Lectures on Robust Mechanism Design at BU

Lectures on at BU Stephen Morris January 2009

Introduction I Mechanism Design and Implementation literatures are theoretical successes I mechanisms seem to complicated to use in practise... I successful applications of theory include ad hoc restrictions... I simplicity, parametric, detail free, ex post equilibrium, Wilson doctrine...

Wilson Doctrine Game theory has a great advantage in explicitly analyzing the consequences of trading rules that presumably are really common knowledge; it is de cient to the extent that it assumes other features to be common knowledge, such as one agent s probability assessment about another s preferences or information. I foresee the progress of game theory as depending on successive reductions in the base of common knowledge required to conduct useful analyses of practical problems. Only by repeated weakening of common knowledge assumptions will the theory approximate reality. Wilson (1987)

Relaxing Common Knowledge Assumptions I in game theory, Harsanyi (1967/68), Mertens and Zamir (1985) established that relaxing common knowledge assumptions is equivalent to adding types... I environments with incomplete information can be modeled as a Bayesian game where wlog there is common knowledge among players of (i) each player s type spaces (ii) each type s beliefs over types of other players I economic analysis assumes smaller type spaces than universal type space yet maintains common knowledge of (i) and (ii) I are the implicit common knowledge assumptions that come from working with small type spaces problematic? perhaps especially in mechanism design (Neeman (2004))

Our Agenda (circa 1998) I introduce rich (higher order belief) types into mechanism design literature. I relax (implicit) common knowledge assumptions by comparative statics on the type space, going from "naive" type space to "universal" type space I in particular 1. brie y establish a few easy benchmark "abstract" results 2. develop a close link between this robust approach and applications I ten years and seven papers/notes later, we are kind of done with (1)

Seven Papers 1. 2. Ex Post Implementation 3. Robust Implementation in Direct Mechanisms 4. An Ascending Auction for Interdependent Values 5. The Role of the Common Prior Assumption in Robust Implementation 6. Robust Implementation in General Mechanisms 7. Robust Virtual Implementation

Allocating a Single Object I I agents I Agent i has type θ i 2 [0, 1] I Agent i s valuation of the "object" is v i (θ 1,.., θ I ) I All agents have quasi-linear utility I Don t know anything about agent i s beliefs and higher order beliefs about θ i

Private Values v i (θ) = θ i I "Second Price Sealed Bid Auction" I Agents bids b i 2 [0, 1] I Object allocated to agent with the highest bid I Winner pays the second highest bid

Private Values I This is "direct mechanism" I Truthful reporting leads to e cient allocation of the object: ( qi 1 (θ) = #fj:θ j θ k for all kg, if θ i θ k for all k 0, otherwise I Dominant strategy to truthfully report type

Interdependent Values I Simplest example, Maskin 92 etc... v i (θ) = θ i + γ θ j j6=i I "Generalized VCG Mechanism" I Agents bids b i 2 [0, 1] I Object allocated to agent with the highest bid I Winner pays the second highest bid PLUS γ times the bid of others, i,e, if i wins with bid b i, he pays max j6=i b j + γ b j j6=i

Interdependent Values I This is "direct mechanism" I Truthful reporting leads to allocation q, which is e cient if γ 1. I Truthful reporting is an ex post equilibrium of the direct mechanism if γ 1.

Detour for De nitions I "Ex post equilibrium": each type of each agent has an incentive to tell truth if he expects all other agents to tell the truth. I Under private values, ex post equilibrium is equivalent to dominant strategies equilibrium I If truthtelling is an ex post equilibrium of the direct mechanism for an allocation rule (including transfers), then the allocation rule "ex post incentive compatible" [EPIC]

Algebra I Suppose your type is θ i, others types are θ i and you report (bid) b i : I Your value of the object is θ i + θ j j6=i I I You get the object (for sure) only if b i > max j6=i θ j If you get the object, you pay max j6=i I Thus payo is 8 >< u i (θ i, θ i, b i ) = >: θ i θ i # max j6=i max j6=i θ j k :θ k =max j6=i 0, if b i < max j6=i θ j + j6=i θ j θ j, if b i > max j6=i θ j θ j, if b i = max j6=i θ j θ j

General Question 1 () I When does there exist a mechanism such that, whatever each player s beliefs and higher order beliefs about other players types, there is an equilibrium with outcomes consistent with a social choice correspondence (SCC)? I In example, consider "e cient correspondence" generated by e cient allocation of the good q (and any transfers). I Truthtelling is an equilibrium under the generalized VCG mechanism for any beliefs and higher order beliefs. I Thus the answer is "yes" in our single object environment.

