Robust Mechanism Design∗

Dirk Bergemann† Stephen Morris‡

First Version: December 2000 This Version: November 2001 Preliminary and incomplete

Abstract The implementation literature is often criticized because the mechanisms do not seem to be robust to assumed features of the underlying environment. This paper explores one way of formalizing robustness: fixing the set of pay- off relevant types of each player, we ask whether it is possible to implement a given object on large type spaces (reflecting a lack of common knowledge the environment). We show how equilibrium implementation conditions on the universal type space translates into strong solution concepts on the original types space of payoff relevant types.

Keywords: Mechanism Design, Common Knowledge, Universal Type Space, Interim Equilibrium, Ex-Post Equilibrium. Jel Classification:

∗This research is support by NSF Grant #SES-0095321. We are grateful for comments from seminar participants at the Institute for Advanced Study, the Stony Brook Game Theory Conference: Seminar on Contract Theory, and the University of Pennsylvania who heard very preliminary versions of this work. We thank Matt Jackson, Jon Levin, Zvika Neeman, Andrew Postlewaite and especially Bart Lipman for valuable discussions about the universal type space. †Department of Economics, Yale University, 28, Hillhouse Avenue, New Haven, CT 06511, [email protected]. ‡Department of Economics, Yale University, 30, Hillhouse Avenue, New Haven, CT 06511, [email protected]. “Game theory has a great advantage in explicitly analyzing the con- sequences of trading rules that presumably are really common knowl- edge; it is deficient to the extent it assumes other features to be com- mon knowledge, such as one agent’s probability assessment about an- other’s preferences or information. I foresee the progress of game theory as depending on successive re- ductions in the base of common knowledge required to conduct use- ful analyses of practical problems. Only by repeated weakening of common knowledge assumptions will the theory approximate reality.” Wilson (1987)

1. Introduction

Two different strands of thought motivate this research. First, the implementation literature is often thought to be fragile because optimal mechanisms seem to be too sensitive to the assumed structure of the environment. This leads researchers to look for mechanisms that are insensitive to certain features of the environment. However, there is not a clear theoretical rationale for the notions of robustness used. We would like to develop a better language for discussing the robustness of mechanisms. Second, the work of Harsanyi (1967/1968) and Mertens and Zamir (1985), establishes that we can, without loss of generality, model environments with in- complete information by a Bayesian game. That is, without loss of generality, one can assume that there is common knowledge among the players of each player’s type spaces and each type’s beliefs over the types of other players. However, as a practical matter, applied economic analysis tends to assume much smaller type spaces than the universal type space, and yet maintain the assumption that there is common knowledge among the players of each player’s type spaces and each type’s beliefs over the types of other players. Of course, the latter assump- tion is certainly not without loss of generality. It is then useful to check if and when the results of applied economic analysis are robust to the inclusion of a rich type spaces with little common knowledge. A recent important paper by Neeman (1999) suggests that the small type space assumption may be especially problematic in the context of mechanism design. We would like to examine the robustness of the implementation literature to higher order uncertainty. Bringing these two strands together, we propose to introduce rich type spaces (with non-trivial higher order beliefs) into the implementation literature; in doing so, we hope to provide a natural language for talking about robustness. More precisely, our approach to robustness is to weaken the base of common knowledge by allowing agents to have richer type spaces and in particular richer higher- order beliefs. This effectively reduces the common knowledge among agents and designer. In consequence, our investigation encompasses, but is not restricted to earlier notions of robustness. In this context, we may mention briefly two of the

2 most prominent positions taken in the literature. Wilson (1985) states that a desirable property of a trading rule is that it “does not rely on features of the agents’ common knowledge, such as their probability assessments.” By gradually extending the type space, and asking whether a social choice function can still be implemented, we obtain a precise notion to what extent a trading rule relies on common knowledge among the agents. Similarly, Maskin and Dasgupta (2000) “seek auction rules that are independent of the details - such as functional forms or distribution of signals - of any particular application and that work well in a broad range of circumstances”. Finally, a frequently raised criticism to the mechanism design literature is that the assumption that the designer shares the common prior with the agents. While this is certainly one direction in which common knowledge can be weakened and it is a special case of our analysis, it is perhaps the easiest one to solve, as any information which is non-exclusive in the sense of Postlewaite and Schmeidler (1986), can always be truthfully elicited from the agents in an interim equilibrium. We proceed as follows. First, we can fixasetofpayoff-relevant types for each player and social choice function mapping profiles of payoff-relevant types to outcomes. The objective of the mechanism designer will be to implement this social choice function. We will say that a social choice function is truth- fully implemented if a truthtelling equilibrium in a direct mechanism yields the outcome prescribed by the social choice function. The social choice function is simply implemented if there exists a mechanism under which the prescription is always followed under every equilibrium. While holding fixed this environment, we can construct many type spaces, where a player’s type specifies both his payoff- relevant type and his belief about other players’ types. Crucially, there may be manytypesofaplayerwiththesamepayoff relevant type. The larger the type space we construct, the harder it will be to implement the social choice function, and so the more “robust” the resulting mechanism will be. Thesmallesttypespacewecanworkwithisthe“naivetypespace,”where we set the possible types of each player equal to the set of payoff-relevant types, and assume a common knowledge prior over this type space. This is the usual exercise performed in the mechanism design literature. The largest type space we can work with is the “universal type space,” where we allow players to have any beliefs or higher order beliefs about other players’ payoff relevant types. This space can be constructed from the players’ payoff relevant types along the lines of Mertens and Zamir (1985). There are many interesting type spaces in between the naive type space and the universal type space that are also interesting to study. For example, we can look at the union of all naive type spaces (so that the players have common knowledge of a prior over payoff relevant types but the mechanism designer does not); and we can look at the subset of types in the universal type space for whom the common prior assumption holds. In the current work, we present three benchmark results. 1. We show that requiring equilibrium implementation on the universal type

3 space is equivalent to requiring ex post equilibrium implementation on the naive type space.

2. We provide a characterization of when the social choice function can be equilibrium implemented on an arbitrary finite type space satisfying the common prior assumption. The condition shows that it is enough to check if equilibrium implementation on the naive type space would have been possible for certain independent types.

3. We show that requiring full equilibrium implementation on the universal type space is equivalent to requiring an incomplete information version of dominance solvability on the naive type space.

In the concluding section, we discuss other issues we hope to address using this framework.

2. Framework

We introduce the payoff relevant environment in Subsection 2.1, then define a variety type spaces in Subsection 2.2 and finally discuss and solution and imple- mentationconceptsinSubsection2.3.

2.1. The Payoff Relevant Environment We consider a finite set of agents = 1, 2, ..., I . Agent i’s payoff relevant type is I { } θi Θi, where Θi is a finite set. We write θ Θ = Θ1 ... ΘI . A social outcome ∈ ∈ × × is denoted x X, where X is a finite set. A vector of transfers is denoted y RI . ∈ ∈ Every agent has a utility function ui : X Θ R,defined on social outcomes and a vector of payoff relevant types. Each× agent→ has quasi-linear utility, so that his utility if the outcome is x,thepayoff relevant type vector is θ and his transfer is yi is ui (x, θ)+yi.

2.2. Type Spaces We are interested in studying type spaces that are richer than Θ,thesetofpayoff relevant types. This is important because we want to allow for the possibility that two types of a player may be identical from a payoff relevant perspective, but they have different beliefs about, say, the payoff relevant types of other players. In addition, we will want to allow for interim type spaces, where there are no restrictions on a type’s interim belief about other player’s types. Requiring that types’ interim beliefs be derived from some prior probability distribution on the type space will then represent an important special case. Agent i’s type is ti Ti. A type of agent i must include a description of his ∈ payoff relevant type. Thus there is a function θi : Ti Θi with θi (ti) being agent → 4 b b i’s payoff relevant type when his type is ti. A type of agent i must also include a description of his beliefs about the types of the other player. Write ∆ (Z) for the space of probability measures on the Borel field of measurable space Z, there is afunctionπi : Ti ∆ (T i) with πi (ti) being agent i’s belief type when his type → − is ti.Thusπi (ti)[E] is the probability that type ti of player i assigns to other players’ types,b t i, being an elementb of a measurable set E T i.Inthespecial − ⊆ − case where each Tj is finite, we will abuse notation slightly by writing πi (ti)[t i] b − for the probability that type ti of player i assigns to other players having types t i.Nowatype space is a collection − b I = Ti, θi, πi . T i=1 ³ ´ We will be interested in some particularb typeb spaces.

2.2.1. The Naive Type Space The standard approach in the mechanism design literature is to assume a common knowledge prior, p ∆ (Θ),onthesetofpayoff relevant types. For convenience, we will assume that∈ all types have strictly positive probability, i.e.,

pi (θi) p θi, θ0 i > 0, ≡ − θ0 i Θ i −X∈ − ¡ ¢ for all i and θi. The naive type space can be described in our general language by setting each Ti = Θi,lettingeachθi ( ) be the identity map, and setting: · p (θ) πi [θi](θb i)= . − pi (θi) Thisnaivetypespacecanbethoughtofthesmallesttypespaceembeddingtheb payoff relevant environment described above. We will denote the naive type space with common prior p by N (p). Where the common prior is not relevant, we may T abbreviate to N . T 2.2.2. The Universal Type Space At the other extreme, we might want to ask what is the largest type space embed- ding the payoff relevant environment described above, placing no restrictions on agents’ beliefs or higher order beliefs about other players’ payoff relevant types. The most straightforward way to do this is to consider a universal type space that is the union of all type spaces we could possibly construct. We will refer to this as the universal type space, U . With some topological assumptions,T one can also construct a canonical uni- versal type space from a hierarchical description of types, following Mertens and Zamir (1985). The only non-standard feature in our description of the

