) θ n ²² A feasible allocation is truthfully implementable if c n = g c n ( θ 1 ,...,θ N ) , y n = g y n ( θ 1 ,...,θ N ) In a direct mechanism, players are basically asked to report their skill type. We are interested in equilibria in which type is revealed truthfully. It turns out there is not loss of generality in doing that. 32...
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