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Journal of Economic Theory 148 (2013) 689–719

www.elsevier.com/locate/jet

Finite-order type spaces and applications ✩

Cheng-Zhong Qin a,∗, Chun-Lei Yang b

a Department of Economics, University of California at Santa Barbara, Santa Barbara, CA 93106, United Statesb Research Center for Humanities and Social Sciences, Academia Sinica, Taipei, Taiwan

Received 25 October 2010; final version received 6 May 2012; accepted 19 August 2012

Available online 7 January 2013

Abstract

We propose a framework of consistent finite-order priors to facilitate the incorporation of higher-orderuncertainties into Bayesian game analysis, without invoking the concept of a universal type space. Severalrecent models, which give rise to stunning results with higher-order uncertainties, turn out to operate withcertain consistent order-2 priors. We introduce canonical representations of consistent finite-order priors,which we apply to establish a criterion for determining the orders of strategically relevant beliefs for abstractHarsanyi type spaces. We derive finite-order projections of type spaces and discuss convergence of BNEsbased on them as the projection order increases. Finally, we introduce finite-order total variation distancesbetween priors, which are suitable for analyzing the issues on equilibrium continuity and robustness. Werevisit recent advancements of Bayesian game theory and develop new insights based on our framework.© 2013 Elsevier Inc. All rights reserved.

JEL classification: C72; D82

Keywords: Bayesian game; Common prior; Complete information; Consistent prior; Finite-order type space; Payofftype space; Projection type space; Robustness; Total variation distance

✩ We thank Yi-Chun Chen, Kim-Sau Chung, Matthew Jackson, Atsushi Kajii, Qingmin Liu, Dov Monderer, andShmuel Zamir for helpful discussions and comments during the development of this paper. We also thank an associateeditor and two anonymous referees for constructive comments that helped to greatly improve the paper. We gratefullyacknowledge research support from RCGEB at Shandong University.

* Corresponding author. Fax: +1 805 893 8830.E-mail addresses: [email protected] (C.-Z. Qin), [email protected] (C.-L. Yang).

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690 C.-Z. Qin, C.-L. Yang / Journal of Economic Theory 148 (2013) 689–719

1. Introduction

Popular applications of Bayesian games assume that the players’ types are their payoff typesand it is common knowledge that their payoff types are jointly drawn by nature according to somecommon prior. The resulting type space is known as a payoff type space, which is the smallesttype space we can work with [2, p. 1773]. There is an extreme implication with a payoff typespace: each payoff type of a player uniquely determines his belief about the other players’ types.Despite its virtue of simplicity, payoff type spaces are restrictive for certain applications. Forexample, consider a sealed-bid, first-price auction for a high-tech patent between two potentialbuyers. In addition to the uncertainty about each other’s reservation prices (payoff types), onebidder might have reason to suspect that the other bidder can acquire information about hisreservation price with some positive probability, say, by hiring a super-talented computer hacker.Payoff type spaces cannot deal with such information settings.

There is, however, a great deal of additional complexity attached to type spaces beyond payofftype spaces. Though they may look simple in the implicit model a la Harsanyi [18], their cor-respondences in Mertens and Zamir’s [26] explicit model of universal type spaces can be verycomplicated. Given a Bayesian game, neither the implicit nor the explicit model provides anycriterion for determining the upper bounds on the orders of strategically relevant beliefs. As wediscuss later in this paper, characterizing type spaces in terms of these upper bounds turns out tobe crucial in applications. We introduce the concepts of consistent finite-order priors and theircanonical type spaces, which we apply to explicitly model type spaces with finite-order beliefhierarchies without invoking the concept of a universal type space, and to determine the upperbounds on the orders of strategically relevant beliefs for abstract type spaces. We also intro-duce finite-order projections of an abstract type space and finite-order total variation distancesbetween priors, which we apply to develop new insights into an array of theory and appliedproblems associated with higher-order uncertainties.

Our analysis begins with the private-value order-2 common prior case. In this case, naturedraws for each player a payoff type and a first-order belief about the others’ payoff types ac-cording to a common prior. Such a common prior is of order-2 by virtue of being a probabilitymeasure over payoff types and first-order beliefs. But, unlike with types in payoff type spaceswhich are de facto of order 1, nature may draw multiple first-order beliefs for a given payoff typeof a player. The consistency of the prior then requires that for each player, each of his first-orderbeliefs drawn by nature coincides with the posterior belief he can derive by updating the commonprior using his private information (the payoff type and first-order belief).1 We show that givena payoff environment, consistent order-2 priors form a convex class that contains all order-1 andcomplete information priors as proper subsets. It follows that the class of consistent order-2 pri-ors is richer than both the class of order-1 and the class of complete information priors. Indeed,as we discuss later, substantially different results can be established for several familiar Bayesiangames once the information structure changes from an order-1 or complete information commonprior to a consistent order-2 common prior.

The model of a consistent order-2 prior with private value can be extended to the model ofa consistent order-k prior with common value for any finite positive integer k: nature draws apayoff relevant parameter and for each player, a coherent belief hierarchy of order k − 1. As

1 For reasons that will become clear after Theorem 1, we use the terms consistent priors and common-prior type spacesinterchangeably.

C.-Z. Qin, C.-L. Yang / Journal of Economic Theory 148 (2013) 689–719 691

before, players’ order-(k − 1) belief hierarchies drawn by nature coincide with the order-(k − 1)belief hierarchies they can obtain from updating the prior using their private information. Re-interpreting such order-(k − 1) belief hierarchies in the support of the prior as abstract Harsanyitypes, the prior then induces a canonical common prior type space. The consistency of the priorguarantees that each of these abstract Harsanyi types generates the first (k − 1) orders of beliefsin the canonical type space, which are exactly the same as those consisting of the (k − 1)-beliefhierarchy taken as the abstract Harsanyi type.

We say that a common prior type space is of order k if it can be mapped by a type morphisminto the canonical type space of a consistent order-k prior. We show that order-k common priortype spaces are those such that any two types of each player must either be separable by theirorder-(k − 1) belief hierarchies, or have identical belief hierarchies of any order. As such, k is anupper bound on the orders of strategically relevant beliefs in an order-k type space. In this sense,a consistent order-k prior is equivalent to an order-k common prior type space.

In a seminal paper, Rubinstein [34] shows that equilibrium behavior can be very sensitive tohigher-order beliefs in a famous example known as the electronic mail game. The players cancoordinate on the efficient outcome of the game if there is common knowledge about payoffs,but this efficient coordination fails should there be any lack of common knowledge. His studyhas been very influential in the development of the new literature on implications of higher-orderbeliefs, and timely sensitized important issues of equilibrium convergence with converging in-formation structures. Our framework of finite-order type spaces is applicable to several branchesof this literature. First, by truncating players’ belief hierarchies at finite orders and restrictingthe common prior to the truncated belief hierarchies, our framework can be applied to derivefinite-order projections of a type space, to be referred to as projection type spaces. Finite-orderprojection type spaces can be regarded as bounded-rational realizations of type spaces, in thesense that nature only informs the players of their belief hierarchies up to some finite orderinstead of their entire belief hierarchies. Projection type spaces are suitable for analyzing im-plications of imposing finite upper bounds on the orders of players’ beliefs. With the help of anexample of a Cournot duopoly game with incomplete information, we demonstrate in Section 3.3that as projection order increases, the global stability under uncertainty condition in [37] ensuresthe convergence of the BNEs based on projection type spaces to the BNE of the Cournot duopolygame.

Second, Carlsson and van Damme [5] show that only the risk-dominant equilibrium of thestag-hunt game survives a class of incomplete information perturbations of the game, whichresult in global games. Morris and Shin [31] show that allowing for different classes of perturba-tions may lead to the selection of the Pareto-dominant equilibrium instead. Weinstein and Yildiz[36] further show that ambiguity like this is the rule with the usual product topology and gener-ally, any rationalizable strategy can be the limit point of rationalizable strategies of some weaklyconvergent sequence of types in the universal type space. In Section 4.3 we show that contrastingto the Rubinstein [34] model which relies on unbounded orders of strategically relevant beliefs,the type spaces of the global games are of order 2. This prompts the open question of when thecontinuity of rationalizability result in [36] can be obtained with type spaces of a given finiteorder.

Third, as demonstrated in [36], the product topology is too coarse to yield convergence resultsof any predictive power. Dekel, Fudenberg, and Morris [11] introduce the strategic topologyas the coarsest topology that makes interim correlated rationalizability a continuous solution.Chen, di Tillio, Faingold, and Xiong [6] develop uniform weak topology, which is finer and moretransparent than the strategic topology. However, these topologies are equivalent to the product


topology for the spaces of belief hierarchies truncated at finite orders.2 In this paper, we introducean order-k total variation distance between priors based on our notion of order-k projection ofa type space for each finite integer k � 1. The topology induced by the order-k total variationdistance is stronger than the weak topology and, as illustrated later in this paper, it is suitable foranalyzing issues related to continuity and robustness of BNEs against incomplete informationperturbations.

Fourth, Monderer and Samet [27] show that if payoffs of a complete information game arecommon-p belief for probability p close to 1, then all strict Nash equilibria are robust againstlimit common knowledge elaborations. Kajii and Morris [21] propose a general approach to an-alyzing the robustness of Nash equilibrium of a complete information game against incompleteinformation perturbations. Under their definition, a Nash equilibrium is robust against incom-plete information perturbations, if every nearby elaboration of the complete information gamehas a BNE that generates behavior close to the Nash equilibrium on average. We demonstratethat our order-k total variation distance is a natural candidate for a stronger topology to analyzethe robustness of BNE against finite-order perturbations of type spaces. We apply this distanceto generalize the robustness notion of Kajii and Morris [21] to Bayesian games. We also char-acterize conditions that ensure the robustness of BNEs, a result which can be understood as ageneralization of Monderer and Samet’s [27] robustness result for NEs.

Aside from various theoretical advances, the applied fields have also witnessed a recentsurge of work that exploits higher-order beliefs to achieve some refreshing and striking results.One strand starts with allowing a player’s payoff type to have multiple first-order beliefs thatlead to significantly different outcomes. For example, Neeman [33] shows that the well-knownfull-surplus-extraction (FSE) result in the mechanism design literature crucially hinges on theproperty that agents’ beliefs uniquely determine their payoff types (the BDP property). By im-plicitly constructing a consistent order-2 prior, Neeman shows that the FSE fails without the BDPproperty. Moreover, Heifetz and Neeman [19] show that the BDP is non-generic in general typespaces, in contrast to the claim by Cremer and McLean [8,9] that the BDP, and hence the FSE, isgeneric in the class of payoff type spaces.

Bergemann and Välimäki [3] have an example that demonstrates the strategic implications ofintroducing order-2 priors to auction models. Once again, allowing a payoff type of a player tohave two different first-order beliefs leads to altered bidding strategies, which result in inefficientoutcomes for the first-price auction and make the revenue equivalence result invalid.

Feinberg and Skrzypacz [14] show that the one-sided repeated offers bargaining game be-tween the buyer and seller of an indivisible good results in a failure of the Coase conjecture, inthe sense that delay occurs with positive probability in a class of equilibria satisfying the intu-itive criterion. We demonstrate in Section 4.3 that their information structures are of order 2 andrepresentable by convex combinations of order-1 priors. As an implication of the convexity ofconsistent order-2 priors and the insights from [19], we show that the Feinberg and Skrzypaczconjecture that the failure of the Coase conjecture is generic holds within the class of consistentorder-2 priors. We also show that the Coase conjecture is not a continuous property with respectto the order-2 total variation distance, while it is continuous under the weak topology. In con-trast, the failure of the Coase conjecture is a continuous property with respect to the order-2 totalvariation distance.