5 universal type space is that we want to assume that it is common knowledge that each player knows his own payoff relevant type. We will build this fea- ture into our construction. Player i’s 0th level type is his payoff relevant type 0 0 ti = θi Θi.LetTi Θi be player i’s set of 0th level types. Player i’s 1st level∈ type must specify≡ his payoff relevant type and his belief about other 1 1 0 players’ 0th level types. Thus ti Ti Θi ∆ T i .Playeri’s 2nd level type must specify his payoff relevant∈ type≡ and× his belief− about other players’ 2 2 1 ¡ ¢ 1st level types. Thus ti Ti Θi ∆ T i . Iterating this construction, we ∈ ≡ × − k k k 1 have ti Ti Θi ∆ T −i , and we¡ obtain¢ an infinite hierarchy of beliefs ∈ ≡ × − 0 1 2 ti ,ti ,ti , .... . We want to³ require´ that high level types, which intuitively contain more information than lower level types, are consistent with lower levels. For- ¡mally, an infi¢nite hierarchy is coherent if all higher level types have the same payoff relevant type as lower level types and if the projection of their beliefs over other players’ types onto lower level type spaces is consistent with lower level types’ beliefs. Now if we impose some topological structure on the belief spaces, we can let player i’s possible types, Ti, be the set of all infinite hierarchies of beliefs. The crucial property of such type spaces is that the set of types, constructed as infinite hierarchies, can be identified with pairs of payoff relevant types and beliefs types, so that, for each i, there exists a homeomorphism fi : Ti Θi ∆ (T i).For example, Brandenburger and Dekel (1993) show that if we→ topologize× the− belief spaces with a complete separable metric, then this follows from Kolmogorov’s Existence Theorem. Now letting θi be the projection of fi onto Θi and letting πi be the projection of fi onto ∆ (T i), this canonical universal type space fits into − the language for type spaces describedb above. b As an aside, we remark that the canonical universal type space based on the infinite hierarchy construction may not be quite rich enough for our purposes. Consider the following type space:

T1 = t1,t10 , T2 = ©t2,t20 ª , with a single payoff relevant type for each© player:ª

θ1 (t1)=θ1 t10 = θ1,

θ2 (t2)=θ2 ¡t20 ¢ = θ2, b b ¡ ¢ and the following associatedb belief typesb 2 π (t )[t ]= , 1 1 2 3 1 π t [t ]= , b1 10 2 3 2 π ¡(t ¢)[t ]= , b 2 2 1 3

b 6 1 π t [t ]= . 2 20 1 3 ¡ ¢ Since all types have the same payob ff relevant type, the infinite hierarchy of beliefs 0 1 is degenerate. That is, for type t1,wewouldhavet1 = θ1; t1 =(θ1, [θ2]), where [θ2] is the degenerate probability distribution putting probability 1 on 2 2 2 2 θ2; t1 = θ1, t1 , where t1 =(θ2, [θ1]) and t1 is the degenerate probability distribution putting probability 1 on t2 ;andsoon.Buttypet would then ¡ £ ¤¢ 1 £ ¤ 10 have the same degenerate hierarchies of beliefs. The problem here is that in £ ¤ the universal type space, types with the same infinite hierarchy are (implicitly) identified with each other. However, in strategic settings types with different belief types should be distinguished despite having the same infinite hierarchy of beliefs (since private correlating devices matter). This type space does not exist as a subset of the canonical construction of the universal type space. To incorporate it, we could have constructed a hierarchy of types where, in addition to beliefs and higher order beliefs about payoff relevant states Θ, individuals also had beliefs and higher order beliefs about some rich payoff irrelevant space, say, [0, 1].

2.2.3. The Union-Naive Type Space

The naive type space N (p) and the universal type space U can be seen as the smallest and largest typeT spaces we might be interested in.T For many questions, we will be interested in intermediate type spaces. Suppose that there is a true prior p over the payoff relevant types, but the mechanism designer does not know what it is. We can represent this as follows. Thetypespaceis Ti = ∆ (Θ) Θi, × with typical element ti =(pi, θi) . The payoff relevant type is defined in the natural way:

θi (pi, θi)=θi.

The belief type is defined on theb assumption that there is common knowledge of thetrueprioramongtheplayers:

pi (θ i θi ) ,ifpj = pi for all j = i, πi [(pi, θi)] (pj, θj)j=i = − | 6 6 0,otherwise. ³ ´ ½ Notice thatb this type space is smaller than the universal type space because of the very strong assumption that there is common knowledge among the players of a true prior over their payoff relevant types. We will label this type space UN. T

7 2.2.4. The Common Prior Universal Type Space In the universal type space, there is no requirement that agents’ beliefs be derived from some common prior. But it makes sense to discuss that subset of the uni- versal type space where a common prior assumption holds. Formally, following Mertens and Zamir (1985), say that a subset of the belief type space is “belief- closed” if all types in the subset assign probability one to other players being in that subset. A belief closed subset satisfies the common prior if there exists a probability distribution on the belief closed subset such that player’s interim beliefs can be obtained by Bayes rule from the prior in the usual way. The com- mon prior universal type space is the union of such belief closed subsets with the common prior property. Unfortunately, we know very little about the structure of the common prior universal type space (see Gul (1998), Feinberg (2000) and Lipman (1997)). How- ever, we make some brief comments about implementation on the common prior universal type space in what follows. We will label this type space CPU. T 2.3. Mechanism Design, Solution Concepts and Implementation 2.3.1. Game Forms We have described a type space and an outcome space. Now we need to define a game form or mechanism for the agents to play in order the determine the social x outcome. Let Ai be the set of actions available to player i.Letg (a) be the outcome if action profile a is chosen and gy (a) be the vector of transfers of action profile a is chosen. For simplicity, we focus on pure mechanisms that do not involve randomization contingent on the action profile. In the settings we study, allowing for such randomization would not alter the analysis. Thus a mechanism is a collection

=(A1, ..., AI ,g( )) , G · where g : A X RI and g (a) (gx (a) ,gy (a)). Now holding→ fi×xed the payoff≡relevant environment, we can combine a type space with a game form to get an incomplete information game ( , ).Inthis incompleteT information game,G the payoff of player i if action profileT aGis chosen and type profile t is chosen is

x y vi (a, t) ui g (a) , θ (t) + g (a) . (2.1) ≡ i ³ ´ 2.3.2. Solution Concepts b

A pure strategy for player i is a function si : Ti Ai. →

8 Definition 2.1. Pure strategy profile s∗ is an interim equilibrium if

πi (ti)[t i] vi (s∗ (t) ,t) πi (ti)[t i] vi ai,s∗ i (t i) ,t , − ≥ − − − t i T i t i T i −X∈ − −X∈ − ¡¡ ¢ ¢ b b for all i, ti, ai.

This is equivalent to the standard interim definition of Bayesian Nash equi- librium in an incomplete information game, if beliefs are derived from a common prior. The ex ante definition of Bayesian Nash equilibrium implies interim equi- librium behavior for almost all types.1 A stronger solution concept requires each player’s strategy is optimal, given that he expects others to follow their equilibrium strategies, whatever his beliefs about others’ types.

Definition 2.2. Pure strategy profile s∗ is an ex post equilibrium if

vi (s∗ (t) ,t) vi ai,s∗ i (t i) ,t , ≥ − − for all i, t, ai. ¡¡ ¢ ¢

Such a solution concept was studied by D’Aspremont and Gerard-Varet (1979); they called it uniform equilibrium. It has been used in the recent literature on auctions with interdependent values, e.g., Dasgupta and Maskin (2000). Perry and Reny (2000) dubbed this solution concept ex post equilibrium. We will be interested in making comparisons to interim and ex post dominant strategy solution concepts.

Definition 2.3. Pure strategy profile s∗ is interim strategy-proof if

πi (ti)[t i] vi ((si∗ (ti) ,s i (t i)) ,t) πi (ti)[t i] vi ((ai,s i (t i)) ,t) , − − − ≥ − − − t i T i t i T i −X∈ − −X∈ − b b for all i, ti, ai and s i. −

Definition 2.4. Pure strategy profile s∗ is ex post strategy-proof if

vi ((si∗ (ti) ,a i) ,t) vi ((ai,a i) ,t) , − ≥ − for all i, t, a.

1 Dekel, Fudenberg and Levine (2001) note that it is difficult to provide a learning justification for equilibrium in the absence of common prior. We nonetheless find it useful to examine equilibria on the universal type space as a benchmark in our analysis.

9 We will also be interested in strategies that are rationalizable under incom- plete information (we allow correlation in players’ beliefs in constructing rational- izability, and thus rationalizability is equivalent to surviving iterated deletion of strictly dominated strategies). A standard notion of rationalizability for incom- plete information games is interim rationalizability. At each round of an iterative process, we will delete those actions for each type that are not a best response for some beliefs about the opponents’ play in which already deleted strategies have to be assigned zero probability. 0 Let Ai (ti)=Ai and define recursively: k+1 Ai (ti)= k k λi Λi such that a Ak (t ) ∃ ∈ k i i i ai arg max πi (ti)[t i] λi (a i t i ) vi ((ai0 ,a i) ,t)  ∈ ¯ ∈ − − | − −  ¯ ai0 t i T i a i A i  ¯ − ∈ − − ∈ −  ¯ P P where ¯ b  ¯  k k k k Λi = λi : T i ∆ (A i) λi (a i t i )=0if aj / Aj (tj) for some j = i . − → − − | − ∈ 6 n ¯ o Let ¯ ¯ k IRi (ti)= Ai (ti) . k 0 \≥ Definition 2.5. Pure strategy profile s is interim rationalizable if s (ti) ∗ i∗ ∈ IRi (ti) for all i and ti. The notion of interim equilibrium was strengthened to ex post equilibrium by insisting that the strategy of each player is a best response even after knowing the exact type profile of his opponents. We introduce the analogous change to the definition of interim rationalizability. Note that in this case, we weaken the definition, i.e., ex post rationalizability will be a more permissive solution concept than interim rationalizability, because it becomes harder to delete a strategy in 0 any round. Now let Ai (ti)=Ai and define recursively: k+1 Ai (ti) =

k k k k ai Ai λi Λi s.th. ai arg max λi (a i t i ) vi ai0 ,a i ,t ,  ∈ ¯∃ ∈ ∈ a − | − −  ¯ i0 t i T i a i A i  ¯ −X∈ − −X∈ − ¡¡ ¢ ¢ ¯ where ¯  ¯  k k k k Λi = λi ∆ (T i A i) λi (a i t i )=0if aj / Aj (tj) for some j = i . ∈ − × − − | − ∈ 6 n ¯ o Let ¯ ¯ k EPRi (ti)= Ai (ti) k 0 \≥

10 Definition 2.6. Pure strategy profile s is ex post rationalizable if s (ti) ∗ i∗ ∈ EPRi (ti) for all i and ti.

The following standard characterization of rationalizable actions will be useful.

