Uniform topologies on types

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1 Theoretcal Economcs 5 (00), / Unform topologes on types Y-Chun Chen Department of Economcs, Natonal Unversty of Sngapore Alfredo D Tllo IGIER and Department of Economcs, Unverstà Lug Boccon Eduardo Fangold Department of Economcs, Yale Unversty Syang Xong Department of Economcs, Rce Unversty We study the robustness of nterm correlated ratonalzablty to perturbatons of hgher-order belefs. We ntroduce a new metrc topology on the unversal type space,called unform-weak topology, under whch two types are close f they have smlar frst-order belefs, attach smlar probabltes to other players havng smlar frst-order belefs, and so on, where the degree of smlarty s unform over the levels of the belef herarchy. Ths topology generalzes the now classc noton of proxmty to common knowledge based on common p-belefs (Monderer and Samet 989). We show that convergence n the unform-weak topology mples convergence n the unform-strategc topology (Dekel et al. 006). Moreover, when the lmt s a fnte type, unform-weak convergence s also a necessary condton for convergence n the strategc topology. Fnally, we show that the set of fnte types s nowhere dense under the unform strategc topology. Thus, our results shed lght on the connecton between smlarty of belefs and smlarty of behavors n games. Keywords. Ratonalzablty, ncomplete nformaton, hgher-order belefs, strategc topology, Electronc Mal game. JEL classfcaton. C70, C7. Y-Chun Chen: Alfredo D Tllo: Eduardo Fangold: Syang Xong: We are very grateful to a co-edtor and three anonymous referees for ther comments and suggestons, whch greatly mproved ths paper. We also thank Perpaolo Battgall, Martn Crpps, Edde Dekel, Jeffrey C. Ely, Amanda Fredenberg, Drew Fudenberg, Qngmn Lu, George J. Malath, Stephen Morrs, Marcn Pesk, Dov Samet, Marcano Snscalch, Aaron Sojourner, Tomasz Strzaleck, Satoru Takahash, Jonathan Wensten, and Muhamet Yldz for ther nsghtful comments. Chen and Xong gratefully acknowledge fnancal support from the NSF (Grant SES ) and the Northwestern Unversty Economc Theory Center. Copyrght 00 Y-Chun Chen, Alfredo D Tllo, Eduardo Fangold, and Syang Xong. Lcensed under the Creatve Commons Attrbuton-NonCommercal Lcense 3.0. Avalable at DOI: 0.398/TE46

2 446 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00). Introducton The Bayesan analyss of ncomplete nformaton games requres the specfcaton of a type space, whch s a representaton of the players uncertanty about fundamentals, ther uncertanty about the other players uncertanty about fundamentals, and so on, ad nfntum. Thus the strategc outcomes of a Bayesan game may depend on entre nfnte herarches of belefs. Crtcally, n some games ths dependence can be very senstve at the tals of the herarches, so that a mspecfcaton of hgher-order belefs, even at arbtrarly hgh orders, can have a large mpact on the predctons of strategc behavor, as shown by the Electronc Mal game of Rubnsten (989). As a matter of fact, ths phenomenon s not specal to the E-Mal game. Recently, Wensten and Yldz (007) have shown that n any game satsfyng a certan payoff rchness condton, f a player has multple actons that are consstent wth nterm correlated ratonalzablty the soluton concept that embodes common knowledge of ratonalty then any of these actons can be made unquely ratonalzable by sutably perturbng the player s hgher-order belefs at any arbtrarly hgh order. Ths phenomenon rases a conceptual ssue: f predctons of strategc behavor are not robust to mspecfcaton of hgherorder belefs, then the common practce n appled analyss of modelng uncertanty usng small type spaces often fnte may gve rse to spurous predctons. A natural approach to study ths robustness problem s topologcal. Consder the correspondence that maps each type of player nto hs set of nterm correlated ratonalzable (ICR) actons. The fraglty of strategc behavor dentfed by Rubnsten (989) and Wensten and Yldz (007) can be recast as a certan knd of dscontnuty of the ICR correspondence n the product topology over herarches of belefs,.e., the topology of weak convergence of k-order belefs, for each k. Whle n every game the ICR correspondence s upper hemcontnuous n the product topology, lower hemcontnuty can fal even for the strct ICR correspondence a refnement of ICR that requres the ncentve constrants to hold wth strct nequalty. Strctness rules out ncentves that hnge on a knfe edge, whch can always be destroyed by sutably perturbng the payoffs of the game. Indeed, nonstrct soluton concepts are known to fal lower hemcontnuty n other contexts, e.g., n complete nformaton games, Nash equlbrum, and, n fact, even best-reply correspondences fal to be lower hemcontnuous wth respect to payoff perturbatons. By contrast, the strct Nash equlbrum and the strct best-reply correspondences are lower hemcontnuous. It s, therefore, surprsng that ths form of contnuty breaks down when t comes to perturbatons of hgher-order belefs. There exst, of course, fner topologes under whch the ICR correspondence s upper hemcontnuous and the strct ICR correspondence s lower hemcontnuous n all games. The coarsest such topology s the strategc topology ntroduced by Dekel et al. (006); t embodes the mnmum restrctons on the class of admssble perturbatons of hgher-order belefs necessary to render ratonalzable behavor contnuous. Thus See Dekel et al. (007, Proposton ) and Battgall et al. (008, Theorem4). Here, the noton of strctness s actually qute strong: the slack n the ncentve constrants s requred to be bounded away from zero unformly on a best-reply set. Despte ths, the strct ICR correspondence fals to be lower hemcontnuous n the product topology.

