Interim Rationalizability. Eddie Dekel, Drew Fudenberg and Stephen Morris. Working Paper No September, 2005

Transcription

1 Interm Ratonalzablty By Edde Dekel, Drew Fudenberg and Stephen Morrs Workng Paper No September, 2005 The Foerder Insttute for Economc Research and The Sackler Insttute of Economc Studes

2 Interm Ratonalzablty Edde Dekel Northwestern Unversty and Tel Avv Unversty Drew Fudenberg Harvard Unversty Stephen Morrs Yale Unversty Frst Draft: May Ths Draft: August 2005 We are grateful to Avad Hefetz and Muhamet Yldz for helpful dscussons, Marcn Pesk for detaled comments and Satoru Takahash for expert proofreadng. We acknowledge nancal support from the Natonal Scence Foundaton under grants and

3 Abstract Ths paper proposes the soluton concept of nterm ratonalzablty, and shows that all type spaces that have the same herarches of belefs have the same set of nterm ratonalzable outcomes. Ths soluton concept characterzes common knowledge of ratonalty n the unversal type space. JEL Class caton and keywords: C70, C72, ratonalzablty, ncomplete nformaton, common knowledge, unversal type space.

4 Introducton Harsany (967-8) proposes solvng games of ncomplete nformaton usng type spaces, and Mertens and Zamr (985) show how to construct a unversal type space, nto whch all other type spaces (satsfyng certan techncal regularty assumptons) can be mapped. The unversal type space s the set of all n nte herarches of belefs satsfyng common knowledge of coherency. However, Bergemann and Morrs (200, secton 2.2.2) and Battgall and Snscalch (2003) emphasze that type spaces may allow for more correlaton than s captured n the belef herarches, so dentfyng types that have dentcal herarches may lead to a loss of nformaton, and soluton concepts can d er when appled to two d erent type spaces even f they are mapped nto the same subset of the unversal type space. In response, ths paper proposes the soluton concept of correlated nterm ratonalzablty. We show that the concept s well-de ned, that ts teratve and xed-pont de ntons concde, and that all type spaces that have the same herarches of belefs have the same set of correlated nterm ratonalzable outcomes. Thus ths s a soluton concept whch can be characterzed by workng wth the unversal type space, and more generally permts dentfyng those type spaces that have the same herarches of belefs. We also show that the soluton concept has smlar propertes to ts complete nformaton counterpart. Frst, n lemma 3 we establsh that the process of teratve elmnaton of strctly domnated strateges yelds the same soluton set. Second, recall that Brandenburger and Dekel (987a) showed that the set of actons that survve terated deleton of strctly domnated strateges n a complete nformaton game s equal to the set of actons that could be played n a subjectve correlated equlbrum. Proposton 3 reports a straghtforward extenson of Brandenburger and Dekel s observaton to games wth ncomplete nformaton; ths shows that nterm ratonalzablty characterzes common knowledge of ratonalty. We now sketch the man constructs n the paper. Fx a type space, where players have belefs and hgher order belefs about some payo -relevant state space. A game conssts of payo functons mappng from acton pro les and to the real lne. We dscuss two de ntons of nterm ratonalzablty. To nd the set of ndependently nterm ratonalzable strateges, teratvely delete for each type all actons that are not best responses gven that type s belefs over others types and and gven any pro le of strateges of all other players, Further detals and references are provded n the next secton.

5 where under those strateges, each type puts postve probablty only on survvng actons. The set of correlated nterm ratonalzable actons results from teratvely deletng for each type all actons that are not best responses gven that type s belefs over others types and and gven any, perhaps correlated, condtonal belefs about whch survvng actons are played by at a gven type pro le and payo relevant state. In ths de nton, a player s belefs allow for correlaton between one player s actons and the payo -relevant state and other players actons. In the complete nformaton case (.e., when s a sngleton), these de ntons reduce to the standard de ntons of ndependent and correlated ratonalzablty, respectvely (e.g., as n Brandenburger and Dekel (987a)). In the complete-nformaton case, ndependent and correlated ratonalzablty are equvalent when there are two players but not necessarly wth three or more players. We wll see that wth ncomplete nformaton, they may d er even n the two-person case (because of the possble correlaton n a player s conjecture between the opponent s actons and the payo -relevant state). We use the concept of (correlated) nterm ratonalzablty n our ongong work on de nng strategc topologes on the unversal type space (Dekel, Fudenberg and Morrs (2005)). 2 For ths exercse, t s mportant for us to know that the soluton concept depends only on herarches of belefs and not on "redundant" elements of the type space. Another contrbuton of ths note s to dentfy and analyze the dstncton between ndependent and correlated nterm ratonalzablty. Battgall and Snscalch (2003) de ne an umbrella noton of "-ratonalzable" actons n ncomplete nformaton envronments, where can be vared to capture common-knowledge restrctons on the rst order of belefs n the herarchy. They show that there s an equvalence between actons survvng an teratve procedure capturng common knowledge of and the set of actons that mght be played n equlbrum on any type space where s common knowledge. Correlated nterm ratonalzable actons are exactly -ratonalzable actons, where we let consst of a complete descrpton of the n nte herarches of belefs. Wth ths, proposton 3 below corresponds to ther proposton 4.3; they do not explctly menton ether correlated or ndependent nterm ratonalzablty. 3 Forges (993) explores the related queston of how to de ne correlated equlbrum for 2 For ths purpose, we also verfy below that all the results extend to "-ratonalzablty and "-equlbrum. 3 As ther analyss deals wth restrctons on rst order belefs, our result corresponds to an extenson of ther approach to allow for restrctons on the entre herarchy of belefs. 2

