Building an IC Dynamic Mechanism Instead of E θ S , calculate γ B using S’s reported ˆ θ S : � ˆ � = � c � � θ S , ˆ χ 2 ( ˆ θ S , ˆ θ B ) , ˆ γ B θ B θ S ? But then S, who pays γ B , would lie to manipulate it! Let B’s γ B = change in S’s expected [CP] cost induced by B’s report: � ˆ � = � c � � + E ˜ � � �� θ S , ˆ χ 2 ( ˆ θ S , ˆ θ B ) , ˆ χ 2 ( ˆ θ S , ˜ θ B ) , ˆ γ B θ B θ S c θ S . θ B Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 5 / 17
Building an IC Dynamic Mechanism Instead of E θ S , calculate γ B using S’s reported ˆ θ S : � ˆ � = � c � � θ S , ˆ χ 2 ( ˆ θ S , ˆ θ B ) , ˆ γ B θ B θ S ? But then S, who pays γ B , would lie to manipulate it! Let B’s γ B = change in S’s expected [CP] cost induced by B’s report: � ˆ � = � c � � + E ˜ � � �� θ S , ˆ χ 2 ( ˆ θ S , ˆ θ B ) , ˆ χ 2 ( ˆ θ S , ˜ θ B ) , ˆ γ B θ B θ S c θ S . θ B γ B lets B internalize S’s cost ) B will not lie regardless of what θ S he infers Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 5 / 17
Building an IC Dynamic Mechanism Instead of E θ S , calculate γ B using S’s reported ˆ θ S : � ˆ � = � c � � θ S , ˆ χ 2 ( ˆ θ S , ˆ θ B ) , ˆ γ B θ B θ S ? But then S, who pays γ B , would lie to manipulate it! Let B’s γ B = change in S’s expected [CP] cost induced by B’s report: � ˆ � = � c � � + E ˜ � � �� θ S , ˆ χ 2 ( ˆ θ S , ˆ θ B ) , ˆ χ 2 ( ˆ θ S , ˜ θ B ) , ˆ γ B θ B θ S c θ S . θ B γ B lets B internalize S’s cost ) B will not lie regardless of what θ S he infers θ B γ B ( ˜ θ B , θ S ) � 0 ) having S pay γ B does not alter S’s incentives E ˜ if B is truthful Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 5 / 17
Building an IC Dynamic Mechanism Instead of E θ S , calculate γ B using S’s reported ˆ θ S : � ˆ � = � c � � θ S , ˆ χ 2 ( ˆ θ S , ˆ θ B ) , ˆ γ B θ B θ S ? But then S, who pays γ B , would lie to manipulate it! Let B’s γ B = change in S’s expected [CP] cost induced by B’s report: � ˆ � = � c � � + E ˜ � � �� θ S , ˆ χ 2 ( ˆ θ S , ˆ θ B ) , ˆ χ 2 ( ˆ θ S , ˜ θ B ) , ˆ γ B θ B θ S c θ S . θ B γ B lets B internalize S’s cost ) B will not lie regardless of what θ S he infers θ B γ B ( ˜ θ B , θ S ) � 0 ) having S pay γ B does not alter S’s incentives E ˜ if B is truthful Thus letting ψ S ( θ B , θ S ) = � ψ B ( θ B , θ S ) = γ S ( θ S ) � γ B ( θ B , θ S ) yields a BIC balanced-budget mechanism Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 5 / 17
Generalizing Example: Add Another Period of Trade Seller type constant across repetitions, buyer type serially correlated 1a. Seller learns θ S 1b. Buyer buys x 1 from Seller 2a. Buyer learns θ B , 2 2b. Buyer buys x 2 from Seller 3a. Buyer learns θ B , 3 3b. Buyer buys x 3 from Seller Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 6 / 17
Generalizing Example: Add Another Period of Trade Seller type constant across repetitions, buyer type serially correlated 1a. Seller learns θ S 1b. Buyer buys x 1 from Seller 2a. Buyer learns θ B , 2 2b. Buyer buys x 2 from Seller 3a. Buyer learns θ B , 3 3b. Buyer buys x 3 from Seller Pay buyer γ B = change in S’s expected cost induced by B’s report in each repetition. Implies t = 3 incentive payment to buyer is: � ˆ � � � θ S , ˆ θ B , 3 , ˆ χ 3 ( ˆ θ S , ˆ θ B , 3 ) , ˆ γ B , 3 θ B , 2 = � c θ S �� � � � � ˆ χ 1 ( ˆ θ S , ˜ θ B , 3 ) , ˆ + E ˜ c θ S θ B , 2 . θ B , 3 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 6 / 17
Generalizing Example: Add Another Period of Trade Seller type constant across repetitions, buyer type serially correlated 1a. Seller learns θ S 1b. Buyer buys x 1 from Seller 2a. Buyer learns θ B , 2 2b. Buyer buys x 2 from Seller 3a. Buyer learns θ B , 3 3b. Buyer buys x 3 from Seller Pay buyer γ B = change in S’s expected cost induced by B’s report in each repetition. Implies t = 3 incentive payment to buyer is: � ˆ � � � θ S , ˆ θ B , 3 , ˆ χ 3 ( ˆ θ S , ˆ θ B , 3 ) , ˆ γ B , 3 θ B , 2 = � c θ S �� � � � � ˆ χ 1 ( ˆ θ S , ˜ θ B , 3 ) , ˆ + E ˜ c θ S θ B , 2 . θ B , 3 In t = 2, buyer sees add’l e¤ect of reporting ˆ θ B , 2 : a¤ects beliefs Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 6 / 17
