Bounded Reasoning and Higher-Order Uncertainty

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1 Bounded Reasonng and Hgher-Order Uncertanty Wllemen Kets February 1, 2012 Abstract Standard models of games wth ncomplete nformaton assume that players form belefs about ther opponents belefs about ther opponents belefs and so on, that s, that players have an nfnte depth of reasonng. Ths paper generalzes the type spaces ntroduced by Harsany ( ) so that players can have a fnte depth of reasonng. The nnovaton s that players can have a coarse percepton of the hgher-order belefs of other players, whch formalzes the small world dea of Savage (1954) n a type space context. Unlke n other models of fnte-order reasonng, players wth a fnte depth of reasonng can have nontrval hgher-order belefs about certan events. Intutvely, some hgher-order events are generated by events of lower orders, makng t possble for players to reason about them, even f they have a fnte depth. Kellogg School of Management, Northwestern Unversty. E-mal: w-kets@kellogg.northwestern.edu. Phone: Ths paper supersedes Kets (2009). I am grateful to Adam Brandenburger, Yoss Fenberg, and Matthew Jackson for ther gudance and support, and to Adam Brandenburger, Alfredo D Tllo, Amanda Fredenberg, and Avad Hefetz for mportant nput. I benefted from dscussons wth Songz Du, Matt Ellott, Yoss Fenberg, Ben Golub, Joe Halpern, Peter Hammond, Ta-We Hu, Yar Lvne, Martn Meer, Rosemare Nagel, Andrés Perea, Tomasz Sadzk, Burkhard Schpper, Elas Tsakas, and Jonathan Wensten, as well as from nput from numerous semnar audences. Much of ths research was carred out durng vsts to Stanford Unversty and the NYU Stern School of Busness, and I thank these nsttutons for ther hosptalty. Fnancal support from the Ar Force Offce for Scentfc Research under Grant FA s gratefully acknowledged.

2 1. Introducton Analyzng games of ncomplete nformaton requres takng nto account not ust the belefs of players, but also ther hgher-order belefs. If a manager of a frm does not observe ts compettors costs, for example, he needs to form belefs about the cost structure of all the frms n the ndustry the state of nature to predct ts compettors prcng decsons, so as to optmally set ts own prce. But the prcng decsons of the frm s compettors n turn depends on ther belefs about nature. To decde on hs optmal acton, the manager therefore needs to form a belef not only about nature (a so-called frst-order belef), but also about hs compettors belefs about nature (a second-order belef,.e., a belef about a frst-order belef). And because hs compettors lkewse consder hs belefs about nature, the manager has to form a belef about ther belefs about hs belefs about nature (a thrd-order belef), and so on, ad nfntum (cf. Harsany, ). Are real players capable of such hgher-order reasonng? The answer to ths queston s not so clear cut. A statement such as John Dean dd not know that Nxon knew that Dean knew that Nxon knew that McCord had burgled O Bren s offce n the Watergate Apartments s nherently dffcult to reason about (Clark and Marshall, 1981). At the same tme, other types of hgher-order reasonng seem unproblematc. If two players, say Ann and Bob, st across the table from each other and have eye contact, then clearly each of them knows that they have eye contact, knows that the other knows that, knows that the other knows that they know, and so on. That s, t s common knowledge between Ann and Bob that they have eye contact (Lews, 1969; Chwe, 2001). These examples suggest that some hgher-order events are easer to reason about than others. Exstng models do not take ths nto account. On the one hand, standard game-theoretc models assume that players have hgher-order belefs about every possble event, at all possble orders. That s, these models assume that players have an nfnte depth of reasonng. On the other hand, n models developed n the expermental lterature, such as cogntve herarchy models or models of level-k reasonng, 1 players can have a fnte depth of reasonng. However, these models assume that a player wth a fnte depth of reasonng cannot reason about any event at suffcently hgh order, rulng out, for example, that t can be common knowledge between Ann and Bob that they have eye contact. Because belefs at arbtrarly hgh order can have a sgnfcant mpact on economc outcomes, 2 t s mportant to carefully model what hgher-order events players wth bounded 1 See, e.g., Nagel (1995), Stahl and Wlson (1995) Ho et al. (1998), Costa-Gomes et al. (2001), Camerer et al. (2004), Crawford and Irberr (2007), Strzaleck (2009), and Hefetz and Kets (2011). 2 An acton that s optmal for a player gven her kth-order belef, for example, may no longer be optmal 2

3 reasonng abltes can hold belefs about. Ths paper provdes a framework that does ust that. We propose a class of type spaces, called extended type spaces, startng from the dea that a player can have a coarse percepton of the state of the world, where the state of the world specfes not only the physcal realty the state of nature, but also players belefs and hgher-order belefs. A player who has a coarse percepton does not dstngush states of the world that dffer only n certan detals, such as the belefs of other players at very hgh orders. We show that a player wth a coarse percepton has a fnte depth of reasonng, n the sense that there are events beyond a certan fnte order that a player wth that percepton cannot form belefs about. We show that a player wth a fnte depth of reasonng can nevertheless hold nontrval belefs about certan hgher-order events, provded that these events are generated by events of suffcently low order. Ths result explans why the statement about Dean s and Nxon s hgher-order belefs s hard to reason about, whle t can be common knowledge players that they have eye contact, even f they have a fnte depth of reasonng. The ntuton s smple. That Dean knew that Nxon knew of the burglary, and that Nxon knew that Dean knew does not tell us whether or not Dean knew that Nxon knew that Dean knew that Nxon knew of the break-n. That s, there s no low-order event that pns down Dean s belef about the relevant hgherorder event. Ths s dfferent n the case where players have eye contact, or where they hear a publc announcement: the smple fact that they have eye contact (respectvely, hear the announcement) pns down ther belefs about the stuaton at all orders. As the name suggests, extended type spaces generalze the standard type spaces ntroduced by Harsany ( ): a Harsany type space s smply an extended type space n whch each type has an nfnte depth of reasonng. Extended type spaces can also be seen as a generalzaton of cogntve herarchy and level-k models: Kets (2012) constructs an extended type space such that there s no hgher-order event such that players belef about that event are completely determned by ther belefs about some lower-order event. That means that f a player has a fnte depth d, then there are no hgher-order events she can reason about, as n the cogntve herarchy and level-k models. To better understand the framework, t wll be helpful to consder a smple example. Suppose Ann and Bob are uncertan about the state of nature θ, whch can ether be hgh (H) or low (L). If Ann reasons about Bob s belefs about θ, then she s able to dstngush states of the world that dffer only n Bob s belefs about θ, as n Fgure 1(a). If she does not ask gven her (k + 1)th-order belef, for any fnte k (Rubnsten, 1989; Carlsson and van Damme, 1993). Belefs at arbtrarly hgh order may also determne whether players wth a common pror can have dfferent posterors (Aumann, 1976; Geanakoplos and Polemarchaks, 1982). 3