6 games of ncomplete nformaton. Forges allows correlatng devces that enable a player s own actons to depend on the payo states even when the player cannot dstngush between the states; as we dscuss at the end of subsecton 3.4 ths s smlar to what we do. Furthermore, lke Battgall and Snscalch (2003), Forges studes the soluton concept correspondng to allowng the type space to vary over all spaces (wth a common pror). Forges proposton 3, that relates common knowledge of ratonalty (wth a common pror and a gven type space) to agent-normal form correlated equlbrum s analogous to our dscusson followng our proposton 3 showng the sense n whch our results demonstrate an equvalence between common knowledge of ratonalty and (correlated) nterm ratonalzablty. Wensten and Yldz (2003) prove that the concluson of ths proposton can be strengthened n certan drectons. Pck any type t n the unversal type space, and any "rch" equlbrum s of a game played on the unversal type space (meanng that every acton s played by some type n the equlbrum ) and any acton a that s nterm ratonalzable for t : Then for generc payo s, for any k, there s a type t 0 that has the same nte-order belefs as t up to level k, and such that s (t ) = s (t 0 ). A recent paper by Ely and Pesk (2005) also notes that the set of nterm ratonalzable outcomes depend on more than just the standard unversal type space. They characterze how the standard unversal type space must be expanded to deal wth ths ssue. Thus, whle we nd the soluton concept whch depends on types only va ther (standard) herarches of belefs, Ely and Pesk provde an extended noton of herarches of belefs for whch a d erent soluton concept depends on types only va those extended herarches. 2 Type Spaces We base our development of type spaces on Hefetz and Samet s (998) topology-free constructon. For measurable X we denote by (X) the set of (probablty) measures on X; all product spaces are endowed wth the product -algebra. The prmtves of our model are a nte set of states of Nature a nte set I of players, and a type space T = (T ; ) I =, where each T s a measurable space, and each : T! (T ) s a measurable functon. 4 Followng Hefetz and Samet, we assume that for every measurable space X, the set (X) 4 Hefetx and Samet consder a general measurable space. Fnte sets n ths paper are endowed wth the obvous -algebra. 3

7 of measures on X s endowed wth the -algebra generated by ff : (E) pg : p 2 [0; ] and E a measurable subset of Xg () Ponts t 2 T are called player 0 s types, and we say that each type t of player has belef (t ) about the jont dstrbuton of the opponent s type and the state of Nature. The above setup de nes what Hefetz and Samet call a measurable type space. There s a belef-preservng morphsm from one measurable type space nto another measurable type space f t can be mapped nto that space whle preservng the belef structure. Formally, there s a belef-preservng morphsm from (T ; ) nto ~T ; ~ f for each there exsts measurable ' : T! ~ T wth ~ (' (t )) (F ) = (t ) ; t : ; ' (t ) 2 F for F ~ T ; we call ' = (' ; :::; ' n ) the morphsm. A partcularly useful type space s a "unversal type space" that we descrbe next. Let X 0 =, and de ne X k = X k [ (X k )] I, where (X k ) s the set of probablty measures on the algebra descrbed above, and each X k s gven the product algebra over ts two components. An element ( ; 2 ; :::) 2 k=0 (X k), H s called a herarchy (of belefs). For the topology-free model we descrbe here, Hefetz and Samet (998) prove the exstence of a unversal type space comprsed of a subset of herarches, T H, and a measurable belef functon, : T! T, for all. The type space s unversal n that there s a unque belef-preservng morphsm of any other measurable type space nto ths unversal type space. Spec cally for any herarchy t 2 T,we wrte k (t) for the k th component of t and we wrte Tk for the (measurable) set of kth -level belefs for all types n T ; Tk (X k ). (Where no confuson results we drop the subscrpt of for notatonal smplcty.) Gven any measurable type space type t s margnal belefs about the state of Nature are ^ [t ] () = [t ] (f(t ; ) : t 2 T g). For each k = 2; 3:::, the translaton mples the exstence of measurable b k : T! T k such that for measurable E X k, n b k [t ] (E) = [t ] t 2 T : b (t ) ; ::::; b k o (t ) ; 2 E. Let b (t ) = b k (t ), and then b : T! T s the morphsm ' dscussed above. k= The connecton between ths and the topologcal constructon of the unversal type space (Mertens and Zamr (985); see also Brandenburger and Dekel (993), Hefetz (993), 4