Generalizing Example: Add Another Period of Trade Seller type constant across repetitions, buyer type serially correlated 1a. Seller learns θ S 1b. Buyer buys x 1 from Seller 2a. Buyer learns θ B , 2 2b. Buyer buys x 2 from Seller 3a. Buyer learns θ B , 3 3b. Buyer buys x 3 from Seller Pay buyer γ B = change in S’s expected cost induced by B’s report in each repetition. Implies t = 3 incentive payment to buyer is: � ˆ � � � θ S , ˆ θ B , 3 , ˆ χ 3 ( ˆ θ S , ˆ θ B , 3 ) , ˆ γ B , 3 θ B , 2 = � c θ S �� � � � � ˆ χ 1 ( ˆ θ S , ˜ θ B , 3 ) , ˆ + E ˜ c θ S θ B , 2 . θ B , 3 In t = 2, buyer sees add’l e¤ect of reporting ˆ θ B , 2 : a¤ects beliefs “Correction term” was there to neutralize seller’s incentive to manipulate γ B , 3 through report of ˆ θ S Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 6 / 17
Generalizing Example: Add Another Period of Trade Seller type constant across repetitions, buyer type serially correlated 1a. Seller learns θ S 1b. Buyer buys x 1 from Seller 2a. Buyer learns θ B , 2 2b. Buyer buys x 2 from Seller 3a. Buyer learns θ B , 3 3b. Buyer buys x 3 from Seller Pay buyer γ B = change in S’s expected cost induced by B’s report in each repetition. Implies t = 3 incentive payment to buyer is: � ˆ � � � θ S , ˆ θ B , 3 , ˆ χ 3 ( ˆ θ S , ˆ θ B , 3 ) , ˆ γ B , 3 θ B , 2 = � c θ S �� � � � � ˆ χ 1 ( ˆ θ S , ˜ θ B , 3 ) , ˆ + E ˜ c θ S θ B , 2 . θ B , 3 In t = 2, buyer sees add’l e¤ect of reporting ˆ θ B , 2 : a¤ects beliefs “Correction term” was there to neutralize seller’s incentive to manipulate γ B , 3 through report of ˆ θ S But in period 2, this correction distorts buyer’s incentives Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 6 / 17
The Model In each period t = 1 , 2 , . . . Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 7 / 17
The Model In each period t = 1 , 2 , . . . Each agent i = 1 , . . . , N privately observes signal θ i , t 2 Θ i , t 1 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 7 / 17
The Model In each period t = 1 , 2 , . . . Each agent i = 1 , . . . , N privately observes signal θ i , t 2 Θ i , t 1 Agents send simultaneous reports 2 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 7 / 17
The Model In each period t = 1 , 2 , . . . Each agent i = 1 , . . . , N privately observes signal θ i , t 2 Θ i , t 1 Agents send simultaneous reports 2 Each agent i makes private decision x i , t 2 X i , t 3 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 7 / 17
The Model In each period t = 1 , 2 , . . . Each agent i = 1 , . . . , N privately observes signal θ i , t 2 Θ i , t 1 Agents send simultaneous reports 2 Each agent i makes private decision x i , t 2 X i , t 3 Mechanism makes public decision x 0 , t 2 X 0 , t , transfers y i , t 2 R to 4 each i Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 7 / 17
The Model In each period t = 1 , 2 , . . . Each agent i = 1 , . . . , N privately observes signal θ i , t 2 Θ i , t 1 Agents send simultaneous reports 2 Each agent i makes private decision x i , t 2 X i , t 3 Mechanism makes public decision x 0 , t 2 X 0 , t , transfers y i , t 2 R to 4 each i Histories: θ t = ( θ 1 , . . . , θ t ) 2 Θ t = ∏ t Θ i , t ; similarly x t 2 X t τ = 1 ∏ i Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 7 / 17
The Model In each period t = 1 , 2 , . . . Each agent i = 1 , . . . , N privately observes signal θ i , t 2 Θ i , t 1 Agents send simultaneous reports 2 Each agent i makes private decision x i , t 2 X i , t 3 Mechanism makes public decision x 0 , t 2 X 0 , t , transfers y i , t 2 R to 4 each i Histories: θ t = ( θ 1 , . . . , θ t ) 2 Θ t = ∏ t Θ i , t ; similarly x t 2 X t τ = 1 ∏ i � �j x t � 1 , θ t � 1 � Technology: θ t � ν t Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 7 / 17
The Model In each period t = 1 , 2 , . . . Each agent i = 1 , . . . , N privately observes signal θ i , t 2 Θ i , t 1 Agents send simultaneous reports 2 Each agent i makes private decision x i , t 2 X i , t 3 Mechanism makes public decision x 0 , t 2 X 0 , t , transfers y i , t 2 R to 4 each i Histories: θ t = ( θ 1 , . . . , θ t ) 2 Θ t = ∏ t Θ i , t ; similarly x t 2 X t τ = 1 ∏ i � �j x t � 1 , θ t � 1 � Technology: θ t � ν t Preferences: Agent i ’s utility ∞ δ t � � u i , t ( x t , θ t ) + y i , t ∑ t = 1 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 7 / 17
The Model In each period t = 1 , 2 , . . . Each agent i = 1 , . . . , N privately observes signal θ i , t 2 Θ i , t 1 Agents send simultaneous reports 2 Each agent i makes private decision x i , t 2 X i , t 3 Mechanism makes public decision x 0 , t 2 X 0 , t , transfers y i , t 2 R to 4 each i Histories: θ t = ( θ 1 , . . . , θ t ) 2 Θ t = ∏ t Θ i , t ; similarly x t 2 X t τ = 1 ∏ i � �j x t � 1 , θ t � 1 � Technology: θ t � ν t Preferences: Agent i ’s utility ∞ δ t � � u i , t ( x t , θ t ) + y i , t ∑ t = 1 0 < δ < 1 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 7 / 17
