Interim Correlated Rationalizability 1

Transcription

1 Interm Correlated Ratonalzablty Edde Dekel Northwestern Unversty and Tel Avv Unversty Drew Fudenberg Harvard Unversty Stephen Morrs Prnceton Unversty Frst Draft: May Ths Draft: November 2006 Ths paper began as part of our work on "Topologes on Types" but took on a lfe of ts own n We thank semnar partcpants n many audences for helpful comments, and are partcularly grateful to Avad Hefetz and Muhamet Yldz for helpful dscussons, Paolo Battgall, Bart Lpman, Marcn Pesk and the referees for detaled comments, and Satoru Takahash for expert proofreadng. We acknowledge nancal support from the Natonal Scence Foundaton under grants 080, and Morrs s grateful for nancal support from the John Smon Guggenhem Foundaton and the Center for Advanced Studes n the Behavoral Scences.

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

3 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. However, 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 the type spaces are mapped nto the same subset of the unversal type space. In response, ths paper proposes the soluton concept of nterm correlated ratonalzablty. We show that the concept s well-de ned, that ts teratve and xed-pont de ntons concde, and that any two types wth the same herarchy of belefs have the same ntermcorrelated-ratonalzable actons, regardless of whether they resde n the same type space. Thus ths s a soluton concept that can be characterzed by workng wth the unversal type space. 2 We also show that the soluton concept has smlar propertes to ts completenformaton counterpart. Frst, n clam, we note that the process of teratve elmnaton of nterm strctly domnated strateges yelds the same soluton. Second, proposton 2 shows that nterm correlated ratonalzablty s characterzed by common certanty of ratonalty. 3 Thrd, we extend a result of Brandenburger and Dekel (987). They 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; remark 2 reports a straghtforward extenson of Brandenburger and Dekel s observaton to games wth ncomplete nformaton. See Bergemann and Morrs (2005) and Battgall and Snscalch (2003) for a dscusson of ths ssue. 2 We use the concept of nterm correlated ratonalzablty n our study of topologes on the unversal type space (Dekel, Fudenberg and Morrs (2006)). For that purpose, t s mportant to know that the soluton concept depends only on herarches of belefs (and not on other, redundant, elements of the type space), as we establsh here. 3 We follow a recent conventon n the epstemc foundatons of game theory lterature of usng "certanty" to mean "belef wth probablty one." An earler conventon - that we followed n Dekel, Fudenberg and Morrs (2006) and other earler work - uses "knowledge" nstead to refer belef wth probablty one. But the conventon n phlosophy and other closely related dscplnes has been to reserve "knowledge" for true belef. The truth of players belefs does not play a role n our analyss, so n ths paper we swtch to the newer conventon.

4 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. Our focus s on the concept of nterm correlated ratonalzablty, but we also de ne the concept of nterm ndependent ratonalzablty; we use comparsons between the two concepts to help explan and motvate the correlated verson. These two soluton concepts are ncomplete-nformaton analogs of the complete-nformaton concepts of correlated ratonalzablty and ndependent ratonalzablty, and reduce to them when s a sngleton. To understand these concepts, recall that all ratonalzablty notons nvolve teratvely deletng every acton that s not a best reply to some player s belefs, where at each stage of the deleton the belefs are restrcted to assgn postve probablty only to actons that have not yet been deleted. Our de ntons of nterm ratonalzablty teratvely delete actons for all types that are not best reples to some probablty dstrbuton over actons and states that s consstent wth the belefs of each type of each player about and the other players types, and wth the restrctons on conjectures about the opponents actons that were obtaned at earler stages of the teraton. We call such probablty dstrbutons "forecasts." In the case of nterm ndependent ratonalzablty, the allowed forecasts for a player of type t are gven by combnng (ndependent) conjectures of strategy pro les for each opponent s types, wth that type t s belefs over opponents types and over. In the case of nterm correlated ratonalzablty, the allowed forecasts are generated by combnng t s belefs over types and wth any, perhaps correlated, condtonal conjectures about whch (survvng) actons are played at a gven type pro le and payo relevant state. In ths latter de nton, a type s forecast can allow for correlaton among the payo -relevant state, other players types, and other players actons. 4 Much work n ths paper s devoted to establshng that our results hold on general type spaces. Ths generalty s mportant for evaluatng the clam that all types that map to the same pont n the unversal type space have the same set of nterm-correlated-ratonalzable 4 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, nterm ndependent and correlated ratonalzablty may d er even n the two-person case, because of the possble correlaton n a player s forecast of the opponent s actons and the payo -relevant state, condtonal on the opponent s type. 2

5 outcomes, so that nterm correlated ratonalzablty can be analyzed usng the unversal type space. However, workng on general type spaces ntroduces a number of techncal complcatons, startng wth the queston of what sorts of type spaces to consder (we use the non-topologcal de nton of Hefetz and Samet (998)) and proceedng to the queston of whether the set of best responses s measurable, whether trans nte nducton s requred to equate the teratve and xed-pont de ntons of ratonalzablty, and so on. These ssues are mportant for a general analyss, but they shed lttle lght on ether the motvaton for the de nton of nterm correlated ratonalzablty or ts nvarance property. For ths reason we restrct attenton to nte type spaces n the rst part of the paper, whch allows us to gve less techncal de ntons and statements of some of our results. We then proceed to the more general analyss. We now consder an example to llustrate some of these deas. The example llustrates our concluson that ths concept corresponds to common certanty of ratonalty and that t depends only on the types (herarches of belefs) and not on other (redundant) aspects of the type space, and that the latter ndependence s not true for nterm ndependent ratonalzablty. It also emphaszes the form of correlaton allowed by our man concept; a more detaled dscusson of ths correlaton appears n subsecton 3.2. Example (The E ect of Correlaton wth Nature) Consder the followng two-player game wth ncomplete nformaton,. Player chooses the row, player 2 chooses the column and Nature chooses whether payo s are gven by the left hand matrx (n state ) or the rght hand matrx (n state 0 ). L R U ; 0 0; 0 D 3 5 ; ; 0 0 L R U 0; 0 ; 0 D 3 5 ; ; 0 We assume that each player beleves that each state s equally lkely, and that ths s common certanty. 5 Clearly, ether acton s ratonal for player 2; as she s nd erent between all actons. Now suppose that player beleves that wth probablty, the true state wll be 2 and player 2 wll choose L, and wth probablty, the true state wll be 2 0 and player 2 wll 5 Formally, ths means that the event that each player assgns equal probablty to the states s common certanty, as de ned n secton