Lemma 2.7. For each ai EPRi (ti), there exists λi ∆ EPRj (tj) A i ∈ ∈ j×=i × − such that µµ 6 ¶ ¶

ai arg max λi (a i t i ) vi ai0 ,a i ,t . ∈ a − | − − i0 t i T i a i A i −X∈ − −X∈ − ¡¡ ¢ ¢ This solution concept was studied by Battigalli (1999) (who called it incom- plete information rationalizability). Chung and Ely (2000) studied a notion of ex post rationalizability with weak rather than strict dominance.

2.3.3. (Truthful) Implementation A social choice function (SCF) is a mapping ξ : Θ X. We thus assume that the social planner does not care about transfers between→ the agents, although the framework can be extended to deal with this case. Thus if we are concerned with the efficient allocation of a single object among the agents, and vi (θ) is player i’s valuation of the object if the payoff relevant type profile is θ,wewouldhave X = 1, ..., I and (assuming no ties) { }

ξ (θ)= argmaxvi (θ) . i 1,...,I ∈{ } When is it possible for a social planner to choose a mechanism in such a way that some or all equilibria of the resulting incomplete information game have ξ (θ) chosen whenever the payoff-relevant types of the players are given by the vector θ? We will say that ξ is truthfully implemented if ξ is followed under a truthtelling equilibrium in direct mechanism; and that ξ is implemented if ξ is always followed under every equilibrium. Whether an SCF can be (truthfully) implemented also depends on the equilibrium solution concept being employed. Our generic definition of implementation concepts is now:

Definition 2.8. SCF ξ is (truthfully) implementable in type space under so- lution concept α if there exists a mechanism such that every (truthtelling)T strategy profile s satisfying solution concept α Gin incomplete information game ( , ) also satisfies: T G gx (s (t)) = ξ θ (t) , for all t T. ³ ´ ∈ b

11 3. Some Key Properties of Type Spaces

Before we turn to our results, we examine some critical properties of the type space for implementation. Neeman (1999) emphasized that a key property in the surplus extraction results of Cremer and McLean (1988) is the following:

Definition 3.1. Type ti Ti satisfies belief extraction with respect to Ti if ∈ πi (t )=πi (ti) and t Ti θi (t )=θi (ti). i0 i0 ∈ ⇒ i0 b Thisb property says impliesb that ifb the mechanism designer can find out the beliefs of a player about other player’s types, then the mechanism designer can deduce his payoff relevant type. A more intuitive assumption is that while beliefs about other players’ types and your own payoff relevanttypemaywellbecorre- lated, there is no a priori restriction on how they might be linked together. This is formalized in the following property:

Definition 3.2. Type ti Ti satisfies diffuse beliefs with respect to Ti if for all ∈ θi Θi, there exists t Ti such that πi (t )=πi (ti) and θi (t )=θi. ∈ i0 ∈ i0 i0

Note that both these properties areb propertiesb of a particularb type ti,but whethertheyholdornotdependsonthesetofpossibletypes,Ti, for that player. Consider first the type spaces that we introduced in the previous section. Suppose we endowed the union-naive type space with a non-atomic common prior q ∆ (∆ (Θ)) with full support, then with probability 1 under q, the belief extraction∈ property would hold. This is equivalent to the observation that all types in the naive type space will satisfy the belief extraction property (with respecttothatnaivetypespace)ifthepriorp is chosen “generically” in the space ∆ (∆ (Θ)). If the types are independent on the naive type space, or

I p (θ)= pi (θi) , i=1 Y then all types will satisfy the diffuse belief property (with respect to that naive type space). These observations imply the classical results that there are infor- mational rents with independent types, but no informational rents for generic correlated types. However, observe that all types in the universal type space satisfies the diffuse belief property (with respect to the universal type space), by construction. One way to see this is to note the homeomorphism between a player’s types and set the of payoff relevant type / belief type pairs. This is exactly what we require to show the diffuse beliefs property. The common prior universal type space is a more delicate case. The diffuse belief property does not hold for all types with respect to the common prior universal type space. To see this, consider the following simple example. Suppose

12 that π1 is a belief for agent 1 that assigns probability 1 to agent 2 assigning probability 1 to agent 1 being payoff relevant type θ1. Withacommonprior, there can be no type t1 of agent 1 with π1 (t1)=π1 and θ1 (t1) = θ1.Inthe remainder of this section, we discuss in more detail the belief extraction6 property forcommonpriortypes. b b First, observe that for every common prior type ti and every θi Θi, there ∈ is a type ti0 in the set of universal types for i, with the property that πi (ti0 )= πi (ti) and θi (ti0 )=θi. The only problem is that type ti0 may not be a common priortype(astheaboveexampleshows).Wecanfind subsets of theb set of commonb priorb types with the property that diffuse beliefs property holds for all types in that subset with respect to that subset. Trivially, we could look at a naive type space with independent types. However, it is also straightforward to construct such subsets where types are highly correlated. Neeman (1999) considered the following construction. First, nature draws a profile of “belief types” for the players, (b1, ..., bI ), according to some prior. Then, each player’s payoff relevant type is drawn, with the distribution depending of that player’s belief type but independent of other players’ belief type and payoff relevant type. Thus we construct a type space with two dimensional type Ti = Bi Θi,with × I p ((b1, θ1) ,...,(bI , θI )) = β (b1, ..., bI ) νi (θi bi ) , | i=1 Y where β ∆ (B1 ... BI ) and νi : Bi ∆ (Θi). Clearly, if the probability distributions∈ have× full× support, then all→ types on this type space will satisfy diffuse beliefs with respect to this type space. In fact, one can show that under the common prior assumption, one can always decompose types into a belief type and payoff relevanttype,insuchawaythat conditional independence holds. The following result is essentially Lemma 2 from Neeman (1999), stated in our language. For simplicity, we state the result for finite type spaces, but this results should extend straightforwardly to arbitrary type spaces. Let Πi and Π be the range of πˆi ( ) and πˆ ( ), respectively. · · Lemma 3.3 (Conditional Independence). If is a finite type space with a T common prior p ( ), then there exists β ∆ (Π) and, for each i, νi : Πi ∆ (Θi), such that · ∈ → I p (t)=β (π) νi (θi πi ) . (3.1) | t T :π(t)=π and θ(t)=θ i=1 { ∈ X } Y b Proof. Write φi (θi tb i, πi ) for the probability that player i has payoff relevant | − type θi, contingent on player i having belief type πi and other players having types t i. Abusing notation, write: −

p (πi) p (t) , ≡ t i T i ti:πi(ti)=πi −X∈ − { X } 13 b p (πi) p (t) , ≡ t i T i ti:πi(ti)=πi −X∈ − { X } p (t) b t i T i t :π (t )=π and θ (t )=θ − ∈ − i i i i i i i p (θi πi) P { P } , | ≡ p (t) b b t i T i ti:πi(ti)=πi − P∈ − { P } p (ti,t i) b − ti:πi(ti)=πi and θi(ti)=θi p (t i θi, πi ) { P } . − | ≡ p t ,t b b i 0 i − t0 i T i ti:πi(ti)=πi and θi(ti)=θi − P∈ − { P } ¡ ¢ We notice that by the definition of beliefb type: b

p (t i θi, πi )=πi (t i) , − | − for all θi and t i.So − p (πi) p (θi πi) p (t i θi, πi ) φi (θi t i, πi )= | − | , | − p (πi) p θi0 πi p t i θi0 , πi − θi0 Θi P∈ ¡ ¯ ¢ ¡ ¯ ¢ p (πi) p (θi πi¯) πi (t i) ¯ = | − , p (πi) p θi0 πi πi (t i) − θi0 Θi P∈ ¡ ¯ ¢ = p (θi πi) . ¯ | Thus p (t) t:π(t)=π and θ(t)=θ p (θ π ) { P } , | ≡ p (t) b b t:π(t)=π { } I P b = p (θi πi) . | i=1 Y By setting β (π) p (t) ≡ t:π(t)=π { X } and b νi (θi πi ) p (θi πi) , | ≡ | we obtain the representation given in (3.1).¥ This lemma gives us a convenient characterization of when the belief extrac- tion property fails on any common prior type space. The belief extraction prop- erty fails exactly if there is some conditional independence built into the type space.

