Micro II Benny Moldovanu University of Bonn Summer Term 2015 Benny Moldovanu (University of Bonn) Micro II Summer Term 2015 1 / 109
Quasi-Linear Utility x = (k, t 1,.., t I ), where: k K (physical outcomes, projects ), t i R (money) u i (x, θ i ) = v i (k, θ i ) + t i f (θ) = f (θ 1,..θ I ) = (k(θ), t 1 (θ),.., t I (θ)) Definition Efficient SCF f (θ) = (k (θ), t 1 (θ),.., t I (θ)) : 1 Value maximization: θ, k (θ) arg max k i v i(k, θ i ) 2 Budget Balance: θ, i t i (θ) = 0 Benny Moldovanu (University of Bonn) Micro II Summer Term 2015 2 / 109
Example: Allocation of indivisble good Indivisible good owned by seller I buyers k = (y 1,.., y I ) where y i {0, 1} and y i 1 v i (k, θ i ) = v i ((y 1,.., y I ), θ i ) = y i θ i Efficient allocation: y i = 1 if θ i arg max j θ j ; all monetary transfers from buyers go to seller Benny Moldovanu (University of Bonn) Micro II Summer Term 2015 3 / 109
The Vickrey-Clarke-Groves (VCG) Mechanism Direct Revelation Mechanism k(θ) = k (θ) (value maximization) t i (θ) = j i v j(k (θ), θ j ) + h i (θ i ), where h i is arbitrary Theorem The VCG mechanism truthfully implements the value maximizing SCF in dominant strategies Benny Moldovanu (University of Bonn) Micro II Summer Term 2015 4 / 109
The Pivot Mechanism Problem VCG mechanism requires huge transfers to the agents. Solution Appropriate definition of the h i functions Denote by k i (θ i) the value maximizing project in the absence of i Define t i (θ) = j i v j (k (θ), θ j ) + h i (θ i ) = j i v j (k (θ), θ j ) j i v j (k i(θ i ), θ j ) Exercise: Prove that : θ, i t i (θ) 0 Benny Moldovanu (University of Bonn) Micro II Summer Term 2015 5 / 109
Example: Allocation of Indivisible Object Efficient allocation: y i = 1 if θ i arg max j θ j In VCG mechanism: t i (θ) = In pivot mechanism: { 0 + hi (θ i ), if i = arg max θ j arg max θ j + h i (θ i ), otherwise } { ti arg maxj i θ (θ) = j, if i = arg max θ j 0, otherwise } Second price auction! Benny Moldovanu (University of Bonn) Micro II Summer Term 2015 6 / 109
The Green-Laffont Theorem for Bilateral Bargaining I Agent 1 is seller, owns indivisble object, value for object θ 1 Agent 2 is buyer, value for object θ 2 Values are distributed independently on interval [0, 1] according to densities φ 1, φ 2. VCG Mechanism: k (θ) = k (θ 1, θ 2 ) = { 1, if θ1 θ 2 2, otherwise { } t1(θ) 0 + h1 (θ = 2 ), if θ 1 θ 2 θ 2 + h 1 (θ 2 ), otherwise { } t2(θ) θ1 + h = 2 (θ 1 ), if θ 1 θ 2 0 + h 2 (θ 1 ), otherwise } Benny Moldovanu (University of Bonn) Micro II Summer Term 2015 7 / 109
The Green-Laffont Theorem for Bilateral Bargaining II Budget Balance: 1 1 0 H 1 + H 2 + 0 1 1 t 1 (θ) + t 2 (θ) = 0 [t 1 (θ) + t 2 (θ)]φ 1 (θ 1 )φ 2 (θ 2 )dθ 1 dθ 2 = 0 0 0 max[θ 1, θ 2 ]φ 1 (θ 1 )φ 2 (θ 2 )dθ 1 dθ 2 = 0 where H i = E θ i h i. Noting that θ 1 < max[θ 1, θ 2 ] a.e., this yields: H 1 + H 2 < E θ 1 With positive probability B. Moldovanu () April 2014 8 / 19
The Green-Laffont Theorem for Bilateral Bargaining III Participation Constraints: Highest Seller Type : 1 + H 1 1 H 1 0 Lowest Buyer Type : Eθ 1 + H 2 0 H 2 Eθ 1 This yields: H 1 + H 2 Eθ 1 a contradiction! Benny Moldovanu (University of Bonn) Micro II Summer Term 2015 8 / 109
Bayesian Implementation and Payoff Equivalence Benny Moldovanu University of Bonn Summer Term 2015 Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 9 / 109
First-Price Auction n bidders; i.i.d types F (θ i ), ϕ(θ i ) > 0 Symmetric Equilibrium b i (θ i ) = b(θ i ) increasing and differentiable. Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 10 / 109
First-Price Auction U i (θ i, ˆθ i ) = [θ i b(ˆθ i )] F (ˆθ i ) n 1 FOC: b (ˆθ i )F (ˆθ i ) n 1 + (n 1)(θ i b(ˆθ i ))F (ˆθ i ) n 2 ϕ(ˆθ i ) = 0 b (θ i ) = (n 1)(θ i b(θ i ) ϕ(θ i ) F (θ i ) b(θ i ) = θ i Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 11 / 109
First-Price Auction Example n = 2, θ i U[0, 1] b (θ i ) = 1 θ i (θ i b(θ i )); b(0) = 0 Solution: b(θ i ) = 1 2 θ i Expected Utility: (θ i 1 2 θ i) θ i = 1 2 θ2 i Second Price: (θ i E[θ j θ j θ i ])θ i = 1 2 θ2 i Revenue= 1 3 Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 12 / 109
