Uniform Topology on Types and Strategic Convergence

Transcription

1 Unform Topology on Types and Sraegc Convergence lfredo D Tllo IGIER and IEP Unversà Boccon alfredo.dllo@unboccon. Eduardo Fangold Deparmen of Economcs Yale Unversy eduardo.fangold@yale.edu ugus Prelmnary draf. Commens welcome. bsrac We sudy he connuy of he correspondence of nerm ε-raonalzable acons n ncomplee nformaon games. We nroduce a opology on ypes called unform-wea opology under whch wo ypes of a player are close f hey have smlar frs-order belefs aach smlar probables o oher players havng smlar frs-order belefs and so on where he degree of smlary s unform over he levels of he belef herarchy. Ths noon of proxmy of ypes s an exenson of he concep of common p-belef due o Monderer and Same 989. We show ha gven any fne game every acon ha s nerm raonalzable for a fne ype remans nerm ε-raonalzable for all ypes suffcenly close o n he unform-wea opology. Conversely gven any fne ype here exs ε > 0 and a fne game such ha some nerm raonalzable acon for fals o be nerm ε-raonalzable for every ype ha s no close o n he unform-wea opology. Our resuls hus esablsh he equvalence beween he unform-wea opology and he sraegc opology of Deel Fudenberg and Morrs 2006 around fne ypes. Inroducon Incomplee nformaon games are games n whch some payoff-relevan saes are no common nowledge among he players. Harsany observes ha he Bayesan analyss of ncomplee nformaon games requres a model n whch each player s We han Jeffrey Ely Drew Fudenberg George J. Malah Marcn Pes and Qngmn Lu for helpful commens.

2 equpped wh an nfne herarchy of belefs: a belef abou he payoff-relevan saes a belef abou hs opponens belefs abou he payoff-relevan saes and so on. Followng hs observaon Harsany nroduces ype spaces as a parsmonous model ha encodes he belef herarches and s suable for game heorec analyss n ha nerm bes-reply ses can be appropraely defned. Merens and Zamr 985 provde a foundaon for he use of ype spaces showng ha he space T of coheren belef herarches s a unversal ype space. Tha s T s a ype space self and moreover every ype space can be embedded n T va a belef-preservng morphsm. Hence he unversal ype space T capures he rchness of any absrac ype space and no more. The Merens-Zamr unversal ype space comes wh a naural opology: he produc opology. dsncve feaure of he produc opology s ha s nsensve o he als of belef herarches: wo ypes are close n he produc opology f and only f her h -order belefs are close for some large fne. Sraegc behavor however can be very sensve o hgh order belefs. Ths s rue even for nerm raonalzably see Deel Fudenberg and Morrs 2007 he mos permssve soluon concep conssen wh common nowledge of raonaly. In effec n Rubnsen 989 s elecronc mal game an acon aac s srcly raonalzable for a ype bu fals o be raonalzable for all ypes n a sequence ha converges o n he produc opology. Hence o he exen ha sraegc behavor s wha one ulmaely cares abou he produc opology yelds an nadequae noon of proxmy of ypes. From hs pon of vew he appropraeness of a opology on ypes depends on wha s mean by sraegc behavor. Bu gven a soluon concep s naural o consder he coarses opology under whch he correspondence ha maps ypes no soluons s connuous n every game. For he soluon concep of nerm ε-raonalzably hs yelds he sraegc opology on ypes nroduced by Deel Fudenberg and Morrs 2006 hereafer DFM. The sraegc opology whle beng srong enough o render ε- raonalzable behavor connuous s remarably wea: DFM show ha fne ypes are dense. Gven he mporance of he sraegc opology 2 and he fac ha s a opology on ypes ha s ndependen of he sraegc suaon.e. acon ses and payoffs we fnd concepually mporan o gve a characerzaon n erms of properes of he belef herarches wh no drec reference o such conceps as behavor sraeges I s only when T s equpped wh he measurable srucure nduced by he produc opology ha T can be shown o be a unversal ype space. Ths s he sense n whch he produc opology s naural. 2 One reason why he sudy of sraegc convergence seems mporan s ha appears o be a useful sep for he examnaon of robusness quesons n mechansm desgn. 2

