CMS-EMS Center for Mathematical Studies in Economics And Management Science. Discussion Paper #1586

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1 CMS-EMS Center for Mathematcal Studes n Economcs And Management Scence Dscusson Paper #1586 "When Do Types Induce the Same Belef Herarchy?" Andres Perea and Wllemen Kets Northwestern Unversty December 13, 2015

2 When Do Types Induce the Same Belef Herarchy? Andrés Perea Wllemen Kets Frst verson: January 2014 Ths verson: December 13, 2015 Abstract Type structures are a smple devce to descrbe hgher-order belefs. But how can we check whether two types generate the same belef herarchy? Ths paper generalzes the concept of a type morphsm and shows that one type structure s contaned n another f and only f the former can be mapped nto the other usng a generalzed type morphsm. Hence, every generalzed type morphsm s a herarchy morphsm and vce versa. Importantly, generalzed type morphsms do not make reference to belef herarches. We use our results to characterze the condtons under whch types generate the same belef herarchy. JEL Classfcaton: C72 Keywords: Types, belef herarches, epstemc game theory, morphsms. Ths paper s a substantally revsed verson of an earler paper wth the same ttle by Andrés Perea. We would lke to thank Perpaolo Battgall, Edde Dekel, Amanda Fredenberg, Chrstan Nauerz, Mklós Pntér and Elas Tsakas for very useful comments on the earler verson. EpCenter and Department of Quanttatve Economcs, Maastrcht Unversty. E-mal: a.perea@maastrchtunversty.nl. Web: MEDS, Kellogg School of Management, Northwestern Unversty. E-mal: w-kets@kellogg.northwestern.edu. Web: 1

3 1 Introducton Hgher-order belefs play a central role n game theory. Whether a player s wllng to nvest n a project, for example, may depend on what he thnks that hs opponent thnks about the economc fundamentals, what he thnks that hs opponent thnks that he thnks, and so on, up to arbtrarly hgh order (e.g. Carlsson and van Damme, 1993). Hgher-order belefs can also affect economc conclusons n settngs rangng from barganng (Fenberg and Skrzypacz, 2005; Fredenberg, 2014) and speculatve trade (Geanakoplos and Polemarchaks, 1982) to mechansm desgn (Neeman, 2004). Hgher-order belefs about actons are central to epstemc characterzatons, for example, of ratonalzablty (Brandenburger and Dekel, 1987; Tan and Werlang, 1988), Nash equlbrum (Aumann and Brandenburger, 1995; Perea, 2007) and forward nducton reasonng (Battgall and Snscalch, 2002). In prncple, hgher-order belefs can be modeled explctly, usng belef herarches. For applcatons, the type structures ntroduced by Harsany (1967) provde a smple, tractable modelng devce to represent players hgher-order belefs. Whle type structures provde a convenent way to represent hgher-order belefs, t may be dffcult to check whether types generate the same belef herarchy. The lterature has consdered the followng queston: gven two type structures, T and T, s t the case that for every type n T, there s a type n T that generates the same belef herarchy? That s, s the type structure T contaned n T? The lterature has consdered two dfferent tests to address ths queston, one based on herarchy morphsms, and one based on type morphsms. Herarchy morphsms can be used to gve a complete answer to ths queston: a type structure T s contaned n T f and only f there s a herarchy morphsm from the former to the latter. A problem wth ths test s that herarchy morphsms make reference to belef herarches, as we shall see. So, ths test requres us to go outsde the purvew of type structures. The second test uses type morphsms. Type morphsms are defned solely n terms of the propertes of the type structures. However, the test based on type morphsms only provdes a suffcent condton: f there s a type morphsm from T to T, then T s contaned n T (Hefetz and Samet, 1998b). But, as shown by Fredenberg and Meer (2011), the condton s not necessary: t may be that T contans T, yet there s no type morphsm from T to T. Fredenberg and Meer also provde a range of condtons under whch the condton s both necessary and suffcent. However, they do not drectly address the queston whether there mght be an alternate test (whch provdes condtons that are both necessary and suffcent) that does not requre us to descrbe the belef herarches explctly. Ths paper provdes such a test, by generalzng the noton of a type morphsm. We show that a type structure s contaned n another f and only f there s a generalzed type morphsm from the former to the latter. So, a generalzed type morphsm s a herarchy morphsm and vce 2

4 versa. Unlke the defnton of herarchy morphsms, the defnton of generalzed type morphsms does not make reference to belef herarches. So, ths test can be carred out wthout leavng the purvew of type structures. Usng ths result, t s straghtforward to verfy whether two types generate the same belef herarchy, as we show. Herarchy morphsms are used n a number of dfferent settngs. For example, they can be used to check whether types have the same ratonalzable actons (Dekel et al., 2007), and play an mportant role n the lterature on the robustness to msspecfyng the parameter set more generally; see, e.g., Ely and Pesk (2006) and Lu (2009). Herarchy morphsms are also used to study the robustness of Bayesan-Nash equlbra to msspecfcatons of players belef herarches (Fredenberg and Meer, 2007; Yldz, 2009) and n epstemc game theory. The current results make t possble to study these ssues wthout descrbng players belef herarches explctly, usng that every herarchy morphsm s a generalzed type morphsm and conversely. A crtcal ngredent n the defnton of a generalzed type morphsm s the σ-algebra on a player s type set whch separates hs types f and only f they dffer n the belef herarchy that they generate. Mertens and Zamr (1985, p. 6) use ths σ-algebra to defne nonredundant type structures, and ths σ-algebra also plays an mportant role n the work of Fredenberg and Meer (2011), where t s used to characterze the condtons under whch herarchy morphsms and type morphsms concde. Mertens and Zamr provde a nonconstructve defnton of ths σ-algebra, and Fredenberg and Meer show that the σ-algebra defned by Mertens and Zamr s the σ-algebra generated by the functons that map types nto belef herarches. We provde a constructve defnton of ths σ-algebra, by means of a type parttonng procedure, that does not make reference to belef herarches. Whle many of the ngredents that underle our results are known n some form or another, we vew the contrbuton of ths paper as combnng these deas n a new way to generalze the concept of a type morphsm so that t provdes a necessary and suffcent condton for a type structure to be contaned n another that does not refer to belef herarches. A number of papers has shown that the measurable structure assocated wth type structures can mpose restrctons on reasonng (Brandenburger and Kesler, 2006; Fredenberg, 2010; Fredenberg and Meer, 2011; Kets, 2011; Fredenberg and Kesler, 2011). Ths paper contrbutes to that lterature n two ways. Frst, we elucdate the connecton by constructng the measurable structure on type sets that s generated by players hgher-order belefs. Second, we provde tools to easly go from the doman of type structures to the doman of belef herarches and vce versa. The outlne of ths paper s as follows. The next secton ntroduces basc concepts. Secton 3