8 Mertens, Sorn and Zamr (994)) s clar ed by the followng lemma, whch s due to Avad Hefetz, and by ts mmedate corollary. The lemma states that the -algebra de ned by () s the Borel algebra for the weak topology. Hence we also use (X) to denote the Borel algebra when t s equvalent. Lemma For a topologcal space X, the -algebra on (X) generated by () when X s endowed wth the Borel algebra, s the same as the Borel algebra of (X) when (X) s endowed wth the weak topology. Proof. For a collecton C of sets n X let C be the -algebra on (X) generated by ff : (E) pg : p 2 [0; ] and E n Cg, and for any topologcal space X let B X denote the Borel algebra on X. The collecton of closed subsets of X; F, s a sub-bass for the weak* topology (Bllngsley (968), App. III), so F = B (X). Lemma 2 n Vglzzo (2005) states that for any -system on X, say G, that generates a -algebra S on X, G = S. Snce F generates B X we have F = (B X ). Combnng these we have B (X) = (B X ) as clamed. Therefore the herarches of belefs constructed above concde wth the herarches that are constructed n Brandenburger and Dekel (993) for any set that s Polsh (a fortor nte). Snce the unversal type space s unque (Hefetz and Samet (998, Proposton 3.5)), the topologcal constructon of the unversal type space and the measure theoretc constructon result n the same unversal type space (up to belef-preservng morphsms). A smlar concluson was obtaned ndependently by Ely and Pesk (2005). Corollary For nte there s a belef-preservng somorphsm between the measuretheoretc unversal type space constructed n Hefetz and Samet (998) and the earler topologcal constructons referred to above. 2. Examples of Redundant Types A player s type captures everythng about hs belefs and hgher order belefs about. However, type spaces also contan types that cannot be dstngushed on the bass of ther belefs and hgher order belefs about. Whle Mertens and Zamr (985) labelled these redundant types, they may nonetheless be strategcally relevant. Ths ssue, and ts sgn cance for the nterpretaton of the unversal type space, has been dscussed by 5

9 Bergemann and Morrs (200) and Battgall and Snscalch (2003). In partcular, t s not true that b (t 0 ) = b (t ) ) (t 0 ) = (t ). As noted by the above authors, ths s most easy to see n the case of complete nformaton, where s a sngleton. Example Let I = 2, T = ft ; t 0 ; t 00 g, T 2 = ft 2 ; t 0 2; t 00 2g and = fg. generated by the common pror below. Call ths type space T. Let belefs be t 2 t 0 2 t 00 2 t 6 t t Now (for each ) b (t 00 ) = b (t 0 ) = b (t ) but (t 0 ) 6= (t ), (t 00 ) 6= (t ) and (t 00 ) 6= (t 0 ). We wll see that n de nng ratonalzablty, ths partcular type of redundancy s relatvely easy to deal wth. But the redundancy n the followng type space turns out to be trcker. Example 2 Let I = 2, T = ft ; t 0 ; t 00 g, T 2 = ft 2 ; t 0 2; t 00 2g and = f; 0 g. Let belefs be generated by a common pror below. Call ths type space T 2. t 2 t 0 2 t 00 2 t 2 t t t 2 t 0 2 t 00 2 t 24 t t Agan (for each ) b (t 00 ) = b (t 0 ) = b (t ) but (t 0 ) 6= (t ), (t 00 ) 6= (t ) and (t 00 ) 6= (t 0 ). 3 Games and soluton concepts Each player has a nte set of possble actons A. A game g conssts of, for each player, a payo functon g, where g : A! [0; ]. Wrte G for the set of possble games. The 6

10 soluton concepts we study are appled to a par (g; T ), and specfy possble acton pro les for such a game of ncomplete nformaton, where an acton pro le s an element of the set. A T Our man soluton concept s " (correlated) nterm ratonalzablty, where " s a measure of sub-optmzaton. To clarfy the role of correlaton we also provde a de nton of ndependent nterm ratonalzablty. We also de ne other soluton concepts and equvalences among them n a manner that s analogous to what s known for the case of complete nformaton (.e., beng a sngleton). Throughout we hold the game g and the number " 0 xed; hence to smplfy notaton and termnology we do not explctly wrte that varous functons depend on these parameters, e.g. the phrase "best reply" wll mean a reply that gves wthn " of the maxmum payo. The man queston s whether the soluton depends on the type space, so we do specfy the dependence on T. 3. Best reples and undomnated actons For any subset of actons for all types, we rst de ne the best reples when belefs over opponents strateges are restrcted to those actons. For any measurable strategy pro le of the opponents, : T! (A ), and any belef over opponents types and the state of Nature, (t ) 2 (T ), denote the nduced belef over the space of types, Nature and actons by ( (t ) ; ) 2 (T A ), where for measurable F T ; ( (t ) ; ) (F f; a g) = R F (t ; ) [a ] (t ) [dt ; ]. De nton The correspondence of best reples for all types gven a subset of actons for all types s denoted BR T : 2 A T! 2 A T and s de ned as follows. Frst, 2I gven a spec caton of a subset of actons for each possble type of opponent, denoted by E = E tj t j 2T j j6=, wth E tj A j for all t j and j 6=, we de ne the " best reples for t n game g as measurable : T! (A ) such that >< () (t ; ) [a ] > 0 ) a j 2 E tj for all j 6= and t j 2 T BR T " # j >= (t ; E ) = a 2 A R g (a ; a ; ) () d ( (t ) ; ) " (t ;;a ) g (a 0 ; a ; ) >: >; for all a 0 2 A (2) 7 2I

