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Mechanism

# Mechanism - Mechanisms for Provisions of Public Projects...

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Mechanisms for Provisions of Public Projects Econ 210C UCSB June 1, 2011

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Mechanisms Z : A set of collective choices (outcomes). u i ( z , t i ) , z Z : Agent i ’s utility function, where t i T i is privately known to agent i . Parameter t i is known as a “payoff type” (valuation type) of agent i . p : A probability distribution over T = T 1 × · · · T n (a common prior). f : T -→ Z : A social choice function. Γ = ( { M i } i , g ) : A mechanism where M i is the message set of agent i , g : M -→ Z is the outcome function . Bayesian Game Induced by a Mechanism: ( { M i , U i , T i } i , p ) where U i ( m , t ) = u i ( g ( m ) , t i ) . Econ 210CUCSB Paper
Dominant Strategies A strategy s i : T i -→ M i is (weakly) dominant for player i if for all t i T i , t - i T - i u i ( g ( s i ( t i ) , s - i ( t - i )) , t i ) p ( t - i | t i ) t - i T - i u i ( g ( m i , s - i ( t - i )) , t i ) p ( t - i | t i ) , m i M i , s - i : T - i -→ M - i . (1) Theorem A strategy s i : T i -→ M i is dominant if and only if for all t i T i , m i M i , m - i M - i , u i ( g ( s i ( t i ) , m - i ) , t i ) u i ( g ( m i , m - i ) , t i ) . (2) Econ 210CUCSB Paper

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Implementation Implementation in Bayesian-Nash Equilibrium: A mechanism Γ = ( { M i } n i = 1 , g ) implements social choice function f : T -→ Z if there is a BNE s * = ( s * 1 , · · · , s * n ) for the induced Bayesian game such that g ( s * ( t )) = f ( t ) for all t T . Implementation in Dominant Strategies: A mechanism Γ = ( { M i } n i = 1 , g ) implements social choice function f : T -→ Z in dominant strategies if there is a strategy profile s * = ( s * 1 , · · · , s * n ) for the induced Bayesian game such that g ( s * ( t )) = f ( t ) for all t T and s * i is dominant for each player i . Econ 210CUCSB Paper
Direct Mechanism, Truthful Implementation Direct Mechanism: A direct mechanism is one with M i = T i for all i and g ( t ) = f ( t ) , t T . Truthful Implementation in BNE ( Bayesian Incentive Compatibility ): A social choice function f : T -→ Z is truthfully implementable in BNE if the direct mechanism Γ = ( { T i } n i = 1 , f ) has a BNE s * = ( s * 1 , · · · , s * n ) such that s * i ( t i ) = t i , t i T i , i = 1, 2, · · · , n . (3) Truthful Implementation in Dominant Strategies ( Dominant Strategy Incentive Compatibility, Strategy-Proofness, Straightforwardness ): A social choice function f : T -→ Z is truthfully implementable in dominant strategies if f is truthfully implementable in dominant strategy BNE.

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Mechanism - Mechanisms for Provisions of Public Projects...

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