6 choose R. Ths makes U optmal for player. As we wll see n secton 3.4, ths means that U s consstent wth common certanty of ratonalty. Acton U s also consstent wth nterm correlated ratonalzablty. To llustrate ths we consder two type spaces that capture the same assumptons as above about players hgherorder belefs. In type space T, each player = ; 2 has two possble types, T = ft 0 ; t 00 g and belefs are generated by the followng common pror over T T 2 f; 0 g: t 0 2 t 00 2 t 0 t t 0 2 t 00 2 t 0 t In ^T each player has one possble type, and the belefs are gven by the followng common pror. ^t 2 ^t 2 0 ^t 2 ^t 2 Notce that n both type spaces, for every type of both players, there s common certanty that each player assgns probablty =2 to the true state beng. The types n T are redundant n the sense of Mertens and Zamr (985): there are multple copes of types that agree wth respect to ther belefs and hgher-order belefs about. But these types nonetheless d er n ther conjectures about ther opponents and ths s potentally mportant dependng on the choce of soluton concept. Redundant types can serve as a correlatng devce, and so these types are not truly redundant unless the addton of correlatng devces has no e ect. To nd the nterm correlated ratonalzable actons of wth the above type spaces we teratvely elmnate actons for each type t of player that are not best responses to some forecast for the player over the trples (t j ; ; a j ) that puts probablty on type acton pars (t j ; a j ) that have not been deleted and that are consstent wth type t s belefs over (t j ; ). In the example, no acton wll be elmnated for any type n ether type space by the argument that we gave above. Now consder the alternatve soluton concept of nterm ndependent ratonalzablty, where we add the addtonal requrement that at each round, for an acton to survve, type t s forecast over (t j ; ; a j ) must treat the choce of player j s acton as ndependent of, condtonal on hs type. Wth ths soluton concept, acton U wll not be nterm ndependent ratonalzable for type ^t : there s no condtonally ndependent forecast over actons, states 4

7 and types that supports play of acton U. Thus D s the only nterm ndependent ratonalzable acton for type ^t. On the other hand, f type t 0 conjectures that type t 0 2 wll play acton L and type t 00 2 wll play acton R, then he wll attach probablty to each of acton-state 3 pro les (L; ) and (R; 0 ). Ths s enough to make acton U a best response. Thus both U and D are nterm ndependent ratonalzable for types t 0 and t 00. The example shows how redundant types n the type space have no mpact on the nterm correlated ratonalzable actons for a gven type, but do matter for nterm ndependent ratonalzablty. Ths result s proved and dscussed n greater generalty n the paper. The example also helps see the ntuton for why our soluton concept depends only on the types, and not the detals of the type space: The concept allows players to have correlated forecasts over other players actons, ther types, and the state, so the ablty of redundant types to support such correlaton s, truly, redundant. In ths sense, the classcal unversal type space of Mertens and Zamr (985) s the rght type space for our correlated verson of nterm ratonalzablty, for whch the only part of a player s type that matters s hs belefs and hgher-order belefs about. There are three papers that study closely related ssues. Battgall and Snscalch (2003) de ne an umbrella noton of -ratonalzable actons n ncomplete-nformaton envronments, where can be vared to capture common-certanty restrctons on players forecasts. They show that there s an equvalence between actons survvng an teratve procedure capturng common certanty of and the set of actons that mght be played n equlbrum on any type space where s common certanty. Correlated nterm ratonalzable actons are exactly -ratonalzable actons, where s set equal to a complete descrpton of the n nte herarches of belefs. Wth ths, our proposton 2 corresponds to ther proposton 4.3. They do not analyze ths partcular and therefore do not address the ssue of the dstncton between correlated and ndependent nterm ratonalzablty. 6 Forges (993) examnes d erent ways of de nng correlated equlbrum for games of ncomplete nformaton. Her unversal Bayesan approach (n secton 6) allows a player s 6 The common certanty restrctons n Battgall and Snscalch (2003) are assumed to restrct only rst-order belefs about, not hgher-order belefs. Thus whle our exercse s conceptually a specal case of Battgall and Snscalch (2003), a slghtly extended class of restrctons would be requred to formally ncorporate t. 5

8 own actons to depend on the payo states even when the player cannot dstngush between the states; ths s analogous to the correlaton n forecasts that we use n de nng our soluton concept (we dscuss ths further n subsecton 3.2). Thus our approach s the non-common pror analogue of Forges unversal Bayesan approach. A recent paper by Ely and Pesk (2006) also notes that the set of nterm ndependent ratonalzable outcomes n two-player games depend on more than just the herarchy of belefs over the payo -relevant states of nature. In response, they provde an extended noton of herarches of belefs for two-player games, and show that nterm ndependent ratonalzablty n two-player games depends on types only va those extended herarches. 2 Setup and Soluton Concepts The prmtves of our model are a nte set of states of Nature, a nte set of players, I, and for each player 2 I a nte set of actons A and a payo functon g, where g : A! [0; ], and A = (A ) 2I. 7 For the rst part of the paper, we restrct attenton to a nte type space T = (T ; ) 2I, where each T s a nte set, and each maps T to the set (T ) of probablty measures on the nte set T. 8 We later relax the assumpton that T s nte (and we do not repeat the restrcton untl then); ts role here s to smplfy ssues regardng measurablty and the choce of a sgma eld. Our vew of ths type space s that t s an exogenously gven part of the model. Ths could be because the type space corresponds to some actual nformaton structure, (but not necessarly one that s a complete descrpton of the world just whatever the modeler vews as the pertnent parts) or s the modeler s (partal) descrpton of the players percepton of the envronment (the players vews of the belefs about belefs...about ). We dscuss ths further n secton 3.4. In our descrpton of the type space, and n the belefs allowed n the soluton concept descrbed next, we do not restrct to common prors. Thus we call player s belef. The man soluton concept that we study s that of nterm correlated ratonalzablty, 7 Henceforth we use analogous notaton,.e., Q = (Q c ) c2c for the ordered collecton of any set of C sets fq c : c 2 Cg. Also, we use the ndex j 6= for fj 2 I : j 6= g and wrte Q for (Q j ) j6=. Elements of these are wrtten as usual as q c 2 Q c, q 2 Q, and q 2 Q. 8 Throughout the paper, every nte set s gven the obvous sgma eld. 6

