Hierarchies of Beliefs and the Belief-invariant. Bayesian Solution

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1 Herarches of Belefs and the Belef-nvarant Bayesan Soluton Qanfeng Tang March 26, 2015 Abstract The belef-nvarant Bayesan soluton s a noton of correlated equlbrum n games wth ncomplete nformaton proposed by Forges (1993), and herarchy of belefs over condtonal belefs s ntroduced by Ely and Pesk (2006) n ther study of nterm ndependent ratonalzablty. We study the connecton between the two concepts. We partally characterze the correlatons embedded among type spaces wth the same set of herarches of belefs over condtonal belefs wth partally correlatng devces, whch send correlated sgnals to players n a way that preserves each player s belef about others types. Snce the belef-nvarant Bayesan soluton s also mplemented by such correlatng devces, we then establsh that t s nvarant on equvalent type space. The author thanks Marcn Pesk for hs nvaluable gudance and detaled comments, Syang Xong for very benefcal dscussons, and semnar partcpants at Tsnghua Unversty and conference partcpants at 2012 Game Theory Socety World Congress for helpful comments. The author s also grateful to an edtor and two anonymous referees for valuable suggestons that substantally mprove the exposton of the paper. All errors are my own. School of Economcs, Shangha Unversty of Fnance and Economcs. Emal address: tangqanfeng198@gmal.com. 1

2 JEL Classfcaton: C70; C72 Keywords: Games wth ncomplete nformaton, Correlated equlbrum, Herarches of belefs 1 Introducton Harsany ( ) proposes type spaces to model players belefs and hgher-order belefs n games wth ncomplete nformaton, and later Mertens and Zamr (1985) construct a unversal type space whch ncorporates all herarches of belefs. These works provde the foundatons for strategc analyss of games wth ncomplete nformaton. One phenomenon that recently attracts game theorsts attenton s that, for a gven soluton concept, type spaces and herarches of belefs are not always strategcally equvalent. To be more precse, multple type spaces can represent the same set of herarches of belefs and are hence equvalent n ths respect. However, equvalent type spaces may dffer n the amount of correlatons ncorporated among types, and consequently, for a gven soluton concept, they may nduce dfferent predctons on the actons played by types wth the same herarchy of belefs. In partcular, type spaces that represent the same set of Mertens-Zamr herarches of belefs always nduce the same nterm correlated ratonalzable outcomes (Dekel et al., 2006; Dekel et al., 2007), but may nduce dfferent nterm ndependently ratonalzable outcomes (Ely and Pesk, 2006) and Bayes Nash equlbrum outcomes. The mplct correlatons, or hdden nformaton, that herarches of belefs fal to ncorporate are exactly those embedded among equvalent type spaces. Lu (2015) shows how to dstll such correlatons and explctly characterzes them va state-dependent correlatng devces. Accordng to the characterzaton, a type space has the same set of 2

3 Mertens-Zamr herarches of belefs as a non-redundant type space f and only f t can be generated from the conjuncton of the non-redundant type space and a state-dependent correlatng devce. 1 Therefore, the characterzaton decomposes the nformaton n type space nto a correlatng devce and the prmtve nformaton the herarches of belefs. Herarchy of belefs over condtonal belefs, also called -herarchy of belefs, s ntroduced by Ely and Pesk (2006) and then extended by Tang (2015) n the study of nterm ratonalzablty. It ncorporates rcher nformaton than Mertens-Zamr herarchy of belefs. As a result, t dentfes a stronger noton of soluton concept Ely and Pesk show that two types wth the same herarchy of belefs always have the same set of nterm ndependently ratonalzable actons. In the same sprt of Lu (2015) s work, we are nterested n understandng the correlatons embedded among type spaces wth the same set of -herarches of belefs and ther connecton to correlated equlbrum n games wth ncomplete nformaton. We begn wth provdng a partal characterzaton of such correlatons. For our purpose, two type spaces are sad to be equvalent f they have the same set of -herarches of belefs. A partally correlatng devce sends correlated sgnals to players based on nterm stage nformaton and preserves each player s belef about others types. We show that any type space s equvalent to ts conjuncton wth a partally correlatng devce, and that f two type spaces are equvalent, then they can always generate the same type space va conjunctons wth partally correlatng devces. The correlatons ncorporated n partally correlatng devces and the belef-nvarant Bayesan soluton (Forges, 1993;Forges, 2006) take the same form. When the sgnals n the correlatng devce are smply recommendatons of actons to the players, the conjuncton of the type space and the correlatng devce generates exactly an epstemc model for the 1 In a closely related work, Lu (2009) shows that under mld assumptons, a redundant type space can be represented by a non-redundant type space defned on an extended space of basc uncertantes. 3