5 3 dscusses type morphsms and herarchy morphsms. Secton 4 defnes our generalzaton of a type morphsm, and proves the man result. Secton 5 apples ths result to characterze the condtons under whch types generate the same belef herarchy. Secton 6 consders the specal case where players have fntely many types. Proofs are relegated to the appendx. 2 Belef Herarches and Types In ths secton we show how belef herarches can be encoded by means of a type structure, and how every type wthn a type structure can be decoded by dervng a full belef herarchy from t. 2.1 Encodng Belef Herarches by Type Structures Consder a fnte set of players I. Assume that each player faces a basc space of uncertanty (X, Σ ) where X s a set and Σ a σ-algebra on X. That s, X = (X, Σ ) s a measurable space. We assume that X s countably generated: there s a countable collecton of subsets E n X, n = 1, 2,..., such that Σ s the coarsest σ-algebra that contans these subsets. Examples of countably generated σ-algebras nclude the dscrete σ-algebra on a fnte or countable set and the Borel σ-algebra on a fnte-dmensonal Eucldean space. The combnaton X = (X, Σ ) I of basc uncertanty spaces s called a mult-agent uncertanty space. The basc space of uncertanty for player could, for nstance, be the set of opponents choce combnatons, or the set of parameters determnng the utlty functons of the players, or even a combnaton of the two. A belef herarchy for player specfes a probablty measure on X the frst-order belef, a probablty measure on X and the opponents possble frst-order belefs the second-order belef, and so on. As s standard, we encode such nfnte belef herarches by means of type structures. For any measurable space (Y, ˆΣ), we denote by (Y, ˆΣ) the set of probablty measures on (Y, ˆΣ). We endow (Y, ˆΣ) wth the coarsest σ-algebra that contans the sets {µ (Y, ˆΣ) µ(e) p} : E ˆΣ, p [0, 1]. Ths s the σ-algebra used n Hefetz and Samet (1998b) and many subsequent papers; t concdes wth the Borel σ-algebra on (Y, ˆΣ) (nduced by the weak convergence topology) f Y s metrzable and ˆΣ s the Borel σ-algebra. Product spaces are endowed wth the product σ- algebra. Gven a collecton of measurable spaces (Y, Y ), I, wrte Y for the product σ-algebra j I Y j and Y for the product σ-algebra j Y j, where I. 4

6 Defnton 2.1. [Type Structure] Consder a mult-agent uncertanty space X = (X, Σ ) I. A type structure for X s a tuple T = (T, Σ T, b ) I where, for every player, (a) T s a set of types for player, endowed wth a σ-algebra Σ T, and (b) b : T (X T, ˆΣ ) s a measurable mappng that assgns to every type t a probablstc belef b (t ) (X T, ˆΣ ) on ts basc uncertanty space and the opponents type combnatons, where ˆΣ = Σ Σ T s the product σ-algebra on X T. Fnally, f f : Y (Y, Σ ) s a functon from Y to the measurable space (Y, Σ ), then σ(f) s the σ-algebra on Y generated by f, that s, t s the coarsest σ-algebra that contans the sets {y Y : f(y) E} for E Σ. 2.2 From Type Structures to Belef Herarches In the prevous subsecton we have ntroduced a type structure as a way to encode belef herarches. We now show how to decode a type wthn a type structure, by dervng the full belef herarchy t nduces. Consder a type structure T = (T, Σ T, b ) I for X. Then, every type t wthn T nduces an nfnte belef herarchy h T (t ) = (µ T,1 (t ), µ T,2 (t ),...), where µ T,1 (t ) s the nduced frst-order belef, µ T,2 (t ) s the nduced second-order belef, and so on. We wll nductvely defne, for every n, the nth order belefs nduced by types t n T, buldng upon the (n 1)th order belefs that have been defned n the precedng step. We start by defnng the frst-order belefs. For each player, defne H 1 := (X, Σ ) to be the set of belefs about X, and for every type t T, defne ts frst-order belef µ T,1 (t ) by µ T,1 (t )(E ) := b (t )(E T ) for all E Σ. Clearly, µ T,1 (t ) (X, Σ ) for every type t. Defne h T,1 (t ) := µ T,1 (t ). The mappng µ T,1 from T to H 1 s measurable by standard arguments. For n > 1, suppose the set H n 1 has been defned and that the functon h T,n 1 σ-algebra on X j H n 1 j, and defne from T to H n 1 s measurable. Let ˆΣ n 1 be the product H n := H n 1 (X H n 1 n 1, ˆΣ ). 5