2 This seems to suggest that the discussion of critical types in [13] necessarily implicates infinite orders of beliefs.


Another strand of the new applied literature aims at robust implementation in the mechanismdesign literature. Bergemann and Morris [2] show that, under some conditions, the Bayesianimplementability of a two-player budget-balanced social choice function on all complete infor-mation type spaces implies its ex post implementability. As complete information type spaces arenot of order 1 but are of order 2, their results imply that for robust implementation, it is sufficientto have Bayesian implementability with respect to all consistent order-2 priors. It is interesting tonote that in the proof of the necessity of monotonicity for undominated Nash equilibrium-closureimplementability, Chung and Ely [7] consider perturbations of a complete information type spacewhich also turn out to be of order 2.

Methodological justifications of heterogeneous priors have caused many debates in the lit-erature [28]. Due to the lack of an explicit move by nature to generate the interim privateinformation, heterogeneous priors are often criticized for being too ad hoc, making trivial theexplanation for how differences in information result in differences in behavior. As Myerson[32] argues, there are strong modeling methodological reasons why the common prior assump-tion (CPA) has become standard in economic analysis. Nevertheless, heterogeneous priors areoften encountered in applications. We show in Section 2.4 that two-person private-value mod-els with order-1 heterogeneous priors having common supports can be embedded into order-2consistent priors. This may be interpreted as a partial justification of heterogeneous informationmodeling.

Finally, a critical issue discussed in [12] concerns strategic relevance of redundancy in typespaces. Liu [25] establishes an interesting equivalence between BNEs in type spaces with redun-dancy and correlated equilibrium in redundancy-free type spaces. We illustrate with an examplein Section 4.4 that our framework can be extended to model strategic redundancy using addi-tional signals as equilibrium correlation devices. In other words, even though our model has thefeature of being free of redundancy, this feature is not essential and can be bypassed to allow fora richer strategic environment.

The rest of the paper is organized as follows. In Section 2, we introduce the notion of consis-tent order-2 priors for the private value case and discuss their properties. In Section 3, we extendthe notion to common-value consistent finite-order priors, link them with finite-order universaltype spaces, and introduce finite-order projections of type spaces and total variation distances be-tween priors. Applications of different natures are discussed throughout the paper but particularlyin Section 4. Conclusions are given in Section 5, proofs of results are organized in Appendix A,and further miscellaneous material is presented in Appendix B.

2. Consistent order-2 priors with private value

We adopt the following notations. Given a subset C of a topological space, �(C) denotes theset of probability measures on the σ -algebra of Borel sets of C. Endow �(C) with the weaktopology and the product space with product topology. Given P ∈ �(C), let suppP denote thesupport of P . For a ∈ C, δa denotes the Dirac measure at a. For C = X × Y , we use PX ormargXP to denote the marginal probability measure of P ∈ �(C) over X, and we use BX orprojXB to denote the projection of subset B ⊆ C into X. The marginal probability measure of P

over Y and projection of B into Y are similarly defined.A game with incomplete information is a collection Γ = {(Ai, ui, Ti,πi)i∈N,Θ} where N =

{1,2, . . . , n} is the player set; Θ is the space of payoff relevant parameters; and for each i ∈ N ,Ai is the action set of player i, ui : A × Θ −→ � his payoff function, Ti his type set, andπi : Ti −→ �(Θ × T−i ), with T−i = ∏

Tj , is his belief function. For ti ∈ Ti , πi(ti) specifies
j �=i


player i’s belief about the others’ types t−i ∈ T−i and the payoff parameter θ ∈ Θ (i.e., π(ti) ∈�(Θ ×T−i )). The collection T = (Θ, (Ti,πi)∈N) is the type space a la Harsanyi based on payoffenvironment Θ (fundamental uncertainty). When the payoff environment is unambiguous, wewrite T simply as T = (Ti,πi)i∈N . We assume that Θ is a Polish space (i.e. a complete separablemetric space), as is the case when Θ is finite and endowed with the discrete topology. A typespace T is finite if Ti is finite for all i. A finite type space T has a common prior if there exists aprobability measure P ∈ �(Θ × T ) such that margTi

P (ti) > 0 and πi(ti)[θ, t−i] = P(θ, t−i |ti )for all i and for all (θ, t) ∈ Θ × T . A strategy for player i in game Γ is a mapping σi : Ti −→�(Ai). A strategy profile σ = (σi)i∈N is a BNE for Γ if for all i ∈ N, ti ∈ Ti , and ai ∈ Ai ,

Eti

[ui

((σi(ti), σ−i (t−i )

), θ

)]�Eti

[ui

((ai, σ−i (t−i )

), θ

)],

where the expectation is taken over Θ × T−i with respect to πi(ti) on both sides. For tractability,we assume that Θ is finite and confine attention to finite type spaces throughout this paper unlessnoted otherwise. When Θ = ∏

i∈N Θi and ui(a, (θi, θ−i )) = ui(a, (θi, θ′−i )) for all a ∈ A and

for θ−i , θ′−i ∈ Θ−i = ∏

j �=i Θj , the payoff environment is known as having private value. In thatcase, θi ∈ Θi is called a payoff type of player i.

2.1. Consistency and convexity

We begin with a private-value environment Θ = ∏i∈N Θi , which is often assumed in appli-

cations. Imagine that nature draws a profile (θ,φ) ∈ Θ × ∏i∈N �(Θ−i ) according to a prior

Φ ∈ �(Θ × ∏i∈N �(Θ−i )) and then informs each player i of both his payoff type θi and first-

order belief φi .3 Since player i can use Φ in combination with his private information (θi, φi) toderive a posterior belief about θ−i via the Bayes rule, this derived posterior belief must coincidewith φi to avoid any contradiction for considering the latter as his first-order belief. Unless notedotherwise, in this paper we consider priors with finite supports for notational convenience.

Definition 1. We say that Φ ∈ �(Θ ×∏ni=1 �(Θ−i )) is a consistent order-2 prior if for all i and

(θi, φi) ∈ supp margΘi×�(Θ−i )Φ ,

φi(·) = margΘ−iΦ(· | θi, φi). (1)

The following proposition establishes convexity of the class of consistent order-2 priors. Con-vexity is useful for the analysis of generic properties, among other applications.

Proposition 1. Any convex combination of two consistent order-2 priors is a consistent order-2prior.

A convex combination of two priors can be interpreted as the result of a random choice bynature. Example 1 below illustrates that richer information structures can be generated by havingnature randomly select among simpler ones.

Example 1. Think of a bilateral trade game with an indivisible good between a buyer (player 1)and a seller (player 2). Assume that the buyer’s reservation value is either high or low, denoted

3 Since Θ is Polish, �(Θ) is also Polish (see Theorem 15.15 in [1]). It follows that singletons in �(Θ) are Borelmeasurable. Since most of the present paper focuses on finite type spaces, it suffices to consider probabilities assigned tosingle payoff type and belief profiles instead of Borel measurable sets.


by θ1h and θ1l , while the seller has a single reservation value, denoted by θ2. Assume furtherthat there are two possible order-1 priors P A,P B ∈ �(Θ) with Θ = {(θ1h, θ2), (θ1l , θ2)} dueto different product origins, say location A and location B. It is commonly known among theplayers that the probability for A (resp. B) being the true origin is λ (resp. 1 − λ). As privateinformation, the buyer is informed of his realized reservation value and the seller of the productorigin. What would be an appropriate type space to model this information structure?

To answer the question, consider order-2 priors Φz with Φz(θ,φ) = P z(θ)δφz(φ) for θ ∈ Θ

and φ ∈ �1(Θ2)×�2(Θ1) where φz1 = δθ2 , φz

2(θ1) = P z(θ1, θ2), and z = A,B . By construction,ΦA and ΦB generate exactly the same first-order beliefs for the players as the priors P A and P B

do. Further, they both satisfy the consistency condition (1).By Proposition 1, Φλ = λΦA + (1 − λ)ΦB is a consistent order-2 prior. Furthermore,

suppΦλ = suppΦA ∪ suppΦB

and

Φλ(θ,φ) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

λpAh if θ = (θ1h, θ2), φ = (φA

1 , φA2 ),

(1 − λ)pBh if θ = (θ1h, θ2), φ = (φB

1 , φB2 ),

λpAl if θ = (θ1l , θ2), φ = (φA

1 , φA2 ),

(1 − λ)pBl if θ = (θ1l , θ2), φ = (φB

1 , φB2 ),

0 otherwise

for λ ∈ (0,1), where pzh = P z(θ1h, θ2) and pz

l = P z(θ1l , θ2) for z = A,B . A simple analysisshows that the information structure in this example can be generated by having Φλ as the com-mon prior. Note that Φλ is not equivalent to an order-1 prior because the same payoff type θ2of the seller is associated with two different first-order beliefs φA

2 and φB2 . Interpreting a convex

combination as nature’s move to choose among order-1 priors enriches the information structureby one belief order higher. It turns out, as we discuss in Section 4.1, that Feinberg and Skrzypacz[14] exploited such a class of information structures with suitable choices of λ, which leads tothe generic failure of the Coase conjecture.

2.2. Complete information priors

Games of complete information are often benchmarks for modeling economic problems.For example, Bergemann and Morris [2, Proposition 5] show that under certain conditions, theBayesian implementability of a budget-balanced social choice function on all complete informa-tion type spaces implies its ex post implementability. Therefore, it is important to properly sortout complete information priors within the universal type space. Intuitively, players have com-plete information about θ̄ ∈ Θ if player i believes with probability 1 that the others’ payoff typesare θ̄−i for all i.

Definition 2. Given an order-2 prior Φ ∈ �(Θ × ∏ni=1 �(Θ−i )) and θ̄ ∈ supp margΘΦ , we say

that information about payoff-type profile θ̄ is complete if

φi(·) = δθ̄−i(·)

for all i and for all φ such that (θ̄ , φ) ∈ suppΦ; we say that Φ is a complete information prior ifthe above condition holds for all θ̄ ∈ suppΘ Φ .


Observe that the definition implies that Φ is consistent whenever Φ is a complete informationprior. To explore further special properties, for each θ̄ ∈ Θ let Φθ̄ be the order-2 prior defined by

Φθ̄(θ,φ) = δθ̄ (θ)δ(δθ̄−i)ni=1

(φ), ∀θ ∈ Θ, φ ∈n∏

i=1

�(Θ−i ). (2)

Then, Φθ̄ is consistent and (θ,φ) ∈ suppΦθ̄ if and only if θ = θ̄ and φ = (δθ̄−i)ni=1. That is, the

one-point prior Φθ̄ in (2) is a degenerate complete information prior.

Proposition 2. An order-2 prior Φ ∈ �(Θ × ∏ni=1 �(Θ−i )) is a complete information prior if

and only if Φ is in the convex hull of {Φθ̄ }θ̄∈Θ .

Remark 1. An alternative way to model complete information works as follows. First, manipu-late the set of fundamental uncertainties by setting Θ̄i = Θ as the new set of private signals forplayer i. Next, consider a common prior P ∈ �(

∏ni=1 Θ̄i) = �(Θn). Information is complete

under prior P if and only if (θ̄i)ni=1 ∈ suppP implies θ̄1 = θ̄2 = · · · = θ̄n.

2.3. Convex hull of order-1 priors

Complete information priors are special consistent order-2 priors in that players’ first-orderbeliefs are degenerate under such priors. Nevertheless, multiple first-order beliefs can be associ-ated with a payoff type for a player. Next, observe that order-1 priors are also special consistentorder-2 priors in that each payoff type of a player is associated with a single first-order belief.4

Indeed, an order-2 prior Φ ∈ �(Θ × ∏i �(Θ−i )) is equivalent to an order-1 prior if and only if

it is consistent and for all (θ,φ), (θ ′, φ′) ∈ suppΦ , θi = θ ′i implies φi = φ′

i for all i. The neces-sity of the condition is obvious. To see the sufficiency, for θ ∈ suppΦ , let φθi

denote the uniquefirst-order belief that corresponds to player i’s payoff type θi and let φθ = (φθ1 , . . . , φθn). Then,

Φ(θ,φ) = P(θ)δP (·|θ)(φ), ∀(θ,φ) ∈ suppΦ, (3)

where P(θ) = Φ(θ,φθ ) ∈ �(Θ) and P(·|θ) denotes the profile of conditional probabilitiesP(·|θi), i = 1,2, . . . , n. It follows easily that Φ and P generate the same first-order beliefs;hence, they are equivalent. By (3), P can be regarded as the “representing” order-1 prior for Φ .