The Model In each period t = 1 , 2 , . . . Each agent i = 1 , . . . , N privately observes signal θ i , t 2 Θ i , t 1 Agents send simultaneous reports 2 Each agent i makes private decision x i , t 2 X i , t 3 Mechanism makes public decision x 0 , t 2 X 0 , t , transfers y i , t 2 R to 4 each i Histories: θ t = ( θ 1 , . . . , θ t ) 2 Θ t = ∏ t Θ i , t ; similarly x t 2 X t τ = 1 ∏ i � �j x t � 1 , θ t � 1 � Technology: θ t � ν t Preferences: Agent i ’s utility ∞ δ t � � u i , t ( x t , θ t ) + y i , t ∑ t = 1 0 < δ < 1 u i , t uniformly bounded Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 7 / 17
Direct Mechanisms Measurable decision plan: χ t : Θ t ! X t Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 8 / 17
Direct Mechanisms Measurable decision plan: χ t : Θ t ! X t χ 0 , t are prescribed public decisions Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 8 / 17
Direct Mechanisms Measurable decision plan: χ t : Θ t ! X t χ 0 , t are prescribed public decisions χ i , t are recommended private decisions for agent i � 1 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 8 / 17
Direct Mechanisms Measurable decision plan: χ t : Θ t ! X t χ 0 , t are prescribed public decisions χ i , t are recommended private decisions for agent i � 1 Decision plan induces stochastic process µ [ χ ] on Θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 8 / 17
Direct Mechanisms Measurable decision plan: χ t : Θ t ! X t χ 0 , t are prescribed public decisions χ i , t are recommended private decisions for agent i � 1 Decision plan induces stochastic process µ [ χ ] on Θ Transfers: ψ i , t : Θ t ! R ; PDV Ψ i ( θ ) = ∑ ∞ t = 0 δ t ψ i , t ( θ t ) Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 8 / 17
Direct Mechanisms Measurable decision plan: χ t : Θ t ! X t χ 0 , t are prescribed public decisions χ i , t are recommended private decisions for agent i � 1 Decision plan induces stochastic process µ [ χ ] on Θ Transfers: ψ i , t : Θ t ! R ; PDV Ψ i ( θ ) = ∑ ∞ t = 0 δ t ψ i , t ( θ t ) Measurable, uniformly bounded Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 8 / 17
Direct Mechanisms Measurable decision plan: χ t : Θ t ! X t χ 0 , t are prescribed public decisions χ i , t are recommended private decisions for agent i � 1 Decision plan induces stochastic process µ [ χ ] on Θ Transfers: ψ i , t : Θ t ! R ; PDV Ψ i ( θ ) = ∑ ∞ t = 0 δ t ψ i , t ( θ t ) Measurable, uniformly bounded Budget balance: ∑ i ψ i , t ( θ ) � 0 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 8 / 17
Direct Mechanisms Measurable decision plan: χ t : Θ t ! X t χ 0 , t are prescribed public decisions χ i , t are recommended private decisions for agent i � 1 Decision plan induces stochastic process µ [ χ ] on Θ Transfers: ψ i , t : Θ t ! R ; PDV Ψ i ( θ ) = ∑ ∞ t = 0 δ t ψ i , t ( θ t ) Measurable, uniformly bounded Budget balance: ∑ i ψ i , t ( θ ) � 0 Information Disclosure: All announcements are public Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 8 / 17
Direct Mechanisms Measurable decision plan: χ t : Θ t ! X t χ 0 , t are prescribed public decisions χ i , t are recommended private decisions for agent i � 1 Decision plan induces stochastic process µ [ χ ] on Θ Transfers: ψ i , t : Θ t ! R ; PDV Ψ i ( θ ) = ∑ ∞ t = 0 δ t ψ i , t ( θ t ) Measurable, uniformly bounded Budget balance: ∑ i ψ i , t ( θ ) � 0 Information Disclosure: All announcements are public Disclosing less less would preserve equilibrium as long as agents can still infer recommended private decisions Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 8 / 17
Strategies Agent i ’s strategy de…nes Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 9 / 17
Strategies Agent i ’s strategy de…nes Reporting plan β i , t : Θ t i � Θ t � 1 ! Θ i , t � i Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 9 / 17
Strategies Agent i ’s strategy de…nes Reporting plan β i , t : Θ t i � Θ t � 1 ! Θ i , t � i Private action plan α i , t : Θ t i � Θ t � i ! X i , t Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 9 / 17
Strategies Agent i ’s strategy de…nes Reporting plan β i , t : Θ t i � Θ t � 1 ! Θ i , t � i Private action plan α i , t : Θ t i � Θ t � i ! X i , t Strategy also de…nes behavior following agent’s own deviations, but this is irrelevant for the normal form Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 9 / 17
