Dynamic Unawareness and Rationalizable Behavior

Transcription

1 Dynamc Unawareness and Ratonalzable Behavor Avad Hefetz Martn Meer Burkhard C. Schpper February 18, 2010 Abstract We defne generalzed extensve-form games whch allow for mutual unawareness of actons. We extend Pearce s (1984) noton of extensve-form (correlated) ratonalzablty to ths settng, explore ts propertes and prove exstence. We defne also a new varant of ths soluton concept, prudent ratonalzablty, whch refnes the set of outcomes nduced by extensve-form ratonalzable strateges. We apply prudent ratonalzablty to the analyss of verfable communcaton wth unawareness. Fnally, we defne the normal form of a generalzed extensve-form game, and characterze n t extensve-form ratonalzablty by teratve condtonal domnance. Keywords: Unawareness, extensve-form games, extensve-form ratonalzablty, prudent ratonalzablty, teratve condtonal domnance. JEL-Classfcatons: C70, C72, D80, D82. We are grateful to Perpaolo Battgall for numerous nsghtful comments and suggestons and to Elchanan Ben Porath for helpful dscussons and exchange of deas. We thank Paulo Barell, Andy McLennan, Yoram Halevy as well as to semnar partcpants at Barcelona, Boccon, Caltech, Maryland, Maastrcht, Pttsburgh, USC, UC Davs, Tel Avv, Stony Brook 2007, LOFT 2008, Games 2008 and NSF/NBER/CEME Avad s grateful for fnancal support from the Open Unversty of Israel s Research Fund grant no Martn was supported by the Spansh Mnstero de Educacón y Cenca va a Ramon y Cajal Fellowshp (IAE-CSIC) and a Research Grant (SEJ ). Burkhard s grateful for fnancal support from the NSF SES The Economcs and Management Department, The Open Unversty of Israel. Emal: avadhe@openu.ac.l Insttut für Höhere Studen, Wen, and Insttuto de Análss Económco - CSIC, Barcelona. Emal: martn.meer@hs.ac.at Department of Economcs, Unversty of Calforna, Davs. Emal: bcschpper@ucdavs.edu

2 1 Introducton In real-lfe dynamc nteractons, unawareness of players regardng the relevant actons avalable to them s at least as prevalent as uncertanty regardng other players strateges, payoffs or moves of nature. Players frequently become aware of actons they (or other players) could have taken n retrospect, when they can only re-evaluate the past actons chosen by partners or rvals who were aware of those actons from the start, and hence re-assess ther lkely future behavor. Yet, whle uncertanty can be captured wthn the standard framework of extensve-form games wth mperfect nformaton, unawareness and mutual uncertanty regardng awareness requre an extenson of ths framework. Such an extenson s the frst task of the current paper. At frst, one may wonder why the standard framework would not suffce. After all, f a player s unaware of an acton whch s actually avalable to her, then for all practcal purposes she cannot choose t. Why wouldn t t be enough smply to truncate from the tree all the paths startng wth such an acton? The reason s that the strategc mplcatons of unawareness of an acton are dstnct from the unavalablty of the same acton. To see ths, consder the followng standard battle-of-the-sexes game (where Bach and Stravnsky concerts are the two avalable choces for each player) B II S B 3, 1 0, 0 I S 0, 0 1, 3 augmented by a domnant Mozart concert for player II: II B S M B 3, 1 0, 0 0, 4 I S 0, 0 1, 3 0, 4 M 0, 0 0, 0 2, 6 The new game s domnance solvable, and (M,M) s the unque Nash equlbrum. Suppose that the Mozart concert s n a dstant town, and II can go there only f player I gves hm her car n the frst place: Here, f player I doesn t gve the car to player II, player II may conclude by forward nducton that player I would go to the Bach concert 2

3 Fgure 1: I I II B S B 3, 1 0, 0 S 0,0 1,3 M 0, 0 0, 0 not gve car to player II gve car to player II II B S M B 3, 1 0, 0 0, 4 I S 0, 0 1, 3 0, 4 M 0, 0 0, 0 2, 6 wth the hope of gettng the payoff 3 (because by gvng the car to II, player I could have acheved the payoff 2). The best reply of player II s to follow sut and attend the Bach concert as well. Hence, n the unque ratonalzable outcome, player I s not to gve the car to player II and to go to the Bach concert. 1 But what f, nstead, the Mozart concert s n town but player II s ntally unaware of the Mozart concert, whle player I can enable player II to go to the concert smply by tellng hm about t? If player II remans unaware of the Mozart concert, then nether does he conceve that player I could have told hm about the Mozart concert, and n partcular he cannot carry out any forward-nducton calculaton. For hm, the game s a standard battle-of-the-sexes game, where both actons of player I are ratonalzable. Ths strategc stuaton s depcted n Fgure 2. Fgure 2: I I II B S B 3, 1 0, 0 S 0, 0 1, 3 M 0, 0 0, 0 not tell player II about the Mozart concert tell player II about the Mozart concert II B S M B 3, 1 0, 0 0, 4 I S 0, 0 1, 3 0, 4 M 0, 0 0, 0 2, 6 I II B S B 3, 1 0, 0 S 0, 0 1, 3 1 For a dscusson of forward nducton n battle-of-the-sexes games see van Damme (1989). 3