Another way to look at the specialty of complete information and order-1 priors is to apply themethod of genericity analysis in [19]. By Proposition 1, a convex combination of an order-1 anda complete information prior is a consistent order-2 prior. However, such a convex combinationis neither an order-1 nor a complete information prior. Hence, using the terminologies in [19],order-1 and complete information priors are non-generic in the class of consistent order-2 priors.

Proposition 2 shows that the convex hull of degenerate complete information priors coincideswith the class of all complete information priors. It is natural to guess that, as a parallel result,the convex hull of order-1 priors coincides with the class of consistent order-2 priors. We showby example that this is not true.

4 Degenerate complete information priors are order-1 priors. But, not all non-degenerate complete information priorsare order-1 priors, because a payoff type of a player may have multiple first-order beliefs.


(θ21, φ21) (θ22, φ22) (θ23, φ23) (θ23, φ24)

(θ11, φ11) 19

19

19 0

(θ12, φ12) 19

19

227

127

(θ13, φ13) 19

227

127 0

(θ13, φ14) 0 127 0 2

27

Fig. 1. An example of a type space not in the convex hull of order-1 priors.

Example 2. Let Θi = {θi1, θi2, θi3} for i = 1,2. For i �= j and k = 1, . . . ,4, define first-orderbeliefs for player i by

φik = (φik(θj1),φik(θj2),φik(θj3)

) =

⎧⎪⎨⎪⎩

( 13 , 1

3 , 13 ) if k = 1,2,

( 12 , 1

3 , 16 ) if k = 3,

(0, 13 , 2

3 ) if k = 4.

Now consider order-2 prior Φ in Fig. 1.Notice that Φ is an order 2 prior because the third payoff type θi3 of player i has two first-

order beliefs φi3 and φi4. However, Φ is not in the convex hull of order-1 priors. A proof isgiven in Appendix B. Example 2 is taken from [3, Section 3.2]. The authors use this example toillustrate important implications of higher-order uncertainties. In particular, they find that unlikewith order-1 type spaces, first-price auctions with the type space in this example do not alwaysyield efficient allocations.

2.4. Embedding heterogeneous order-1 priors

We say that a profile (P1,P2) of heterogeneous order-1 priors in �(Θ) can be embedded inan order-2 consistent prior Φ if for all θ ∈ Θ with θi ∈ supp margΘi

Pi for all i,(θ,P (·|θ)

) = (θ,

(P1(·|θ1),P2(·|θ2)

)) ∈ suppΦ. (4)

Notice that the support of the common prior Φ may contain elements that are not possible underthe given heterogeneous priors. Consequently, the embedding common prior may add beliefs notpresent in the original heterogeneous type space.

Proposition 3. Suppose n = 2. Then, players’ order-1 priors in �(Θ) can be embedded in aconsistent order-2 prior in �(Θ × ∏2

i=1 �(Θ−i )) whenever they have a common support.

This embedding result may be regarded as a reconciliation of the CPA controversy for the two-person case, to the extent that players’ heterogeneous priors could be justified by their inducedposteriors being simultaneously generated from a common prior of exactly one order higher. Itcould be conceived as if nature made its move according to the higher-order common prior, butsystematically failed to (privately) inform the players about certain realizations. With this failurebeing common knowledge, the resulting information structure behaves as if every player simplyupdates his own heterogeneous prior to get the appropriate posteriors. The reader is referredto [28] for an elaborate discussion on the CPA in economic theory.


Proposition 3 can be extended to arbitrary finite order cases. For the n-person case, we cancharacterize conditions for players’ order-1 heterogeneous priors to be embedded in an order-2consistent prior. These extensions are left out for expositional brevity.

3. Finite-order type spaces: The general case

In this section we extend the notion of a consistent order-2 prior and the results in Section 2 toconsistent priors of arbitrary finite orders with common value. As illustrated in Appendix B, theprivate-value case can be viewed as a special form of the common-value case. For these reasons,we will focus the paper on the common-value case in later sections.

We introduce canonical representations of consistent finite-order priors and establish a cri-terion for determining upper bounds on the orders of strategically relevant beliefs for abstractHarsanyi type spaces. We also introduce finite-order projection type spaces and discuss conver-gence of BNEs based on them as the projection order increases. Finally, for each integer k � 1,we introduce an order-k total variation distance between priors, which can be applied to analyzeequilibrium continuity and robustness.

3.1. Finite-order belief hierarchies, priors, and canonical type spaces

Let k � 2 be an integer and let Θ be a given payoff environment. For each player i, thefollowing recursive construction of the spaces of belief hierarchies of orders h = 1, . . . , k − 1 isstandard:

T 1i = �(Θ), T 2

i = T 1i × �

(Θ × T 1−i

), . . . ,

T k−1i = T k−2

i × �(Θ × T k−2

−i

), k > 2. (5)

An element tk−1i ∈ T k−1

i is composed of player i’s first-order belief φ1i ∈ �(Θ), second-order

belief φ2i ∈ �(Θ × T 1−i ), . . . , and (k − 1)-th order belief φk−1

i ∈ �(Θ × T k−2−i ). We refer to

tk−1i = (φ1

i , φ2i , . . . , φk−1

i ) as an order-(k − 1) belief hierarchy of player i. As usual, tk−1i is

coherent if

margΘφ2 = φ1 and margΘ×T h−2

−iφh

i = φh−1i , h = 3, . . . , k − 1. (6)

Set T k−1 = ∏ni=1 T k−1

i . Elements in �(Θ ×T k−1) are order-k priors. As will become clear later,k is also an upper bound on the orders of strategically relevant beliefs whenever consistency asdefined below is satisfied.

Definition 3. For k � 2, an order-k prior Φ ∈ �(Θ × T k−1) is consistent if for all tk−1 ∈supp margT k−1 Φ and for all i, tk−1

i = (φ1i , . . . , φk−1

i ) is coherent and

φk−1i

(θ, t̃k−2

−i

) = margΘ×T k−2

−iΦ

(θ, t̃k−2

−i

∣∣tk−1i

), ∀(

θ, t̃k−2−i

) ∈ T k−2−i .

Notice that the consistency of Φ in Definition 3 together with the coherency condition (6) oftk−1i ∈ supp marg

T k−1i

Φ implies consistency at lower orders; that is,⎧⎨⎩

φ1i (θ) = margΘΦ

(θ |tk−1

i

), θ ∈ Θ,

φhi

(θ, t̃h−1

−i

) = margΘ×T h−1

−iΦ

(θ, t̃h−1

−i

∣∣tk−1i

), t̃h−1

−i ∈ T h−1−i , 2 � h � k − 1.

(7)

By sorting terms carefully, a parallel result to Proposition 1 can be similarly established.


Proposition 4. Consistent order-k priors consist of a convex subset of �(Θ × T k−1).

Recall that a type space is a collection T = (Ti,πi)i∈N , where πi : Ti −→ �(Θ × T−i ) is thebelief mapping for player i. For k � 2, a type ti ∈ Ti generates recursively an order-(k − 1) beliefhierarchy, denoted by τ k

i (ti |T ) = (ψ1i (ti |T ), . . . ,ψk−1

i (ti |T )) ∈ T k−1i , via the standard method

as follows5:

ψ1i (ti |T )[θ ] =

∑t−i∈T−i

πi(ti)[θ, t−i], ∀θ ∈ Θ, (8)

and for 2 � h� k − 1,

ψhi (ti |T )

[θ, th−1

−i

]=

∑t−i∈T−i : th−1

−i

=(ψ1−i (t−i |T ),ψ2−i (t−i |T ),...,ψh−1−i (t−i |T ))

πi(ti)[θ, t−i], ∀(θ, th−1

−i

) ∈ Θ × T h−1−i . (9)

Here, ψhi (ti |T )[θ, th−1

−i ] is the probability with which player i believes that the payoff parameter

value is θ and the other players’ order-(h-1) belief hierarchies are prescribed by th−1−i ∈ T h−1

−i ,given that his own type is ti ∈ Ti . By construction, τ k

i (ti |T ) is coherent. Furthermore, it is ameasurable transformation from T to T k−1.

Let ρki denote the inverse of τ k

i . Then,

ρki

(tk−1i |T ) = {

ti ∈ Ti

∣∣ τ ki (ti |T ) = tk−1

i

}(10)

for all tk−1i ∈ T k−1

i .6 Notice that

τ∞i (ti |T ) = lim

k−→∞ τ ki (ti |T )

is the belief hierarchy in the universal type space that corresponds to player i’s type ti ∈ Ti . Wewill suppress T from τ k

i and ρki to save space whenever there is no ambiguity.

An order-k prior Φ ∈ �(Θ × T k−1) uniquely determines a common-prior type space, T Φ =(T Φ

i ,πΦi )i∈N , where

T Φi = supp marg

T k−1i

Φ (11)

is the set of types for player i and

πΦi

(tΦi

)[θ, tΦ−i

] = Φ(θ, tΦ−i

∣∣tΦi ), ∀(

θ, tΦ−i

) ∈ Θ × T Φ−i , tΦi ∈ T Φi . (12)

By viewing T Φ as an abstract type space, to be referred to as the canonical type space of Φ , (8)and (9) can be applied to derive players’ finite-order belief hierarchies τh

i (tΦi ) for all tΦi ∈ T Φi

and all i. It can be sown that players’ order-(k − 1) belief hierarchies in T Φ coincide with thosein suppΦ , provided Φ is a consistent order-k prior. See Lemma A.1 in Appendix A.

5 Since Θ is Polish, singletons are Borel measurable. We use summation instead of integration because we primarilyfocus on finite type spaces for tractability.

6 See [4, pp. 182, 537] for a formal definition of measurable mappings and properties of their inverse images.


3.2. Orders of type spaces

Let T = (Ti,πi)i∈N and T ′ = (T ′i , π

′i )i∈N be two type spaces based on Θ . A mapping f =

(f0, f1, . . . , fn) from Θ × T to Θ × T ′, with f0 : Θ �→ Θ and fi : Ti �→ T ′i , is a type morphism

from T to T ′ if f0 is the identity mapping on Θ and

π ′i

(fi(ti)

)[E′−i

] = πi(ti)[f −1

−i

(E′−i

)](13)

for all i = 1,2, . . . , n, ti ∈ Ti , and E′−i ⊆ Θ ×T ′−i . A type morphism between T and T ′ preservesuniversal types (see [20], for details).

Definition 4. We say that a type space T = (Ti,πi)i∈N is of order k for some integer k � 1, ifthere is a type morphism from T to the canonical type space T Φ of some consistent order-k priorΦ ∈ �(Θ × T k−1).

The following theorem establishes necessary and sufficient conditions for a common priortype space to be of order k.

Theorem 1. A common prior type space T = (Ti,πi)i∈N is of order k for some k � 1 if and onlyif for all i and ti , t

′i ∈ Ti ,

τ ki (ti) = τ k

i

(t ′i) ⇒ τ∞(ti) = τ∞(

t ′i). (14)

Theorem 1 states that a type space is of order k if and only if any two types of a player withthe same order-(k − 1) belief hierarchy must have the same universal type. In particular, all non-redundant types of an order-k type space have different order-(k − 1) belief hierarchies. The fullseparability of players’ types by the associated order-(k − 1) belief hierarchies implies that toassess opponents’ strategy choices, it suffices for each type of a player to resort to the k-th orderbeliefs only. Thus, k is an upper bound on the orders of strategically relevant beliefs.