7 For every type t, defne ts n-th-order belef µ T,n (t ) by for all E ˆΣ n 1 : µ T,n (t )(E) = b (t )({(x, t ) X T (x, h T,n 1 (t )) E}), wth h T,n 1 (t ) = (h T,n 1 j (t j )) j. Snce h T,n 1 j a probablty measure on (X H n 1, ˆΣ n 1 ). Defne h T,n that h T,n (t ) H n. Moreover, ht,n s measurable. Fnally, for every type t T, we denote by h T (t ) := (µ T,n (t )) n N s measurable for every player j, µ T,n (t ) := (h T,n 1 (t ), µ T,n s ndeed (t )). It follows the belef herarchy nduced by type t n T. Also, defne H to be the set (X ) n 1 (X H n ) of all belef herarches. We say that two types, t and t, of player generate the same belef herarchy f h T (t ) = h T (t ). Types t and t generate the same nth-order belef f µt,n (t ) = µ T,n (t ).1 3 Herarchy and Type Morphsms The lterature has consdered two concepts that map type structures nto each other, type morphsms and herarchy morphsms. Throughout the remander of the paper, fx two type structures, T = (T, Σ T, b ) I and T = (T, ΣT, b ) I on X. The functons that map types from T and T nto belef herarches are denoted by h T and h T, respectvely. Defnton 3.1. [Herarchy Morphsm] For each player I, let ϕ be a functon from T to T such that for every type t T, h T (ϕ (t )) = h T (t ). Then, ϕ s a herarchy morphsm (from T to T ). Wth some abuse of notaton, we refer to the profle (ϕ ) I as a herarchy morphsm. So, f there s a herarchy morphsm between T and T, then every type n T can be mapped nto a type n T n a way that preserves belef herarches. We say that the type structure T contans T f, and only f, there s a herarchy morphsm from T to T. Type morphsms are mappngs between type structures that preserve belefs. Defnton 3.2. [Type Morphsm] For each player I, let ϕ be a functon from T to T that s measurable wth respect to Σ T and Σ T. 2 Suppose that for each player, type t T, 1 Clearly, t and t generate the same nth-order belef f and only f h T,n (t ) = h T,n (t ). 2 That s, for each E Σ T, we have {t T ϕ (t ) E} Σ T. 6

8 and E Σ Σ T, b (t )({(x, t ) X T (x, ϕ (t )) E}) = b (ϕ (t ))(E). (1) Then, ϕ := (ϕ ) I s a type morphsm (from T to T ). Hefetz and Samet (1998b) have shown that one type structure s contaned n another whenever there s a type morphsm from the former to the latter. Proposton 3.3. (Hefetz and Samet, 1998b, Prop. 5.1) If ϕ s a type morphsm from T to T, then t s a herarchy morphsm. So, f there s a type morphsm from T to T, then T contans T. Unlke herarchy morphsms, type morphsms do not make reference to belef herarches. So, to check whether there s a type morphsm from one type structure to another, we need to consder only the type structures. However, the condton that there be a type morphsm from one type structure to another provdes only a suffcent condton for the former to be contaned n the latter. Indeed, Fredenberg and Meer (2011) show that the condton s not necessary: there are type structures such that one s contaned n the other, yet there s no type morphsm between the two. 4 Generalzed Type Morphsms Type morphsms requre belefs to be preserved for every event n the types σ-algebra. However, for two types to generate the same belef herarchy, t suffces that ther belefs are preserved only for events that can be descrbed n terms of players belef herarches. We use ths nsght to defne generalzed type morphsms and show that a type structure contans another f and only f there s a generalzed type morphsm from the latter to the former. The frst step s to defne the relevant σ-algebra. Mertens and Zamr (1985, p. 6) provde the relevant condton. We follow the presentaton of Fredenberg and Meer (2011). Defnton 4.1. (Fredenberg and Meer, 2011, Def. 5.1) Fx a type structure T and fx a sub-σ algebra Σ T Σ T for each player I. Then, the product σ-algebra Σ T s closed under T f for each player, {t T b (t )(E) p} Σ T for all E Σ Σ T and p [0, 1]. 7

9 The coarsest (sub-)σ algebra that s closed under T s of specal nterest, and we denote t by F T = I F T. It s the ntersecton of all σ-algebras that are closed under T.3 Mertens and Zamr use ths σ-algebra to defne nonredundant type spaces, and Fredenberg and Meer use t to characterze the condton under whch a herarchy morphsm s a type morphsm. Fredenberg and Meer (2011) provde a characterzaton of the σ-algebra F T n terms of the herarchy mappngs. Recall that σ(h T ) s the σ-algebra on T generated by the mappng h T. That s, σ(h T ) s the coarsest σ-algebra that contans the sets {t T h T (t ) E} : E H measurable. Lemma 4.2. (Fredenberg and Meer, 2011, Lemma 6.4) Let the product σ-algebra F T be the coarsest σ-algebra that s closed under T. Then, for each player, F T = σ(h T ). We are now ready to defne generalzed type morphsms. Defnton 4.3. [Generalzed Type Morphsm] For each player I, let ϕ be a functon from T to T that s measurable wth respect to ΣT and F T. 4 Suppose that for each player, type t T, and E Σ F T, b (t )({(x, t ) X T (x, ϕ (t )) E}) = b (ϕ (t ))(E). Then, ϕ := (ϕ ) I s a generalzed type morphsm (from T to T ). Lke type morphsms, generalzed type morphsms are defned usng the language of type structures alone; the defnton does not make reference to belef herarches. The dfference between type morphsms and generalzed type morphsms s that the former requres belefs to be preserved for all events n the σ-algebra Σ Σ T for player, whle the latter requres belefs to be preserved only for events n the σ-algebra Σ F T, and ths σ-algebra s a coarsenng of Σ Σ T (Defnton 4.1 and Lemma 4.2). Our man result states that one structure s contaned n another f and only f there s a generalzed type morphsm from the former to the latter. Theorem 4.4. A mappng ϕ s a herarchy morphsm from T to T f and only f t s a generalzed type morphsm from T to T. Hence, a type structure T contans T f and only f there s a generalzed type morphsm from T to T. 3 Snce Σ T s closed under T (by measurablty of the belef maps b ), the ntersecton s nonempty. It s easy to verfy that the ntersecton s a σ-algebra. 4 That s, for each E F T, we have {t T ϕ (t ) E} Σ T. 8