14 How reasonable is this conditional independence property? There is a certain sense in which the property is non-generic. Suppose that we fix,foreachplayer i,afinite type space Ti and a payoff relevance function θi : Ti Θi.Suppose → that we endow the type space T = T1 .... TI with a common prior p ∆ (T ), × × ∈ and let players’ belief types be derived from the common prior,b with πi (ti)[t i]= I − p (t i ti). This gives us a type space = Ti, θi, πi . It remains true that if p − | T i=1 b is chosen according to Lebesgue measure on³ the set´∆ (T), then with probability 1, all types will satisfy the belief extraction property.b b That is, we noted above that it was true when Ti = Θi and now we note that it remains true even if Ti is a much larger set. Indeed, even if we allowed the type spaces to be infinite, analogues of this claim would continue to hold (see McAfee and Reny (1992)). We don’t believe that this settles the issue. When you fixanarbitrarysetof types (finite or infinite) and vary the prior on that type space, you are implicitly assuming that the prior on the type space remains common knowledge as you perturb it. Why would such a perturbation make sense? The common knowledge of the prior assumption is without loss of generality only iftheoriginaltypespace was the universal type space. But if the original type space was the universal type space, it makes no sense at all to vary the beliefs holding the types fixed! A better way to think about genericity is to ask what is typically true of types in the universal type space. It is not clear what the right topology on the universal type space is. However, we will suggest one formal argument why types failing the belief extraction property (Neeman types for short) are more generic than they may appear at first. We hope to improve on this analysis in later work. We want to argue that for any reasonable topology on types in the universal type space, the Neeman types are dense. Fix a finitesetoftypesforeachplayer, Ti,apayoff relevant mapping for each player, θi : Ti Θi, and a full support → prior p ∆ (T ).Nowfix ε > 0 and, for each player, let δi : Ti ∆ (Θi), where ∈ → δi θi (ti) ti 1 ε. Now construct a newb type space with T = Ti Θi, ≥ − i0 × 0 ³ ¯ ´ θi (ti, θi)=¯ θi and: b ¯ I b p0 ((t1, θ1) ,...,(tI, θI )) = p (t1, ...., tI ) δi (θi ti) . | i=1 Y Thus this type space is identical to the original type space, expect that with small probability each player is switched to a different payoff relevant type. The diffuse belief property holds on this type space by construction. But also observe that if we think of the higher order belief types constructed in section

2.2.2, type ti0 = ti, θi (ti) in the new type space is close to type ti in the original type space, if ε is³ small. To´ see why, observe first that they share the same payoff relevant type. Theirb beliefs over the payoff relevant types of the opponent will be close in the weak topology. Their beliefs over the product of their opponents’ payoff relevant types and their opponents’ beliefs about their payoff relevant types

15 will also be close in the weak topology. Since the types come from a finite type space, types ti0 and ti can be made close uniformly at all levels by choosing ε small. By a similar argument, one can also show that the non-Neeman types are also dense in the space of finitely generated common prior types. However, we conjecture that if in the subset of the common prior universal type space, where all types have full support beliefs (ruling out the example mentioned above), the belief extraction property would always fail. It should be emphasized that it is also easy to come up with asymmetric information justifications for Neeman’s conditional independence property. For example, in McLean and Postlewaite (2001), there are multidimensional types. A player’s type includes an idiosyncratic component and a noisy signal of a com- mon component. This similarly gives a correlated type space where the belief extraction property fails. However, we would like to argue that we should expect the belief extraction property to fail simply because of players’ lack of common knowledge, rather than because of some preference assumption about idiosyn- cratic components.

4. Truthful Implementation

After the preliminary discussion on the implementation properties of different type spaces, we now suggest a precise notion for robust implementation. Namely, we relate the four natural type spaces identified in section 2.2 through different equilibrium implementation notions. For each of the richer type spaces, we look at interim equilibrium implementation and ask what stronger solution concept on the naive type space is equivalent to interim equilibrium implementation on the larger space. As it is more difficult to implement a given social choice function on a larger/richer type space, requiring implementation on a larger type space is a way of securing robust implementation. For our first result, it is useful to introduce one more type space: the union of all naive type spaces where players have independent beliefs and probability at least ε to each type; i.e., the common prior p ∆ (Θ) satisfies ∈ I p (θ)= pi (θi) , i=1 Y and pi (θi) ε, ≥ for all θi Θi. We label this type space UN (ε). Now we have the following lemma: ∈ T

Lemma 4.1. Fix the payoff relevant environment and social choice function ξ. There exists ε > 0 such that if ξ is interim equilibrium implementable in UN (ε), then ξ is ex post equilibrium implementable on the naive type space. T

16 Proof. Suppose that ξ is not ex post equilibrium implementable. Then for some i, there does not exist yi : Θ R such that θ : → ∀

ui (ξ (θi, θ i) , θ) yi (θi, θ i) ui ξ θi0 , θ i , θ yi θi0 , θ i , θi0 Θi.(4.1) − − − ≥ − − − ∀ ∈ ¡ ¡ ¢ ¢ ¡ ¢ Thus there exists θ i = θ∗ i such that no transfer functions can satisfy (4.1). − − Thus fixing θ i = θ∗ i, the solution to the following minimization problem: − − u ξ θ , θ , θ , θ y θ , θ min max i i0 ∗ i i ∗ i i i0 ∗ i , (4.2) u ξ θ , θ− , θ , θ− − y θ , θ− yi( ,θ∗ i) θi,θi0 i i ∗ i i ∗ i i i ∗ i { · − } ½ ½ −¡ ¡ ¡ −¢ ¡ −¢¢ − ¡ −¢¢ ¾¾ has a strictly positive solution,¡ ¡ say¡ δ. Without¢ ¡ loss¢¢ of generality,¡ ¢¢ we may assume the positive solution arises locally from the incentive constraint at θi versus θi0 . Now suppose that ξ is interim equilibrium implementable on the naive type space with independent prior p. There must exist yi : Θ R, such that after we define → yi (θi) yi (θi, θ i) p i (θ i) , ≡ − − − θ i Θ i −X∈ − and yi θi0 yi θi0 , θ i p i (θ i) , ≡ − − − θ i Θ i ¡ ¢ −X∈ − ¡ ¢ we have:

ui ξ θi, θ∗ i , θi, θ∗ i p i θ∗ i yi (θi)+ (ui (ξ (θi, θ i) , θ)) p i (θ i) − − − − − − − − ≥ θ i Θ i θ∗ i ¡ ¡ ¢ ¡ ¢¢ ¡ ¢ − ∈ P− \ − ui ξ θi0 , θ∗ i , θi, θ∗ i p i θ∗ i yi θi0 + ui ξ θi0 , θ i , θ p i (θ i) . − − − − − − − − θ i Θ i θ∗ i − ∈ − \ ¡ ¡ ¢ ¡ ¢¢ ¡ ¢ ¡ ¢ P − ¡ ¡ ¡ ¢ ¢¢(4.3) But for a given social choice function ξ, we can always find p i (θ i) such that − −

p i (θ i) (ui (ξ (θi, θ i) , θ)) ui ξ θi0 , θ i , θ − − < δ. − − − p i θ∗ i θ i Θ i θ∗ i − − − ∈X− \ − ¡ ¡ ¡ ¡ ¢ ¢¢¢ ¡ ¢ But the hypothesis of (4.2), it then follows that we cannot find y (θ ) i i θi Θi which satisfy the set of inequalities represented by (4.3). Finally given{ δ,wecan} ∈ always find ε > 0 to satisfy the requirements of the lemma.¥

The current model is based on discrete type space, and a similar robustness result can be obtained with a continuous type space, provided that the utility function of each agent is continuous in the type. Based on the previous theorem, we can make some additional observations. First, we observe that if we require interim implementation to hold for all priors rather than all strictly positive priors, then the results hold a fortiori. As a matter of fact, under the former condition

17 we can then always choose to set the prior probabilities of all agents but i as follows: 0, if θ i = θ∗ i, p i (θ i)= − 6 − (4.4) − − 1, if θ i = θ∗ i. ½ − − As a consequence under the prior as in (4.4), the I agent implementation problem is reduced to a single agent implementation problem, namely of agent i.Butwith a single agent problem, the notion of dominant and interim implementation are equivalent, and a fortiori interim implementation implies ex-post implementation as observed earlier in Bergemann and Valimaki (2001).

Corollary 4.2. If ξ is interim equilibrium implementable in UN,thenξ is ex post equilibrium implementable on the naive type space. T

Withonemorelemma,wecanproveourmainresultofthissection.

Lemma 4.3. If ξ is ex post equilibrium implementable in the naive type space, then ξ is interim equilibrium implementable on any type space.

Proof. This follows immediately from the definitions. Since ξ is ex post equi- librium implementable in the naive type space, there exists a mechanism and x G an ex post equilibrium, s,of( N , ) such that g (s (θ)) = ξ (θ) for all θ.Now consider incomplete informationT gameG ( , ) for any type space , and suppose T G T that each player i follows the strategy si0 (ti)=si θi (ti) . The strategy profile s is an interim equilibrium, since we know that s³ θ (t ´) is an optimal action 0 i b i i regardless of type ti’s belief type, as long as he expects³ others´ to follow s0.But by construction, b x g s0 (t) = ξ θ (t) , ¡ ¢ ³ ´ which completes the proof.¥ b

We thus obtain the following equivalence relations between the type spaces.

Proposition 4.4. The following are equivalent:

1. ξ is interim implementable in the universal type space;

2. ξ is interim implementable in the common prior universal type space;

3. ξ is interim implementable in the union-naive type space;

4. ξ is ex post equilibrium implementable in the naive type space.

18 Proof. By lemma 4.3, claim (4) in the proposition implies (1), (2) and (3). Since UN is contained in the common prior universal type space and the universal type T space, corollary 4.2 implies that any of (1), (2) and (3) imply (4).¥

Combining proposition 4.4 and lemma 4.1, we obtain Ledyard’s (1978) ob- servation that, with private values, implementation for all priors is equivalent to dominant strategy implementation. He noted that if there are private values, then ξ is ex post equilibrium implementable on the naive type space if and only if ξ is ex post strategy proof implementable on the naive type space. Proposition 4.4 provides a useful benchmark result, confirming that the so- lution concept of ex post equilibrium on the naive type space exactly captures a formal notion of robustness to all possible beliefs over payoff relevant types. However, implementation on the universal type space is clearly a very strong property. We also established that implementation on the union of all naive type spaces is also equivalent. However, that argument relied on the inclusion of naive type spaces with independent types. What if the mechanism designer knows that types are correlated? If the type space is reasonably rich, it will still be the case that nothing less than ex post equilibrium will do. To show this, we first establish a more precise characterization of when implementation is possible. This result builds on the main argument in Neeman (1999). The basic idea is that, given risk neutral types, it is always possible to identify a player’s belief type via a set of bets, along the lines of Cremer and McLean (1988). Thus the implementation of the social choice function is equivalent to satisfying a set of incentive constraints for distinguishing types with the same belief type but different payoff relevant types. This problem is equivalent to incentive conditions for independent types on the naive type space. To state our characterization, we require some additional notation. For sim- plicity, we restrict attention to finite type spaces. We will be interested in agent i’s beliefs about other agents’ payoff relevant types, ψi ∆ (Θ i). Letting Πi be ∈ − the range of πi,wecandefine ψi : Πi ∆ (Θ i) as follows: → −

ψi (πi)[θ i]b πi (ti)[t i] . b − ≡ − t i:θ i(t i)=θ i { − −X− − } b b Define b vi θi, θi0 , ψi ψi (θ i) ui ξ θi0 , θ i , (θi, θ i) , (4.5) ≡ − − − θ i Θ i ¡ ¢ −X∈ − ¡ ¡ ¢ ¢ as player i’s expected utility from the social outcome x given that he has beliefs ψi over his opponent’s payoff relevant type, he has payoff relevant type θi,he reports himself to have payoff relevant type θi0 , and he expects the social planner to behave as if there is truth-telling.

