Backward Induction Reasoning in Games with Incomplete Information

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1 Backward Inducton Reasonng n Games wth Incomplete Informaton Antono Penta y Unversty of Wsconsn - Madson, Dept. of Economcs Ths verson: September 26, 2011 Abstract Backward Inducton s one of the central notons n game theory. A number of soluton concepts, such as subgame perfect and sequental equlbrum, are thought of as beng based on backward nducton reasonng. Yet, t s not clear what ths means precsely, partcularly n stuatons wth ncomplete nformaton, where the game cannot even be solved backwards. Ths paper ntroduces a soluton concept for games wth ncomplete nformaton, backward extensve form ratonalzablty ( BR for short), and proves several propertes that show how, n a precse sense, BR characterzes the mplcatons of backward nducton reasonng n games wth ncomplete nformaton. These results reconcle n a untary framework several deas tradtonally (though only nformally) assocated to the logc of backward nducton, such as the dea of devatons as mstakes, tremble-based equlbrum concepts, the belef-persstence hypothess, the noton of subgame consstency and the possblty of solvng the game backwards. JEL Codes: C72; C73; D82. 1 Introducton Backward nducton s one of the basc notons of game theory. Although backward nducton s only de ned for games wth perfect and complete nformaton, the logc of backward nducton has a much broader scope n game theory. So, for nstance, subgame perfect equlbrum s certanly vewed as the natural extenson of ths logc to games wth Some of the materal contaned n ths manuscrpt prevously crculated as part of my job market paper Robust Dynamc Mechansm Desgn. I m ndebeted to my advsor, George Malath. I also thank partcpants of several semnars and conferences. y emal: apenta@ssc.wsc.edu. 1

2 mperfect nformaton. But there s a sense n whch also soluton concepts for ncomplete nformaton games, such as sequental or tremblng-hand perfect equlbrum, are often thought of as havng a backward nducton avor. Yet, t s not clear what backward nducton means n games wth ncomplete nformaton. More broadly: What do we mean by Backward Inducton Reasonng? Despte ts central poston n game theory, there s no comprehensve, formal answer to ths queston. Fllng ths gap s the am of ths paper. In pursut of an answer, a good startng pont s to nspect the concepts that are normally assocated to the dea of backward nducton reasonng. Subgame Perfect Equlbrum (SPE) s certanly one of these. SPE s probably the sngle most common soluton concept for dynamc games wth complete nformaton. An n uental argument n support of SPE s provded by Harsany and Selten s (1988) noton of subgame consstency: It s natural to requre that a soluton functon for extensve games s subgame consstent n the sense that the behavor prescrbed on a subgame s nothng else than the soluton to the subgame (bd., p.90) Subgame consstency warrants SPE the recursve structure of backward nducton,.e. the possblty of determnng the soluton concept s predctons for a subgame by lookng at the subgame n solaton. Hence the possblty (n games wth nte horzon) to solve for the subgame perfect equlbra startng from the termnal nodes and proceedng backwards. Ths s extremely convenent, and certantly one of the man reasons for the promnence of SPE n appled work. SPE assumes that, even after observng a devaton, agents mantan ther belefs n the equlbrum contnuaton strateges. Ths assumpton, whch can be referred to as the belef persstence hypothess, often consttuted the man target for the numerous crtques to SPE and ts mplct assumptons on counterfactuals. 1 Several soluton concepts extend the deas of SPE to games wth ncomplete nformaton, and many of these nvolve trembles (e.g., tremblng-hand perfect equlbrum (Selten, 1975), sequental equlbrum (Kreps and Wlson, 1982), etc.). In these soluton concepts, trembles are a shortcut to formalze another dea typcally assocated to the logc of backward nducton: that o -equlbrum moves are mstakes, unntended devatons. 2 Beng the ncomplete nformaton counterparts of SPE, t s commonly accepted that these soluton concepts share a backward nducton avor. But, f ths s true at an ntu- 1 See e.g. Stalnaker (1996, 1998). 2 The vew of devatons as mstakes contrasts wth the logc of forward nducton, whch requres nstead that unexpected moves be ratonalzed (f possble) as purposeful devatons. 2

3 tve level, ts precse meanng s not clear, as no formal de nton of backward nducton s avalable for games wth ncomplete nformaton. One mportant d erence between SPE and ts ncomplete nformaton counterparts s that the latters lack the recursve structure of SPE. Under ncomplete nformaton, an equlbrum requres a spec caton of agents belefs about the opponents types at each nformaton set. But such belefs are endogenous, equlbrum objects, and must be jontly determned wth the equlbrum strateges. Wth ncomplete nformaton, contnuaton games cannot be consdered n solaton, and the equlbrum analyss requres the soluton of xed pont problems, often d cult to compute. Thus, on the one hand, the tremble-based soluton concepts supposedly embody the logc of backward nducton; but, on the other hand, they lack the recursve structure of SPE, whch seems almost a de nng feature of backward nducton reasonng. Ths paper puts forward a soluton concept for belef-free dynamc games called Backwards Extensve Form Ratonalzablty (BR for short). BR conssts of an terated deleton procedure for games n extensve form that at each round elmnates strateges that are not sequental best responses to conjectures that, at each pont n the game, must be concentrated on opponents contnuaton strateges that are consstent wth the prevous rounds of deleton. Through the followng results, t s further argued that BR characterzes the behavoral mplcatons of Backward Inducton Reasonng n games wth ncomplete nformaton: Result 1: BR can be computed by a convenent backwards procedure that combnes the logc of (normal form) ratonalzablty and backward nducton. The backwards procedure conssts of the terated applcaton of (normal form) ratonalzablty to the contnuaton games from each nformaton set consdered n solaton, startng from the end of the game and then proceedng backwards. Besdes smplfyng the computaton of the set of BR strateges, ths result mples that BR sats es a property analogous to subgame consstency, whch we may call contnuatongame consstency: the predctons of BR for each contnuaton game are nothng but the BR strateges of the contnuaton game. I ntroduce next an equlbrum concept for dynamc Bayesan games, nterm perfect equlbrum (IPE). Bayesan games are obtaned appendng a model of agents belefs,.e. a type space, to the belef-free game. IPE s the weakest equlbrum noton consstent wth sequental ratonalty and Bayesan updatng, and concdes wth SPE n complete nformaton games. I show that: Result 2: IPE s consstent wth a tremble-based re nement of Bayesan equlbrum, n whch trembles may be correlated wth anythng. 3