10 Ths result establshes an equvalence between generalzed type morphsms and herarchy morphsms. It thus provdes a test that can be used to verfy whether one type structure s contaned n the other that does not refer to belef herarches. Whle the characterzaton n Theorem 4.4 does not make reference to belef herarches, the result may not be easy to apply drectly. The σ-algebras F T are defned as the ntersecton of σ-algebras that are closed under T, and there can be (uncountably) many of those. We next defne a smple procedure to construct ths σ-algebra. Procedure 1. [Type Parttonng Procedure] Consder a mult-agent uncertanty space X = (X, Σ ) I, and a type structure T = (T, Σ T, b ) I for X. Intal step. For every player, let S T,0 = {T, } be the trval σ-algebra of hs set of types T. Inductve step. Suppose that n 1, and that the sub-σ algebra S T,n 1 on T has been defned for every player. Then, for every player, let S T,n be the coarsest σ-algebra that contans the sets {t T b (t )(E) p} for all E Σ S T,n 1 n ST,n. and all p [0, 1]. Also, let S T, A smple nductve argument shows that S T,n be the σ-algebra generated by the unon refnes S T,n 1 for all players and all n; clearly, for any n. The next result shows that the type parttonng procedure delvers S T, refnes S T,n the σ-algebras that are generated by the herarchy mappngs. Proposton 4.5. Fx a type structure T and let I. Then, S T, for all n 1. So, S T, = F T. = σ(h T ) and ST,n = σ(h T,n ) Hence, we can use the type parttonng procedure to construct the σ-algebras whch we need for our characterzaton result (Theorem 4.4). Hefetz and Samet (1998a) consder a smlar procedure n the context of knowledge spaces to show that a unversal space does not exst for that settng. The procedure also has connectons wth the constructon n Kets (2011) of type structures that descrbe the belefs of players wth a fnte depth of reasonng. In the next secton, we use Theorem 4.4 and the type parttonng procedure to characterze the types that generate the same belef herarches. 5 Characterzng Types Wth the Same Belef Herarchy We can use the results n the prevous secton to provde smple tests to determne whether two types from the same type structure or from dfferent structures generate the same belef 9

11 herarchy. To state the result, recall that an atom of a σ-algebra Σ on a set Y s a set a Σ such that Σ does not contan a nonempty proper subset of a. That s, for any a Σ such that a a, we have a = a or a =. 5 Lemma 5.1. Let I and n 1. The σ-algebras S T,n and S T, are atomc. That s, for each t T, there are atoms a n (t ) and a (t ) n S T,n and S T,, respectvely, such that t a n (t ) and t a (t ). Ths result motvates the name type parttonng procedure : the procedure constructs a σ-algebra that parttons the type sets nto atoms. Proposton 5.2 shows that these atoms contan precsely the types that generate the same hgher-order belefs. Proposton 5.2. For every player, every n 1, and every two types t, t T, we have that (a) for every n 0, types t and t generate the same nth-order belef f and only f there s an atom a n S n such that t, t an ; (b) types t and t generate the same belef herarchy f and only f there s an atom a such that t, t a. S T, There s a connecton between Proposton 5.2 and the work of Mertens and Zamr (1985). Mertens and Zamr defne a type structure T to be nonredundant f for every player, the σ- algebra F T separates types; see Lu (2009, Prop. 2) for a result that shows that ths defnton s equvalent to the requrement that there are no two types that generate the same belef herarchy. So, Mertens and Zamr already note the connecton between the separatng propertes of F T and the queston whether types generate the same belef herarchy. The contrbuton of Proposton 5.2 s to provde a smple procedure to construct the σ-algebra F T, and to show that the separatng sets can be taken to be atoms (as long as the σ-algebra on X s countably generated). Proposton 5.2 can also be used to verfy whether two types from dfferent type structures generate the same hgher-order belefs, by mergng the two structures. Specfcally, consder two dfferent type structures, T 1 = (T 1, Σ1, b1 ) I and T 2 = (T 2, Σ2, b2 ) I, for the same mult-agent nduce the uncertanty space X = (X, Σ ) I. To check whether two types t 1 T 1 and t 2 T 2 same belef herarchy, we can merge the two type structures nto one large type structure, and then run the type parttonng procedure on ths larger type structure. That s, defne the type T 2, and defne the structure T = (T, Σ, b ) I as follows. For each player, let T := T 1 5 Clearly, for any y Y, f there s an atom a that contans y (.e., y a), then ths atom s unque. 10