4 belef-nvarant Bayesan soluton. Based on the partal characterzaton of correlatons obtaned, we establsh that for any game-form, the set of belef-nvarant Bayesan soluton payoffs on a type space s the unon of ts Bayesan Nash equlbra payoffs across equvalent type spaces. And n an mmedate corollary, we show that the belef-nvarant Bayesan soluton s nvarant across equvalent type spaces. Therefore, we have obtaned a belef foundaton for the belef-nvarant Bayesan soluton. To dentfy the set of all such solutons, t s wthout loss of generalty for us to consder the unversal type space of -herarches of belefs. 2 Ths paper s closely related to and s n parallel wth Lu (2015). We can fnd counterparts for varous concepts and results of our paper n Lu s work. Snce -herarchy of belefs ncorporates rcher nformaton than Mertens-Zamr herarchy of belefs, t dentfes stronger notons of soluton concepts (nterm ndependent ratonalzablty and the belef-nvarant Bayesan soluton versus nterm correlated ratonalzablty and Lu s noton of correlated equlbrum), and leaves less nformaton embedded among equvalent type spaces (partally correlatng devces versus state-dependent correlatng devces). Lehrer et al. (2010) and Lehrer et al. (2013) study the relatonshp between garblngs and the equvalence of type spaces. The non-communcatng garblngs that they use have smlar features as partally correlatng devces. Bergemann and Morrs (2014) propose and study the Bayes correlated equlbrum. They focus on correlatng devces that are dependent on ex post stage nformaton but free from belef-nvarance restrctons. The rest of ths paper s organzed as follows. We present the basc notatons Secton 2, and formulate herarches of belefs n Secton 3. In Secton 4, we characterze the correlatons embedded n equvalent type spaces and then apply the result on the belef- 2 An applcaton s on the robustness of soluton concepts to ncomplete nformaton, a study ntated by Kaj and Morrs (1997). The belef foundaton of the belef-nvarant Bayesan soluton allows us to study ts robustness by perturbng the -herarchy of belefs of a type, usng the nterm approach taken by Wensten and Yldz (2007) n ther study of the robustness of nterm correlated ratonalzablty. 4

5 nvarant Bayesan soluton. Secton 5 concludes. 2 Notatons For any metrc space X, let X denote the space of probablty measures on the Borel σ-algebra of X endowed wth the weak -topology. Let the product of two metrc spaces be endowed wth the product Borel σ-algebra. For any probablty measure µ X, let supp µ denote the support of µ, and for any measure µ (X Y), let marg X µ denote the margnal of µ on X. We study games wth ncomplete nformaton wth n players. The set of players s N = {1, 2,..., n}. For each N, let denote the set of s opponents. Players play a game n whch the payoffs are uncertan and parameterzed by a fnte set Θ. Each element θ Θ s called a state of nature. For each N, let A be the set of actons for player and let A N A be the set of acton profles. A (strategc-form) game s a profle G = (g, A ) N. For each N, we assume the payoff functon s bounded: g : A Θ [ M, M], for some postve real number M. The set of fnte bounded games s denoted by G. A type space over Θ s defned as T = (T, π ) N, where for each, T s a fnte set of types for player and π : T (T Θ) s a belef mappng such that π (t )[(t, θ)] descrbes player s belef on the event that the others type profle s t and the state of nature s θ. 3 3 Throughout, gven arbtrary x X and y Y, we use the notaton π (x)[y] to denote player s belef about y condtonal on x. More precsely, the object n the round bracket always denotes the object player that condtons on, and the object n the square bracket always denotes the object that assgns probablty to. 5