Let Zi Θi Πi be the range of the function θi ( ) , πi ( ) : Ti Θi Πi. ⊆ × · · → × ³ ´ b b 19 Proposition 4.5. ξ is interim equilibrium implementable if and only if for each i, there exists a function y∗ : Zi R such that for each (θi, πi) , θi0 , πi Zi, i → ∈ ¡ ¢ vi θi, θi, ψ (πi) + y∗ (θi, πi) vi θi, θ0 , ψ (πi) + y∗ θ0 , πi . (4.6) i i ≥ i i i i ³ ´ ³ ´ ¡ ¢ Proof. (Necessity).b Define yi : Ti Πi R by b × →

yi (ti, πi) πi (t i) yi∗ (ti,t i) . e ≡ − − t i T i −X∈ − e Thus yi (ti, πi) is the expected transfer to player i if he reports himself to be type ti and his belief type is πi. Now observe that the payoff to type ti of player i reportinge himself to be type ti0 is:

ζi ti,ti0 πi (ti)[t i] ui ξ ti0 ,t i , θ (ti,t i) + yi ti0 ,t i , ≡ − − − − t i T i ¡ ¢ −X∈ − h ³ ¡ ¢ ´ ¡ ¢i b vi θi (tbi) , θi t0 , ψ (πi (ti)) + yi t0 , πi (ti) . ≡ i i i ³ ¡ ¢ ´ ¡ ¢ Now incentive compatibilityb b requiresb that ζ (ti,ti) ζ (ti,t ) for all ti,t Ti. b i e≥ ib i0 i0 ∈ This implies that if θi (ti)=θi (ti0 ) and πi (ti)=πi (ti0 ),then y (t , π (t )) = y t , π t . b i bi i i b i i0 ib i0 So y (t , π ) candependonlyonthebeliefandpayo¡ ¡ff¢¢relevant types corresponding i i0 i e e to ti0 .Writingyi θi0 , πi0 , πi for the expected transfer to a type of player i whose true belief over the opponent’s type is πi,butwhoreportshimselftobeatype e ¡ ¢ with belief πi0 andb payoff relevant type θi0 , the incentive compatibility conditions become, for all ti,t Ti, i0 ∈

vi θi (ti) , θi (ti) , ψi (πi (ti)) + yi θi (ti) , πi (ti) , πi (ti) ³ ´ ³ ´ vi θi (ti) , θi t0 , ψ (πi (ti)) + yi θi t0 , πi t0 , πi (ti) . ≥ b b i b i b b b i b i b ³ ¡ ¢ ´ ³ ¡ ¢ ¡ ¢ ´ This is equivalentb to theb requirementb that for all (bθi, πi) , θ0 , π Zi, b b b i i0 b∈ ¡ ¢ vi θi, θi, ψ (πi) + yi (θi, πi, πi) vi θi, θ0 , ψ (πi) + yi θ0 , π0 , πi . i ≥ i i i i ³ ´ ³ ´ ¡ ¢ Thus if (θi, πi) ,b θ0 , πi Zi,wemusthave b i ∈ b b ¡ ¢ vi θi, θi, ψ (πi) + yi (θi, πi, πi) vi θi, θ0 , ψ (πi) + yi θ0 , πi, πi . i ≥ i i i ³ ´ ³ ´ ¡ ¢ So setting b b b b yi∗ (θi, πi)=yi (θi, πi, πi) , we must have (4.6) satisfied. b 20 (Sufficiency) Suppose that there exists a function yi∗ : Zi R satisfying (4.6). Let → vi θi, θi, ψi (πi) + yi∗ (θi, πi) K =max . (4.7) (θ ,π ), θ ,π Z ¯ ³ ´ ¯ i i ( i0 i0 ) i ¯ vi θi, θi0 , ψi (πi) yi∗ θi0 , πi0 ¯ ∈ ¯ − b − ¯ ¯ ³ ´ ¯ For each i,choosehi : T i Πi ¯ R such that ¡ ¢ ¯ − × ¯→ b ¯

πi (t i) hi (t i, πi) πi (t i) hi t i, πi0 >K, (4.8) − − − − − t i T i t i T i −X∈ − −X∈ − ¡ ¢ for all π = πi. This is possible by standard separation arguments. Let i0 6

yi (t)=yi∗ θi (ti) , πi (ti) + hi (t i, πi (ti)) . − ³ ´ Now if player i is a type ti withbθi (ti)=b θi and πi (ti)=bπi, and he reports himself to be a type ti0 with θi (ti0 )=θi0 and πi (ti0 )=πi0 , his expected payoff is b b vi θi, θi0 , ψbi (πi) + yi∗ θi, πi0 + πi (t i) hi t i, πi0 . b − − t i T i ³ ´ ¡ ¢ −X∈ − ¡ ¢ b If he tells the truth, his payoff is

vi θi, θi, ψi (πi) + yi∗ (θi, πi)+ πi (t i) hi (t i, πi) . − − t i T i ³ ´ −X∈ − b His has an incentive to tell the truth if the latter expression exceeds the former, or:

πi (t i) hi (t i, πi) πi (t i) hi (t i, πi0 ) − − − −  "t i T i − t i T i #  0.  − P∈ − − P∈ −  ≥  + vi θi, θi, ψ (πi) + y∗ (θi, πi) vi θi, θ0 , ψ (πi) y∗ θ0 , π0  i i − i i − i i i  h ³ ´ ³ ´ ¡ ¢i  Ifπ = πi,thefirstb term is strictly greater than Kb(by (4.8)) and the second term i0 6 is less than or equal to K (by (4.7)), so the inequality holds. If πi = πi0 ,thefirst term is zero and the second term holds by (4.6).¥ Notice that the “only if” part of the result will continue to hold if there are an infinite set of types. For the “if” type, one could try to deal with infinite type spaces using the techniques of McAfee and Reny (1992), although one would probability require additional structure on the infinite type space. Neeman (1999) employed a similar decomposition in finding the optimal mech- anism in a public good provision problem and he noted how a similar construction couldbeusedinfinding the revenue-maximizing mechanism in the problem of allocating a private good. Notice that property (4.6) is automatically satisfied if the type space satisfies the belief extraction property. If the belief extraction property fails, we see that

21 the conditions for implementing ξ reduce to conditions that would have arisen with independent types in the naive type space. If enough beliefs over the payoff relevant types of the opponent are consistent with a given belief type, then this condition will reduce to the ex post implementability conditions.

5. Implementation

The preceding results were established by considering truthful rather than imple- mentation ‘tout court’. We now extend out investigation to consider implementa- tion. While a more permissive equilibrium notion allowed for truthful implemen- tation under a larger type space, the permissiveness becomes a handicap when we seek full implementation. Thus, we can now expect the equilibrium notion to become tighter as the type space becomes larger. In this section we start with the universal type space in Subsection 5.5 and then present the results for the remaining type spaces in Subsection 5.2

5.1. Universal Type Space Proposition 5.1. ξ is interim equilibrium implementable in the universal type space if and only if ξ is ex post rationalizable implementable in the naive type space. This proposition says that the most permissible solution concept (equilibrium) imposes few restrictions if one seeks unique implementation and simultaneously wishes to allow for any possible higher order belief. N Proof. Fix any mechanism .LetEPR (θi) be the set of ex post rationalizable G i actions for agent i in the incomplete information game ( N , ).Byconstruction, we have that for any type space , including the universalT typeG space, the set of T N ex post rationalizable actions for ( , ) is equal to EPRi (θi). Since all actions players will take in an interim equilibriumT G will be ex post rationalizable, we immediately have the result that ex post rationalizable unique implementation in the naive type space implies interim equilibrium unique implementation in the universal type space. To prove the other direction, it is enough to show that if ξ cannot be ex post rationalizable implemented in the naive type space, then ξ cannot be interim equilibrium implemented in the universal type space. To see this, again fixany N mechanism and let EPR (θi) be the set of ex post rationalizable actions for G i agent i in the incomplete information game ( N , ). Suppose that there exists T G θ∗ Θ and I ∈ N a∗ (a1∗, ...., aI∗) EPRi (θi∗) , ≡ ∈i×=1 x ξ such that g (a ) = ξ (θ∗). We will construct a type space and an equilibrium ∗ 6 T of the incomplete information game ( N , ) where ξ is not followed. Let T G N T ∗ = (θi,ai):ai EPR (θi) ; i ∈ i © ª 22 Thus Ti∗ consists of pairs of payoff relevant types (θi) and ex post rationalizable actions (ai) for that payoff relevant type. Similarly, define:

N T ∗i = (θj,aj)j=i aj EPRj (θj) for all j = i . − 6 ∈ 6 n ¯ o By lemma 2.7, we know that for¯ any (θi,ai) T , we can construct beliefs ¯ ∈ i∗ πi = πi (θi,ai) ∆ T ∗i , such that ai is a best response for payoff-relevant ∈ − type θi of player i. We naturally identify type (θi,ai) with payoff relevant type ¡ ¢ θi (andb set θi (θi,ai)=θi). The type space we have constructed must be a subset of the universal type space and we can label some subset of types in the universal typesb space such that there is an isomorphism between that subset and our type space. But now there is an equilibrium of the universal type space game with si (θi,ai)=ai. Consequently, we know that there exists t T ∗ such that x x ∈ g (s (t)) = g (a ) = ξ (θ∗)=ξ θ (t) . ∗ 6 ¥ ³ ´ This argument is an incompleteb information analogue of the Brandenburger and Dekel (1987) complete information result showing the equivalence of the set of actions played in any subjective correlated equilibrium and the set of actions sur- viving iterated deletion of strictly dominated strategies. Battigalli (1999) showed that common knowledge of rationality on the universal type space implies ex post rationalizable behavior, and thus this is the appropriate generalization of rationalizability to incomplete information in this context.