12 σ-algebra Σ on T by Also, defne b by E Σ f and only f E T 1 Σ 1 and E T 2 Σ 2. b (t ) := { b 1 (t ), f t T 1 b 2 (t ), f t T 2 for all types t T. 6 Applyng the type parttonng procedure on T gves a σ-algebra S, on T for each player. If t 1 T 1 and t 2 T 2 belong to the same atom of S,, then t 1 and t 2 nduce the same belef herarchy. The converse also holds, and hence we obtan the followng result. Proposton 5.3. Consder two type structures T 1 = (T 1, Σ1, b1 ) I and T 2 = (T 2, Σ2, b2 ) I. Let T = (T, Σ, b ) I be the large type structure defned above, obtaned by mergng the two type structures, and let S,, for a gven player, be the σ-algebra on T generated by the type parttonng procedure. Then, two types t 1 T 1 and t 2 T 2 nduce the same belef herarchy, f and only f, t 1 and t2 belong to the same atom of S,. The type parttonng procedure s thus an easy and effectve way to check whether two types, from possbly dfferent type structures, generate the same belef herarchy or not. We expect our man results to apply more broadly. The proofs can easly be modfed so that the man results extend to condtonal probablty systems n dynamc games (Battgall and Snscalch, 1999), lexcographc belefs (Blume et al., 1991), belefs of players wth a fnte depth of reasonng (Kets, 2011; Hefetz and Kets, 2013), and the -herarches ntroduced by Ely and Pesk (2006). 6 Fnte Type Structures When type structures are fnte, our results take on a partcularly smple and ntutve form. Say that a type structure T s fnte f the type set T s fnte for every player. For fnte type structures, we can replace σ-algebras by parttons. We frst defne the type parttonng procedure for the case of fnte type structures. A fnte partton of a set A s a fnte collecton P = {P 1,..., P K } of nonempty subsets P k A such 6 Ths s wth some abuse of notaton, snce b s defned on X T, whle b 1 and b 2 are defned on X T 1 and X T 2, respectvely. By defnng the σ-algebra Σ T j on T j as above, the extenson of b 1 and b 2 to the larger doman s well-defned. 11

13 that K k=1 P k = A and P k P m = whenever k m. We refer to the sets P k as equvalence classes. For an element a A, we denote by P(a) the equvalence class P k to whch a belongs. The trval partton of A s the partton P = {A} contanng a sngle set the full set A. For two parttons P 1 and P 2 on A, we say that P 1 s a refnement of P 2 f for every set P 1 P 1 there s a set P 2 P 2 such that P 1 P 2. In the procedure we recursvely partton the set of types of an agent nto equvalence classes startng from the trval partton, and refnng the prevous partton wth every step untl these parttons cannot be refned any further. We show that the equvalence classes produced n round n contan exactly the types that nduce the same n-th order belef. In partcular, the equvalence classes produced at the end contan precsely those types that nduce the same (nfnte) belef herarchy. Procedure 2 (Type Parttonng Procedure (fnte type structures)). Consder a mult-agent uncertanty space X = (X, Σ ) I, and a fnte type structure T = (T, Σ T, b ) I for X. Intal step. For every agent, let P 0 be the trval partton of hs set of types T. Inductve step. Suppose that n 1, and that the parttons P n 1 agent. Then, for every agent, and every t T, have been defned for every P n (t ) = {t T b (t )(E P n 1 ) = b (t )(E P n 1 ) (2) for all E Σ, and all P n 1 P n 1 }. The procedure termnates at round n whenever P n = P n 1 for every agent. In ths procedure, P n 1 s the partton of the set T nduced by the parttons Pj n 1 on T j. Agan, t follows from a smple nductve argument that P n s a refnement of P n 1 for every player and every n. Note that f the total number of types, vz., I T, equals N, then the procedure termnates n at most N steps. We now llustrate the procedure by means of an example. Example 1. Consder a mult-agent uncertanty space X = (X, Σ ) I where I = {1, 2}, X 1 = {c, d}, X 2 = {e, f}, and Σ 1, Σ 2 are the dscrete σ-algebras on X 1 and X 2, respectvely. Consder the type structure T = (T 1, T 2, Σ 1, Σ 2, b 1, b 2 ) n Table 1. Here, b 1 (t 1 ) = 1 2 (c, t 2) (d, t 2 ) means that type t 1 assgns probablty 1 2 to the par (c, t 2 ) X 1 T 2, and probablty 1 2 to the par (d, t 2 ) X 1 T 2. Smlarly for the other types n the table. We wll now run the Type Parttonng Procedure. 12

14 Type structure T = (T 1, T 2, b 1, b 2 ) T 1 = {t 1, t 1, t 1 }, T 2 = {t 2, t 2, t 2 } b 1 (t 1 ) = 1 2 (c, t 2) (d, t 2 ) b 1 (t 1 ) = 1 6 (c, t 2) (c, t b 1 (t 1 ) = 1 2 (c, t 2 ) (d, t 2 ) 2 ) (d, t 2 ) b 2 (t 2 ) = 1 4 (e, t 1) (e, t 1 ) (f, t 1 ) 1 ) b 2 (t 2 ) = 1 8 (e, t 1) (e, t 1 ) (f, t b 2 (t 2 ) = 3 8 (e, t 1) (e, t 1 ) (f, t 1 ) Table 1: The type structure from Example 1. Intal Step. Let P1 0 be the trval partton of the set of types T 1, and let P2 0 partton of the set of types T 2. That s, be the trval P 0 1 = {{t 1, t 1, t 1}} and P 0 2 = {{t 2, t 2, t 2}}. Round 1. By equaton (2), P 1 1 (t 1 ) = {τ 1 T 1 b 1 (τ 1 )({c} T 2 ) = b 1 (t 1 )({c} T 2 ) = 1 2, b 1 (τ 1 )({d} T 2 ) = b 1 (t 1 )({d} T 2 ) = 1 2 } = {t 1, t 1, t 1}, whch mples that P 1 1 = P 0 1 = {{t 1, t 1, t 1}}. 13