6 3 Herarches of belefs 3.1 Formulaton We frst present Mertens and Zamr s standard formulaton of herarches of belefs (see also Brandenburger and Dekel, 1993), and based on that present Ely and Pesk s constructon of -herarches of belefs. Let X 0 = Θ, and for each k 1, let X k = X k 1 ( (X k 1 )) n 1. Let h 1 (t ) = marg Θ π (t ) be player s belef over Θ at type t. For each k 2, let h k (t )[S] = π (t )[{(θ, t ) : (θ, (h l (t )) 1 l k 1 ) S}], for any measurable subset S X k 1. In ths constructon, h k (t ) (X k 1 ) represents player s k-th order belef at t. The profle h(t ) = (h 1 (t ),..., h k (t ),...) k=0 X k s called player s Mertens-Zamr herarchy of belefs at type t. As extensons of Mertens-Zamr herarches of belefs, -herarches of belefs descrbe players belefs and hgher-order belefs about the condtonal belefs on states of nature. Ths concept s ntroduced by Ely and Pesk (2006) n ther study of nterm ndependent ratonalzablty. We begn wth defnng condtonal belefs. Gven belef π (t ) (T Θ), the condtonal belef of type t over Θ, condtonng on the others types beng t, s denoted by π (t )(t ) (Θ), also wrtten as π (t, t ). For any type space T, let B (t ) = {π (t, t ) (Θ) : t T } be the set of all possble condtonal belefs at t. Type t s belef over T then nduces a belef over B (t ) (Θ) : for any measurable subset S (Θ), π (t )[S] = π (t )[{t : π (t, t ) S}]. The defnton of -herarchy of belefs at t treats the set of possble condtonal belefs,.e., (Θ), as the set of basc uncertantes. In a -herarchy of belefs, the frst-order belef of a player s her belef over the set of condtonal belefs, the second-order belef s her belef over the others belefs over the set of condtonal belefs, and so on. 6

7 For any type space T = (T, π ) N on Θ, we can transform t nto a type space T = (T, π ) N on (Θ). In the new type space, players types are unchanged, and player s belef at type t s π (t ) (T (Θ)), such that π (t )[S] = π (t )[{t : (t, π (t, t )) S}], for any measurable subset S T (Θ). In the type space T on (Θ), let the Mertens-Zamr herarchy of belefs at t be denoted by h(t T ). Defnton 1. For any type space T and nteger k 1, let the k-th order -herarchy of belefs at t T be h k (t T ) and denote t by δ k (t ). Also, let δ(t ) = (δ 1 (t ),..., δ k (t ),...) denote the -herarchy of belefs at t. By defnton, δ(t ) = h(t T ). For player, we use δ to denote the profle of the others -herarches of belefs. 3.2 Equvalence of type spaces For any type space T, let the set of all -herarches of belefs of player be Λ (T) = {δ(t ) : t T }. Just lke Mertens-Zamr herarches of belefs, the set of -herarches of belefs does not unquely pn down a type space. Instead, multple type spaces may nduce the same set of -herarches of belefs. Defnton 2. Two type spaces T and T are equvalent, wrtten as T T, f they have the same set of -herarches of belefs for all players, that s, f Λ (T) = Λ (T ), N. 7

8 A type space n whch dfferent types of a player always have dfferent herarches of belefs s called a reduced type space (Aumann, 1998), or a non-redundant type space (Mertens and Zamr, 1985). For any Mertens-Zamr herarchy of belefs, we are able to construct such a type space that generates t. However, ths s not true for -herarches of belefs. We llustrate ths wth a smple type space taken from Ely and Pesk (2006). Example 1. Consder a type space T n whch Θ = T 1 = T 2 = {+1, 1}, and players belefs are updated from a common pror π (Θ T 1 T 2 ) such that π(t 1, t 2, θ) = 1 4 f t t 2 = θ, 0 otherwse. In ths type space, the set of condtonal belefs for each type contans pont mass on θ = +1 and pont mass on θ = 1, and both types of player 1 (or player 2) have the same -herarchy of belefs common certanty of equal probablty on the pont masses. However, type space T s the most compact one that supports ths -herarchy of belefs. 4 Man results 4.1 A partal characterzaton of correlatons Wthout dstngushng non-redundant and redundant type spaces, we can acheve a partal characterzaton of the correlaton embedded across equvalent type spaces. As we shall see, ths partal characterzaton s suffcent for the nvarance of the belef-nvarant soluton. Defnton 3. For any type space T, a partally correlatng devce on T s a profle Q = (q, S ) N, where for each N, S s a fnte set of sgnals and q : T S, 8