4 Result 3: The set of BR strateges n the belef-free game concdes wth the set of strateges played as part of some IPE for some type space. Result 3 says that BR characterzes the robust predctons of IPE, that s the IPE predctons that do not depend on assumptons on players exogenous belefs. Furthermore, whle for a gven type space the computaton of the IPE strateges remans a xed pont problem, Results 1 and 3 together mply that a property analogous to subgame consstency holds for the set IPE strateges: the robust predctons are contnuaton-game consstent. At a practcal level, these results show that rather than computng the set of IPE by solvng a large (possbly n nte, n fact) number of xed pont problems, the set of all IPE strateges can be computed by means of a tractable backwards procedure. These results therefore can be partcularly useful n applcatons. 3 Fnally, two epstemc characterzatons of BR are provded: Result 4: BR s characterzed by Ratonalty and the followng assumptonson agents belefs: Common Certanty of Future Ratonalty. That s, players share common certanty of ratonalty at the begnnng of the game. If an unexpected move s observed, players are wllng to accept the dea that the devaton was a mstake and mantan common certanty of ratonalty n the contnuaton game; or Common Certanty of Full Ratonalty and Belef Persstence. That s, at the begnnng of the game, players share common certanty of ratonalty and they never change ther belefs about the opponents contnuaton strateges. Full Ratonalty s a stronger noton than ratonalty, n that t refers to statements about agents ratonalty condtonal on counterfactual hypothess. (To accommodate such counterfactual propostons, t wll be necessary to nnovate on the exstng lterature by ntroducng rcher epstemc models.) Overall, the results above (formally) reconcle all the features that are (nformally) assocated to backward nducton reasonng: the recursve structure, the noton of contnuatongame consstency, the belef persstence hypothess, the dea of trembles and of devatons as unntended mstakes. There s thus a precse sense n whch IPE s the ncomplete nformaton counterpart of SPE embodyng the backwards nducton logc and nothng more. 3 See Penta (2009) for an applcaton to problems of robust dynamc mechansm desgn. 4

5 I thus argue that the epstemc assumptons of Common Certanty of Future Ratonalty (or, alternatvely, Common Certanty of Ratonalty and Belef Persstence) provde a comprehensve formal answer to the openng queston: What do we mean by Backward Inducton Reasonng? From an appled perspectve, ths paper provdes foundatons to a tractable backwards procedure that characterzes the set of equlbra of dynamc games wth ncomplete nformaton. The tractablty of the algorythm may prove useful n overcomng the d cultes typcally faced n appled and emprcal works. 2 Belef-Free Dynamc Games The analyss that follows concerns multstage games wth observable actons. 4 de ned by an extensve form and agents preferences and nformaton. Extensve Form. These are The game has L stages, ndexed by l = 1; 2; :::; L. Let h 0 denote the empty hstory, and for every player 2 N = f1; :::; ng, let A denote the ( nte) set of actons avalable to player throughout the game. At stage l = 1, agents 2 N smultaneously choose actons a 1 from the nte sets A (h 0 ) A (for each 2 N). The chosen acton pro le s publcly observed, hence the set H 1 = 2N A (h 0 ) denotes the set of hstores of length one. For every h 2 H 1, let A (h) A denote the set of actons avalable to player at hstory h. 5 For every l = 2; :::; L, the set H l of publc hstores of length l s de ned recursvely as follows: for any l and h l 1 2 H l 1, let A h l 1 A denote the set of player s actons avalable at hstory h l 1, and let A h l 1 = 2N A h l 1 and A h l 1 = j2nnfg A j h l 1. 6 H l = n h l 1 ; a l 2 H l 1 A l : a l 2N 2 A h l 1 o for every 2 N : The set of publc hstores s de ned as H = S L 1 t=0 Hl (where H 0 = fh 0 g), whle the set of termnal hstores s Z = H L. Wthout loss of generalty, sets A (h) are assumed non-empty for each h 2 H: player s nactve at h f ja (h) j = 1; he s actve otherwse. Ths setup allows ntely repeated games as a specal case, or games wth perfect nformaton f H s such that only one player s actve at each h. If H = fh 0 g, the game s statc. It wll be convenent to ntroduce the precedence relaton on H: h l h l+k f and only f there exsts h l+k = h l ; a k. k=1;:::;k a k k=1;:::;k such that 4 See Fudenberg and Trole, 3.2 and 8.2. At the expense of heaver notaton, the analyss can be easly adapted to all nte dynamc games wth perfect recall. 5 The set A (h) may vary wth hstory h, hence the game need not be a repeated game. 6 Smlar notaton wll be adopted for other product sets. 5

6 Preferences and Informaton. To model stuatons wth ncomplete nformaton, players preferences over the termnal nodes are parametrzed on a fundamental space of uncertanty = 0 1 ::: n. 7 Elements of are referred to as payo states, and payo functons are denoted by u : Z! R, for each 2 N. For each = 1; :::; n, s the set of player s payo types. 0 s referred to as the set of states of nature. For each, = j2nnfg j, so that = 0. (To avod unnecessary techncaltes, the set s assumed nte throughout.) When the true state s ( 0 ; 1 ; :::; n ), player s payo type s, prvately observed at the begnnng of the game. Hence, payo types represent agents nformaton about the payo o state: f s payo type s ^, knows that the true state belongs to the set 0 n^. The set 0 represents the resdual uncertanty that s left after poolng everybody s nformaton. The tuple 0 ; ( ; u ) 2N thus represents agents nformaton about payo s. It s assumed common knowledge and referred to as preference-nformaton structure (PIstructure). Specal cases of nterest are: complete nformaton ( k s a sngleton for all k = 0; 1; :::; n); prvate values (f, for all 2 N, u s constant n ( 0 ; )); no nformaton (f u s are constant on 0 ); dstrbuted knowledge (f u s are constant n 0 ) Belef-Free Games A belef-free dynamc game s thus de ned by a tuple = N; H; Z; 0 ; ( ; u ) 2N : Notce that ths s not a Bayesan game, as does not nclude a model of agents nteractve belefs over. Bayesan games wll be ntroduced n Secton 4.1. Strategc Forms. Pure strateges n the belef-free game are functons s : H! A such that for each h 2 H, s (h) 2 A (h). The set of player s strateges s denoted by S, and as usual we de ne the sets S = 2N S and S = j2nnfg S j. To dstngush them from those that wll be ntroduced for Bayesan games, elements of S are referred to as nterm (pure) strateges. Any strategy pro le s 2 S nduces a termnal hstory z () 2 Z. Hence, we can de ne strategc-form payo functons U : S! R as U (s; ) = u (z (s) ; ) for each s and. 7 General nformaton parttons on could be consdered, at the expense of heaver notaton. Restrctng attenton to product structures entals no essental loss of generalty. 8 Product structures wth dstrbuted knowledge are common n the lterature on robust mechansm desgn. (see, e.g., Bergemann and Morrs 2005, 2009) 6