15 At the same tme, whch mples that P2 1(t 2 ) = {t 2 }, and hence P 1 2 (t 2 ) = {τ 2 T 2 b 2 (τ 2 )({e} T 1 ) = b 2 (t 2 )({e} T 1 ) = 3 4, b 2 (τ 2 )({f} T 1 ) = b 2 (t 2 )({f} T 1 ) = 1 4 } = {t 2, t 2} P 1 2 = {{t 2, t 2}, {t 2}. Round 2. By equaton (2), whch mples that P1 2(t 1 ) = {t 1 }, and hence P 2 1 (t 1 ) = {τ 1 T 1 b 1 (τ 1 )({c} {t 2, t 2}) = b 1 (t 1 )({c} {t 2, t 2}) = 1 2, b 1 (τ 1 )({c} {t 2}) = b 1 (t 1 )({c} {t 2}) = 0, b 1 (τ 1 )({d} {t 2, t 2}) = b 1 (t 1 )({d} {t 2, t 2}) = 0, b 1 (τ 1 )({d} {t 2}) = b 1 (t 1 )({d} {t 2}) = 1 2 } = {t 1, t 1}, P 2 1 = {{t 1, t 1}, {t 1}}. Snce P1 1 = P0 1, we may mmedately conclude that P 2 2 = P 1 2 = {{t 2, t 2}, {t 2}}. Round 3. As P2 2 = P1 2, we may mmedately conclude that P 3 1 = P 2 1 = {{t 1, t 1}, {t 1}}. 14

16 By equaton (2), whch mples that P2 3(t 2 ) = {t 2 }, and hence P 3 2 (t 2 ) = {τ 2 T 2 b 2 (τ 2 )({e} {t 1, t 1}) = b 2 (t 2 )({e} {t 1, t 1}) = 3 4, b 2 (τ 2 )({e} {t 1}) = b 2 (t 2 )({e} {t 1}) = 0, b 2 (τ 2 )({f} {t 1, t 1}) = b 2 (t 2 )({f} {t 1, t 1}) = 0, b 2 (τ 2 )({f} {t 1}) = b 2 (t 2 )({f} {t 1}) = 1 4 } = {t 2, t 2}, P 3 2 = {{t 2, t 2}, {t 2}} = P 2 2. As P1 3 = P2 1 and P3 2 = P2 2, the procedure termnates at round 3. The fnal parttons of the types are thus gven by P 1 = {{t 1, t 1}, {t 1}} and P 2 = {{t 2, t 2}, {t 2}}. The reader may check that all types wthn the same equvalence class ndeed nduce the same belef herarchy. That s, t 1 nduces the same belef herarchy as t 1, and t 2 nduces the same belef herarchy as t 2. Moreover, t 1 and t 1 nduce dfferent belef herarches, and so do t 2 and t 2. Our characterzaton result for the case of fnte type structures states that the type parttonng procedure characterzes precsely those groups of types that nduce the same belef herarchy. We actually prove a lttle more: we show that the parttons generated n round n of the procedure characterze exactly those types that yeld the same n-th order belef. Proposton 6.1 (Characterzaton Result (fnte type structures)). Consder a fnte type structure T = (T, Σ, b ) I, where Σ s the dscrete σ-algebra on T for every player. For every agent, every n 1, and every two types t, t T, we have that (a) h T,n (t ) = h T,n (t ), f and only f, t Pn (t ); (b) h T (t ) = h T (t ), f and only f, t P (t ). The proof follows drectly from Proposton 5.2, and s therefore omtted. As before, ths result can be used to verfy whether two types from dfferent type structures generate the same belef herarches, by frst mergng the two type structures, and then runnng the type parttonng procedure on ths large type structure. 15

17 A Proofs A.1 Proof of Theorem 4.4 By defnton, T contans T f and only f there s a herarchy morphsm from T to T. So, t suffces to show that every generalzed type morphsm s a herarchy morphsm and vce versa. Part I. Every herarchy morphsm s a generalzed type morphsm. To show that every herarchy morphsm s a generalzed type morphsm, we need to show two thngs. Frst, we need to show that any herarchy morphsm s measurable wth respect to the approprate σ-algebra. Second, we need to show that belefs are preserved for the relevant events. Let us start wth the measurablty condton. Suppose ϕ s a herarchy morphsm. Let I and E F T. We need to show that {t T ϕ (t ) E} Σ T. Recall that F T = σ(h T ) (Lemma 4.2). So, there s a measurable subset B of the set H of belef herarches such that E = {t T h T (t ) B}. Hence, {t T ϕ (t ) E} = {t T h T (ϕ (t )) B} = {t T h T (t ) B}, where the second equalty follows from the assumpton that ϕ s a herarchy morphsm. By Lemma 4.2, we have {t T h T (t ) B} F T. Snce Σ T F T (Defnton 4.1 and Lemma 4.2), the result follows. We next ask whether herarchy morphsms preserve belefs for the relevant events. Agan, let ϕ be a herarchy morphsm. Let I, t T, and E Σ F T. We need to show that b (t ) (Id X, ϕ ) 1 (E ) = b (ϕ (t ))(E ), where Id X s the dentty functon on X, and where we have used the notaton (f 1,..., f m ) for the nduced functon that maps (x 1,..., x m ) nto (f 1 (x 1 ),..., f m (x m )), so that b (t ) 16