3 Theoretcal Economcs 5 (00) Unform topologes on types 447 the strategc topology gves a tght measure of the robustness of strategc behavor: f the analyst consders any larger set of perturbatons, he s bound to make a nonrobust predcton n some game. Gven ths sgnfcance, we beleve the strategc topology deserves closer examnaton. Indeed, Dekel et al. (006) only defne t mplctly n terms of proxmty of behavor n games, as opposed to explctly usng some noton of proxmty of probablty measures. Ths leaves open the mportant queston as to what proxmty n the strategc topology means n terms of the belefs of the players. To address ths queston, we ntroduce a new metrc topology on types, called unform-weak topology, under whch a sequence of types (t n ) n converges to a type t f the k-order belef of t n weakly converges to that of t and the rate of convergence s unform over k. More precsely, for each k, we consder the Prohorov metrc, d k,overk-order belefs a standard metrc that metrzes the topology of weak convergence of probablty measures and then defne the unform-weak topology as the topology of convergence n the metrc d UW sup k d k. Our frst man result, Theorem, s that convergence n the unform-weak topology mples convergence n the unform-strategc topology. The latter, also ntroduced by Dekel et al. (006), s the coarsest topology on types under whch the ICR correspondence s upper hemcontnuous and the strct ICR correspondence s lower hemcontnuous, where the contnuty s now requred to hold unformly across all games. 3 In partcular, Theorem mples that convergence n the unform-weak topology s a suffcent condton for convergence n the strategc topology. The unform-weak topology s nterestng n ts own rght, as t generalzes the classc noton of approxmate common knowledge due to Monderer and Samet (989). Gven a payoff-relevant parameter θ, say that a type of a player has common p-belef n θ f he assgns probablty no smaller than p to θ, assgns probablty no smaller than p to the event that θ obtans and the other players assgn probablty no smaller than p to θ,and so forth, ad nfntum. A sequence of types (t n ) n has asymptotc common certanty of θ f for every p<, t n has common p-belef n θ for all n large enough. Monderer and Samet (989) use ths noton of proxmty to common knowledge to study the robustness of Nash equlbrum to small amounts of ncomplete nformaton. Although they focus on an ex ante noton of robustness and consder only common pror perturbatons, ther man result has the followng counterpart n our nterm, noncommon pror, nonequlbrum framework. If a sequence of types (t n ) n has asymptotc common certanty of θ, then, for every game, every acton that s strctly nterm correlated ratonalzable when θ s common certanty remans nterm correlated ratonalzable for type t n, for all n large enough. It turns out that asymptotc common certanty of θ s equvalent to unform-weak convergence to the type that has common certanty of θ (.e., common -belef). Thus, our Theorem s a generalzaton of Monderer and Samet s (989) man result to envronments where the lmt game has ncomplete nformaton. 3 See Secton 3 for the precse defnton of the modulus of contnuty on whch the unformty s based.