Observe that when a type space is finite, there exists a finite integer k such that any two typesof each player can be separated at the (k − 1)-th order beliefs or have the same universal type.Thus, a direct application of Theorem 1 establishes the following corollary.

Corollary 1. Any finite common-prior type space is of finite order.

In an order-k type space, the belief hierarchy τ∞i (ti) in the universal type space that corre-

sponds to type ti contains no additional strategically relevant information not already containedin τ k

i (ti). That is, the tails of the belief hierarchies can be ignored for strategic analysis. Thecentral message of Theorem 1 is that any order-k type space has the canonical representationof a consistent order-k prior as its non-redundant equivalence. Consequently, convergence offinite-order priors can be handled using topologies for the space of finite-order belief hierarchies.Notice also that if T is of order k, then it is of order l for all l > k.

3.3. Order-k projections and total variation distance

For any k � 2, the mappings τ k(·) in (8) and (9) and ρk in (10) can be applied to projecta common prior type space T = (Ti,πi)i∈N into T k−1 as follows. Define T̃ k−1

i ⊂ T k−1i and

Φ̃k ∈ �(Θ × T k−1) by

T̃ k−1 = τ k(Ti) and Φ̃k(θ, tk−1) = P

(θ,ρk

(tk−1)), (

θ, tk−1) ∈ Θ × T̃ k−1, (15)
i i


where P is the common prior for T . Next, let π̃ ki : T̃ k−1

i −→ �(Θ × T̃ k−1−i ) be given by

π̃ ki

(tk−1i

)[θ, tk−1

−i

] = Φ̃k(θ, tk−1

−i

∣∣tk−1i

),

(θ, tk−1

−i

) ∈ Θ × T̃ k−1−i , tk−1

i ∈ T k−1i . (16)

By (15) and (16), T̃ k = (T̃ k−1i , π̃ k

i )i∈N is a type space of order k with common prior Φ̃k , to bereferred to as the order-k projection of T . With these notations in place, iterated applications ofthe mappings in (8) and (9) yield the following result.7

Lemma 1. Let T be a type space. Then, τ k(τh(t)) = τ k(t) for all type profile t ∈ T and h �k � 2.

Given an order-k type space, projections of orders h > k do not carry more information thanthe order-k projection. Thus, Lemma 1 and Theorem 1 together yield the following corollary.

Corollary 2. Let T be an order-k type space. Then, for all h > k and th−1 ∈ T̃ h−1,

π̃hi

(th−1i

)[θ, th−1

−i

] = π̃ ki

(τ ki

(th−1i

))[θ, τ k−i

(th−1−i

)].

Suppose that it is common knowledge that the true model of information structure is capturedby type space T . One way to model bounded rationality is perhaps to have an “imperfect na-ture”, which draws a combination (θ, t) and informs the players of their corresponding beliefs ofsome (randomly determined) finite order k instead of the whole belief hierarchies. Our consistentorder-k prior approach reflects the realizations of such imperfect nature.8

The order-k projection type space T̃ k converges to T in product topology as k increases.A natural question is whether the BNEs with projection type spaces converge to those with Tas the projection order approaches to infinity. Rubinstein [34] shows that this is not the case ingeneral. However, Weinstein and Yildiz [37] show that under the global stability condition, thesensitivity of BNEs with respect to variations of higher order beliefs vanishes as the order ofbeliefs increases. Incorporating their arguments into our framework of projection type spaces,this condition ensures the convergence of BNEs with projection type spaces. We omit a formaldiscussion here to avoid the complexity of too many new definitions. Instead, we illustrate theequilibrium convergence by revisiting the following example of Weinstein and Yildiz [37] interms of our notion of finite-order projection type spaces.

Example 3. Consider a Cournot duopoly with demand function p(Q) = θ − Q, where θ ∈ Θ isan unknown demand parameter. Assuming zero production costs, firm i’s profit at quantities qi ,qj and parameter θ is given by

vi(qi, qj , θ) = qi · (θ − qi − qj ).

7 To derive the order-(k − 1) belief hierarchy profile τk(τh(t)), we view τh(t) as an abstract type profile in the order-hprojection type space.

8 Kets [23] explicitly models finite depth of reasoning as part of a player’s type description, which can be interpretedas a form of this “imperfect nature” condition. Finite depth of reasoning is modeled with a coarse sigma algebra thatcannot separate other players’ types beyond some given order of beliefs. This is similar in spirit to our characterizationof finite-order type spaces. While her model is an extension of the standard Harsanyi model of a type space by addinganother element to the usual term of infinite belief hierarchy, ours is a reduction of the complexity associated with thestandard Harsanyi model, which enables us to “close” the belief model at a finite order for ease of application.


Let T = (Ti,πi)i∈N denote a common prior type space based on Θ . A strategy profile (q∗1 , q∗

2 )

with q∗i : Ti �→ Ai ⊂ �+ is a BNE if for each type ti , q∗

i (ti ) maximizes qi(E[θ |ti] − qi −E[q∗

j |ti]). This yields the following first-order condition:

q∗i (ti ) = 2−1(E[θ |ti] − E

[q∗j |ti

]).

For a random variable X with appropriate measurability, let Ethi[X] denote the expectation of X

with respect to thi and let Ehi [X] = E[X|B(T h

i )], where B(T hi ) denotes the sigma-algebra of

Borel sets of T hi with the product topology. As in [37], iterated applications of both firms’ first-

order conditions and the law of iterated expectation establish

q∗i (ti ) =

∞∑m=1

(−1)m+12−mEτmi (ti )E

m−1j Em−2

i · · ·E1im

[θ ],

where im = i whenever m is odd and im = j otherwise. If T is of order k, then, by Corollary 2,Eh

i = Eki for all h > k; hence, the equilibrium strategy depends only on belief hierarchies up to

order k. If q∗(k) denotes the BNE with the order-k projection of T , then its first k terms are thesame as those with the original type space. If Ai is compact for all i, then the terms in the tailconverge to zero at the exponential rate of 2−k . It follows that q∗(k) → q∗.

The approach by Weinstein and Yildiz [37] is to first identify the best response in terms ofuniversal beliefs and show that the weight for all beliefs higher than the k-th order exponentiallyvanishes, without a definitive equilibrium response characterization for any finite k. In contrast,for each k, we explicitly find the BNE using the order-k projection and show that it differs fromthe BNE of the original Cournot game with the difference vanishing with k.

We end this section with an introduction to an order-k total variation distance between priorsbased on the same payoff environment. First, given two measures P and Q on a measurablespace (Ω,B), the total variation distance between them is given by

‖P − Q‖TV = 2 supB∈B

∣∣P(B) − Q(B)∣∣.

The reader is referred to [35, Section 11.3] for an introduction and applications of this distancein economics. By applying this distance to order-k projections of priors, an order-k distance forthe space of priors with given payoff environment can be defined as follows.

Definition 5. Let P ′ and P ′′ be priors based on Θ . We call ‖P ′ − P ′′‖k = ‖Φ̃ ′k − Φ̃ ′′k‖TVthe order-k total variation distance (henceforth, the TV-k distance) between P ′ and P ′′, whereΦ̃ ′k, Φ̃ ′′k ∈ �(Θ × T k−1) are the order-k projections of P ′ and P ′′. We say that a sequence of{P m}m of priors converges to a prior P in TV-k distance if ‖Φ̃m,k − Φ̃k‖TV −→ 0 as m −→ ∞,where Φ̃k and Φ̃m,k are the order-k projections of P and P m, respectively.

Observe that, although two priors P ′ and P ′′ are not necessarily defined on the same sigma-algebra, their order-k projections, Φ̃ ′k and Φ̃ ′′k , are both in �(Θ × T k−1). Thus, the TV-kdistance ‖P ′ − P ′′‖k = ‖Φ̃ ′k − Φ̃ ′′k‖TV between P ′ and P ′′ is well-defined.9 We can now statetwo results that will be used later for robustness analysis.

9 By taking the supremum with respect to the order of projections, our definition of finite-order total variation distancebetween priors can be extended to the infinite belief hierarchy setting. More precisely, define the total variation distance‖P ′ − P ′′‖TV between P ′ and P ′′ by ‖P ′ − P ′′‖TV = sup1�k<∞ ‖P ′ − P ′′‖k .


Lemma 2. Fix k � 2 and Θ . If a sequence {P m}m of priors converges to prior P in TV-k distance,then supp Φ̃k ⊆ supp Φ̃m,k for large enough m.

Lemma 3. Fix k � 2 and Θ . If a sequence {P m}m of priors converges to prior P in TV-k distance,then for any γ > 0, there exists an integer mγ such that

Φ̃m,k(supp Φ̃m,k \ supp Φ̃k

)< γ

for all m > mγ .

Lemma 2 shows that the order-k projections of the converging priors must have larger supportsthan the limit prior does. Lemma 3 further shows that the difference in supports between the con-verging and the limit priors converges to zero in probability with respect to the converging prior.

4. Applications: New insights and beyond

Recent development in the Bayesian game literature has made many new stunning discoveriesbased on the introduction of higher-order uncertainties in familiar models, some of which werealready mentioned in the Introduction. In this section, we first revisit the existing results of sucha model and answer some related open questions. We then proceed to tackle robustness of BNEand show that the type spaces of global games are of order 2. We end this section with an exampleillustrating how to deal with the issue of strategic redundancy within our framework, which isformally redundancy free.

4.1. Coase conjecture with order-2 consistent priors

Consider the one-sided repeated-offer bargaining game between the buyer and seller of anindivisible good. In each round the seller makes an offer that the buyer can either accept orreject. There is a common discount factor. When the information structure is modeled by a payofftype space with one-sided private information and common knowledge of positive gains fromtrade, this simple bargaining procedure is known to yield no delay. Here, no delay means that asoffers become increasingly frequent, the actual time for reaching agreement converges to zero.This convergence is known as the Coase conjecture (e.g., [16] and [17]). Recently, Feinbergand Skrzypacz [14] show that introducing higher-order uncertainties dramatically changes theoutcome of the bargaining. Specifically, the presence of the buyer’s second-order uncertaintiesabout the seller’s first-order beliefs can lead to a failure of the Coase conjecture, in the sense thatdelay occurs with positive probability in equilibrium satisfying the intuitive criterion.

In [14], the buyer’s value for the indivisible good is either θ1h or θ1l with θ1h > θ1l > 0, butthe value of the good to the seller is θ2 = 0. The buyer’s private information is his value of thegood. Though the seller’s value is commonly known, the seller has as private information twofirst-order beliefs, φA

2 and φB2 , that assign probabilities 0 < pB

h < pAh to the buyer having the

high value. The information setting is the same as our Example 1 with P A(θ1h, θ2) = pAh and

P B(θ1h, θ2) = pBh .10 Their information structure (6), which we denote by Φ6, is the same as Φλ

in our Example 1 with

λ = pBh (1 − β)

pAh β + pB

h (1 − β). (17)

10 These two probabilities are denoted by α0 and αN with αN < α0 in [14].


Feinberg and Skrzypacz also consider an alternative information structure (information struc-ture (1) in their paper), denoted here by Φ1, in which the seller has an informed type in the sensethat he is sure that the buyer has the high value. It is the same as Φλ with pA

1 = 1 and

λ = pBh (1 − β)

β + pBh (1 − β)

. (18)

They show that, imposing the intuitive criterion, the Coase conjecture is valid with informationstructure Φ6, but fails with Φ1 in that delay in reaching agreement occurs with positive probabil-ity.