3 and bes-reples whch are ed o fxed games. In hs paper we ae a frs sep owards such characerzaon and show ha around fne ypes he sraegc opology concdes wh he unform-wea opology. The laer s he opology nduced by he merc d defned as follows: for each order le d be a merc ha nduces he opology of wea convergence of -order belefs; merc d s defned as he supremum of d over all orders. The connecon beween unform opologes on ypes and he sraegc opology was frs suggesed by Morrs 2002 who sudes a parcular class of nfne-acon games called hgher-order expecaon games HOE and shows ha a ceran opology on ypes dfferen from ours s equvalen o he weaes opology under whch he ε-raonalzably correspondence s connuous n every game of he HOE class. Ths unform opology s oo srong for our purposes: here exss a sequence of ypes n whch fals o converge n hs unform opology o a ype and ye n every fne game every raonalzable acon for remans ε-raonalzable for n for all n large enough. Hence he sraegc separaon of ypes ha are no close n hs unform opology requres an nfne game. The connecon beween unform and sraegc convergence of ypes also underles he man resul n Monderer and Same 989. They show ha a suffcen condon for he correspondence of Bayesan-Nash ε-equlbrum o be connuous a a compleenformaon ype profle s ha he sequence of approxmang ype profles converges o s complee nformaon lm n he common p-belef sense. Tha s for every p > 0 a every ype profle suffcenly far n he al of he sequence here s common p-belef of he sae ha s common cerany n he lm. Moreover hey show ha hs noon of convergence of ype profles yelds sraegc connuy n every game. Ka and Morrs 997 prove he converse: If a sequence of ype profles fals o converge o a complee nformaon ype n he common p-belef sense hen a fne game exss such ha for some ε > 0 some equlbrum of he complee nformaon game wll fal o be an ε-equlbrum a every ype profle n he al of he sequence. I s neresng o noe ha a sequence of ypes converges o a complee nformaon ype n he unform-wea opology f and only f converges n he common p-belef sense. Hence he opology of unform-wea convergence exends he noon of common p-belef convergence o ncomplee nformaon lm ypes. Ths paper s also closely relaed o conemporaneous wor by Ely and Pes Followng her ermnology a ype s called regular f for every fne game he ε- raonalzably correspondence s connuous n he produc opology. Ely and Pes 2007 provde an nsghful characerzaon of regular ypes n erms of properes of he belef herarches and show ha he se of regular ypes s generc n a opologcal 3

4 sense. They prove: Theorem Ely and Pes ype s regular f and only f for every p > 0 and every closed n he produc opology proper subse W of he unversal ype space W s no common p-belef a. Furhermore he se of regular ypes s resdual ha s conans a counable nersecon of open and dense ses. Thus n a opologcal sense around almos all ypes he sraegc opology concdes wh he produc opology. Whle opologcal genercy s neresng we hn should no be he end of he sory. We fnd concepually mporan o characerze he sraegc opology around crcal ypes namely hose ypes whch are no regular. In fac gven Ely and Pes 2007 s resul appears o us ha every ype space ever consdered n applcaons consss enrely of crcal ypes. We ae a frs sep owards such characerzaon by provng he equvalence beween he sraegc opology and he unform-wea opology around fne ypes. ll fne ypes are crcal bu no conversely. 3 2 Prelmnares Hereafer we fx a wo-player se I and a fne space of basc uncerany Θ. Gven a player I le denoe he oher player n I. Gven a opologcal space X wre X for he se of probably measures on he Borel subses of X endowed wh he opology of wea convergence of probably measures. Unless explcly noed all produc spaces wll be endowed wh he produc opology and subspaces wh he relave opology. 2. The Merens-Zamr Unversal Type Space Le Y 0 = Θ and Y = Y 0 Y 0. Then for 2 defne recursvely { Y = µ... µ Y 0 Y 0 Y } : marg Y l 2µ l = µ l l = 2... By he coherency condons on margnal dsrbuons from he defnon of Y an elemen of Y s unquely denfed by s frs and las coordnaes. Thus wh slgh abuse of noaon gven Θ and µ Y we wll somemes wre µ Y. 3 We conecure our characerzaon s vald for arbrary crcal ypes bu do no have a proof ye. 4