Strategies Agent i ’s strategy de…nes Reporting plan β i , t : Θ t i � Θ t � 1 ! Θ i , t � i Private action plan α i , t : Θ t i � Θ t � i ! X i , t Strategy also de…nes behavior following agent’s own deviations, but this is irrelevant for the normal form Strategy is truthful-obedient if for all θ t , i , θ t � 1 β i , t ( θ t � i ) = θ i , t , � θ t � α i , t ( θ t ) = χ i , t Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 9 / 17
Balanced Team Mechanism � � t � � U i ( χ � ( θ ) , θ ) = ∑ ∞ t = 1 δ t u i , t ˜ , ˜ χ t θ θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 10 / 17
Balanced Team Mechanism � � t � � U i ( χ � ( θ ) , θ ) = ∑ ∞ t = 1 δ t u i , t ˜ , ˜ χ t θ θ E¢cient decision χ � : max χ E µ [ χ ] [ ∑ i U i ( χ � ( θ ) , θ )] ˜ θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 10 / 17
Balanced Team Mechanism � � t � � U i ( χ � ( θ ) , θ ) = ∑ ∞ t = 1 δ t u i , t ˜ , ˜ χ t θ θ E¢cient decision χ � : max χ E µ [ χ ] [ ∑ i U i ( χ � ( θ ) , θ )] ˜ θ Balanced Team Transfers: � θ i , t , θ t � 1 � � 1 i , t ( θ t ) γ j , t ( θ j , t , θ t � 1 ) , where ψ B I � 1 ∑ = γ i , t j 6 = i 0 1 � � @ E µ j t [ χ ] j θ j , t , θ t � 1 ∑ i 6 = j U i ( χ � ( θ ) , θ ) γ j , t ( θ j , t , θ t � 1 ) δ � t ˜ A = θ � � � E µ t [ χ ] j θ t � 1 ∑ i 6 = j U i ( χ � ( θ ) , θ ) ˜ θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 10 / 17
Balanced Team Mechanism � � t � � U i ( χ � ( θ ) , θ ) = ∑ ∞ t = 1 δ t u i , t ˜ , ˜ χ t θ θ E¢cient decision χ � : max χ E µ [ χ ] [ ∑ i U i ( χ � ( θ ) , θ )] ˜ θ Balanced Team Transfers: � θ i , t , θ t � 1 � � 1 i , t ( θ t ) γ j , t ( θ j , t , θ t � 1 ) , where ψ B I � 1 ∑ = γ i , t j 6 = i 0 1 � � @ E µ j t [ χ ] j θ j , t , θ t � 1 ∑ i 6 = j U i ( χ � ( θ ) , θ ) γ j , t ( θ j , t , θ t � 1 ) δ � t ˜ A = θ � � � E µ t [ χ ] j θ t � 1 ∑ i 6 = j U i ( χ � ( θ ) , θ ) ˜ θ Theorem Assume independent types: conditional on x t 0 , agent i’s private information θ t i , x t i does not a¤ect the distribution of θ j , t , for j 6 = i. Also � x t , θ t � does not depend on θ t i , x t assume private values: u j , t i for all t , i 6 = j. Then balanced team mechanism is BIC. Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 10 / 17
Balancing: Example In initial example: � � = c � � + δ c � � + δ 2 c � χ ( ˆ θ ) , ˆ χ 1 ( ˆ θ S ) , ˆ χ 2 ( ˆ θ S , ˆ θ B , 2 ) , ˆ χ 3 ( ˆ θ S , ˆ � U S θ S θ S θ S θ � � γ B , 3 ( ˆ θ B , 2 , ˆ θ B , 3 , ˆ χ 3 ( ˆ θ S , ˆ θ B , 3 ) , ˆ θ S ) = � c θ S �� � � � χ 3 ( ˆ θ S , ˜ θ B , 3 ) , ˆ � ˆ + E ˜ c θ S θ B , 2 θ B , 3 � � � δ E ˜ � � �� � ˆ γ B , 2 ( ˆ θ B , 2 , ˆ χ 2 ( ˆ θ S , ˆ θ B , 2 ) , ˆ χ 3 ( ˆ θ S , ˜ θ B , 3 ) , ˆ θ S ) = � c c θ S θ S θ B θ B , 3 � � � + δ c � �� χ 2 ( ˆ θ S , ˜ θ B , 2 ) , ˆ χ 3 ( ˆ θ S , ˜ θ B , 3 ) , ˆ + E ˜ c θ S θ S θ B , 2 , ˜ θ B , 3 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 11 / 17
Balancing: Proof Sketch � ˜ � = ∑ i 6 = j U i ( χ � ( θ ) , θ ) , pv of j ’s payments: Let Ψ j θ � � ˜ �� � � ˜ �� t � 1 t � 1 � j ) = E µ j t [ χ ] j ˆ θ j , t , ˆ � E µ t [ χ ] j ˆ t t � 1 θ θ δ t γ j , t ( ˆ j , ˆ Ψ j Ψ j θ θ θ θ ˜ ˜ θ θ | {z } | {z } t � 1 ) t � 1 ) γ + j , t ( ˆ θ j , t , ˆ γ � j , t ( ˆ θ θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 12 / 17
Balancing: Proof Sketch � ˜ � = ∑ i 6 = j U i ( χ � ( θ ) , θ ) , pv of j ’s payments: Let Ψ j θ � � ˜ �� � � ˜ �� t � 1 t � 1 � j ) = E µ j t [ χ ] j ˆ θ j , t , ˆ � E µ t [ χ ] j ˆ t t � 1 θ θ δ t γ j , t ( ˆ j , ˆ Ψ j Ψ j θ θ θ θ ˜ ˜ θ θ | {z } | {z } t � 1 ) t � 1 ) γ + j , t ( ˆ θ j , t , ˆ γ � j , t ( ˆ θ θ � ˜ � Two terms are expectations of the same function Ψ j θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 12 / 17
Balancing: Proof Sketch � ˜ � = ∑ i 6 = j U i ( χ � ( θ ) , θ ) , pv of j ’s payments: Let Ψ j θ � � ˜ �� � � ˜ �� t � 1 t � 1 � j ) = E µ j t [ χ ] j ˆ θ j , t , ˆ � E µ t [ χ ] j ˆ t t � 1 θ θ δ t γ j , t ( ˆ j , ˆ Ψ j Ψ j θ θ θ θ ˜ ˜ θ θ | {z } | {z } t � 1 ) t � 1 ) γ + j , t ( ˆ θ j , t , ˆ γ � j , t ( ˆ θ θ � ˜ � Two terms are expectations of the same function Ψ j θ t � 1 ) uses only period t � 1 information γ � j , t ( ˆ θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 12 / 17