4 θ = H; Bob beleves that θ = H. θ = H; Bob beleves that θ = L. θ = H. (a) (b) Fgure 1: (a) Ann can reason about Bob s belefs about the state of nature θ. percepton s too coarse to reason about Bob s belefs about θ. (b) Ann s herself what belefs Bob holds about θ, on the other hand, then she does not dstngush states of the world that dffer Bob s belefs, but are dentcal n other respects (cf. Fgure 1(b)). To her, t s as f there s only one state, whch s n fact the collecton of the states n Fgure 1(a). In that case, Ann can form a belef about θ, but not about Bob s belefs. In Fgure 1(b), for example, Ann assgns probablty 1 to the event that the state of nature s hgh (because θ s hgh n the unque state she thnks possble), but she cannot reason about Bob s belefs. That s, Ann has a frst-order belef, but no belefs at hgher orders, and we say that her depth of reasonng s equal to one. Coarse perceptons thus model small worlds, ntroduced by Savage (1954) n the context of one-person decson stuatons. 3 To obtan a tractable model of hgher-order belefs, we translate ths dea to the context of type spaces. Ths gves an extended type space: a set of types for each player, and for each type, a belef (probablty measure) over nature and the types of the other players, as n the type spaces ntroduced by Harsany ( ). Unlke n a Harsany type space, the belefs of dfferent types of a player can be defned on dfferent σ-algebras. 4 As a type s belef assgns a probablty only to those subsets of her opponents types that are n the type s σ-algebra, a type wth a coarse σ-algebra has a coarse percepton of the other players types. And because types generate hgher-order belefs, the coarseness of a type s σ-algebra therefore determnes what features of the other players hgher-order belefs the type can reason about. In partcular, f the type cannot form a belef about others belefs at all orders, t has a fnte depth of reasonng. To llustrate, let us translate the example where Ann and Bob are uncertan about the 3 Savage (1954) defnes a world to be the obect about whch a decson-maker s uncertan; a state n a small world descrbes the possble uncertantes n less detal than a state n a large world, so that a smaller world s derved from a larger by neglectng some dstnctons between states, and a state of the smaller world corresponds not to one state of the larger, but to a set of states (p. 9, emphass added). 4 A σ-algebra F on a set X s a collecton of subsets of X that contans X and s closed under complements and countable unons. Importantly, f a probablty measure on X s defned on a σ-algebra F, t can assgn a probablty only to those subsets of X that belong to F. 4

5 state of nature θ to a type space settng. Bob has two types, t b and t b. Type t b beleves (wth probablty 1) that the state of nature s hgh, whle t b beleves t s low. If the σ-algebra of a type for Ann contans the sngletons {t b } and {t b }, then the type can formulate a belef about Bob s belefs about the state of nature: A type for Ann that assgns probablty p to the sngleton {t b } puts probablty p on the event that Bob beleves that the state of nature s hgh, and 1 p on the event that Bob beleves that t s low. Ths s the type-space analogue of the state space n Fgure 1(a). On the other hand, f the type has a σ-algebra that contans no nontrval subsets of Bob s type set {t b, t b }, then the type does not have a well-artculated belef about events such as that Bob beleves that the state of nature s hgh, as n Fgure 1(b). A frst queston that arses s whether types nduce well-defned belef herarches, as n the Harsany case. Ths s ndeed the case, as we demonstrate n Theorem 4.3. We go on to characterze the depth of reasonng of types. We say that a type has depth (of reasonng) k < f the type nduces a kth-order belef about every event, but does not have belefs about at least some events at hgher orders; a type has an nfnte depth f t nduces a kthorder belef about all events for each k. We show that each type has a well-defned depth (Proposton 4.6). For applcatons, t can be mportant to know the depth of reasonng of each type. In prncple, the depth of reasonng of a type can be determned by wrtng out the belef herarchy t nduces, and checkng whether there s some fnte k such that the belef herarchy nduced by the type does noes not specfy kth-order belefs about certan events, but ths can of course be tedous. Theorem 4.12 therefore characterzes the depth of reasonng of a type n terms of the propertes of the type space alone, wthout makng reference to belef herarches, under a mld condton on the type space. Together, Theorems 4.3 and 4.12 demonstrate that extended type spaces provde an mplct descrpton of players fnte and nfnte herarches of belefs, ncludng hgher-order uncertanty about others depth of reasonng, by specfyng types, belefs about types, and a collecton of σ-algebras on each type set, ust lke the Harsany type spaces model players nfnte belef herarches mplctly, by specfyng types and belefs about types. Havng characterzed the bounds on reasonng for types wth a coarse percepton, we turn to the queston what hgher-order events a type wth a fnte depth can reason about. We show that a type wth a fnte depth d can form belefs about a kth-order event for k > d f and only f the event s expressble n terms of an event of suffcently low order (Corollary 5.1). Gong back to our earler examples, n a stuaton where Dean and Nxon have eye contact, Nxon may not be able to form belefs about whether Dean knew or dd not know that Nxon knew that Dean knew that Nxon knew of the break-n (a ffth-order event), even f he beleves 5