18 (Id X, ϕ ) 1 s the mage measure nduced by (Id X, ϕ ). By a smlar argument as before, there s a measurable subset B of the set X H such that E = {(x, t ) X T (x, h T (t )) B }. If E s an element of Σ j {T j, }, then the result follows drectly from the defntons. So suppose E Σ j {T j, }. Then, for every n 1, defne and B n := {(x, µ 1,..., µ n ) X H n (x, µ 1,..., µ n, µ n+1,...) B E n := {(x, t ) X T (x, h T,n (t )) B n }. for some (µ n+1, µ n+2,...)} Then, E n E and E n E. Also, we have E n Σ j σ(ht,n j ) and thus E n Σ F T (Lemma 4.2). For every n, b (t ) (Id X, ϕ ) 1 (E n 1 ) = b (t ) (Id X, ϕ ) 1 (Id X, h T,n 1 ) 1 (B n 1 ) = b (t ) (Id X, h T,n 1 ϕ ) 1 (B n 1 ) = µ T,n (t )(B n 1 ) = b (ϕ(t )) (Id X, h T,n 1 ) 1 (B n 1 ) = b (ϕ(t ))(E n 1 ), where the penultmate equalty uses the defnton of a herarchy morphsm. By the contnuty of the probablty measures b (t ) and b (ϕ (t )) (e.g., Alprants and Border, 2005, Thm. 10.8), we have b (t ) (Id X, ϕ ) 1 (E ) = b (ϕ (t ))(E ), and the result follows. Part II. Every generalzed type morphsm s a herarchy morphsm. For the other drecton, that s, to show that every generalzed type morphsm s a herarchy morphsm, suppose that ϕ s a generalzed type morphsm from T = (T, Σ T, b ) I to T = (T, ΣT, b ) I. We can use an nductve argument to show that t s a herarchy morphsm. Let I and t T. Then, for all E Σ, µ T,1 (ϕ (t ))(E) = b (ϕ (t ))(E T ) = b (t )(E T ) = µ T,1 (t ), 17

19 where the frst and the last equalty use the defnton of a frst-order belef nduced by a type, and the second uses the defnton of a generalzed type morphsm. So, µ T,1 (t ) = µ T,1 (ϕ (t )) and thus h T,1 (t ) = h T,1 (ϕ (t )) for each player and every type t T. For n > 1, suppose that for each player and every type t T, we have h T,n 1 (t ) = h T,n 1 (ϕ (t )). We wll use the notaton (f 1,..., f m ) for the nduced functon that maps (x 1,..., x m ) nto (f 1 (x 1 ),..., f m (x m )), so that µ (f 1,..., f m ) 1 s the mage measure nduced by a probablty measure µ and (f 1,..., f m ). Let E be a measurable subset of X H n. By Lemma 4.2, we have F T j = σ(h T j ); and, clearly, σ(h T j ) σ(ht,n 1 j ). So, f we wrte Id X for the dentty functon on X, we have (Id X, h T,n 1 ) 1 (E) Σ F T. Then, for every player and type t T, µ T,n (ϕ (t ))(E) = b (ϕ (t )) (Id X, h T,n 1 ) 1 (E) = b (t ) (Id X, h T,n 1 ) 1 (Id X, ϕ ) 1 (E) = b (t ) (Id X, ϕ h T,n 1 ) 1 (E) = b (t ) (Id X, h T,n 1 ) 1 (E) = µ T,n (t )(E), where the frst equalty uses the defnton of an nth-order belef, the second uses the defnton of a generalzed type morphsm, the thrd uses the defnton of the composton operator, the fourth uses the nducton hypothess, and the ffth uses the defnton of an nth-order belef agan. Conclude that µ T,n (t ) = µ T,n (ϕ (t )) and thus h T,n (t ) = h T,n (ϕ (t )) for each player and every type t T. So, for each player I and each type t T, we have h T (t ) = h T (ϕ (t )), whch shows that ϕ s a herarchy morphsm. A.2 Proof of Proposton 4.5 Let I. It wll be convenent to defne h T,0 {ν }. So, the σ-algebra σ(h T,0 ) generated by h T,0 consder n = 1. Fx player I. By defnton, σ(h T,1 the sets to be the trval functon from T nto some sngleton s just the trval σ-algebra S 0 = {T, }. Next ) s the coarsest σ-algebra that contans {t T h T,1 (t ) E} : E H 1 measurable. It suffces to restrct attenton to the generatng sets E of the σ-algebra on H 1 = (X ) (e.g., Alprants and Border, 2005). So, σ(h T,1 ) s the coarsest σ-algebra that contans the sets {t T h T,1 (t ) E} 18

20 where E s of the form {µ (X ) µ(f ) p} for F Σ and p [0, 1]. Usng that for each type t, h T,1 (t ) s the margnal on X of b (t ), we have that σ(h T,1 ) s the coarsest σ-algebra that contans the sets {t T b (t )(E) p} : E Σ S 0, p [0, 1]. That s, σ(h T,1 ) = S 1. In partcular, ht,1 s measurable wth respect to S T,1. For n > 1, suppose, nductvely, that for each player I, σ(h T,n 1 s measurable wth respect to S T,n 1. Fx N. By defnton, σ(h T,n h T,n 1 σ-algebra that contans the sets n σ(h T,n 1 ) and the sets {t T µ T,n (t ) E} : E (X H n 1 ) measurable. ) = S T,n 1, so that ) s the coarsest Agan, t suffces to consder the generatng sets of the σ-algebra on (X H n 1 ). Hence, σ(h T,n ) s the coarsest σ-algebra that contans the sets {t T µ T,n (t )(F ) p} : F X H n 1 measurable and p [0, 1]. (Note that ths ncludes the generatng sets of σ(h T,n 1 ), gven that the nth-order belef nduced by a type s consstent wth ts (n 1)th-order belef.) Usng the defnton of µ T,n and the nducton assumpton that S T,n 1 T that contans the sets That s, σ(h T,n = σ(h T,n 1 ), we see that σ(h T,n ) s the coarsest σ-algebra on {t T b (t )(F ) p} : F Σ S T,n 1, p [0, 1]. ) = S T,n, and h T,n s measurable wth respect to S T,n So, for each player and n 1, σ(h T,n σ-algebra on T generated by the cylnders σ(h T,n A.3 Proof of Lemma 5.1. ) = S T,n. It follows mmedately that σ(h T ), as the ), s equal to S T,. Let I. Recall that X s countably generated, that s, there s a countable subset D 0 of Σ such that D 0 generates Σ (.e., Σ s the coarsest σ-algebra that contans D 0 ). Throughout ths proof, we wrte σ(d) for the σ-algebra on a set Y generated by a collecton D of subsets of Y. The followng result says that a countable collecton of subsets of a set Y generates a countable algebra on Y. For a collecton D of subsets of a set Y, denote the algebra generated by D by A(D). So, A(D) s the coarsest algebra on Y that contans D. 19