Balancing: Proof Sketch � ˜ � = ∑ i 6 = j U i ( χ � ( θ ) , θ ) , pv of j ’s payments: Let Ψ j θ � � ˜ �� � � ˜ �� t � 1 t � 1 � j ) = E µ j t [ χ ] j ˆ θ j , t , ˆ � E µ t [ χ ] j ˆ t t � 1 θ θ δ t γ j , t ( ˆ j , ˆ Ψ j Ψ j θ θ θ θ ˜ ˜ θ θ | {z } | {z } t � 1 ) t � 1 ) γ + j , t ( ˆ θ j , t , ˆ γ � j , t ( ˆ θ θ � ˜ � Two terms are expectations of the same function Ψ j θ t � 1 ) uses only period t � 1 information γ � j , t ( ˆ θ t � 1 ) uses, in addition, agent j ’s period- t report γ + j , t ( ˆ θ j , t , ˆ θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 12 / 17
Balancing: Proof Sketch � ˜ � = ∑ i 6 = j U i ( χ � ( θ ) , θ ) , pv of j ’s payments: Let Ψ j θ � � ˜ �� � � ˜ �� t � 1 t � 1 � j ) = E µ j t [ χ ] j ˆ θ j , t , ˆ � E µ t [ χ ] j ˆ t t � 1 θ θ δ t γ j , t ( ˆ j , ˆ Ψ j Ψ j θ θ θ θ ˜ ˜ θ θ | {z } | {z } t � 1 ) t � 1 ) γ + j , t ( ˆ θ j , t , ˆ γ � j , t ( ˆ θ θ � ˜ � Two terms are expectations of the same function Ψ j θ t � 1 ) uses only period t � 1 information γ � j , t ( ˆ θ t � 1 ) uses, in addition, agent j ’s period- t report γ + j , t ( ˆ θ j , t , ˆ θ For any deviation by agent i , if the others are truthful-obedient: Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 12 / 17
Balancing: Proof Sketch � ˜ � = ∑ i 6 = j U i ( χ � ( θ ) , θ ) , pv of j ’s payments: Let Ψ j θ � � ˜ �� � � ˜ �� t � 1 t � 1 � j ) = E µ j t [ χ ] j ˆ θ j , t , ˆ � E µ t [ χ ] j ˆ t t � 1 θ θ δ t γ j , t ( ˆ j , ˆ Ψ j Ψ j θ θ θ θ ˜ ˜ θ θ | {z } | {z } t � 1 ) t � 1 ) γ + j , t ( ˆ θ j , t , ˆ γ � j , t ( ˆ θ θ � ˜ � Two terms are expectations of the same function Ψ j θ t � 1 ) uses only period t � 1 information γ � j , t ( ˆ θ t � 1 ) uses, in addition, agent j ’s period- t report γ + j , t ( ˆ θ j , t , ˆ θ For any deviation by agent i , if the others are truthful-obedient: Claim 1: Expected present value of γ i , t equals, up to a constant, that of ψ i , t Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 12 / 17
Balancing: Proof Sketch � ˜ � = ∑ i 6 = j U i ( χ � ( θ ) , θ ) , pv of j ’s payments: Let Ψ j θ � � ˜ �� � � ˜ �� t � 1 t � 1 � j ) = E µ j t [ χ ] j ˆ θ j , t , ˆ � E µ t [ χ ] j ˆ t t � 1 θ θ δ t γ j , t ( ˆ j , ˆ Ψ j Ψ j θ θ θ θ ˜ ˜ θ θ | {z } | {z } t � 1 ) t � 1 ) γ + j , t ( ˆ θ j , t , ˆ γ � j , t ( ˆ θ θ � ˜ � Two terms are expectations of the same function Ψ j θ t � 1 ) uses only period t � 1 information γ � j , t ( ˆ θ t � 1 ) uses, in addition, agent j ’s period- t report γ + j , t ( ˆ θ j , t , ˆ θ For any deviation by agent i , if the others are truthful-obedient: Claim 1: Expected present value of γ i , t equals, up to a constant, that of ψ i , t Claim 2: Expected present value of γ j , t is zero for each j 6 = i Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 12 / 17
Proof of Claim 2 For any possible deviation of agent i , expected present value of γ j , t is zero for each j 6 = i : θ j , 1 θ � j , 1 θ j , 2 θ j , t . . . . . . - γ � γ + - γ � γ + - γ � γ + j , 1 j , 1 j , 2 j , 2 j , t j , t δ 2 γ j , 2 δ t γ j , t ! δγ j , 1 ! ! Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 13 / 17
Proof of Claim 2 For any possible deviation of agent i , expected present value of γ j , t is zero for each j 6 = i : θ j , 1 θ � j , 1 θ j , 2 θ j , t . . . . . . - γ � γ + - γ � γ + - γ � γ + j , 1 j , 1 j , 2 j , 2 j , t j , t δ 2 γ j , 2 δ t γ j , t ! δγ j , 1 ! ! � � θ t i , x t � 1 Independent types ) agent i ’s private history does not i a¤ect beliefs over ˜ θ j , t Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 13 / 17
Proof of Claim 2 For any possible deviation of agent i , expected present value of γ j , t is zero for each j 6 = i : θ j , 1 θ � j , 1 θ j , 2 θ j , t . . . . . . - γ � γ + - γ � γ + - γ � γ + j , 1 j , 1 j , 2 j , 2 j , t j , t δ 2 γ j , 2 δ t γ j , t ! δγ j , 1 ! ! � � θ t i , x t � 1 Independent types ) agent i ’s private history does not i a¤ect beliefs over ˜ θ j , t t � 1 ) before time t If agent j is truthful, the expectation of γ + j , t ( ˜ θ j , t , ˆ θ t � 1 ) , for any report history ˆ t � 1 j , t ( ˆ equals γ � θ θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 13 / 17
Proof of Claim 2 For any possible deviation of agent i , expected present value of γ j , t is zero for each j 6 = i : θ j , 1 θ � j , 1 θ j , 2 θ j , t . . . . . . - γ � γ + - γ � γ + - γ � γ + j , 1 j , 1 j , 2 j , 2 j , t j , t δ 2 γ j , 2 δ t γ j , t ! δγ j , 1 ! ! � � θ t i , x t � 1 Independent types ) agent i ’s private history does not i a¤ect beliefs over ˜ θ j , t t � 1 ) before time t If agent j is truthful, the expectation of γ + j , t ( ˜ θ j , t , ˆ θ t � 1 ) , for any report history ˆ t � 1 j , t ( ˆ equals γ � θ θ LIE: ex ante expectation of γ j , t equals zero Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 13 / 17
