Uniform topologies on types
Yi-ChunChen
Department of Economics, National University of Singapore
AlfredoDiTillio
IGIER and Department of Economics, Università Luigi Bocconi
EduardoFaingold Department of Economics, Yale University
SiyangXiong
Department of Economics, Rice University
We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We introduce a new metric topology on the universal type space, calleduniform-weak topology, under which two types are close if they have similar first-order beliefs, attach similar probabilities to other players having sim-ilar first-order beliefs, and so on, where the degree of simsim-ilarity is uniform over the levels of the belief hierarchy. This topology generalizes the now classic no-tion of proximity to common knowledge based oncommonp-beliefs(Monderer and Samet 1989). We show that convergence in the uniform-weak topology im-plies convergence in theuniform-strategic topology(Dekel et al. 2006). Moreover, when the limit is a finite type, uniform-weak convergence is also a necessary con-dition for convergence in the strategic topology. Finally, we show that the set of finite types is nowhere dense under the uniform strategic topology. Thus, our re-sults shed light on the connection between similarity of beliefs and similarity of behaviors in games.
Keywords. Rationalizability, incomplete information, higher-order beliefs, strategic topology, Electronic Mail game.
JELclassification. C70, C72.
Yi-Chun Chen:[email protected]
Alfredo Di Tillio:[email protected]
Eduardo Faingold:[email protected]
Siyang Xiong:[email protected]
We are very grateful to a co-editor and three anonymous referees for their comments and suggestions, which greatly improved this paper. We also thank Pierpaolo Battigalli, Martin Cripps, Eddie Dekel, Jeffrey C. Ely, Amanda Friedenberg, Drew Fudenberg, Qingmin Liu, George J. Mailath, Stephen Morris, Marcin Peski, Dov Samet, Marciano Siniscalchi, Aaron Sojourner, Tomasz Strzalecki, Satoru Takahashi, Jonathan Weinstein, and Muhamet Yildiz for their insightful comments. Chen and Xiong gratefully acknowledge financial support from the NSF (Grant SES 0820333) and the Northwestern University Economic Theory Center.
Copyright©2010 Yi-Chun Chen, Alfredo Di Tillio, Eduardo Faingold, and Siyang Xiong. Licensed under the
Creative Commons Attribution-NonCommercial License 3.0. Available athttp://econtheory.org. DOI:10.3982/TE462
1. Introduction
The Bayesian analysis of incomplete information games requires the specification of a type space, which is a representation of the players’ uncertainty about fundamentals, their uncertainty about the other players’ uncertainty about fundamentals, and so on, ad infinitum. Thus the strategic outcomes of a Bayesian game may depend on entire infinite hierarchies of beliefs. Critically, in some games this dependence can be very sensitive at the tails of the hierarchies, so that a mispecification of higher-order beliefs, even at arbitrarily high orders, can have a large impact on the predictions of strategic behavior, as shown by the Electronic Mail game ofRubinstein(1989). As a matter of fact, this phenomenon is not special to the E-Mail game. Recently,Weinstein and Yildiz (2007) have shown that in any game satisfying a certain payoff richness condition, if a player has multiple actions that are consistent withinterim correlated rationalizability— the solution concept that embodies common knowledge of rationality1—then any of these actions can be madeuniquelyrationalizable by suitably perturbing the player’s higher-order beliefs at any arbitrarily high order. This phenomenon raises a conceptual issue: if predictions of strategic behavior are not robust to mispecification of higher-order beliefs, then the common practice in applied analysis of modeling uncertainty using small type spaces—often finite—may give rise to spurious predictions.
A natural approach to study this robustness problem is topological. Consider the correspondence that maps each type of player into his set of interim correlated ratio-nalizable (ICR) actions. The fragility of strategic behavior identified byRubinstein(1989) andWeinstein and Yildiz(2007) can be recast as a certain kind of discontinuity of the ICR correspondence in theproduct topologyover hierarchies of beliefs, i.e., the topology of weak convergence ofk-order beliefs, for eachk≥1. While in every game the ICR corre-spondence is upper hemicontinuous in the product topology, lower hemicontinuity can fail even for thestrictICR correspondence—a refinement of ICR that requires the incen-tive constraints to hold with strict inequality.2 Strictness rules out incentives that hinge on a “knife edge,” which can always be destroyed by suitably perturbing the payoffs of the game. Indeed, nonstrict solution concepts are known to fail lower hemicontinu-ity in other contexts, e.g., in complete information games, Nash equilibrium, and, in fact, even best-reply correspondences fail to be lower hemicontinuous with respect to payoff perturbations. By contrast, the strict Nash equilibrium and the strict best-reply correspondencesarelower hemicontinuous. It is, therefore, surprising that this form of continuity breaks down when it comes to perturbations of higher-order beliefs.
There exist, of course, finer topologies under which the ICR correspondence is up-per hemicontinuous and the strict ICR correspondence is lower hemicontinuous in all games. The coarsest such topology is thestrategic topologyintroduced byDekel et al. (2006); it embodies the minimum restrictions on the class of admissible perturbations of higher-order beliefs necessary to render rationalizable behavior continuous. Thus
1SeeDekel et al.(2007, Proposition 2) andBattigalli et al.(2008, Theorem 4).
2Here, the notion of strictness is actually quite strong: the slack in the incentive constraints is required to be bounded away from zero uniformly on a best-reply set. Despite this, the strict ICR correspondence fails to be lower hemicontinuous in the product topology.
the strategic topology gives a tight measure of the robustness of strategic behavior: if the analyst considers any larger set of perturbations, he is bound to make a nonrobust prediction in some game. Given this significance, we believe the strategic topology de-serves closer examination. Indeed,Dekel et al.(2006) only define itimplicitlyin terms of proximity of behavior in games, as opposed toexplicitlyusing some notion of proximity of probability measures. This leaves open the important question as to what proximity in the strategic topology means in terms of the beliefs of the players.
To address this question, we introduce a new metric topology on types, called uniform-weak topology, under which a sequence of types(tn)n≥1 converges to a type t if the k-order belief of tn weakly converges to that oft and the rate of convergence
is uniform over k≥1. More precisely, for eachk≥1, we consider theProhorov met-ric,dk, overk-order beliefs—a standard metric that metrizes the topology of weak
con-vergence of probability measures—and then define the uniform-weak topology as the topology of convergence in the metricdUW≡supk≥1dk. Our first main result, Theo-rem 1, is that convergence in the uniform-weak topology implies convergence in the uniform-strategic topology. The latter, also introduced byDekel et al.(2006), is the coars-est topology on types under which the ICR correspondence is upper hemicontinuous and the strict ICR correspondence is lower hemicontinuous, where the continuity is now required to hold uniformly across all games.3In particular,Theorem 1implies that con-vergence in the uniform-weak topology is a sufficient condition for concon-vergence in the strategic topology.
The uniform-weak topology is interesting in its own right, as it generalizes the classic notion of approximate common knowledge due toMonderer and Samet (1989). Given a payoff-relevant parameterθ, say that a type of a player hascommonp-belief inθif he assigns probability no smaller thanptoθ, assigns probability no smaller thanpto the event thatθobtains and the other players assign probability no smaller thanptoθ, and so forth, ad infinitum. A sequence of types(tn)n≥1hasasymptotic common certainty of
θif for everyp <1,tn has commonp-belief inθfor allnlarge enough. Monderer and
Samet(1989) use this notion of proximity to common knowledge to study the robust-ness of Nash equilibrium to small amounts of incomplete information. Although they focus on an ex ante notion of robustness and consider only common prior perturba-tions, their main result has the following counterpart in our interim, noncommon prior, nonequilibrium framework.
If a sequence of types(tn)n≥1has asymptotic common certainty ofθ, then, for every game, every action that is strictly interim correlated rationalizable whenθis common certainty remains interim correlated rationalizable for typetn, for allnlarge enough.
It turns out that asymptotic common certainty ofθisequivalentto uniform-weak con-vergence to the type that has common certainty ofθ(i.e., common 1-belief ). Thus, our Theorem 1is a generalization ofMonderer and Samet’s (1989) main result to environ-ments where the limit game has incomplete information.