9 or ICR. 9 As wth correlated ratonalzablty n complete-nformaton games, ICR s de ned by an teratve deleton procedure. At each round, an acton survves for a gven type only f t s a best response to a forecast over T A that () puts postve probablty only on type-acton pars of the opponents that have not yet been deleted; and (2) s consstent wth that type s belefs about T. Formally, we have R;0 T (t ) = A ; 8 9 there exsts 2 (T A ) such that >< () [(t ; ; a )] > 0 ) a 2 Rj;k T (t j) j6= >= R;k+ T (t ) = a 2 A (2) a 2 arg max g ((a 0 ; a ) ; ) [(t ; ; a )] ; a 0 t ;;a >: (3) [(t ; ; a )] = (t )[(t ; )] >; a and R T (t ) = \ k=r T ;k (t ). To better explan ICR we wll compare t to a related soluton concept, nterm ndependent ratonalzablty, or IIR. The latter concept mposes the addtonal restrcton that type t s forecast supportng an acton corresponds to ndependent conjectures,.e., that the conjecture about actons condtonal on type and state s a product measure that does not depend on the state. Let IIR;0 T (t ) = A ; 8 there exsts 2 (T A ) such that () [(t ; ; a )] > 0 ) a 2 IIRj;k T (t j) >< j6= (2) a IIR;k+ T (t ) = a 2 A 2 arg max [(t ; ; a )]g ((a 0 ; a ) ; ) a 0 t ;;a for each j 6= there exsts j : T j! (A j ) such that >: (3) [(t ; ; a )] = (t )[(t ; )] Y j (t j ) [a j ] j6= and IIR T (t ) = \ k=iir T ;k (t ). Thus ICR and IIR can be seen as polar cases wth respect to the amount and knd of correlaton that s allowed. An ntermedate concept, that we menton below but do not de ne formally, could allow for correlaton among players actons but not wth nature, by specfyng : T! (A ) nstead of ( j ) j6= n (2) above. 9 We dscuss the sense n whch these concepts are nterm and how they relate to an ex ante concept n the concludng remarks. 7 9 >= ; >;

10 We want to show that for ICR t s only the players belefs and hgher-order belefs about states of nature ther Mertens-Zamr types that matter. For ths we need to de ne, for each type t n a nte type space T = (T ; ) 2I, that type s belefs and hgher-order belefs about. Let b (t ) [] = (t ) [f(t ; ) jt 2 T g]. For each k = 2; 3; :::, let b k (t ) e j; :::; e k j j6= ; = (t ) (t ; ) b j (t j ) j6= k = = e j j6= k =. Fnally, let b (t ) = b k (t ). k= 3 Propertes of the soluton concept 3. Dependence on types but not on type spaces Proposton If t s a type n a nte type space T, t 0 s a type n nte type space T 0 and b (t ) = b (t 0 ), then R T (t ) = R T 0 (t 0 ). Proof. We establsh by nducton for each k that f b k (t ) = b k (t 0 ) then R;k T (t ) = R T 0 ;k (t0 ). Suppose that ths holds for k, that b (t ) = b (t 0 ) and that a 2 R;k T (t ). Thus there exsts 2 (T A ) such that () [(t ; ; a )] > 0 ) a 2 Rj;k T (t j) j6= (2) a 2 arg max g ((a 0 ; a ) ; ) [(t ; ; a )] a 0 t ;;a (3) a [(t ; ; a )] = (t ) [(t ; )] We now construct a 0 2 T 0 A such that the above three condtons hold when 0 and (t 0 ) replace and (t ), respectvely. Let D k = b k j (t ) t 2 T. j6= For e k 2 D k, let e k ; = [(t ; ; a )]. Then for e k ; n o (t ;a ):(b k j (t j )) =e k j6= 8

11 such that e k and for all other ; > 0, set e k e k ; [a ] = ; set [(t ; ; a )] n o t :(b k j (t j )) =e k j6= e k ; e k ( =# R T j;k (t j ) j6= ; [a ] = 0 otherwse f a 2 R T j;k (t j) j6= Next, let where b k 0 t 0 ; ; a = (t 0 ) t 0 ; b k t 0 2 D k snce b (t ) = b (t 0 ). Then e k ; = n o (t ;a ):(b k j (t j )) =e k j6= = (t ) = (t 0 ) b k j (t j ) t : t : b k j (t j ) t 0 ; [a ], [(t ; ; a )] j6= j6= = e k fg = e k fg. Hence we obtan the followng. t 0 [(t ; ; a )] = = = (t 0 ) [(t ; )] t e k 2D k e k 2D k b k (t ) ; [a ] (t 0 ) t : b k j (t j ) = e k fg j6= e k ; n o t :(b k j (t j )) =e k j6= e k [(t ; ; a )] ; b k (t ) ; [a ] = t [(t ; ; a )]. () So and 0 have the same margnal dstrbutons on A. 9

12 Now we clam () 0 [(t ; ; a )] > 0 ) a 2 Rj;k T (t j) j6= (2) a 2 arg max g ((a 0 ; a ) ; ) 0 [(t ; a ; )] a 0 t ;;a (3) a 0 [(t ; ; a )] = (t 0 ) [(t ; )] () s true by the nductve hypothess and the constructon, (2) because and 0 have the same margnal dstrbutons on A, and (3) by constructon. So a 2 R T ;k (t0 ). The ntuton for the proposton s as follows. The rst-level ratonalzable actons R are those that are best responses to arbtrary conjectures about the opponents; because conjectures allow correlaton wth the forecast then depends only on rst-order belefs ^ about. Second-level ratonalzablty depends on belefs about and the opponents rstlevel ratonalzable sets; these n turn depend only on the opponent s rst-order belefs, so second-level ratonalzablty s determned by second-order belefs, ^ 2, and so on. In the course of the proof we demonstrated the corollary below, whch gves an equvalent de nton of R T ;k+ (t ). It states that t s forecast can be decomposed nto conjectures about opponents strateges, and the belefs. Corollary 8 9 there exsts 2 (T A ) such that () [(t ; ; a )] > 0 ) a 2 Rj;k T (t j) j6= >< (2) a 2 arg max g ((a 0 ; a ) ; ) [(t ; ; a )] >= R;k+ T a 0 t (t ) = a 2 A ;;a (3) there exsts ; : T! (A ) such that [(t ; ; a )] = (t )[(t ; )] (t ; ) [a ] >: [(t ; ; a )] = (t ) [(t ; )] >; a Note that s conjecture about other players actons,, allows for j s acton to be correlated wth other players actons, the state, and other player s types. Contrastng ths wth the de nton of IIR makes clear where ICR allows addtonal correlaton. 3.2 Dscusson The correlaton allowed by ICR can have surprsng consequences, as n the next example. 0