Proof of Claim 1 For any possible deviation of agent i , expected present value of γ i , t equals, up to a constant, that of ψ i , t : θ i , 1 θ � i , 1 θ � i , t � 1 θ i , t . . . . . . - γ � γ + - γ � γ + - γ � γ + i , 1 i , 1 i , 2 i , t � 1 i , t i , t = 0 ! = 0 ! Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 14 / 17
Proof of Claim 1 For any possible deviation of agent i , expected present value of γ i , t equals, up to a constant, that of ψ i , t : θ i , 1 θ � i , 1 θ � i , t � 1 θ i , t . . . . . . - γ � γ + - γ � γ + - γ � γ + i , 1 i , 1 i , 2 i , t � 1 i , t i , t = 0 ! = 0 ! � � θ t i , x t � 1 Independent types ) agent i ’s private history does not i a¤ect beliefs over ˜ θ � i , t Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 14 / 17
Proof of Claim 1 For any possible deviation of agent i , expected present value of γ i , t equals, up to a constant, that of ψ i , t : θ i , 1 θ � i , 1 θ � i , t � 1 θ i , t . . . . . . - γ � γ + - γ � γ + - γ � γ + i , 1 i , 1 i , 2 i , t � 1 i , t i , t = 0 ! = 0 ! � � θ t i , x t � 1 Independent types ) agent i ’s private history does not i a¤ect beliefs over ˜ θ � i , t If the others are truthful, agent i ’s time- t expectation of t � 1 ) equals γ + t � 1 ) for any ˆ t � 1 i , t + 1 ( ˜ θ � i , t , ˆ θ i , t , ˆ i , t ( ˆ θ i , t , ˆ θ i , t , ˆ γ � θ θ θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 14 / 17
Proof of Claim 1 For any possible deviation of agent i , expected present value of γ i , t equals, up to a constant, that of ψ i , t : θ i , 1 θ � i , 1 θ � i , t � 1 θ i , t . . . . . . - γ � γ + - γ � γ + - γ � γ + i , 1 i , 1 i , 2 i , t � 1 i , t i , t = 0 ! = 0 ! � � θ t i , x t � 1 Independent types ) agent i ’s private history does not i a¤ect beliefs over ˜ θ � i , t If the others are truthful, agent i ’s time- t expectation of t � 1 ) equals γ + t � 1 ) for any ˆ t � 1 i , t + 1 ( ˜ θ � i , t , ˆ θ i , t , ˆ i , t ( ˆ θ i , t , ˆ θ i , t , ˆ γ � θ θ θ LIE: the two terms have the same ex ante expectations as well Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 14 / 17
Proof of Claim 1 For any possible deviation of agent i , expected present value of γ i , t equals, up to a constant, that of ψ i , t : θ i , 1 θ � i , 1 θ � i , t � 1 θ i , t . . . . . . - γ � γ + - γ � γ + - γ � γ + i , 1 i , 1 i , 2 i , t � 1 i , t i , t = 0 ! = 0 ! � � θ t i , x t � 1 Independent types ) agent i ’s private history does not i a¤ect beliefs over ˜ θ � i , t If the others are truthful, agent i ’s time- t expectation of t � 1 ) equals γ + t � 1 ) for any ˆ t � 1 i , t + 1 ( ˜ θ � i , t , ˆ θ i , t , ˆ i , t ( ˆ θ i , t , ˆ θ i , t , ˆ γ � θ θ θ LIE: the two terms have the same ex ante expectations as well t δ τ ˜ ∑ γ + γ � Thus, expectation of γ i , τ equals to that of ˜ i , t � ˜ i , 1 τ = 1 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 14 / 17
Proof of Claim 1 For any possible deviation of agent i , expected present value of γ i , t equals, up to a constant, that of ψ i , t : θ i , 1 θ � i , 1 θ � i , t � 1 θ i , t . . . . . . - γ � γ + - γ � γ + - γ � γ + i , 1 i , 1 i , 2 i , t � 1 i , t i , t = 0 ! = 0 ! � � θ t i , x t � 1 Independent types ) agent i ’s private history does not i a¤ect beliefs over ˜ θ � i , t If the others are truthful, agent i ’s time- t expectation of t � 1 ) equals γ + t � 1 ) for any ˆ t � 1 i , t + 1 ( ˜ θ � i , t , ˆ θ i , t , ˆ i , t ( ˆ θ i , t , ˆ θ i , t , ˆ γ � θ θ θ LIE: the two terms have the same ex ante expectations as well t δ τ ˜ ∑ γ + γ � Thus, expectation of γ i , τ equals to that of ˜ i , t � ˜ i , 1 τ = 1 � ˜ � γ � γ + i , 1 is una¤ected by reports; ˜ i , t ! Ψ i θ as t ! ∞ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 14 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Agents send simultaneous reports 2 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Agents send simultaneous reports 2 Each agent i chooses private action x i , t 3 Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Agents send simultaneous reports 2 Each agent i chooses private action x i , t 3 Each agent i chooses public action x 0 , i , t , makes public payment 4 z i , j , t � 0 to each agent j Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Agents send simultaneous reports 2 Each agent i chooses private action x i , t 3 Each agent i chooses public action x 0 , i , t , makes public payment 4 