An important corollary of Theorem 1is that the strategic, uniform-strategic, and product topologies generate the same σ-algebra.4 Indeed, a fundamental result of Mertens and Zamir(1985), which is the Bayesian foundation ofHarsanyi’s (1967–1968) model of types, is that the space of hierarchies of beliefs, called theuniversal type space, exhausts all the relevant uncertainty of the players when endowed with the productσ -algebra. It is reassuring to know that this universality property remains valid when the players can reason about any strategic event.5
Our second main result,Theorem 2, is that uniform-weak convergence is also a nec-essary condition for strategic convergence when the limit is a finite type, i.e., a type belonging to a finite type space. Indeed, for any finite typet and for any sequence of (possibly infinite) types(tn)n≥1that fails to converge totuniform-weakly, we construct a game in which an action is strictly interim correlated rationalizable fort, but not in-terim correlated rationalizable for tn, infinitely often along the sequence.6 Thus, the
uniform-weak topology fully characterizes the strategic topology around finite types. Moreover, the assumption that the limit is a finite type cannot be dispensed with. Un-der the uniform-weak topology, the universal type space is notseparable, i.e., it does not contain a countable dense subset; by contrast,Dekel et al.(2006) show that a countable set of finite types is dense under the strategic topology.7This implies the existence of in-finite types to which uniform-weak convergence is not a necessary condition for strate-gic convergence. (We explicitly construct such an example inSection 4.) While this fact imposes a natural limit to our analysis, finite type spaces play a prominent role in both applied and theoretical work, so it is important to know that our sufficient condition for strategic convergence is also necessary in this case.
Finite types are also the focus of our third main result,Theorem 3. We show that, un-der the uniform-strategic topology, the set of finite types isnowhere dense, i.e., its closure has an empty interior. To understand the conceptual implications of this result, recall thatDekel et al.(2006) demonstrate the denseness of finite types under the nonuniform version of the strategic topology.8 Arguably, this result provides a compelling justifica-tion for why it might be without loss of generality to model uncertainty with finite type spaces: Irrespective of how large the “true” type spaceT is, for any given game there is always a finite type spaceTwith the property that the predictions of strategic behavior 4This is because uniform-weak balls are countable intersections of finite-order cylinders and the strate-gic topologies are sandwiched between the uniform-weak and the product topologies, byTheorem 1.
5Morris(2002, Section 4.2) raises the question of whether the Mertens–Zamir construction is still mean-ingful when strategic topologies are assumed.
6This complements the main result ofWeinstein and Yildiz(2007), whofix a game(satisfying a payoff-richness assumption) and a finite typet, and thenconstruct a sequence of typesconverging totin the prod-uct topology such that the behavior oftis bounded away from the behavior of all types in the sequence. By way of contrast, wefix a sequence of typesthat fails to converge to a finite typetin the uniform-weak topology and thenconstruct a gamefor which the behavior oftis bounded away from the behavior of the types in the sequence infinitely often.
7WhileDekel et al.(2006) state only the weaker result that the set ofallfinite types is dense in the strategic topology, their proof actually establishes the stronger result above.
8Mertens and Zamir(1985) prove the denseness of finite types under the product topology.Dekel et al. (2006) argue that this result does not provide a sound justification for restricting attention to finite types, for strategic behavior is not continuous in the product topology.
based onTare arbitrarily close to those based onT. Our nowhere denseness result im-plies that such finite type spaceTcannot be chosen independently of the game. This is particularly relevant for environments such as those of mechanism design, where the game—both payoffs and action sets—is not a priori fixed. More generally, our result implies that the uniform-strategic topology is strictly finer than the strategic topology. Thus, while a priori these two notions of strategic continuity seem equally compelling, assuming one or the other can have a large impact on the ensuing theory.
The exercise in this paper is similar in spirit to that ofMonderer and Samet(1996) andKajii and Morris(1998), who, like us, consider perturbations of incomplete infor-mation games. These papers provide belief-based characterizations of strategic topolo-gies for Bayesian Nash equilibrium in countable partition models à laAumann(1976). However, since both of these papers assume a common prior and adopt an ex ante ap-proach, while we adopt an interim approach without imposing a common prior, it is difficult to establish a precise connection.9Another important difference between their approach and ours is in the distinctpayoff-relevance constraintsadopted: we fix the set of payoff-relevant states, so our games cannot have payoffs that depend directly on play-ers’ higher-order beliefs;Monderer and Samet(1996) andKajii and Morris(1998) have no such payoff-relevance constraint.
The connection between uniform and strategic topologies first appears in Morris (2002), who studies a special class of games, called higher-order expectation (HOE), games, and shows that the topology of uniform convergence of higher-orderiterated expectationsis equivalent to the coarsest topology under which a certain notion of strict ICR correspondence—different from the one we consider—is lower hemicontinuous in every game of the HOE class.10Compared to the uniform-weak topology, the topology of uniform convergence of iterated expectations is neither finer nor coarser, even around finite types. We further elaborate on this relationship inSection 5.
This paper is also related to contemporaneous work byEly and P˛eski(2008). Fol-lowing their terminology, a typetiscriticalif, under the product topology, the strict ICR correspondence is discontinuous attin some game.Ely and P˛eski(2008) provide an in-sightful characterization of critical types in terms of a common belief property: a type is critical if and only if, for somep >0, it has commonp-belief in some closed (in product topology) proper subset of the universal type space.11 Conceptually, this result shows that the usual type spaces that appear in applications consist almost entirely of critical types, as these type spaces typically embody nontrivial common belief assumptions. For instance, all finite types are critical and so are almost all types belonging to a common 9Monderer and Samet(1996) fix the common prior and consider proximity of information partitions, whereasKajii and Morris(1998) vary the common prior on a fixed information structure. For this reason, the precise connection between these papers is already unclear.
10Morris(2002) defines his strategic topology for HOE games using a distance that makes no reference to ICR. But, as we claimed above, it can be shown that his strategic topology coincides with the coarsest topology under which a certain notion of strict ICR correspondence is continuous in every HOE game. The notion of strictness implicit inMorris(2002) analysis, unlike ours, does not require the slack in the incentive constraints to be uniform.
11Moreover, they show that under the product topology theregulartypes, i.e., those types which are not critical, form aresidualsubset of the universal type space—a standard topological notion of a “generic” set.
prior type space. ThusEly and P˛eski’s (2008) result tells us when—based on the common beliefs of the players—there will be some game and some product-convergent sequence along which strategic behavior is discontinuous, whereas we identify a condition for an arbitrary sequence to display continuous strategic behavior in all games.
The rest of the paper is organized as follows. Section 2 introduces the standard model of hierarchies of beliefs and type spaces, and reviews the solution concept of ICR. Section 3reviews the strategic and uniform-strategic topologies ofDekel et al.(2006), introduces the uniform-weak topology, and presents our two main results concerning the relationship between these topologies (Theorems1and2). Section 4examines the nongenericity of finite types under the uniform-strategic and uniform-weak topologies, and presents the nowhere denseness result (Theorem 3).Section 5discusses the relation with some other topologies. Section 6concludes with some open questions for future research.
2. Preliminaries
Throughout the paper, we fix a two-player setIand a finite setof payoff-relevant states with at least two elements.12 Given a playeri∈I, we write−ito designate the other player in I. All topological spaces, when viewed as measurable spaces, are endowed with their Borelσ-algebra. For a topological spaceS, we write(S)to designate the space of probability measures overSequipped with the topology of weak convergence. Unless explicitly noted, all product spaces are endowed with the product topology and subspaces are endowed with the relative topology.
2.1 Hierarchies of beliefs and types
Our formulation of incomplete information followsMertens and Zamir(1985).13Define X0=, andX1=X0×(X0), and, for eachk≥2, define recursively
Xk= (θ μ1 μk)∈X0× k
×
=1(X−1):margX−2μ=μ−1∀=2 kBy virtue of the above coherency condition on marginal distributions, each element of Xk is determined by its first and last coordinates, so we can identify Xk with ×(Xk−1). For eachi∈I and k≥1, we let Tk
i =(Xk−1) designate the space of
k-order beliefsof playeri, so thatTik=(×T−ki−1). The spaceTiofhierarchies of beliefs
of playeriis Ti= (μk)k≥1∈
×
k≥1 (Xk):margXk−2μk=μk−1∀k≥212We restrict attention to two-player games for ease of notation. Our results remain valid with any finite number of players.
