Maskin Meets Abreu and Matsushima

Transcription

1 Maskin Meets Abreu and Matsushima August 2, 2017 Yi-Chun Chen (National University of Singapore) Takashi Kunimoto (Singapore Management University) Yifei Sun (UIBE, Beijing) Siyang Xiong (Bristol)

2 What is Nash Implementation? A social choice function (SCF) is given as a single-valued mapping from states to outcomes. An SCF is Nash implementable if there exists a mechanism (or game-form) that satisfies the following two requirements: Existence: there always exists a Nash equiilbrium whose outcome is socially desirable; and Uniqueness: every Nash equilibrium results in the socially desirable outcome. 2

3 Maskin (1977, 1999) Maskin shows that his monotonicity condition is a necessary and almost sufficient condition for Nash implementation. However, the sufficiency part suffers from the following deficiencies: 1. Maskin s mechanism uses the integer game construction; 2. Maskin ignores mixed strategy NE; 3. Maskin cannot handle the case of two agents. 3

4 Abreu and Matsushima (1992, 1994) AM (1992, 94) dispense with Maskin monotonicity, while resolving the first two deficiencies of Maskin s characterization. In doing so, they twist the model as follows: They consider an environment with lotteries and transfers AM (1992) rather appeal to a different notion of implementation, i.e., virtual implementation and use rationalizability rather than NE; and AM (1994) appeal to a somewhat non-standard solution concept, i.e., iterated weak dominance. 4

5 We unify Maskin and Abreu and Matsushima We construct a simple mechanism that resolves all the problems. We achieve this in an economic environment with lotteries and transfers. More formally, Theorem: An SCF is Nash implementable by a finite mechanism with no obviously questionable features if and only if it satisfies Maskin monotonicity. 5

6 Additional Motivation: Robustness to Information Perturbations Chung and Ely (2003) and Aghion, Fudenberg, Holden, Kunimoto, and Tercieux (2012) show Maskin monotonicity is a necessary and almost sufficient condition for robust implementation under information perturbations in undominated Nash equilibrium and subgame perfect equilibrium, respectively. Since the Nash equilibrium correspondence is upper hemi-continuous in a finite game, our result shows that Maskin monotonicity is a necessary and sufficient condition for robust implementation under information perturbations. 6

7 Model I = {1,..., I}: finite set of players A = ( X R I) : set of allocations/lotteries Θ i : finite set of types each θ i induces a quasilinear EU u i (, θ i ) : A R; u i (, 0, θ i ) is bounded 7

8 Model Cont. θ Θ i I Θ i is called a state f : Θ A: a social choice function (SCF) θ is commonly known among the agents but unknown to the designer. 8

9 Maskin Monotonicity Define the strict lower-counter set of lottery l for type θ i as L i (l, θ i ) = { l A : u i (l, θ i ) > u i ( l, θ i )}. Define the strict upper-contour set of lottery l for type θ i as, U i (l, θ i ) = { l A : u i ( l, θ i ) > ui (l, θ i )}. Definition: θ, θ Θ, An SCF f satisfies strict Maskin monotonicity if, f (θ) f(θ ) i I s.t. L i (f (θ), θ i ) U i (f (θ), θ i ). 9

10 Best Challenge Scheme Whenever L i (f( θ), θ i ) U i (f( θ), θ i ), fix some l( θ, θ i ) L i (f( θ), θ i ) U i (f( θ), θ i ). The best challenge scheme for type θ i against θ Θ is defined as B θi ( θ) = f( θ), if L i (f( θ), θ i ) U i (f( θ), θ i ) = ; l( θ, θ i ), if L i (f( θ), θ i ) U i (f( θ), θ i ). 10

11 Dictator Lotteries The Assumption: θ i θ i u i( ; θ i ) u i ( ; θ i ); AM (1992) show under the Assumption, there exists a menu of dictator lotteries l i : Θ i A such that for any θ i, θ i with θ i θ i, u i ( l i (θ i ), θ i ) > ui (l i (θ i ), θ i). 11

12 Mechanism Consider I = 2. Each agent i reports m i = ( m 1 i, m2 i, m3 i ) Θi Θ Θ i. where we write m 2 i = ( m 2 i,i, m2 i,j) Θ. The outcome is either [1 checks 2] or [2 checks 1] with equal probability. Two key properties we wish to exploit: consistency: m 2 i = m2 j. no challenge: B m 3 i ( m 2 j ) = f(m 2 j ). 12

13 i Checks j: Outcome Function If there is neither inconsistency nor challenge, then implement f(m 2 j ). If there is either inconsistency or challenge, then implement the lottery: 1 2 ( l i (m 1 i ) + l j (m1 j )) with probability ε B m 3 i (m 2 j ) with probability 1 ε 13