Myerson s Auction Model 1 object, n bidders independent, private values θ i distributed: F i (θ i ), ϕ i (θ i ) > 0 Revelation Mechanism p i (θ) [0, 1], i; n i=1 p i(θ) 1 t i (θ) R, i Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 13 / 109
Myerson s Auction Model q i (θ i ) = E θ i p i (θ i, θ i ) T i (θ i ) = E θ i t i (θ i, θ i ) U i (θ i, ˆθ i ) = θ i q i ( ˆθ i ) + T i ( ˆθ i ) Bidder i s problem: max ˆθ i U i (θ i, ˆθ i ) = U i (θ i, θ i ) Truthtelling condition! Denote Ū i (θ i ) = U i (θ i, θ i ) Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 14 / 109
Myerson s Auction Model Theorem (Myerson) A mechanism {p i, t i } n i=1, is Bayesian incentive compatible if and only if: 1 q i is increasing in θ i 2 Ū i (θ i ) = Ū i (θ i ) + θ i θ i q i (s)ds, where Ū i (θ i ) = U i (θ i, θ i ) Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 15 / 109
1) ˆθ i > θ i ˆθ i q i (ˆθ i ) + T i (ˆθ i ) ˆθ i q(θ i ) + T i (θ i ) Myerson s Auction Model Proof. θ i q i (θ i ) + T i (θ i ) θ i q i (ˆθ i ) + T i (ˆθ i ) (ˆθ i θ i )q i (ˆθ i ) (ˆθ i θ i )q i (θ i ) Ū i (θ i ) is maximum of affine functions convex, equal integral of its derivative. Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 16 / 109
Myerson s Auction Model Proof. (cont.) 2) assume w.l.o.g. θ i > ˆθ iūi (θ i ) U i (θ i, ˆθ i ) = Ū i (θ i ) Ū i (ˆθ i ) q(ˆθ i )[θ i ˆθ i ] = θ i ˆθ i q(s)ds q(ˆθ i )[θ i ˆθ i ] = θ i ˆθ i [q(s) q(ˆθ i )]ds 0 Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 17 / 109
Myerson s Auction Model Consequence: Ū i (θ i ) = Ūi(θ i ) + θ i θ = θ i q i (θ i ) + T i (θ i ) q(s)ds T i (θ i ) = θ i q i (θ i ) + θ i θ q i(s)ds + Ū i (θ i ) Payoff and revenue equivalence! Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 18 / 109
The Optimal Auction ( n ) max {pi,t i } n E θ T i=1 i (θ i ) 1 p i (θ) [0, 1]; n i=1 p i(θ) 1, θ 2 q i (θ i ) is monotone i=1 Observation: Ū i (θ i ) = 0 is optimal Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 19 / 109
The Optimal Auction θ i θ i T i (θ i )ϕ i (θ i )dθ i = θi θ [θ i q i (θ i ) θ i i θ q i (s)ds]ϕ i (θ i )dθ i j = E θ [θ i p i (θ)] θ i θ [ θ i i θ q i (s)ds]ϕ i (θ i )dθ i i Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 20 / 109
The Optimal Auction = θi θ i θ i θ [ θ i i θ q i (s)ds]ϕ i (θ i )dθ i i q i (s)ds θi θ i q i (θ i )F i (θ i )dθ i = θ i θ q i (θ i )[ 1 F i (θ i ) ]ϕ j i(θ i )dθ i ϕ i (θ i ) Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 21 / 109
The Optimal Auction = E θ (p i (θ)[ 1 F i (θ i ) ϕ(θ i ) ]) to conclude: E θ (T i (θ i )) = E θ [p i (θ i )(θ i 1 F i (θ i ) ϕ i (θ i ) )] E θ ( n i=1 T i(θ i )) = E θ [ n i=1 p i(θ)(θ i 1 F i (θ i ) ϕ(θ i ) )] Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 22 / 109
The Optimal Auction Assumption: J i (θ i ) = θ i 1 F i (θ i ) ϕ i (θ i ) increasing Revenue Maximization: 1, {i arg max J j (θ j )} {J i (θ i ) 0} p i (θ) = j 0, otherwise Satisfies monotonicity constraint! Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 23 / 109
The Optimal Auction Assumption: F i = F, i Result: second-price auction with reservation price that satisfies R 1 F (R ) ϕ(r ) = 0 is optimal! Benny Moldovanu (University of Bonn) Bayesian Implementation and Payoff Equivalence Summer Term 2015 24 / 109
Equivalence between Bayesian and Dominant Strategy Incentive Compatible Mechanisms Benny Moldovanu University of Bonn Summer Term 2015 Benny Moldovanu (University of Bonn) Equivalence between Bayesian and Dominant Strategy Summer IncentiveTerm Compatible 2015 Mechanisms 25 / 109
A Problem in Discrete Tomography Problem When does a 0 1 matrix with given row and column sums exist? Consider row sum (3, 2, 2, 1, 1) and two different column sums: 1 1 0 0 1 3 1 1 0 0 0 2 1 1 0 0 0 2 1 0 0 0 0 1 0 1 0 0 0 1 4 4 0 0 1 1 2! 0 0 0 3 1 1 0 0 0 2 1 1 0 0 0 2 1 0 0 0 0 1 1 0 0 0 0 1 5 4 0 0 0 Matrix exist if the vector of column sums is "less diverse" than the vector (5, 3, 1, 0, 0). See Gale (1957), and Ryser (1957) for the general result. Variations (continuous case, densities) are in Kellerer (1961) and Strassen (1965). A. Gershkov, B. Moldovanu, X.Shi () Computed Tomography & Mechanism Design January 2012 3 / 19