Sinceis finite,Tiis a compact metrizable space. Moreover, there is a unique mapping
μi:Ti→(×T−i)that isbelief preserving, i.e., for allti=(ti1 ti2 )∈Tiandk≥1,
μi(ti)[θ×(π−ki)−1(E)] =tik+1[θ×E] for allθ∈and measurableE⊆T−ki
whereπik is the natural projection ofTi ontoTik. Furthermore, the mappingμi is a
homeomorphism, so to save on notation, we identify each hierarchy of beliefti∈Tiwith
its corresponding beliefμi(ti)over×T−i. Similarly, for eachti∈Ti, we writetik∈Tik
instead of the more cumbersomeπik(ti).
Hierarchies of beliefs can be implicitly represented using atype space, i.e., a tuple (Ti φi)i∈I, where eachTi is a Polish space oftypesand eachφi:Ti→(×T−i)is a
measurable function. Indeed, every typeti∈Ti is mapped into a hierarchy of beliefs
νi(ti)=(νki(ti))k≥1in a natural way:νi1(ti)=margφi(ti)and, fork≥2,
νki(ti)[θ×E] =φi(ti)[θ×(ν−k−1i )−1(E)] for allθ∈and measurableE⊆T−ki−1
The type space(Ti μi)i∈Iis called theuniversal type space, since for every type space
(Ti φi)i∈Ithere is a unique belief-preserving mapping fromTiintoTi, namely the
map-pingνi above.14 When the mappings (νi)i∈I are injective, the type space(Ti φi)i∈I is
callednonredundant. In this case, (νi)i∈I are measurable embeddings onto their
im-ages(νi(Ti))i∈I, which are measurable and can be viewed as a nonredundant type space,
since we haveμi(νi(ti))[×ν−i(T−i)] =1for alli∈I andti∈Ti. Conversely, any(Ti)i∈I
such thatTi⊆Tiandμi(ti)[×T−i] =1for alli∈Iandti∈Tican be viewed as a
nonre-dundant type space.
2.2 Bayesian games and interim correlated rationalizability
Agameis a tupleG=(Ai gi)i∈I, whereAiis a finite set ofactionsfor playeriandgi:
Ai×A−i×→ [−M M]is hispayoff function, withM >0an arbitrary bound on
pay-offs that we fix throughout.15 We write Gto denote the set of all games and, for each integerm≥1, we writeGmfor the set of games with|Ai| ≤mfor alli∈I.
The solution concept ofinterim correlated rationalizability(ICR) was introduced in Dekel et al.(2007). Given aγ∈R, a type space(Ti φi)i∈I, and a gameG, for each player
i∈I, integerk≥0, and typeti∈Ti, we letRki(ti G γ)⊆Aidesignate the set ofk-order
γ-rationalizable actionsofti. These sets are defined as:
R0i(ti G γ)=Ai
and recursively for each integerk≥1, Rki(ti G γ)is the set of all actions ai∈Ai for
which there is aconjecture, i.e., a measurable functionσ−i:×T−i→(A−i)such that16
suppσ−i(θ t−i)⊆Rk−−i1(t−i G γ) ∀(θ t−i)∈×T−i (1)
14To say thatν
iis belief-preserving means thatμi(νi(ti))[θ×E] =φi(ti)[θ×(ν−i)−1(E)]for allθ∈and
measurableE⊆T−i.
15We will also denote byg
ithe payoff function in the mixed extension ofG, writinggi(αi α−i θ)with the
obvious meaning for anyαi∈(Ai)andα−i∈(A−i).
16Relaxing condition (1) by requiring it to hold only forφ
i(ti)-almost every(θ t−i)would not alter the
definition of rationalizability. Indeed, any conjecture that has a(k−1)-order rationalizable supportφi(ti)
and for allai∈Ai, ×T−i gi(ai σ−i(θ t−i) θ)−gi(ai σ−i(θ t−i) θ) φi(ti)(dθ×dt−i)≥ −γ (2)
For future reference, a conjectureσ−i:×T−i→(A−i)that satisfies the former
condi-tion will be called a(k−1)-orderγ-rationalizable conjecture. The set ofγ-rationalizable actions of typetiis then defined as
Ri(ti G γ)= k≥1
Rki(ti G γ)
Finally, followingEly and P˛eski(2008), an actionai∈Aiisstrictly interim correlatedγ
-rationalizablefor typetiand we writeai∈
◦
Ri(ti G γ)ifai∈Ri(ti G γ)for someγ< γ.
As shown in Dekel et al.(2007), Ri(ti G γ)is nonempty for every gameG, typeti
andγ≥0.17
Interim correlated rationalizability has a characterization in terms ofbest-reply sets. A pair of measurable functions ςi:Ti→2Ai, i∈I, has theγ-best-reply property if for
eachi∈I andti∈Ti, each actionai∈ςi(ti)is aγ-best reply fortito a conjectureσ−i:
×T−i→(A−i)with
suppσ−i(θ t−i)⊆ς−i(t−i) ∀(θ t−i)∈×T−i
If(ςi)i∈Ihas theγ-best-reply property, thenςi(ti)⊆Ri(ti G γ)for alli∈Iandti∈Ti. As
shown inDekel et al.(2007), the pair(Ri(· G γ))i∈I is themaximalpair of
correspon-dences with the γ-best-reply property. This means there is no other pair(ςi)i∈I with
theγ-best-reply property such thatRi(ti G γ)⊆ςi(ti) for eachi∈I andti∈Ti, with
strict inclusion for somei∈I andti∈Ti. Therefore, an action isγ-rationalizable for a
typetiif and only if it is aγ-best reply to aγ-rationalizable conjecture, i.e., a conjecture
σ−i:×T−i→(A−i)such that
suppσ−i(θ t−i)⊆R−i(t−i G γ) ∀(θ t−i)∈×T−i
Dekel et al.(2007) also show that the set ofγ-rationalizable actions of a type is deter-mined by the induced hierarchy of beliefs. Indeed, for anyk≥1, any two types (possibly belonging to different type spaces) mapping into the samek-order belief must have the same set ofk-orderγ-rationalizable actions. This has two implications. First, for interim correlated rationalizability, it is without loss of generality to identify types with their cor-responding hierarchies. Thus, in what follows we restrict attention to type spaces(Ti)i∈I
withTi⊆Tiandti[×T−i] =1for alli∈Iandti∈Ti.18Accordingly, we take the
univer-sal type spaceTito be the domain of the correspondenceRi(· G γ):Ti⇒Ai. Second,
everywhere. This is possible because the correspondenceRk−−i1is upper hemicontinuous, and hence it ad-mits a measurable selection by the Kuratowski–Ryll–Nardzewski selection theorem (see, e.g.,Aliprantis and Border 1999).
17Note that forγ <−2M, we haveR
i(ti G γ)=∅, and forγ >2Mwe haveRi(ti G γ)=Ai.
18Recall that we identify each typet
to establish whether an action isk-orderγ-rationalizable for a typeti, we can restrict
attention to(k−1)-orderγ-rationalizable conjecturesσ−i, which are measurable with
respect to(k−1)-order beliefs.19
Finally, the following result shows that, similar to rationalizability in complete in-formation games, interim correlated rationalizability has a characterization in terms of iterated dominance, where the notion of dominance now becomes an interim one. Proposition1. Fixγand a gameG=(Ai gi)i∈I. For eachk≥1, playeri∈I, typeti∈Ti,
and actionai∈Ai, we haveai∈Rki(ti G γ)if and only if, for eachαi∈(Ai\ {ai}), there
exists a measurableσ−i:×T−i→(A−i)with
suppσ−i(θ t−i)∈Rk−−i1(t−i G γ) ∀(θ t−i)∈×T−i (3) such that ×T−i gi(ai σ−i(θ t−i) θ)−gi(αi σ−i(θ t−i) θ)ti(dθ×dt−i)≥ −γ
The proof of this proposition, relegated to theAppendix, uses a separation argument analogous to that which establishes the equivalence between strictly dominated and never best-reply strategies in complete information games. Here, too, the usefulness of the result comes from the fact that to check whether an action is rationalizable for a type, we are able to reverse the order of quantifiers and seek a possibly different conjecture for each possible (mixed) deviation.