5 The Merens-Zamr unversal ype space T s defned as { µ T = µ 2... Y 0 Y } : marg Y 2µ = µ 2. For each le : T Y denoe he naural proecon. For every I and le T and Y denoe copes of T and Y respecvely wre : T Y for and defne T = T. n elemen T s a ype of player and s s assocaed -order belef. Each ype of unquely deermnes a belef over Θ T. More precsely for each T here exss a unque probably measure µ Θ T whose margnal on concdes wh for all. Conversely for every such probably measure n Θ T here exss a unque ype T such ha he laer belef-preservaon Y propery holds. Moreover he map µ : T Θ T s a homeomorphsm. fne ype space s a collecon T I wh T a fne subse of T for all I such ha he suppor of µ s conaned n Θ T for all T and I. ype T s called a fne ype f T for some fne ype space T I. 2.2 Inerm Correlaed Raonalzably and he Topologes on Types fne game s a uple g I wh each a fne se and g : Θ [ where = X I. For a mxed acon profle α wre gα for he expecaon of g under α. Gven a fne game G = g I and a ype T for each 0 we denoe by R ε ; G he se of -order ε-raonalzable acons of ype. These ses are defned recursvely as follows see Deel Fudenberg and Morrs 2007: R 0 ε ; G = and for acon a belongs o R ε ; G f here exss a measurable funcon σ : Θ T such ha: a supp σ R ε ; G for µ -almos every and b for all a Θ T [ g a σ g a σ µ d d ε 5

6 The se of ε-raonalzable acons of ype s hen defned as R ε ; G = R ε ; G Noe ha he se R ε ; G only depends on va he -order belefs of. Thus wh slgh abuse of noaon and whenever convenen gven any T we wll wre R ε ; G o ndcae he -order ε-raonalzable sraeges of any ype of wh assocaed -order belefs. Defnon 2.. The sraegc opology s he weaes opology on T such ha for every fne game G he correspondence ε R ε ; G s connuous. Deel Fudenberg and Morrs 2006 nroduce a dsance d S he sraegc opology. on T ha merzes Gven a merc space X ρ he Prohorov dsance beween any wo µ µ X s { nf δ > 0 : µ µ δ } δ for every Borel X where δ denoes he se of all x X such ha nf y ρx y < δ. Now le d 0 be he dscree merc on Θ and wre d for he Prohorov dsance on Θ. Then recursvely for every 2 le d be he Prohorov dsance on Y when Y s gven he produc merc nduced by d 0 d... d. Defnon 2.2. The unform-wea opology on T s he opology nduced by he merc d = sup d for all T. Two ypes are close n he unform-wea opology f and only f hey have smlar frs-order belefs aach smlar probables o he oher player havng smlar frsorder belefs and so on where he degree of smlary s unform over he levels of he belef herarchy. Ineresngly f s a complee nformaon ype ha s a ype a whch here s common nowledge of some sae hen for all δ > 0 and T d < δ s common δ-belef a. Hence he unform-wea opology s an exenon of he noon of common p-belef Monderer and Same 989 o perurbaons of ncomplee nformaon envronmens. 6