6 that Dean beleves that Nxon beleves that Dean beleves that Nxon beleves they have eye contact (another ffth-order event). We then focus on the condtons under whch there can be (approxmate) common belef or hgh-order mutual belef. Ths case s of specal nterest, as predctons that hold under the assumpton that certan events are common belef can change dramatcally when belefs are perturbed at arbtrarly hgh order. 5 We establsh two sets of results. On the one hand, we show that common belef of nontrval events can be attaned even f players have a bounded depth of reasonng. On the other hand, t can be taxng for players to have hgh-order mutual belef n an event f common belef s unattanable. The ntuton s precsely that common belef can be nduced by a lower-order event (as n the case where Ann and Bob have eye contact), whle that s not the case f an event s hgh-order mutual belef, but not common belef (also see Clark and Marshall, 1978, 1981, for a dscusson). The dea that smple events can nduce (almost) common knowledge s not new; t s central to the conceptualzaton of common knowledge by the phlosopher Davd Lews (1969) and t underles the formalzaton of common knowledge and approxmate common belef n Aumann (1976) and Monderer and Samet (1989), respectvely. Indeed, speakng of a belef herarchy such as the one descrbed above, where Ann knows she and Bob have eye contact, knows that Bob knows that, and so on, Lews wrtes: ths s a chan of mplcatons, [t does not represent] steps n anyone s actual reasonng. Therefore, there s nothng mproper about ts nfnte length (p. 53). Our contrbuton here s to pont out that ths dea apples beyond the context of common knowledge, and, more fundamentally, to formalze t n the context of players wth bounded reasonng abltes, buldng on Savage s small world dea, and to use ths nsght to characterze the hgher-order events that bounded reasoners can have belefs about. One could also argue that the abundant use of smple type spaces n applcatons can be motvated by the assumpton that players typcally do not have very complcated hgher-order belefs. In a smple type space, there s some fnte k (typcally equal to 1) such that the hgher-order belefs of players (about any event) are common knowledge condtonal on ther kth-order belefs (see Secton 6.1 for a more formal dscusson). For nstance, the standard assumpton that there s a common pror and that players receve prvate sgnals mples that players hgher-order belefs are completely determned by ther belef about nature. There are two reasons why a smple Harsany type space cannot fully capture bounded reasonng. Frst, n a smple Harsany type space all events are equally easy to reason about. But ths cannot capture a stuaton where Dean and Nxon st across the table from each other and have eye contact (so that that s common knowledge between them), whle Nxon 5 See the references n footnote 2, as well as Ely and P esk (2011). 6

7 s unable to reason about the queston whether Dean knew that Nxon knew that Dean knew that Nxon knew of the break-n at the Watergate complex. Second, as shown by Ely and P esk (2011), the predctons from these type spaces are senstve to small msspecfcatons of players belefs, even at arbtrarly hgh order. Ths lack of robustness s caused precsely by the assumpton that players have well-defned belefs at all orders. Intutvely, ths makes t easy to perturb players hgher-order belefs. The remander of ths paper s organzed as follows. The next secton llustrates our man results usng smple examples. Secton 3 formally ntroduces the noton of an extended type space, and Secton 4 characterzes the depth of reasonng of types. Secton 5 nvestgates the hgher-order events players wth a fnte depth can reason about. Secton 6 dscusses the related lterature All proofs are relegated to the appendces. 2. Examples 2.1. Extended type spaces We present some examples to ntroduce our framework, and to llustrate the man results. Throughout ths secton, we consder a settng n whch two players, Ann (a) and Bob (b), are uncertan about the state of nature θ, whch can be ether hgh (H) or low (L). We represent the uncertanty faced by the players, ncludng ther uncertanty about the belefs of the other player, by an extended type space. As n the type spaces of Harsany ( ), each player = a, b s endowed wth a type space T, and each type t T s assocated wth a belef (probablty measure) β (t ) over the state of nature and the other player s type set. Unlke n a Harsany type space, the belefs of types n an extended type space can be defned on dfferent σ-algebras. That s, Ann s type set T a s endowed wth a collecton S a of σ-algebras, and a type t b T b for Bob over Ann s type s defned on some σ-algebra Σ b (t b ) n S a ; lkewse for Bob s type set and the belefs of Ann s types. The dea s that the σ-algebra on whch a type s belef s defned reflects the extent to whch the type thnks through the belefs of the other player. An extended type space s thus a tuple (T, S, Σ, β ) =a,b (we also requre that extended type spaces satsfy some addtonal condtons, but wll gnore that n ths nformal treatment) Infnte depth We llustrate how types generate hgher-order belefs usng the type space n Fgure 2. The collecton S a of σ-algebras on Ann s type set smply conssts of the σ-algebra that contans the sngletons; lkewse for S b. Snce type sets are fnte, t suffces to specfy the belef β a (t a ) 7

8 for a type t a for Ann on the partton of Bob s type set that ts σ-algebra Σ a (t a ) nduces, and smlarly for the types for Bob. For example, specfyng the belef for type t 1 a on the pars (θ, t b ), wth θ = H, L and t b = t 1 b, t2 b, t3 b, t4 b, specfes ts belef on the full σ-algebra. β a (t 1 a) H L β a (t 2 a) H L β b (t 1 b ) H L β b(t 2 b ) H L t 1 1 b 0 t 1 2 b 0 0 t 1 1 a 2 0 t 1 a 0 0 t 2 1 b 0 t 2 2 b 0 0 t 2 1 a 2 0 t 2 b 0 0 t 3 b 0 0 t 3 1 b 0 t 3 2 a 0 0 t 3 1 a 2 0 t 4 b 0 0 t 4 1 b 0 t 4 2 a 0 0 t 4 1 a 2 0 β a (t 3 a) H L β a (t 4 a) H L β b (t 3 b ) H L β b(t 4 b ) H L t 1 b 0 0 t 1 1 b 0 t 1 2 a 0 0 t 1 1 a 0 2 t 2 b 0 0 t 2 1 b 0 t 2 2 a 0 0 t 2 1 a 0 2 t 3 b t 3 b 0 0 t 3 a t 3 a 0 0 t 4 b t 4 b 0 0 t 4 a t 4 a 0 0 Fgure 2: A Harsany type space, wth the belefs for types for Ann on the left, and those for Bob on the rght; we wrte x for the sngleton {x}. The types and ther belefs determne players hgher-order belefs. For example, type t 1 a for Ann beleves (wth probablty 1) that the state of nature s H and that Bob beleves that the state of nature s H, as t assgns probablty 1 to types t 1 b and t2 b for Bob, whch both beleve that the state of nature s H. Usng that Bob s types have a belef about Ann s type, we see that t 1 a assgns probablty 1 to the event that Bob beleves that Ann beleves that the state 2 s L (as t 1 a assgns probablty 1 to type 2 t2 b for Bob, whch n turn assgns probablty 1 to the event that Ann has type t 3 a or t 4 a, whch both put probablty 1 on θ = L). Gong further, we can derve Ann s belefs about Bob s belefs about Ann s belefs about Bob s belefs about θ, and so on. Thus, each type nduces an nfnte belef herarchy: a belef about nature, a belef about the other player s belef about nature, and so on; we say that types have an nfnte depth (of reasonng) n ths case. Indeed, t can be checked that the type space n Fgure 2 s a regular Harsany type space. In general, a Harsany type space s an extended type space n whch the belefs of the types for a player are all defned on a σ-algebra that s suffcently fne for types to nduce belefs at all orders. Extended type spaces thus generalze the Harsany framework. 8