9 where S = N S, s a belef mappng such that 1. For any = j, t T, supp q (t) = supp q j (t). 2. belef nvarance s satsfed: at dfferent types t, t of the others, player receves s wth the same probablty,.e., q (t, t )[s ] = q (t, t )[s ],, t, s. {s S:s =s } {s S:s =s } From the defnton, player beleves that at (t, t ) T, the partally correlatng devce selects a sgnal profle (s, s ) S accordng to the dstrbuton q (t, t ) S, and for each j N, s j wll be reported by a medator to player j. Belef nvarance ensures that from the sgnals that the players receve, they cannot nfer any extra nformaton about the others types. Also note that the correlated sgnals depend only on the nterm stage nformaton players types not on states of nature. The partally correlatng devce here and the state-dependent correlatng devce of Lu (2015) take smlar forms, whle dfferng from each other manly n two ways: () n the former, correlated sgnals are dependent on the nterm stage nformaton (the type profle of players), whle n the latter, they are dependent on the ex post stage nformaton (the state of the world,.e., both the type profle of players and the true state of nature); and () the belef-nvarance restrcton of the former requres that the sgnals do not change each player s belef about the others types, whle that of the latter requres that they do not change each player s belef about the states of the world. Every partally correlatng devce s a state-dependent correlatng devce, but the converse s not true. When the partally correlatng devce uses actons as sgnals and therefore the sgnals are smply recommendatons of play, we say that the correlatng devce s canoncal. Defnton 4. If for all N, S = A, then Q s a canoncal partally correlatng devce. 9

10 Let q (t, t )[s s ] be player s belef on the others recevng the sgnal profle s, condtonal on her recevng s. Ths condtonal belef can be easly calculated gven q : T S. Defnton 5. For any type space T = (T, π ) N and any partally correlatng devce Q = (q, S ) N, let T Q be the enlarged type space generated from T through operatng Q on T. More precsely, T Q = (T Q, π Q ) N such that T Q = {(t, s ) : t T, q (t)[s ] > 0, for some t T }, and for all (t, s ) T Q, θ Θ and (t, s ) T Q, π Q ((t, s ))[((t, s ), θ)] = π (t )[(t, θ)] q (t, t )[s s ]. Intutvely, we can vew T Q as the type space faced by players at the ex ante stage, when n addton a correlatng devce s expected to send sgnals to players after the realzaton of players types. The followng theorem provdes a partal characterzaton of the correlaton embedded n equvalent type spaces. Proposton 1. Let T be any type space. Then 1. for any partally correlatng devce Q, T Q T; more specfcally, for any (t, s ) T Q, δ((t, s )) = δ(t ). 2. for any type space ˆT satsfyng T ˆT, there exst partally correlatng devces Q and ˆQ such that T Q = ˆT ˆQ. The frst part of the proposton states that the enlarged type space T Q generated by type space T and any partally correlatng devce Q always has the same set of - 10

11 herarches of belefs as T does. In partcular, after recevng any sgnal s at type t, player s -herarchy of belefs does not change. The second part of the proposton states that for any par of type spaces T and ˆT whch have the same set of -herarches of belefs, we can fnd two partally correlatng devce Q and ˆQ through whch T and ˆT generate the same enlarged type space. Our characterzaton of correlaton s partal because unlke Lu (2015), we do not show that any redundant type space can be enlarged from the nonredundant type space wth a correlatng devce. Instead, we establsh the connecton between two equvalent type spaces wthout dstngushng redundant and non-redundant type spaces. The proof of ths proposton s relegated nto Appendx A. The proof of part 1 s by nducton, and the key s the belef nvarance property of partally correlatng devces. Next, we brefly sketch the ntuton for the proof of part 2. We transform the type space ˆT nto a partally correlatng devce Q and transform the type space T nto a partally correlatng devce ˆQ. Then we show that T Q and ˆT ˆQ both represent some verson of the product type space T ˆT and hence are the same. Specfcally, when the correlatng devce Q s operated on type space T, from player s vew, at any type profle (t, t ), the set of sgnals she may receve from Q s exactly the set of types n the type space ˆT that have the same -herarchy of belefs as t, and when she receves any such sgnal ˆt, the set of profles of sgnals that her opponents may receve from Q s exactly the set of type profles ˆt n ˆT that nduce the same profle of -herarches as t and the same condtonal belef at ˆt as π (t, t ). 11