A. Gershkov, B. Moldovanu, X.Shi () Computed Tomography & Mechanism Design January 2012 4 / 19
The Monotone Lift Problem When unique reconstruction is not possible, are there solutions with special properties? Theorem (Gutmann et al. (1991)) Let φ = φ(x 1, x 2,..x n ) be measurable on [0, 1] n with 0 φ 1.Assume that the one-dimensional marginals Φ i (x i ) = φ(x 1, x 2,..x n )dx i are non-decreasing in x i, i = 1, 2,..n. Then there exists ψ measurable on [0, 1] n such that 0 ψ 1, ψ has the same marginals as φ, and moreover, ψ is non-decreasing in each coordinate. A. Gershkov, B. Moldovanu, X.Shi () Computed Tomography & Mechanism Design January 2012 5 / 19
Monotone Lift: Example Example φ = 2 4 4 10 4 2 6 12 4 6 4 14 10 12 14 = ψ = 2 4 4 10 4 4 4 12 4 4 6 14 10 12 14 Note that i,j (ψ ij ) 2 i,j (φ ij ) 2. A. Gershkov, B. Moldovanu, X.Shi () Computed Tomography & Mechanism Design January 2012 6 / 19
The Independent Private Values Model with Linear Utility K social alternatives and N agents. The utility of agent i in alternative k is given by ai k x i + ci k + t i where x i [0, 1] is agent i s private type, where ai k, ck i R with ai k 0, and where t i R is a monetary transfer. Types are drawn independently of each other, according to strictly increasing distributions F i. Type x i is private information of agent i. Manelli and Vincent assume: K = N ; a i i = 1, a j i = 0 for any j = i; c k i = 0 for any i, k. A. Gershkov, B. Moldovanu, X.Shi () Computed Tomography & Mechanism Design January 2012 7 / 19
Incentive Compatible Mechanisms I Definition A direct revelation mechanism (DRM) M is given by K functions q k : [0, 1] N [0, 1] and N functions t i : [0, 1] N R where q k (x 1,..., x N ) is the probability with which alternative k is chosen, and t i (x 1,..., x N ) is the transfer to agent i if the agents report types x 1,..., x N. Definition A DRM M is Dominant-Strategy Incentive Compatible (DIC) if truth-telling constitutes a dominant strategy equilibrium in the game defined by M and the given utility functions. A DRM M is Bayes-Nash Incentive Compatible (BIC) if truth-telling constitutes a Bayes-Nash equilibrium in the game defined by M and the given utility functions. A. Gershkov, B. Moldovanu, X.Shi () Computed Tomography & Mechanism Design January 2012 8 / 19
Incentive Compatible Mechanisms II Fact A necessary condition for M to be DIC is that, for each agent i, and for any signals of others, the function K k=1 a k i qk (x 1,..., x N ) is non-decreasing in x i. Moreover, any K functions q k that satisfy this condition are part of a DIC mechanism. Fact A necessary condition for M to be BIC is that, for each agent i, the function K k=1 a k i Qk i (x i ) is non-decreasing, where i, k, Qi k ( x i ) = q k (x 1,..., x i, x i, x i+1,..., x N )df i, [0,1] N 1 is the expected probability that alternative k is chosen if agents j = i report truthfully while agent i reports type x i.moreover any K functions q k that satisfy this condition are part of a BIC mechanism. A. Gershkov, B. Moldovanu, X.Shi () Computed Tomography & Mechanism Design January 2012 9 / 19
Equivalent Mechanisms Definition 1 Two mechanisms M and M are P-equivalent if, for each i, k and x i, it holds that Q k i (x i ) = Q k i (x i ), where Q k i and Q k i are the conditional expected probabilities associated with M and M, respectively. 2 Two mechanisms M and M are U-equivalent if they provide the same interim utilities for each agent i and each type x i of agent i. For each agent i, interim utility is obtained (up to a constant) by integrating the function K k=1 a k i Qk i (x i ) with respect to x i - this is the Payoff Equivalence Theorem. Thus P-equivalence implies U-equivalence. A. Gershkov, B. Moldovanu, X.Shi () Computed Tomography & Mechanism Design January 2012 10 / 19