3. Topologies on types
Thestrategic(or simply S)topologyintroduced inDekel et al.(2006) is the coarsest topol-ogy on the universal type spaceTiunder which the ICR correspondence is upper
hemi-continuous and the strict ICR correspondence is lower hemihemi-continuous in all games. More explicitly, following a formulation due toEly and P˛eski(2008), the S topology is the topology generated by the collection of all sets of the form
{ti∈Ti:ai∈/Ri(ti G γ)} and {ti∈Ti:ai∈
◦
Ri(ti G γ)}
whereG=(Ai gi)i∈I,ai∈Ai, andγ∈R.20
The S topology onTiis metrizable by the distancediS, defined as follows.21For each
gameG=(Ai gi)i∈I, actionai∈Ai, and typeti∈Ti, let
hi(ti|ai G)=inf{γ:ai∈Ri(ti G γ)}
19This means thatσ
−i(θ s−i)=σ−i(θ t−i)for allθand all typess−i t−iwith the same(k−1)-order
be-liefs.
20The strategic topology can be given an equivalent definition that makes no direct reference toγ -rationalizability forγ=0. Indeed, byEly and P˛eski(2008, Lemma 4), a subbasis of the strategic topology is the collection of all sets of the form{ti:ai∈/R(ti G0)}and{ti:ai∈
◦
Ri(ti G0)}.
21Dekel et al.(2006) define the S topology directly using the distancedS
i, rather than using the topological
Then, for eachsiandti∈Ti, diS(si ti)= m≥1 2−m sup G=(Aigi)i∈I∈Gm max ai∈Ai hi(si|ai G)−hi(ti|ai G)
In terms of convergence of sequences,Dekel et al.(2006) show that for everyti∈Tiand
every sequence(tin)n≥1inTi, we havedSi(tin ti)→0if, and only if, for every gameG=
(Ai gi)i∈I, actionai∈Ai, andγ∈R, the following upper hemicontinuity (u.h.c.) and
lower hemicontinuity (l.h.c.) properties hold: For every sequenceγn→γ,
ai∈Ri(tin G γn) ∀n≥1 ⇒ ai∈Ri(ti G γ) (u.h.c.)
and for some sequenceγnγ,
ai∈Ri(ti G γ) ⇒ ai∈Ri(tin G γn) ∀n≥1 (l.h.c.)
Dekel et al. (2006) also introduce the uniform-strategic (US) topology, which strengthens the definition of the strategic topology by requiring the convergence to be uniform over all games. More precisely, the US topology is the topology of convergence under the metricdUSi , which is defined as
diUS(ti si)= sup G=(Aigi)i∈I∈G
max
ai∈Ai
hi(ti|ai G)−hi(si|ai G)
This uniformity renders the US topology particularly relevant for environments where the game—both payoffs and action sets—is not fixed a priori, such as in a mechanism design environment.
We now introduce a metric topology on types, which we call uniform-weak(UW) topology, under which two types of player are close if they have similar first-order be-liefs, attach similar probabilities to other players having similar first-order bebe-liefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. Thus, unlike the S and US topologies, which arebehavior-based, the UW topology is a belief-basedtopology, i.e., a metric topology defined explicitly in terms of proximity of hierarchies of beliefs. The two main results of this section, Theorems1and2below, establish a connection between these behavior- and belief-based topologies.
Before we present the formal definition of the UW topology, recall that for a complete separable metric space(S d), the topology of weak convergence on(S)is metrizable by theProhorovdistanceρ, defined as
ρ(μ μ)=inf{δ >0 :μ(E)≤μ(Eδ)+δfor each measurableE⊆S} ∀μ μ∈(S) whereEδ= {s∈S: infs∈Sd(s s) < δ}. The UWtopology is the metric topology on Ti
generated by the distance
diUW(si ti)=sup k≥1
dki(si ti) ∀si ti∈Ti
whered0is the discrete metric onand recursively fork≥1,dikis the Prohorov distance on(×T−ki−1)induced by the metricmax{d0 dk−−1i }on×T−ki−1.
In the remainder ofSection 3we explore the relationship between the UW topology and the S and US topologies. First, we show that the UW topology is finer than the US topology (Theorem 1). Second, we prove a partial converse, namely that aroundfinite types, i.e., types belonging to a finite type space, the S topology (and hence also the US topology) is finer than the UW topology (Theorem 2).
3.1 UW convergence implies US convergence Theorem1. For each playeri∈Iand for all typessi ti∈Ti,
diUS(si ti)≤4MdiUW(si ti)
Thus the UW topology is finer than the US topology.
This theorem is a direct implication of the following proposition. Proposition2. Fix a gameG,γ≥0andδ >0. For each integerk≥1,
dik(si ti) < δ ⇒ Rki(ti G γ)⊆Rki(si G γ+4Mδ) ∀i∈I∀si ti∈Ti
The main challenge in proving this result is due to the fact that(k−1)-order ratio-nalizable conjecturesσ−i:×T−i→(A−i)need not be continuous under the topology
of weak convergence of(k−1)-order beliefs. This implies that, keeping the conjecture fixed, the incentive constraints of playerifork-orderγ-rationalizability (cf. (2)) may be discontinuous in his type under the topology of weak convergence ofk-order beliefs. Our proof overcomes this issue by endowing close-by types with similar, but not iden-tical, conjectures. Indeed, the characterization of ICR fromProposition 1implies that for a given actionai∈Ai and a given mixed deviationαi∈(Ai), there always exists
a(k−1)-order rationalizable conjecture that is optimaltoγ-rationalize ai againstαi
at orderk.22 Following this observation, in our proof we endow type tiwith an
opti-mal conjecture forγ-rationalizability and endow typesiwith an optimal conjecture for
(γ+4Mδ)-rationalizability. Using these optimal conjectures, we then prove, using an integration-by-parts type argument, that every action that is k-orderγ-rationalizable fortiremainsk-order(γ+4Mδ)-rationalizable forsi.
Proof ofProposition2. Fix a gameG=(Ai gi)i∈I,γ≥0andδ >0. The proof is by
induction onk. Fork=1, letsiandti∈Tibe such thatdi1(si ti) < δ. Fix an arbitrary
ai∈R1i(ti G γ)and let us show thatai∈R1i(si G γ+4Mδ)usingProposition 1. Fix
αi∈(Ai\ {ai})and letσ−i:→(A−i)be a conjecture such that23 θ∈ gi(ai σ−i(θ) θ)−gi(αi σ−i(θ) θ) ti1[θ] ≥ −γ (4)
22To be precise, when we say thatσ
−iis anoptimal conjecture toγ-rationalizeaiagainstα−iat orderk,
we mean thatσ−iis a(k−1)-orderγ-rationalizable conjecture that satisfies the following property: for any
typeti, the expected payoff difference betweenaiandαifor typetiis at least−γundersome(k−1)-order
γ-rationalizable conjecture if and only if this expected payoff difference is at least−γunderσ−i.
23Recall thatt1
(Note that condition (3) is trivial fork=1.) Pick any function a−i:→A−isuch that a−i(θ)∈arg max a−i∈A−i [gi(ai a−i θ)−gi(αi a−i θ)] ∀θ∈ and define h(θ)=gi(aia−i(θ) θ)−gi(αia−i(θ) θ) ∀θ∈ so that h(θ)≥gi(ai σ−i(θ) θ)−gi(αi σ−i(θ) θ) ∀θ∈ (5)
To conclude the proof fork=1, we now show thatθ∈h(θ)si1[θ] ≥ −γ−4Mδ. Indeed, let{θn}Nn=1be an enumeration ofsuch thath(θn)≥h(θn+1)for all1≤n≤N−1. Thus,
it follows fromd1i(si ti) < δand|h(θ)| ≤2Mfor allθthat θ∈ h(θ)(s1i[θ] −t1i[θ])= N−1 n=1 (h(θn)−h(θn+1)) n m=1 (si1[θm] −ti1[θn]) = N−1 n=1 (h(θn)−h(θn+1)) ≥0 s1i[{θm}nm=1] −ti1[{θm}nm=1] ≥−δ ≥ −δ N−1 n=1 h(θn)−h(θn+1) = −δ(h(θ1)−h(θN)) ≥ −4Mδ hence θ∈ h(θ)s1i[θ] = θ∈ h(θ)(si1[θ] −ti1[θ])+ θ∈ h(θ)ti1[θ] ≥ −4Mδ+ θ∈ h(θ)ti1[θ] ≥ −4Mδ+ θ∈ gi(ai σ−i(θ) θ)−gi(αi σ−i(θ) θ) ti1[θ] ≥ −γ−4Mδ
where the penultimate inequality follows from (5) and the last inequality follows from (4). Thus,ai∈R1i(si G γ+4Mδ)byProposition 1, which proves the desired result for
k=1.