7 It s useful to ntroduce notaton for the nterm mxed strateges (elements of (S )) and the nterm behavor strateges (elements of h2h (A )), whch wll be used for the analyss of Bayesan games (Secton 4.1). To save on notaton, we wll take advantage of Kuhn s (1953) equvalence theorem and use the same symbol, 2 P, to denote both knds of strateges (ts nterpretaton wll be clear from the context). Notaton [s ] refers to the probablty that mxed strategy attaches to strategy s ; whle (a jh) refers to the probablty of acton a at prvate hstory h accordng to the behavor strategy : For each publc hstory h and player, let S (h) denote the set of player s strateges that are consstent wth hstory h beng observed. It s also convenent to de ne strateges and payo functons for the contnuatons games: For each publc hstory h 2 H, let S h denote the set of strateges n the contnuaton game startng from h, and for each s 2 S, let s jh 2 S h denote the contnuaton of s from hstory h. The notaton z (sjh; ) refers to the termnal hstory nduced by strategy pro le s from the publc hstory h, when the realzed state s. Strategc-form payo functons can be de ned for contnuatons from a gven publc hstory: for each h 2 H and each (s; ) 2 S, let U (s; ; h) = u (z (sjh) ; ). (For the ntal hstory h 0, U (s; ) wll be wrtten nstead of U (s; ; h 0 ).) 3 Backwards Extensve Form Ratonalzablty Backwards Extensve Form Ratonalzablty (BR) s a soluton concept for belef-free games n extensve form, concept: n game not be consstent wth each other. Endogenous Belefs: Conjectures.. Smlar to ratonalzablty, BR s a non-equlbrum soluton agents form conjectures about everyone s behavor, whch may or may At every hstory, players hold conjectures about the state of nature, the opponents payo types and (everybody s) behavor. These are represented by condtonal probablty systems (CPS),.e. arrays of condtonal belefs, one for each hstory. These belefs d er from those that wll be ntroduced n Secton 4.1 n that they concern and depend on endogenous varables such as the opponents behavor. Hence, these are endegenous belefs. To avod confuson, we thus refer to ths knd of belefs as conjectures, retanng the term belefs for those ntroduced n Secton 4.1. For each hstory h 2 H, de ne the event [h] 0 S as: [h] = 0 S (h) : (Notce that, by de nton, [h] [h 0 ] whenever h follows h 0.) De nton 1 A conjecture for agent s a condtonal probablty system (CPS hereafter), that s a collecton = ( (h)) h2h of condtonal dstrbutons (h) 2 ( 0 S) 7

8 that satsfy the followng condtons: C.1 For all h 2 H, supp ( (h)) [h] ; C.2 For every measurable A [h] [h 0 ], (h) [A] (h 0 ) [h] = (h 0 ) [A]. The set of CPS over 0 S s denoted by H ( 0 S). Condton C.1 states that agents are always certan of what they know; condton C.2 states that agents conjectures are consstent wth Bayesan updatng whenever possble. Notce that n ths spec caton agents entertan conjectures about the payo state, the opponents and ther own strateges. The latter pont s not entrely standard: n smlar non-equlbrum soluton concepts for games n extensve form t s common practce to model conjectures on the opponents behavor only (see, e.g. Battgall and Snscalch, 2007, or Penta, 2010). We wll dscuss ths pont n some detal n Secton 4. Sequental Ratonalty. Strategy s s sequentally ratonal for type wth respect to conjectures f, at each hstory h 2 H, t prescrbes optmal behavor n the contnuaton game wth respect to conjectures (h). Formally: for any ^ 2, gven a CPS 2 H ( 0 S) and a hstory h, ^ s expected payo from s at h, gven, s de ned as: U s ; ; h; ^ = P marg 0 S (h) [ 0 ; ; s ] 0 ; ;s U s ; s ; 0 ; ^ (1) ; ; h De nton 2 Strategy s s sequentally ratonal for payo -type wth respect to 2 H ( 0 S), wrtten s 2 r ( ; ), f and only f for each h 2 H and each s 0 2 S the followng nequalty s sats ed: U s ; ; h; U s 0 ; ; h; : (2) If s 2 r ( ; ), we say that conjectures justfy strategy s for type. 3.1 Backwards Ratonalzablty n the Extensve Form We ntroduce next the soluton concept that, t wll be argued, characterzes backward nducton reasonng n ncomplete nformaton games: Backwards Extensve Form Ratonalzablty (BR). 8

9 De nton 3 For each 2 N let BR 0 = S. Recursvely, for k = 1; 2; :::, let BR k 1 = j2nnfg BR k 1 j, and for each 2, let, H ( 0 S) s.t. >< BR k ( ) = ^s 2 BR k 1 ( ) : (1) ^s 2 r ( ; ) (2) supp ( (h 0 )) 0 BR k 1 f^s g (3) for each h 2 H : ( ; s) 2 supp marg S (h) mples: (3.1) s jh = ^s jh, and >: (3.2) 9s 0 2 BR k 1 ( ) : s 0 jh = s jh >; BR k = ( ; s ) 2 S : s 2 BR k ( ), BR k = k 2N BR, and nally BR := T BR k. BR conssts of an terated deleton procedure. At each round, strategy ^s survves for type f t s just ed by conjectures that satsfy two condtons: condton (2) states that at the begnnng of the game, the agent must be certan of hs own strategy ^s and have conjectures concentrated on pars of the opponents type and strateges consstent wth the prevous rounds of deleton; condton (3) restrcts the agent s conjectures at unexpected hstores: condton (3.1) states that agent s always certan of hs own contnuaton strategy; condton (3.2) requres conjectures to be concentrated on opponents contnuaton strateges that are consstent wth the prevous rounds of deleton. Notce however that agents conjectures about 0 at unexpected hstores are unrestrcted. Thus, condton (3) embeds two conceptually dstnct knds of assumptons: the rst concernng agents conjectures about 0 ; the second concernng ther conjectures about the contnuaton behavor. For ease of reference, they are summarzed as follows: Unrestrcted-Inference Assumpton (UIA): At unexpected hstores, agents conjectures about 0 are unrestrcted. In partcular, agents are free to nfer anythng about the opponents prvate nformaton from the publc hstory. For example, condtonal conjectures may be such that marg (h) s concentrated on opponents types for whom some of the prevous moves n h would be rratonal, or mstakes. Nonetheless, condton (3.2) mples that t s beleved that such types wll behave ratonally n the future. From an epstemc vewpont, t can be shown that BR can be nterpreted as common certanty of future ratonalty at every hstory (for the formal statement, see Secton 6.3) Common Certanty n Future Ratonalty (CCFR): at every hstory (expected or not), agents share common certanty n future ratonalty. 9 >= ; k0