13 Example 2 There are two states of Nature and 0 ; and t s common certanty that each player assgns probablty =2 to each state. Thus n the unversal type space each player = ; 2 has a sngle type t : Each player decdes whether to bet (acton B) or not (acton N). If both players chose B, they transfer 3 or 3 from one to the other dependng on the state, and choosng B ncurs a cost of regardless of the opponent s acton. Ths generates the followng payo functons: B N B 2; 4 ; 0 N 0; 0; 0 0 B N B 4; 2 ; 0 N 0; 0; 0 In ths game, t s ICR for each player to choose B: To see ths, note that N s a best response to the forecast that assgns probabltes =2 to (t 2; ; N) and =2 to (t 2; 0 ; N) whch mples that the opponent always plays N), that B s a best reply for player to the forecast =2 on (t 2; ; B) and =2 on (t 2; 0 ; N), and symmetrcally B s a best reply for player 2 to the forecast =2 on (t ; ; N) and =2 on (t 2; 0 ; B). Thus the ICR set for each player s fb; Ng. Note that the forecast that supports B for player supposes that the opposng player j bets exactly when ths s good for and bad for j. Thus each player expects there to be costly speculatve trade (and ndeed usng the epstemc set-up of the next secton, there s common certanty of trade wth probablty bounded above zero) even though there s a common pror. Ths possblty reles on each player belevng n correlaton between the other player s actons and the state. 0 To justfy the orgnal de nton of ndependent ratonalzablty n Bernhem (984) and Pearce (984), t s necessary to add addtonal condtonal ndependence assumptons. 0 Although there s a common pror over, ths observaton s not nconsstent wth no-trade theorems because there s not common certanty of the condtonal probablty of trade n each state. Ths lack of common certanty s possble because ICR allows belefs about strategc behavor that are not consstent wth a common pror, and ths, as n complete-nformaton games, allows each player to thnk that he s outguessng the other. Note that f we set the payo to choosng acton B when the opponent chooses N to be 4 nstead of, then acton B s no longer ratonalzable for any common-pror type, although t remans ratonalzable for some non-common-pror types. Thus common-pror and non-common-pror types can be dstngushed n some no-trade games. Dekel, Fudenberg and Morrs (2006) use ths observaton to show that nte common-pror types are not dense n the unversal type space n the strategc topology that they de ne.

14 The queston of whether or not to mpose the assumptons parallels an older debate n the complete-nformaton envronment. Brandenburger and Dekel (987) showed that correlated ratonalzablty (allowng players to have correlated conjectures over others actons) corresponds to common certanty of ratonalty. To nterpret ths correlaton, t s mportant to remember that a player s conjectures represent hs subjectve belefs about the dstrbuton of play; any correlatons n these belefs need not correspond to objectve correlaton that would be seen by an outsde observer. The correlatons we consder n ths paper should be nterpreted n the same way. There has been ncreasng acceptance of usng the correlated verson of ratonalzablty, n part based on the n uental argument of Aumann (987):...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 has no connecton wth any overt or even covert colluson between and 2; they may be actng entrely ndependently... Interm correlated ratonalzablty extends ths vew, by 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? One mght argue that any correlaton about players or about Nature should be made explct. We take the opposng, small-worlds, vew that such correlaton may not be an nherent part of the nteracton beng studed, and hence s best ncorporated nto the soluton concept and not the model. One mght also argue n favor of a hybrd soluton concept n between ICR and IIR that allows arbtrary correlaton n conjectures about other players but nssts that the correlaton wth Nature s explctly captured n the type space. 2 A d culty wth such hybrd notons s that the resultng soluton concept wll be senstve to the addton of a Of course, f one wants to explctly model and study the e ect of d erent forms of correlaton one would need to use a d erent soluton concept (such as IIR) that does not mplctly allow all such correlaton. 2 Ely and Pesk (2006) study a de nton of nterm ratonalzablty n two-player ncomplete-nformaton games that s equvalent to our de nton of nterm ndependent ratonalzablty (n two-player games). They suggest that, n many-player games, one mght want to examne hybrd notons of nterm ratonalzablty such as the one we crtcze here. 2

15 dummy player who any sngle other player beleves s omnscent. That s, the exstence of a player k whom thnks knows more about the state of Nature than does j, enables to beleve j s actons are correlated wth Nature va k. Hence, f one allows for correlaton wth opponents but not wth Nature then games must completely specfy all agents, even f ther actons are not payo relevant. Example 3 (Example 2 contnued) In the precedng bettng game the only IIR acton s N. Now add a thrd player to the game who chooses an acton a 3 2 A 3 = fb; Ng. The payo s to players and 2 are unchanged, and una ected by player 3 s choce, whle player 3 payo s constant. Player 3 has two possble types t 3 and t 0 3 who know whether the state s or 0. IIR requres ndependence across opponents and nature, and hence ths has no e ect. However, f one allows arbtrary correlatons n forecasts about players actons but not wth Nature, then the resultng nterm hybrd ratonalzablty soluton concept (whch we have not formally de ned) would allow for B (as well as N), as player could beleve that player 2 s play s correlated wth player 3 s, and that player 3 s play s correlated wth. 3 A recent paper of Brandenburger and Fredenberg (2006) suggests a soluton concept ntermedate between correlated ratonalzablty and ndependent ratonalzablty n completenformaton games. They requre that players hold condtonally ndependent conjectures about ther opponents play, contngent on ther belefs and hgher-order belefs about players actons, and also that a player s conjecture about another player s actons does not change f he learns a thrd player s belefs and hgher-order belefs about players actons. They show that most, but not all, correlated-ratonalzable actons satsfy common certanty of ratonalty and these restrctons. Intutvely, hgher-order uncertanty about players actons ntroduces ntrnsc correlaton nto the game envronment. One could presumably extend ther soluton concept to ncomplete nformaton settngs to obtan yet another soluton concept ntermedate between nterm correlated ratonalzablty and nterm ndependent ratonalzablty. 3 The concluson about the e ect of dummy players also holds n a model where player 3 has a thrd possble type t 00 3 and player 2 s certan that player 3 s t 00 3: what s mportant s only that player s certan that player 3 s certan of. Unlke the example n the text, ths verson does not reproduce the entre set of ICR actons. 3