More generally... I [1] veri es that anytime there is an ex post incentive compatible allocation consistent with SCC, the answer is "yes" I But is this necessary? [1] shows yes in the case of "separable" environments (e.g., the one above), not necessarily in non-separable environments, e.g., e cient level of a public good with budget balance.

General Question 2 (Ex Post Implementation) I When does there exist a mechanism such that truthtelling is the unique ex post equilibrium of some mechanism? If yes, we say there is "ex post implementation". I [2] shows that a social choice function must be EPIC and "ex post monotonic"; these conditions are "almost" su cient. I In example, with three or more agents, truthtelling is the only ex post equilibrium in the generalized VCG mechanism (if γ 6= 0) I Birulin (2003) showed not true with two agents.

General Question 3 (Robust Implementation) When does there exist a mechanism such that, whatever each player s beliefs and higher order beliefs about other players types, every equilibrium outcome is consistent with a social choice function? If yes, we say there is "robust implementation".

Back to the Single Object Example... I Robust implementation fails even in the private values case, since truthtelling is only a weak best response and there are many equilibria leading to ine cient outcomes in second price sealed bid auctions. I Robust implementation of the e cient allocation is not possible in the single object example (with private or independent values) even if augmented (but well-behaved) mechanisms are allowed. I But robust implementation is achievable for a nearly e cient allocation under additional restrictions...

The Modi ed Sealed Bid Second Price Auction I with probability 1 ε, I I allocate object to highest bidder winner pays second highest bid I for each i, with probability εb i I I i gets object I pays 1 2 b i I truth-telling is a strictly dominant strategy I thus this ε-e cient allocation is robustly implemented

Algebra I Suppose your type is θ i, others report their types to be (bid) b i and you report (bid) b i : I Your payo is u i θ i, θ i, θ 0 i = 8 >< >: ε 1 I b i θ i 2 b i + (1 ε 1 I b i θ i 2 b i ε) θ i max j6=i + (1 ε) θ i max # j6=i b j k :b k =max j6=i ε I b i θ i 1 2 b i + (1 ε), if bi < max j6=i b j, if b i > max j6=i b j, if b i = max j6=i b j b j b j

The Modi ed Generalized VCG Mechanism I with probability 1 ε, I I allocate object to highest bidder i winner pays max b j + γ b j j6=i j6=i I for each i with probability εb i I I i gets object pays 1 2 b i + γ b j j6=i I, I truth telling is a strict ex post equilibrium

The Modi ed Generalized VCG Mechanism I but existence of strict ex post equilibrium does not imply robust implementation I in fact, this mechanism robustly implements the e cient outcome if jγj < 1 I 1 I and no mechanism robustly implements the e cient outcome if jγj 1 I 1 I c.f. Chung and Ely (2001)

A Key Epistemic Result A message m i can be sent by an agent with payo type θ i in an equilibrium on some type space if and only if m i is "incomplete information rationalizable" in the following sense: 1. First, suppose that every payo type θ i could send any message m i 2. Delete those messages m i that are a best response to some conjecture over payo type - message pairs of the opponents that have not yet been deleted. 3. Repeat step 2 until you converge

Illustrate this notion of rationalizability in the modi ed generalized VCG mechanism I β 0 (θ i ) = [0, 1] I β k (θ i ) is set of reports i might send for some conjecture λ i θ 0 i, θ i over his opponents types θ i and reports θ 0 i, with restriction on conjecture λ i θ 0 i, θ i that each type θj of agent j sends a message in β k 1 (θ j ).