10 Fgure 1: Example 1 CCFR can be nterpreted as a condton of belef persstence on the contnuaton strateges. (A formal connecton between BR and the belef persstence hypothess s provded n Secton 6.4). Remark 1 Snce the game s nte, the terated deleton procedure stops after ntely many rounds: there exsts K < 1 such that BR K = BR K+1. It s also trval to show that BR sats es the followng xed pont characterzaton: Lemma 1 Strategy ^s 2 BR ( I ) f and only f 9 2 H ( 0 I S) s.t. (1) ^s 2 r ( ; I ); (2) supp( (h 0 )) 0 BR f^s g and (3) for each h 2 H: s 2supp(marg S (h)) mples: (3.1) s jh = ^s jh, and (3.2) 9 ; s 0 2 BR : s 0 jh = s jh. Example 1 Consder the game n gure 1, and let 1 = f10; 10g, whle 0 and 2 are sngletons (hence, player 1 knows the true state, whle player 2 has no nformaton). Let s apply BR: at the rst round, strateges nvolvng L 3 and R 3 are deleted for all types of player 1, whle strateges nvolvng l (resp. r) are deleted for type 1 = 10 (resp. 1 = 10). So, for nstance, after the rst round strategy rl 1 R 2 survves for type 1 = 10. Now, suppose that 2 s ntal conjectures are concentrated on payo state-strategy par ( 1 ; s 1 ) = ( 10; rl 1 R 2 ): then, l s unexpected, so after observng t we are free to specfy 2 s belefs, who could for example assgn probablty one on par ( 1 ; s 1 ) = (10; ll 1 R 2 ), hence play a 2, or on par ( 1 ; s 1 ) = ( 10; ll 1 R 2 ), hence play a 1. Smlarly, both b 1 and b 2 n the rght-most contnuaton game can be just ed f 2 s ntal conjectures assgn probablty one to 1 = 10. Hence, strateges that survve BR are fa 1 b 2 ; a 1 b 3 ; a 2 b 2 ; a 2 b 3 g for player 2, fll 1 R 1 ; ll 1 R 2 ; ll 2 R 1 ; ll 2 R 2 g for type 1 = 10 and frl 1 R 1 ; rl 1 R 2 ; rl 2 R 1 ; rl 2 R 2 g for type 1 =

11 4 Backwards Ratonalzablty and Equlbra Ths secton explores the connecton between BR and equlbrum predctons. Pursung an equlbrum approach n games wth ncompelte nformaton requres a spec caton of agents herarches of belefs. We wll follow the tradtonal approach (Harsany, ) of modelng such herarches of belefs mplctly, by means of type spaces. Appendng a type space to a belef-free game delvers a Bayesan game. 4.1 Bayesan Games De nton 4 A (-based) type space s a tuple T = (T ; ; ) 2N such that for each 2 N, T s a nte set of types, : T! s an onto functon assgnng a payo -type to each type, and : T! ( T ) assgns to each type a belef about the payo state and the opponents types. 9 The Bayesan game obtaned appendng type space T = (T ; ; ) 2N to the belef-free game s de ned as the tuple T = N; ; H; Z; (T ; ; ; ^u ) 2N where ^u : Z 0 T! R s such that that for each (z; 0 ; t) 2 Z 0 T, ^u (z; 0 ; t) = u (z; 0 ; (t)). To avod unnecessary notaton, n the followng we wll use u to denote both payo functons. Strateges n a Bayesan game are functons b : T! assgnng an nterm (mxed or behavor) strategy to each type n the type space. The notaton b a l ; t ; h l 1 refers to the probablty that behavor strategy b (t ) 2 assgns to acton a l at h l Interm Perfect Equlbrum De ne the set of nformaton sets of player n the Bayesan game as H = T H. 10 system of belefs conssts of collectons (p (h )) h 2H for each agent, such that p (h ) 2 ( 0 T ) for each h 2 H : a belef system represents agents condtonal belefs about the state of nature and the opponents types at each nformaton set. A strategy pro le and a belef system (b; p) form an assessment. For each agent, a strategy pro le b 9 The restrcton to nte type spaces s only made for smplcty of exposton. 10 Behavor strateges n the Bayesan game could be de ned as functons : H! A s.t. (t ; h) 2 (A (h)) for each h 2 H. Clearly, ths s equvalent to the de nton of strateges as functons b : T!. A 11

12 and condtonal belefs p nduce, at each prvate hstory h l 1 = t ; h l 1, a probablty dstrbuton P ;p a l jh l 1 over acton pro les at stage l: X Y P b;p ^a l ; h l 1 = b ^a l ; h l 1 p 0 ; t jh l 1 b j ^a l j; t j ; h l 1 (3) ( 0 ;t )2 0 T : De nton 5 Assessment (b; p) s weakly preconsstent f for each 2 N : 8t 2 T ; p t ; h 0 = (t ) (4) j6= p 8h l = h l 1 ; a l 2 H 0 ; t jh l = p 0 ; t jh l 1 P b;p ^a l ; h l 1 : (5) Condton (4) requres each agent s belefs condtonal on observng type t to agree wth that type s (exogenous) belefs as spec ed n the type space T ; condton (5) requres that the belef system p s consstent wth Bayesan updatng whenever possble. From the pont of vew of each, for each h = (t ; h) 2 H and strategy pro le b, the nduced termnal hstory s a random varable that depends on the opponents type pro le. Ths s denoted by z (bjh ; t ). As done for belef-free games (Secton 2), we can de ne strategc-form payo functons for the contnuaton games: U (b; 0 ; t ; t ; h ) = u (z (bjh ; t ) ; (t)). De nton 6 Fx a belef system p. Strategy pro le s sequentally ratonal wth respect to p f for every 2 N and every h t 2 H n fg, the followng nequalty s sats ed for every b 0 : T! : X p 0 ; t jh l U b 0 ; b ; 0 ; t ; t ; h l 0 T X p U b; 0 ; t ; t ; h l 0 T 0 ; t jh l. De nton 7 An assessment (; p) s an Interm Perfect Equlbrum (IPE) f t s weakly preconsstent and sequentally ratonal. If nequalty (6) s only mposed at prvate hstores of length zero, the soluton concept concdes wth nterm equlbrum (Bergemann and Morrs, 2005). IPE re nes nterm equlbrum mposng two natural condtons: rst, sequental ratonalty; second, preconsstency of the belef system. (6) 12