7 3 Equvalence Beween he Sraegc and he Unform-wea Topologes on Types Proposon 3.. round fne ypes he unform-wea opology s sronger han he sraegc opology. More precsely for every player I fne ype T and ε > 0 here exss δ > 0 such ha for all T d < δ d S < ε. The proposon s a drec mplcaon of he followng: Lemma 3.. Le G = g I be a fne game and T I a fne ype space. For every ε > 0 here exss δ > 0 such ha for every I and T T d < δ R 0 ; G R ε ; G. Proof. For each I and le T T η mn { d T \ { } T } and η mn I mn η. Snce T I s a fne ype space η > 0 for all I and. Moreover here exss 0 such ha η = η 0 for all 0 and hence we have η > 0. Choose any 0 < δ < 2 mn { η ε Θ }. The proof proceeds by nducon n. Fx T and T wh d < δ. For each a R 0 ; G here exss a behavor sraegy b : Θ such ha l ; a a b Θ [ 0 where l ; a a b g a a g a a b [ a. a Then l ; a a b Θ [ l ; a a b Θ [ [ > δ Θ > ε Θ where he second follows from g and he penulmae nequaly follows from d < δ. Hence a R ε ; G whch proves our clam for =. 7

8 Now le 2 and assume he clam holds rue for. Le T and T be a par of ypes wh d < δ. Fx an acon a R 0 ; G and le b : Θ T be a -order behavor sraegy for ype such ha: 4 for all Θ T b R 0 3. for all a Θ T l ; a a b [ where l ; a a b s he expeced payoff loss under b of he devaon from a o a condonal on. Tha s l ; a a b = g a a g a a b a [ a. For each le T { T : R 0 = } so ha { T : T } s a paron of T. For each C T wre B δ C for he δ-open-ball around C n T d wh he convenon ha B δ δ =. Snce δ < η/2 we have B T δ B T for every such ha T and T. Consder he -order behavor sraegy b : Θ T for ype defned as follows: If b B δ T [ where T { } { T }. for some hen for each Θ se b T [ T s he condonal probably of on he even 4 Snce Θ T s a fne se The requremen ha a hold for all Θ T raher han only for Θ T supp s whou loss of generaly. 8

9 If T \ { B δ T : } hen for each Θ se b equal o an arbrary measurable selecon from he se-valued map R ε where he choce of he selecon s mmaeral for he ensung argumen. 5 Snce he nonempy B δ For all Θ and T s are parwse dson b s well defned. T we clam: b R ε. For Θ and T \ { B δ T : } he clam follows from he defnon of b. Fx Θ wh T and B δ T. By consrucon supp b supp b. Snce Hence by he nducon hypohess and herefore B δ T whch proves our clam. T we have d = R supp b 0 R R < δ for some ε ε T. I remans o show ha acon a s ε-opmal under b for ype. Fx a and abbrevae l l ; a a b. Snce d < δ and supp T we have [ Θ B δ T δ > ε/2 and herefore Θ T l [ d d Θ l B δ T [ { } d ε 2. 5 Snce R ε s upper hem-connuous he exsence of a measurable selecon follows from he Kuraows-Ryll-Nardzews Theorem see lprans and Border

10 Bu snce b s consan on each B δ T for every fxed Θ we have Θ T l where l l Θ [ d d l [ { } B δ for any and hence all T B δ ε T. On he oher hand follows from he defnon of b eraed expecaons and 3.2 ha Θ Therefore Θ l l Θ [ { } T l Θ T [ { } B δ = l T [ { } B δ ; a a b T where he penulmae nequaly follows from d Combnng 3.3 and 3.4 yelds l Θ T and herefore a R ε ; G as requred. [ d d [ [ { } T 0. > Θ 2 δ > ε/2 3.4 ε < δ and g. Proposon 3.2. round fne ypes he unform-wea opology s weaer han he sraegc opology. More precsely for every player I fne ype T and δ > 0 here exss ε > 0 such ha for all T d > δ d S > ε. The proposon s a drec mplcaon of he followng lemma whch reles on Lemma. from appendx. 0