11 Next, gven E = (E t ) t 2T 2I AT, we de ne 2I BRT (E) = BR T (t ; E ) t 2T. 5 2I Remark Because A s nte, and utlty depends only on actons and belefs, the set of best responses gven some E, BR T (t ; E ), s non-empty provded there exsts at least one measurable that sats es (). Such exst whenever E s non-empty and measurable, and more generally whenever E admts a measurable selecton. Remark 2 Below we use the fact that whether condton () s sats ed depends only on marg A ( (t ) ; ) 2 (A ). Remark 3 Instead of usng belefs over the opponent s behavoral strategy,, one can work wth dstrbutonal strateges. 6 Spec cally, the de nton above s equvalent to the followng Tj A j such that >< ( 0 ) (t j ; ; a j ) : a j 2 Rj k (t j ; G; ") = >= BR (t ; E j ) = a 2 A ( 0 ) marg T = (t ) " # >: ( 0 R g (a ; a j ; ) ) d " for all a 0 g (a 0 2 A >; ; a j ; ) (t j ;;a j ) Ths follows n one drecton by settng = ( (t ) ; j ), and n the other by settng j = jtj. The latter only need satsfy () of 2 a.e., but, as we argue n the proof of lemma 2 below, j can then be mod ed to satsfy () everywhere. To de ne the ndependent nterm best reples we requre n addton that 8j 6= ; 9 j : T j! (A j ) s.t. = j6= j. We denote ths functon and the resultng xed pont by addng the pre x I, thus the best reply correspondence s denoted IBR T. the added requrement embodes two forms of ndependence: belefs over opponents jont actons are the product of the belefs over each opponent, and opponents actons are ndependent of condtonal on ther type. 5 We abuse notaton and wrte BR both for the corresondence specfyng best reples for a type and for the corresondence specfyng these actons for all types. 6 Mlgrom and Weber (985) proposed the noton of dstrbutonal strateges for studyng equlbra n games of ncomplete nformaton; our noton d ers n that they consdered dstrbutons for each over A T whle we consder dstrbutons over A T that allow correlaton across players and the state of nature. 8

12 It s convenent to consder belefs on the nte set A. Gven E = wth non-empty E tj A tj for all t j and j 6=, let 8 >< (t ; E ) = >: 2 (A ) : = R T ( (t ) ; ) [dt ; ; a ] E tj for some measurable : T! (A ) s.t. (t ; ) [a ] > 0 ) a j 2 E tj t j 2T j j6=, for all j 6= and t j 2 T j The followng techncal lemma s useful for establshng that the set of best responses s measurable, and for relatng the set of best responses to the set of undomnated strateges. Note that f E tj = E t 0 j whenever ^ j (t j ) = ^ j t 0 j, the correspondence E depends only on the herarchy of belefs and hence can be translated nto the unversal type space. Spec cally, for any type space we can use the belef-preservng morphsm of that type space nto the unversal type space as de ned n Secton 2, namely ' =, to de ne Et = E j ' (t j). So when E sats es ths condton, we denote by E ths translaton. Lemma 2 If E tj = E t 0 j whenever ^ j (t j ) = ^ j t 0 j and E vewed as a subset of T A s closed (accordng to the product of the weak topologes for T ) then s compact and convex. Proof. Consder n!. There exsts n as n the de nton above. Let n 2 (T A) be de ned by n (t ) ; n (t ; ). De ne n 2 T A to be the mage of v n, accordng to the belef-preservng morphsm, n the unversal type space. [Snce A s nte ths can be de ned usng the mappng ' ^ from T to T. For F T let n F fg fa g = n ' F fg fa g.] As argued earler, the -algebra on T s the Borel algebra generated by the weak topology, and as shown n Mertens and Zamr (985), ths s compact when s nte. Therefore there s a subsequence of n that converges to 2 T A. Moreover, by compactness we also have regular condtonal probabltes, jt 2 (A ), whch, by regularty are measurable as a functon of T. Hence we can de ne a measurable : T! (A ). Snce, as noted above, we have a (belef-preservng) measurable map ' from T nto T, ths nduces a measurable strategy : T! (A ). 7 7 Spec cally, (t ; ) = (' (t ) ; ), where ' s the belef-preservng functon from T to T. We need t : (t ; ) =, for each 2 (due to nteness we can treat each separately) and each 2 (A ), to be measurable. By de nton t 2 T : (t ; ) = = t 2 T : (' (t ) ; ) = = ft 2 T : ' (t ) 2 S (; )g = ' (S (; )), where S (; ) = 9 >=. >; 9

13 Next we clam that for all t, (t ) ; ( E) =. As noted, snce Etj = E t 0 j whenever ^ j (t j ) = ^ j tj 0 we de ne E t = E j ' (t j). Snce n ( E) =, also n ( E ) =, so ( E ) =. Therefore ^ (t ) ; ( E ) =, and hence, by the de nton of E, (t ) ; ( E) =. Fnally, we clam we can strengthen ths concluson to (t ; ) [a ] > 0 ) a tj 2 E tj for all j 6= and t j 2 T j. The support condton obtaned n the precedng paragraph can be wrtten as marg T (N) = 0, where N (t ; ) : supp (t ; ) 6 E t changng on N has no e ect on any expect payo s or other calculatons, and can be done so long as measurablty of contnues to be sats ed. Fx 2 for the remander of the argument. For each (of the ntely many) B A, let B E t : E t = B B t : supp (t ; ) B. Both sets are measurable, hence B E. So and B s measurable, and snce (N) = 0 also B E B = 0. So rede ne (t ; ) on B E B to equal any a 2 B. Snce fa g s measurable, so s fa g [ B E B, so after ths rede nton s stll measurable and B E B s empty. Dong ths process for all B A we obtan a measurable s.t. (t ; ) 2 E t for every (not only a.e.) t. Fudenberg and Trole (99, page 226) demonstrate the mportant dstncton between nterm and ex ante (strctly) domnated strateges. As one would expect, t s the nterm verson that s related to nterm ratonalzablty. Spec cally, gven E = E tj t j 2T j j6=, wth non-empty E tj A tj for all t j and j 6=, the " nterm undomnated actons for t n game g are U T (t ; E ) de ned below. 8 >< U T (t ; E ) = a 2 A >: There does not exst 2 (A ) such that for all measurable : T! A wth (t ; ) 2 E tj for all j 6= and t j 2 T j ; " # R g ( ; a ; ) d (^ (t ) ; ) > ". g (a ; a ; ) (t ;;a ) 9 >= >; Usng the de nton of ; ths s equvalent to t 2 T : t ; =. Snce s measurable, we have that S (; a) s a measurable set, and snce ' s measurable we have that ' (S (; )) s a measurable set as clamed. 0