We say that an order-2 prior Φ has certainty belief (i.e. informed type) for player i if thereis a pair (θ,φi) ∈ supp margΘ×�(Θ−i )

Φ such that φi = δθ−i. It can be shown that the subset of

consistent order-2 priors free of certainty beliefs for a player is contained in a proper face of thefamily of all consistent order-2 priors. Thus, using the terms in [19], having a certainty belief isa generic property in geometric sense. Feinberg and Skrzypacz [14] also ask whether delay isgeneric when the order of beliefs is not uniformly bounded.11 The following lemma provides ananswer by showing that the Coase conjecture fails generically in the geometric sense of Heifetzand Neeman [19] with respect to consistent order-2 priors. Furthermore, the failure of the Coaseconjecture is a continuous property with respect to TV-2 distance, although it is discontinuous inconsistent order-2 priors with respect to the weak topology.

Lemma 4. The failure of the Coase conjecture is generic with respect to consistent order-2priors. Furthermore, prior Φ6 does not converge to Φ1 in TV-2 distance as pA

h −→ 1.

Note that by (17) and (18), Φ6 converges to Φ1 in distribution as pAh −→ 1. Thus, the Coase

conjecture is discontinuous in consistent order-2 priors with respect to the weak topology, in thesense that the conjecture is not always valid with the limit information structure. Consideringthat within the class of consistent order-2 priors Φ1 is generic while Φ6 is not, this discontinuityindicates that weak topology is too weak. In contrast, Lemma 4 implies that the failure of theCoase conjecture is a continuous property with respect to the TV-2 distance.

Remark 2. The information structures in this section can be alternatively modeled by order-1priors on an enlarged set Θ̄ = Θ × {A,B} of the payoff environment. However, this would glossover higher-order beliefs that are intrinsic to the primitive fundamental uncertainties in the model.In general, any type space T = (Ti,πi)i∈N can be treated as an order-1 type space based on thenew payoff environment Θ̄i = Ti for all i. This procedure, however, obscures the higher-orderbelief structure based on the primitive fundamental uncertainties.

4.2. Robustness analysis

A BNE of a game specifies for each type profile a type-specific distribution over the actionprofiles. Kajii and Morris [21] consider the equilibrium action distribution corresponding to a

11 To quote Feinberg and Skrzypacz [14, p. 89], “Another possible extension is the consideration of yet higher-orderuncertainties or even infinite order uncertainties. It is unclear which properties of more general information structuresinduce delay. Furthermore, in our analysis the information structure leading to delay requires the existence of a seller typethat knows the valuation of the buyer. It is natural to ask whether in a general type space delay is a generic phenomena(e.g., in the sense of Morris [29]).”


BNE as the average of the type-specific action distributions across all type profiles [21, Defi-nition 2.3, p. 1289]. The authors define a Nash equilibrium for a complete information gameto be robust against incomplete information perturbations, if for every convergent sequence ofelaborations of the game, there is a sequence of BNEs of the elaborations whose equilibrium ac-tion distributions converge to the Nash equilibrium action distribution [21, Definitions 2.4, 2.5,p. 1289]. It turns out that if a sequence of priors converges in TV-2 distance, then it qualifies as aconvergent sequence of elaborations in the sense of Kajii and Morris, while convergence in theirsense implies convergence in TV-1 distance. Below we generalize Kajii and Morris’s [21] ideaof equilibrium robustness to general Bayesian games based on the TV-k distance.

Let Γ = {(Ai, ui, Ti,πi)i∈N,Θ} be a given Bayesian game. Assume that the type spaceT = (Ti,πi)i∈N has a common prior and Ai is finite for all i. By (15) and (16), each order-(k−1)belief hierarchy tk−1

i ∈ T̃ k−1i of player i in the order-k projection type space T̃ k of T induces

a conditional distribution over types ti ∈ Ti , all having tk−1i as their common order-(k − 1) be-

lief hierarchy. Using these conditional distributions, we can derive an “order-k projection” of astrategy profile σ into T k−1

i , denoted by σ̃ k where

σ̃ ki

(tk−1i

) =∑

ti∈ρk(tk−1i )

σi(ti)margTiP (ti)∑

ti∈ρk(tk−1i )

margTiP (ti)

(19)

for all tk−1i ∈ T k−1

i such that ρk(tk−1i ) �= ∅ and all i, where P is the common prior for T . If T is

of order k and has no redundant types, then it follows from (10), (19), and Lemma 1 that

σ̃ k(τh(t)

) = σ(t), t ∈ T , h� k.

Consequently, if T is of order k, then σ is a BNE with type space T if and only if σ̃ h is a BNEwith type space T̃ h for all h � k.12

Definition 6. Let T = (Ti,πi)i∈N be a type space with common prior P and C a collection ofcommon prior type spaces such that T ∈ C. Let Γ = {(Ai, ui)

ni=1,T } be a Bayesian game. We

say that a BNE σ ∗ for Γ is C-strong-k robust if for any sequence of type spaces T m ∈ C withcommon priors P m converging to P in TV-k distance, there is a sequence of BNEs, σ ∗m forΓ m = {{Ai,ui}ni=1,T m}, such that

limm−→∞

[ ∑tk−1∈supp Φ̃k

∥∥σ̃ ∗m,k(tk−1) − σ̃ ∗k

(tk−1)∥∥margT k−1Φ̃

m,k(tk−1)

+∑

tk−1∈supp Φ̃m,k\supp Φ̃k

∥∥σ̃ ∗m,k(tk−1)∥∥margT k−1Φ̃

m,k(tk−1)] = 0,

where ‖ · ‖ denotes the Euclidean norm on �A and Φ̃m,k the order-k projection of P m.

In the preceding definition, the closeness between two BNEs, one with type space T andone with a perturbed type space T m, is measured by the average of the Euclidean distancesbetween the type-specific order-k projections of these two equilibrium strategies. The averageis taken with respect to projections of the perturbed priors. By Lemmas 2 and 3, the support of

12 Notice that ρk(τh(t)) is a singleton for all h � k if T is of order k and has no redundant types.


each perturbed prior contains that of the limit prior under the TV-k distance convergence, butthe difference becomes smaller as the perturbation reduces. Thus, weighting the type-specificdistance by the perturbed priors is appropriate. In comparison, Kajii and Morris [21] measurethe closeness by the distance in the max norm between the equilibrium action distributions of theequilibrium strategies. For non-degenerate incomplete information games, however, two differentstrategy profiles may specify the same action distribution. Thus, the limit of a sequence undertheir convergence may be consistent with a class of strategy profiles, many of which are notBNEs. Of course, in the degenerate case of complete information, such limit does lead to aunique strategy profile as there is only one type for each player.

Definition 7. We say that type space T = (Ti,πi)i∈N is belief-closed in another type space T ′ =(T ′

i , π′i )i∈N , if Ti ⊆ T ′

i and πi(ti) = π ′i (ti ) when considering π(ti) as an element in �(Θ ×T ′−i ),

for all ti ∈ Ti and for all i.

The belief-closedness of type space T in T ′ means that the players’ beliefs are preserved astheir type sets are enlarged. An implication of the belief-closedness is that, fixing the players’action sets and the payoff environment, any BNE with type space T remains a “component” of aBNE as the players’ type sets are enlarged. This allows us to extend a BNE from a smaller typespace to a BNE with a larger type space, as shown in the following lemma.13

Lemma 5. Let T = (Ti,πi)i∈N be a non-redundant common prior type space, Γ ={(Ai, ui)

ni=1,T } a Bayesian game, and let σ ∗ be a BNE for Γ . Suppose T ′ = (T ′

i , π′i )i∈N is an-

other type space in which T is belief-closed. Then, there is a BNE σ ′ ∗ for Γ ′ = {(Ai, ui)ni=1,T ′}

such that

σ ′ ∗(t) = σ ∗(t), ∀t ∈ T .

We now apply Lemma 5 to establish the robustness of BNEs.

Theorem 2. Let T = (Ti,πi)i∈N be a common prior order-k non-redundant type space and letΓ = {(Ai, ui)

ni=1,T } be a Bayesian game. Then, all BNEs for Γ are C-strong-(k + 1) robust for

any collection C of order-k common prior type spaces such that T ∈ C.

Recall that a type space is of order k if k is an upper bound on the orders of strategicallyrelevant beliefs. Notice also that convergence in TV-h distance implies convergence in TV-h′distance for all h′ � h. As such, Theorem 2 shows that provided there exists a finite uniformbound k on the orders of the target space and its allowable perturbations, BNEs based on thetarget space are robust with respect to TV-(k + 1) distance. Alternatively, the order imposed on

13 Our Lemma 5 is inspired by the Basic Lemma in [22]. Friedenberg and Meier [15] consider games in which playersknow more than the modeler, in the sense that the type space the modeler uses contains the type space used by the playersas a proper subspace. One of the issues they analyze is whether each BNE of the game with the smaller type space cancorrespond to a BNE of the game with the larger type space, a property they refer to as the equilibrium extension property.A result in their paper shows that if (i) there is an injective bimeasurable type morphism from the players’ type spaceto that of the modeler, (ii) the modeler’s type space has a common prior, and (iii) the game with the modeler’s typespace has a BNE, then the equilibrium extension property holds (see Corollary 8.1 in [15]). With T ⊂ T ′, the identitymapping on T is both injective and bimeasurable, our Lemma 5 can be derived from their result. We provide a proof forself-completeness.


perturbations of the target space can be interpreted as resulting from bounded rational perturba-tions. For example, let T be a type space of order k. Then, as a corollary of Theorem 2, BNEsbased on T are robust against order-h projections of perturbations of the target space with respectto TV-(h + 1) distance for all h � k.

Theorem 2 implies that a genuine selection among BNEs of a game with the robustness ap-proach is only possible, if the required strong convergence is of orders no greater than the orderof the type space of the game or the projection order of the type space on which the BNEs arebased. A interesting special case is that of complete information.

Corollary 3. Every Nash equilibrium of a complete information game is C-strong-3 robust forany collection C of order-2 common prior type spaces.

Corollary 3 parallels in spirit to the robustness result for strict Nash equilibria of completeinformation games in [27]. In that paper, the authors introduce a notion of approximate commonknowledge which is sufficient to make all strict Nash equilibria of a complete information gamerobust, in the sense that both their strategies and payoffs are close to those that would be playedand received if players had common knowledge about the game.

4.3. Global games and order-2 type spaces

Carlsson and van Damme [5] introduce global games as those obtained via incomplete in-formation perturbations to a complete information game. For the stag hunt game, they showthat as the perturbation diminishes, iterated elimination of dominated strategies forces the play-ers to conform to Harsanyi and Selten’s risk dominance criterion, which echoes the result byRubinstein [34].14 Morris and Shin [31] show that the selection in favor of risk-dominance istechnically a result of the particular form of the perturbations. In a more general form of perturba-tions where there is a public signal about the mean of the payoff parameter, the Pareto-dominantNE can be uniquely selected if the public signal is sufficiently more accurate than the privateones. Weinstein and Yildiz [36] show that in fact any rationalizable strategy can be selected byvarying the form of perturbations.

Let θ denote the payoff parameter for the stag-hunt game, which is unknown to the players andis drawn according to some prior. Carlsson and van Damme [5] consider a form of incompleteinformation perturbations of the stag-hunt game, under which each player i receives a privatesignal

xi = θ + εηi, (20)

where η1 and η2 are i.i.d. random variables with mean 0 and are independent of θ .15 From (20)it follows that as ε becomes smaller, xi converges to θ , meaning that information about θ iscomplete in the limit. Given xi , the mean of player i’s first-order belief about θ is xi . As a result,the first-order beliefs of player i are completely separable by their means.