13 Notce that weak preconsstency mposes no restrctons on the belefs held at hstores that receve zero probablty at the precedng node. 11 Hence, even f agents ntal belefs admt a common pror, IPE s weaker than Fudenberg and Trole s (1991) perfect Bayesan equlbrum. Also, notce that any player s devaton s a zero probablty event, and treated the same way. In partcular, f hstory h l s precluded by b h l 1 alone, h l =2suppP ;p h l 1, and agent s belefs at h l are unrestrcted the same way they would be after an unexpected move of the opponents. Ths feature of IPE s not entrely standard, but t s key to the result that the set of IPE strateges (takng the unon over all type spaces) can be computed by means of a convenent backwards procedure : Treatng own devatons the same as the opponents s key to the possblty of consderng contnuaton games n solaton, necessary for the result. 12 Tremble-Based Formulaton. It can be shown that IPE s consstent wth a tremblnghand vew of unexpected moves, n whch no restrctons on the possble correlatons between trembles and other elements of uncertanty are mposed. 4.3 Characterzaton of the set of IPE As emphaszed above, n BR agents hold conjectures about both the opponents and ther own strateges. Frst, notce that condtons (2) and (3.2) n the de nton of BR mantan that agents are always certan of ther own strategy; furthermore, the de nton of sequental best response (def. 2) depends only on the margnals of the condtonal conjectures over 0 S. Hence, ths partcular feature of BR does not a ect the standard noton of ratonalty. The fact that conjectures are elements of H ( 0 S) rather than H ( 0 S ) corresponds to the assumpton that IPE treats all devatons the same; ts mplcaton s that both hstores arsng from unexpected moves of the opponents and from one s own devatons represent zero-probablty events, allowng the same set of condtonal belefs about 0 S, wth essentally the same freedom that IPE allows after anyone s devaton. Ths s the man nsght behnd the followng result (the proof s n Appendx A.1). 11 Unlke other notons of weak perfect Bayesan equlbrum, n IPE agents belefs are consstent wth Bayesan updatng also o -the-equlbrum path. In partcular, n complete nformaton games, IPE concdes wth subgame-perfect equlbrum. 12 In Penta (2009b) I consder a mnmal strengthenng of IPE, n whch agents belefs are not upset by unlateral own devatons, and I show how the analyss that follows adapts to that case: The backwards procedure to compute the set of equlbra across models of belefs must be mod ed, so to keep track of the restrctons the extensve form mposes on the agents belefs at unexpected nodes. The possblty of envsonng contnuaton games n solaton s thus lost. 13

14 Proposton 1 Fx a belef-free game. For each, s 2 BR ^ f and only f 9T, ^t 2 T and ^b; ^p such that: () ^b; ^p s an IPE of T ; () ^t = ^ and () ^s 2supp ^b ^t : An analogous result can be obtaned for the more standard re nement of IPE, n whch unlateral own devatons leave an agents belefs unchanged, applyng to a mod ed verson of BR: such mod caton entals assumng that agents only form conjectures about 0 S (that s, 2 H ( 0 S )) and by consequently adaptng condtons (2) and (3) n the de nton of BR. (See Penta, 2009b.) Hence, the assumpton that IPE treats anyone s devaton the same (and, correspondngly, that n BR agents hold conjectures about ther own strategy as well) s not crucal to characterze the set of equlbrum strateges across models of belefs. It s crucal nstead for the next result, whch shows that such set can be computed applyng a procedure that extends the logc of backward nducton to envronments wth ncomplete nformaton (proposton 2 below). 5 Contnuaton-game Consstency Ths Secton provdes a characterzaton of BR n terms of a recursve procedure, that solves the game backwards. Together wth proposton 1, the result n ths secton mples that the robust predctons of IPE satsfy a property analogous to Harsany and Selten s (1988) subgame consstency: for each h, the set of IP E contnuaton strateges from h concdes wth the set of IP E strateges of the contnuaton game consdered n solaton. The Backwards Procedure. The backwards procedure s descrbed as follows: Fx a publc hstory h L 1 of length L 1. For each payo -type 2 of each agent, the contnuaton game s a statc game, to whch we can apply belef-free ratonalzablty (e.g., Bergemann and Morrs, 2009). For each h L 1, let R hl 1 denote the set of pars ; s jh L 1 such that contnuaton strategy s jh L 1 s ratonalzable n the contnuaton game from h L 1 for type. We now proceed backwards: for each publc hstory h L 2 of length L 2, we apply agan ratonalzablty to the contnuaton game from h L 2 (n normal form), restrctng contnuaton strateges s jh L 2 2 S hl 2 to be ratonalzable n the contnuaton games from hstores of length h L 1. R hl 2 denotes the set of pars ; s jh L 2 such that contnuaton strategy s jh L 2 s ratonalzable n the contnuaton game from h L 2 for type. Inductvely, ths s done for each h l 1, l = L; :::; 1, untl the ntal node s reached. Before ntroducng the procedure formally, consder the followng example: Example 2 Consder the game n gure 1 agan. If we apply belef-free ratonalzablty to the contnuaton game followng r, R 3 s deleted at the rst round for both types of 14

15 player 1, and b 1 at the second round for player 2. In ths contnuaton game, the procedure selects contnuatons fb 2 ; b 3 g for player 2 and contnuatons fr 1 ; R 2 g for both types of player 1. Smlarly, after l, belef-free ratonalzablty selects fa 1 ; a 2 g for player 2, and fl 1 ; L 2 g for both types of player 1. So, now we apply belef-free ratonalzablty to the normal form n whch t s mantaned that contnuaton strateges are ratonalzable n the correspondng contnuatons,.e. the relevant strategy sets now are fa 1 b 2 ; a 1 b 3 ; a 2 b 2 ; a 2 b 3 g for player 2, and fll 1 R 1 ; ll 1 R 2 ; ll 2 R 1 ; ll 2 R 2 g [ frl 1 R 1 ; rl 1 R 2 ; rl 2 R 1 ; rl 2 R 2 g for (both types of) player 1: at ths stage, type 1 = 10 deletes all strateges nvolvng r at the rst round, and so does type 1 = 10 wth those nvolvng l, but player 2 doesn t delete anythng: f he expects 1 = 10 (hence 1 to play l), then both a 2 b 2 and a 2 b 3 are best responses n the normal form; smlarly, f he expects 1 = 10 (.e. 1 to play r), then both a 2 b 3 and a 1 b 3 are optmal. Notce that the resultng strateges are precsely those selected by BR n example 1. As proposton 2 shows, ths nsght has general valdty.} The backwards procedure s de ned recursvely, startng from the last stage of the game and proceedng backwards: [l = L 1] For each h L 1 2 H L 1, and for each 2, let R 0 ; h L 1 = S hl 1 For each k = 1; 2; :::, let h L 1 n = ; s hl 1 8 R k 1 : s hl 1 2 R k 1 ; h L 1o, s hl 1 2 R k 1 ; h L 1 : (R.1): R k 1 h L 1 >< R k ; h L 1 (R.2): for all s 0 2 S hl 1 = P 0 ; ;s hl 1 hu s hl 1 ; s hl 1 ; ; ; h L 1, >: U s 0 ; s hl 1 ; ; ; h T 1 R h L 1 1\ = R k h L 1 k=1 0 ; ; s hl 1 9 >= 0 >; 15