4 448 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) An mportant corollary of Theorem s that the strategc, unform-strategc, and product topologes generate the same σ-algebra. 4 Indeed, a fundamental result of Mertens and Zamr (985), whch s the Bayesan foundaton of Harsany s ( ) model of types, s that the space of herarches of belefs, called the unversal type space, exhausts all the relevant uncertanty of the players when endowed wth the product σ- algebra. It s reassurng to know that ths unversalty property remans vald when the players can reason about any strategc event. 5 Our second man result, Theorem, s that unform-weak convergence s also a necessary condton for strategc convergence when the lmt s a fnte type,.e., a type belongng to a fnte type space. Indeed, for any fnte type t and for any sequence of (possbly nfnte) types (t n ) n that fals to converge to t unform-weakly, we construct a game n whch an acton s strctly nterm correlated ratonalzable for t, but not nterm correlated ratonalzable for t n, nfntely often along the sequence. 6 Thus, the unform-weak topology fully characterzes the strategc topology around fnte types. Moreover, the assumpton that the lmt s a fnte type cannot be dspensed wth. Under the unform-weak topology, the unversal type space s not separable,.e.,t does not contan a countable dense subset; by contrast, Dekel et al. (006) show that a countable set of fnte types s dense under the strategc topology. 7 Ths mples the exstence of nfnte types to whch unform-weak convergence s not a necessary condton for strategc convergence. (We explctly construct such an example n Secton 4.) Whle ths fact mposes a natural lmt to our analyss, fnte type spaces play a promnent role n both appled and theoretcal work, so t s mportant to know that our suffcent condton for strategc convergence s also necessary n ths case. Fnte types are also the focus of our thrd man result, Theorem 3. We show that, under the unform-strategc topology, the set of fnte types s nowhere dense,.e.,ts closure has an empty nteror. To understand the conceptual mplcatons of ths result, recall that Dekel et al. (006) demonstrate the denseness of fnte types under the nonunform verson of the strategc topology. 8 Arguably, ths result provdes a compellng justfcaton for why t mght be wthout loss of generalty to model uncertanty wth fnte type spaces: Irrespectve of how large the true type space T s, for any gven game there s always a fnte type space T wth the property that the predctons of strategc behavor 4 Ths s because unform-weak balls are countable ntersectons of fnte-order cylnders and the strategc topologes are sandwched between the unform-weak and the product topologes, by Theorem. 5 Morrs (00, Secton 4.) rases the queston of whether the Mertens Zamr constructon s stll meanngful when strategc topologes are assumed. 6 Ths complements the man result of Wensten and Yldz (007), who fx a game (satsfyng a payoffrchnessassumpton) andafntetype t, andthen construct a sequence of types convergng to t n the product topology such that the behavor of t s bounded away from the behavor of all types n the sequence. By way of contrast, we fx a sequence of types thatfalstoconvergetoafntetypet n the unform-weak topology and then construct a game for whch the behavor of t s bounded away from the behavor of the types n the sequence nfntely often. 7 Whle Dekel et al. (006) state only the weaker result that the set of all fnte types s dense n the strategc topology, ther proof actually establshes the stronger result above. 8 Mertens and Zamr (985) prove the denseness of fnte types under the product topology. Dekel et al. (006) argue that ths result does not provde a sound justfcaton for restrctng attenton to fnte types, for strategc behavor s not contnuous n the product topology.