9 Depth-1 types We now turn to the other extreme, where nether player asks hmself what the other player thnks. Fgure 3 depcts an extended type space that models such a stuaton, wth the types for Ann and Bob gven by qa, 1 qa 2 and qb 1, q2 b, respectvely. β a (qa) 1 H L β b (qb 1) H L {qb 1, q2 b } 1 0 {q1 a, qa} β a (q 2 a) H L β b (q 2 b ) H L {q 1 b, q2 b } 0 1 {q1 a, q 2 a} 0 1 Fgure 3: An extended type space n whch each type has depth 1. As before, there s a type q 1 a for Ann that beleves that the state of nature s H, but n ths case, the type does not dstngush between types for Bob that have dfferent belefs about nature: qa 1 assgns probablty 1 to the event that Bob has type qb 1 or q2 b, but does not reason about the ndvdual types (.e., ts σ-algebra Σ a (qa) 1 contans {qb 1, q2 b }, but no nonempty subset thereof). Snce the types qb 1 and q2 b have dfferent belefs about nature, wth q1 b belevng that θ s hgh, and qb 2 belevng that θ s low, type q1 a cannot form belefs about Bob s belef about nature. The same holds for the other types, ncludng the types for Bob. Thus, the types n ths extended type space generate belefs about all possble frst-order events (the state of nature), but there are hgher-order events the types cannot reason about. For example, types for Bob cannot reason about the second-order event Ann assgns probablty 1 to θ = H, and lkewse for Ann s types. We say that the types have depth of reasonng equal to 1 n ths case Uncertanty about depth In the prevous example, all types had the same depth of reasonng. Players can also be uncertan about the depth of reasonng of the other players, as the extended type space n Fgure 4 llustrates. Ann has two types, s 1 a, s 2 a and lkewse for Bob. Here, the types s 1 a and s 1 b are endowed wth the fnest σ-algebra on the type set of the other player, and s 2 a and s 2 b have the coarsest one. As before, type s2 a cannot assgn a probablty to a specfc belef for Bob about nature. That s, ts σ-algebra lumps together types for Bob that have dfferent belefs about nature. Thus, the depth of type s 2 a equals 1; lkewse for type s 2 b for Bob. Types s1 a and s 1 b, on the other hand, do dstngush the dfferent types, and can reason about the other player s belefs at arbtrarly hgh order. Type s 1 a, for example, assgns probablty 1 2 to the event that θ = H and that Bob puts probablty on θ = H and the

10 β a (s 1 a) H L β b (s 1 b ) H L s 1 b s 2 b s 1 a s 2 a β b (s 2 a) H L β b (s 2 b ) H L {s 1 b, s2 b } 0 1 {s1 a, s 2 a} 0 1 Fgure 4: An extended type space wth types of dfferent depth. event that Ann beleves that, and so on, and t can be checked that s 1 a has an nfnte depth of reasonng. Types s 1 a and s 2 b are uncertan about the depth of the other player: each of those types assgns equal probablty to the other player havng depth 1 or nfnty Depth of reasonng In the type spaces consdered thus far, the depth of reasonng of each type could easly be checked. How do we determne the depth of reasonng of a type n general? It wll be nstructve to consder an extended type space where types have a σ-algebra of ntermedate coarseness, as n the extended type space n Fgure 5. The σ-algebra Σ a (r a ) assocated wth each type r a for Ann s generated by the partton {{rb 1, r2 b }, {r3 b, r4 b }} of Bob s type set, and lkewse for the σ-algebra assocated wth the types for Bob. β a (r 1 a) H L β a (r 2 a) H L β b (r 1 b ) H L β b(r 2 b ) H L {r 1 b, r2 b } 1 0 {r1 b, r2 b } 0 0 {r1 a, r 2 a} 1 0 {r 1 a, r 2 a} 0 0 {r 3 b, r4 b } 0 0 {r3 b, r4 b } 1 0 {r3 a, r 4 a} 0 0 {r 3 a, r 4 a} 1 0 β a (r 3 a) H L β a (r 4 a) H L β b (r 3 b ) H L β b(r 4 b ) H L {r 1 b, r2 b } 0 0 {r1 b, r2 b } 0 1 {r1 a, r 2 a} 0 0 {r 1 a, r 2 a} 0 1 {r 3 b, r4 b } 0 1 {r3 b, r4 b } 0 0 {r3 a, r 4 a} 0 1 {r 3 a, r 4 a} 0 0 Fgure 5: An extended type space wth σ-algebras of ntermedate coarseness. Each type r a for Ann nduces a well-artculated second-order belef, that s, a belef about Bob s belefs about θ. Consder type r 1 a, for example. Ths type assgns probablty 1 to the event that Bob has type rb 1 or r2 b (.e., to {r1 b, r2 b }), and thus to the event that Bob beleves that the state of nature s hgh (as both rb 1 and r2 b assgn probablty 1 to H). It can be checked that the other types also generate a belef about Bob s belef about nature. What makes ths possble s that the σ-algebra F b := Σ a (r a ) for each type r a for Ann 10