12 4.2 The belef-nvarant Bayesan soluton The belef-nvarant Bayesan soluton s a noton of correlated equlbrum for games wth ncomplete nformaton proposed by Forges (1993) 4 and later dscussed n Forges (2006) and Bergemann and Morrs (2014). A belef-nvarant Bayesan soluton on a type space T s defned as an epstemc model that embeds T n a belef-nvarant way and satsfes Bayesan ratonalty at each state of the world. As Forges (2006) ponts out, each such epstemc model s an enlarged type space T Q for some partally correlatng devce Q, and each belef-nvarant soluton corresponds to a Bayes Nash equlbrum (BNE) of the game (G, T Q ). 5 Therefore, the set of belef-nvarant Bayesan soluton payoffs on T s the unon of BNE payoffs across enlarged type spaces. Let Q denote the set of all partally correlatng devces. For each game (G, T) and a BNE σ of ths game, let g (t σ) be player s nterm payoff at type t, and let g(t σ) be the nterm payoff vector at type profle t T. Denote the set of nterm BNE payoffs of the game (G, T) as NE(G, T) = {g(t σ) : t T, σ s a BNE of (G, T)}. Let B I (G, T) denote the set of nterm belef-nvarant Bayesan soluton payoffs, then B I (G, T) = {Q:Q Q}NE(G, T Q ). The result below states that the set of players nterm payoffs from belef-nvarant Bayesan solutons at a type space s exactly the unon of nterm BNE payoffs on equva- 4 Forges defnton of the Bayesan soluton s restrcted to two-player games for type spaces wth common prors; what we present here s the n-player non-common pror analogue of her defnton. 5 From a pont of vew analogous to the "revelaton prncple" n the mechansm desgn lterature, Forges (2006) also characterzes belef-nvarant Bayesan soluton payoffs wth canoncal partally correlatng devces. 12

13 lent type spaces. Proposton 2. B I (G, T) = { ˆT: ˆT T} NE(G, ˆT). Proof. By Proposton 1, for all Q Q, T T Q. Therefore, t s straghtforward that {Q:Q Q}NE(G, T Q ) { ˆT: ˆT T} NE(G, ˆT). We also know that for any ˆT such that ˆT T, there exst ˆQ and Q such that ˆT ˆQ = T Q. Therefore, NE(G, ˆT) NE(G, ˆT ˆQ ) NE(G, ˆT) NE(G, T Q ). That s, we also have { ˆT: ˆT T} NE(G, ˆT) {Q:Q Q}NE(G, T Q ). Lehrer et al. (2010) and Lehrer et al. (2013) study the payoff equvalence of nformaton structures wth respect to belef-nvarant Bayesan solutons. They show that two nformaton structures are equvalent f and only f there exst non-communcatng garblng transformatons between them. Non-communcatng garblngs share smlar features as partally correlatng devces. The man dfference between them s that a garblng garbles orgnal sgnals, whle a correlatng devce ntroduces new sgnals nto the structure. It s mmedate from Proposton 2 that f two type spaces represent the same set of - herarches of belefs, they must nduce the same set of belef-nvarant Bayesan soluton payoffs n any game. In other words, the belef-nvarant Bayesan soluton s nvarant on the equvalent class of type spaces. Corollary 1. If ˆT T, then B I (G, T) = B I (G, ˆT). 13