P- and U-Equivalence for 2 Alternatives Since q 2 (x 1,..., x N ) = 1 q 1 (x 1,..., x N ), we have 2 ai k Qi k (x i ) = ai 2 + (ai 1 ai 2 )Qi 1 (x i ), k=1 and therefore U-equivalence implies P-equivalence (the two notions coincide). Theorem Assume that K = 2. Then for any BIC mechanism there exists a P-equivalent (and thus U-equivalent) DIC mechanism. A. Gershkov, B. Moldovanu, X.Shi () Computed Tomography & Mechanism Design January 2012 11 / 19
U-Equivalence for Symmetric Settings Theorem Assume that a k i = a k j = a k for all k, i, j, and that F i = F for all i. Moreover, assume that 0 = a 1 a 2 a K = 1. Then for any symmetric, BIC mechanism there exists an U-equivalent symmetric DIC mechanism Proof shows how to achieve U-equivalence using only the 2 alternatives with highest and lowest slope, respectively. Thus, U-equivalence does not necessarily ensure that the ex-ante probabilities of different alternatives are preserved. A. Gershkov, B. Moldovanu, X.Shi () Computed Tomography & Mechanism Design January 2012 12 / 19
Multidimensional Types Benny Moldovanu University of Bonn Summer Term 2015 Benny Moldovanu (University of Bonn) Multidimensional Types Summer Term 2015 26 / 109
Multidimensional Types v i (k, θ i ) = θi k Types - ind. distributed Vector Field: qi k(θ i) = E θ i [pi k(θ)] T i (θ i ) = E θ i [t i (θ)] U i (θ i, ˆθ i ) = θ i q i (ˆθ i ) + T i (ˆθ i ) Benny Moldovanu (University of Bonn) Multidimensional Types Summer Term 2015 27 / 109
Multidimensional Types + Bayesian Implementation Theorem (Jehiel, Moldovanu, Stacchetti, JET (1999)) The mechanism { {p k i } k K, t i }i I is Bayes-Nash incentive compatible if and only if: 1 [q i (θ i ) q i ( ˆθ i )] [θ i ˆθ i ] 0 2 Û i (θ i ) = Û i (θ i ) + θ i θ i Path independence! q i (s) ds, i, θ i Benny Moldovanu (University of Bonn) Multidimensional Types Summer Term 2015 28 / 109
Multidimensional Types Example two objects 1 buyer, valuations θ A, θ B Mechanism: P A.P B, P AB < P A + P B Benny Moldovanu (University of Bonn) Multidimensional Types Summer Term 2015 29 / 109
Multidimensional Types + Private Values Observation Lemma k(θ i, θ i ) = k(ˆθ i, θ i ) t i (θ i, θ i ) = t i (ˆθ i, θ i ) k(θ i, θ i ) = l, k(ˆθ i, θ i ) = ˆl ˆθˆl i θˆl i ˆθ l i θ l i θ i, ˆθ i, θ i Weak monotonicity! Benny Moldovanu (University of Bonn) Multidimensional Types Summer Term 2015 30 / 109
Multidimensional Types + Private Values Proof. θi l + t i (θ i, θ i ) θˆl i + t i (ˆθ i, θ i ) ˆθˆl i + t i (ˆθ i, θ i ) ˆθ i l + t i (θ i, θ i ) θi l + ˆθˆl i θˆl i + ˆθ i l ˆθˆl i θˆl i ˆθ i l θi l Benny Moldovanu (University of Bonn) Multidimensional Types Summer Term 2015 31 / 109
Multidimensional Types + Private Values Theorem (Saks-Yu) Assume types spaces are convex, and consider k(θ). There exist t 1 (θ),, t I (θ) such that f (θ) = [k(θ), t 1 (θ),, t I (θ)] is truthfully implementable in dominant strategies if and only if k(θ) is weakly monotone. Benny Moldovanu (University of Bonn) Multidimensional Types Summer Term 2015 32 / 109
Affine Maximizers α 1,, α I R + R 1,, R K R k(θ) arg max k [ I ] α i V i (k, θ i ) + R k i=1 Benny Moldovanu (University of Bonn) Multidimensional Types Summer Term 2015 33 / 109
Affine Maximizers Theorem (Roberts) Let Θ i = R K and K 3. Then f (θ) is truthfully implementable in dominant strategies if and only if the associated k(θ) is an affine maximizer. Benny Moldovanu (University of Bonn) Multidimensional Types Summer Term 2015 34 / 109
Interdependent Values Benny Moldovanu University of Bonn Summer Term 2015 Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 35 / 109
Akerlof s Model Sellers with cars of quality θ [θ, θ] F (θ); F (θ) = f (θ) > 0 Utility = { θ P, R(θ) + P, buyer seller Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 36 / 109
Complete Information Equilibrium Condition P(θ) = θ S(P) = {θ R(θ) θ} = D(P) EfficientTrade! Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 37 / 109
Incomplete Information S(P) = {θ R(θ) P} E(P) E[θ θ S(P)] 0, E(P) < P D(P) = [θ, θ], E(P) = P [θ, θ], E(P) > P Equilibrium : P = E(P ) Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 38 / 109
Incomplete Information Example R(θ) = 2 θ; F (θ) = θ on [0, 1] 3 E[θ 2 3 θ P] = E[θ θ 3 2 P] = 3 4 P P = 3 4 P P = 0 No car is sold at all! Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 39 / 109