Rationalizability I agent i conjectures that other agents have type θ i and report θ 0 i I agent i with type θ i has best response θ 0 i = θ i + γ j6=i θ j θ 0 j. linear best response allows us characterize β k (θ i ): h i β k (θ i ) = β k (θ i ), β k (θ i )

Rationalizability β k (θ i ) = min ( = min 1, θ i + γ max f(θ 0 i,θ i):θ 0 j 2β k (θ j ) for all j6=ig j6=i ( 1, θ i + γ max θ i β k (θ i ) = minf1, θ i + γ max θ i ) θ j β k 1 (θ j ) j6=i θ j θ j β k 1 (θ j ) g j6=i θj 0 )

Rationalizability rewriting: and likewise n o β k (θ i ) = min 1, θ i + (γ (I 1)) k, n o β k (θ i ) = max 0, θ i (γ (I 1)) k. thus θ 0 i 6= θ i ) θ 0 i /2 β k (θ i ) for su ciently large k, provided that γ < 1 I 1

On the other hand... I but now suppose that γ 1 I 1 I each type θ i convinced that others types are θ j = 1 2 + 1 γ(i 1) 1 2 θ i I now θ i + γ (I 1) h 1 2 + 1 γ(i 1) i 1 2 θ i = 1 2 [1 + γ (I 1)] I types cannot be distinguished in direct or any other mechanism...

In the example: I robust implementation possible (using the modi ed generalized VCG mechanism) if jγj < 1 I 1 I robust implementation impossible (in any mechanism) if γ 1 I 1

In general... I Each Θ i is a compact subset of the real line I Agent i s preferences depend on θ through h i : Θ! R I Preferences are single crossing in h i (θ) I Robust implementation is possible in the direct mechanism if strict EPIC and the "contraction property" hold. I Robust implementation is impossible in any mechanism if either strict EPIC or the "contraction property" fails.

Contraction Property I "deception": β = (β 1,.., β I ); β i : Θ i! 2 Θ i? with θ i 2 β i (θ i ) I "truth-telling": β = (β 1,.., β I ) I The aggregator functions h satisfy the strict contraction property if, for all β 6= β, there exists i and θ 0 i 2 β i (θ i ) with θ 0 i 6= θ i, such that sign θ i θ 0 i = sign hi (θ i, θ i ) h i θ 0 i, θ 0 i, for all θ i and θ 0 i 2 β i (θ i ). I this is equivalent to jγj < 1 I 1

Contraction Property 2 I if h i (θ) = θ i + γ ij θ j j6=i I contraction property satis ed if largest eigenvalue of 2 3 0 jγ 12 j jγ 1I j jγ 21 j 0 Γ, 6 4.... 7 5 jγ I 1 j 0 is less than 1.

General Question 4: Can we do better with dynamic mechanisms? I Static Cournot Oligopoly I Linear Demand P = 2 I Zero Cost Q I If there are three or more rms, every outcome between 0 and monopoly output 1 is rationalizable I best response if you think all opponents will choose 1 is 0 I best response if you think all opponents will choose 0 is 1 I therefore every outcome between 0 and 1 can be played in subjective correlated equilibrium

Dynamic Mechanisms 2 I Dynamic Cournot Oligopoly I Auctioneer has continuous clock from 0 and 1 I Each rm decides when to drop out from auction; he is then committed to the quantity at the time he dropped out I Uniquely sequentially rationalizable action: stay until I symmetric Nash output of I +1 2

Dynamic Mechanisms 3 I But with 0 < γ < 1, direct mechanism in single object example is a linear best response game with strategic substitutes: b i = θ i + γ (θ j b j ). j6=i I If γ 1 I 1, many rationalizable outcomes I But truthful revelation in increasing clock mechanism.