14 As discussed in [29], by working out the higher-order beliefs that the information structure of a global game generates,it can be seen that the analytical logic is closely related to what occurs in the email game of Rubinstein [34]. See [30] fora survey of global games and applications.15 In the stag hunt game of Carlsson and van Damme [5], each player chooses between attack and no-attack. The payofffor each player is θ < 1 when both choose attack and is 0 when both choose no-attack. The attacker gets θ −1 and the no-attacker 0 at other choice combinations. In this game, there are two pure-strategy NEs with both choosing attack or bothchoosing no-attack. The risk-dominant NE is the one with no-attack by both players and the other NE is Pareto-dominant.


Morris and Shin [31] assume that players receive a public signal about the mean of θ as wellas private signals about θ itself in (20). They assume that θ , η1, and η2 are independently andnormally distributed with θ ∼ N(μθ ,σ

2θ ) and ηi ∼ N(0,1), for i = 1,2. With the presence of the

public signal, the mean of θ is set equal to the realized value of the public signal while the vari-ance is revised to incorporate the noise of the public signal.16 The normality and independencetogether with (20) implies

xi ∼ N(μθ ,σ

2θ + ε2), i = 1,2. (21)

Furthermore, (θ, x1, x2) jointly have a multivariate normal distribution:

(θ, x1, x2) ∼ N3(μ,Σ), μ =(

μθ

μθ

μθ

), Σ =

⎛⎝σ 2

θ σ 2θ σ 2

θ

σ 2θ ε2 + σ 2

θ σ 2θ

σ 2θ σ 2

θ ε2 + σ 2θ

⎞⎠ . (22)

By (21) and (22), given xi , the conditional distribution of (θ, xj ) is also normal. That is,

(θ, xj )|xi ∼ N

⎛⎝

⎛⎝ ε2μθ+σ 2

θ xi

ε2+σ 2θ

ε2μθ+σ 2θ xi

ε2+σ 2θ

⎞⎠ ,

⎛⎝ ε2σ 2

θ

ε2+σ 2θ

,ε2σ 2

θ

ε2+σ 2θ

ε2σ 2θ

ε2+σ 2θ

, ε2 + ε2σ 2θ

ε2+σ 2θ

⎞⎠

⎞⎠ .

Thus, since singleton sets in �3 are Borel measurable, N3(μ,Σ) can be used as the priordistribution of (θ, x1, x2) that generates the preceding conditional distribution as player i’s pos-terior belief about (θ, xj ). Note that the posterior distribution of θ given xi is also normal (see[10, p. 269]):

θ |xi ∼ N

(ε2μθ + σ 2

θ xi

ε2 + σ 2θ

,ε2σ 2

θ

ε2 + σ 2θ

).

It follows that as with the case without any public signal, player i’s first-order beliefs are com-pletely separable by their means.

Lemma 6. The type spaces of global games are of order 2.

The standard iterative construction of finite-order belief hierarchies in (5) and the notion ofcoherency (6) are equally applicable for cases where Θ = (−∞,∞) endowed with Euclideandistance. Notice that Θ is Polish. By considering beliefs about Borel sets of payoff relevantparameters and belief hierarchies instead of just singleton sets, we can extend Definition 3, map-pings τ k and ρk in (8)–(10), and our construction of the canonical type space of an order-kprior Φ ∈ �(Θ × T k−1) to allow for Θ = (−∞,∞). Finally, the definition of a type morphismin [20] allows Θ to be a continuum. Consequently, the proofs of Lemma A.1 and Theorem 1 canbe adapted to provide a proof of Lemma 6. For this reason, we omit the proof.

It is surprising that the order-2 formulation of global games generates the same impact forequilibrium selection as the seemingly complex structure with unbounded orders of strategicallyrelevant beliefs as in [34]. Intuitively, since complete information type spaces are of order 2, itseems that perturbations of the same order might offer a sufficient range of variation. This alsoleads to the speculation that Weinstein and Yildiz’s [36] results on continuity of rationalizabilitymight still hold by restricting weakly convergent perturbations to type spaces of the same order.

16 Under the specification in [36, p. 367], θ is normally distributed with mean y, the public signal, and variance ε/√

2π .


This line of thinking echoes what we discussed earlier about robustness. In fact, Lemmas 2 and 3also apply to general measurable spaces. Similarly, Lemma 5 applies to the type spaces of globalgames. Thus, by replacing summation with integration in (19), the proof of Theorem 2 can beadapted to show that both the Pareto- and the risk-dominant NEs of the stag hunt game are C-strong-3 robust for the collection C of the (perturbation) type spaces of global games.

4.4. Correlation devices and strategic relevance of redundancy

Ely and Peski [12] point out that redundancy may have strong strategic implications. Inparticular, they show that strategic relevant primitives ought to include information about corre-lations between beliefs across players not captured in the formulation of the universal type space.Liu [24,25] investigates the relationship between redundant type spaces and their non-reducibleforms. His analysis shows that, with respect to BNE, the added redundancy can be interpreted ascorrelation devices to generate correlated equilibrium with incomplete information.

Our finite-order consistent prior approach is by construction redundancy-free. However, themodel can be extended to deal with the issue of “strategic redundancy” in finite-order type spaces.For example, in the order-2 case, we can enlarge the set of fundamental uncertainties Θ by addinga space S = ∏

Si of profiles of private signals si ∈ Si . Nature jointly draws according to commonprior Φ ∈ �(Θ ×S ×∏n

i=1 �(Θ ×S−i )) an element θ ∈ Θ , a profile s ∈ S = ∏ni=1 Si of private

signals, and a profile ξ = (ξ1, ξ2, . . . , ξn) of first-order beliefs in∏n

i=1 �(Θ ×S−i ). Next, natureinforms player i of both signal si and belief ξi . Player i can then derive a posterior belief byupdating prior Φ using his private information (si , ξi) via the Bayes rule. Accordingly, Φ isconsistent if for all i and (si , ξi) ∈ supp margSi×�(S−i )

Φ ,

ξi(θ, s−i ) = margΘ×S−iΦ(θ, s−i |si , ξi), ∀(θ, s−i ) ∈ Θ × S−i . (23)

Given a prior Φ ∈ �(Θ × S × ∏ni=1 �(Θ × S−i )), we can derive an order-2 prior

margΘ×∏ni=1 �(Θ−i )

Φ based on Θ only by (canonical projection):

margΘ×∏ni=1 �(Θ−i )

Φ(θ,φ) =∑

(θ,ξ): (θ,si ,ξi )∈supp margSi×�(S−i )Φ,

margΘξi=φi , ∀i

Φ(θ, s, ξ), (24)

for all (θ,φ) ∈ Θ × ∏ni=1 �(Θ). It turns out that such a projection preserves consistency.

Lemma 7. If Φ ∈ �(Θ × S × ∏ni=1 �(Θ × S−i )) is consistent, then margΘ×∏n

i=1 �(Θ−i )Φ is a

consistent prior in �(Θ × ∏ni=1 �(Θ)).

Consequently, the signal-enriched extension of a consistent order-2 prior serves as an implicitdevice to build redundancy in Bayesian games. It can accommodate correlation of equilibriumstrategies not feasible in redundancy-free models. To illustrate, let Θ = {θ ′, θ ′′} where θ ′ and θ ′′are associated with the payoff matrices in Fig. 2.17

Assume that there is a common order-1 prior with which the probability for having θ ′ isp � 1/2. Then, the type space has one type for each player given by the first-order belief thatassigns probability p to θ ′ and 1 − p to θ ′′. It is straightforward to show that the only BNE

17 This is the basic motivating example in [12] and [25].


Player 1

Player 2L R

U 1,1 0,0

D 0,0 1,1

(θ ′)

Player 1

Player 2L R

U 0,0 1,1

D 1,1 0,0

(θ ′′)

Fig. 2. A coordination game with common value.

outcome has each player receive payoff p. Let S1 = {sD, sU } and S2 = {sL, sR} be the sets ofprivate signals for player 1 and player 2. Set S = S1 × S2 and define Ξ ∈ �(Θ × S) by

Ξ(θ, s) ={ p

2 , (θ, s) = (θ ′, sU , sL) or (θ ′, sD, sR),

1−p2 , (θ, s) = (θ ′′, sU , sR) or (θ ′′, sD, sL).

Next, for si ∈ Si and i = 1,2, define ξ si ∈ �(Θ × S−i ) by

ξsii (θ, s−i ) = Ξ(θ, s−i |si).

Finally, define Φ ∈ �(Θ × S × ∏2i=1 �(Θ × S−i )) by

Φ(θ, s, ξ) = Ξ(θ, s)δξs (ξ),

where ξ s = (ξs11 , ξ

s22 ) for s = (s1, s2) ∈ S. By construction, Φ satisfies consistency (23). Further-

more, ξsii (θ ′, S−i ) = p for all si ∈ Si and for all i, implying that the two different types of each

player share the same first-order belief about θ ∈ Θ . By construction, it is also straightforward toshow that all their higher-order beliefs are identical. In fact, the canonical projection yields theprior with a single-element support. However, given Φ , the strategies a1 = (a1(s

U ), a1(sD)) =

(U,D) and a2 = (a2(sL), a2(s

R)) = (L,R) constitute a BNE that yields a payoff of 1 for eachplayer as a result of perfect correlation via private signals s ∈ S.18

5. Conclusion

The complete information type spaces are not in the closure of payoff type spaces underfamiliar topologies. This is because a payoff type of a player may be associated with multiplefirst-order beliefs in the former but with only one first-order belief in the latter. Hence, to extendanalysis beyond the complete information settings and payoff type spaces, one normally resortsto the conceptually rather complex notion of universal type space. In this paper, we showed byconstruction that there is actually a much smaller common denominator for both the completeinformation and the payoff type spaces; namely, the class of incomplete information settingsrepresented by consistent order-2 priors. Indeed, as illustrated in this paper, many refreshingfindings with higher-order uncertainties or incomplete information perturbations of completeinformation games turn out to operate with certain consistent order-2 priors.

Within the top–down structure of the universal type space, finite-order beliefs are not givenany strategic role that is independent of the infinite belief hierarchies they pertain to. In compar-ison, we depart from the payoff type spaces by having nature draw a first-order belief as well

18 Liu [25] also shows that given a redundant type space, a signal-enriched non-redundant type space can be constructedin such a way that a type morphism between the two exits. For a partition model, Liu [24] provides an abstract conditionto generate redundant equivalents to a non-redundant universal type space.


as a payoff type for each player according to an order-2 prior. The consistency requirement call-ing for the compatibility between updated posterior beliefs and private information closes themodel, in that players do not need to resort to beliefs of orders higher than 2 to assess one anoth-er’s strategic behavior. An iterative repetition of the procedure can extend the model to arbitraryfinite orders.

Several new topologies have been proposed in recent advancements of Bayesian game theory.While these topologies are defined over the space of universal types that are suitable for analyzingsolutions such as interim correlated rationalizability, the topology induced by our order-k totalvariation distance is on the space of priors. As we illustrated in several applications, this newtopology is suitable for analyzing issues related to equilibrium continuity and robustness.

Appendix A. Proofs

Proof of Proposition 1. Let Φ ′ and Φ ′′ be consistent order-2 priors and let λ be any weight in(0,1). First, Φ = λΦ ′ + (1 − λ)Φ ′′ ∈ �(Θ × ∏n

i=1 �(Θi)) is an order-2 prior and suppΦ =suppΦ ′ ∪ suppΦ ′′. Next, fix player i and consider (θi, φi) ∈ supp margΘi×�(Θ−i )

Φ . If

(θi, φi) ∈ supp margΘi×�(Θ−i )Φ ′ \ supp margΘi×�(Θ−i )

Φ ′′,

then for all θ−i ∈ Θ−i , margΘ×�(Θ−i )Φ(θi, θ−i , φi) = λmargΘ×�(Θ−i )

Φ ′(θi, θ−i , φi) andmargΘi×�(Θ−i )

Φ(θi, φi) = λmargΘi×�(Θ−i )Φ ′(θi, φi). Therefore, margΘ−i

Φ(θ−i |θi, φi) =margΘ−i

Φ ′(θ−i |θi, φi). The consistency of Φ ′ then implies φi(θ−i ) = margΘ−iΦ(θ−i |θi, φi).