11 contans the subsets of Bob s types that hold a partcular belef about θ. That s, the σ- algebra F b lumps together the types for Bob whenever they concde n ther belefs about θ, and separates the types otherwse. For example, t separates the types for Bob that put probablty 1 on H (types r 1 b and r2 b ) from those that put probablty 1 on L (types r3 b and r4 b ). More formally, a type for Ann generates a belef about all second-order events f ts σ- algebra on Bob s type set contans the subset F E,p := { r b {r 1 b, r 2 b, r 3 b, r 4 b} : the margnal of β b (r b ) on {H, L} assgns prob. at least p to E } for every event E {H, L} and p [0, 1]: If ts σ-algebra contans these sets F E,p, then a type for Ann can assgn a probablty to the event that Bob holds a certan belef about θ. Snce the σ-algebra F b satsfes ths condton, any type for Ann has depth at least 2. What s needed, then, for a type to nduce a thrd-order belef, so that t has depth at least 3? Extendng the argument above, a type for Bob nduces a thrd-order belef f ts σ-algebra separates the types for Ann that dffer n ther second-order belef,.e., n ther belef about Bob s belef about θ. Fortunately, we can easly characterze these types for Ann, gven our characterzaton above: the types for Ann that share the same belef about Bob s belefs about θ are precsely the types whose belefs concde on the σ-algebra F b, as ths σ-algebra lumps together types for Bob that have the same belefs about θ. To make ths more formal, say that a subset E {H, L} {rb 1, r2 b, r3 b, r4 b } s expressble n the σ-algebra F b f t belongs to the product of the usual σ-algebra on {H, L} and F b. A type for Bob then nduces a thrd-order belef f ts σ-algebra F a on Ann s type set contans the subset { ra {r 1 a, r 2 a, r 3 a, r 4 a} : β a (r a ) assgns probablty at least p to E } for every event E that s expressble n F b and p [0, 1]. We say that the σ-algebra F a domnates the σ-algebra F b n ths case. Thus, a type for Bob has depth at least 3 f ts σ-algebra domnates a σ-algebra, vz., F b, for a type for Ann that has depth at least 2. Theorem 4.12 shows that ths condton s also necessary, gven a mld condton on the type space (Condton 4), and that t holds for any depth of reasonng: A type for Bob has depth k < f and only f ts σ-algebra (on Ann s type set) domnates a σ-algebra of depth k 1 (on Bob s type set), but does not domnate a σ-algebra of depth k. Lkewse for the types for Ann. Ths result gves us a smple recursve procedure to determne a type s depth from the propertes of the type space alone: we only need to consder the domnance relatons between 11

12 σ-algebras on the type spaces. In partcular, we do not need to wrte out the belef herarchy nduced by a type to determne ts depth. Gong back to the extended type space n Fgure 5, we see that a σ-algebra F a on Ann s type set domnates F b only f t contans the sngletons. For example, f we take E = (H, {rb 1, r2 b }) and p = 1, then the only type that assgns probablty at least p to E s ra, 1 so that a σ-algebra that domnates F b needs to contan the sngleton {ra}. 1 Snce there s no type for Bob whose σ-algebra contans the sngletons, all types have depth 2: they nduce a second-order belef, but not a thrd-order belef; the same, n fact, holds for Ann s types Extended type spaces We now begn the formal treatment. Secton 3.1 defnes the class of extended type spaces, and Secton 3.2 demonstrates that each Harsany type space can be seen as an extended type space Defnton There s a set of two players, denoted by N; the results can be extended to any fnte number of players. Players are uncertan about the state of nature whch s drawn from a set Θ. A state of nature θ Θ could for nstance specfy the payoff functons of players, or ther actons. The set Θ of states of nature s endowed wth some σ-algebra F Θ, and, to rule out trvaltes, we assume that Θ contans at least two elements. Throughout ths paper, f we fx a player, then the player other than s denoted by and vce versa,.e.,. A (Θ-based) extended type space s a structure (T, S, Σ, β ) N, that satsfes Condtons 1 3 below, where for each player, T s a nonempty set of types for player, and S s a collecton of σ-algebras on T, assumed to be nonempty and countable. The functon Σ maps the types n T to a σ-algebra Σ (t ) S on T, and β maps each type t to a probablty measure on the product σ-algebra F Θ Σ (t ). The functon β s the belef map for player, and the probablty measure β (t ) s the belef of t T over the set of states of nature Θ and the other s type set T. The σ-algebra Σ (t ) for a type t T for player on the type set T of the other player represents the percepton 6 By Condton 3 n the formal defnton of extended type spaces, the collecton S of σ-algebras has to nclude the trval σ-algebra {T, } for each player, but ths s not a substantve requrement n ths case. 12

13 of type t of the other s type set. A state (of the world) s a tuple (θ, t), where t := (t ) N s a type profle, wth t T for all, and θ Θ s a state of nature. As noted above, extended type spaces satsfy Condtons 1 3 below. The frst condton states that the σ-algebras assocated wth the types of a player can be completely ordered n terms of set ncluson. That s, the collecton of σ-algebras for a player form a fltraton: Condton 1. [Fltratons] ether F F or F F. For each player N and each par F, F S of σ-algebras, To state the other two condtons, we need some more notaton. Gven a player N, say that a σ-algebra F on T domnates a σ-algebra F on T f for each E F Θ F and p [0, 1], { t T : E F Θ Σ (t ), β (t )(E) p } F. If F domnates F, then we wrte F F ; we wrte F F f F does not domnate F. If F s the coarsest σ-algebra that domnates F, we wrte F * F. Two σ-algebras F S and F S that domnate each other wll be called a mutual-domnance par. Condton 2 states that there s at most one one such par: Condton 2. [Unque par] If (F, F ) s a mutual-domnance par, then there are no F S and F S wth F F or F F such that (F, F ) s a mutual-domnance par. The last condton says that any nontrval σ-algebra domnates some σ-algebra on the type set of the other player. Condton 3. [Domnance] {T, }, there s F S such that ones: (F, F ) form a mutual-domnance par; or For any player N and σ-algebra F S such that F F s the coarsest σ-algebra that domnates F,.e., F * F. It turns out that fner σ-algebras domnate a strctly larger set of σ-algebras than coarser Lemma 3.1. Fx a player N and let F, F S. Then, F F f and only f there s F S such that F F and F F. Moreover, for all F S, f F F, then F F. In that sense, there are no redundant σ-algebras: f a σ-algebra F s a refnement of another σ-algebra F, then F domnates a σ-algebra F that F does not domnate. In turn, F s fner than any σ-algebra n S that s domnated by F, and thus domnates a strctly larger set of σ-algebras n S than any σ-algebra that s domnated by F, and so on. space. The next secton shows that any Harsany type space can be vewed as an extended type 13