General Question 5: What if we know the common prior assumption holds? I Cournot Oligopoly example has unique correlated equilibrium (=Nash equilibrium) I If γ 1 I 1, direct mechanism has many rationalizable actions I But every "incomplete information correlated equilibrium" has each player following the unique ex post equilibrium and telling the truth. I With strategic complementarities, there are multiple equilibria if and only if there are multiple rationalizable actions (famous general observation) I Strategic complementarities in direct mechanism if "negative interdependence," i.e., γ < 0

General Question 6: But what about implementation in general environments using general mechanisms? I "Maskin monotonicity" required to rule out bad equilibria in complete information settings I In (classical) incomplete information settings, Bayesian incentive compatibility su cient for existence of a desirable equilibrium (by revelation principle, truthtelling in the direct mechanism); additional "Bayesian monotonicity" condition required to rule out bad equilibria (Jackson etc...) I We show EPIC and "robust monotonicity" necc. and (almost) su. for robust implementation I robust monotonicity = contraction property = jγj < 1 I 1 =Bayesian monotonicity on all type spaces I like earlier literature, must use badly behaved in nite mechanisms

General Question 7: Does it help if "virtual" is enough? I Abreu-Matsushima papers interpreted as very permissive results if "approximate" implementation is enough. I they also did it using iterated deletion of strictly dominated strategies and nite (or well-behaved mechanisms) I We show EPIC and "robust measurability" necc. and (almost) su. for virtual robust implementation (with nite mechanism) I In example, e cient allocation can be virtually robustly implemented I robust measurability = contraction property in special environment I more tomorrow!

Environment I agents i 2 I = f1, 2,..., I g I i s "payo type" θ i 2 Θ i I payo type pro le θ 2 Θ = Θ 1... Θ I I social outcome y 2 Y I utility function u i : Y Θ! R I social choice correspondence F : Θ! 2 Y / f?g

Back to the Example I agents i 2 I = f1, 2,..., I g I i s "payo type" θ i 2 Θ i [0, 1] I payo type pro le θ 2 Θ = Θ 1... Θ I I social outcome y = (q, x) 2 (f?, 1,..., I g) R I = Y! I utility function u i (q, x) = q i θ i + γ θ j + x i j6=i I e cient social choice correspondence F (θ) = (q, x) q i (θ) > 0 ) θ i = max j θ j

Type Space I richer type spaces than set of payo types Θ I i s type is t i 2 T i I t i includes description of I payo type: bθ i : T i! Θ i I bθ i (t i ) is i s payo type of t i I beliefs about types of other players: bπ i : T i! (T i ) I type space is a collection bπ i (t i ) is i s belief type of t i T = n T i, bθ i, bπ i o I i=1

Properties of Type Spaces I Type Space T is payo type space if T i = Θ i and each bθ i is the identity map. I Finite Type Space T satis es the common prior assumption if there exists π 2 (T ) such that and t i π (t i, t i ) > 0 for all i and t i π (t i, t i ) bπ i (t i ) [t i ] = π t i, t 0. t 0 i > 0 i I Type Space T is universal if T i (T i ) I Universal type space union of all type spaces

Mechanism A mechanism M is a collection ((M i ) I i=1, g) I each M i is a message set ( nite/compact/anything) I outcome function g : M! Y

Equilibrium incomplete information game (T, M) I i s strategy: σ i : T i! (M i ) I strategy pro le σ is an equilibrium if σ i (m i jt i ) > 0 implies m i is in! arg max bπ i [t i ] (t i ) σ j (m j jt j ) u i g m 0 i, m i, bθ (t) mi 0 t i,m i j6=i

Questions Does there exists a mechanism M (with property...) such that for all type spaces T (satisfying property...), and for all (some) equilibrium σ of (T, M), σ (mjt) > 0 ) g (m) 2 F b θ (t).

DEFINITION. Social choice function f : Θ! Y is ex post incentive compatible [EPIC] if u i (f (θ i, θ i ), (θ i, θ i )) u i f θ 0 i,θ i, (θi, θ i ) for all i, θ i, θ i and θ 0 i. DEFINITION. Social choice correspondence F is ex post weakly implementable if there exists social choice function f such that f is EPIC and f (θ) 2 F (θ) for all θ.