14 8 9 There does not exst 2 (A ) such that for all >< 2 (t ; E ) U T " # >= (t ; E ) = a 2 A R g ( ; a ; ) d > ". (;a ) g (a ; a ; ) >: >; The resultng correspondence, U T : to BR T. We beleve that s s equvalent to BR T able to prove the followng more lmted result. 2 A T! 2 A T s de ned analogously 2I 2I n general, but so far we have only been Lemma 3 If for all j E tj = E t 0 j whenever ^ j (t j ) = ^ j tj 0, and the correspondng E T A s closed, then U T (; E) = BR T (; E). Ths s true n partcular for E =. Proof. >From lemma 2, (t ; E ) s a compact convex nte dmensonal set, so we can apply the mnmax theorem. A T 3.2 Iteratve de ntons Let R0 T = A T ; R T k = BR T Rk T, and R T = \ k= RT k. Let R 0 = (A ) T Rk = BR T Rk, and R = \ k= R k : Let U 0 T = A T ;U T k = U T Uk T, and U T = \ k= U k T. T Let ;k (t ) = (t ; Rk; T ); and ;k (t ) = (t ; Rk; ). Lemma 4 R;k (t ) = R;k (t0 ) f ^ k (t ) = b k (t 0 ). That s, types n the unversal type space wth the same k th -order belefs have the same k th -order ratonalzable sets. Moreover, U k = R k. Proof. Because all types n ths proof le n the unversal type space, we drop the from T and t. Let ^ k (t ) be the k th -order belef of t, an element of (X k ). Let T ;k (t ) T be the subset of types wth the same k th -order belefs as t (T ;k (t ) ~t : ^ k ~t = ^ k (t ) ). Recall that R 0 = (A ) T, R ;k (t ) = BR (t ; R ;k ) and R k = (R ;k ) 2I. The proof s by nducton. The clam s obvous for k = 0. Assume t s true for k. Then we need to show that the two types have the same ratonalzable sets, and that

15 R ;k+ = U ;k+ : The latter clam follows from the fact that the kth -order sets depend only on k th -order belefs and the prevous lemma. For the rst clam we need to show 8 9 measurable : T! (A ) such that >< () (t ; ) [a ] > 0 ) a tj 2 R j;k (t j ) for all j 6= and t j 2 T " # j a 2 A R g (a ; a ; ) () d ( (t ) ; ) " (t ;;a ) g (a 0 ; a ; ) >: for all a 0 2 A 8 9 measurable : T! (A ) such that >< () (t ; ) [a ] > 0 ) a tj 2 R j;k (t j ) for all j 6= and t j 2 T " # j = a 2 A R g (a ; a ; ) () d ( (t 0 ) ; ) " (t ;;a ) g (a 0 ; a ; ) >: for all a 0 2 A 9 >= >; 9 (3) >= :(4) PESKI: YOU USE SOMEWHERE BEFORE THAT REMARK 3 ABOUT DISTRIBU- TIONAL STRATEGIES, IT COULD BE MENTIONED. ED: I don t thnk we need to do anythng By hypothess R j;k (t j ) = R j;k t 0 j f tj 2 T j;k t 0 j. That s, Rj;k s constant for all opponents types that have the same k th -level belefs. Assume t 0 2 T ;k+ (t ).We clam that f a s n R ;k (t ), the set de ned n equaton (3), due to the exstence of satsfyng () and () n (3), then there exsts satsfyng () and () n equaton (4) s.t. n R ;k (t 0 ), the set de ned by (4). Wrte for ( (t ) ; ). For every t and de ne (t ; ) = (jt ;k (t ) fg). (Intutvely, s the average of on types of players other than who have the same k th -level belefs.) >; a s As n the proof of lemma 2 we have that condton () holds a.e., and can be extended to hold everywhere. For condton () we wll show that for each a and, R T A g (a ; a ; )d ( (t ) ; R T A g (a ; a ; )d ( (t 0 ) ; ). For xed a and we can gnore g () and for notatonal smplcty drop a and. We need to show that R T ( (t ) ; (t )) [dt ] = R T ( (t 0 ) ; (t )) [dt ]. That s, R T (t ) (t ) [dt ] = R T (t ) (t 0 ) [dt ]. 2

16 We have = = = Z (t ) (t ) [dt ] T Z (t ) (t ) [dt ] T Z (t ) ^ k+ (t ) [dt ] T Z (t ) ^ k+ (t 0 ) [dt ] T where the rst equalty follows from from the de nton of, the second from the fact that depends only on k th order belefs of t (hence the ntegral depends only on the (k + )th order belefs of t ), and the thrd from the fact that t and t 0 have the same k + th -level belefs. Proposton For every k, and all, R;k T : T! 2 A =; s a measurable functon, wth R;k T (t ) = R;k (^ T (t )) and ;k (t ) = ;k (^ (t )). Moreover, ft : a 2 Rk (^ (t))g s closed. Proof. The proof s by nducton on k. As a part of the proof, we wll also show by nducton that ;k (t ) s a contnuous functon of t n the product topology. As part of the undctve step we also prove that ft : a 2 Rk (^ (t))g s closed. The clams are trvally sats ed for k = 0. Now endow the unversal type space wth the product topology, where each level of the belefs s gven the weak topology (as n the usual topologcal constructon of the unversal type space), and suppose the clam has been shown for all k 0 < k: By nductve hypothess, T ;k (t ) = ;k (^ (t )), and moreover ths set depends only on ^ k (t ). Thus, R;k T (t ) = R;k (^ (t )) and ths set depends only on ^ k (t ). Moreover snce Rj;k T (t T j) s measurable, the set ;k (t ) s nonempty, so R;k T (t ) s non-empty as well. We clam next that the set t : a 2 R T ;k (t ) s closed and therefore (from lemma ) measurable. To see ths, consder a sequence t n that converges to t and such that a 2 R;k T (tn ) : Then for each t n there s a k ;n such that a s a best response. And snce ;k s compact (due to RT j;k beng closed) and contnuous n t ; and utlty u (a ; ; t ) = 3