Interdependent Values u i (x, θ 1,, θ J ) = V i (k, θ 1,, θ J ) + t i t i (θ) = j i VCG Transfer: V j (k, θ) + h i (θ i ) Problem: t i depends on θ i! Solution? Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 40 / 109
Interdependent Values Example (1 Object, 2 Bidders) V i (k, θ) = k = 1, 2 { 0, k i aθ i + bθ i, k = i where a > b > 0 k (θ) = { 1, θ 1 θ 2 2, θ 1 < θ 2 Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 41 / 109
Interdependent Values Example (1 Object, 2 Bidders) Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 42 / 109
Important Condition V i (θ) θ i > V j(θ), i, j θ i Equilibrium Notion Nash Equilibrium (ex-post) Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 43 / 109
Bilateral Bargaining Seller :V S (θ S, θ B ) θ S Buyer :V B (θ S, θ B ) θ B V S θ S > V B θ S ; V B θ B > V S θ B Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 44 / 109
Bilateral Bargaining Definition V B (θ S, θ B (θ S)) = V S (θ S, θ B (θ S)) V S (θ S (θ B), θ B ) = V B (θ S (θ B), θ B ) k (θ) = { S, V S (θ) V B (θ) B, V B (θ) > V S (θ) Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 45 / 109
Bilateral Bargaining Definition (cont.) { tb (θ) = 0, k = B V S (θ S, θb (θ S), k = S { ts (θ) = 0, k = S V B (θs (θ S), θ B ), k = B Modified VCG payments Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 46 / 109
Conditions for efficient trade (Fieseler, Kittsteiner, Moldovanu, JET (2003)) E θs [V B (θ S, θ B )] P E θb [V S ( θ S, θ B )] Akerlof s case: E θs [V B (θ S )] P V S ( θ S ) 1 2 < 2 3 Benny Moldovanu (University of Bonn) Interdependent Values Summer Term 2015 47 / 109
Multidimensional Types+ Interdependent Values Benny Moldovanu University of Bonn Summer Term 2015 Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 48 / 109
Multidimensional Types + Interdependent Values Example 2 agents, i = 1, 2 2 alternatives, k = A, B Vi k (θ1 k ), i = 1, 2; k = A, B (Only agent 1 is informed) V i (k, θ 1,, θ J ) Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 49 / 109
Multidimensional Types + Interdependent Values Example (cont.) Value Maximization k(θ 1 ) arg max k 2 V i (k, θ1 k ) i=1 k(θ 1 ) arg max k Incentives for agent 1: [V 1 (A, θ A 1 ) + t A 1, V 1 (B, θ B 1 ) + t B 1 ] Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 50 / 109
Multidimensional Types + Interdependent Values Example (cont.) microii.png Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 51 / 109
Multidimensional Types + Interdependent Values Example (cont.) V 1 (A,θ A 1 ) θ A 1 V 1 (B,θ B 1 ) θ B 1 Congruence Condition [ 2 θ1 A = θ B 1 ] i=1 V i(a, θ1 A [ ) 2 ] i=1 V i(b, θ1 B) Non-generic condition! Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 52 / 109
Theorem Theorem (Jehiel et al., Econometrica (2001)) For generic utility functions only constant social choice functions are truthfully implementable in ex-post equilibrium. Robust implementation is impossible! Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 53 / 109
Multidimensional Types + Conservative Vector Fields Incentive compatibility the vector field q i is 1) monotone 2) conservative 2 q k i (θ i) θ k i θk i = 2 qi k (θ i ) θi k θi k Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 54 / 109
Mechanism Design Without Money Definition L i (x, i ) = {y y i x} A social choice function f is monotonic if θ, L i (f (θ), θ i ) L i (f (θ), θ i ) i f (θ) = f (θ ) = (θ i, θ i ) Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 55 / 109
Mechanism Design Without Money Proof GS (Sketch). 1 R i = p and f DIC f is monotonic 2 R i = p and f DIC and onto f is pareto efficient 3 f is monotonic and efficient f is dictatorial Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 56 / 109
Mechanism Design Without Money Usual setting with general utility function U i (x, θ i ) - private values let R i = { i i = i (θ i ) for some θ i } ordinal preferences Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 57 / 109