PROPOSITION. If F is ex post weakly implementable, then there exists a mechanism M such that for all type spaces T, there exists an equilibrium σ of (T, M), with σ (mjt) > 0 ) g (m) 2 F b θ (t). PROOF. Direct mechanism i.e., each agent announces a type. Consider "truthtelling strategy" pro le σ i (θ i jt i ) = 1 if θ i = bθ i (t i ) CONVERSE IS FALSE.

Example 2 in Paper I Two agents, 1 and 2 I Θ 1 = θ 1, θ 0 1, θ 00 1 I Θ 2 = θ 2, θ 0 2 I Y = (fa, b, c, d, a 0, b 0, c 0, d 0 g).

(Private Values) Payo s u 1 a b c d a 0 b 0 c 0 d 0 1 1 θ 1 1 1 2 1 1 1 1 2 θ1 0 0 0 1 0 0 0 1 0 θ1 00 0 0 0 1 0 0 0 1 u 2 a b c d a 0 b 0 c 0 d 0 θ 2 0 1 0 0 0 1 1 1 θ 0 2 1 0 1 1 1 0 0 0

Social Choice Correspondence F θ 2 θ2 0 θ 1 fa, bg fa 0, b 0 g θ1 0 fcg fc 0 g θ1 00 fdg fd 0 g

NOT dominant strategies implementable F θ 2 θ2 0 θ 1 fa, bg fa 0, b 0 g θ1 0 fcg fc 0 g θ1 00 fdg fd 0 g I write q for prob. a chosen at (θ 1, θ 2 ) I write q 0 for prob. a 0 is chosen at θ 1, θ2 0

NOT dominant strategies implementable I EPIC for θ 1 on θ 0 1 given θ 2 q (1 q) 1 2, q 3 4. I EPIC for θ 1 on θ 00 1 given θ 0 2 q 0 + 1 q 0 1 2, q 0 1 4. I EPIC for θ 2 on θ 0 2 given θ 1 1 q 1 q 0 ) q 0 q I these constraints give contradiction

But we can achieve desirable outcome on every type space in equilibrium... Mechanism m2 1 m2 2 m1 1 a a 0 m1 2 b b 0 m1 3 c c 0 m1 4 d d 0

But we can achieve desirable outcome on every type space in equilibrium... Strategies I agent 1 I I type θ 1 of agent 1 sends message m1 1 if he believes agent 2 is type θ 2 with probability at least 1 2 and message m2 1 if he believes agent 2 is type θ 2 with probability less than 1 2 ; type θ1 0 always sends message m3 1 ; and type θ00 1 always sends message m1 4. I type θ 2 of agent 2 sends message m 1 2 and type θ0 2 sends message m 2 2

An Old Debate (with Private Values) I If social choice correspondence must be implemented without knowing the prior, dominant strategies is required [Dasgupta, Hammond + Maskin (1979), Ledyard (1978, 1979), Groves + Ledyard (1987)] I Counterargument: if the agents know the prior, you can always ask them to reveal it as part of the mechanism (e.g., Choi and Kim (1999)) Our response I "Wrong" question, since common knowledge of a common prior being maintained NOT wlog I With "right" question... I I dominant strategies is not always required but it is in many special cases

Separability DEFINITION. Environment and SCF can be written as Y = Y 0 Y 1... Y I ; eu i : Y 0 Y i Θ! R u i ((y 0, y 1,..., y I ), θ) = eu i (y 0, y i, θ) for all i and y f 0 : Θ! Y 0 F (θ) = f 0 (θ) F 1 (θ)... F I (θ) Includes: I F single valued I quasi-linear environment where transfers do not e ect F

Result In separable environments, the following are equivalent: 1. F is ex post weakly implementable 2. There exists a mechanism M such that for all type spaces T, there exists an equilibrium σ of (T, M), with σ (mjt) > 0 ) g (m) 2 F b θ (t).