14 3.2. Harsany type spaces As the name suggests, extended type spaces generalze Harsany type spaces. A (Θ-based) Harsany type space s a structure (T H, F H, β H ) N, where T H s a nonempty set of types for player, and F H s a σ-algebra on T H. The functon β H maps each type t T H nto a probablty measure β H (t ) on the product σ-algebra F Θ F H. The set of probablty measures on F Θ F H, denoted (Θ T, F Θ F H ), s endowed wth the σ-algebra F (Θ T,F Θ F H ) that s generated by sets of the form { µ (Θ T, F Θ F H ) : E F Θ F H, β H (E) p }, where E F Θ F H s an event, and p [0, 1]. The belef maps β H are assumed to be measurable wth respect to the σ-algebras F H and F (Θ T,F Θ F H ). (Ths specfcatons covers most of the alternatve defntons n the lterature, such as where the type sets are requred to be separable metrzable or Polsh, and the belef maps to be Borel measurable or contnuous.) Gven a Harsany type space (T H, F H, β H ) N, we can defne a structure (T, S, Σ, β ) N, where for each player, T := T H, and β := β H. Also, let S := {F H }, so that Σ (t ) = F H for all t T. Then: Proposton 3.2. The structure (T, S, Σ, β ) N derved from a Harsany type space satsfes Condtons 1 3. Hence, any Harsany type can be seen as an extended type space. As there are clearly extended type spaces that are not Harsany type spaces, such as the one presented n Secton 2.1.2, extended type spaces generalze the Harsany framework. 4. The depth of reasonng of types Ths secton demonstrates how the σ-algebra assocated wth a type determnes ts depth of reasonng. The depth of reasonng of a type s nherently a property of the belef herarchy that the type nduces. Secton 4.1 therefore constructs the space that contans the belef herarches nduced by types from extended type spaces, and shows that each type nduces a well-defned herarchy of belefs. Secton 4.2 dscusses the depth of reasonng of types, and Secton 4.3 characterzes a type s depth of reasonng drectly n terms of the propertes of the type space Belef herarches The frst step s to construct the space of belef herarches, where a belef herarchy specfes a kth-order belef for each k. When we construct the set of belef herarches, we have to 14

15 take nto account what (hgher-order) belefs a player thnks her opponent may have, as the followng two examples demonstrate: Example 4.1. In the extended type space dscussed n Secton 1, a type for Bob ether assgns probablty 0 to the event that the state of nature s H, or probablty 1. Any type for Ann therefore rules out any belef for Bob that assgns probablty p 0, 1 to H, and a type for Ann that does not reason about Bob s belefs about nature can assgn a probablty to the event that Bob puts probablty 0 or 1 on H, though not to the event that he assgns a specfc probablty to H. Example 4.2. Consder an extended type space n whch Bob s types ether assgn probablty 1 2 or 1 to H. As n the prevous example, a type for Ann rules out any belef for Bob that does not put probablty 1 or 1 on H. Thus, f a type for Ann does not reason about Bob s belefs 2 about nature, t can assgn a probablty to the event that Bob beleves that θ = H wth probablty 1 or 1 to H. But, unlke n the prevous example, t cannot assgn a probablty 2 to the event that Bob assgns probablty 0 or 1 to H. Lkewse, a type for Ann that does not reason about Bob s belefs about nature n the type space n Example 4.1 cannot assgn a probablty to the event that Bob assgns probablty 1 2 or 1 to H. These examples demonstrate that the belefs nduced by types from dfferent type spaces can be defned on dfferent σ-algebras, even f they have the same reasonng powers : If a type for Ann n the extended type space n Example 4.1 does not reason about Bob s frstorder belefs, then she has a (trval) belef about the event that Bob assgns probablty 0 or 1 to H, but not about the event that he assgns probablty 1 or 1 to H; the reverse holds for 2 the extended type space n Example 4.2. More generally, Ann s percepton of Bob s kth-order belef depends both on the extent to whch she reasons about Bob s belefs, and on the context, as gven by the extended type space: the (hgher-order) belefs she thnks Bob may have. A space of belef herarches can accommodate ths f for each k, the set of kth-order belefs for a player s endowed wth a collecton of σ-algebras that capture the dfferent perceptons of s kth-order belefs for another player, where the perceptons depend both on s reasonng powers and the context, as gven by the extended type space that generates the kth-order belefs. To construct such a space of belef herarches, we need some more notaton. Gven a set X and a (nonempty) collecton S of σ-algebras on X, let (X, S ) be the set of probablty measures on some σ-algebra F n S. If µ s a probablty measure n (X, S ), then Σ(µ) S s the σ-algebra on whch µ s defned. The set (X, S ) s endowed wth the σ-algebra F (X,S ) generated by the sets { µ (X, S ) : E Σ(µ), µ(e) p } 15

16 for E F for some σ-algebra F S, and p [0, 1]. Ths σ-algebra naturally separates belefs (probablty measures) accordng to the probablty they assgn to events; ths choce of σ-algebra makes t possble to talk about belefs about belefs, and so on (cf. Hefetz and Samet, 1998b). We are now ready to construct the space of belef herarches. The frst step s to construct a sequence B 0, B 1,... of spaces for each player that descrbe the hgher-order belefs for that player. Formally, for each player N, fx an arbtrary seed µ 0. Then defne B 0 := {µ }, and let F B 0 := {B 0, } be the trval σ-algebra on B 0. For every extended type space T = (T n, S n, Σ n, β n ) n N, defne the functon h T,0 from T nto B 0 n the obvous way: h T,0 (t ) := µ 0 for t T. For k > 0, suppose that for each N and l k 1 the set B l has been defned, and that F B l s a σ-algebra on B l. Also assume that h T,l s a functon from T nto B l for each extended type space T. Then, for each player, defne S k := { F Θ A k 1 : A k 1 } s a sub-σ algebra of F B k 1, and B k = B k 1 (Θ B k 1, S k ). Defne the σ-algebra F B k on B k to be the product σ-algebra F B k 1 F (Θ B k 1 for each extended type space T = (T n, S n, Σ n, β n ) n N, defne the functon h T,k by: B k h T,k (t ) = ( h T,k 1 Theorem 4.3 below shows that h T,k (t ), β (t ) ( ) Id Θ, h T,k 1 1 ). s well defned. The nterpretaton s that for any k 1, h T,k ). Fnally, from T nto,s k (t ) s the belef herarchy of order k nduced by t : t specfes the hgher-order belefs for t up to order k. The frst term, h T,k 1 (t ) s the belef herarchy of order k 1 generated by t (whenever k > 1). The second term, the ) 1, s the kth-order belef for type t : t gves the belef probablty measure β (t ) ( Id Θ, h T,k 1 of type t over Θ and the set B T,k 1 of belef herarches of order k 1 nduced by types for. The lower-order belefs of a type can be obtaned from ts kth-order belef by approprate margnalzaton, owng to the recursve constructon). Wth some abuse of termnology, the terms kth-order belef wll therefore sometmes be used to refer to a kth-order belef herarchy. A (full) belef herarchy for player s then smply a sequence of probablty measures that specfy the hgher-order belefs for player at each order. Formally, defne the functon h T 16