4 The strategc stuaton s not a standard extensve-form game (more on ths n Secton 2.6 below). If player I chooses not to tell player II about the Mozart concert, then player II s nformaton set (depcted n blue) conssts of a node n a smpler game namely the one-shot battle-of-the-sexes wth no precedng move by player I. Ths s a smple example of the general novel framework that we defne n Secton 2 for dynamc nteracton wth possbly mutual unawareness of actons, generalzng standard extensve-form games. The framework wll not only allow modelng of stuatons n whch one player s certan that another player s unaware of portons of the game tree, as n the above example, but also of stuatons n whch a player s uncertan regardng the way another player vews the game tree, as well as stuatons n whch the player s uncertan regardng the uncertantes of the other player about yet other players vews of the game tree, and so forth. In fact, ths framework allows not just for unawareness but also for other forms of msconcepton about the structure of the game. Secton 6 specfes further propertes obtanng n generalzed extensve-form games where the only source of players msconcepton s unawareness and mutual unawareness of avalable actons and paths n the game. Snce we focus on ths type of unawareness, most of the examples n the paper satsfy the further propertes specfed n Secton 6. Nevertheless, modelng awareness of unawareness does requre the general framework n Secton 2, as explaned at ts end. In ths new framework, for each nformaton set of a player her strategy specfes from the pont of vew of the modeler what the player would do f and when that nformaton set of hers s ever reached. In ths sense, a player does not necessarly own her full strategy at the begnnng of the game, because she mght not be ntally aware of all of her nformaton sets. That s why a sensble generalzaton of Pearce s (1984) noton of extensve-form ratonalzablty s non-trval. In Secton 3 we put forward a modfed defnton, prove exstence, and show the sense n whch t concdes wth extensve-form ratonalzablty n standard extensveform games. We focus here on a ratonalzablty soluton concept rather than on some noton of equlbrum. Whle an equlbrum s deally nterpreted as a rest-pont of some dynamc learnng or adaptaton process, or alternatvely as a pre-medtated agreement or expectaton, we fnd t dffcult to carry over such nterpretatons to a settng n whch every ncrease of awareness s by defnton a shock or a surprse. Once a player s vew of the game tself s challenged n the course of play, t s hard to justfy the dea that a conventon or an agreement for the contnuaton of the game are readly avalable. 4

5 We chose to focus on extensve-form ratonalzablty because t embodes forward nducton reasonng. If an opponent makes a player aware of some relevant aspect of realty, t s mplausble to dsmss the ncreased level of awareness as an unntended consequence of the opponent s behavor. Rather, the player should try to ratonalze the opponent s choce, re-nterpret the opponent s past behavor, and try to nfer from t the opponent s future moves. Extensve-form ratonalzablty ndeed captures a best ratonalzaton prncple (Battgall, 1997). Wth ratonalzablty, generalzed games are necessary for properly modelng unawareness; tryng to model unawareness by havng the unaware player assgnng probablty zero to the contngency of whch she s unaware mght gve rse to a completely dfferent ratonalzable behavor, whch does not square wth unawareness n the proper sense of the word. To see ths consder the followng example. A Decson Maker (DM) has to choose between two polces, a 0 and a 1. Before choosng she gets a recommendaton from an expert va a narrow communcaton channel, through whch the expert can recommend ether 0 or 1. The expert makes the recommendaton after observng the state of nature, whch may be ether γ 0 or γ 1, and whch the DM does not see. The nterests of the expert and the DM are completely algned: They each bear a cost of 1 f a 1 s mplemented when the state of nature s γ 0 or vce versa. The expert furthermore bears a cost of 10 from lyng,.e. from recommendng 0 when the state of nature s γ 1 or recommendng 1 when the state of nature s γ 0. Assume the DM s aware only of the state γ 0 and unaware of γ 1. The dynamc nteracton s hence modeled by the generalzed game n Fgure 3. E Fgure 3: c γ 0 γ 1 E 0 1 DM 0 1 DM DM DM a 0 a 1 a 0 a 1 a 0 a 1 a 0 a 1 0, 0 1, 1 0, 10 1, 11 1, 11 0, 10 1, 1 0, 0 E DM 0 1 γ 0 DM c a 0 a 1 a 0 a 1 0, 0 1, 1 0, 10 1, 11 5

6 In ths generalzed game the only extensve-form ratonalzable strategy of the DM s to always mplement the polcy a 0 : she does not conceve of a contngency that would make the polcy a 1 superor to a 0 even f she hears from the expert the recommendaton 1 ; n such a case she regrettably concludes that the expert behaved n an rratonal way and bore the cost of lyng. However, f we were to model the DM alternatvely as beng aware of γ 1 but assgnng probablty zero to t, the strategc nteracton would be modeled by the standard game n Fgure 4. Fgure 4: c γ 0 γ 1 E E DM DM a 0 a 1 a 0 a 1 a 0 a 1 a 0 a 1 0, 0 1, 1 0, 10 1, 11 1, 11 0, 10 1, 1 0, 0 In ths game the unque extensve-form ratonalzable strategy of the DM s to choose a 0 upon hearng 0 from the expert, but to mplement a 1 upon hearng the recommendaton 1. Indeed, extensve-form ratonalzablty requres the DM to base her choce on a system of belefs about the expert s strateges wth whch at every nformaton set of hers she mantans a belef that best ratonalzes the choces of the expert whch could have led to that nformaton set. In partcular, upon hearng the recommendaton 1 from the expert, the only way for the DM to ratonalze t s to assume that the state of nature s nevertheless γ 1, where recommendng 1 s strctly domnant for the expert; and n γ 1 the optmal choce for the DM s a 1. Conceptually, upon hearng the surprsng recommendaton 1 both choces of the DM have ther nternal logc. The former gves prorty to only γ 0 s concevable, the latter to the ratonalty of the expert. But n the latter case, f ntally the DM s genunely unaware of γ 1, there s no reason why the DM would conceve precsely of γ 1 and not of some alternatve descrpton γ 1 of nature that would also ratonalze the expert s recommendaton 1 ; some such conceptualzatons γ 1 need not necessarly nduce the DM to adopt the expert s recommendaton. Generalzed games lend themselves also to modelng such msconceptons that may arse upon a surprse, as demonstrated n Fgure 5. Here, the DM s ratonalzable strategy s to choose a 0 also upon hearng the (surprsng) recommendaton 1, because the DM beleves ths recommendaton was 6