5.2. Common Prior Type Spaces We first present necessary and sufficient conditions for a social choice function ξ to be ex-post equilibrium implementable in the naive type space. We then proceed to relate the appropriate equilibrium conditions across different type spaces. The conditions for uniqueness are extensions of Bayesian monotonicity conditions (see Jackson (1991)) to the ex-post equilibrium conditions. In this paper, we restrict our analysis to the implementation of a social choice function rather than a social choice correspondence or set.2 As we maintain the quasilinear environment, the model satisfies the ‘economic environment’ assumption frequently made in the implementation literature. In a separate note, Bergemann and Morris (2001), we present necessary and sufficient conditions for unique ex-post equilibrium imple- mentation of social choice sets in general environments. We call deception a mapping : αi : Ti Ti and let α =(α1, ..., αI ) and → α (t)=(α1 (t1) , ..., αI (tI )). In a direct revelation game αi would indicate i’s reported type as a function of his true type. The notion of a deception is meant to represent the possibility of multiple equilibria, in which agents do not necessarily report truthfully, but rather misreport systematically as represented by α. Again,

2 With the restriction to a social choice function ξ, the closure condition regarding the social choice set appearing in the literature is trivially satisfied.

23 for a direct revelation mechanism, if agents report the deception α rather than truthfully, then the resulting social outcome is given by ξ [α (t)] rather than ξ (t).

Definition 5.2 (Ex-post incentive compatibility). ξ satisfies ex-post incentive compatibility (IC) if for all i and t T : ∈ vi (ξ (t) ,t) vi ξ ti,t i ,t , ti Ti. (5.1) ≥ − ∀ ∈ ¡ ¡ ¢ ¢ Definition 5.3 (Ex-post monotonicity).e e Consider ξ and deception α. ξ satisfies ex-post monotonicity (M) if, when ξ = ξ α, there exists i, t and λ Ξ such that 6 ◦ ∈ vi (λ (α (t)) ,t) >vi (ξ (α (t)) ,t) and

vi ξ ti, α i (t i) , ti, α i (t i) vi λ (α (t)) , ti, α i (t i) , ti Ti. − − − − ≥ − − ∀ ∈ (5.2) ¡ ¡ ¢ ¡ ¢¢ ¡ ¡ ¢¢ e e e e The ex-post monotonicity conditions allows for a similar interpretation as suggested by Jackson for the Bayesian monotonicity condition. For ξ to be ex- post implementable, it has to be that for every deception α, there exists an agent i and a type profile t, such that at type profile t agent i has a strict incentive to denounce the deception by altering the social choice function to λ. However it has to be guaranteed that this particular agent i doesn’t denounce the remaining agents when their announced type profile α i (t i) is in fact their true type profile. The later incentive constraint is represented− − by the second set of (weak) inequalities. The conditions are different only insofar as the Bayesian condition evaluates the utilities at the interim stage whereas in the ex-post condition the utilities are evaluated after the resolution of uncertainty and hence under complete infor- mation. For this reason, the condition of ex-post monotonicity is closely related to the monotonicity condition introduced by Maskin (1999). The monotonicity notion suggested by Maskin can be rephrased, to allow for a direct comparison with ex -post monotonicity as follows.3

Definition 5.4 (Maskin monotonicity). Consider ξ and deception α. (ξ, α) satisfy Maskin monotonicity hypothesis if there exists i, t, λ Ξ such that ∈ vi (ξ (α (t)) ,t) >vi (λ (α (t)) ,t) , while vi (ξ (α (t)) , (α (t))) vi (λ (α (t)) , α (t)) . (5.3) ≥ 3 Palfrey and Srivastava (1989) first suggested such a reformulation for the Bayesian mono- tonicity condition to explore the relationship between Bayesian monotonicity and monotonicity under complete information.

24 The only difference in the ex-post monotonicity condition and the Maskin monotonicity arises in the weak inequality. The difference here is easy to locate. In the complete information environment, all agents have symmetric information about the types. Hence agent i cannot credibly issue a report distinct from the other agents. In consequence, the only type ti of agent i which is of concern to the designer is the one which he shares with the other agents, or ti = α i (t i). − − Thus ex-post monotonicity implies Maskin monotonicity.e We can now state the necessary and sufficient conditions for uniquee ex-post implementation, supposing that #I 3. ≥ Theorem 5.5 (Unique Ex-Post Implementation). ξ is ex-post implementable in the naive type if and only if ξ satisfies (IC) and (M).

Proof. The following mechanism implements the social choice function x if the conditions (IC) and (M) are met. Let

V = 0, 1 2N . { } Thus v = v1,v2, ..., v2N V is a 2N dimensional vector such that each entry is either 0 or 1.Letthemessagespacebe:∈ ¡ ¢

Mi = Ti V Ξ Ξ , × {∅∪ }× ×{∅∪ } with z }| { 1 2 3 4 mi = m ,m ,m ,m Mi, i i i i ∈ and ¡ ¢ M = M1 ... MN . × × Partition M into four sets:

j d0 = m M m =(, , , ) , j , ∈ · ∅ · ∅ ∀ © ¯ ª which will contain the unique equilibrium¯ strategy;

1 m/d0,mj =(, , , ) , j = i, and di = m M ∈ · ∅ · ∅ ∀ 6 , ∈ mi =(, , , ) , ½ ¯ · · · ∅ ¾ ¯ ¯ in which exactly one agent¯ chooses to report a modulo number;

2 mj =(, , , ) , j = i, and di = m M · ∅ · ∅ ∀ 6 , ∈ mi =(, , , λ) ½ ¯ · · · ¾ ¯ ¯ in which exactly one agent chooses¯ to suggest a social choice function λ,different from the default, and which controls the outcome whenever the suggesting agent

25 1 2 can not possible benefit from it in equilibrium. Let d1 = i di and d2 = i di . Finally, the fourth set is given by: S S d3 = m M m/d0 d1 d2 , { ∈ | ∈ ∪ ∪ } with multiple deviations from the equilibrium strategy and where the modulo game is played. Define the outcome function g : M X by: → ξ (m1) , if m d0 d1 m ∈ di ,and∪  ∈ 2  λ (m1) , if ti Ti,vi ξ ti, α i (t i) , ti, α i (t i)  ∀ ∈ − − − −  vi λ (α (t)) , ti, α i (t i) g (m)= ¡≥ ¡ ¢ ¡ − − ¢¢  e em di ,and e  ¡ ∈ 2 ¡ ¢¢  ξ (m1) , if ti Ti, vi ξ ti, α i (t i) ,e ti, α i (t i) ∃ ∈ − − − −  J(k) for all k I ,theni = i,otherwise ∈ ∗ ∈ ∗ ∗ i∗ =1.Thesufficiency part is then proved through a sequence of lemmata.¥

3 Lemma 5.6. For any i, t and s, there exists vi V such that si, where s (ti)=vi ∈ i for all ti and si = si otherwise, is such that i∗ = i whenever [si,s i](t) d3. − ∈ e e Proof. Consider a type t and fixthevectorvk for all agents but i and look at i e i e the entry vk (tk) of agent k.Enterthenumbervk (tk) in the N + k-th entry of i vector vi.Agenti can therefore match every entry v of k I .Itremainsto k ∈ ∗ find a strategy, i.e. digits for the first N entries of vi so that agent i is never matched,whichinturnguaranteesthatJ (i) >J(k). But this is guaranteed by thefactthatagenti has a binary choice of 0 and 1 and hence i is guaranteed to always find a number to preclude a match.¥ Lemma 5.7. If ξ satisfies (IC), then there is a set of strategies s which form an equilibrium to (M,g),suchthatg (s)=ξ.

Proof. Consider si (ti)=(ti, , , ). Notice that g [s (t)] = ξ (t) for all t T .We verify that s is an equilibrium∅ by· showing∅ that there are no improving deviations.∈ Consider a deviation mi by i at ti Ti where ti misreports and states ti. ∈ If mi = ti, , , ,then[mi,s i (t i)] d0 d1. The resulting allocation is · · ∅ − − ∈ ∪ ξ ti,t i .From(IC),e we know that this is not improving. e − ¡ ¢ e e e ¡ ¢ e 26 If mi = ti, , , λ ,then[mi,s i (t i)] d2.If · · − − ∈ ¡ ¢ vi (ξ (ti,eα i (t i)) , (ti, α i (t i))) vi λ ti, α i (t i) , (ti, α i (t i)) e − − e− − ≥ − − − − ¡ ¡ ¢ ¢ for all ti Ti, then the allocation is λ ti, , whiche is not improving by the above ∈ · inequality. Otherwise the allocation is ξ ti, which is not improving by (IC). ¡ ¢ · Notice that all possible mi have been considered.e ¡ ¢ ¥ e Lemma 5.8. If the environmente is economic, then any strategy s which forms an equilibrium to (M,g) must have the property that t, s (t) d0. ∀ ∈ Proof. The proof is by contradiction. Suppose not, then there must be equi- librium strategy profiles at some point t such that s (t) / d0. Suppose then that j j ∈ there exists j and t such that s (t) d1 d2 and for all t0 T , s (t0) d0 d1 d2. But this cannot be an equilibrium either,∈ ∪ as there then exists∈ a second∈ agent∪ i∪who coulddeviatedfromaprofile ( , , , ) to one such that (si0 (ti) ,s i (t i)) d3. By Lemma 5.6, agent i can always· ∅ · guarantee∅ himself to be the decisive− − agent,∈ or i∗ = i. As the environment is economic, and more specifically satisfies quasilinear utility, the winning agent i∗ can always insist on the implementation of the same social outcome ξ as in the hypothetical equilibrium, but with an increased and balancing monetary transfer of ε from every other agent to himself. A similar argument holds for the case that there exists t such that s (t) d3.Asevery agent k I can guarantee himself to be the decisive agent i ,∈ there does not ∈ ∗ ∗ exist an ex post equilibrium at t with s (t) d3. ∈ ¥ Lemma 5.9. If the environment is economic and ξ satisfies (M) and (IC),then foreachsetofstrategiess which form an equilibrium to (M,g), ξ = g (s).