5 Theoretcal Economcs 5 (00) Unform topologes on types 449 based on T are arbtrarly close to those based on T. Our nowhere denseness result mples that such fnte type space T cannot be chosen ndependently of the game. Ths s partcularly relevant for envronments such as those of mechansm desgn, where the game both payoffs and acton sets s not a pror fxed. More generally, our result mples that the unform-strategc topology s strctly fner than the strategc topology. Thus, whle a pror these two notons of strategc contnuty seem equally compellng, assumng one or the other can have a large mpact on the ensung theory. The exercse n ths paper s smlar n sprt to that of Monderer and Samet (996) and Kaj and Morrs (998), who, lke us, consder perturbatons of ncomplete nformaton games. These papers provde belef-based characterzatons of strategc topologes for Bayesan Nash equlbrum n countable partton models à la Aumann (976). However, snce both of these papers assume a common pror and adopt an ex ante approach, whle we adopt an nterm approach wthout mposng a common pror, t s dffcult to establsh a precse connecton. 9 Another mportant dfference between ther approachandourssnthedstnctpayoff-relevance constrants adopted: we fx the set of payoff-relevant states, so our games cannot have payoffs that depend drectly on players hgher-order belefs; Monderer and Samet (996) andkaj and Morrs (998) have no such payoff-relevance constrant. The connecton between unform and strategc topologes frst appears n Morrs (00), who studes a specal class of games, called hgher-order expectaton (HOE), games, and shows that the topology of unform convergence of hgher-order terated expectatons s equvalent to the coarsest topology under whch a certan noton of strct ICR correspondence dfferent from the one we consder s lower hemcontnuous n every game of the HOE class. 0 Compared to the unform-weak topology, the topology of unform convergence of terated expectatons s nether fner nor coarser, even around fnte types. We further elaborate on ths relatonshp n Secton 5. Ths paper s also related to contemporaneous work by Ely and Pęsk (008). Followng ther termnology, a type t s crtcal f, under the product topology, the strct ICR correspondence s dscontnuous at t n some game. Ely and Pęsk (008) provde an nsghtful characterzaton of crtcal types n terms of a common belef property: a type s crtcal f and only f, for some p>0, t has common p-belef n some closed (n product topology) proper subset of the unversal type space. Conceptually, ths result shows that the usual type spaces that appear n applcatons consst almost entrely of crtcal types, as these type spaces typcally embody nontrval common belef assumptons. For nstance, all fnte types are crtcal and so are almost all types belongng to a common 9 Monderer and Samet (996) fx the common pror and consder proxmty of nformaton parttons, whereas Kaj and Morrs (998) vary the common pror on a fxed nformaton structure. For ths reason, the precse connecton between these papers s already unclear. 0 Morrs (00) defnes hs strategc topology for HOE games usng a dstance that makes no reference to ICR. But, as we clamed above, t can be shown that hs strategc topology concdes wth the coarsest topology under whch a certan noton of strct ICR correspondence s contnuous n every HOE game. The noton of strctness mplct n Morrs (00) analyss, unlke ours, does not requre the slack n the ncentve constrants to be unform. Moreover, they show that under the product topology the regular types,.e., those types whch are not crtcal, form a resdual subsetoftheunversaltypespace astandardtopologcalnoton ofa generc set.

6 450 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) pror type space. Thus ElyandPęsk s (008) resulttellsus when basedon thecommon belefs of the players there wll be some game and some product-convergent sequence along whch strategc behavor s dscontnuous, whereas we dentfy a condton for an arbtrary sequence to dsplay contnuous strategc behavor n all games. The rest of the paper s organzed as follows. Secton ntroduces the standard model of herarches of belefs and type spaces, and revews the soluton concept of ICR. Secton 3 revews the strategc and unform-strategc topologes of Dekel et al. (006), ntroduces the unform-weak topology, and presents our two man results concernng the relatonshp between these topologes (Theorems and ). Secton 4 examnes the nongenercty of fnte types under the unform-strategc and unform-weak topologes, and presents the nowhere denseness result (Theorem 3). Secton 5 dscusses the relaton wth some other topologes. Secton 6 concludes wth some open questons for future research.. Prelmnares Throughout the paper, we fx a two-player set I and a fnte set of payoff-relevant states wth at least two elements. Gven a player I, wewrte to desgnate the other player n I. All topologcal spaces, when vewed as measurable spaces, are endowed wth ther Borel σ-algebra. For a topologcal space S, wewrte (S) to desgnate the space of probablty measures over S equpped wth the topology of weak convergence. Unless explctly noted, all product spaces are endowed wth the product topology and subspaces are endowed wth the relatve topology.. Herarches of belefs and types Our formulaton of ncomplete nformaton follows Mertens and Zamr (985). 3 Defne X 0 =,andx = X 0 (X 0 ),and,foreachk, defne recursvely { } k X k = (θ μ μ k ) X 0 (X l ) : marg X l μ l = μ l l = k l= By vrtue of the above coherency condton on margnal dstrbutons, each element of X k s determned by ts frst and last coordnates, so we can dentfy X k wth (X k ). For each I and k, welett k = (X k ) desgnate the space of k-order belefs of player,sothatt k = ( T k ). ThespaceT of herarches of belefs of player s { } T = (μ k ) k (X k ) : marg X k μ k = μ k k k We restrct attenton to two-player games for ease of notaton. Our results reman vald wth any fnte number of players. 3 An alternatve, equvalent formulaton s found n Brandenburger and Dekel (993).