16 3.3 Equvalent formulatons We provde some obvous equvalent de ntons that further llustrate the analoges to the complete-nformaton envronment Iterated Undomnance As one mght expect from earler work, teratvely deletng strateges that are not nterm best reples s equvalent to terated deleton of strctly nterm domnated strateges (where belefs n both are allowed to be correlated). Let U;0 T (t ) = A ; 8 9 there does not exst 2 (A ) such that g ((a ; a ) ; ) [(t ; ; a )] < t ;;a >< (a 0 U;k+ T ) g ((a 0 ; a ) ; ) [(t ; ; a )] >= (t ) = a 2 A a t ;;a ; for all 2 (T A ) such that () [(t ; ; a )] > 0 ) a 2 Uj;k T (t j) j6= >: (2) [(t ; ; a )] = (t ) [(t ; )] >; a and U T (t ) = \ k=u T ;k (t ). Clam R T (t ) = U T (t ) Best-reply sets Smlarly, there s an obvously equvalent best-reply set (Pearce (984)) de nton of ICR. Let S T : T! 2 A? be a spec caton of possble actons for each type, and S T = S T 2I.4 De nton S T s a best-reply set f for each t and a 2 S T (t ), there exsts : T! (A ) such that () (t ; ) [a ] > 0 ) a 2 S T (t ) (2) a 2 arg max (t ) [(t ; )] (t ; ) [a ] g ((a 0 ; a ) ; ) a 0 t ;a ; 4 We abuse notaton by callng S T a set to emphasze the lnk to the complete nformaton case; t s a correspondence. 4

17 Clam 2. If Sc T for all c n some ndex set C are best-reply sets then [ c Sc T s a best-reply set. 2. The unon of all best-reply sets s equal to (t ) t 2T. Property s mmedate. That the unon ncludes R T follows from the observaton that (t ) t 2T s a best-reply set. To see the converse, note that no acton n a best-reply R T 2I set can be deleted at any stage of the teraton, snce at each pont n the teraton each such acton s a best reply to actons n the best-reply set, and hence remans. R T 2I Fxed ponts of a best-reply correspondence Lastly we provde a xed-pont de nton of R T. The best-reply correspondence takes as gven a feasble subset of actons for each type of each opponent of, and for each type t of, determnes the set of best reples. De nton 2 The correspondence of best reples for all types gven subsets of actons for all types s denoted BR T : 2 A! 2 t2t A and s de ned as follows. Frst, 2I t2t 2I gven F = (F t ) t 2T 2 2 A the best reples for t 2I are t2t 2I 8 there exsts : T! (A ) such that >< BR T (t ; F ) = a 2 A () (t ; ) [a ] > 0 ) a 2 F t >: (2) a 2 arg max g ((a 0 ; a ) ; ) (t ; ) [a ] (t ) [(t ; )] a 0 t ;;a 9 >= >; Next we de ne 5 BR T (F ) = Clam 3 The largest xed pont of BR T s BR T (t ; F ) t 2T R T (t ) t 2T Ths follows from the fact that any xed pont s a best-reply set and the prevous clam regardng best-reply sets. 5 We abuse notaton and wrte BR both for the correspondence specfyng best reples for a type and for the correspondence specfyng these actons for all types. 2I 2I.. 5

18 3.4 Epstemc Foundatons To better understand the two soluton concepts, ICR and IIR, we relate them to common certanty of ratonalty. In order to do ths, we ntroduce a rcher language an epstemc model to model the certanty of the players. We are able to provde an epstemc foundaton for the soluton concepts n the sprt of the exstng epstemc-foundatons lterature. 6 We note that at least n our epstemc formulaton addtonal common-certanty assumptons are necessary to provde an epstemc foundaton for nterm ndependent ratonalzablty. We dscuss ths further at the end of ths subsecton. Throughout ths secton, we x a type space T = (T ; ) 2I. We wll sometmes refer to ths object as a standard type space and to elements of T as standard types. As dscussed, we vew ths type space as exogenously gven, for example descrbng the perspectve of the modeler of the envronment, and not necessarly a complete descrpton (n partcular not necessarly ncludng all possble correlaton). We then assume only that ths type space (and the game and ratonalty) s common certanty. That s, we embed the standard type space n an arbtrary larger space, the epstemc space whch can be any extenson to a more complete descrpton of the players percepton of the world, specfyng at least ther actons at any state and we assume that the (orgnal) type space s common certanty n ths epstemc space. Then we ask what can we say about play n the game de ned by the orgnal type space;.e., what soluton concept de ned on games wth the orgnal type space s characterzed by common certanty of ratonalty. 7 Let E be a nte set of epstemc types for player ; and let E = (E ) 2I. An epstemc model spec es for each how e determnes. belefs over the types of others and the payo states, : E! (E ); 2. s acton, a : E! A ; 6 Aumann (987), Brandenburger and Dekel (987), Tan and Werlang (988), Aumann and Brandenburger (995). 7 In general t would be reasonable to allow for larger space of states of Nature 0, where 0 are payo rrelevant, as s also a partal, small worlds, descrpton of the payo relevant parameters. It s easy to see that allowng such an enlargement would not a ect our results. The same remark holds for addng the complete unverse of dummy players whose actons do not a ect the payo s of the small worlds set I n the game studed. As noted, addng payo rrelevant states, dummy players, or just alternatve epstemc states as below, wll not a ect the set of ICR actons but wll change whch actons are IIR. 6