14 5 Concluson We study the correlatons embedded n type spaces wth the same set of herarches of belefs over condtonal belefs, t turns out that such correlatons can be expressed explctly wth partally correlatng devces, whch operate n the nterm stage of the game. Wth these results, we see a complete pcture of the connectons among -herarches of belefs, nterm partally correlated ratonalzablty, and the belef-nvarant Bayesan soluton. We already know that Partally correlatng devces characterze correlatons embedded n type spaces wth the same set of -herarches of belefs, and mplement the Bayesan soluton. Furthermore, Tang (2015) shows that -herarches of belefs fully dentfy nterm partally correlated ratonalzablty. As a result, nterm partally correlated ratonalzablty and the belef-nvarant Bayesan soluton are payoff equvalent. A Appendx: Proof of Proposton 1 Part I. We use nducton to show that for any (t, s ) T Q, δ((t, s )) = δ(t ). Frst note that for any (t, s ) T Q, (t, s ) T Q, and θ Θ, π Q ((t, s ), (t, s ))[θ] = πq ((t, s ))[(t, s ), θ] π Q ((t, s ))[(t, s )] = π (t )[(t, θ)] q (t, t )[(s, s )] π (t )[t ] q (t, t )[(s, s )] = π (t, t )[θ]. Therefore, for any (t, s ) T Q, the set of condtonal belefs at (t, s ) s the same as 14

15 that at t. Furthermore, for any condtonal belef β B (t ), π Q ((t, s ))[β] = π Q ((t, s ))[{(t, s ) : π Q ((t, s ), (t, s )) = β}] = π Q ((t, s ))[{(t, s ) : π (t, t ) = β}] = π Q ((t, s ))[{t : π (t, t ) = β}] = π (t )[{t : π (t, t ) = β}] = π (t )[β]. The fourth equaton above comes from belef nvarance. We have proved that for all (t, s ) T Q, δ 1 ((t, s )) = δ 1 (t ). For hgher-order belefs, we prove by nducton. Now suppose for all 0 < l k and (t, s ) T Q, δ l ((t, s )) = δ l (t ), we show that for all (t, s ) T Q, δ k+1 ((t, s )) = δ k+1 (t ). Let the support of the l-th order belef at type t be B l (t ). As a result, the set of condtonal belefs s relabeled as B 1 (t ). By the premses of nducton, for all (t, s ) T Q and 0 < l k, B l ((t, s )) = B l (t ). Indeed, for any (β, δ 1,..., δ k ) 0<l k B l (t ), δ k+1 ((t, s ))[(β, δ 1,..., δ k )] = π Q ((t, s ))[{(t, s ) : (π Q ((t, s ), (t, s )), (δ l ((t, s ))) k l=1 ) = (β, δ1,... δ k )}] = π Q ((t, s ))[{(t, s ) : π (t, t ) = β, δ 1 (t ) = δ 1,..., δ k (t ) = δ k }] = π (t )[{t : π (t, t ) = β, δ 1 (t ) = δ 1,..., δ k (t ) = δ k }] = δ k+1 (t )[(β, δ 1,..., δ k )]. By nducton, for all (t, s ) T Q, δ((t, s )) = δ(t ). Naturally, T Q and T have the same set of -herarches of belefs, T Q T. Part II. Fx a par of type spaces T = (T, π ) N and ˆT = ( ˆT, ˆπ ) N. Suppose T ˆT, 15