21 Lemma A.1. Let D be a countable collecton of subsets of a set Y. Then the algebra A(D) generated by D s countable. Proof. We can construct the algebra generated by D. Denote the elements of D by D λ, λ Λ, where Λ s a countable ndex set. Defne { } A(D) = D l, F a fnte subset of N, L m Λ, D l = A k or D l = A c k for some k, (3) m F l L m where E c s the complement of a set E. That s, A(D) s the collecton of fnte unons of fnte ntersecton of elements of D and ther complements. We check that A(D) s an algebra. Clearly, A(D) s nonempty (t contans D) and A(D). We next show that A(D) s closed under fnte ntersectons. Let A 1 := D l, A 2 := D l, l L 1 m 1 l L 2 m 2 be elements of A(D). Then, m 1 F 1 A 1 A 2 = (m 1,m 2 ) F 1 F 2 l 1 L 1 m 1 m 2 F 2 l 2 L 2 m 2 D l1 D l2. Clearly, F 1 F 2 s fnte, and so are the sets L 1 m and L 2 m. We can thus rewrte A 1 A 2 so that t s of the form as the elements n (3). We can lkewse show that A(D) s closed under complements: let A := m F l L m D l A(D), so that A c = m F l L m Dl c; then, snce l L m Dl c A(D) for every m, we have A c A(D). So, A(D) s an algebra that contans D, and t s n fact the coarsest such one (by constructon, any proper subset of A(D) does not contan all fnte ntersectons of the sets n D and ther complements). As D s countable, so s the collecton of the elements n D and ther complements; the collectons of the fnte ntersectons of such sets are also countable. Hence, A(D) s countable. Note that for any p [0, 1], the set {t T : b (t )(E) p} can be wrtten as the countable ntersecton of sets {t T : b (t )(E) p l } for some ratonal p l, l = 1, 2,.... So, by Proposton 4.5, the σ-algebra σ(h T,n ), n = 1, 2,..., on the type set T, I, s the coarsest σ-algebra that contans the sets {t T : b (t )(E) p} : E Σ j σ(h T,n 1 j ), p Q We are now ready to prove Lemma 5.1. Fx I. By Lemma A.1, the set D 0 generates a countable algebra A(D 0) on X. Then, by Proposton 4.5 and by Lemma 4.5 of Hefetz and 20

22 Samet (1998b), we have that the σ-algebra σ(h T,1 ) s generated by the sets {t T : b (t )(E) p} : E D 0 j σ(h T,0 j ), p Q. Denote ths collecton of these sets by D 1 A(D 1) σ(ht,1 ) (so that σ(h T,1 ) = σ(a(d 0)))., so that σ(ht,1 ) = σ(d 1); clearly, D1 For m > 1, suppose that for every I, the σ-algebra σ(h T,m 1 a countable collecton D m 1 of subsets of T such that A(D m 1 s countable and ) on T s generated by ). Fx I. By ) s generated ) σ(h T,m 1 Proposton 4.5 and Lemma 4.5 of Hefetz and Samet (1998b), the σ-algebra σ(h T,m by the sets {t T : b (t )(E) p} : E D 0 j A(D m 1 j ), p Q. Denote ths collecton of these sets by D m; as before, Dm s clearly countable and A(D m) σ(h T,m ). Agan, we have σ(h T,m ) = σ(d m) = σ(a(dm )). So, we have shown that for every I and m = 1, 2,..., the σ-algebra σ(h T,m ) s generated by a countable collecton D m of subsets of T. The σ-algebra σ(h T ) s generated by the algebra m σ(ht,m ) = m σ(dm ), or, equvalently, by the unon m Dm (Proposton 4.5). Snce the latter set, as the countable unon of countable sets, s countable, the σ-algebra σ(h T ) s countably generated. It now follows from Theorem V.2.1 of Parthasarathy (2005) that for each player, the σ- algebras σ(h T,m ), m = 1, 2,... are atomc n the sense that for each t T, there s a unque atom a m t n σ(h T,m ) contanng t ; the analogous statement holds for σ(h T ). A.4 Proof of Proposton 5.2 Fx a player I. By Proposton 4.5, we have S T, = σ(h T ) and ST,n = σ(h T,n ) for each n 1. By Lemma 5.1, the σ-algebras σ(h T, ) and σ(h T,n ) are atomc for every n 1. Let n 1. Let t, t T. Snce σ(h T,n ) s atomc, there exst a unque atom a n (t ) σ(h T,n ) such that t a n (t ) and a unque atom a n (t ) σ(ht,n ) such that t a n (t ). Suppose h T,n (t ) = h T,n (t ). Then for every generatng set E of the σ-algebra σ(ht,n ), ether t, t E or t, t E. So, an (t ) = a n (t ). Suppose ht (t ) h T (t ). Then there s a generatng set E of σ(h T,n ) that separates t and t, that s, t E, t E. So, an (t ) a n (t ). The proof of the clam that there s a unque atom a n σ(h T ) that contans both t and t f and only f h T (t ) = h T (t ) s analogous, and therefore omtted. 21

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