16 [l = L 2; :::0] For each h l 2 H l, and for each 2, let R 0 ; h l n = s hl 2 S hl : 8a l+1 2 A l+1 h l, s hl j h l ; a l+1 2 R ; h l ; a l+1o : For each k = 1; 2; :::, let 8 s hl 2 R k 1 ; h l : (R.1): R k 1 h l >< R k ; h l (R.2): for all s 0 2 S hl, = P hu s hl ; s hl ; ; ; h l R h l = >: 1\ k=1 0 ; ;s hl U s 0 ; s hl ; ; ; h l R k h l Proposton 2 BR = R (h 0 ) for each. 0 ; ; s hl 9 >= 0 >; The propertes UIA and CCFR dscussed n Secton 3 provde the basc nsght behnd ths result. Frst, notce that under UIA, the set of belefs agents are allowed to entertan about the state of nature and the opponents payo -types s the same at every hstory ( 0 ). Hence, n ths respect, ther nformaton about the opponents types n the contnuaton game from hstory h s the same as f the game started from h. Also, CCFR mples that agents assumptons about everyone s behavor n the contnuaton s also the same at every hstory. Thus, UIA and CCFR combned mply that, from the pont of vew of BR, a contnuaton from hstory h s equvalent to a game wth the same space of uncertanty and strategy spaces equal to the contnuaton strateges, whch just es the possblty of analyzng contnuaton games n solaton. 6 Epstemc Characterzatons Ths Secton provdes two alternatve epstemc characterzatons of BR, one n terms of Common Certanty of Future Ratonalty, the other n terms of Common Certanty of Full Ratonalty and n Belef Persstence. The epstemc models adopted here have one mportant non-standard feature. exstng epstemc models (e.g., Battgall and Snscalch 2002, 2007), a state of the world s a tuple! = 0 ; ( ; s ; ) 2N,.e. a descrpton of a state of nature (0 ), and for each player, hs payo type, strategy s and epstemc type. Each epstemc type nduces a CPS over the states of world. state of the world s of the form! = In the epstemc models consdered here, a 0 ; ( ; x ; ) 2N, where x :! S denotes player s ex-ante strategy. Ex-ante strateges represent a full theory of s behavor: 16 In

17 n state! =! 0 ; (! ; x! ;! ) 2N 2, player s actual dsposton to act s gven by strategy x! (! ), but x! also represents player s dsposton to act under the hypothess (counterfactual at!) that hs payo type s d erent from!. Epstemc types stll nduce a CPS over the states of the world!, but ths means that now they express rcher nformaton. For nstance, consder a state at whch s belefs are concentrated on a par ^ ; ^x. Ths means that beleves that the opponents type s ^, and that hs actual strategy wll be ^x ^. But t also expresses s vew of how the opponents behavor would be under the hypothess that 0 6=. Ths enrchment of the state space allows to dsentangle epstemc counterfactuals concernng the opponents behavor from counterfactuals concernng the opponents prvate nformaton, necessary for the characterzaton of BR n terms of Common Certanty of Full Ratonalty and Belef Persstence. 13 The characterzaton n terms of Common Certanty of Future Ratonalty doesn t explot the nformaton contaned n such counterfactual statements. The epstemc characterzatons below wll be of the form BR = f( ; s ) : 9! 2 E s.t. (! ; x! (! )) = ( ; s )g, (7) for some measurable event E. Snce each X can be seen as a subset of S, equaton (7) s wrtten for short as BR = proj S E. 6.1 Condtonal Probablty Systems Let be a metrc space and A ts Borel sgma-algebra. Fx a non-empty collecton of subsets C An;, to be nterpreted as relevant hypothess. A condtonal probablty system (CPS hereafter) on (; A; C) s a mappng : A C! [0; 1] such that: Axom 1 For all B 2 C, (B) [B] = 1 Axom 2 For all B 2 C, (B) s a probablty measure on (; A). Axom 3 For all A 2 A, B; C 2 C, f A B C then (B) [A] (C) [B] = (C) [A]. The set of CPS on (; A; C), denoted by C (), can be seen as a subset of [ ()] C (.e. mappngs from C to probablty measures over (; A)). CPS s wll be wrtten as = ( (B)) B2C 2 C (). The subsets of n C are the condtonng events, each nducng belefs over ; () s endowed wth the topology of weak convergence of measures and 13 In envronments wth complete nformaton, Stalnaker (1996, 1998) dscusses the necessty of ncorporatng d erent knds of counterfactual propostons for the analyss of backward nducton reasonng. The ntroducton of ex-ante strateges n the de nton of the states of the world s one way of accomodatng the rcher set of counterfactuals requred by the presence of ncomplete nformaton. 17

18 [ ()] C s endowed wth the product topology. In the belef-free games of Secton 2.1, player s CPS were obtaned settng = 0 S, and the set of condtonng events was naturally provded by the set of hstores H: for each publc hstory h 2 H, the correspondng event [h] was de ned as [h] = 0 S (h). 6.2 Epstemc Models Ex ante strateges assgn an nterm strategy to every payo type. The set of ex ante strateges s denoted by X = S. For each x 2 X, let x jh denote the contnuaton strategy x jh :! S h s.t. for each 2, x jh ( ) = x ( ) jh. De nton 8 A (dynamc) type space on s a tuple D = N; H; ( ; X ; ; ; g ) 2N s.t. for every 2 N, s a Polsh space, and 1. X s a closed subset such that proj X = X 2. g :! H ( 0 ) s a contnuous mappng. For any 2 N, the elements of the set are referred to as Player s epstemc types. A type space s compact f all the sets ( ) 2I are compact. Let = 0 1 ::: n denote the set of states of the world, and let A be ts Borel sgma-algebra. Thus, at any possble world! 2 = 0 1 ::: n, we specfy a payo state (! 0 ;! 1 ; :::;! n) as well as each player s dspostons to act (hs strategy s! = x! (! )) and hs dspostons to beleve (hs CPS g (! ) = (g ;h (! )) h2h ). As dscussed above, x! also spec es s belefs about hs own strategy under the counterfactual (at!) hypothess that 0 6=!. Furthermore, notce that these dspostons also nclude what a player would do and thnk at hstores that are nconsstent wth! (.e. h =2 H (s! )). As standard n nteractve epstemology lterature, we assume that players also have belefs about ther own strategy, and complete a player s system of condtonal belefs by assumng that he s certan of hs true ex ante strategy, nformaton type and epstemc type. More spec cally, we assume that for every state of the world! and every hstory h, player 2 N would be certan of! and! gven h, and would also be certan of x! gven h, provded that x! (! ) 2 S (h). We also assume that f x! (! ) =2 S (h), player would stll be certan that hs contnuaton strategy agrees wth x! () jh for each. Formally, player s condtonal belefs on (; H) are gven by a contnuous mappng g = g;h : h2h! H () derved from g by the followng formula: for all ( ; x ; ) 2, for all h 2 H and E 2 A, g ;h ( ; x ; ) = g ;h ( ) (f( 0 ;! ) 2 0 : ( 0 ;! ; ( ; x 0 ; )) 2 E; for some x 0 2 X s.t. x 0 ( 0 ) jh = x ( 0 ) jh for all 0 2 g) : (8) 18