The same reasoning applies for (θi, φi) ∈ supp margΘi×�(Θ−i )Φ ′′ \ supp margΘi×�(Θ−i )

Φ ′.Finally, if (θi, φi) ∈ supp margΘi×�(Θ−i )

Φ ′ ∩ supp margΘi×�(Θ−i )Φ ′′, we have φi(θ−i ) =

margΘ−iΦ ′(θ−i |θi, φi) = margΘ−i

Φ ′′(θ−i |θi, φi). Applying the fact of a/b = c/d implying[λa + (1 − λ)c]/[λb + (1 − λ)d] = a/b, we get φi(θ−i ) = Φ(θ−i |θi, φi). �Proof of Proposition 2. Suppose that Φ ∈ �(Θ × ∏n

i=1 �(Θ−i )) is a complete informationprior. From Definition 2,∑

θ̄∈Θ

Φ(θ̄ , (δθ̄−i

)ni=1

) = 1.

Thus, from (2),

Φ(θ,φ) =∑θ̄∈Θ

Φ(θ̄ , (δθ̄−i

)ni=1

)Φθ̄(θ,φ).

This shows that Φ is in the convex hull of {Φθ̄ }θ̄∈Θ .Conversely, given complete information priors Φ ′,Φ ′′ and given a number α ∈ (0,1), it fol-

lows from Proposition 1 that the convex combination Φ = αΦ ′ + (1 − α)Φ ′′ is consistent. Since

suppΦ = suppΦ ′ ∪ suppΦ ′′,

(θ̄ , φ̄) ∈ suppΦ implies (θ̄ , φ̄) ∈ suppΦ ′ or (θ̄ , φ̄) ∈ suppΦ ′′. In either case, since Φ ′ and Φ ′′ areboth complete information priors, φ̄ = (δθ̄−i

)ni=1. Hence, Φ is also a complete information prior.

This shows that priors in the convex hull of {Φθ̄ }θ̄∈Θ are all complete information priors. �Lemma A.1. Let k � 2 and let Φ ∈ �(Θ × T k−1) be consistent. Then, τ k

i (tΦi ) = tΦi for alltΦ ∈ T Φ and all i.
i i


Proof. By (8), (11), and (12), for all θ ∈ Θ and all tΦi = (φ1i , . . . , φk−1

i ) ∈ T Φi ,

ψ1i

(tΦi

)[θ ] =∑

tΦ−i∈T Φ−i

πi

(tΦi

)[θ, tΦ−i

] =∑

tΦ−i∈T Φ−i

Φ(θ, t

φ−i

∣∣tφi ) = margΘΦ(θ |tΦi

).

Thus, by (7), ψ1i (tΦi ) = φ1

i . Consequently, by the definition of τhi in (8), (9), and (11),

τ 2i

(tΦi

) = (ψ1

i

(tΦi

)) = projT 1itΦi , ∀tΦi ∈ T F

i , ∀i.

Proceeding now with induction, suppose for all i and 2 � h � k − 1,

τhi

(tΦi

) = (φ1

i , φ2i , . . . , φh−1

i

) = projT h−1

itΦi . (A.1)

Then, by (8)–(12), for all (θ, th−1−i ) ∈ Θ × T h−1

−i ,

ψhi

(tΦi

)[θ, th−1

−i

] =∑

tΦ−i∈T Φ−i : tΦ−i∈ρh−i (th−1−i )

πΦi (ti)

[θ, tΦ−i

]

=∑

tΦ−i∈T Φ−i : tΦ−i∈ρh−i (th−1−i )

Φ(θ, tΦ−i

∣∣tΦi ).

By (A.1), τh−i (tΦ−i ) = proj

T h−1−i

tΦ−i for all tΦ−i ∈ T Φ−i . It follows that∑tΦ−i∈T Φ−i : tΦ−i∈ρh−i (t

h−1−i )

Φ(θ, tΦ−i

∣∣tΦi ) = Φ(θ, th−1

−i

∣∣tΦi ).

Putting the preceding equations together with consistency (7) shows

ψhi

(tΦi

)[θ, th−1

−i

] = φhi

(θ, th−1

−i

) ⇒ τh+1i

(tΦi

) = projT hitΦi .

Thus, by induction, τ ki (tΦi ) = proj

T k−1i

tΦi holds. Since projT k−1

itΦi = tΦi , the proof is com-

plete. �Proof of Theorem 1. Consider first the case with k � 2. Suppose that T is of order k. Then,by Definition 4, there exist a consistent order-k prior Φ ∈ �(Θ × T k−1) and a type morphismf = (f0, f1, . . . , fn) from T to T Φ . Hence,

τ ki (Ti) = τ k

i

(T Φ

i

). (A.2)

Since, by construction of T Φi , tΦi �= t ′Φi for all tΦi , t ′Φi ∈ T Φ

i , (A.2) together with Lemma A.1implies either τ k(ti) �= τ k(t ′i ) or τ∞(ti) = τ∞(t ′i ) for all ti , t

′i ∈ Ti and all i. This establishes the

necessity of (13).To prove the sufficiency, let P ∈ �(Θ × T ) be the common prior for T = (Ti,πi)

ni=1. Define

Φ ∈ �(Θ × T k−1) by

Φ(θ, tk−1) = P

(θ,ρk

(tk−1)) (A.3)

for (θ, tk−1) ∈ Θ × T k−1, where ρk is the inverse of τ k as in (10). Then, since P ∈ �(Θ × T ),it follows from (14) and (A.3) that Φ ∈ �(Θ × T k−1). By the construction of τ k

i and ρki , the

consistency of P implies that Φ is consistent. Finally, by (A.3), the mapping f = (f0, f1, . . . , fn)

with f0 the identity mapping on Θ and fi(ti) = τ ki (ti) for all ti ∈ Ti and all i is a type morphism

from T to T Φ . This establishes the sufficiency.


Consider now the case with k = 1. Then, �(Θ × T k−1) reduces to �(Θ). In this case, theonly first-order belief of each player is the prior P itself. In this case, we can take Φ to be P . �Proof of Lemma 2. Suppose not. Then, supp Φ̃k \ supp Φ̃m,k �= ∅ for infinitely many integers m.Without loss of generality, assume it is true for all m. It follows∥∥Φ̃m,k − Φ̃k

∥∥TV � min

(θ,tk−1)∈supp Φ̃kΦ̃k

(θ, tk−1), ∀m.

Since supp Φ̃k is finite in number, the right-hand side of the preceding inequality is well-definedand positive. Consequently, {P m}m does not converge to P in TV-k distance, which is a contra-diction. �Proof of Lemma 3. Since P m converges to P in TV-k distance, for any γ > 0 there exists aninteger mγ such that∣∣Φ̃m,k(B) − Φ̃k(B)

∣∣ < γ

for all Borel sets B ⊆ Θ × T k−1 and for all m > mγ . Thus, the proof is complete by consideringBm = supp Φ̃m,k \ supp Φ̃k in the preceding inequality for each m > mγ . �Proof of Lemma 4. Let Φ ′ and Φ ′′ be two priors in �(Θ × ∏2

i=1 �(Θ−i )). Then, supp(λΦ ′ +(1 − λ)Φ ′′) = suppΦ ′ ∪ suppΦ ′′ for all λ ∈ (0,1). It follows that λΦ ′ + (1 − λ)Φ ′′ is free ofcertainty types if and only if both Φ ′ and Φ ′′ are free of certainty types. This shows that thesubset of consistent order-2 priors free of certainty beliefs for a player is contained in a properface of the family of all consistent order-2 priors. Thus, having a certainty belief is generic inthe geometric sense of [19]. On the other hand, according to [14], delay occurs with a consistentorder-2 prior if and only if the prior has a certainty belief type for the seller. Putting together, thefailure of Coase conjecture is generic with respect to consistent order-2 priors.

By Lemma 3, for Φλ to converge in TV-2 to a common prior with certainty belief, say, Φ1,it is necessary that suppΦ1 ⊂ suppΦλ for λ close to the number in (18). Hence, Φ6 does notconverge to Φ1 in TV-2 distance. �Proof of Lemma 5. For i ∈ N and ti ∈ T ′

i , set

Σi =∏ti∈T ′

i

�(Ai) and Σ̂i =∏

ti∈T ′i \Ti

�(Ai).

For ti ∈ T ′i \ Ti and σ̂−i ∈ Σ̂−i , define B̂Ri (ti )[σ̂−i] to be the set of xi(ti) ∈ �(Ai) such that

Eti

[ui

((xi, f−i

(σ̂−i , σ

∗−i

)), θ

)] = maxai∈Ai

Eti

[ui

((ai, f−i

(σ̂−i , σ

∗−i

)), θ

)],

where f−i (σ̂−i , σ∗−i ) = (fj (σ̂j , σ

∗j ))j �=i ∈ Σ−i is given by

fj

(σ̂j , σ

∗j

)(tj ) =

{σ ∗

j (tj ) if tj ∈ Tj ;σ̂j (tj ) if tj ∈ T ′

j \ Tj .

Now, define B̂R : Σ̂ −→ Σ̂ by

B̂R(σ̂ ) =∏i∈N

∏t ∈T ′\T

B̂Ri (ti )[σ̂−i].
i i i


By construction and by the finiteness of Ti and compactness of Ai , B̂R together with Σ̂ satisfiesall the conditions for Kakutani’s fixed point theorem. Thus, B̂R has a fixed point which we denoteby σ̂ ∗. Finally, define σ ′ ∗ ∈ Σ by

σ ′ ∗i (ti ) =

{σ ∗

i (ti ), ti ∈ Ti,

σ̂ ∗i (ti ), ti ∈ T ′

i \ Ti,

for all i. By construction, σ ′ ∗i (ti) is a best response against σ ′ ∗−i for all t ′i ∈ T ′

i \ Ti . On the otherhand, by the belief-closedness of T and the assumption that σ ∗ is a BNE for Γ , σ ′ ∗(ti) is also abest response against σ ′−i for ti ∈ Ti ; hence, σ ′ ∗ is a BNE for Γ ′. �Proof of Theorem 2. Let σ ∗ be a BNE for Γ and let {T m}m ⊆ C be a sequence of type spacesT m with common priors P m converging to P in TV-(k + 1) distance as m −→ ∞. By Lemma 2,we may assume that for all m,

supp Φ̃k+1 ⊆ supp Φ̃m,k+1, (A.4)

where Φ̃k+1 and Φ̃m,k+1 are the order-(k + 1) projections of P and P m, respectively. Sinceboth T and T m are of order k, it follows from Theorem 1 that Φ̃k+1 and Φ̃m,k+1 are also oforder k (i.e., T Φ̃k+1

and T Φ̃m,k+1are both of order k). Now putting the consistency (7) together

with (A.4), we have19

margΘ×T k−1

−iΦ̃k+1(θ, tk−1

−i

∣∣tki ) = φki

(θ, tk−1

−i

) = margΘ×T k−1

−iΦ̃m,k+1(θ, tk−1

−i

∣∣tki )for all tk ∈ supp Φ̃k+1 and for all i, where φk

i (·) is the k-th order belief of player i as prescribedin tki about θ and the other players’ order-(k − 1) belief hierarchies. Since Φ̃k+1 and Φ̃m,k+1 areof order k, by Corollary 2, (A.4), and the preceding equation,

supp Φ̃k ⊆ supp Φ̃m,k and

Φ̃k(θ, tk−1

−i

∣∣tk−1i

) = Φ̃m,k(θ, tk−1

−i

∣∣tk−1i

), ∀(

θ, tk−1) ∈ supp Φ̃k, ∀i,

where Φ̃k and Φ̃m,k are the order-k projections of P and P m, respectively. This shows that T̃ k

(the order-k projection of T ) is belief-closed in T̃ m,k (the order-k projection of T m).Since T is non-redundant and of order k, ρk

i (tk−1i ) is singleton for all tk−1

i ∈ τ ki (Ti) and

for all i. Hence, by (19), σ̃ ∗k(τ k(t)) = σ ∗(t) for all t ∈ T , which in turn implies that σ̃ ∗k isa BNE for Γ̃ k = {(Ai, ui)i∈N, T̃ k}. Next, by Lemma 5, there exists a BNE σ ∗m,k for Γ̃ m,k ={{Ai,ui}i∈N, T̃ m,k} such that

σ ∗m,k(τ k(t)

) = σ̃ ∗k(τ k(t)

), ∀t ∈ T . (A.5)

Now, set σ ∗m(t) = σ ∗m,k(τ k(t)) for t ∈ T m. Then, since T m is of order-k and σ ∗m,k is a BNEfor Γ̃ m,k , σ ∗m is a BNE for Γ m and the order-k projection, σ̃ ∗m,k , of σ ∗m,k is the same as σ ∗m,k .By (A.5),∑

tk−1∈supp Φ̃k

∥∥σ̃ ∗m,k(tk−1) − σ̃ ∗k

(tk−1)∥∥margT k−1Φ̃

m,k(tk−1)

19 Notice that since Φ̃k+1 and Φ̃m,k+1 are the order-(k + 1) projections of P and Pm , we need to replace k with k + 1in (7).