16 we now construct Q and ˆQ such that T Q = ˆT ˆQ. To do that, we transform the type space ˆT nto a partally correlatng devce Q and transform T nto a partally correlatng devce ˆQ. We then show that the generated type spaces T Q and ˆT ˆQ are the same. Step 1. Before we start, we need a few ntermedate results. The lemma below says that f two types have the same -herarchy of belefs, then they must have the same belef over condtonal belefs and the others -herarches of belefs. Lemma 1. Fx type spaces T and T. If t T, t T and δ(t ) = δ(t ), then π (t )[(β, δ )] = π (t )[(β, δ )], β, δ. Proof. Wth the basc property of probablty measures, π (t )[(β, δ )] = π (t )[{t : π (t, t ) = β, δ 1 (t ) = δ 1,..., δn (t ) = δ n,...}] = π (t )[ n {t : π (t, t ) = β, δ 1 (t ) = δ 1,..., δn (t ) = δ n }] = lm π (t )[{t : π (t, t ) = β, δ 1 (t ) = δ 1 n,..., δn (t ) = δ n }] = lm δ n+1 (t )[(β, δ 1,..., δ n )] n = lm δ n+1 (t n )[(β, δ1,..., δ n )] = π (t )[(β, δ )]. Also, f two types t T, t T have the same -herarchy of belefs, and f t has a condtonal belef π (t, t ), then t must also have such a condtonal belef, condtonal on some type profle of the others t whch have the same -herarches of belefs as t. Lemma 2. Fx type spaces T and T. Suppose t T, t T, and δ(t ) = δ(t ). Then for any t T that satsfes π (t )[t ] > 0, there exsts t T that satsfes δ(t ) = δ(t ) and π (t )[t ] > 0, such that π (t, t ) = π (t, t ). 16

17 Proof. We prove by contradcton. Suppose t s not the case. Then there exsts a t that satsfes π (t )[t ] > 0 and π (t, t ) = β, such that for all t that satsfes π (t, t ) = β, π (t )[t ] > 0, t must be that δ(t ) = δ(t ). As a result, π (t )[(β, δ (t ))] = 0. However, π (t )[(β, δ (t ))] π (t )[t ] > 0. Gven Lemma 1, ths s n contradcton wth δ(t ) = δ(t ). Step 2. Usng nformaton n type space ˆT, we now construct a partally correlatng devce Q = (q, S ) N whch s to be operated on type space T. For each N, let the set of sgnals for player be S = ˆT, and defne S N S. Defne S (t ) {ˆt ˆT : δ(ˆt ) = δ(t )} and S (t, t ˆt )) {ˆt ˆT : δ(ˆt ) = δ(t ) and ˆπ (ˆt, ˆt ) = π (t, t )}. Intutvely, we are gong to construct q : T S n a way such that the set of sgnals that player could possbly receve when her type s t s restrcted to be S (t ), whch s the set of t s equvalent types n ˆT. Smlarly, S (t, t ˆt )) wll be the restrcted set of sgnals ˆt that players may receve at type profle t from player s vew, when her own type s t and she receves sgnal ˆt. We need the followng result, whch s mmedate from Lemma 1 and Lemma 2, n the constructon of q. It states that f two types are equvalent n ˆT, then ther belefs over condtonal belefs and the others -herarches of belefs must be the same. Lemma 3. If ˆt, û S (t ), then ˆπ (ˆt )[S (t, t ˆt ))] = ˆπ (û )[S (t, t û )]. 17

18 Defne on the type space ˆT a pror ˆp ( ˆT ˆT Θ) for player as follows: ˆp [(ˆt, ˆt, θ)] = 1 ˆT ˆπ (ˆt )[(ˆt, θ)], (ˆt, ˆt, θ) ˆT ˆT Θ. From player s vew, the partally correlatng devce operates only n states of the world (ˆt, ˆt, θ) such that ˆp (ˆt, ˆt, θ) > 0. For each N, we can construct the belef system q : T S as follows: q (t, t )[(ˆt, ˆt )] = ˆp [(ˆt,ˆt )] ˆp [S (t ) S (t,t ˆt )], f (ˆt, ˆt ) S (t ) S (t, t ˆt ); 0, otherwse. Wth Lemma 3, for any (ˆt, ˆt ) S (t ) S (t, t ˆt ), q (t, t )[(ˆt, ˆt )] = = ˆp [ˆt ] ˆπ (ˆt )[(ˆt, θ)] û S (t ) ˆp [û ] ˆπ (û )[S (t, t ) û ] 1/ ˆT 1/ ˆT S (t ) ˆπ (ˆt )[(ˆt, θ)] ˆπ (ˆt )[S (t, t ) ˆt ]. The expresson of q can be rewrtten as q (t, t )[(ˆt, ˆt )] = 1 S (t ) ˆπ (ˆt )[ˆt ] ˆπ (ˆt )[S (t,t ˆt )], f (ˆt, ˆt ) S (t ) S (t, t ˆt ); 0, otherwse. Now we prove that the Q defned above s ndeed a partally correlatng devce. Frst, for any = j, t T, supp q (t) = supp q j (t) = k N S k (t k ). Ths s because from player s vew, each ˆt S (t ) s sent to her wth probablty 1 S (t ), and that for each ˆt k N\{} S k (t k ), there must be ˆt S (t ) such that ˆt 18