Impossibility Theorem Theorem (Gibbard-Satterthwaite) Assume that X is finite with X 3, and that i, P = R i, where P is the set of all rational preferences (without indifference). Let f be onto and dominant strategy incentive compatible; then f is dictatorial! Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 58 / 109
Mechanism Design Without Money Possibility results: 1 X = 2 - simple majority rule 2 single peaked preference Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 59 / 109
Single Peaked Preferences Theorem Let preferences be single-peaked and let the number of agents be odd. Then the SCF that chooses at each profile the median peak is dominant strategy incentive compatible. Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 60 / 109
Single Peaked Preferences Theorem (Moulin 1980) Consider a setting with n agents, and let R i = SP i (with respect to a given order K). Let f be DIC, anonymous and unanimous. Then there exist n 1 phantom voters such that f chooses the median of all voters peaks. Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 61 / 109
Successive Voting With Thresholds Voting on single alternatives in the SP order (say 1, 2,, K). k is chosen if the number of yes votes is at least τ(k), where τ(k) τ(k ), k k. If no alternative passes threshold, K is chosen. Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 62 / 109
Successive Voting With Thresholds Sincere Strategy for voter with peak on k: ( No, No,, No Yes, Yes,, Yes 1, 2,, k 1 k, k + 1,, K Monotone strategy: ) ( No, No,, No Yes, Yes,, Yes 1, 2,, l l + 1,, K ) Sincere monotone Markov strategies Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 63 / 109
Successive Voting With Thresholds Theorem Assume that all players besides i use monotone strategies. Then it is optimal for i to use the sincere strategy. In particular, sincere voting is an ex-post perfect Nash equilibrium. Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 64 / 109
Successive Voting With Thresholds Theorem Equivalence for single-peaked preferences. Anonymous Succsessive Unanimous voting with DIC mechanisms decreasing threshold τ(k) Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 65 / 109
Optimization How many anonymous, unanimous, DIC mechanisms are there? Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 66 / 109
Optimization 2 alternatives, A, B n mechanisms (supermajority) l A - number of phantoms on A l B = n 1 l A utility: u A (x), u B (x), x [0, 1] types x i.i.d., distribution F single crossing: Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 67 / 109
Optimization Definition u A L = E [ u A (x) x x AB] u A H = E [ u A (x) x x AB] u B L = E [ u B (x) x x AB] u B H = E [ u B (x) x x AB] Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 68 / 109
Optimization γ = u A L ub L u A L ub L + ub H ua H > 0 First order conditions: changing l A to l A + 1 or l A 1 should not be beneficial l A = [nγ] Benny Moldovanu (University of Bonn) Multidimensional Types+ Interdependent Values Summer Term 2015 69 / 109
Two-Sided Matching Gale-Shapley 1961 Benny Moldovanu University of Bonn Summer Term 2015 Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 70 / 109
Two-Sided Matching M = {m 1,, m n } W = {w 1,, w p } M W = P(m i ) = w j, w k, m i w l P(w j ) = m i, m l, w j m n Ranked ordinal lists Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 71 / 109
Two-Sided Matching Matching: µ: M W M W 1 µ 2 (x) = x, x M W 2 µ(m) W {m} 3 µ(w) M {w} Individual rationality: µ(x) x x, x M W Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 72 / 109
Two-Sided Matching Stability: µ is stable if: 1 µ is individually rational, and 2 {m, w} such that µ(m) w and w µ(m) m m µ(w) w Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 73 / 109
Two-Sided Matching Theorem (Gale, Shapley (1962)) For any two-sided market, the set of stable matchings is non-empty. Proof. Deferred acceptance algorithm (Gale-Shapley). Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 74 / 109
Two-Sided Matching Example P(m 1 ) = w 2 w 1 w 3 P(w 1 ) = m 1 m 3 m 2 P(m 2 ) = w 1 w 3 w 2 P(w 2 ) = m 3 m 1 m 2 P(m 3 ) = w 1 w 2 w 3 P(w 3 ) = m 1 m 3 m 2 µ 1 = w 1 w 2 w 3 m 1 m 2 m 3 µ 2 = w 2 w 1 w 3 m 1 m 2 m 3 Stable! Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 75 / 109