Proof. Now we apply (M) to show that g (s)=ξ.Supposethatg (s) = ξ.By (M) there exists i, t and λ,suchthat 6

vi (λ (α (t)) ,t) >vi (ξ (α (t)) ,t) and

vi ξ ti, α i (t i) , ti, α i (t i) vi λ (α (t)) , ti, α i (t i) , ti Ti. − − − − ≥ − − ∀ ∈ ¡ ¡ ¢ ¡ ¢¢ ¡ ¡ ¢¢ By Lemmae 5.8, we knowe that for all t, s (t) d0, ande hence there existse α such that ξ α = g (s), and therefore ∈ ◦

vi (λ (α (t)) ,t) >vi (g (s (t)) ,t) while

vi ξ ti, α i (t i) , ti, α i (t i) vi λ (α (t)) , ti, α i (t i) , ti Ti. − − − − ≥ − − ∀ ∈ ¡ ¡ ¢ ¡ ¢¢ ¡ ¡ ¢¢ e e e e 27 Thus i is better off submitting

[αi (ti) ,vi, λ, λ] , where vi is defined as suggested by Lemma 5.6, whenever ti is observed, since the resulting outcome is λ on (ti,T i). This is shown as follows. The deviation i − puts the action in d2 for all t (ti,T i) and the outcome is λ α (t),whichis ∈ − ◦ strictly preferred by i to g (s) on (ti,T i). This contradicts the fact that s is an equilibrium, and so the supposition was− wrong. This completes the sufficiency part. (Necessity) Let (M,g) implement ξ with an equilibrium strategy s (t). Con- sider any i, ti Ti, and the strategy si (ti)=si (ti0 ) for all ti Ti.Sinces is an equilibrium, ∈ ∈ vi (g (s (t)) ,t) vie(g (si (ti) ,s i (t i)) ,t) ≥ − − for all t T. Noting that g (si (ti) ,s i (t i)) = vi (ξ (ti0 ,t i) ,t) establishes (IC). Suppose∈ that for some deception −α, ξ−=e ξ α.Itmustbethat− ξ α is not an 6 ◦ ◦ equilibrium at some t T . Therefore there exists i and ai Ai such that ∈ e ∈

vi (g (si (ti) ,s i (α i (t i))) ,t) >vi (g (s (α (t))) ,t) − − − e where si (ti)=ai for all ti Ti.Letλ = g (si,s i). Then, from above, e ∈ − v (λ (α (t)) ,t) >v (ξ (α (t)) ,t) . e e i i e

Since si is constant, we can set

λ (αi (ti) ,t i)=λ (t)=g (si (ti) ,s i (t i)) . e − − − Thus, since s is an equilibrium it follows that e

vi ξ ti0 ,t i , ti0 ,t i vi λ (αi (ti) ,t i) , ti0 ,t i , ti0 Ti, − − ≥ − − ∀ ∈ ¡ ¡ ¢ ¡ ¢¢ ¡ ¡ ¢¢ and thus λ satisfies the requirement (M).¥ The structure of the proof is similar to Jackson (1991). The mechanism used to prove sufficiency is simpler as we require the strategies to be in an ex-post rather than an interim equilibrium. The entire argument is more compact due to the simplifying assumption of an economic environment and the implementation ofasocialchoicefunctionratherthansocialchoiceset.Wereferthereaderto Bergemann and Morris (2001) for the appropriate uniqueness conditions of an ex-post equilibrium within a general environment.

Theorem 5.10. If ξ is ex post equilibrium implementable in U ,thenξ is T interim equilibrium implementable in U . T

28 Proof. Consider ξ and α. First, suppose the social choice function is ex-post implementable. Then there exists i, t, and λ such that according to definition (5.3): vi (λ (α (t)) ,t) >vi (ξ (α (t)) ,t) and

vi ξ ti, α i (t i) , ti, α i (t i) vi λ (α (t)) , ti, α i (t i) , ti Ti. − − − − ≥ − − ∀ ∈ Based¡ on¡ λ, we then¢ construct¡ a λ0¢¢such that¡ (??)issatis¡ fied as¢¢ well at i, ti and e e e e λ0.Let λ, if t = α (t) , λ = 0 (5.4) ξ, if t = α (t) . ½ 0 6 Evaluating the interim utilityb of agent i at λ α and ξ α and ti, we want to verify whether (??)holds,or ◦ ◦

vi λ (α (t)) ,t pi (t i ti ) > vi (ξ (α (t)) ,t) pi (t i ti ) , − | − | t i T i t i T i −X∈ − ³ ´ −X∈ − and b

vi ξ ti,t i , ti,t i pi t i ti − − − t i S i −X∈ − ¡ ¡ ¢ ¡ ¢¢ ¡ ¯ ¢ e e ¯e vi λ (αi (ti) ,t i) , ti,t i pi t i ti , ti = ti ≥ − − − ∀ 6 s i S i −X∈ − ³ ¡ ¢´ ¡ ¯ ¢ b ¯ By definition of λ,thefirst inequality can bee rewritten as e e

vi (λ (α (t)) ,t) >vi (ξ (α (t)) ,t) , b which is precisely the first part of the ex-post monotonicity condition, which is satisfied by hypothesis. Consider now the second inequality, where we rearrange each side into two terms, or

vi ξ ti,t0 i , ti,t0 i pi t0 i ti − − − t0 i=α i(t i) − 6 X− − ¡ ¡ ¢ ¡ ¢¢ ¡ ¯ ¢ e e ¯e +vi ξ ti, α i (t i) , ti, α i (t i) pi α i (t i) ti − − − − − − ¡ ¡ vi λ α¢i (t¡i) ,t0 i , ti,t¢¢0 i ¡pi t0 i ti¯ ¢ ≥ e e − − − ¯e t0 i=α i(t i) − 6 X− − ³ ¡ ¢ ¡ ¢´ ¡ ¯ ¢ b e ¯e +vi λ (α (t)) , ti, α i (t i) pi α i (t i) ti , − − − − and after some rearranging³ ¡ ¢´ ¡ ¯ ¢ b e ¯e vi ξ ti,t0 i , ti,t0 i − − pi t0 i ti + Ã vi λ¡ α¡i (ti) ,t¢0 ¡i , ti,t¢¢0 i ! − t0 i=α i(t i) − − − − 6 X− − e e ¡ ¯ ¢ ³ ¡ ¢ ¡ ¢´ ¯e vi ξ ti, α i (bt i) , ti, α i (t ei) − − − − pi α i (t i) ti 0. vi λ (α (t)) , ti, α i (t i) − − ≥ Ã −¡ ¡ ¢ ¡ − − ¢¢ ! e e ¡ ¯ ¢ ³ ¡ ¢´ ¯e b e 29 By definition of (5.4), we can rewrite the sum as follows:

vi ξ ti,t0 i , ti,t0 i − − pi t0 i ti , (5.5) vi ξ αi (ti) ,t0 i , ti,t0 i − t0 i=α i(t i) µ − ¡ ¡ ¢−¡ ¢¢− ¶ − 6 X− − e e ¡ ¯ ¢ ¡ ¡ ¢ ¡ ¢¢ ¯e but each term in the brackets only represente the ex-post incentive compatibil- ity conditions, which are satisfied by hypothesis, and hence the sum in (5.5) is nonnegative. Finally consider the remaining term

vi ξ ti, α i (t i) , ti, α i (t i) − − − − v¡i ¡λ (α (t)) , ti,¢α¡ i (t i) , ¢¢ − e −e − which represents precisely the¡ second part¡ of the ex-post¢¢ monotonicity condition, e and hence is nonnegative as well, which concludes this part of the proof.¥ The converse of the result does not hold. In contrast to the case of truth- ful implementation where we could establish equivalence (Theorem 4.4), with implementation a social choice function can be interim implementable for all dis- tributions yet fail to be ex-post implementable. The difference in the result is due to the introduction of a second set of incentive constraints associated with the necessary and sufficient condition for uniqueness. Before we document this difference with an example, it may be informative to recall the conditions for uniqueness of an interim equilibrium. We said that a social choice function ξ satisfies Bayesian monotonicity if for all α, such that ξ = ξ α, i, ti, λ such that 6 ◦ ∃

vi (λ (α (t)) ,t) pi (t i ti ) > vi (ξ (α (t)) ,t) pi (t i ti ) (5.6) − | − | t i T i t i T i −X∈ − −X∈ − while

vi ξ ti0 ,t i , (ti,t i) pi t i ti0 (5.7) − − − t i T i −X∈ − ¡ ¡ ¢ ¢ ¡ ¯ ¢ ¯ vi λ ((αi (ti)) ,t i) , ti0 ,t i pi t i ti0 , ti0 Ti. ≥ − − − ∀ ∈ t i T i −X∈ − ¡ ¡ ¢¢ ¡ ¯ ¢ ¯ The logic behind the earlier equivalence result was to localize the inequality by concentrating probability on a given type profile t i. This essentially amounted − to evaluating the above inequalities at t i,or −

vi (λ (α (t)) ,t) >vi (ξ (α (t)) ,t) (5.8) while

vi ξ ti0 ,t i , (ti,t i) vi λ ((αi (ti)) ,t i) , ti0 ,t i , ti0 Ti. (5.9) − − ≥ − − ∀ ∈ While the¡ ¡ localized¢ condition¢ now¡ appears to be similar¡ with¢¢ the necessary and sufficient conditions for an ex-post equilibrium, observe that the inequalities in

30 (5.9) are evaluated at the true type profile t i, the corresponding ex-post condi- tions, or −

vi ξ ti, α i (t i) , ti, α i (t i) vi λ (α (t)) , ti, α i (t i) , ti Ti, − − − − ≥ − − ∀ ∈ ¡ ¡ ¢ ¡ ¢¢ ¡ ¡ ¢¢ are evaluatede at the truee type profile α i (t i). This distinctione is due toe the fact that in the ex-post conditions, the bene−fit of− denouncing the non-truthful equilib- rium and the incentive condition to denounce the non-truthful equilibrium if and only if it occurs are evaluated at two different true type profiles of the remain- ing agents, namely t i and α i (t i). The localization argument can converge to only one of these two− points− and− hence the previous argument cannot establish equivalence anymore. The following example shows that the failure to establish the equivalence is not due to the proof strategy but inherent to the conditions required for unique implementation. Consider the following example with two agents. Each agent receives a bi- nary signal ti 0, 1 . The states of the world are consequently given by t 00, 01, 10, ∈11 {. The} designer can choose among four different allocations as∈ well,{ which for} simplicity will be labelled exactly as the states of the world, say ξ ( ) 00, 01, 10, 11 .Thepayoff matrix below represents the payoffsof agent 1· in∈ each{ of the four} states. Every submatrix represents the four possible allocations in a given state of the world, where the later is denoted in the square brackets at the southeast corner. The allocation is determined by the report of each agent. The reports of agent 1 appear as rows and the reports of agent 2 as columns.