z i , j , t � 0 to each agent j ) Public action x 0 , t = ( x 0 , i , t ) N i = 1 , total transfer y i , t = ∑ ( z j , i , t � z i , j , t ) to agent i (budget-balanced) j Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Agents send simultaneous reports 2 Each agent i chooses private action x i , t 3 Each agent i chooses public action x 0 , i , t , makes public payment 4 z i , j , t � 0 to each agent j ) Public action x 0 , t = ( x 0 , i , t ) N i = 1 , total transfer y i , t = ∑ ( z j , i , t � z i , j , t ) to agent i (budget-balanced) j Markovian Assumptions: Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Agents send simultaneous reports 2 Each agent i chooses private action x i , t 3 Each agent i chooses public action x 0 , i , t , makes public payment 4 z i , j , t � 0 to each agent j ) Public action x 0 , t = ( x 0 , i , t ) N i = 1 , total transfer y i , t = ∑ ( z j , i , t � z i , j , t ) to agent i (budget-balanced) j Markovian Assumptions: Finite action, type spaces, the same in each period Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Agents send simultaneous reports 2 Each agent i chooses private action x i , t 3 Each agent i chooses public action x 0 , i , t , makes public payment 4 z i , j , t � 0 to each agent j ) Public action x 0 , t = ( x 0 , i , t ) N i = 1 , total transfer y i , t = ∑ ( z j , i , t � z i , j , t ) to agent i (budget-balanced) j Markovian Assumptions: Finite action, type spaces, the same in each period � θ t j θ t � 1 , x t � 1 � Markovian type transitions: ν t = ¯ ν ( θ t j θ t � 1 , x t � 1 ) Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Agents send simultaneous reports 2 Each agent i chooses private action x i , t 3 Each agent i chooses public action x 0 , i , t , makes public payment 4 z i , j , t � 0 to each agent j ) Public action x 0 , t = ( x 0 , i , t ) N i = 1 , total transfer y i , t = ∑ ( z j , i , t � z i , j , t ) to agent i (budget-balanced) j Markovian Assumptions: Finite action, type spaces, the same in each period � θ t j θ t � 1 , x t � 1 � Markovian type transitions: ν t = ¯ ν ( θ t j θ t � 1 , x t � 1 ) � x t , θ t � = ¯ Stationary separable payo¤s u i , t u i ( x t , θ t ) Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Agents send simultaneous reports 2 Each agent i chooses private action x i , t 3 Each agent i chooses public action x 0 , i , t , makes public payment 4 z i , j , t � 0 to each agent j ) Public action x 0 , t = ( x 0 , i , t ) N i = 1 , total transfer y i , t = ∑ ( z j , i , t � z i , j , t ) to agent i (budget-balanced) j Markovian Assumptions: Finite action, type spaces, the same in each period � θ t j θ t � 1 , x t � 1 � Markovian type transitions: ν t = ¯ ν ( θ t j θ t � 1 , x t � 1 ) � x t , θ t � = ¯ Stationary separable payo¤s u i , t u i ( x t , θ t ) ) 9 a “Blackwell policy” χ � - a Markovian decision rule that is e¢cient for all δ close enough to 1, for any starting state Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Decentralized Games (No External Enforcer) In each period t = 1 , 2 , ... Each agent i privately observes signal θ i , t 1 Agents send simultaneous reports 2 Each agent i chooses private action x i , t 3 Each agent i chooses public action x 0 , i , t , makes public payment 4 z i , j , t � 0 to each agent j ) Public action x 0 , t = ( x 0 , i , t ) N i = 1 , total transfer y i , t = ∑ ( z j , i , t � z i , j , t ) to agent i (budget-balanced) j Markovian Assumptions: Finite action, type spaces, the same in each period � θ t j θ t � 1 , x t � 1 � Markovian type transitions: ν t = ¯ ν ( θ t j θ t � 1 , x t � 1 ) � x t , θ t � = ¯ Stationary separable payo¤s u i , t u i ( x t , θ t ) ) 9 a “Blackwell policy” χ � - a Markovian decision rule that is e¢cient for all δ close enough to 1, for any starting state Can we sustain χ � in PBE? Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 15 / 17
Implement the Balanced Team Mechanism When no publicly observed deviation, make payments 1 j , θ t � 1 I � 1 γ j , t ( θ t z i , j , t = � j ) + K i 0 1 � � χ � � ˜ � �� µ j t [ χ � ] j θ t j , θ t � 1 ∞ , ˜ � j 1 @ E u k ¯ θ τ θ τ δ τ � t A + K i I � 1 ∑ ∑ ˜ = θ � � χ � � ˜ � �� � E µ t [ χ � ] j θ t � 1 , ˜ u k ¯ θ τ θ τ τ = t k 6 = j ˜ θ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 16 / 17