11 Lemma 3.2. Le T I be a fne ype space. For every δ > 0 here exs ε > 0 and a fne game G such ha for every I T and T d > δ R 0 ; G R ε ; G. Proof. Fx δ > 0. For each I le µ : T Θ T be he belef mappng. Defne ζ = δ Θ T 2 + T 2. By Lemma. here exs ε > 0 and a game wh fne acon ses T such ha for every T : s a bes-reply o belef µ vewed as a probably over Θ ; for every belef µ Θ s an ε-bes-reply o µ only f µ µ ζ where denoes he maxmum norm. We now clam: Clam. For every I and T T wh d a R 0 ; G ; b R ε ; G. > δ We shall prove he clam by nducon n. Consder = and fx I and T. Le b : Θ and ν Θ be defned as follows: for all Θ and a b [a = [ a and ν [ a = b [ a [. Snce = margθ µ s clear ha ν = µ. Bu hen follows from ha R 0 ; G whch proves par a of he clam for =. To prove par b fx an arbrary T and assume d > δ. The laer means [Θ < [Θ δ for some Θ Θ hence [ < [ δ Θ for some Θ. 3.5 Now fx any b : Θ and defne ν Θ as follows: [ [ ν a = b a [ for all Θ and a.

12 Pc Θ sasfyng 3.5. Then snce marg Θ ν = and marg Θ µ = ν [ a ν [ a snce T a T a < a µ [ a = a T µ [ a δ Θ δ Θ by 3.5 as µ [ {} \ T = 0. Bu hen ν [ a < µ [ a δ Θ T for some a T hence µ ν δ > Θ T > ζ and usng also R ε ; G. Ths concludes he proof of he clam for =. Now le 2 and suppose he clam holds rue for. Fx I and T. Defne he conecure σ : Θ T as follows: for all Θ T b [ [ µ {} {a } a = [ µ {} for all a T f µ [ {} b > 0 and [ a = / R 0 ; G for all a R 0 ; G oherwse. Noe ha b [ a > 0 only f a = ˆ for some ˆ T wh ˆ =. Therefore by he nducon hypohess supp b R 0 ; G for all Θ T. Nex defne ˆµ Θ as [ [ ˆµ a = b a [ for all Θ and a. Consder he behavor sraegy b : Θ T b for all Θ T and a. defned by: [ a = µ [ a 2

13 Behavor sraegy b ogeher wh -order belefs probably ˆµ Θ va: [ ˆµ a = b T for all Θ and a. Snce = margθ T [ ˆµ a = T = µ [ a Θ T nduce a [ a [ µ we have [ µ a marg Θ T [ µ By we have R 0 ; G whch proves par a of he clam. Consder par b. Fx I and T wh d > δ. Le b : Θ T be an arbrary behavor sraegy such ha: b ˆ R ε ˆ ; G 3.6 for all ˆ Θ T. Noe ha by he nducon hypohess for every T we can have b ˆ [ > 0 only f dˆ δ. Behavor sraegy b ogeher wh -order belefs nduce a probably µ Θ va: µ [ a = b T ˆ [ a [ dˆ 3.7 for all Θ and a. Snce d > δ here exss some Θ T [ { } δ B < Le be an arbrary ype n T wh 3.6 for every ˆ Thus by 3.7 hence { T µ [ = T : such ha [ δ Θ T. 3.8 =. By he nducon hypohess and we can have b ˆ [ > 0 only f ˆ B δ. B δ = b [ } µ ˆ < [ [ dˆ [ { } B δ [ δ Θ T 3