Idea of Proof Consider type space where agents other than i are known to be θ i. There must exist g i,θ i : Θ i! Y i such that eu i (f 0 (θ i, θ i ), g (θ i ), (θ i, θ i )) eu i f 0 θ 0 i, θ i, g θ 0 i, (θi, θ i ) for all i, θ i, θi 0. Now let f (θ) = f 0 (θ), g 1,θ 1 1 (θ 1 ),.., g i,θ i i (θ i ),.., g I,θ I I (θ I )

Rationalizable Messages I initialize at S M,0 i (θ i ) = M i, inductive step: I S M,k+1 i (θ i ) = 8 >< 9 µ i 2 (Θ i M i ) s.t.: m i >: (1) µ i (θ i, m i ) > 0 ) m i 2 S M,k i (θ i ) (2) m i 2 arg max µ i (θ i, m i ) u i (g (m mi 0 i 0, m i ), θ) θ i,m i 9 >= >; I limit set I S M i S M i (θ i ) = \ k0 S M,k i (θ i ). (θ i ) are rationalizable actions of type θ i in mechanism M

Epistemic Foundations: Result PROPOSITION. m i 2 S M i 1. a type space T, 2. an equilibrium σ of (T, M) and 3. a type t i 2 T i, such that 3.1 σ i (m i jt i ) > 0 and 3.2 bθ i (t i ) = θ i. (θ i ) if and only if there exist Brandenburger and Dekel (1987), Battigalli (1996), Bergemann and Morris (2001), Battigalli and Siniscalchi (2003).

Epistemic Foundations: Proof If m i 2 S M i (θ i ), then 9 µ m i,θ i i µ i (θ i, m i ) > 0 ) m i 2 S M i (θ i ) m i 2 arg max mi 0 Construct type space: 2 (Θ i M i ) s.t. I T i = (θ i, m i ) 2 Θ i M i mi 2 S M i (θ i ) I bθ i (θ i, m i ) = θ i I µ i (θ i, m i ) u i g m 0 i, m i, θ θ i,m i bπ i ((θ i, m i )) (θ j, m j ) j6=i = µ m i,θ i i (θ i, m i ) Construct equilibrium: I Type (θ i, m i ) sends message m i

Environment I nite set of agents, 1, 2,..., I I agent i s payo type is θ i 2 Θ i, Θ i is a compact set I type pro le θ 2 Θ = Θ 1 Θ I I Y is a compact set of outcomes I agent i s utility: u i (y, θ) I social choice function, f : Θ! Y

Mechanism I m i 2 M i compact set of messages available to agent i I outcome function: g : M! Y, g (m) outcome under message pro le m I mechanism is a collection: M = (M 1,..., M I, g ()), I direct mechanism: M i = Θ i for all i and g (θ) = f (θ)

Deceptions I message pro le of agent i, S i : Θ i! 2 M i? I in direct mechanism, S i : Θ i! 2 Θ i? I deception: β i : Θ i! 2 Θ i? with θi 2 β i (θ i ) I truth-telling: minimal deception β i (θ i ) = fθ i g

Rationalizable Messages I belief of agent i over message and type pro les of remaining agents: λ i 2 (M i Θ i ) I set S 0 i (θ i ) = M i and inductively de ne 8 < Si k (θ i ) = : m i 9λ i s.th.: (1) belief λ i is consistent (2) message m i is best-response i 9 = ;

Rationalizable Messages Si k (θ i ) = m i 2 consistent (1): best response (2): Z 9λ i s.th.: (1) belief λ i is consistent (2) message m i is best-response λ i (m i, θ i ) > 0 ) m j 2 S k 1 j (θ j ), 8j 6= i; λ i (m i, θ i ) u i (g (m i, m i ), (θ i, θ i )) m i,θ i Z λ i (m i, θ i ) u i g m 0 i, m i, (θi, θ i ), 8mi 0 2 M i. m i,θ i

Rationalizable Messages I limit set S M (θ) S M (θ), lim k! S k (θ), for all θ 2 Θ. De nition (Robust Implementation) f is robustly implemented by mechanism M if m 2 S M (θ) ) g (m) = f (θ).