7 Fgure 5: c γ 0 γ 1 E E 0 1 DM 0 1 DM DM DM a 0 a 1 a 0 a 1 a 0 a 1 a 0 a 1 0, 0 1, 1 0, 10 1, 11 1, 11 0, 10 1, 1 0, 0 E DM 0 1 c γ 0 E 0 1 DM DM a 0 a 1 a 0 a 1 0, 0 1, 1 0, 10 1, 11 γ 0 DM c γ 1 DM E 0 1 a 0 a 1 a 0 a 1 0, 11 1, 10 0, 1 1, 0 a 0 a 1 a 0 a 1 0, 0 1, 11 0, 10 1, 11 strctly domnant for the expert but that her nterest and those of the expert are now opposed. In Secton 4 we ntroduce a related soluton concept, prudent ratonalzablty, whch s the drect generalzaton of terated admssblty to dynamc games wth unawareness. Unlke n normal-form games, ths generalzaton s surprsngly not always a refnement of extensve-form ratonalzablty (even for standard extensve-form games). However, we prove that prudent ratonalzable strateges do refne the set of outcomes obtanable by extensve-form ratonalzable strateges. We show how prudent ratonalzablty s effectve n rulng out less plausble ratonalzable outcomes n examples due to Pearce (1984) and Ozbay (2007). Of partcular nterest s the applcaton of prudent ratonalzablty to the Mlgrom- Roberts (1986) communcaton game, n whch a sender sends a verfable (and hence correct) pece of nformaton to a recever who makes a decson on ts bass. Mlgrom and Roberts (1986) showed that the unque sequental equlbrum n ths game features full unravelng of nformaton, and that at equlbrum the recever nterprets each pece of nformaton n the most skeptcal manner. We show that the complete unravelng outcome s also the unque outcome n prudent strateges, and hence that t hnges on ratonalzablty (or, more precsely, on prudence) consderatons and does not requre the full power of equlbrum analyss. Nevertheless, we show that f the certfed nformaton has multple dmensons and the recever s unaware of some of them, then complete unravelng need not occur wth prudent strateges. Thus, ths s yet another example n 7

8 whch unawareness has strategc mplcatons whch are genunely dfferent than those mpled by asymmetrc nformaton. In standard game theory, the extensve form has been consdered as a more complete descrpton of the strategc stuaton than the normal form. Ths has been questoned by Kohlberg and Mertens (1986) who argued that the normal form contans all strategcally relevant nformaton. For standard extensve-form games, Shmoj and Watson (1998) showed how extensve-form reasonng emboded n extensve-form ratonalzablty can be carred out n the normal form. Arguably generalzed extensve-form games contan more tme relevant structure than standard extensve-form games snce they also formalze changes n the awareness of players. It s therefore an ntrgung queston whether a soluton to generalzed extensve-form games can be found when the analyss s carred out n the approprately defned normal form assocated to a generalzed extensve-form game. In Secton 5 we defne the normal form assocated to general extensve-form games. We extend Shmoj and Watson characterzaton of extensve-form ratonalzablty by terated condtonal strct domnance to games wth unawareness. In some applcatons, t may be more practcal to apply terated condtonal strct domnance n the normal form rather than extensve-form ratonalzablty. Our framework for dynamc nteracton under unawareness seems to be smpler than the one proposed by Halpern and Rêgo (2006) and Rêgo and Halpern (2007), n whch they nvestgated the notons of Nash and sequental equlbrum, respectvely. 2 Fenberg (2009) defnes unawareness by explct unbounded sequences of mutual vews of the game, wth analogous propertes both for statc and for dynamc games. In hs dynamc settng, he does not mpose perfect recall, whch mght hamper the extenson of known soluton concepts such as sequental equlbrum or extensve-form ratonalzablty that rely on perfect recall; n contrast, extensve-form ratonalzablty and prudent ratonalzablty are the focal soluton concepts that we extend and defne and analyze n our paper, and to ths effect we extend the defnton of perfect recall to our settng. L (2006) consdered dynamc unawareness wth perfect nformaton, whle our framework allows for both unawareness and mperfect nformaton. Ozbay (2007) studes sender-recever games, n whch an announcer can make an unaware decson maker aware of more states of nature before the decson maker takes 2 The smplfcaton obtaned n our framework s due to the fact that our ntal buldng block s a tree representng physcal moves, wth nformaton sets defned only n the sub-trees whch represent subjectve vews of the game; n contrast, Halpern and Rêgo (2006) had nformaton sets defned already n ther basc tree. As a result, not all sub-trees could be consdered, and Halpern and Rêgo (2006) had to postulate addtonal condtons relatng the nformaton sets n sub-trees to those of the basc tree. 8

9 an acton. Such games can also be naturally formulated as a partcular nstance of our framework. For these games Ozbay studes an equlbrum noton ncorporatng forward-nducton reasonng. Flz-Ozbay (2007) studes a related settng n whch the aware announcer s a rsk neutral nsurer, whle the decson maker s a rsk averse or ambguty averse nsuree. At equlbrum, the nsurer does not always reveal all relevant contngences to the nsuree. 3 Our am s to provde a general framework for modelng msperceptons about the avalablty of actons n dynamc strategc stuatons. Dfferent knds of percepton bases among players n games have been a popular topc n the recent lterature on behavoral game theory. For nstance, n statc games Eyster and Rabn (2005) analyze players wth correct conjectures about opponents actons but msperceptons about how those opponents actons are correlated wth the opponents nformaton. In mult-stage games wth moves of nature, Jehel (2005) studes players that bundle nodes at whch other players choose nto analogy classes, correctly antcpate the average behavor for each analogy class, and thus may have msperceptons about how others behavor s related others nformaton. Recently there has been a renassance of non-equlbrum teratve soluton concepts n behavoral game theory lke level-k thnkng and related models (e.g. Stahl and Wlson, 1995, Camerer, Hu and Chong, 2004, Crawford and Irberr, 2007). Note that our teratve soluton concepts, would-be ratonalzablty and prudent ratonalzablty, do not only provde behavoral predctons n the lmt but also at every fnte level of ratonalzaton. 2 Generalzed extensve-form games To defne a generalzed extensve-form game Γ, consder frst, as a buldng block, a fnte perfect nformaton game wth a set of players I, a set of decson nodes N 0, actve players I n at node n wth fnte acton sets A n of player I n (for n N 0 ), chance nodes C 0, and termnal nodes Z 0 wth a payoff vector (p z ) I R I for the players for every z Z 0. The nodes N 0 = N 0 C 0 Z 0 consttute a tree. 3 Currently we are unaware of further papers focusng drectly and explctly on dynamc games wth unawareness. The lterature on unawareness n general s growng fast see e.g. 9