19 Type spaces encode a collecton of n nte herarches of CPSs for each epstemc type of each player. Battgall and Snscalch (1999) constructed the unversal type space for CPS,.e. a type space whch encodes all concevable herarchcal belefs. Consder the followng de nton: De nton 9 A belef-complete type space s a type space D such that, for every 2 N, g :! H ( 0 ) s onto. Battgall and Snscalch (1999) showed that a belef-complete type space may always be constructed (for all nte games, and also a large class of n nte games) by takng the sets of epstemc types to be the collecton of all possble herarches of condtonal probablty systems that satsfy certan ntutve coherency condtons. Also, every type space may be vewed as a belef-closed subspace of the space of n nte herarches of condtonal belefs. Fnally, snce we assume that the set of external states s nte and hence compact, the belef-complete type space thus constructed s also compact. Sequental h Ratonalty. Fx a type space D. For every player 2 N, let f = (f ;h ) h2h :! ( 0 X S ) H denote hs rst-order belef mappng, that s, for all! 2 and h 2 H, f ;h (! ) [ 0 ; ; x ; s ] = Z!2 s.t. (! 0 ;! ;x! ;x! (! ))=( 0; ;x ;s ) dg ;h (! ) : (9) It s easy to see that f s contnuous and that, for every! 2, f (! ) s ndeed a CPS over X S, where for each h 2 H, the correspondng condtonng event [h] ( 0 X S ) s de ned as [h] = f( 0 ; ; x ; s ) : (x ( ) ; s ) 2 S (h)g (10) For any CPS 2 H ( 0 X S ), de ne the map : H ( 0 X S )! H ( 0 S) so that, for each 2 H ( 0 X S ), () = ;h () h2h and for each h 2 H and [ 0 ; ; s ; s ] 2 0 S, ;h () [ 0 ; ; s ; s ] = x 0 :x0 ( X (h) 0 ; ; x 0 ; s )=s In words, () represents the payo relevant component of, obtaned by gnorng the nformaton on counterfactual belefs encoded by x! 6=!. It s easy to verfy that () thus de ned s a CPS whenever s. (11) 19

20 Fnally, we can de ne ratonalty: we say that player s ratonal at a state! f and only f (! ; x! (! )) 2 r ( (f (! ))). Then the event Rat = f! 2 : (! ; x! (! )) 2 r ( (f (! )))g (12) corresponds to the statement player s ratonal. De ne also the event everybody s ratonal : R = T 2N R : Belefs Operators. The next buldng block s the epstemc noton of (condtonal) probablty one belef (or certanty). Recall that an epstemc type encodes the belefs a player would hold, should any one of the possble non-termnal hstores occur. Ths allows us to formalze statements such as, Player would be certan that Player j s ratonal, were he to observe hstory h. Gven a type space D, for every 2 N, h 2 H and E 2 A, de ne the event B ;h (E) =! 2 : g;h (! ) [E] = 1 whch corresponds to the statement Player would be certan of E, were he to observe hstory h. Observe that ths de nton ncorporates the natural requrement that a player only be certan of events whch are consstent wth her own (contnuaton) strategy and epstemc type (recall how g was de ned, equaton 8). Recallng that h 0 s the empty hstory, B ;h 0 (E) s the event Player beleves E at the begnnng of the game. For each player 2 I and hstory h 2 H, the de nton dent es a set-to-set operator B ;h : A! A whch sats es the usual propertes of fals able belefs (see, for example, Chapter 3 of Fagn et al. (1995)); n partcular, t sats es: Conjuncton: For all events E; F 2 A, B ;h (E \ F ) = B ;h (E) \ B ;h (F ); Monotoncty: For all events E; F 2 A, E F mples B ;h (E) B ;h (F ). The mutual belef operator s de ned as B h (E) = T 2N B ;h (E). We wll be nterested n terated belefs operators. In general, x a self-map on A, : A! A. For any E 2 A, let 1 (E) = (E), and for k = 2; 3; :::, de ne k (E) = k 1 (E). The event common belef n E at h s thus de ned as T k1 Bk h (E). 6.3 Common Certanty of Future Ratonalty For every h, every and 2 H ( 0 S), let r ( jh) denote the set of pars payo type-strategy that are sequental best response to CPS from hstory h onwards. 20

21 That s: for any 2 H ( S), ( ; s ) 2 r jh f and only f 8h 0 h, s 2 arg max s 0 2S U s 0 ; ; h 0 ; : 14 For every and h 2 H, the event s ratonal n the contnuaton from h s de ned as follows: Rat ;h f! 2 : (! ; x! (! )) 2 r ( (f (! )) jh)g ; (13) The event ratonalty n the contnuaton from h s de ned as Rat h = T Rat ;h. (Clearly, Rat = Rat h 0) The event common certanty of future ratonalty (CCFR) corresponds to the assumpton that, at each pont n the game, there s common certanty that everybody s ratonal n the contnuaton game. It s formally de ned as: CCF R = \! \ Bh k (Rat h ) : : h2h k1 The next proposton shows that BR corresponds to the epstemc assumptons of ratonalty and common certanty of future ratonalty. Proposton 3 BR = proj S Rat \ CCF R Proof. (See Appendx ) 6.4 Belef Persstence We ntroduce here an alternatve characterzaton of BR. De nton 10 For each 2 N let BP 0 2, let BP k 1 = j2nnfg BP k 1 8 >< BP k ( ) = >: ^s 2 BP k 1 : j, 2N = S. Recursvely, for k = 1; 2; :::, and 9 2 H ( 0 X S ) s.t. (A) ^s 2 r () ; (B) supp marg X S (h 0 ) BP k 1 f^s g (C) for each h 2 H: supp marg X h S h (h) = supp marg X h S h (h 0 ) BP k = ( ; s ) 2 S : s 2 BP k (t ), BP k = k 2N BP, and nally BP := T 14 For a de nton of U s ; ; h 0 ;, see equaton 1, p. 8. k0 9 >= >; BP k. 21