Aggregator Single Crossing I aggregator h i : Θ! R for each i: u i (y, θ), v i (y, h i (θ)) I h i (θ) is continuous and strictly increasing in θ i I no restriction on behavior of h i () in terms of θ i I h i (θ), possibly distinct for every i

Aggregator Single Crossing The environment satis es strict single crossing (SC) if v i (y, φ) > v i y 0, φ and v i y, φ 0 = v i y 0, φ 0 ) v i y, φ 00 < v i y 0, φ 00 if φ < φ 0 < φ 00. I only require aggregation φ of type pro le θ to be ordered on the real line I does not require allocation y to be one-dimensional or ordered

Example I two allocations, a or b I Y = [0, 1] [ K, K ] I, where y 0 is the probability of allocation a u i (y, θ) = y 0 v a i (θ) + (1 I An equivalent representation is h u i (y, θ) = y 0 vi a (θ) y 0 ) v b i (θ) + y i. v b i i (θ) + y i. We can give this a monotonic aggregator representation by setting h i (θ) = vi a (θ) vi b (θ) and v i (y, h (θ)) = y 0 h (θ) + y i, we have u i (y, θ) = v i (y, h i (θ)), and now v i indeed satis es the strict single crossing condition.

Informational Example I two agents I agent i s utility depends on expected value of ω 0 + ω i I ω 0, ω 1, ω 2 independently normally distributed with zero mean and variance σ 2 i. I agent i observes signal θ i = ω 0 + ω i + ε i, where each ε i is independently normally distributed with zero mean and variance τ 2 i. now E i (ω 0 + ω i jθ i, θ j ) = const h i (θ i, θ j ) σ 2 0 τ2 i = θ i + σ 2 0 τ2 j + σ 2 0 σ2 i + σ 2 0 σ2 j + τ 2 j σ2 i + σ 2 θ j. i σ2 j γ ij = σ 2 0 τ2 i σ 2 0 τ2 j + σ 2 0 σ2 i + σ 2 0 σ2 j + τ 2 j σ2 i + σ 2. i σ2 j

De nition (Strict Ex Post Incentive Compatibility) scf f satis es strict ex post incentive compatibility (EPIC) if u i (f (θ i, θ i ), (θ i, θ i )) > u i f θ 0 i, θ i, (θi, θ i ) for all i, θ and θ 0 i 6= θ i.

De nition (Strict Contraction Property) The aggregator functions h satisfy the strict contraction property if, for all β 6= β, there exists i and θ 0 i 2 β i (θ i ) with θ 0 i 6= θ i, such that sign θ i θ 0 i = sign hi (θ i, θ i ) h i θ 0 i, θ 0 i, for all θ i and θ 0 i 2 β i (θ i ). I for some agent i the direct impact of private signal θ i on aggregator h i (θ) is always su ciently strong to outweigh the reported types of other agents

THEOREM. If strict EPIC and the strict contraction property are satis ed, then there is robust implementation in the direct mechanism. I no conditions on how f varies with type pro le I in particular, f does not have to respond to θ similar to the response of any of the aggregators h i I no condition on the number of agents, such as I > 2

I interaction of single crossing and contraction property I x deception β I contraction property: there exists i and θ 0 i 2 β i (θ i ) with (wlog) θ i > θ 0 i such that h i (θ i, θ i ) > h i θ 0 i, θ 0 i for all θ 0 i 2 β i (θ i ) I now x true type pro le: θ, reported type pro le: θ 0 2 β (θ) I choose θi 00 > θi 0 with h i (θ i, θ i ) > h i θi 00, θ 0 i > hi θi 0, θ 0 i

I so h i (θ i, θ i ) > h i θi 00, θ 0 i > hi θi 0, θ 0 i I θi 0 2 β i (θ i )

so h i (θ i, θ i ) > h i θi 00, θ 0 i > hi θi 0, θ 0 i I θ 0 i 2 β i (θ i ) I now v i f θ 0, h i θi 00, θ 0 i < v i f θi 00, θ 0 i, hi θi 00, θ 0 i EPIC h i θ 0 i v i f θ 0, h i θ 0 > v i f θ 00 i, θ 0 i, hi θ 0 EPIC h i θ

so h i (θ i, θ i ) > h i θi 00, θ 0 i > hi θi 0, θ 0 i I θ 0 i 2 β i (θ i ) I now v i f θ 0, h i (θ) < v i f θi 00, θ 0 i, hi (θ) AGG, SC h i ( " " v i f θ 0, h i θ 0 < v i f θi 00, θ 0 i, hi θi 00, θ 0 i EPIC h i θ 00 v i f θ 0, h i θ 0 > v i f θ 00 i, θ 0 i, hi θ 0 EPIC h i θ 0 i, i "