Proceeding by induction, we now suppose the result is valid for somek≥1and show that it remains valid fork+1. Letsi ti∈Tibe such thatdki+1(si ti) < δ. Fix an arbitrary
ai∈Rki+1(ti G γ)and let us show thatai∈Rik+1(si G γ+4Mδ). Fixαi∈(Ai\ {ai})
and letσ−i:×T−ki→(A−i)be ak-orderγ-rationalizable conjecture such that24
×T−ki
gi(ai σ−i(θ t−ki) θ)−gi(αi σ−i(θ t−ki) θ)tik+1(dθ×dtk−i)≥ −γ (6)
24Recall thattk
Pick any measurable function a−i:×T−ki→A−isuch that
a−i(θ t−ki)∈ arg max a−i∈Rk−i(t−kiGγ+4Mδ)
(gi(ai a−i θ)−gi(αi a−i θ)) ∀(θ t−ki)∈×T−ki
By construction, a−iis ak-order(γ+4Mδ)-rationalizable conjecture. Thus, by
Proposi-tion 1, to conclude thatai∈Rki+1(si G γ+4Mδ), we need show only that
×T−ki
gi(aia−i(θ t−ki) θ)−gi(αia−i(θ t−ki) θ)ski+1(dθ×dt−ki)≥ −γ−4Mδ (7)
LetA¯1 A¯Lbe an enumeration of the nonempty subsets ofA−iand define
h(θ)= max a−i∈ ¯A
[gi(ai a−i θ)−gi(αi a−i θ)] ∀θ∈∀1≤≤L
Next, define a partition{P1 PL}ofT−kias
P= {t−ki∈T−ki:R−ki(t−ki G γ)= ¯A} ∀1≤≤L
Sinceσ−iis ak-orderγ-rationalizable conjecture, we have
h(θ)≥gi(ai σ−i(θ t−ki) θ)−gi(αi σ−i(θ t−ki) θ) ∀(θ t−ki)∈×P and, therefore, θ∈ L =1 h(θ)tik+1[θ×P] (8) ≥ ×Tk −i gi(ai σ−i(θ t−ki) θ)−gi(αi σ−i(θ t−ki) θ) tik+1(dθ×dt−ki) Likewise, define a partition{Q1 QL}as
Q= {t−ki∈T−ki:Rk−i(t−ki G γ+4Mδ)= ¯A} ∀1≤≤L Thus we have ×Tk −i gi(aia−i(θ t−ki) θ)−gi(αia−i(θ t−ki) θ)sik+1(dθ×dt−ki) = θ∈ L =1 h(θ)sik+1[θ×Q]
which, together with (6) and (8), implies
×Tk −i gi(aia−i(θ t−ki) θ)−gi(αia−i(θ tk−i) θ) ski+1(dθ×dt−ki) ≥ ×T−ki gi(ai σ−i(θ t−ki) θ)−gi(αi σ−i(θ t−ki) θ) tik+1(dθ×dt−ki)
+ θ∈ L =1 h(θ)(ski+1[θ×Q] −tik+1[θ×P]) ≥ −γ+ θ∈ L =1 h(θ)(ski+1[θ×Q] −tik+1[θ×P])
Therefore, to prove (7) and conclude thatai∈Rki+1(si G γ+4Mδ), we need only show
that θ∈ L =1 h(θ)(sik+1[θ×Q] −tik+1[θ×P])≥ −4Mδ
To prove this inequality first note that the induction hypothesis implies
Pδ⊆
n: ¯An⊇ ¯A
Qn ∀1≤≤L (9)
Next, letN= ||Land consider an enumeration{(θn n)}Nn=1of× {1 L}such that
for alln,
hn(θn)≥hn+1(θn+1) and for allm,n,
(θm=θnandA¯m⊇ ¯An) ⇒ m≤n
25 (10)
Thus, for eachn=1 N, sik+1 n m=1 θm×Qm ≥ski+1 n m=1 θm×Pδm (by (9) and (10)) =ski+1 n m=1 θm×Pm δ ≥tik+1 n m=1 θm×Pm −δ (bydki+1(si ti) < δ) and, therefore, θ∈ L =1 h(θ)(ski+1[θ×Q] −tik+1[θ×P]) = N n=1 hn(θn)(s k+1 i [θn×Qn] −t k+1 i [θn×Pn])
25To see why an enumeration of× {1 L}that satisfies these two properties exists, note that it fol-lows directly from the definition ofh(θ)thatA¯⊇ ¯Amimpliesh(θ)≥hm(θ).
= N−1 n=1 (hn(θn)−hn+1(θn+1)) n m=1 (ski+1[θm×Qm] −t k+1 i [θm×Pm]) = N−1 n=1 (hn(θn)−hn+1(θn+1) ≥0 ) ski+1 n m=1 θm×Qm −tik+1 n m=1 θm×Pm ≥−δ ≥ −δ N−1 n=1 (hn(θn)−hn+1(θn+1))= −δ[h1(θ1)−hN(θN)] ≥ −4Mδ as required.
Corollary1. The Borelσ-algebras of the UW, US, S, and product topologies coincide. Proof. Theorem 1implies that the Borelσ-algebra of the US topology is contained in the Borelσ-algebra of the UW topology. Moreover, Lemma 4 inDekel et al.(2006) im-plies that the Borelσ-algebra of the strategic topology contains the productσ-algebra. Hence, it suffices to show that the productσ-algebra contains the UWσ-algebra. In ef-fect, every uniform-weak ball is a countable intersection of cylinders, therefore, every uniform-weak ball is product-measurable, which implies that every UW-measurable set
is product measurable.
An important implication of this corollary is that the Mertens–Zamir universal type space(Ti μi)i∈I remains a universal type space when equipped with any of the
topolo-gies S, US, or UW instead of the product topology, a fact that was not known prior to this paper. Indeed these topologies leave the measurable structure unchanged, so μi:Ti→(×T−i)remains the unique belief-preserving mapping and a Borel
isomor-phism, albeit no longer a homeomorphism.
3.2 S convergence to finite types implies UW convergence
Here we provide a partial converse toTheorem 1. We show that, as far as convergence to finite types is concerned, convergence in the S topology implies convergence in the UW topology (and hence also in the US topology).
Theorem2. Around finite types the S topology is finer than the UW topology, i.e., for each playeri∈I, finite typeti∈Tiandδ >0there existsε >0such that for eachsi∈Ti,
dSi(si ti)≤ε ⇒ diUW(si ti)≤δ
This theorem is a direct implication of Proposition3below, which in turn relies on the following result.
Lemma1. Let(Ti)i∈I be a finite type space. For everyδ >0, there existε >0and a game
G=(Ai gi)i∈IwithAi⊇Tifor alli∈I, such that for everyi∈Iandti∈Ti,
ti∈arg max ai∈Ai θ∈ t−i∈T−i gi(ai t−i θ)ti[θ t−i] (11)
and for everyψ∈(×A−i)such thatψ[D] ≤ti[D] −δfor someD⊆×T−i,
min ai∈Ai θ∈ a−i∈A−i (gi(ti a−i θ)−gi(ai a−i θ))ψ[θ a−i]<−ε (12)
The proof of this lemma, given in theAppendix, uses a “report-your-beliefs” game embedded in a “coordination” game. More precisely, we construct a game where each player ichooses a point in a finite grid Ai⊆(×T−i) that includes all types inTi
(viewed as probability distributions over×T−i). If player−ichooses an action inT−i
the payoff to playeriis given by aproper scoring rule,2627which guarantees that coor-dinating on truthful reporting has the best-reply property, as shown in (11). If, instead, player−ichooses an action inA−i\T−i, then the payoff to playeriis no greater than
the minimum payoff under the scoring rule and strictly less when choosing an action in Ti. Thus, if the gridAi⊆(×T−i)is sufficiently fine, no actionti∈Tican be anε-best
reply to a conjectureψ∈(×A−i)that is far fromti(viewed as a probability
distrib-ution over×A−i), as shown in (12). Indeed, eitherψassigns large probability to−i
choosing an action inA−i\T−i, which makes anyai∈Ai\Tia profitable deviation, or it
assigns enough probability to×T−iso that the conditionalψ¯ =ψ(·|×T−i)is close to
ψand hence far fromti. Thus, in both cases, any grid pointai∈Ai\Tisufficiently close
toψ¯ is a profitable deviation.