17 R u (a ; a ; )d (a ) s contnuous n t, we can conclude that there s a 2 ;k (t ) such that a s a best response for t to. Usng the measurable map ^ from T nto T ; we conclude that t : a 2 R;k (^ (t )) s measurable, and thus so s the set t : a 2 R;k T (t )). 3.3 Fxed-pont de ntons Let S T : T! 2 A? be a spec caton of possble actons for each type, and S T = S T ; :::; SI T. De nton 2 S T s a best-reply set f for each t and a 2 S T (t ), there exsts a measurable : T! (A ) such that () (t ; ) [a ] > 0 ) a 2 S T (t ) " # R g (a ; a j ; ) () d ( (t ) ; ) g (a 0 ; a j ; ) (t ;;a ) " for all a 0 2 A The followng propertes are mmedate from the de ntons. Lemma 5. If S T c for all c n some ndex set C are best-reply sets then [ c S T c s a best-reply set. 2. The unon of all best-reply sets s a best reply set. It s also the largest xed pont of BR T. Property follows from a smlar argument to the one gven above regardng xed ponts. To see property 2 denote the unon of all best-reply sets as S and observe that f a 2 BR T (t ; S ), then addng a to S (t ) wll contnue to consttute a best-reply set. De nton 3 R T = R T (t ) t 2T A T s the largest xed pont of 2I 2I BRT. We can smlarly de ne the equvalent xed pont of undomnated actons, U T = R T. The pro le of ndependent nterm ratonalzable actons s analogously the largest xed pont of the decreasng functon IBR, and s denoted IR T = (t ) t 2T. In general, the largest xed pont need not be concde wth the teratve de nton gven above, as reducng the set to the largest xed pont may requre trans nte nducton; see 4 IR T 2I

18 Lpman (994). However, because payo s depend only on dstrbutons over the nte sets of actons and states of nature, we can show that the xed pont de nton s well posed and concdes wth the teratve de nton. Proposton 2 R T equals R T. Proof. It s su cent to prove that R T s a best-reply set. That nothng larger can be a best-reply set s mmedate. For every a 2 R T we have that for every k there s a conjecture k 2 R T ;k aganst whch a s a best reply. Snce all the R;k T are closed, and u s contnuous, we have that a s best reply aganst lm k 2 R ;. Corollary 2 Gven two type spaces, T and T 0, on the set of states of Nature, wth t a type of n T and t 0 a type of n T 0, we have b (t 0 ) = b (t ) ) R T 0 (t 0 ) = R T (t ) : 3.4 Examples In example, we clearly have IR T (t ) = IR T (t 0 ) = IR T (t 00 ) and R T (t ) = R T (t 0 ) = R T (t 00 ) for any two-player game g. In partcular, these sets wll be the "-ratonalzable actons of the underlyng complete-nformaton game and the result s an mplcaton of the equvalence of correlated and ndependent ratonalzablty n two-player complete-nformaton games. But n the type space of example 2, thngs wont be so smple, even n two-player games. Recall that ths type space s generated by the followng common pror: t 2 t 0 2 t 00 2 t 2 t t t 2 t 0 2 t 00 2 t 24 t t Agan (for each ) b (t 00 ) = b (t 0 ) = b (t ) but b (t 0 ) 6= b (t ), b (t 00 ) 6= b (t ) and b (t 00 ) 6= b (t 0 ) Consder the followng game g, where player chooses the row and player 2 chooses the column. 5

19 L R u ; 0 0; 0 d 3 5 ; ; 0 0 L R u 0; 0 ; 0 d 3 5 ; ; 0 Note that all strateges are both ratonalzable and ndependent ratonalzable for player 2. Let " = 0. For all types of player, d s a best response to the player 2 strategy "always L," whle u s a best response for both t and t 0 to the player-2 strategy "L f t 2 ; R f t 0 2: However, u s domnated for t " 2, so IR T 2 (t ) = IR T 2 (t 0 ) = fu; dg, but IR T 2 (t 00 ) = fdg. Next we turn to correlated ratonalzablty. Ths cannot reduce the ratonalzable sets of any type, so once agan R T 2 (t ) = R T 2 (t 0 ) = fu; dg. Moreover, R T 2 (t 00 ) = fu; dg as well, because u s a best response to the dstrbuton placng probablty =2 on (t 00 2; ; L) and =2 on (t 00 2; 0 ; R). Ths example has the same avor as examples showng the non-equvalence of correlated and nterm ratonalzablty n three-player complete-nformaton games. For example, consder the three-player game where player chooses the row, player 2 chooses the column and player 3 chooses the matrx, wth payo s A L R u ; 0; 0 0; 0; 0 d 3 4 ; 0; ; 0; 0 B L R u 0; 0; 0 ; 0; 0 d 3 4 ; 0; ; 0; 0 Here, acton d fals to be ndependently ratonalzable for player but s correlated ratonalzable. In an n uental argument, Aumann (987) wrtes n ths context that...n games wth more than two players, correlaton may express the fact that what 3, say, thnks that wll do may depend on what he thnks 2 wll do. Ths s no connecton wth any overt or even covert colluson between and 2; they may be actng entrely ndependently...(page 62) We propose treatng nature as another player. If player, say, does not know what determnes whch of hs ratonalzable actons player 2 wll play, why should ths subjectve uncertanty be completely ndependent of the uncertanty about the choce of nature? Ths nterpretaton ntroduces the possblty that there are other (payo rrelevant) states of the world that are not modelled n but that lead to these belefs. We explctly explot such an 6