19 3. s standard type, : E! T. Thus an epstemc model conssts of (E ; ; a ; ) 2I ; ts state space s = E. There are some events n whch we are partcularly nterested. For a gven epstemc model, we wrte Rat for the set of states where player s ratonal, 8 9 < Rat = : e ; e = 0 ; 0 a (e ) 2 arg max g ((a ; a (e )) ; ) (e ) [(e ; )] ;, a e ; and Rat for the set of states where all players are ratonal, Rat = \ Rat. We wrte W for the set of states where player has the correct belefs about T gven hs type 8 >< W = >: e ; e 0 ; 0 fe :( j (e j )) j6= =t g 9 >= (e ) [(e ; )] = ( (e )) [(t ; )] >;, W = T W. The set of states where ndvdual s certan of the event H s 8 9 < C (H) = : e ; e = 0 ; 0 (e ) [(e ; )] = ;, f(e ;):((e ;e );)2Hg the set of states where everyone s certan of the event H s C (H) = T C (H), and the set of states where there s common certanty of H s CC (H) = T n=0 (C ) n (H), where (C ) 0 (H) = H. Proposton 2 Interm Correlated Ratonalzablty characterzes common certanty of ratonalty and of the standard type space,.e., 7

20 . n any epstemc model, f ((e ; e ) ; ) 2 CC (Rat \ W ), then a (e ) 2 R T ( (e )); 2. there s an epstemc model such that f a 2 R T (t ), then there s a state ((e ; e ) ; ) such that ((e ; e ) ; ) 2 CC (Rat \ W ), (e ) = t and a (e ) = a Proof. () Suppose e ; e ; 2 CC (Rat). Let Ej be the set of epstemc types of player j where j s certan of CC (Rat). Let S (t ) = fa j for some e 2 E, a (e ) = a and (e ) = t g : Observe that, by constructon, a (e ) 2 S ( (e )). Now for any a 2 S (t ), pck any e 2 E a ;t [(t ; ; a )] = Agan by constructon, a 2 arg max a 0 such that a (e ) = a and (e ) = t. Let f(e ;): (e )=t and a (e )=a g (e ) [(e ; )]. t ;;a a;t [(t ; ; a )] g ((a 0 ; a ) ; ). Common certanty of W ensures that a a ;t [(t ; ; a )] = (t ) [(t ; )] for all t ;. Thus an nductve argument ensures that S (t ) R T ;k (t ) for all k and thus S (t ) R T (t ). So a (e ) 2 S ( (e )) R T ( (e )). (2) We wll construct an epstemc type space. Let E = (t ; a ) : a 2 R T (t ). Let a (e ) = a ((t ; a )) = a (e ) = ((t ; a )) = t. Observe that for each a 2 R T (t ), there exsts a ;t 2 (T A ) such that () a ;t [(t ; ; a )] > 0 ) a 2 Rj T (t j ) j6= a;t [(t ; ; a )] g ((a 0 ; a ) ; ) t ;;a (2) a 2 arg max a 0 (3) a a ;t [(t ; ; a )] = (t ) [(t ; )] for all t ; ; 8

21 Let h (e ) [(e ; )] = (t ; a ) (t j ; a j ) j6= ; = a ;t [(t ; ; a )]. By constructon, Rat = W = for all and thus CC (Rat \ W ) =. Now, also by constructon, for any a 2 R T (t ), there s an epstemc type e = (t ; a ) wth ((e ; e ) ; ) 2 CC (Rat \ W ) =, (e ) = t and a (e ) = a. Remark To see why B s consstent wth common certanty of ratonalty n Example 2 note that each player can beleve that two epstemc types of the opposng player can correspond to the same standard type but take d erent actons, and that the epstemc types are correlated wth. Remark 2 A standard renterpretaton of the result s that f we start wth the standard type space T = (T ; ) 2I, we can construct a larger type space T 0 = (T 0 ; 0 ) 2I and belef preservng morphsms ' : T! T 0 from the orgnal type space to the larger type space, and a Nash equlbrum on that larger type space, such that for each type t n the orgnal type space and each nterm correlated ratonalzable acton for that type, there s a correspondng type t 0 = ' (t ) n the larger space who plays that acton n equlbrum. Remark 3 A referee noted that one could gve a d erent nterpretaton of our epstemc analyss: At states n the epstemc type space where there s common certanty of ratonalty, every player wll be choosng an nterm ndependent ratonalzable acton for hs type n that epstemc type space, so there s a sense n whch one can nterpret the result as yeldng IIR and not ICR. To understand the d erence between these two nterpretatons, consder Aumann s (987) characterzaton of correlated equlbrum. Aumann essentally assumed a sngleton type space (.e. a complete nformaton game) that was embedded n an epstemc space, and then showed that f there s a common pror on the epstemc space, then the dstrbuton of actons corresponds to a correlated equlbrum dstrbuton on the orgnal game. The approach of the referee corresponds to focusng on the actons that are played on the enlarged game. In two player games, ths yelds exactly the Nash equlbra. We follow Aumann n studyng the mplcatons of common-certanty assumptons on any epstemc space n whch a certan game (wth a degenerate type space n Aumann s case, or a general type space n ours) and ratonalty of the players s common certanty. 9

22 Fnally, we bre y note for comparson an epstemc characterzaton of nterm ndependent ratonalzablty n our language. The set of states where player has ndependent belefs,.e., beleves that each other player s type s a su cent statstc for hs behavor, s 8 9 for each j 6=, there exsts j : T j! A j such that >< Y = e ; e (e ) [(e ; )] >= 0 ; 0 0f(e ;): (e )=t and a (e )=a g >: (e ) [(e ; )] A Y j (t j ) [a j ] >; j6= f(e ;): (e )=t g Y = \ Y Proposton 3 Independent Interm Ratonalzablty characterzes common certanty of ratonalty, the standard type space and ndependent belefs,.e.,. n any epstemc model, f ((e ; e ) ; ) 2 CC (Rat \ W \ Y ), then a (e ) 2 IIR ( (e )); 2. f a 2 IIR (t ), then there exsts an epstemc model and a state ((e ; e ) ; ) such that ((e ; e ) ; ) 2 CC (Rat \ W \ Y ), (e ) = t and a (e ) = a. The proof closely follows the proof of proposton 2 and hence s not provded. Ths result s the ncomplete-nformaton analog of Proposton 3. n Brandenburger and Dekel (987). Ths proposton shows that addtonal assumptons beyond common certanty of ratonalty and the type space are needed to justfy restrctng attenton to actons that are nterm ndependent ratonalzable on the type space. The addtonal assumpton of common certanty of ndependent belefs makes explct the key dea underlyng the soluton concept: no unexplaned correlaton n belefs s allowed. 4 In nte Type Spaces 4. The type spaces We now extend our analyss to type spaces that are not necessarly nte. To do so, we base our development on Hefetz and Samet s (998) topology-free constructon. 20