Proposition3. Let(Ti)i∈Ibe a finite type space. For eachδ >0, there existε >0and a
gameGsuch that for each integerk≥1, each playeri∈I, and each(ti si)∈Ti×Ti,
dki(si ti) > δ ⇒ Ri(ti G0)Rki(si G ε)
26Aproper scoring ruleon a measurable spaceis a measurable functionf:×()→Rsuch that
f (ω μ)μ(dω)≥f (ω μ)μ(dω)for allμ,μ∈(), with strict inequality wheneverμ=μ. In the proof of the lemma, we use the scoring rulefi:×T−i×(×T−i)→ [−11]such that(θ t−i ψ)→
2ψ[θ t−i] − ψ2i.
27Dekel et al.(2006) use a report-your-beliefs game to prove their Lemma 4, which states that for every
k≥1andδ >0, there existsε >0such that for allti si∈Ti,dik(si ti)≥δimpliesdSi(si ti)≥ε. However,
it can be shown that, ask→ ∞, the number of actions in their game grows without bound andεshrinks to0. Thus, we cannot use a similar construction to prove our result. The game we construct differs from theirs in two respects: First, in our game the players report infinite hierarchies of beliefs, albeit in afinite type space, whereas in their game players report only finitely many orders; second,Dekel et al.(2006) use a pure report-your-beliefs game, while we embed a report-your-beliefs game in a coordination game. The coordination feature ensures that the rationalizable outcomes of our game hinge on infinitely many levels of the hierarchy. This is important because when types fail to be close underdiUW, there is no upper bound on the lowest order at which the failure of proximity occurs.
Proof. Fix a finite type space(Ti)i∈Iandδ >0. Choose0< η < δsuch that for allk≥1,
i∈Iandti ui∈Ti,28
tik=uki ⇒ dik(ti ui) >2η (13)
ByLemma 1, there existε >0and a gameG=(Ai gi)i∈IwithAi⊇Tisuch that (11) and
(12) hold for everyti∈Tiand everyψ∈(×A−i)such thatψ[D] ≤ti[D] −ηfor some
D⊆×T−i. Thus, for each(ti si)∈Ti×Tiand each measurable functionσ−i:×T−i→
(A−i), if for someD⊆×T−i, (θa−i)∈D T−i σ−i(θ s−i)[a−i]si(θ×ds−i) ψ(θa−i) ≤ti[D] −η
then for someai∈Ai,
×T−i
gi(ti σ−i(θ s−i) θ)−gi(ai σ−i(θ s−i) θ)si(dθ×ds−i) <−ε (14)
We now show that for eachi∈I,
ti∈Ri(ti G0) ∀ti∈Ti (15)
dki(si ti)≥η ⇒ ti∈/Rki(si G ε) ∀k≥1∀(ti si)∈Ti×Ti (16)
Fori∈Iandti∈Ticonsider the conjectureσ−i:×T−i→(A−i)withσ−i(θ t−i)[t−i] = 1for all(θ t−i)∈×T−i. Then actiontiis a best reply for typetito conjectureσ−iby
(11), henceti∈Ri(ti G0)by the characterization of ICR in terms of best-reply sets, thus
proving (15).
To prove (16) fork=1, picksi∈Tiwithd1i(si ti)≥η. Then there existsE⊆such
thats1i[E] ≤ti1[E] −ηand, hence, for everyσ−i:×T−i→(A−i), lettingD=E×T−i, (θa−i)∈D T−i σ−i(θ s−i)[a−i]si(θ×ds−i)= θ∈E T−i σ−i(θ s−i)[T−i]si(θ×ds−i) ≤s1i[E] ≤ti1[E] −η=ti[D] −η It follows from (14) thatti∈/R1i(si G ε).
Proceeding by induction, letk≥2and assume that (16) holds fork−1. Fixi∈Iand ti∈Ti, and picksi∈Tiwithdik(ti si)≥η. Then there exists someE⊆×π−k−1i (T−i)with
ski[Eη] ≤tik[E] −η (17)
DefineD= {(θ t−i)∈×T−i:(θ t−k−i1)∈E}, so thatti[D] =tik[E]. Consider an arbitrary
(k−1)-orderε-rationalizable conjectureσ−i:×T−i→(A−i), i.e.,
suppσ−i(θ s−i)⊆Rk−−i1(s−i G ε) ∀(θ s−i)∈×T−i
28Such positive η exists because, given any finite type space (T
i)i∈I, there exists K≥1 such that
By the induction hypothesis and the condition above, d−k−1i (s−i t−i)≥η ⇒ σ−i(θ s−i)[t−i] =0 ∀(θ s−i t−i)∈×T−i×T−i (18) Thus, (θa−i)∈D T−i σ−i(θ s−i)[a−i]si(θ×ds−i) = (θt−k−i1)∈E T−i σ−i(θ s−i)[T−i∩(π−k−i1)−1(t−k−i1)]si(θ×ds−i) ≤ (θt−k−i1)∈E (π−k−i1)−1({tk−1 −i }η) σ−i(θ s−i)[T−i∩(π−k−1i )−1(t−k−1i )]si(θ×ds−i) ≤ (θt−k−i1)∈E sik[θ× {t−k−1i }η] =ski[Eη] ≤tik[E] −η=ti[D] −η
where the first inequality follows from (18), the second equality follows from (13), and the last inequality follows from (17). By (14), this impliesti∈/Rki(si G ε).
Theorems1and2combined yield the following corollary.
Corollary2. The UW, US, and S topologies are all equivalent around finite types. To end this section, we remark that inTheorem 2we cannot dispense with the as-sumption thattiis a finite type. Indeed, in the next section we prove that the US
topol-ogy is strictly finer than the S topoltopol-ogy. Thus, the UW topoltopol-ogy cannot be equivalent to the S topology, for we have shown that the UW topology is finer than the US topology (Theorem 1).
A more direct way to argue that the UW topology is strictly finer than the S topol-ogy is to note that the universal type space is not separable under the UW topoltopol-ogy (a result that is interesting in its own right), whereas Dekel et al. (2006) show that a countable set of finite types is dense under the strategic topology. To see why the uniform-weak topology is not separable, fix two states θ0 andθ1 in, and consider the nonredundant type space(Xi)i∈I, whereXi= {01}N and each typexi=(xin)n∈N
assigns probability 1 to the pair(θxi1 Li(xi)), whereLi:Xi→X−iis the shift operator, i.e.,L((xi1 xi2 ))=(xi2 xi3 )for eachxi=(xin)n∈N. Clearly, the UW distance
between any two different types inXi is 1 and, hence, under the UW metric, Xi is a
discrete subset of the universal type space. SinceXiis uncountable, it follows that the
universal type space is not separable under the UW topology. 4. Nongenericity of finite types
Dekel et al.(2006) show that finite types are dense under the S topology, thus strength-ening an early result ofMertens and Zamir(1985) that finite types are dense under the
product topology. In contrast, inTheorem 3below we show that under the US topology, finite types arenowhere dense, i.e., the closure of finite types has an empty interior.29An implication of this result andTheorem 1is that the US topology isstrictlyfiner than the S topology.30
The proof ofTheorem 3relies on Lemmas2and3below.Lemma 2states that finite types are not dense under the UW topology. To prove this, we consider an instance of the countably infinite common-prior type space fromRubinstein’s (1989) E-Mail game and show that none of its types can be UW-approximated by a sequence of finite types. In Lemma 3we show that any sequence of types that fails to converge to a type in the E-Mail type space under the UW topology must also fail to converge under the US topology. Together, these lemmas imply that finite types are bounded away from the E-mail type space in US distance, which we state asProposition 4below. This implies that the set of finite types is not dense under the US topology. Using this result, the proof ofTheorem 3 shows that every finite type can be US-approximated by a sequence of infinite types, none of which is the US limit of a sequence of finite types, thereby establishing nowhere denseness.