10 2.1 Partally ordered set of trees Consder now a famly T of subtrees of N0, partally ordered ( ) by ncluson. One of the trees T 1 T s meant to represent the modeler s vew of the paths of play that are objectvely feasble; each other tree represents the feasble paths of play as subjectvely vewed by some player at some node at one of the trees. In each tree T T denote by n T the copy n T of the node n N 0 whenever the copy of n s part of the tree T. However, n what follows we wll typcally avod the subscrpt T when no confuson may arse. Denote by N T the set of nodes n whch player I s actve n the tree T T. We requre two propertes: 1. All the termnal nodes n each tree T T are copes of nodes n Z If for two decson nodes n, n N T (.e. I n I n ) t s the case that A n A n, then A n = A n.4 Property 1 s needed to ensure that each termnal node of each tree T T s assocated wth well defned payoffs to the players. Property 2 means that s actve nodes N T are parttoned nto equvalence classes, such that the actons avalable to player are dentcal wthn each equvalence class and dsjont n dstnct equvalence classes. It wll be needed for the defnton of nformaton sets whch follows shortly. 5 Denote by N the unon of all decson nodes n all trees T T, by C the unon of all chance nodes, by Z the unon of termnal nodes, and by N = N C Z (copes n T of a gven node n n dfferent subtrees T are dstnct from one another, so that N s a dsjont unon of sets of nodes). For a node n N we denote by T n the tree contanng n. 2.2 Informaton sets In standard extensve-form game, an nformaton set π (n) of a player s both (1) the set of nodes that the player consders as possble, and (2) the set of nodes n whch the 4 Sometmes the modeler may want to mpose an addtonal property: If n the orgnal tree the probabltes of reachng n 1,... n k N from the chance node c C are p n1 c > 0,..., p n k c > 0 but some of these nodes do not appear n the subtree, then the probabltes of reachng the remanng nodes emanatng from c are renormalzed so as to sum to 1 n the subtree. We do not mpose ths property here snce t may be natural n some contexts but unnatural n others. 5 The dea wll be that n a gven tree T, each acton wll correspond only to one vew the player can have regardng the way the dynamc nteracton has evolved that far, and wll hence be avalable at (all the nodes of) a unque nformaton set. 10

11 player has the same nformaton as n the nodes whch she consders as possble. In games wth unawareness the two notons need not concde, and our defnton of an nformaton set π (n) below generalzes (1) above; π (n) wll be n a dfferent tree than n the tree T n f at n the player s unaware of some of the paths n T n, and rather envsages the dynamc nteracton as takng place n the tree contanng π (n). Formally, for each decson node n N, defne for each actve player I n an nformaton set π (n) wth the followng propertes: I0 Confnement: π (n) T for some tree T. I1 No deluson: If π (n) T n then n π (n). I2 Introspecton: If n π (n) then π (n ) = π (n). I3 No dvnng of currently unmagnable paths, no expectaton to forget currently concevable paths: If n π (n) T (where T T s a tree) and there s a path n,..., n T such that I n I n then π (n ) T. I4 No magnary actons: If n π (n) then A n A n. I5 Dstnct acton names n dsjont nformaton sets: For a subtree T, f n, n T and A n = A n then π (n ) = π (n). I6 Perfect recall: Suppose that player s actve n two dstnct nodes n 1 and n k, and there s a path n 1, n 2,..., n k such that at n 1 player takes the acton a. If n π (n k ), then there exsts a node n 1 n and a path n 1, n 2,..., n l = n such that π (n 1) = π (n 1 ) and at n 1 player takes the acton a. The followng fgures (Fgure 6) llustrate propertes I0 to I6. Propertes (I1), (I2), (I4), and (I5) are standard for extensve-form games, and propertes (I0) and (I6) generalze other standard propertes of extensve-form games to our generalzed settng. The essentally new property s (I3). At each nformaton set of a player, property (I3) confnes the player s antcpaton of her future vew of the game to the vew she currently holds (even f, as a matter of fact, ths antcpaton s about to be shuttered as the game evolves). We denote by H the set of s nformaton sets n all trees. For an nformaton set h H, we denote by T h the tree contanng h. For two nformaton sets h, h n a gven tree T, we say that h precedes h (or that h succeeds h ) f for every n h there s a path n,..., n such that n h. We denote h h. 11

12 Fgure 6: Propertes I0 to I6 I0 n T T n I1 π (n) π (n) T n T n n n I2 n n n n I3 T n T n n T n T n n n I4 T n T n a b c a b I5 T n T n n n n n a b a b c a ba b a ba b I6 n 1 a n 1 1 a n l n l n k 12

13 Remark 1 The followng property s mpled by I2 and I4: If n, n h where h = π (n) s an nformaton set, then A n = A n. Proof. If n, n h where h = π (n) s some nformaton set, then by ntrospecton (I3) we must have π (n ) = π (n ) = π (n). Hence by (I4) A n A n and A n A n. Remark 2 Propertes I0, I1, I2 and I6 mply no absent-mndedness: No nformaton set h contans two dstnct nodes n, n on some path n some tree. Proof. Suppose by contradcton that there exsts an nformaton set h wth a node n h such that some other node n h precedes n n the tree T n. Denote by n the frst node on the path from the root to n that s also n h. Now apply I6 wth n l := n to get a path n = n 1,..., n l = n, wth π (n ) = π (n 1 ) = π (n ) = h. By I1, we have n h and n s a predecessor of n, a contradcton. The perfect recall property I6 and Remark 2 guarantee that wth the precedence relaton player s nformaton sets H form an arborescence: For every nformaton set h H, the nformaton sets precedng t {h H : h h } are totally ordered by. For trees T, T T we denote T T whenever for some node n T and some player I n t s the case that π (n) T. Denote by the transtve closure of. That s, T T ff there s a sequence of trees T, T,..., T T satsfyng T T T. 2.3 Generalzed games A generalzed extensve-form game Γ conssts of a partally ordered set T of subtrees of a tree N 0 satsfyng propertes 1-2 above, along wth nformaton sets π (n) for every n T, T T and I n, satsfyng propertes I0-I6 above. For every tree T T, the T -partal game s the partally ordered set of trees ncludng T and all trees T n Γ satsfyng T T, wth nformaton sets as defned n Γ. A T - partal game s a generalzed game,.e. t satsfes all propertes 1-2 and I0-I6. We denote by H T the set of s nformaton sets n the T -partal game. 13