23 The prmtves of our model reman a nte set of states of Nature, a nte set I of players, and a type space T = (T ; ) 2. We now assume that each T s a measurable space, set T = j6= T j ; and gve T the product sgma-algebra. For measurable we denote by () the set of (probablty) measures on. 8 Followng Hefetz and Samet, we assume that for every measurable space, the set () of measures on s endowed wth the sgma-algebra generated by ff j (Z) pg jp 2 [0; ] and Z a measurable subset of g : Each (T ) gets the correspondng sgma algebra; we then assume that each : T! (T ) s a measurable functon. Ponts t 2 T are called player 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. 9 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 )) [Z] = (t ) (t ; ) : ' (t ) ; 2 Z for all measurable Z ~ T. We call ' = (' ; :::; ' n ) the morphsm. A partcularly useful type space s a unversal type space that we descrbe next. Let 0 =, and de ne k = k k I, where k s the set of probablty measures on the algebra descrbed above, and each k s gven the product algebra over ts two components. An element ; 2 ; ::: 2 k, H s called a herarchy (of k=0 belefs). For the topology-free model we descrbe here, Hefetz and Samet (998) prove the exstence of a unversal type space T = (T ; ) 2I comprsed of a subset of herarches, T H, 8 The measurable structure on player s belefs s used to model the belefs of other players about s type. The set-up here, whch s standard, mplctly assumes that any two players and j have the same measurable structure on the types of a thrd player k. 9 Hefetz and Samet allowed to be a general measurable space. We contnue to endow and all other nte sets wth the obvous -algebra. 2

24 and a measurable belef functon, : T! T, for all. Note that snce there s a common uncertanty space, the sets T are copes of the same set T. Therefore, where no confuson results, we drop the subscrpt of for notatonal smplcty. 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 T ;k for the (measurable) set of k th -order belefs for all types n T, T ;k ( k belefs about are de ned pontwse by ). Gven any measurable type space, type t s margnal ^ (t ) [] = (t ) [f(t ; ) jt 2 T g]. For each k = 2; 3:: and measurable Z k hn b k (t ) [Z] = (t ) (t ; ), let b (t ) ; ::::; b k o (t ) ; 2 Z ; the morphsm guarantees that b k : T! T ;k s measurable for each k. Let b (t ) = b k (t ), and then b : T! T s the morphsm ' dscussed above. k= We use the topology-free approach because we do not want to restrct ourselves to a partcular topology and t enables us to provde stronger results, as they apply to all measurable type spaces. However, we rely on the fact that n our context there s a belef-preservng somorphsm between the unversal type space (T ; ) 2I dscussed above and the more famlar constructons usng topologcal methods due to Mertens and Zamr (985) (see also Brandenburger and Dekel (993) and Hefetz (993)). These authors construct a unversal type space T = (T ; ) 2I wth a topology on T under whch : T! T are contnuous and under whch T are compact. They show ths type space s unversal for all contnuous type spaces n the sense that for any type space T = (T ; ) 2 for whch s contnuous accordng to a topology on T, there s a contnuous belef-preservng morphsm nto T. Mertens, Sorn and Zamr (994, Theorem.3) show that n fact T s unversal for all measurable type spaces there s a belef-preservng morphsm of any measurable type space nto T. Gven unqueness (up to belef-preservng somorphsms) of the unversal type space constructed by Hefetz and Samet (998, Proposton 3.5) we have that there s a belef-preservng somorphsm between T and T ; we use ths equvalence extensvely n the proofs below and for notatonal smplcty wrte T for both. Even though the proofs therefore use the contnuty propertes of T, because of the somorphsm our results do not nvolve a topology or contnuty. 22

25 4.2 Interm Correlated Ratonalzablty We now restate some of our earler de ntons and prove for ths envronment the key analogous results. In many cases, the only changes the de ntons requre are easy to dentfy: sums need to be replaced by ntegrals, measurablty condtons must be mposed, and nte probabltes must be replaced by measures. We descrbe n detal those few cases where extra care s requred n the notaton; for brevty, we do not repeat the de ntons whose extensons are obvous Best reples For any subset of actons for all types, we rst de ne the best reples when conjectures over opponents strateges are restrcted to those actons. We wrte R T f () () [(dt ; ; a )] when ntegratng wth respect to t only, holdng and a xed. De nton 3 The correspondence of best reples for all types gven subsets of actons for all types s denoted BR T : 2 A T! 2 A T and s de ned as follows. Frst, gven a 2I 2I spec caton of a subset of actons for each possble type, F = F tj t j 2T j j2i, wth F tj A j for all t j and j 2 I, we de ne the best reples for t as 8 there exsts a measurable : T! (A ) such that >< BR T (t ; F ) = a 2 A () (t ; ) [a ] > 0 ) a 2 F t P R >: (2) a 2 arg max g (a 0 ; a ; ) (t ; ) [a ] (t ) [(dt ; )] a 0 ;a t 9 >= >; Next we de ne 20 BR T (F ) = BR T (t ; F ) t 2T 2I. Remark 4 Because A s nte, and expected utlty depends only on actons and belefs, the set of best responses gven some F, BR T (t ; F ), s non-empty provded there exsts at least one measurable that sats es (). Such exst whenever F s non-empty and measurable, and more generally whenever F admts a measurable selecton. 20 We abuse notaton and wrte BR both for the correspondence specfyng best reples for a type and for the correspondence specfyng these actons for all types. 23