7 Theoretcal Economcs 5 (00) Unform topologes on types 45 Snce s fnte, T s a compact metrzable space. Moreover, there s a unque mappng μ : T ( T ) that s belef preservng,.e., for all t = (t t ) T and k, μ (t )θ (π k ) (E)]=t k+ θ E] for all θ and measurable E T k where π k s the natural projecton of T onto T k. Furthermore, the mappng μ s a homeomorphsm, so to save on notaton, we dentfy each herarchy of belef t T wth ts correspondng belef μ (t ) over T. Smlarly, for each t T,wewrtet k T k nstead of the more cumbersome π k(t ). Herarches of belefs can be mplctly represented usng a type space,.e., a tuple (T φ ) I,whereeachT s a Polsh space of types and each φ : T ( T ) s a measurable functon. Indeed, every type t T s mapped nto a herarchy of belefs ν (t ) = (ν k(t )) k n a natural way: ν (t ) = marg φ (t ) and, for k, ν k (t )θ E]=φ (t )θ (ν k ) (E)] for all θ and measurable E T k Thetypespace(T μ ) I s called the unversal type space, snce for every type space (T φ ) I there s a unque belef-preservng mappng from T nto T,namelythemappng ν above. 4 When the mappngs (ν ) I are njectve, the type space (T φ ) I s called nonredundant. In ths case, (ν ) I are measurable embeddngs onto ther mages (ν (T )) I, whch are measurable and can be vewed as a nonredundant type space, snce we have μ (ν (t )) ν (T )]= for all I and t T.Conversely,any(T ) I such that T T and μ (t ) T ]= for all I and t T can be vewed as a nonredundant type space.. Bayesan games and nterm correlated ratonalzablty A game s a tuple G = (A g ) I,whereA s a fnte set of actons for player and g : A A M M] s hs payoff functon, wth M>0 an arbtrary bound on payoffs that we fx throughout. 5 We wrte G to denote the set of all games and, for each nteger m, wewrteg m for the set of games wth A m for all I. The soluton concept of nterm correlated ratonalzablty (ICR) was ntroduced n Dekel et al. (007). Gven a γ R, atypespace(t φ ) I,andagameG, for each player I, nteger k 0, andtypet T,weletR k (t G γ) A desgnate the set of k-order γ-ratonalzable actons of t. These sets are defned as: R 0 (t G γ)= A and recursvely for each nteger k, R k (t G γ) s the set of all actons a A for whch there s a conjecture,.e., a measurable functon σ : T (A ) such that 6 supp σ (θ t ) R k (t G γ) (θ t ) T () 4 To say that ν s belef-preservng means that μ (ν (t ))θ E]=φ (t )θ (ν ) (E)] for all θ and measurable E T. 5 We wll also denote by g the payoff functon n the mxed extenson of G,wrtngg (α α θ)wth the obvous meanng for any α (A ) and α (A ). 6 Relaxng condton () by requrng t to hold only for φ (t )-almost every (θ t ) would not alter the defnton of ratonalzablty. Indeed, any conjecture that has a (k )-order ratonalzable support φ (t )- almost everywhere can be changed nto one that yelds the same expected payoff and satsfes the condton

8 45 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) and for all a A, T g (a σ (θ t ) θ) g (a σ (θ t ) θ) ] φ (t )(dθ dt ) γ () For future reference, a conjecture σ : T (A ) that satsfes the former condton wll be called a (k )-order γ-ratonalzable conjecture. Thesetofγ-ratonalzable actons of type t s then defned as R (t G γ)= k R k (t G γ) Fnally, followng Ely and Pęsk (008), an acton a A s strctly nterm correlated γ- ratonalzable for type t and we wrte a R (t G γ)f a R (t G γ ) for some γ <γ. As shown n Dekel et al. (007), R (t G γ) s nonempty for every game G, typet and γ 0. 7 Interm correlated ratonalzablty has a characterzaton n terms of best-reply sets. A par of measurable functons ς : T A, I, hastheγ-best-reply property f for each I and t T,eachactona ς (t ) s a γ-best reply for t to a conjecture σ : T (A ) wth supp σ (θ t ) ς (t ) (θ t ) T If (ς ) I has the γ-best-reply property, then ς (t ) R (t G γ)for all I and t T.As shown n Dekel et al. (007), the par (R ( G γ)) I s the maxmal par of correspondences wth the γ-best-reply property. Ths means there s no other par (ς ) I wth the γ-best-reply property such that R (t G γ) ς (t ) for each I and t T,wth strct ncluson for some I and t T. Therefore, an acton s γ-ratonalzable for a type t f and only f t s a γ-best reply to a γ-ratonalzable conjecture,.e., a conjecture σ : T (A ) such that supp σ (θ t ) R (t G γ) (θ t ) T Dekel et al. (007)also show that the set of γ-ratonalzable actons of a type s determned by the nduced herarchy of belefs. Indeed, for any k, any two types (possbly belongng to dfferent type spaces) mappng nto the same k-order belef must have the same set of k-order γ-ratonalzable actons. Ths has two mplcatons. Frst, for nterm correlated ratonalzablty, t s wthout loss of generalty to dentfy types wth ther correspondng herarches. Thus, n what follows we restrct attenton to type spaces (T ) I wth T T and t T ]= for all I and t T. 8 Accordngly, we take the unversal type space T to be the doman of the correspondence R ( G γ): T A.Second, everywhere. Ths s possble because the correspondence R k s upper hemcontnuous, and hence t admts a measurable selecton by the Kuratowsk Ryll Nardzewsk selecton theorem (see, e.g., Alprants and Border 999). 7 Note that for γ< M,wehaveR (t G γ)=, and for γ>m we have R (t G γ)= A. 8 Recall that we dentfy each type t T wth hs belef μ (t ) ( T ).