14 2.4 Strateges A (pure) strategy s S h H A h for player specfes an acton of player at each of her nformaton sets h H. Denote by S = S j j I the set of strategy profles n the generalzed extensve-form game. If s = (a h ) h H S, we denote by s (h ) = a h the player s acton at the nformaton set h. If player s actve at node n, we say that at node n the strategy prescrbes to her the acton s (π (n)). In generalzed extensve-form games, a strategy cannot be conceved as an ex ante plan of acton. If h T but T T, then at h player may be nterpreted as beng unaware of her nformaton sets n H T \ H T. Thus, a strategy of player should rather be vewed as a lst of answers to the hypothetcal questons what would the player do f h were the set of nodes she consdered as possble?, for h H. However, there s no guarantee that such a queston about the nformaton set h H T would even be meanngful to the player f t were asked at a dfferent nformaton set h H T when T T. The answer should therefore be nterpreted as gven by the modeler, as part of the descrpton of the stuaton. For a strategy s S and a tree T T, we denote by s T the strategy n the T -partal game nduced by s. If R S s a set of strateges of player, denote by R T the set of strateges nduced by R n the T -partal game, The set of s strateges n the T -partal game s thus denoted by S T. Denote by S T = j I ST j T -partal game. the set of strategy profles n the We say that a strategy profle s S reaches the nformaton set h H f the players actons and nature s moves (f there are any) n T h lead to h wth a postve probablty. (Notce that unlke n standard games, an nformaton set π (n) may be contaned n tree T T n. In such a case, by defnton s (π (n)) nduces an acton to player also n n and not only n the nodes of π (n).) 14

15 We say that the strategy s S reaches the nformaton set h f there s a strategy profle s S of the other players such that the strategy profle (s, s ) reaches h. Otherwse, we say that the nformaton set h s excluded by the strategy s. Smlarly, we say that the strategy profle s S reaches the nformaton set h f there exsts a strategy s S such that the strategy profle (s, s ) reaches h. A strategy profle (s j ) j I reaches a node n T f the players actons s j (π j (n )) j I and nature s moves n the nodes n T lead to n wth a postve probablty. Snce we consder only fnte trees, (s j ) j I reaches an nformaton set h H f and f there s a node n h such that (s j ) j I reaches n. As s the case also n standard games, for every gven node, a gven strategy profle of the players nduces a dstrbuton over termnal nodes n each tree, and hence an expected payoff for each player n the tree. For an nformaton set h, let s s h denote the strategy that s obtaned by replacng actons prescrbed by s at the nformaton set h and ts successors by actons prescrbed by s. The strategy s / s h s called an h -replacement of s. The set of behavoral strateges s 2.5 Awareness of unawareness h H (A h ). In some strategc stuatons a player may be aware of her unawareness n the sense that she s suspcous that somethng s amss wthout beng able to conceptualze ths somethng. Such a suspcon may affect her payoff evaluatons for actons that she knows are avalable to her. More mportantly, she may take actons to nvestgate her suspcon f such actons are physcally avalable. To model awareness of unawareness some of the trees may nclude magnary actons as placeholders for actons that a player may be unaware of and termnal nodes/evaluatons of payoffs that reflect her awareness of unawareness. (The approach of modelng awareness of unawareness by magnary moves was proposed by Halpern and Rêgo, 2006.) Consder the example n Fgure 7. In both rght and left trees, player 1 can decde whether or not to rase the suspcon of player 2. If he does not, then player 2 can decde between two actons. Snce n ths case player 2 s nformaton set s n the lower tree, she does not even realze that player 1 could have rased her suspcon. If player 1 rases 15

16 Fgure 7: Game form wth awareness of unawareness a 2 b 1 rase 2 s suspcon 2 nvestgate 2 2 a 1 rase 2 s suspcon 2 2 b nvestgate t 2 2 A B C D a b a some b thng a b 2 a b player 2 s suspcon, then player 2 s nformaton set s n the left tree. She must decde whether to nvestgate her suspcon or not. If she doesn t, then she can decde between two actons but ths tme she realzes that player 1 rased her suspcon (and could have refraned from dong so); and that she could have chosen to nvestgate, n whch case she may have had somethng else to do, that she cannot conceptualze n advance. Once she nvestgates, she becomes aware of two more actons and her nformaton set s n the rght tree. She also realzes that player 1 ntally rased her suspcon wthout beng explctly aware of those actons of hers by hmself. Note that before she decdes whether or not to nvestgate, she s not modeled as antcpatng to be n the rght tree, because she cannot conceptualze the nature of the actons she reveals f and when she nvestgates. 2.6 The connecton to standard extensve-form games Harsany (1967) showed how to transform games wth asymmetrc nformaton nto games wth mperfect nformaton about a move of nature. Can a smlar dea be used to transform any generalzed extensve-form game nto a standard extensve-form game? Gven a generalzed extensve-form game Γ wth a partally ordered set of trees T, one could defne the transformaton of Γ to be the extensve-form game wth an ntal move of nature, n whch nature chooses one of the trees n T. Notce, however, that the resultng structure would not be a standard extensve-form game. To see ths, notce that every standard extensve-form game has the followng property (E): the equvalence class of nodes n whch a player consders as possble a 16

17 gven possblty set of nodes s dentcal wth that possblty set; ths set s called an nformaton set of the player, and n all of ts nodes the player has the same set of avalable actons. In contrast, n the transformaton consdered above for games wth msperceptons, ths equvalence class may be a strct super-set of the possblty set. For example, when the generalzed game n Fgure 8(a) s transformed so as to have an ntal move of nature, the possblty set for the (unque) player s the rght node, whle the equvalence class contans both the rght and left node. Fgure 8: t a b c nature nature a b a b c a b a b c a b (a) (b) (c) Thus, f after addng the ntal move of nature the nformaton sets are defned to be synonymous wth the possblty sets, the resultng game would be non-standard, because for some nformaton set there may be addtonal nodes outsde t n whch the player consders t as possble (as n Fgure 8(b), where n the left node the player consders only the rght node as possble). If, n contrast, we choose the alternatve defnton, by whch an nformaton set s the equvalence class n whch a player has a partcular set of nodes that she consders as possble, the resultng game would agan be non-standard, ths tme because the actons avalable to the player n the nodes of a gven nformaton set mght not be dentcal across these nodes (as n Fgure 8(c), where n the left node the player has more avalable actons than n the rght node, even though both are wthn the same nformaton set). There s also another aspect that prevents the above transformaton from yeldng a standard extensve-form game. a full-support pror on the moves of nature. 6 In a standard extensve-form game each player has Usng Bayes rule, the player therefore has a well-defned belef about nature at each stage of the game. In contrast, n the above transformaton each player ascrbes probablty 1 only to one of the ntal moves of nature; moreover, along the path of play the player may swtch completely the move 6 Moreover, n the classcal defnton of an extensve-form game the prors of the dfferent players about nature are actually dentcal,.e. the players have a common pror about nature. 17