17 from T to k=1 (Θ Bk 1, S k ) by h T (t ) := ( β (t ) ( Id Θ, h T,0 ) 1, β (t ) ( ) Id Θ, h T,1 1,...). Then, h T (t s the belef herarchy generated by type t n type space T, wth β (t ) ( ) IdΘ, h T,k 1 1 the kth-order belef of type t. The next result shows that every type nduces a well-defned belef herarchy Theorem 4.3. Let T = (T n, S n, Σ n, β n ) n N be an extended type space. Then, for each player N and type t T, (a) for each k, t nduces a kth-order belef herarchy: h T,k (t ) B k ; (b) t nduces a full belef herarchy: h T (t ) k=1 (Θ Bk 1, S k ); Theorem 4.3 allows us to defne the set of belef herarches nduced by types n extended type spaces. Gven an extended type space T = (T n, S n, Σ n, β n ) n N and a player N, let H T be the mage h T (T ) of T. Also, let H be the unon of all such spaces H T of belef herarches. That s, H s the set of belef herarches generated by some type for n some extended type space T. It follows from Theorem 4.3 that H T type space T, so that H s nonempty. s nonempty for every extended A kth-order belef nduced by a type s thus a probablty measure on Θ and the space of all (k 1)th-order belefs, not ust the belef herarches nduced by types n the type s type space. The next result states that the kth-order belef nduced by a type n an extended type space T can nevertheless be vewed as a probablty measure on nature and the (k 1)th-order belefs nduced by types n T. Moreover, f a type lumps together the belef herarches of the other player, then t lumps together precsely the belef herarches nduced by the types n that type space, n lne wth the examples above. To state the result, wrte B T,k for the mage h T,k (T ) of the functon h T,k n B k for a gven extended type space T, player, and k, and denote the relatve σ-algebra on B T,k nduced by F B k by F B T,k. Also, for any Q X Y, let π Q Z be the functon that proects Q nto X: π Q X (A) := {x X : there s y s.t. (x, y) A} for any A Q. Then, { (π A T,k 1,d 1 B := T,k 1 ) 1(B) B : B T,d 1 FB T,d 1 for k = 1, 2,... and d k s the σ-algebra on B T,k nduced by events concernng the (d 1)thorder belefs for. Note that A T,k 1,k 1 = F B T,k 1 }. Proposton 4.4. Let T = (T n, S n, Σ n, β n ) n N be an extended type space, and fx a player N, a type t T, and k = 1, 2,.... Then there s d k such that for every event E F Θ A T,k 1,d 1, the belef β (t ) (Id Θ, h T,k 1 ) 1 (E) about E s well defned. 17.

18 As the support of the kth-order belef β (t ) (Id Θ, h T,k 1 demonstrates that the kth-order belef β (t ) (Id Θ, h T,k 1 ) 1 s n Θ B T,k 1, ths result ) 1 nduced by type t can be vewed as a probablty measure on Θ and the set B T,k 1 of belef herarches nduced by types for player n T. Moreover, the σ-algebra on whch the kth-order belef of a type for player n a type space T s defned lumps together precsely those belef herarches that are generated by a type for n T that concde up to some order d 1: for any event B n F B T,d 1, (π,k 1 BT B T,d 1 ) 1(B) = {( µ 0, β (t ) ( Id Θ, h T,0 ( µ 0, β (t ) ( Id Θ, h T,0 ) 1,..., β (t ) ( ) ) Id Θ, h T,k 1 1 : t T, ) 1,..., β (t ) ( ) Id Θ, h T,d 1 1 ) } B s a subset of belef herarches of order k 1 generated by types n T whch s completely determned by the (d 1)th-order belef herarches generated by types n T. Thus, a secondorder belef for a type for Ann n Example 4.1 that does not reason about Bob s frst-order belef hs belefs about θ s defned on a σ-algebra that lumps together precsely the events that Bob assgns probablty 0 or 1 to H, and lkewse for Example 4.2. The present constructon s closely related to the constructon of Hefetz and Samet (1998b) of the space of belef herarches nduced by types from Harsany type spaces. Indeed, the set H H of belef herarches for player nduced by types from Harsany type spaces s equvalent to the set of belef herarches constructed by Hefetz and Samet for the Harsany case, where H H s a proper subset of H. The crtcal dfference between the present constructon and that of Hefetz and Samet s that the kth-order belefs for player can be defned on dfferent σ-algebras here, to reflect dfferent depths of reasonng. The next secton defnes the depth of reasonng of types, and shows that each type has a well-defned depth Depth of reasonng We frst defne the depth of reasonng of a belef herarchy. Informally, a belef herarchy has an nfnte depth of reasonng f t specfes a well-defned belef over each kth-order event, for every k, where a kth-order event concerns the state of nature and the (k 1)th-order belef of the other player. A belef herarchy has a fnte depth of reasonng d f t has well-defned belefs about every kth-order event for k d, but there exst hgher-order events t cannot assgn a probablty to. Formally, a (k)th-order event for a player s an element of F Θ F B T,k 1, that s, an event that nvolves the state of nature and the (k 1)th-order belef for. To defne the depth of reasonng of belef herarches, fx an extended type space T = (T n, S n, Σ n, β n ) n N, and recall 18