Implement the Balanced Team Mechanism When no publicly observed deviation, make payments 1 j , θ t � 1 I � 1 γ j , t ( θ t z i , j , t = � j ) + K i 0 1 � � χ � � ˜ � �� µ j t [ χ � ] j θ t j , θ t � 1 ∞ , ˜ � j 1 @ E u k ¯ θ τ θ τ δ τ � t A + K i I � 1 ∑ ∑ ˜ = θ � � χ � � ˜ � �� � E µ t [ χ � ] j θ t � 1 , ˜ u k ¯ θ τ θ τ τ = t k 6 = j ˜ θ Can we prevent public deviations (=“quitting”) for any history? Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 16 / 17
Implement the Balanced Team Mechanism When no publicly observed deviation, make payments 1 j , θ t � 1 I � 1 γ j , t ( θ t z i , j , t = � j ) + K i 0 1 � � χ � � ˜ � �� µ j t [ χ � ] j θ t j , θ t � 1 ∞ , ˜ � j 1 @ E u k ¯ θ τ θ τ δ τ � t A + K i I � 1 ∑ ∑ ˜ = θ � � χ � � ˜ � �� � E µ t [ χ � ] j θ t � 1 , ˜ u k ¯ θ τ θ τ τ = t k 6 = j ˜ θ Can we prevent public deviations (=“quitting”) for any history? Can think of this as joint IC-IR constraints Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 16 / 17
Implement the Balanced Team Mechanism When no publicly observed deviation, make payments 1 j , θ t � 1 I � 1 γ j , t ( θ t z i , j , t = � j ) + K i 0 1 � � χ � � ˜ � �� µ j t [ χ � ] j θ t j , θ t � 1 ∞ , ˜ � j 1 @ E u k ¯ θ τ θ τ δ τ � t A + K i I � 1 ∑ ∑ ˜ = θ � � χ � � ˜ � �� � E µ t [ χ � ] j θ t � 1 , ˜ u k ¯ θ τ θ τ τ = t k 6 = j ˜ θ Can we prevent public deviations (=“quitting”) for any history? Can think of this as joint IC-IR constraints Problem: transfers may be unbounded as δ ! 1. Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 16 / 17
Implement the Balanced Team Mechanism When no publicly observed deviation, make payments 1 j , θ t � 1 I � 1 γ j , t ( θ t z i , j , t = � j ) + K i 0 1 � � χ � � ˜ � �� µ j t [ χ � ] j θ t j , θ t � 1 ∞ , ˜ � j 1 @ E u k ¯ θ τ θ τ δ τ � t A + K i I � 1 ∑ ∑ ˜ = θ � � χ � � ˜ � �� � E µ t [ χ � ] j θ t � 1 , ˜ u k ¯ θ τ θ τ τ = t k 6 = j ˜ θ Can we prevent public deviations (=“quitting”) for any history? Can think of this as joint IC-IR constraints Problem: transfers may be unbounded as δ ! 1. But: with limited persistence of ˜ θ , the two expectations may be close as τ ! ∞ Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 16 / 17
Sustaining E¢ciency Theorem Take the Markov game with independent private values, which has a zero-payo¤ belief-free static NE. Suppose that a Blackwell policy χ � induces a Markov process with a unique ergodic set (and a possibly empty transient set), and that the ergodic distribution gives a positive expected total surplus. Then for δ large enough, χ � can be sustained in a PBE using Balanced Team Transfers. Susan Athey () Dynamic Mechanism Design Tutorial July 7, 2009 17 / 17
Dynamic Games – In decentralized games, actions and transfers have to be self- enforcing; not commitment mechanism is available to the agents – In many games, transfers are not available – What is the relationship between the outcomes that can be at- tained WITH commitment and transfers, and what can be at- tained without? – When can e¢ciency be sustained as an eqm? – What do equilibria look like for di¤erent discount factors? – E¢ciency includes BB
Literature in Microeconomics on Dynamic Games and Contracts – Collusion: Athey and Bagwell (series of papers) – Repeated Trade: Athey and Miller – Relational Contracts: Levin, Rayo – Continuous time models, principal agent: Sannikov and coauthors – Cost of ex post as opposed to Bayesian equilibrium: Miller Literature in Dynamic Public Finance, Macro – Amador, Angeletos, and Werning; Tsyvinski; Athey, Atkeson, and Kehoe; others
Focus Today: Hidden Information – Hidden actions impt, techniques and applications often di¤erent – Auctions, collusion, bilateral or multilateral trade, public good provision, resource allocation, favor-trading in relationships, mu- tual insurance Contracts, Games, and Games as Contracts
Mechanism Design Approach to Dynamic Games – In static theory, we are familiar with mechanism design approach to analyzing games such as auctions – Use tools such as envelope theorem, revenue equivalence, etc. to characterize equilibria – Analyze constraints – Take this approach to dynamic games – Combine dynamic programming and mechanism design tools – Frontier of current research: fully dynamic games (not repeated)
A Toolkit for Analyzing Dynamic Games and Contracts Abreu-Pearce-Stacchetti and dynamic programming The mechanism design approach to repeated games with hidden in- formation Sustaining e¢ciency with transfers The folk theorem without transfers Dynamic Programming for Dynamic Games
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