26 Gven F as n the prevous de nton, wth non-empty F tj A tj for all t j and j 6=, we wll wrte T (t ; F ) for the set of belefs on the nte set A, consstent wth type t s belefs and certanty that other players are choosng actons consstent wth F. Thus 8 [(a ; )] = R 9 (t ; ) [a ] (t ) [(dt ; )] >< T T >= (t ; F ) = 2 (A ) for some measurable : T! (A ). >: such that (t ; ) [a ] > 0 ) a 2 F >; t Ths s the set of dstrbutons over A that s consstent wth t s belefs about T and certanty that the play of t s consstent wth F, so 8 T >< there exsts BR T (t ; F ) = >: a 2 (t ; F ) such that 2 A a 2 arg max [(a ; )] g ((a 0 ; a ) ; ) a 0 ;a Iteratve de ntons We now de ne ratonalzablty as the result of teratng the BR map. As n the nte case, let R0 T = (A ) t 2T 2I ; RT k = BRT Rk T, and R T = \ k= RT k. The correspondng objects on the unversal type space are R0 = (A ) t 2T, 2I R k = BRT Rk, and R = \ k= R k : T T Let ;k (t ) = (t ; Rk T ); and ;k (t T ) = (t ; Rk ). Lemma If ' s a belef-preservng morphsm from (T; ) to (T ; ) and ' (t ) = t ; then for all k, R T ;k (t ) = R ;k (t ), T t ; R T ;k = (' (t ) ; R T ;k ), RT ;k : T! 2 A =; s a measurable functon, and t 2 T : a 2 R ;k (t ) s closed n the weak topology. Proof. The proof s by nducton on k: 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. So suppose that for all, and t 2 T, R T ;k (t ) = R ;k (' (t )) and T ;k (t ) = ;k (' (t )), that R;k T : T! 2 A =; s a measurable functon, and that t : a 2 R;k (t ) T s closed. (Part I) The set t : a 2 R;k (t ) s closed and therefore measurable. To see ths, consder a sequence t n that converges to t and such that a 2 R;k (tn ). Then for each t n there s a k ;n P 2 ;k (t n ) such that a 2 arg max a 0 ;a g (a 0 ; a ; ) k ;n [(a ; )]. Moreover, k ;n [(a ; )] = R n t t ; [a ] (t n ) dt ; for some n : T! 24 9 >= >;.

27 (A ) where n t ; [a ] > 0 mples a 2 R ;k t. Let n = n ; (t n ), and by compactness of T A consder a convergent subsequence of n convergng to. Moreover, by compactness we also have a regular verson of condtonal probabltes, denoted [j (t ; )] 2 (A ), whch, by regularty s a measurable functon on T. Hence we can de ne a measurable : T! (A ) by t ; [a ] = a j t ;. Note that = (t ) ;. De ne 2 (A ) by [(a ; )] R [(dt T ; ; a )]. Clearly a 2 arg max g (a 0 ; a ; ) [a ; ] : a 0 ;a It remans to show that (t ; ) [a ] > 0 ) a 2 R ;k. Note rst that t ; ; a a 2 R ;k (t ) =. (2) Ths follows from n t ; ; a : a 2 R ;k = and n!. Equaton (2) can be wrtten as (t ) [N] = 0, where N t ; supp t ; 6 R ;k t : So changng on N has no e ect on expected payo s, 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) non-empty subsets B A, let B t 2 T : R ;k t = B and B t 2 T : supp t ; B. Both sets are measurable, hence B B s measurable, and snce (t ) [N] = 0, also (t ) B B = 0. So rede ne t ; on B B to equal any a 2 B. Snce fa g s measurable, so s fa g [ B B, so after ths rede nton s stll measurable and B B s empty. Dong ths process for all B A we obtan a measurable such that t ; 2 R ;k t for every (not only a.e.) t. (Part II) Snce T ;k (t ) = ;k (' (t )) t s mmedate that R T ;k (t ) = R ;k (' (t )). (Part III) By (part I), (part II) and the measurablty of ' we have that R T ;k : T! 2 A =; s measurable. (Part IVa) We now argue that ;k (t T ) ;k (t ). 8 >< [(a ; )] = R t t ; [a ] (t ) dt ; ;k (t ) = 2 (A ) for some measurable : T! (A ) >: such that t ; [a ] > 0 ) a 2 R ;k t 25 9 >= >;

28 Fx and the n the above expresson, and de ne : T! (A ) by (t ; ) = ' (t ) ;. Snce R T ;k (t ) = R ;k ' (t ) and t ; [a ] > 0 ) a 2 R ;k we have (t ; ) [a ] > 0 ) a 2 R T ;k. So Z [(; a )] = (t ; ) [a ] (t ) [(dt ; )] t s n T t ; R T ;k. From the morphsm we have that Z [(; a )] = t ; [a ] (' (t )) dt ; Thus = t Z t (t ; ) [a ] (t ) [(dt ; )]. ;k (t ) T ;k (t ) : (Part IVb) To prove the converse, suppose 2 T t ; R T ;k and let be the assocated conjecture so [(; a )] = R t (t ; ) [a ] (t ) [(dt ; )]. : T! (A ) as follows. We wll de ne Frst, for every B A let T (B ) = t 2 T : B = R T ;k (t ) and (B ) = t 2 T : B = R ;k (t ) By the nducton hypothess both T (B ) T and (B ) T are measurable, T (B ) = ' (B ), and hence (' (t )) (B ) = (t ) T (B ). We construct t ; [] 2 (A ) as follows. Map (t ; ) nto t ; by takng all t for whom B s k ratonalzable, denoted T (B ), takng the condtonal average of (t ; ) over those t, and assgnng that average conjecture to those t who have that same k ratonalzable set,.e., to (B ). Moreover, each (B ) s a superset of ' T (B ), and these supersets partton T. So we can combne all those averages to get a strategy for all t 2 T. There s a slght ssue for the case where the condtonal sn t well de ned because the condtonng event, T (B ), has probablty zero. In that case the strategy s really rrelevant, but as we requre t to be measurable and to map nto the k ratonalzable set, we add that restrcton by havng the strategy assgn probablty to some k ratonalzable acton for all t 2 (B ) whenever (t ) T (B ) = 0. To do ths, for each B x some a (B ) 2 B. 26