19 S (t, t ˆt ) and ˆπ (ˆt )[ˆt ] > 0, due to Lemma 2. Second, belef nvarance s satsfed: for any (t, t ) T and any û S (t ), the probablty that player wll receve sgnal û s q (t, t )[(û, ˆt )] = {ˆt ˆT:ˆt =û } {ˆt :ˆt S (t,t û )} 1 S (t ) ˆπ (û )[ˆt ] ˆπ (û )[S (t, t û )] = 1 {ˆt :ˆt S (t,t û )} ˆπ (û )[ˆt ] S (t ) ˆπ (û )[S (t, t û )] = 1 S (t ) = 1 S (t ), ˆπ (û )[S (t, t û )] ˆπ (û )[S (t, t û )] whch s ndependent of t. Thus the sgnal does not provde extra nformaton on the others types. Step 3. Gven the partally correlatng devce Q constructed usng nformaton n ˆT, we can generate a new type space T Q = (T Q, π Q ) N from the type space T. In T Q, T Q = {(t, ˆt ) : t T, ˆt S (t )}, and for any (ˆt, ˆt ) S (t ) S (t, t ˆt ), π Q ((t, ˆt ))[((t, ˆt ), θ)] = π (t )[(t, θ)] ˆπ (ˆt )[ˆt ] ˆπ (ˆt )[S (t, t ˆt )]. Smlarly, we can construct another partally correlatng devce ˆQ usng nformaton n the type space T, and generate a new type space ˆT ˆQ from ˆT. In ˆT ˆQ, ˆT ˆQ ˆT, t S (ˆt )}, and for any (t, t ) S (ˆt ) S (ˆt, ˆt t ), = {(ˆt, t ) : ˆt π ˆQ π ((ˆt, t ))[((ˆt, t ), θ)] = ˆπ (ˆt )[(ˆt, θ)] (t )[t ] π (t )[S (ˆt, ˆt t )]. It s straghtforward that T Q = ˆT ˆQ, N. Now we show π Q ((t, ˆt )) = π ˆQ ((ˆt, t )). By the defnton, for any (t, t ) and (ˆt, ˆt ) S (t ) S (t, t ˆt ), we know that 19

20 π (t, t ) = ˆπ (ˆt, ˆt ) = β, δ(t ) = δ(ˆt ) = δ, for some β and δ. We can decompose the belef π Q as follows: π Q ((t, ˆt ))[((t, ˆt ), θ)] = π (t, t )[θ] π (t )[t ] = π (t, t )[θ] π(t )[t ] ˆπ (ˆt )[ˆt ]. π (t )[(β, δ )] ˆπ (ˆt )[ˆt ] ˆπ (ˆt )[{ˆt : δ(ˆt ) = δ(t ), ˆπ (ˆt, ˆt ) = π (t, t )}] Smlarly, π ˆQ ((ˆt, t ))[((ˆt, t ), θ)] can also be decomposed: π ˆQ ((ˆt, t ))[((ˆt, t ), θ)] = ˆπ (ˆt, ˆt )[θ] ˆπ (ˆt )[ˆt ] π (t )[t ]. ˆπ (ˆt )[(β, δ )] We compare π Q and π ˆQ term by term. Frst, π (t, t )[θ] = ˆπ (ˆt, ˆt )[θ]. Second, π (t )[t ] ˆπ (ˆt )[ˆt ] = ˆπ (ˆt )[ˆt ] π (t )[t ]. Thrd, from Lemma 1, π (t )[(β, δ )] = ˆπ (ˆt )[(β, δ )]. Snce for any N, (t, ˆt ) T Q = ˆT ˆQ, π Q ((t, ˆt )) = π ˆQ ((ˆt, t )), we have T Q = ˆT ˆQ. 20

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