In effect, consider the following instance of the E-Mail type space. Let= {θ0 θ1} and let the type space(U1 U2)be31
U1= {u10 u11 u12 } U2= {u20 u21 u22 } whereu10[θ0 u20] =1,u20[θ0 u10] =2/3,u20[θ1 u11] =1/3,
u1n[θ1 u2n−1] =2/3 u1n[θ1 u2n] =1/3 ∀n≥1 u2n[θ1 u1n] =2/3 u2n[θ1 u1n+1] =1/3 ∀n≥1 We have the following result.
Proposition4. For everyi∈I, finite typeti∈Ti, andn≥0,dUSi (ti uin)≥M/6.
The proposition is a direct consequence of the following two lemmas. Lemma2. For everyi∈I, finite typeti∈Ti, andn≥0,dUWi (ti uin)≥1/3.
Lemma3. For everyi∈I,ti∈Ti, andn≥0,dUSi (ti uin)≥(M/2)diUW(ti uin).
29This is equivalent to saying that the complement of the set of finite types contains an open and dense set under the US topology.
30Dekel et al.(2006) state the result that the US topology is strictly finer than the S topology. However, as reported inChen and Xiong(2008), the proof in that paper contains a mistake.
31This type space is an instance of the E-Mail type space where the more informed player1who received
kmessages attaches probabilityp=2/3(resp.1−p=1/3) to player2having receivedk−1(resp.k) mes-sages, and the less informed player2who receivedkmessages attaches probabilityp(resp.1−p) to player1 having receivedk(resp.k+1) messages. Our choice thatp=2/3is immaterial; our results hold true if we assume any other value forp.
Figure1. The game fromLemma 3forN=1andM=4.
In the proof ofLemma 2, given in theAppendix, we first show that the UW distance between any two distinct types of any player in the E-Mail type space above is at least 2/3.32Second, we show that any finite typet2nwhose UW distance fromu2nis less than
1/3must attach positive probability to (and hence implies the existence, in the same finite type space, of ) a typet1n+1whose UW distance fromu1n+1is less than1/3, which
in turn implies the existence in the same finite type space of some type t2n+1whose UW distance fromu2n+1is less than1/3and so on. These two facts together imply the
contradiction that the typesti1 ti2 are all different but belong to the same finite type space, whence the result follows.
Turning toLemma 3, the proof, also in theAppendix, constructs, for eachδ≥0and N ≥0, a game such that for each0≤n≤N, a certain action ain is rationalizable for
uinbut is notδ-rationalizable for any typetiwithdik(ti uin) >2δ/M, where the orderk
grows with the differenceN−n. To provide intuition, we sketch the argument for the caseN=1. The game corresponding to this case is depicted inFigure 1, with the payoff bound normalized toM=4.
It is clear that in this game, for alli=12andn=01, actionain is rationalizable
foruin.33 However,ainis weakly dominated bysi, and the payoffs frombinandcinare
such that whenever the beliefs of a typetiare sufficiently far from those ofuin, then any
δ-rationalizable conjecture about player−ithatδ-rationalizesainagainstsicannot do
so against bothbinandcinas well. Indeed, we have
dik(ti uin) >2δ/M ⇒ ain∈/Rki(ti δ) ∀1≤k≤2−n (19)
To see this fork=1, first note thata10is weakly dominated bys1, hencea10∈/R11(ti δ)
for any typet1withd11(t1 u10) > δ/2. Indeed,u110[θ0] =1and henced11(t1 u10) > δ/2 impliest11[θ0]<1−δ/2, so the highest possible expected payoff fort1undera10is−2δ, whereass1yields 0. By the same token,a11∈/R11(t1 δ)for anyt1withd11(t1 u11) > δ/2 anda21∈/R12(t2 δ)for anyt2withd21(t2 u21) > δ/2. Consider actiona20now and pick anyt2 such that d21(t2 u20) > δ/2. Sinceu210[θ1] =1/3, we must have either t21[θ1]<
32The typeu1
kof player1who receivedkmessages assigns probability2/3to the other player having
receivedk−1messages, whileu1k+1attaches probability 0 to that event, and similarly for player2.
33The pair(ς1 ς2)withς
1/3−δ/2ort21[θ0]<2/3+δ/2. Pick any conjectureσ1thatδ-rationalizesa20, so that the difference in expected payoff betweens1anda20is at mostδ. This requires the induced distribution over×A1to satisfy
Pr[θ0 a10|t2 σ1] +Pr[θ1 a11|t2 σ1] ≥1−δ/4 hence the difference in expected payoffs betweenb20anda20is
Pr[θ0 a10|t2 σ1] −2Pr[θ1 a11|t2 σ1] ≥ −3Pr[θ1 a11|t2 σ1] +1−δ/4
which is greater thanδ whent21[θ1]<1/3−δ/2. Likewise, the difference in expected payoffs betweenc20anda20is
−Pr[θ0 a10|t2 σ1] +2Pr[θ1 a11|t2 σ1] ≥ −3Pr[θ0 a10|t2 σ1] +2−δ/2
which is greater thanδwhent21[θ0]<2/3+δ/2. Thus, in any case,a20∈/R12(t2 δ)and the proof of (19) fork=1is complete. The proof forn=0andk=2uses the arguments just given for the case k=1and is completely analogous—for instance, those arguments show that ifσ2is a first-orderδ-rationalizable conjecture thatδ-rationalizesa10for a typet1, then we must have1−δ/4≤Pr[θ0 a20|t1 σ2] ≤t12[θ0× {u120}2δ/M]and hence the distance between the second-order beliefs oft1andu10is at mostδ.
We are now ready to prove the main result of this section.
Theorem3. Finite types are nowhere dense under the US and the UW topology.
Proof. It suffices to prove that every finite type can be UW-approximated by a se-quence of infinite types, none of which is the US limit of a sese-quence of finite types.34 Fix a finite type space(T1 T2)and a typet2∈T2. For eachn≥1, letδn=1/(n+1)and
define the infinite typet2nby the requirement that, for everyk≥1and every measurable
E⊆×T1k−1,
t2kn[E] =(1−δn)t2k[E] +δnuk20[E] Note that for alln≥1,k≥1, and measurableE⊆×T1k−1, we have
tk2n[E] =(1−δn)tk2[E] +δnuk20[E] ≤t2kn[Eδn] +δn
hencedUW2 (t2n t2)≤δn−→0.
It remains to prove that none of the types in the sequence(t2n)n≥1is in the US
clo-sure of the set of finite types, i.e., for everyn≥1, there existsεn>0such that the US
distance betweent2nand every finite type inT2is at leastεn. Thus, fixn≥1, pick any
0< εn<min{M/6 M/(3n+1)}, any finite type space(S1 S2), and any types2∈S2, and let us show thatd2US(t2n s2)≥εn. UsingLemma 2, chooseN≥1large enough so that
d12(N+1)(t1 u10)≥1/3 ∀t1∈T1∪S1 (20) 34Indeed, byTheorem 1, the sequence also US-approximates the finite type, hence nowhere denseness in the US topology follows. By the same theorem, none of the types in the sequence will be the UW limit of a sequence of finite types, thus nowhere denseness in the UW topology also follows.
and letGN=(AiN giN)i=12be the game defined in the proof ofLemma 3. Now define another gameGN=(AiN giN)i=12as
A1N=A1N A2N=A2N× {01}
and for alla1∈A1N,a2∈A2N,x∈ {01}andθ∈,
g1N(a1 a2 x θ)= 12g1N(a1 a2 θ) g2N(a1 a2 x θ)= 12g2N(a1 a2 θ)+ ⎧ ⎨ ⎩ M/2 ifx=1anda1=a10 −M/(3n+1) ifx=1anda1=a10 0 otherwise.
Note that since all payoffs inGN are between−MandM, the same is true for all payoffs
inGN. Moreover, we have the following lemma, which is proved in theAppendix. Lemma4. For allk≥0and allε≥0,
Rk1(t1 GN2ε)=R1k(t1 GN ε) ∀t1∈T1 (21) Rk2(t2 GN2ε)=projA2NR
k
2(t2 GN ε) ∀t2∈T2 (22) We now prove that(a21)∈R2(t2n GN0)for somea2∈A2N, but(a21) /∈R2(s2 GN εn)for alla2∈A2N, reaching the desired conclusion thatd2US(t2n s2)≥εn.