+∑



m,k(tk−1)

=∑



m,k(tk−1).

By Lemma 3,

limm−→∞

∑tk−1∈supp Φ̃m,k\supp Φ̃k


m,k(tk−1) = 0.

This completes the proof. �Proof of Proposition 3. Let P1 and P2 be the order-1 priors for player 1 and player 2 such thatsuppP1 = suppP2. For each θ̄ ∈ suppPi , let P θ̄ ∈ �(Θ) be an order-1 prior given by

P θ̄ (θ) = P1(θ2|θ̄1)P2(θ1|θ̄2), θ ∈ Θ.

It is clear that P θ̄ is well-defined and θ1 and θ2 are independently distributed under P θ̄ . Thus,for i �= j ,

P θ̄ (θj |θi) = Pi(θj |θ̄i ), for θi ∈ Θi such that (θi, θ̄j ) ∈ suppPj , θj ∈ Θj , i �= j. (A.6)

By construction and the assumption that suppP1 = suppP2,

θ ∈ suppP θ̄ ⇐⇒ (θ1, θ̄2), (θ̄1, θ2) ∈ suppP1. (A.7)

Thus, putting (A.6) and (A.7) together, every θ ∈ suppP θ̄ results in the same first-order beliefsP1(·|θ̄1) for player 1 and P2(·|θ̄2) for player 2. It follows that

Φθ̄ (C,D) = P θ̄(C ∩ suppP θ̄

)δP (·|θ̄ )(D), C × D ⊆ Θ ×

2∏i=1

�(Θ−i ), (A.8)

defines a probability measure on the algebra of Borel sets C × D of the cartesian productΘ × ∏2

i=1 �(Θ−i ). By the Extension Theorem (e.g., Theorem 3.1 in [4]), Φθ̄ can be ex-tended to a unique probability measure on the σ -algebra generated by Borel sets C × D ⊆Θ × ∏2

i=1 �(Θ−i ). We use Φθ̄ itself to denote the extension. By (A.6) and (A.8), Φθ̄ is consis-tent and

suppΦθ̄ = suppPi × {P(·|θ̄ )

}. (A.9)

Now choose αθ̄ > 0 for θ̄ ∈ suppPi such that∑

θ̄∈suppPiαθ̄ = 1 and set

Φ =∑

θ̄∈suppPi

αθ̄Φθ̄ .

Then, Φ is a convex combination of priors Φθ̄ , θ̄ ∈ suppPi . By Proposition 1, Φ is a consistentprior in �(Θ × ∏2

i=1 �(Θ−i )). Furthermore, for all θ̄ ∈ suppPi , since suppP1 = suppP2 and

suppΦθ̄ ⊆ suppΦ , it follows from (A.9) that (θ̄ ,P θ̄ (·|θ̄ )) ∈ suppΦ . This shows that Φ satis-fies (4). �


Proof of Lemma 7. Fix i. For any (θ,φi) ∈ Θ × �(Θ) and for any ξi ∈ �(Θ × S−i ) with(θ, si ,ψi) ∈ supp margΘ×Si×�(Θ×S−i )

Φ , it follows from the consistency (23) that

φi(θ) = margΘξi(θ) =∑

s−i∈S−i

ξi(θ, s−i ) = Φ(θ, si, ξi)

Φ(si, ξi).

Thus, using the fact that ab

= cd

implies a+cb+d

,

φi(θ̃) =

∑ψ : (θ̃ ,si ,ψi )∈supp margSi×�(S−i )

Φ,

margΘψi=φi

Φ(θ̃, si ,ψi)

∑ψ : (θ̃ ,si ,ψi )∈supp margSi×�(S−i )

Φ,

margΘψi=φi

Φ(si,ψi).

The proof is complete by combining the above equation with (24). �Appendix B. Miscellaneous material

B.1. Consistent order-2 priors with common value

Let Θ be the set of payoff relevant parameters not necessarily decomposable into individu-alized payoff types. We continue to assume that Θ is finite. In this case, the information settingmodeled by a common prior Φ ∈ �(Θ × �n(Θ)), with �n(Θ) = ∏n

i=1 �(Θ), is such that na-ture first draws an element θ ∈ Θ and a belief in φi ∈ �(Θ) for each player i according to Φ andthen, nature informs each player i of his first-order belief φi . Unlike the private-value case, nopart of θ is necessarily revealed to him. Thus, prior Φ is consistent if for all (θ,φ) ∈ suppΦ ,

φi(·) = margΘΦ(·|φi), i ∈ N. (B.1)

With (B.1) replacing (1), a parallel extension of our previous analysis and results can benaturally made. In this case, a complete information common prior type space can be written asa consistent order-2 prior Φ such that (θ,φ) ∈ suppΦ if and only if φi = δθ for all i.

B.2. Private value as a special form of common value

A private-value prior can be viewed as a special form of a common-value prior. To see this, letΘ = ∏n

i=1 Θi and let Φ ∈ �(Θ × ∏ni=1 �(Θ−i )) be a private-value prior satisfying (1). Then,

Φ ′(θ,ψ) := Φ(θ, (margΘ−i

ψi)ni=1

) · δ(δθi)ni=1

((margΘi

ψi)ni=1

)for all (θ,ψ) ∈ Θ × �n(Θ) is a common-value consistent order-2 prior with the additional in-formation on payoff types. It is clear that Φ and Φ ′ yield the same information structure.

Proof for Example 2. Suppose the contrary is true. Then, there exist order-1 priors Φ1, . . . ,

Φm ∈ �(Θ × ∏ni=1 �(Θ−i )) and positive numbers a1, . . . , am with m > 1 such that

m∑l=1

al = 1 and Φ(θ,φ) =m∑

l=1

alΦl(θ,φ), ∀(θ,φ) ∈ suppΦ. (B.2)

By (3), for each l = 1,2, . . . ,m, there exists P l ∈ �(Θ) such that

Φl(θ,φ) = P l(θ)δP l(·|θ)(φ), ∀(θ,φ) ∈ suppΦl.


Since ((θ13, θ22), (φ14, φ22)) ∈ suppΦ , there exists a number 1 � l̄ � m such that

P l̄(θ13, θ22) > 0, φ14(·) = P l̄(·|θ13), φ22(·) = P l̄(·|θ22). (B.3)

Notice that φ22 is uniformly distributed over Θ1. Hence, we have from the last equality in (B.3)

P l̄(θ11, θ22) = P l̄(θ12, θ22) = P l̄(θ13, θ22) > 0.

Next, since the payoff type θ11 of player 1 has a single first-order belief φ11 and P l̄(θ11, θ22) > 0as established above, using the representing order-1 priors of Φl , l = 1, . . . ,m, (B.2) implies thatφ11(·) = P l̄(·|θ11). Since φ11 is uniform over Θ2, we have

P l̄(θ11, θ21) = P l̄(θ11, θ22) = P l̄(θ11, θ23) > 0.

Similarly, since P l̄(θ11, θ21) > 0 as implied above and θ21 has a single first-order belief φ21which is uniform, it must be

P l̄(θ11, θ21) = P l̄(θ12, θ21) = P l̄(θ13, θ21) > 0,

which in turn implies P l̄(θ21|θ13) > 0. But, this contradicts the fact that φ14(θ21) =P l̄(θ21|θ13) = 0 as specified in (B.3) and Fig. 1. �Example B.1. To illustrate how to apply Theorem 1 to determine an order for a type space,consider an example with n = 2 and Θ = Θ1 ×Θ2, where Θi = {θi1, θi2} for i = 1,2.20 Let T =(Ti,πi)

2i=1 be a common prior type space based on Θ with Ti = {ti1, ti2, ti3, ti4} and common

prior in Fig. 3.The payoff types associated with player i’s types are specified by mapping θi : Ti −→ Θi

where

θi(ti) ={

θi1 if ti = ti1, ti2;θi2 if ti = ti3, ti4.

By (8) and (9), simple calculations show that21

• Player 1’s first-order beliefs

ψ11 (t11)[θ21] = ψ1

A(t12)[θ21] = 1

2, ψ1

1 (t13)[θ21] = 1, ψ11 (t14)[θ21] = 0.

20 Complete information type spaces can be constructed from this example as those priors that assign non-zero proba-bilities only to the diagonal. This example illustrates that perturbations of complete information type spaces may easilylead to type spaces of orders higher than 2.21 In a private-value type space, θi ∈ Θi becomes part of player i’s private information. That is, each type ti alsospecifies a payoff type θi (ti ) for player i as well as a belief type πi(ti ) (see [2, p. 1776]). Thus, in the description of thebelief hierarchy of player i’s type ti , it is also required that player i believes his payoff type is θi (ti ) with probability 1:

margΘiφh

i (ti ) = δθi (ti ), ∀h.

Thus, the spaces of finite-order belief hierarchies are recursively constructed as: T 0i

= Θi,T1i

= T 0i

×�(T 0−i

), . . . , T k−1i

= T k−2i

× �(T k−2−i

). An element in T k−1i

is then a (k − 1)-belief hierarchy tk−1i

=(θi , φ

1i, . . . , φk−1

i) with θi ∈ T 0

i, φ1

i∈ �(T 0−i

), . . . , φk−1i

∈ �(T k−2−i

). With these in place, concepts and results forcommon-value cases can be established in a parallel fashion for private-value cases.


t21 t22 t23 t24

t1116 0 1

6 0

t12112

112

16 0

t13 0 16 0 0

t14 0 0 0 16

Fig. 3. An example of an order-3 type space.

• Player 2’s first-order beliefs

ψ12 (t21)[θ11]=1, ψ1

2 (t22)[θ11]= 1

3, ψ1

2 (t23)[θ11]=1, ψ12 (t24)[θ11]=0.

• Player 1’s second-order beliefs

φ2(θ11) = 1 ⇒ ψ21 (t11)[θ21, φ2] = 1

2, ψ2

1 (t12)[θ21, φ2] = 1

4.

Thus, τ 22 (t2) �= τ 2

2 (t ′2) if and only if t2 �= t ′2, τ 21 (t11) = τ 2

1 (t12), but τ 31 (t1) �= τ 3

1 (t ′1) if and only ift1 �= t ′1. This shows that (14) is satisfied with k = 3; hence, T is not of order 1 or order 2 but isof order 3.

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