9 Theoretcal Economcs 5 (00) Unform topologes on types 453 to establsh whether an acton s k-order γ-ratonalzable for a type t,wecanrestrct attenton to (k )-order γ-ratonalzable conjectures σ, whch are measurable wth respect to (k )-order belefs. 9 Fnally, the followng result shows that, smlar to ratonalzablty n complete nformaton games, nterm correlated ratonalzablty has a characterzaton n terms of terated domnance, where the noton of domnance now becomes an nterm one. Proposton. Fx γ and a game G = (A g ) I.Foreachk,player I,typet T, and acton a A, we have a R k (t G γ)f and only f, for each α (A \{a }),there exsts a measurable σ : T (A ) wth such that supp σ (θ t ) R k (t G γ) (θ t ) T (3) T g (a σ (θ t ) θ) g (α σ (θ t ) θ) ] t (dθ dt ) γ The proof of ths proposton, relegated to the Appendx, uses a separaton argument analogous to that whch establshes the equvalence between strctly domnated and never best-reply strateges n complete nformaton games. Here, too, the usefulness of the result comes from the fact that to check whether an acton s ratonalzable for a type, we are able to reverse the order of quantfers and seek a possbly dfferent conjecture for each possble (mxed) devaton. 3. Topologes on types The strategc (or smply S) topology ntroduced n Dekel et al. (006) s the coarsest topology on the unversal type space T under whch the ICR correspondence s upper hemcontnuous and the strct ICR correspondence s lower hemcontnuous n all games. More explctly, followng a formulaton due to Elyand Pęsk (008), the S topologys the topology generated by the collecton of all sets of the form {t T : a / R (t G γ)} and {t T : a R (t G γ)} where G = (A g ) I, a A,andγ R. 0 The S topology on T s metrzable by the dstance d S, defned as follows. For each game G = (A g ) I,actona A,andtypet T,let h (t a G)= nf{γ : a R (t G γ)} 9 Ths means that σ (θ s ) = σ (θ t ) for all θ and all types s t wth the same (k )-order belefs. 0 The strategc topology can be gven an equvalent defnton that makes no drect reference to γ- ratonalzablty for γ 0. Indeed, by Ely and Pęsk (008, Lemma 4), a subbass of the strategc topology s the collecton of all sets of the form {t : a / R(t G 0)} and {t : a R (t G 0)}. Dekel et al. (006) defne the S topology drectly usng the dstance d S, rather than usng the topologcal defnton above.