To show that (a21)∈R2(t2n GN0) for somea2∈A2N, it suffices to construct a
rationalizable conjectureσ1 in gameGN under which, for alla2∈A2N, actions(a20) and(a21)givet2nthe same expected payoff. Letσ1:×T1→(A1N)be an arbitrary
rationalizable conjecture inGNand defineσ1:×T1→(A1N)as
σ1(θ t1)[a1] =σ1(θ t1)[a1] ∀t1∈T1\U1∀a1∈A1N
σ1(θ u1k)[a1k] =1 ∀k≥0
From the proof ofLemma 3, it follows, using (21) withε=0, thatσ1 is a rationalizable conjecture inGNand also, using (20) and the fact thatεn< M/6, it follows that
a10∈/R1(t1 GN εn) ∀t1∈T1∪S1 (23) Thus,σ1(θ t1)[a10] =0for allθ∈andt1∈T1, hence for alla2∈A2Nwe have
×T1 g2N(σ1(θ t1) a21 θ)−g2N(σ1(θ t1) a20 θ)t2n(dθ×dt1) =2δn 3 M 2 − 1−2δn 3 M 3n+1=0 This proves that(a20)and(a21)give typet2nthe same expected payoff underσ1 for
Turning to the proof that(a21) /∈R2(s2 GN εn) for alla2∈A2N, consider an
ar-bitraryεn-rationalizable conjectureσ1 in gameGN . By (21) and (23), for allθ∈and
s1∈S1, we must haveσ1(θ s1)[a10] =0. Thus, for alla2∈A2N, (θs1)∈×S1 s2[θ s1] g2N(σ1(θ s1) a21 θ)−g2N(σ1(θ s1) a20 θ) = − M 3n+1<−εn which proves that(a21)is notεn-rationalizable fors2in gameGN.
5. Discussion
5.1 Relation with common p-beliefs
As we mentioned in theIntroduction, the uniform-weak topology is related to the notion ofcommonp-belief due toMonderer and Samet(1989). Fix a stateθ∈andp∈ [01]. For each playeri∈I, define
Bi1p(θ)= {ti1∈Ti1:ti1[θ] ≥p} and Bkpi (θ)=tik∈Tik:tik[θ×Bk−−i1p(θ)] ≥p recursively for allk≥2. A typetihascommonp-belief inθ, and we writeti∈Cip(θ), if
tik∈Bkpi (θ)for allk≥1. A sequence of types(tin)n≥1hasasymptotic common certainty ofθif for everyp <1, we havetin∈Cip(θ)fornlarge enough.
Monderer and Samet (1989) use this notion of proximity to common certainty, i.e., common1-belief, to study the robustness of Nash equilibrium to small amounts of incomplete information. Their main result states that for any game and any sequence of common-prior type spaces, a sufficient condition for Nash equilibrium to be robust to incomplete information (relative to the given sequence of type spaces) is that for some sequencepn1, the prior probability of the event that the players have commonpn
-belief on the payoffs from the complete information game converges to 1 asn→ ∞. A re-lated paper,Kajii and Morris(1997), shows that asymptotic common certainty is actually a necessary condition for robustness in all games. Since both results are formulated for Bayesian Nash equilibrium in common-prior type spaces, to facilitate comparison with our results, we report (without proof ) an analogue of their results for interim correlated rationalizability without imposing common priors.
Proposition 5. A sequence of types(tin)n≥1 has asymptotic common certainty ofθif
and only if for every game and everyε >0, every action that is rationalizable for playeri whenθis common certainty remains interim correlatedε-rationalizable for typetinfor
allnlarge enough.
Thus the “only if” part is an interim version of Monderer and Samet(1989, Theo-rem B∗) and the “if” part is an interim version ofKajii and Morris(1997, Proposition 10). As it turns out, the uniform-weak topology can be viewed as an extension of the concept of asymptotic common certainty: these two notions of convergence coincide when the limit type has common certainty of some state. Indeed, lettingtiθdesignate
Proposition6. A sequence(tin)n≥0has asymptotic common certainty ofθif and only if dUWi (tin tiθ)→0asn→ ∞.
Proof. It suffices to show that for eachi∈I,p∈ [01], andk≥1, we haveBkpi (θ)=
{tiθk }1−p. Fork=1, this follows directly fromtiθ1 [θ] =1. Now suppose this holds fork−1 and let us show that it also holds fork. Indeed,
Bikp(θ)=tik∈Tik:tik[θ×Bk−−1i p(θ)] ≥p
=tik∈Tik:tik[θ× {t−k−1iθ}1−p] ≥p= {tiθk }1−p
where the second equality follows from the induction hypothesis and the third equality
follows from the fact thattiθk [θ t−k−iθ1] =1.
Thus, taken together, Theorems1and2extendProposition 5to environments where the limit type has nondegenerate incomplete information.35
5.2 Other uniform metrics
The Prohorov metric, on which the uniform-weak topology is based, is but one of many equivalent distances that metrize the topology of weak convergence of probability mea-sures. For any such distance, one can consider the associated uniform distance over hierarchies of beliefs. Interestingly, these metrics can generate different topologies over infinite hierarchies, even though the induced topologies overk-order beliefs coincide for eachk≥1. Below we provide such an example.
Given a metric space (S d), let BL(S d)designate the vector space of real-valued, bounded, Lipschitz continuous functions overS, endowed with the norm
fBL=max sup x |f (x)|supx=y |f (x)−f (y)| d(x y) ∀f∈BL(S d) Recall that thebounded Lipschitzdistance over(S d)is
β(μ μ)=sup f dμ− f dμ:f∈BL(S d)withfBL≤1 ∀μ μ∈(S d) This distance metrizes the topology of weak convergence and it relates to the Prohorov metricρas36
(2/3)ρ2≤β≤2ρ
Now define a uniform metricβUWi over hierarchies of beliefs as follows. Letβ0denote the discrete metric overand, recursively, fork≥1, letβki denote the bounded Lip-schitz metric on(×T−ki−1)when×T−ki−1is equipped with the metricmax{β0 βk−1
−i }. Then βUWi =sup k≥1 βki 35Note thatt
iθis a finite type.
For eachk≥1, the metricβki is equivalent todik, as they both induce the weak topol-ogy onk-order beliefs. However, as we now show,βUWi isnotequivalent todiUW.37 Sup-pose that= {θ0 θ1}and for eachn≥1, consider the type space(Tin)i∈I, where
Tin= {ui0 ui1 tin} ∀i∈I and beliefs are
ui0[θ0 u−i0] =1 ui1[θ1 u−i1] =1 ∀i∈I and
tin[θ0 u−i0] =1/n tin[θ1 t−in] =1−1/n ∀i∈I
Thusdki(tin ui1)=1/n for allk≥1and, therefore, diUW(tin ui1)→0asn→ ∞. We now show thatβUWi (tin ui1)→0. Letfbe the indicator function of{θ1}, i.e.,f (θm)=m
form∈ {01}. Then define thek-orderiterated expectationoff for eachk≥1and each playeri, denotedfik:Tik→R, as fi1(ti1)= f dti1=ti1[θ1] and fik(tik)= f−ki−1dtik fork≥2 Thus, we have f−k−1i duki1=1 and f−k−1i dtink =(1−1/n)k
Since it can be shown thatfik∈BL(Tik βki)andfikBL≤1, we haveβki(tin ui1)≥1− (1−1/n)kand henceβUWi (tin ui1)≥1for everyn≥1.
This example is also relevant for the comparison between our work and Morris (2002), who shows that the topology of uniform convergence of iterated expectations is equivalent to the strategic topology associated with a restricted class of games, called higher-order expectations(HOE) games. By this result and the example above, uniform-weak convergence is not sufficient for convergence in the strategic topology for HOE games. This might seem puzzling at first, given that uniform-weak convergence has been shown to imply convergence inDekel et al.(2006) strategic topology, which is de-fined by requiring lower hemicontinuity of the strict ICR correspondence inallgames, not just HOE games. To reconcile these facts, we note that the notion of strict ICR cor-respondence implicitly used inMorris(2002) is different from the one we use, in that it does not require the slack in the incentive constraints to hold uniformly in a best-reply set. Thus, for a given game, continuity ofMorris(2002) notion of strict ICR is more demanding than ours.
37The example below actually shows that the two metrics are not equivalent even around complete in-formation types. In particular, asymptotic common certainty does not guarantee convergence underβUWi .