The Roommate Problem P(a)=bcd P(b)=cad P(c)=abd P(d)=abc µ 1 = ab cd µ 2 = a b c d µ 3 = ab dc None is stable! Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 76 / 109
Lattice Operator µ(m), if µ(m) µ (m) µ M µ (m) = m µ (m), otherwise µ(w), if µ(w) µ (w) µ M µ (w) = w µ (w), otherwise and analogously for M. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 77 / 109
Lemma Lemma (Conway) Let µ, µ be stable matchings. Then µ M µ and µ M µ are also stable matchings. Consequence: M-Optimal and W-Optimal stable matchings. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 78 / 109
Strategic Questions Theorem There is no mechanism such that: 1. It is a dominant strategy for all agents to state preferences truthfully. 2. A stable matching is chosen for every report. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 79 / 109
Strategic Questions Proof. M = {m 1 m 2 } W = {w 1 w 2 } P(m 1 ) = w 1 w 2 P(w 1 ) = m 2 m 1 P(m 2 ) = w 2 w 1 P(w 2 ) = m 1 m 2 µ M = m 1 m 2 w 1 m 2 µ W = m 1 m 2 w 2 w 1 Suppose Γ(p) = µ M Consider strategy Q(w 2 ) = m 1 then Γ(p w2, Q(w 2 )) = µ W Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 80 / 109
Strategic Questions Theorem Let Γ = Γ M be the mechanism that assigns to each profile of reports the M-optimal stable matching. Then truthful reporting is a dominant strategy for all men. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 81 / 109
Strategic Questions Theorem Consider a Nash equilibrium of Γ M where men use their dominant strategies (and women respond optimally). Then the outcome is a stable matching for the true preferences! Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 82 / 109
Strategic Questions Proof. P - True Preferences. P stated preferences. Let µ = Γ(P ). Assume {m,w} block µ. Then, in Γ M (p ) m must have proposed to w, and she rejected him. Consider P (w) = m then {m, w} would be matched. Contradiction to P being an equilibrium. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 83 / 109
Many-to-one Matching n firms: F i = {F 1,, F n } m workers: W = {W 1,, W m } matching: µ : F W 2 F W 1 µ(w) = 1, w W 2 µ(w) = w, if µ(w) / F 3 µ(f ) W [µ(f ) = ] 4 µ(w) = F i w µ(f i ) Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 84 / 109
Many-to-one Matching P(w i ) = F i, F j,, F k, w i, P(F j ) = S 1.S 2.,, where S i W (sets of workers) Definition Ch F (S) = S S S, it holds that S S. F Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 85 / 109
Many-to-one Matching Matching is blocked by {w, F } if: 1 µ(w) F 2 F w µ(w) 3 w Ch F (µ(f ) {w}) Individual rationality: µ(w) w w µ(f ) = Ch F (µ(f )) F Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 86 / 109
Many-to-one Matching Definition Firms have substitutable preferences if: S, w, w S w Ch F (S) w Ch F (S \ w ). Theorem If all firms have substitutable preferences, then the set of stable matchings is not empty. Proof. Run G-S algorithm with firms proposing. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 87 / 109
The Assignment Game M W = V (S) = 0 if S M, or S W V (S) = max µ S m S v(m, µs (m)) where µ S is a matching on coalition S. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 88 / 109
The Assignment Game µ is optimal if V (M W ) = m M v(m, µ (m)) w 1 w 2 w 3 m 1 10 11 7 m 1 6 8 2 m 3 5 5 9 Optimal matching generically unique. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 89 / 109
The Assignment Game Interpretation: M-sellers, each has indivisible object; reserve price: c m. W-buyers, each wants at most one object; value for object m: r wm. v(m, w) = max{0, r wm c m } Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 90 / 109
The Assignment Game Stable payoff vectors 1 feasible: i M W x i V (M W ) 2 ind. rational: x i 0, i M W 3 no blocking: x m + x w v(m, w), m, w Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 91 / 109
The Assignment Game Example w 1 w 2 m 1 5 3 m 2 7 8 (x m1, x w1, x m2, x w2 ) (2.5, 2.5, 4, 4) (1, 4, 3, 5) Stable Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 92 / 109