14 where he las nequaly s us 3.8. Bu snce [ = { T : = } µ [ we have and herefore T µ [ [ δ µ > Θ T µ µ δ > Θ T 2 > ζ. I follows from ha R ε ; G. Ths concludes he proof of he clam. ppendx for Secon 3 Lemma.. For each I le T be a fne se and µ : T Θ T a funcon. For every 0 < ζ < here exs ε > 0 and a game wh fne acon ses T such ha for every T : s a bes-reply o belef µ vewed as a probably over Θ ; For every belef µ Θ s an ε-bes-reply o µ only f µ µ ζ where denoes he maxmum norm. Proof. Fx ζ 0. Le f : Θ T Θ T R denoe he funcon defned by f ; µ = 2µ µ 2 Θ T for all µ Θ T Θ T and le F : Θ T Θ T R be he funcon defned by F µ µ = Θ T f ; µ µ for all µ µ Θ T Θ T. { Le η = 2 mn F µ µ F µ µ : µ µ 2 Θ T µ µ } ζ 2. We have η > 0 for F s connuous and µ = µ s he unque maxmzer of F µ on Θ T for all µ. 4

15 By he unform connuy of F here exss γ > 0 such ha for all µ µ 2 Θ T µ µ < γ F µ µ F µ µ < η. The compac se Θ T can be covered by a fne unon of open balls of radus γ. Choose one pon n whch of hese balls and le Θ T denoe he fne se of chosen pons. Enlarge f necessary o ensure T. We denfy each T wh µ. F µ µ F a µ < η. Thus for every µ Θ here exss a such ha Defne he payoff funcon g : Θ R f a ; a : a a T g a a = 4 ζ : a T a \ T : a \ T a \ T. We are now n a poson o prove par of he lemma. Suppose player s belef over Θ s gven by µ for some T. I follows drecly from he defnon of g and he fac ha µ [ Θ T = ha each acon a yelds player an expeced payoff of F a µ. Snce F µ µ F a µ for all a we conclude ha s a bes-reply o µ. Ths proves par. Fx any 0 < ε < mn{η ζ 2 ζ 2 }. We shall prove par now. Fx T and µ Θ wh µ µ > ζ. Suppose µ ζ Θ T < 2. The complemenary case wll be handled n he nex paragraph. Consder a devaon from o an arbrary acon a \ T. Snce F maps no [ he gan from hs devaon s bounded below by ζ ζ ζ + = ζ 2 > ε and herefore s no an ε-bes-reply o µ whch concludes he proof of par n he case µ Θ T < ζ 2. Now suppose µ Θ T ζ 2. Snce µ µ > ζ here exss Θ T such ha µ [ µ [ > ζ.. Consder he condonal probably µ µ Θ T. We have µ [ µ [ = µ [ µ Θ T µ [ ζ 2.2 and herefore µ [ µ [ < ζ 2. 5

16 Hence by.2 and. µ [ [ µ µ [ [ µ µ [ [ µ > ζ 2 whch mples F µ µ F µ 2η by he defnon of η. Now pc any a wh µ a < γ so ha F a µ F µ µ > η and herefore F a µ F µ > η. Hence he payoff gan of he devaon from o a s bounded below by µ ζ Θ T η η > ε 2 and herefore s no an ε-bes-reply o µ as requred. References lprans C. D. and K. C. Border 999: Infne Dmensonal nalyss. Sprnger second edn. Deel E. D. Fudenberg and S. Morrs 2006: Topologes on Types Theorecal Economcs : Inerm Correlaed Raonalzably Theorecal Economcs Ely J. C. and M. Pes 2007: Crcal Types mmeo. Norhwesern Unversy. Harsany J : Games wh Incomplee Informaon Played by Bayesan Players I-III Managemen Scence Ka. and S. Morrs 997: Refnemens and Hgher Order Belefs: Unfed Survey mmeo. Prnceon Unversy. Merens J.-F. and S. Zamr 985: Formulaon of Bayesan nalyss for Games wh Incomplee Informaon Inernaonal Journal of Game Theory Monderer D. and D. Same 989: pproxmang Common Knowledge wh Common Belefs Games and Economc Behavor Morrs S. 2002: Typcal Types mmeo. Prnceon Unversy. Rubnsen. 989: The Elecronc Mal Game: Game wh lmos Common Knowledge mercan Economc Revew