18 of nature n whch she confdes even f nothng n the path of play tself mposed such a swtch. Such a swtch corresponds to a node n the generalzed game n whch the player s def ned as becomng aware of new aspects of the dynamc nteracton; such an ncrease of awareness may occur even when the physcal path of play per se dd not mply a surprse, and may have also been compatble wth the player s prevous concepton of the game. Thus, f we do add an ntal move of nature to connect the trees of the generalzed game, the player s (evolvng) belef about nature cannot be encapsulated wthn an ntal probablstc belef about nature, and must be represented explctly by a belef system as part of the defnton of the game. Addng an ntal move of nature has a further conceptual drawback. In classcal extensve-form games the mplct assumpton s that the players understand the entre structure of the dynamc nteracton as emboded n the game tree. 7 Assgnng probablty zero to some move of nature s stll compatble wth realzng what could have happened f ths zero-probablty move were nevertheless to materalze. Ths s conceptually dstnct from beng completely unaware of a subset of paths n the game, and t s the latter concept that we want to model here. Moreover, as we have seen n the example of the ntroducton (Fgures 3 and 4), t may lead to behavoral predctons dfferent from unawareness. Thus, standard extensve-form games are nether techncally ft (wthout further generalzaton) for modelng behavor under dynamc msperceptons and unawareness, nor do they convey the approprate conceptual apparatus for modelng such nteractons, hence the need for our defnton of generalzed games. 8 3 Extensve-form ratonalzablty Pearce (1984) defned extensve-form (correlated) ratonalzable strateges by a procedure of an teratve elmnaton of strateges. The dea behnd the defnton nvolves a noton of forward nducton. In generc perfect-nformaton games, ratonalzable strategy pro- 7 For nstance, Myerson (1991, p. 4) puts forward explctly the tenet that game theory deals wth ntellgent players, where a player n the game s ntellgent f he knows everythng that we know about the game and he can make any nference about the stuaton that we can make. 8 Even f one nevertheless prefers to model such nteractons usng an ntal move of nature and generalzng accordngly the notons of nformaton sets and belefs about nature n standard extensveform games, the propertes (I0)-(I6) of our defnton consttute restrctons on the structure of such extended standard games that are needed n order to guarantee e.g. that the expectatons of each player about future paths are dynamcally consstent (property I3) and perfect recall s well-defned (property I6). 18

19 fles yeld the backward nducton outcome, though they need not be subgame-perfect equlbrum strateges (Reny 1992, Battgall 1997). In what follows we extend ths defnton to generalzed extensve-form games. A belef system of player b = (b (h )) h H ( s a profle of belefs - a belef b (h ) S T h h H ) ( S T h ) about the other players strateges n the T h -partal game, for each nformaton set h H, wth the followng propertes b (h ) reaches h,.e. b (h ) assgns probablty 1 to the set of strategy profles of the other players that reach h. If h precedes h (h h ) then b (h ) s derved from b (h ) by Bayes rule whenever possble. Denote by B the set of player s belef systems. For a belef system b B, a strategy s S and an nformaton set h H, defne player s expected payoff at h to be the expected payoff for player n T h gven b (h ), the actons prescrbed by s at h and ts successors, and condtonal on the fact that h has been reached. 9 We say that wth the belef system b and the strategy s player s ratonal at the nformaton set h H f ether s doesn t reach h n the tree T h, or f s does reach h n the tree T h expected payoff n T h then there exsts no h -replacement of s whch yelds player a hgher gven the belef b (h ) on the other players strateges S T h. We say that wth the belef system b and the strategy s player would be ratonal at the nformaton set h H f there exsts no acton a h A h such that only replacng the acton s (h ) by a h results n a new strategy s whch yelds player a hgher expected payoff at h gven the belef b (h ) on the other players strateges S T h. The dfference between these two defntons s as follows. The defnton of ratonalty of a strategy s at an nformaton set h takes a global perspectve. It s mute regardng nformaton sets whch the strategy s tself rules out. Also, at an nformaton set h 9 Even f ths condton s counterfactual due to the fact that the strategy s does not reach h. The condtonng s thus on the event that nature s moves, f there are any, have led to the nformaton set h, and assumng that player s past actons (n the nformaton sets precedng h ) have led to h even f these actons are dstnct than those prescrbed by s. 19