19 that A T,k 1,d 1 = {(π,k 1 BT B T,d 1 ) (B) : B F B T,d 1 for d k s the σ-algebra on the space B T,k 1 of (k 1)th-order belef herarches for player, formed by lumpng together the belef herarches that concde up to order d 1. That s, F Θ A T,k 1,d 1 s the σ-algebra on the kth-order uncertanty doman Θ B T,k 1 for player generated by dth-order events. As a player s mth-order belef specfes hs belefs at all lower orders (see Secton 4.1), A T,k 1,m F B T,k 1 s a refnement of any A T,k 1,m s a refnement of A T,k 1,m 1. In partcular, A T,k 1,k 1 =. The depth of reasonng d of a belef herarchy for player s the order up to whch t dstngushes between dfferent belef herarches for player, as gven by the σ-algebras A T,k 1,d 1 : Defnton 4.5. Let (µ 1, µ 2,...) H be a belef herarchy for player. Then: } (a) (µ 1, µ 2,...) has depth d = f for each extended type space T = (T n, S n, Σ n, β n ) n N and type t T such that h T (t ) = (µ 1, µ 2,...), ( ) IdΘ, h T,k 1 1(E) FΘ Σ (t ) for every k = 1, 2,... and E F Θ F B T,k 1; (b) (µ 1, µ 2,...) has depth d = 1, 2,... f for each extended type space T = (T n, S n, Σ n, β n ) n N and type t T such that h T (t ) = (µ 1, µ 2,...), the followng hold: (b1) ( Id Θ, h T,k 1 (b2) ( Id Θ, h T,k 1 every m > d 1, ) 1(E) FΘ Σ (t ) for k = 1,..., d 1 and event E F Θ F B T,k 1; ) 1(E) FΘ Σ (t ) for k > d 1 and E F Θ A T,k 1,d 1 ; and for A T,k 1,d 1 A T,k 1,m. The last part of (b2) s mportant: t mples that there are kth-order events that µ k cannot assgn a probablty to, for any k > d. We wll say that a type t for player has depth (of reasonng) d (t ) = d f t generates a belef herarchy of depth d, where d = 1, 2,... or d =. Proposton 4.6. For every belef herarchy (µ 1, µ 2,...) H, there s a unque d N { } such that (µ 1, µ 2,...) has depth d. Proposton 4.6 mples that the depth of reasonng of each type s well defned. But, gven a type, how do we determne ts depth of reasonng? One easy result s that types wth a fner σ-algebra have a greater depth of reasonng: 19

20 Proposton 4.7. For any par of types t, t for player N n T, Σ (t ) Σ (t ) f and only f d (t ) < d (t ). Intutvely, a type of depth k nduces a kth-order belef µ k ; and any lth-order belef µ l for l k can be obtaned from µ k by takng the approprate margnal (see the dscusson n Secton 4.1). Thus, a kth-order belef contans more nformaton than a belef of lower order, and ths translates nto a fner σ-algebra for types that have a kth-order belef as opposed to types that only have belefs at lower orders. Another straghtforward result s that types from Harsany type spaces have an nfnte depth of reasonng, as we would expect: Proposton 4.8. Let (µ 1, µ 2,...) H H be a belef herarchy for player nduced by a type from a Harsany type space. Then (µ 1, µ 2,...) has an nfnte depth of reasonng. For arbtrary extended type spaces, the depth of reasonng of a type can n prncple be determned by wrtng out the belef herarchy nduced by the type: f a type has a well-defned belef over each kth-order event for each k, then the type has an nfnte depth of reasonng; otherwse, there s some fnte k such that the type has well defned belefs over each lthorder event for l k, but there exst (k + 1)th-order events (and thus mth-order events for m > k + 1) t cannot reason about. Ths method of dentfyng a type s depth can of course be cumbersome. In the next secton, we therefore develop the tools that allow us to determne the depth of reasonng of a type drectly from the propertes of the type space Rank and depth In ths secton we show that the depth of reasonng of a type can be determned n terms of the propertes of the type space. Ths characterzaton s complete provded that the set of σ-algebras s suffcently rch. To characterze a type s depth n ths way, we frst need to ntroduce some new concepts. We defne the rank of a σ-algebra, whch characterzes the set of σ-algebras (on the other type set) that the σ-algebra domnates. Fx an extended type space T = (T n, S n, Σ n, β n ) n N, and for N, defne For l = 2, 3,..., let R 1 := { F S : there s no F S such that F F }. R l := { F S : there s F R l 1 such that F F, and there s no F S \ m<l R m such that F F }. 20

21 That s, R 1 s the set of σ-algebras on T that do not domnate any σ-algebras on T ; R l for l > 1 s the set of σ-algebras that domnate some σ-algebra n R l 1, but no σ-algebra n S that does not belong to any R m for m < l. Also, let R := { F S : for each l N, there s F m l R m such that F F }. The subsets R, R 1, R 2,... thus classfy the σ-algebras n S accordng to the σ-algebras they domnate. Each σ-algebra n S belongs to precsely one of these subsets, as the next result shows: Lemma 4.9. For each player N, the sets R, R 1, R 2,... partton S. Say that a σ-algebra F S for player N has rank k f F R k, where k = 1, 2,... or k =. Lemma 4.9 guarantees that the rank of each σ-algebra s well defned. Wth some abuse of termnology, we say that a type t for has rank r (t ) = k f ts σ-algebra Σ (t ) S has rank k. The rank of a type s σ-algebra can be used to characterze a type s depth of reasonng. The key result s that a σ-algebra of rank k = 1, 2,... contans precsely those sets of types that can be characterzed n terms of ther belef herarches up to order k 1: Lemma For each player N, and each σ-algebra F k R k of rank k, F k = { {t T : h T,k 1 } (t ) B} : B F B T,k 1. That s, F k conssts of subsets of T of the form {t T : h T,k 1 (t ) B}, where B s an event n F B T,k 1. The result s ntutve. As noted above, the rank of σ-algebra characterzes the σ-algebras that the σ-algebra domnates. The key nsght provded by Lemma 4.10 s that the set of σ-algebras that are domnated by a gven σ-algebra determnes the depth of reasonng of a type wth that σ-algebra. For example, suppose a σ-algebra F on the type set T for player domnates only the trval σ-algebra {T, } on T, whle there ar σ-algebras on T that are not domnated by F, then F s assgned rank r = 1. As F domnates {T, }, t separates the types for that dffer n ther belefs on F Θ {T, },.e., the types for that dffer n ther frst-order belef. A type t for wth Σ (t ) = F can therefore form a belef about any frst-order belef for. In turn, a σ-algebra F on T that domnates F (but no fner σ-algebras n S ) has rank r = 2, and separates the types for that dffer n ther belef as expressble n F Θ F. A type t wth that σ-algebra can therefore form a belef about any second-order belef for, and so on. 21