10 454 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) Then, for each s and t T, d S (s t ) = m m sup G=(A g ) I G m max a A h (s a G) h (t a G) In terms of convergence of sequences, Dekel et al. (006) show that for every t T and every sequence (t n ) n n T,wehaved S (t n t ) 0 f, and only f, for every game G = (A g ) I,actona A,andγ R, the followng upper hemcontnuty (u.h.c.) and lower hemcontnuty (l.h.c.) propertes hold: For every sequence γ n γ, and for some sequence γ n γ, a R (t n G γ n ) n a R (t G γ) (u.h.c.) a R (t G γ) a R (t n G γ n ) n (l.h.c.) Dekel et al. (006) also ntroduce the unform-strategc (US) topology, whch strengthens the defnton of the strategc topology by requrng the convergence to be unform over all games. More precsely, the US topology s the topology of convergence under the metrc d US, whch s defned as d US (t s ) = sup G=(A g ) I G max h (t a G) h (s a G) a A Ths unformty renders the US topology partcularly relevant for envronments where the game both payoffs and acton sets s not fxed a pror, such as n a mechansm desgn envronment. We now ntroduce a metrc topology on types, whch we call unform-weak (UW) topology, under whch two types of player are close f they have smlar frst-order belefs, attach smlar probabltes to other players havng smlar frst-order belefs, and so on, where the degree of smlarty s unform over the levels of the belef herarchy. Thus, unlke the S and US topologes, whch are behavor-based, the UW topology s a belef-based topology,.e., a metrc topology defned explctly n terms of proxmty of herarches of belefs. The two man results of ths secton, Theorems and below, establsh a connecton between these behavor- and belef-based topologes. Before we present the formal defnton of the UW topology, recall that for a complete separable metrc space (S d), the topology of weak convergence on (S) s metrzable by the Prohorov dstance ρ, defnedas ρ(μ μ ) = nf{δ>0:μ(e) μ (E δ ) + δ for each measurable E S} μ μ (S) where E δ ={s S :nf s S d(s s )<δ}. generated by the dstance The UW topology s the metrc topology on T d UW (s t ) = sup d k (s t ) k s t T where d 0 s the dscrete metrc on and recursvely for k, d k s the Prohorov dstance on ( T k ) nduced by the metrc max{d 0 d k } on T k.

11 Theoretcal Economcs 5 (00) Unform topologes on types 455 In the remander of Secton 3 we explore the relatonshp between the UW topology and the S and US topologes. Frst, we show that the UW topology s fner than the US topology (Theorem ). Second, we prove a partal converse, namely that around fnte types,.e., types belongng to a fnte type space, the S topology (and hence also the US topology) s fner than the UW topology (Theorem ). 3. UW convergence mples US convergence Theorem. For each player I and for all types s t T, d US (s t ) 4Md UW (s t ) Thus the UW topology s fner than the US topology. Ths theorem s a drect mplcaton of the followng proposton. Proposton. Fx a game G, γ 0 and δ>0. Foreachntegerk, d k (s t )<δ R k (t G γ) R k (s G γ+ 4Mδ) I s t T The man challenge n provng ths result s due to the fact that (k )-order ratonalzable conjectures σ : T (A ) need not be contnuous under the topology of weak convergence of (k )-order belefs. Ths mples that, keepng the conjecture fxed, the ncentve constrants of player for k-order γ-ratonalzablty (cf. ()) may be dscontnuous n hs type under the topology of weak convergence of k-order belefs. Our proof overcomes ths ssue by endowng close-by types wth smlar, but not dentcal, conjectures. Indeed, the characterzaton of ICR from Proposton mples that for a gven acton a A and a gven mxed devaton α (A ),therealwaysexsts a (k )-order ratonalzable conjecture that s optmal to γ-ratonalze a aganst α at order k. Followng ths observaton, n our proof we endow type t wth an optmal conjecture for γ-ratonalzablty and endow type s wth an optmal conjecture for (γ + 4Mδ)-ratonalzablty. Usng these optmal conjectures, we then prove, usng an ntegraton-by-parts type argument, that every acton that s k-order γ-ratonalzable for t remans k-order (γ + 4Mδ)-ratonalzable for s. Proof of Proposton. Fx a game G = (A g ) I, γ 0 and δ>0. The proof s by nducton on k. For k =, lets and t T be such that d (s t )<δ. Fx an arbtrary a R (t G γ) and let us show that a R (s G γ + 4Mδ) usng Proposton. Fx α (A \{a }) and let σ : (A ) beaconjecturesuchthat 3 ( g (a σ (θ) θ) g (α σ (θ) θ) ) t θ] γ (4) θ To be precse, when we say that σ s an optmal conjecture to γ-ratonalze a aganst α at order k, we mean that σ s a (k )-order γ-ratonalzable conjecture that satsfes the followng property: for any type t, the expected payoff dfference between a and α for type t s at least γ under some (k )-order γ-ratonalzable conjecture f and only f ths expected payoff dfference s at least γ under σ. 3 Recall that t desgnates the frst-order belef of type t.

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