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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## This note was uploaded on 12/10/2011 for the course MAT 121 taught by Professor Wong during the Fall '10 term at SUNY Stony Brook.

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