22 Condton (C) can be nterpreted as a belef persstence hypothess: at each hstory (whether t s reached wth postve probablty or not), the support of the conjectures about the contnuaton strateges (one s own and the opponents ) must not change. Condton (C) thus conforms to what n the phlosophy lterature s known as the conservatvty prncple. Such prncple states that When changng belefs n response to new evdence, you should contnue to beleve as many of the old belefs as possble (Harman, 1986, p. 46). 15 Here, agent s conjectures over X h represent hs theory of everyone s (future) behavor n the game, both n the states ( 0 ; ) n the support of and those that are not (smlarly, conjectures over S h regard s belefs about hs own contnuaton). De nton 11 requres that upon observng hstory h, agent revses hs belefs so to accomodate the new nformaton but wthout changng hs theory about evreyone s behavor n the contnuaton game. Notce though that no restrcton are mposed on the agent s belefs about the opponents prvate nformaton,.e. on marg (h). For example, t may concentrate belefs on a payo type pro le for whch the observed hstory s rratonal, but the restrcton on the belefs on the contnuaton strateges entals that t s beleved that type wll behave ratonally n the future. In ths sense, belef persstence here only refers to the endogenous varables (strateges), not the exogenous ones (payo -types). Proposton 4 For each, and k, BR k = BP k. Proof. (See Appendx) Common Certanty of full Ratonalty and Belef Persstence. The event player s ratonal, de ned n equaton 12, conssts of the set of states of the world! n whch s actual behavor x! (! ) s a (sequental) best response to s belefs at that state. No restrctons are mposed on the counterfactual behavor, x! ( 0 ) for 0 6=!. We ntroduce next a stronger noton of ratonalty, whch also restrcts s counterfactual behavor. Rat = f! 2 R : 8 0 2, ( 0 ; x! ( 0 )) 2 proj S Rat g (14) Rat can thus be nterpreted as the set of states of the world n whch not only s ratonal, but also hs dsposton to act n the (counterfactual) hypothess that he observed a d erent payo -type s consstent wth ratonalty. The event Rat thus descrbes states of the world n whch s ex-ante strategy s ratonal. Rat s referred to as the event s fully ratonal. As above, the event everyone s fully ratonal s de ned as Rat = T Rat : 15 The conservatvty prncple, or the prncple of belef persstence, s one of the gudng prncples adopted n the phlosophy lterature on theores of ratonal belef changes (see, e.g., Gardenfors, 1988). Battgall and Bonanno (1991) provde a semantc and synctatc characterzaton of the prncple. 2N 22

23 De nton 11 Say that -restrctons satsfy the belef persstence hypothess f: 8 2 N, 8 2 BP = 2 H ( 0 X S ) : 8h 2 H, o supp marg (h) X h = supp marg S h X h S h (h0 ) : For gven collecton = (( ) 2 ) 2N where H ( 0 X S ), de ne the event [ ] = f( 0 ; ; x ; ;! ) 2 : f ( ; x ; ) 2 g ; [ ] = \ \ [ ] and [] = [ ] 2 2N Proposton 5 For any belef-complete type space: 1. BR 1 = proj S Rat \ BP 2. for every k 1, BR k+1 = proj S Rat \ BP \ k T =1 B h 0 Rat \ BP ; 3. f the type space s also compact, then BR = proj S Rat\ BP \ T 7 Further Remarks on the Soluton Concepts. Backwards procedure, Subgame-Perfect Equlbrum and IPE. k=1 B k h 0 Rat \ BP. In games wth complete and perfect nformaton, the backwards procedure R (h 0 ) concdes wth the backward nducton soluton, hence wth subgame perfecton. 16 The next example (borrowed from Perea, 2009) shows that f the game has complete but mperfect nformaton, strateges played n Subgame-Perfect Equlbrum (SPE) may be a strct subset of R (h 0 ): Example 1 Consder the game n the followng gure: 16 For the specal case of games wth complete nformaton, Perea (2009) ndependently ntroduced a procedure that s equvalent to R, and showed that R concdes wth the backward nducton soluton f the game has perfect nformaton. 23

24 R 1 (h 0 ) = fbc; bd; acg and R 2 (h 0 ) = ff; gg. The game though has only one SPE, n whch player 1 chooses b: n the proper subgame, the only Nash equlbrum entals the mxed (contnuaton) strateges 1 2 c d and 3 4 f g, yeldng a contnuaton payo of 11 4 for player 1. Hence, player 1 chooses b at the rst node. In games wth complete nformaton, IPE concdes wth SPE, but R (h 0 ) n general s weaker than subgame perfecton. At rst glance, ths may appear n contradcton wth propostons 1 and 2, whch mply that R (h 0 ) characterzes the set of strateges played n IPE across models of belefs. The reason s that even f the envronment has no payo uncertanty ( s a sngleton), the complete nformaton model n whch B s a sngleton for every s not the only possble: models wth redundant types may exst, for whch IPE strateges d er from the SPE-strateges played n the complete nformaton model. The source of the dscrepancy s analogous to the one between Nash equlbrum and subjectve correlated equlbrum (Aumann, 1974). We llustrate the pont constructng a model of belefs and an IPE n whch strategy (ac) s played wth postve probablty by some type of player Let payo s be the same as n example 1, and consder the model T such that T 1 = n o t bc 1 ; t bd 1 ; t ac 1 and T 2 = t f 2; t g 2, wth the followng belefs: ( h 1 (t 1 ) t f 1 f t 1 = t bc 1 ; t ac 1 2 = 0 otherwse and 2 (t g 2) t ad 1 = 1, 2 t f 2 t bc 1 = 1 The equlbrum strategy pro le s such that 8; 8t, b (t s ) = s. The system of belefs agrees wth the model s belefs at the ntal hstory, hence the belefs of types t g 2 and t ac 1 17 It s easy to see that such d erence s not merely due to the possblty of zero-probablty types. Also the relaxaton of the common pror assumpton s not crucal. 24

25 are unquely determned by Bayesan updatng. For types t s 6= t g 2; t ac 1, t s su cent to set p (t s ; a ) = (t s ) (.e. mantan whatever the belefs at the begnnng of the game were) Then, t s easy to verfy that (; p) s an IPE. On the other hand, f jj = 1 and the game has perfect nformaton (no stage wth smultaneous moves), then R (h 0 ) concdes wth the set of SPE-strateges. Hence, n envronments wth no payo uncertanty and wth perfect nformaton, only SPE-strateges are played n IPE for any model of belefs. 8 Concludng Remarks. On the Soluton Concepts. Proposton 1 can be seen as the dynamc counterpart of Brandenburger and Dekel s (1987) characterzaton of correlated equlbrum. The weakness of IPE (relatve to other notons of perfect Bayesan equlbrum) s key to that result: the heart of proposton 2 s BR s property of contnuaton-game consstency, whch allows to analyze contnuaton games n solaton, n analogy wth the logc of backward nducton. The CCFR and UIA assumptons (p. 9) provde the epstemc underpnnngs of the argument. To understand ths pont, t s nstructve to compare BR wth Battgall and Sncalsch s (2007) weak and strong versons of extensve form ratonalzablty (EFR), whch correspond respectvely to the epstemc assumptons of (ntal) common certanty of ratonalty (CCR) and common strong belef n ratonalty (CSBR): BR s stronger than the rst, and weaker than the latter. The strong verson of EFR fals the property of subgame consstency because t s based on a forward nducton logc, whch nherently precludes the possblty of envsonng contnuatons n solaton : by takng nto account the possblty of counterfactual moves, agents may draw nferences from ther opponents past moves and re ne ther conjectures on the behavor n the contnuaton. The weak verson of EFR fals contnuaton-game consstency for opposte reasons: an agent can make weaker predctons on the opponents behavor n the contnuaton than he would make f he envsoned the contnuaton game n solaton, because no restrctons on the agents belefs about ther opponents ratonalty are mposed after an unexpected hstory. Thus, the form of backward nducton reasonng mplct n IPE (whch generalzes the dea of subgame perfecton) s based on stronger (respectvely, weaker) epstemc assumptons than CCR (respectvely, CSBR). Appendx 25