20 expanson of the space n the next secton to prove the equvalence of nterm ratonalzablty wth more famlar soluton concepts. 4 Interm Ratonalzablty, Equlbrum on Large Type Spaces and Common Knowledge of Ratonalty One message from Brandenburger and Dekel (987a) was that equlbrum has no bte when there are large type spaces and the common pror assumpton s dropped. We can state the analogous result for ths ncomplete nformaton settng. Spec cally, we prove that gven any type space and game, any nterm ratonalzable acton s also played n an equlbrum of that same game but wth an expanded type space. Brandenburger and Dekel prove that any ratonalzable acton of a complete nformaton game s played n some subjectve correlated equlbrum, whch s just an equlbrum of a game wth an expanded type space that functons as a subjectve correlatng devce. Our constructon below s very smlar, we expand the type spaces by addng to each player s type a sgnal that corresponds to a recommended acton. Fx type space T. We wll consder an enlarged type space ( T e I = ~T ; ~ = ) whch can be translated nto T (wth translaton ' : e T! T ). Gven a g and the type space e T, we have an ncomplete nformaton game. A strategy pro le s = (s ; :::; s I ), each s : T e! A, measurable, s a pure strategy "-nterm equlbrum of the game g; T e f and only f Z g s et ; s et ; de et et ; Z g a ; s et ; de et " et ; for all, et 2 e T and a 2 A. Proposton 3 a 2 R T (t ) f and only f there exsts an enlarged type space T e and an "-nterm equlbrum of the game g; T e, such that s et = a and ' et = t for some et 2 e T. 7

21 Proof. For each a 2 R T (t ) by lemma 5. there exsts a ;t : T! (A ) such that and a ;t Z (t ;;a ) (t ; ) [a ] > 0 ) a j 2 R T (t j ) for all j 6=, and all t j " g (a ; a ; ) g (a 0 ; a ; ) Now consder the followng enlarged type space wth # d (t ) ; a ;t " for all a 0 2 A. (5) Consder the strategy pro le wth By constructon, f et = (t ; a ), Z ~T = T A e ((t ; a )) = (t ) ; a ;t ' ((t ; a )) = t. s ((t ; a )) = a. g s et ; s et ; de et = = et ; Z et ; Z et ; Z et ; g s et ; s et ; d (t ) ; a ;t g a 0 ; s et ; d (t ) ; a ;t g a 0 ; s et ; de et " " for all a 0. Conversely, suppose that there exsts an enlarged type space T e and an "-nterm equlbrum of the game g; T e, s. Let S (t ) = a : s et = a and ' et = t for some et 2 T. Suppose s et = a and ' et = t for some et 2 T. Snce S sats es the "-best-response property for game g, we have by lemma 5.2 that a 2 S (t ) R T (t ). 8

22 Followng Bernhem (984) and Pearce (984), and subsequently Aumann (987), Brandenburger and Dekel (987a) and Tan and Werlang (988), we now study the sense n whch the set of nterm ratonalzable actons for type t are the set of actons that are consstent wth common knowledge of ratonalty (whle mantanng the mplct assumpton that there s common knowledge of the game g and type t s belefs and hgher order belefs about ). Formally, for a gven type space, T, game g, and measurable strategy pro le s = (s ; :::; s I ), each s : T! A, the event that s ratonal s 8 8 >< >< Z " # [Rat (g)] = >: t : s (t ) 2 >: a g (a ; s (t ) ; ) (t ) (d (t ; )) g (a 0 ; s (t ) ; ) (t ;) 99 >= >= ", 8a 0 >; >;. It follows from propostons 2 and 3 that common knowledge of ratonalty mples that players choose nterm ratonalzable actons. That s, f for all, [Rat (g)] s common knowledge at some t then s (t ) s nterm ratonalzable for t. 8 Proposton 3 also mples a converse. Fx a type space T, a game g, and an nterm ratonalzable pro le of actons a 2 R T (t) for some pro le of types t 2 T. Then there s a type space ~ T, wth a belefpreservng morphsm ' from ~ T nto T, a strategy pro le, s : ~ T! A, and a type pro le ~t 2 ~ T at whch ratonalty s common knowledge, such that s ~t = a and ' ~t = t. Note that the sense n whch we obtan that common knowledge of ratonalty and nterm ratonalzablty concde does depend on the noton of belef-morphsm used on type spaces. Ely and Pesk (2005) use a more demandng noton of equvalence for type spaces and hence end up studyng a more re ned soluton concept, ndependent nterm ratonalzablty. References [] Aumann, R. (987) Correlated Equlbrum as an Expresson of Bayesan Ratonalty, Econometrca 55, -8. [2] Battgall, P. and M. Snscalch (2003) Ratonalzaton and Incomplete Informaton, Advances n Theoretcal Economcs 3, Artcle Usng, e.g., Brandenburger and Dekel s (987b) extenson of Aumann s (976) noton of common knolwedge. 9