14 i Checks j: Transfer Rule We choose η > 0 large enough so that Transfer to agents m 2 i,j = m2 j,j m 2 i,j m2 j,j m 2 i,j = m1 j or m2 i,j m1 j m 2 i,j = m1 j m 2 i,j m1 j (τ i (m), τ j (m)) (0, 0) (η, η) ( η, η) To sum up, m 1 i controls dictator lotteries l i ; m 2 i controls consistency and transfers; m 3 i controls whether to challenge m 2 j. 14

15 Step 1: Contagion of Truth 1st report 2nd report Agent 1 θ 1 β 1, β 2 Agent 2 α 2 γ 1, γ 2 1st report 2nd report Agent 1 θ 1 β 1, β 2 Agent 2 α 2 θ 1, γ 2 1st report 2nd report Agent 1 θ 1 θ 1, β 2 Agent 2 α 2 θ 1, γ 2 15

16 Step 1: Contagion of Truth Continued 1st report 2nd report Agent 1 θ 1 β 1, β 2 Agent 2 θ 2 γ 1, γ 2 1st report 2nd report Agent 1 θ 1 θ 1,θ 2 Agent 2 θ 2 θ 1,θ 2 Therefore, truth-telling constitutes a pure-strategy NE. 16

17 Step 2: No Randomization on 2nd Report Assume both agents randomize on their 2nd reports. Inconsistency (and dictator lotteries) occurs with positive probability. Hence, for some (β 1, β 2 ) (γ 1, γ 2 ), 1st report 2nd report Agent 1 θ 1 β 1, β 2 Agent 2 θ 2 γ 1, γ 2 A contradiction to Step 1 (contagion of truth). 17

18 Step 2: No Randomization on 2nd Report Continued. Suppose that only agent 1 (say) randomizes on the 2nd report. Agent 2 believes with positive probability that his dictator lottery is triggered. 1st report 2nd report Agent 1 β 1, β 2 Agent 2 θ 2 γ 1,γ 2 Again, a contradiction to Step 1 (contagion of truth). 1st report 2nd report Agent 1 γ 1,θ 2 Agent 2 θ 2 γ 1,θ 2 18

19 Step 3: Consistency No randomization on 2nd report means 1st report 2nd report Agent 1 β 1,β 2 Agent 2 γ 1,γ 2 If β 1 γ 1, then agent 1 s dictator lottery is triggered with positive probability and hence 1st report 2nd report Agent 1 θ 1 θ 1,β 2 Agent 2 θ 1,γ 2 Contradiction! Hence, β 1 =γ 1, and similarly, β 2 =γ 2. 19

20 Step 4: No Challenge Consistency on 2nd report implies 2st report 3nd report Agent 1 θ 1, θ 2 δ 1 Agent 2 θ 1, θ 2 δ 2 WANT: L i (f( θ), θ i ) U i (f( θ), θ i ) =, i {1, 2}. This implies no challenge is invoked and f( θ) = f (θ) (by Maskin monotonicity). 20

21 Step 4: No Challenge Continued. We prove this by considering the following two cases: Case 1: Each agent i tells the truth in his 1st report. By Step 1 (Contagion of Truth), everyone announces θ in his second report. Then, challenging the truth leads to an allocation in L i (f(θ), θ i), which makes agent i strictly worse off. 21

22 Step 4: No Challenge Continued. Case 2: At least one agent i tells a lie in his first report. Let m i supp(σ i ) be such a message with m 1 i θ i. Under m i, agent i must believe the dictator lotteries are not triggered: and L j (f( θ), θ j ) U j (f( θ), m 3 j ) =, m j supp(σ j ); L i (f( θ), θ i ) U i (f( θ), m 3 i ) =. 22

23 Step 4: No Challenge Continued. Let m i = (m 1 i, m2 i, θ i). Since m i is a best response to σ i, m i cannot be worse than m i against σ i : L i (f( θ), θ i ) Ui (f( θ), θ i ) =. WANT: L i (f( θ), θ i ) U i (f( θ), m 3 i ) =, m i supp(σ i ). If m 3 i = θ i, we are done. So, fix m i supp(σ i ) with m 3 i θ i. Suppose by way of contradiction that L i (f( θ), θ i ) U i (f( θ), m 3 i ). 23

24 Step 4: No Challenge Continued. Then, the dictator lotteries are triggered with positive probability for both i and j. Step 1 (contagion of truth) implies that θ = θ and L i (f(θ), θ i) U i (f(θ), m i). m i generates an allocation in L i (f(θ), θ i), which makes agent 1 worse off than ( m 1 i, m2 i, θ i) This contradicts m i supp(σ i ). 24

25 Extensions Rationalizable implementation of any Maskin monotonic SCF (Bergemann, Morris, Tercieux (2011)); Continuous implementation of any Maskin monotonic SCF (Oury and Tercieux (2012)) Probabilistically sophisticated agents; Social choice correspondences; Small transfers (AM (1994)). 25