01 01 2+ε 1 1+ε 1 0 ∗ 0 ∗ (00) (01) (00) (01) 1 2 2 0 1 1 (10) (11) (10) (11) [00] [01] (5.10) 01 01 0 2 2 1 0 0 (00) (01) (00) (01) 1 1+ε 1 2 + ε 1 ∗ 1 ∗ (10) (11) (10) (11) [10] [11]

In the following it will be sufficient to consider the payoffsofagent1 as the discrepancy in the conditions for one agent imply that we can always find payoff matrices for the other agents with similar deficiencies. Suppose then that we would like to implement the following social choice rule ξ (t)=t for all t. With the payoff matrix as above, it can be shown that truthtelling forms an ex-post and hence interim equilibrium strategy for agent 1. However there is another ex-post

31 equilibrium in which every agent always misrepresents his type, or si (ti)=ti with ti = ti. Again, it can be verified that given the supposed strategy by 6 agent 2, always misrepresenting is an equilibrium strategy for agent 1. Whilee this concernse equilibria in the direct mechanism we now discuss the necessary and sufficient conditions for uniqueness when we permit indirect or augmented mechanism. Here we observe that for agent 1 we cannot find a state t and mechanism λ which would eliminate the second and nondesirable equilibrium. To see this notice firstthatinthestates[01] and [10],agent1 receives more in the “bad” equilibrium than he can receive through any other allocations. Thus if the designer wants to offer incentives for agent 1 to denounce the “bad” equilibrium, they have to giveneitherinstate[00] or [11].Asthepayoff matrix is symmetric it will be sufficient to discuss the case of [00]. The new mechanism λ has to set λ (1, 1) = λ (α (0, 0)) = (0, 0) to reward agent 1, as this is the only allocation which provides sufficient incentives for agent 1. But besides the positive incentives:

v1 (λ (α (0, 0)) , (0, 0)) >v1 (ξ (α (0, 0)) , (0, 0)) (5.11) the new mechanism λ must be undesirable for agent i when all agents are reporting truthfully: v1 (ξ (0, 1) , (0, 1)) v1 (λ (1, 1) , (0, 1)) , (5.12) ≥ and v1 (ξ (1, 1) , (1, 1)) v1 (λ (1, 1) , (1, 1)) , (5.13) ≥ butitcanbeverified that (5.12) does not hold. We complete the discussion of this example by demonstrating that despite the failure of the guarantee for a unique ex-post equilibrium, we can guarantee a unique interim equilibrium. The reason appears can be traced back to the difference between interim and ex-post. Consider the payoff matrix in state [01] and we consider again the mechanism λ which λ (1, 1) = (0, 0), but in all other instances assigns λ (t)=t.Theinterim incentives can now be satisfied for in the case of t1 =0,ifitsufficiently likely that agent 2 will sometimes display t2 =0. In this case, a switch to the mechanism λ would not be beneficial for agent 1,asitwouldinduceapayoff of 1 rather than 2+ε. As the gains from misrepresenting in the state t1 =1are only ε it follows that the probability of t1 =0does not have to be large to reestablish the incentive. As we can tailor the mechanism to the prior probabilities under consideration it 1 is easy to see that for p =Pr(t2 =0) 2 ,themechanismλ suggested above ≥ 1 satisfies the full set of incentives. Likewise for p< 2 ,themechanismλ (t)=t for all t with the exception of (0, 0) at which λ (0, 0) = (1, 1) establishes in an entire symmetric fashion the desired inequalities. Matsushima (1993) showed that with transferable utility, sufficient conditions for unique implementation can be given which are easily stated and generically (in terms of prior probability distribution) satisfied. He requires that the incen- tive compatibility conditions are strictly satisfied and a no consistent deception

32 condition, which holds if the conditional probabilities satisfy

pi (t i ti ) = pi t0 i ti0 for all s = s0. − | 6 − 6 ¡ ¯ ¢ 6. Discussion ¯

6.1. Robustness Our current analysis of robustness focused on implementation problems which can (i) be represented by a social choice function, ξ : Θ X independent of the distribution of types and (ii) allowed implementation with→ probability one. Many optimal design problems, such as revenue maximizing auctions depend on the distribution of (payoff-relevant) types and hence our current approach seems less suitable as it is focused on the distribution free properties of the de- sign problem.4 Recent work by Hansen, Sargent and various co-authors, (see Hansen and Sargent (2001)) on robust control in macroeconomic models might be suggestive for modelling robust mechanisms when there is uncertainty about the type environment. In settings where we have a common prior over a type space or a subset thereof, we would like to consider virtual implementation (Abreu and Matsushima (1992)) where we only require implementation with high ex ante probability (rather than for every possible type). The discussion about the prevalence and genericity of certain types in Section 3 suggests yet a different approach to the robustness question. For a given type space, we could ask when is it the case that for every type there is a nearby type (in an appropriate topology) which can be implemented with probability one. Finally, we can also understand complete information implementation as a special case of our framework, where we focus on types in the universal type space among whom there is common belief of their payoff relevant types. One would like to understand how robust existing mechanisms are to their complete information assumptions. One way to do this would be to consider subsets of the universal type space where there is almost complete information. For example, suppose we looked at subsets of the universal type space where the common prior assumption holds. Suppose that it is possible to label each (rich) type of each player with his own payoff relevant type and a payoff relevant type profile for the rest of the population in such a way that with ex ante probability 1 ε, each type has the same label. This type space is intuitively ε-close to complete− information subsets of the universal type space (if ε were equal to zero, we would

4 In fact, with a common prior over the payoff-relevant types, the social choice function of an optimal auction depends on the 0—th order and 1-st order beliefs of the agent, and can be represented: ξ : Θ ∆ (Θ) X. × → A natural extension of the previous analysis would then be to ask when is the implementation of ξ robust to richer tpes space beyond the 1-st order beliefs.

33 have complete information of payoff relevant types). Now we can examine virtual implementation on this subset of the universal type space. This approach is the natural mechanism design analogue of Kajii and Morris’ (1997) examination of the robustness of complete information equilibria to incomplete information, in general strategic settings. A starting point for our analysis would be Abreu and Matsushima’s positive results using only iterated deletion of dominated strategies. Since such strategies are robust to incomplete information in the sense of Kajii and Morris, they can be used to construct equilibria on the “nearly complete information” larger type spaces that virtually implement the desired outcomes. Theworkplanintheprevioussectionentailedaninvestigationofthecondi- tions for implementation as we varied the type space, leaving the environment, including the sets of possible payoff-relevant types, fixed. The second set of ques- tions for us to pursue is to see how existing results for the naive type space in particular mechanism design settings will change as the type space changes. As we discussed in the introduction, it was exactly such questions that led us to pursue our approach to analyzing robustness.

6.2. Ex-Post Equilibrium Following Maskin (1992), a number of papers have examined ex post equilibrium implementation for environments with interdependent types.5 We formalized above the idea that ex post equilibrium implementation is actually required if we were to require implementation on the universal type space. In fact, ex post equilibrium implementation is also required simply to ensure implementation on the naive type space for all independent distributions over types. But it should also be possible to show, using the p-generated type spaces discussed above, that even if the mechanism designer knew that the payoff relevant types were correlated according to a particular distribution, ex post equilibrium implementation would be required. Our ex post rationalizability notion involved the iterative deletion of strictly dominated strategies. Strict dominance has no bite in the ex post mechanisms in this literature, so there exist equilibria (in the naive type space, and thus in the universal type space) that do not achieve an efficient allocation. However, in general those equilibria involve weakly dominated strategies. An open question iswhatwouldhappenifwedeletedweaklydominatedstrategiesintheexpost rationalizability concept. While their version of ex post rationalizability does not quite match ours, Chung and Ely (2000) have some results along these lines. For naive spaces, Dasgupta and Maskin (2000) identify a single crossing prop- erty that is sufficient for efficient ex post equilibrium implementation. In their leading example, they consider a two player case where player 1 observes a signal θ1,player2 observes a signal θ2 and their valuations of an object are v1 = aθ1 +θ2

5 Dasgupta and Maskin (2000), Jehiel and Moldovanu (2001), Perry and Reny (1999), Berge- mann and Valimaki (2001).

34 and v2 = aθ2 + θ1, respectively. Their single crossing sufficient condition in this example is that a 1. We would like to examine the analogue to this property on larger type spaces.≥ Their are two ways of doing this (and we wish to pursue both). One is to take the signals θ1 and θ2 as the payoff-relevant types in our larger type space constructions. But it is also natural to ask what would happen if we took players’ valuations to be their payoff-relevant types in our construction. Now to capture the signal approach of Dasgupta and Maskin (2000), we would need to impose common knowledge restrictions on players’ beliefs about valuations. In the leading example, it is (implicitly) assumed that player 1 knows for certain the 2 value of av1 v2 (it is always equal to a 1 θ1). Just as a recent literature has tried to understand− the meaning of the common− prior assumption when expressed in the language of players’ higher order¡ beliefs¢ in the universal types space (see, e.g., Feinberg (2000)), we can also try and express implementability conditions in terms of players’ higher order beliefs about valuations. The current results thus merely represent some initial steps to spell out how far the requirements of common knowledge can be weakened in the pursuit of mechanism design solutions. In this sense they represent an attempt to make the “Wilson doctrine”, which prefaced this paper, precise and operational.

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