20 whch s does reach, t consders h -replacements, whch may alter s not only at h, but also smultaneously at h and/or at some of the succeedng nformaton sets of player. In contrast, the second defnton takes a local perspectve. It takes serously the reasonng about ratonalty assumng that h has been reached, whether ths assumpton s realstc (when h can n fact be reached wth a postve probablty gven the actons prescrbed by s at precedng nformaton sets) or counterfactual (when h s ruled out by s own actons wth the strategy s at precedng nformaton sets). Moreover, t consders alternatve actons a h only at h tself. Ths s motvated by the mplct assumpton that at h, player s certan that at future nformaton sets she wll be actng accordng to the strategy s, but at the same tme she also realzes that at each such future nformaton set she wll have the opportunty to re-consder her acton, and that at h she has no way to commt herself to the acton she wll be takng at such a future nformaton set. We fnd the second defnton more appealng n the context of unawareness. Wth unawareness, a player does not necessarly conceve of her entre strategy. Rather, she mght be aware only of a subset of her nformaton sets. She may plan what to do f and when such an nformaton set s reached. However, once her level of awareness gets ncreased along the path of play, she may suspect that a smlar revelaton can happen agan. She may then realze that whatever she plans to do, wth her current level of awareness, s n fact subject to reconsderaton. That s why wth unawareness, what a strategy specfes for future nformaton sets should better be conceptualzed as expressng current belefs about one s future actons rather than as a rgd plan to whch the player s bound to conform. The followng lemma descrbes the close connecton between the two defntons when all of the nformaton sets h are consdered. The lemma follows from the prncple of optmalty n dynamc programmng. The explct proof appears n the appendx. Lemma 1 Wth a belef system b of player, () f a strategy s of player would be ratonal at all nformaton sets h H then t s ratonal at all nformaton sets h H ; and () f a strategy s of player s ratonal at all nformaton sets h H, then there exsts a strategy ŝ whch concdes wth s at all nformaton sets reached by s, such that ŝ would be ratonal at all nformaton sets h H. The connecton between the two defntons descrbed n Lemma 1 s related to the noton of a plan of acton (Rubnsten 1991, Reny 1992). A plan of player specfes her 20

21 acton when she s called to play, and does not specfy what she would do at nformaton sets whch are ruled out by that plan. Formally, a plan of acton for player s an equvalence class of strateges P S such that two strateges s, ŝ are n P f and only f for every strategy profle s of the other players, (s, s ) and (ŝ, s ) nduce the same dstrbuton over termnal nodes n each of the trees of the game Γ. If s P we say that the strategy s nduces the plan of acton P. Wth ths termnology, Lemma 1 mples: Lemma 2 For a gven belef system b of player, there exsts a strategy s whch s ratonal at all nformaton sets h H and nduces the plan of acton P f and only f there exsts a strategy ŝ whch would be ratonal at all nformaton sets h H and nduces the plan of acton P. We now turn to defne ratonalzablty n generalzed extensve-form games. Defnton 1 (Would-be ratonalzable strateges) Defne, nductvely, the followng sequence of belef systems and strateges of player. B 1 = B S 1 = {s S : there exsts a belef system b B 1 wth whch for every nformaton set h H player s would-be ratonal at h }. B k = {b B k 1 players strateges s S k 1 : for every nformaton set h, f there exsts some profle of the other = j Sk 1 j b (h ) assgns probablty 1 to S k 1,T h } such that s reaches h n the tree T h, then S k = {s S : there exsts a belef system b B k wth whch for every nformaton set h H player would be ratonal at h } The set of player s would-be ratonalzable strateges s S = S k. k=1 Remark 3 S k S k 1 for every k > 1. Proof. Consder s S k. By defnton, s would-be ratonal at each of player s nformaton sets gven some belef system b B k. Snce B k B k 1, s would also be 21

22 ratonal at each of player s nformaton sets gven a belef system n B k 1, namely gven b. Hence s S k 1. The generalzaton of Pearce s (1984) noton of extensve-form correlated ratonalzable strateges s ntroduced next. The nductve defnton below generalzes Defnton 2 n Battgall (1997), whch he proved to be equvalent to Pearce s orgnal defnton. Defnton 2 (Extensve-form correlated ratonalzable strateges) For k 1 let ˆB k, Ŝk be defned nductvely as B k, S k above, respectvely, the only change beng that the phrase for every nformaton set h H player would be ratonal at h n the defnton of S k s changed to for every nformaton set h H player s ratonal at h n the defnton of Ŝk. The set of player s extensve-form correlated ratonalzable strateges s Ŝ = Ŝ k. k=1 Remark 4 Ŝk Ŝk 1 for every k > 1. Proof. Analogous to the proof of Remark 3 above. Lemma 2 above mples the followng proposton. Proposton 1 The set of strateges S k s contaned n Ŝk, but S k nduces a set of plans of acton dentcal to the set of plans of acton nduced by Ŝk. Consequently, the set of would-be ratonalzable strateges s contaned n the set of extensve-form correlated ratonalzable strateges, S = S k Ŝ = k=1 k=1 but both sets nduce the same set of plans of actons. Ŝ k The ncluson mentoned n the proposton may be strct. For nstance, n our frst example n the ntroducton (Fgure 1), t s ratonalzable for player 1 not to gve the car to player 2 and to subsequently go to the Bach concert, but to have gone to the Stravnsky concert (or to the Bach concert, or to the Mozart concert) had he gven the car to 2. In contrast, the only would-be ratonalzable strategy of player 1 s not to gve the car to player 2 and subsequently attend the Bach concert, but to have gone to the 22

23 Mozart concert had he gven the car to player 2. As the proposton asserts, no dfference arses between ratonalty and would-be ratonalty along the unque realzed path. Proposton 2 The set of would-be ratonalzable strateges s non-empty. Consequently, the set of extensve-form correlated ratonalzable strateges s non-empty. The proof s n the appendx. What are the would-be ratonalzable strateges n our battle-of-the-sexes example from the ntroducton (Fgure 2)? Remark 5 In the Bach-Stravnsky-Mozart example wth unawareness from the ntroducton (Fgure 2), no player has a unque would-be ratonalzable strategy. Proof. At the frst level, any strategy would-be ratonal for player I except all strateges that prescrbe gong to the Mozart concert after don t tell. For player II, both the Bach concert and the Stravnsky concert would-be ratonal f he s unaware of the Mozart concert. If he s aware of the Mozart concert, then only ths concert s ratonal snce t s a domnant acton. Thus, SII 1 = {(B, M), (S, M)}. Not tellng player II about the Mozart concert and gong to the Bach concert would-be ratonal for player I assumng that she beleves wth probablty at least 1 that player II wll go to the Bach concert 2 under such crcumstances. Tellng player II about the Mozart concert and gong to the Mozart concert would-be ratonal for player I f she beleves wth probablty at least 1 2 that player II would go to the Stravnsky concert f not told about the Mozart concert. To summarze, S 2 I = { ( don t tell, B, M, B), ( don t tell, B, M, S), ( tell, B, M, B), ( tell, B, M, S), ( tell, S, M, B), ( tell, S, M, S) where the second (resp. thrd) component of the strategy vector refers to player I s choce after hstory don t tell (resp. tell ), and the last component denotes the acton n the lower subtree. Fnally, note that S k II = S1 II for k 1 and Sk I = S2 I for k 2. } When we compare ths example to the game n whch both players are aware of the Mozart concert but player I has the opton of not provdng her car for gong to ths concert (Fgure 1), we note that the strategc mplcatons of unawareness of actons are dstnct from a stuaton n whch both players are aware of the actons but some acton may not always be avalable. The reason s that f player I keeps player II unaware 23