I preference aggregator h i (θ) is linear for each i: h i (θ) = θ i + γ ij θ j. j6=i I γ ij represent in uence of signal of j on the value of i I interdependence matrix Γ: 2 3 0 jγ 12 j jγ 1I j jγ 21 j 0 Γ, 6 4.... 7 5 jγ I 1 j 0

I consider the class of deceptions of the form: β i (θ i ) = [θ i εc i, θ i + εc i ] \ Θ i Lemma (Linear Aggregator) Linear aggregator functions h satisfy the strict contraction property if and only if, for all c 2 R I + with c 6= 0, there exists i such that c i > γ ij c j. j6=i I if the set of deceptions by i is too large, then there is an overhang which can be nipped and tucked

Lemma (Duality) The following two properties of Γ are equivalent: 1. for all c 2 R I + with c 6= 0, there exists i such that: c i > γ ij c j ; j6=i 2. there exists d 2 R I + such that: d i > γ ji d j, j6=i (P) (D) for all i.

Theorem (Contraction Property via Eigenvalue) The interdependence matrix Γ has the contraction property if and only if its eigenvalue λ < 1. I contraction property if and only if we can nd a vector d, the eigenvector of the matrix Γ and eigenvalue λ < 1 with λγ T d = d I for nonnegative matrices, left and right hand side eigenvalues coincide, hence also for some c λγc = c I λ rate at which the iterative procedure contracts

I symmetric preferences: γ ij = 1, if j = i, γ, if j 6= i. and hence I moderate interdependence γ < 1 I 1 cyclic preferences: γ ij = γ (i j)mod I and γ (i j) < 1 j6=i

I two bidders: I su cient condition is: Γ = 0 γ12 γ 21 0 γ 12 γ 21 < 1.

I public good example with general Γ: I single private good example 1 + γ ji 0 j6=i γ ij 1

De nition (Contraction Property) The aggregator functions h satisfy the contraction property if, for all β 6= β, there exists i and θ 0 i 2 β i (θ i ) with θ 0 i 6= θ i, such that either h i (θ i, θ i ) = h i θ 0 i, θ 0 i, or sign θ i θ 0 i = sign hi (θ i, θ i ) h i θ 0 i, θ 0 i, for all θ i and θ 0 i 2 β i (θ i ). I "nongeneric" weakening by allowing equality: h i (θ) = h i θi 0, θ 0 i

De nition (Strict Ex Post Incentive Compatibility*) scf f satis es strict ex post incentive compatibility (EPIC) if u i (f (θ i, θ i ), (θ i, θ i )) > u i f θ 0 i, θ i, (θi, θ i ) for all i, θ and θ 0 i with f θ i, θ 0 i 6= f θ 0 i, θ 0 i for some θ 0 i.

De nition If f is robustly implementable, then f satis es strict EPIC* and the contraction property. I necessity holds for all mechanisms, direct or augmented I implementation in direct mechanism cannot be improved by augmented mechanism The social choice function satis es robust monotonicity if, for all β 6= β, there exists i, θ i and θi 0 2 β i (θ i ) such that, for all θ 0 i and ψ i 2 β 1 i θ 0 i, there exists y such that ψ i (θ i ) u i (y, (θ i, θ i )) > ψ i (θ i ) u i f θi 0, θ 0 i θ i 2Θ i θ i 2Θ i while u i f θi 00, θ 0 i, θi, θ 0 i > ui y, θi 00, θ 0 i Robust for Mechanism all θdesign i 00 2 Θ i., (θi, θ i )