The Assignment Game Theorem (Shapley-Shubik) The set of stable payoff vectors is not empty. Proof. ( min x m + ) x w m M w W s.t. 1 x i 0 i M W 2 x m + x w v(m, w), m, w Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 93 / 109
The Assignment Game Proof. (cont.) Program has solution. Let R be the minimum. Need to show feasibility R V (M W ). Dual program max i,j p ij v(i, j) s.t. 1 2 i p ij 1, j j p ij 1, i 3 p ij 0, i, j Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 94 / 109
The Assignment Game Proof. (cont.) Dual has solution with p ij {0, 1}, p ij = 1 i M, j W or vice-versa. The max achieved equals R (duality theorem). Obvious now that R V (M W ) Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 95 / 109
The Assignment Game Lemma Let x, y be stable payoff vectors. Define: u m = max(x m, y m ), m M u w = min(x w, y w ), w W Then u is also a stable payoff vector (and analogously for women). Proof. Exercise! Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 96 / 109
The Assignment Game Consequence: The set of stable payoff vectors is a lattice with a maximal and minimal element. Proof. Above Lemma + compactness Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 97 / 109
The Assignment Game Example (homogeneous objects) W M 25 11 23 16 20 19 17 21 10 22 6 24 5 competitive prices 19 P 20 19 20 w 1 6 5 w 2 46 3 w 3 1 0 m 1 8 9 m 2 3 4 m 3 0 1 Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 98 / 109
The Assignment Game Example (m 1, m 2, w 1, w 2 ) P 1, P 2 (1, 0, 2, 3) (1, 0) (3, 2, 0, 1) (3, 2) w 1 w 2 m 1 3 4 m 2 1 3 Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 99 / 109
The Assignment Game Example (cont.) Vickrey Prices P 1 = 1; P 2 = 0 (each buyer pays externality) w 1 w 2 m 1 3 4 m 2 1 3 Outcome equivalent to minimal stable payoff vector (buyer optimal). Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 100 / 109
The Simultaneous Ascending Clock Auction M - objects; W - buyers c m r wm (c m and r wm are integers) Assume M contains dummy object m 0 : c m0 = 0 r wm0 = 0, w Dummy can be assigned to several buyers Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 101 / 109
The Simultaneous Ascending Clock Auction Let P be a vector of prices, one for each object in M Demand of w at P D w (P) = {m M r wm P m max m M (r wm P m )} Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 102 / 109
The Simultaneous Ascending Clock Auction Definition P is quasi-competitive if there exists a matching µ such that: 1 µ(w) = m m D w (P) 2 µ(w) = w m 0 D w (P) Competitive equilibrium (P, µ): 1 quasi-competitive + 2 P m = c m if m / µ(w ) Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 103 / 109
The Simultaneous Ascending Clock Auction Lemma 1 Every competitive equilibrium yields a stable payoff vector. 2 Every stable payoff vector can be obtained via a competitive equilibrium. Proof. Exercise. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 104 / 109
Hall s Theorem Let B,C be disjoint sets b B, D b C Can we find a matching µ : B C such that 1 b, µ(b) D b 2 µ(b) µ(b ), b, b Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 105 / 109
Hall s Theorem Necessary condition: B B, b B D b B Theorem The above condition is also sufficient! Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 106 / 109
Auction P m (i) = c m, m Bidders announce D w (P(1)) Find µ such that µ(w) D w (P(1)) If possible stop otherwise W such that W > w W D w (P(1)) There exists an overdemanded set of objects Choose M M to be a minimal over-demanded set. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 107 / 109
Auction P m (2) = Continue... { P m (1) + 1, m M P m (1), otherwise Algorithm must stop since at high prices only m 0 D w (P(K)), w Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 108 / 109
Auction Theorem The ascending auction stops at the minimum quasi-competitive price vector. In the corresponding direct revelation mechanism, truthfully revealing valuations is a dominant strategy for each buyer. Benny Moldovanu (University of Bonn) Two-Sided Matching Gale-Shapley 1961 Summer Term 2015 109 / 109