24 of the Mozart concert, then player II can not nfer the ntenton of player II to go to the Bach concert. In other words, awareness of an avalable acton (provdng the car for gong to the Mozart concert) and certanty that t hasn t been taken has stronger strategc mplcatons than unawareness of the very same acton. In the Bach-Stravnsky-Mozart example wth unawareness from the ntroducton (Fgure 2), the would-be ratonalzable outcome s not unque. Ths s n contrast to the example wth unavalablty of actons nstead, where there s a unque would-be ratonalzable outcome. However, there exst also games where wth unavalablty of actons there are more would-be ratonalzable outcomes than wth unawareness of the same actons, as the example n Remark 8 n Secton 5 demonstrates. 4 Prudent ratonalzablty In normal-form games, terated admssblty (.e. teratve elmnaton of weakly domnated strateges) s a refnement of ratonalzablty. Van Damme (1989) and more generally Ben-Porath and Dekel (1992) showed that terated elmnaton of weakly domnated strateges sngles out the forward nducton outcome n money-burnng games. 10 One nterpretaton of terated admssblty s that n every round of elmnaton, each player s prudent and hence does not exclude completely any strategy profle of the other players whch has not been thus far elmnated. In ths secton, we use the dea of prudence to defne an analogous noton of ratonalzablty for dynamc games: Defnton 3 (Prudent ratonalzablty) Let S 0 = S For k 1 defne nductvely B k = b B : for every nformaton set h, f there exsts some profle S k 1 S k 1 j s = j of the other players strateges such that s reaches h n the tree T h, then the support of b (h ) s the set of strategy profles s S k 1,T h that reach h 10 A smlar result was shown by Herngs and Vannetelbosch (1999) who defned terated admssblty n terms of full support belefs and called t tremblng-hand perfect ratonalzablty. 24

25 S k = { s S k 1 : there exsts b B k would be ratonal at h such that for all h H player } The set of prudent ratonalzable strateges of player s S = Proposton 3 The set of player s prudent ratonalzable strateges s non-empty. Proof. Frst, observe that B k for every k 1, because f an nformaton set k 1 h H s reached by some s S, then s reaches also all of s nformaton sets that precede h n the tree T h. We proceed by nducton. S0 = S and hence non-empty. Notce also that for every b B 1, a standard backward nducton procedure on the arborescence of nformaton sets H yelds a strategy s S 1 wth whch player would be ratonal h H gven b. S k 1 k=1 Suppose, nductvely, we have already shown that I 0), and also that for every b B k 1 player would be ratonal h H gven b. S k Sk 1 there exsts a strategy s 0 (and hence that S k 1 wth whch Let b B k. Let Ḣ H be the set of s nformaton sets not reached by any profle k 1 k 2 s S but reached by some profle s S. If Ḣ, for every h Ḣ wth no predecessor n Ḣ, modfy (f necessary) b (h ) so as to have full support on the profles n S k 2 that reach h, and n succeedng nformaton sets modfy b by Bayes rule whenever possble. Denote the modfed belef system by ḃ. Then by constructon also ḃ B k. Consder a sequence of belef systems b,n ḃ = and gven ths sequence 11 b,n B k 1 such that ) ( ) (ḃ (h ) lm b h H,n (h ) n B k 1 let s,n S k 1 h H be a correspondng sequence of strateges wth the property that gven b,n, t s the case that wth the strategy s,n player would be ratonal at every h H. Snce player has fntely many strateges, some strategy s appears nfntely often n the sequence s,n. Snce expected utlty s B k 1 11 To construct such a sequence b,n, for every nformaton set h H not reached by any k 1 s S defne b,n (h ) = ḃ (h ) for every n 1; and for every h H wth no predecessors but k 1 reached by some profle s S defne b,n (h ) ( ) Sk 1 to be any convergng sequence of belefs such that for every n 1 the support of b,n (h k 2 ) s the subset of profles n S that reach h, whle lm n b,n (h ) = ḃ (h ). In succeedng nformaton sets reached by some s k 1 S defne b,n (h ) by Bayes rule whenever possble. 25

26 lnear n belefs and hence contnuous, also gven ḃ t s the case that wth the strategy s player would be ratonal at every h H. Hence s S k as well. Now, snce player s set of strateges S s fnte and by defnton S k for every k 1, for some l we eventually get S l l+1 l+1 l+2 = S I and hence B = B I. Inductvely, S l = S l+1 = S l+2 =... S k+1 and therefore S = S k = S l k=1 as requred. 4.1 A frst tenson between ratonalzaton and prudence: Dvnng the opponent s past behavor In normal-form games, terated admssblty s a refnement of ratonalzablty. Somewhat surprsngly, n extensve-form games prudent ratonalzablty s not a refnement of would-be ratonalzablty, as the followng example (Fgure 9) demonstrates. Fgure 9: I a b c 6, 6 II d e f d e f 3, 4 4, 3 10, 0 5, 5 6, 6 0, 0 In ths example, player 1 can guarantee herself the payoff 6 by choosng a and endng the game. If player 2 s called to play, should he beleve that player 1 chose b or c? If player 1 s certan that player 2 s ratonal, she s certan that player 2 wll not choose f. Hence, f player 2 s certan that player 1 s certan that he (player 2) s ratonal, then at hs nformaton set player 2 s certan that player 1 chose c. The reason s that among player 1 s actons leadng to 2 s nformaton set, c s the only acton whch, assumng 2 beleves c was chosen and that 2 s ratonal and wll hence choose e, yelds player 1 the payoff 6, whch s just as hgh as the payoff she could guarantee herself wth the outsde opton a. Hence (a, e) and (c, e) are the profles of extensve-form (correlated) ratonalzable strateges (as well as would-be ratonalzable strateges) n ths game. 26