23 [3] Bernhem, B. D. (984) Ratonalzable Strategc Behavor, Econometrca 52, [4] Bergemann, D. and S. Morrs (200) Robust Mechansm Desgn, [5] Bllngsley, P. (995) Probablty and Measure, 3rd ed. John Wley and Sons, New York, NY, USA. [6] Brandenburger, A. and E. Dekel (987a) Ratonalzablty and Correlated Equlbra, Econometrca 55, [7] Brandenburger, A. and E. Dekel (987b) Common Knowledge wth Probablty, Journal of Mathematcal Economcs 6, [8] Brandenburger, A. and E. Dekel (993) Herarches of Belefs and Common Knowledge, Journal of Economc Theory 59, [9] Dekel, E., D. Fudenberg and S. Morrs (2005) Topologes on Types. [0] Ely, J. and M. Pesk (2005) Herarches of Belef and Interm Ratonalzablty, [] Forges, F. (993) Fve Legtmate De ntons of Correlated Equlbrum n Games wth Incomplete Informaton, Theory and Decson 35, [2] Fudenberg, D. and J. Trole (99). Game Theory Cambrdge MA: MIT Press. [3] Harsany, J. C. (967-8) Games wth Incomplete Informaton Played by Bayesan Players, parts I, II, and III, Management Scence 4, 59-82, , and [4] Hefetz, A. (993) The Bayesan Formulaton of Incomplete Informaton The Non- Compact Case, The Internatonal Journal of Game Theory 2, [5] Hefetz A., and D. Samet (998) Topology-Free Typology of Belefs, Journal of Economc Theory 82, [6] Lpman, B. (994) A Note on the Implcatons of Common Knowledge of Ratonalty, Games and Economc Behavor 6,

24 [7] Mertens, J.-F., S. Sorn and S. Zamr (994) Repeated Games: Part A Background Materal, CORE Dscusson Paper #9420. [8] Mertens, J.-F. and S. Zamr (985) Formulaton of Bayesan Analyss for Games wth Incomplete Informaton, Internatonal Journal of Game Theory 4, -29. [9] Mlgrom, P. R. and R. J. Weber (985) Dstrbutonal strateges for games wth ncomplete nformaton, Mathematcs of Operatons Research 0, [20] Pearce, D. Ratonalzable Strategc Behavor and the Problem of Perfecton, Econometrca 52, [2] Tan, T. and S. Werlang (988), The Bayesan Foundaton of Soluton Concepts of Games, Journal of Economc Theory 45, [22] Vglzzo, I (2005) A Lemma about the Generators of (X), mmeo. [23] Wensten, J. and M. Yldz (2003). Fnte Order Implcatons of Any Equlbrum, 2

25 THE FOERDER INSTITUTE FOR ECONOMIC RESEARCH and THE SACKLER INSTITUTE FOR ECONOMIC STUDIES The Etan Berglas School of Economcs Tel-Avv Unversty Recent Lst of Workng Papers Elhanan Helpman Gene M. Grossman Adam Szedl Elhanan Helpman Gene M. Grossman Bernhard Eckwert Itzhak Zlcha Cham Fershtman Arel Pakes Tomer Blumkn Yoram Margaloth Efram Sadka Complementartes between Outsourcng and Foregn Sourcng A Protectonst Bas n Majortaran Poltcs Improvement n Informaton: ncome Inequalty and Human Captal Formaton Fnte State Dynamc Games wth Asymmetrc Informaton: A Framework for Appled Work On Vertcally Challenged and Horzontal Equty Antdscrmnaton Rules vs. Income Taxaton Tomer Blumkn Efram Sadka Gene M. Grossman Elhanan Helpman Daron Acemoglu Pol Antras Elhanan Helpman Assaf Razn Prakash Loungan Arel Rubnsten Yuval Salant Enrqueta Aragones Itzhak Glboa Andrew Wess Assaf Razn Yona Rubnshten Efram Sadka Edde Dekel Jeffrey C. Ely Okan Ylankaya Itay Fanmesser Cham Fershtman Nel Gandal A Case for Taxng Educaton Party Dscplne and Pork-Barrel Poltcs Contracts and the Dvson of Labor Globalzaton and Inflaton-Output Tradeoffs A Model of Choce From Lsts Makng Statements and Approval Votng Corporate Taxaton and Blateral FDI wth Threshold Barrers Evoluton of Preferences A Consstent Weghted Rankng Scheme: Wth an Applcaton to NCAA College Football Rankngs Foerder Insttute for Economc Research, Tel-Avv Unversty, Tel-Avv, Israel, Tel: ; fax: ; e-mal: foerder@post.tau.ac.l Papers from 2000 onwards (and some from 998 and999) can be downloaded from our webste as follows:

26 Recent Lst of Workng Papers - contnues Edde Dekel Drew Fudenberg Stephen Morrs Edde Dekel Drew Fudenberg Stephen Morrs Topologes on Type Interm Ratonalzablty Foerder Insttute for Economc Research, Tel-Avv Unversty, Tel-Avv, Israel, Tel: ; fax: ; e-mal: foerder@post.tau.ac.l Papers from 2000 onwards (and some from 998 and999) can be downloaded from our webste as follows: 2