RATIONALIZABILITY AND FINITE-ORDER IMPLICATIONS OF EQUILIBRIUM

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1 RATIONALIZABILITY AND FINITE-ORDER IMPLICATIONS OF EQUILIBRIUM JONATHAN WEINSTEIN AND MUHAMET YILDIZ Abstract. Present economc theores assume common nowledge of the type structure after specfyng the frst or the second orders of belefs. We analyze the set of equlbrum predctons that can be deduced from the nowledge of equlbrum and players belefs at fnte orders. For generc fnte-acton games and the games wth undmensonal acton spaces and sngle-peaed preferences, we show that, f the space of underlyng uncertanty s suffcently rch, then these equlbrum predctons wll be equvalent to the predctons that follow from ratonalzablty at every nstance n whch the players frst-order belefs are common nowledge. In partcular, unless the game s domnance solvable, the equlbrum wll be hghly senstve to hgh orders of belefs, and present economc theores may be msleadng. Key words: hgher-order uncertanty, ratonalzablty, ncomplete nformaton, equlbrum. JEL Numbers: C72, C73. Game theory... s defcent to the extent t assumes other featurestobecommonnowledge,suchasoneplayer sprobablty assessment about another s preferences or nformaton. I foresee the progress of game theory as dependng on successve reductons n the base of common nowledge requred to conduct useful analyses of practcal problems. Only by repeated weaenng of common nowledge assumpton wll the theory approxmate realty. Wlson (1987) 1. Introducton Most present economc theores are based on equlbrum analyss of models that are closed after specfyng the frst and second-order belefs,.e., the belefs about underlyng uncertanty and the belefs about other players belefs about underlyng uncertanty. These models assume that the specfed belef structure s common nowledge,.e., condtonal on the frst and the second-order belefs, all of the players hgher-order belefs are common nowledge. There s generally no reason to beleve that these assumptons are satsfed n the n the actual Date: September,

2 2 JONATHAN WEINSTEIN AND MUHAMET YILDIZ ncomplete-nformaton stuaton modeled. Hence, these theores wll be msleadng when the mpact of hgher-order belefs on equlbrum behavor s large. There are examples that suggest that ths mght be the case. To overcome ths fundamental defcency, one may want to close the model at hgher orders and hence weaen the common nowledge assumpton by specfyng more orders of belefs. As Wlson (1987), one may hope that by specfyng more and more orders of belefs, the theory would approxmate realty. We wll demonstrate that ths s not the case. For generc normal-form games, 1 actual predctve power of any closed model wll be no more than that of ratonalzablty, no matter how many orders of belefs are specfed; any predcton that does not follow from ratonalzablty wll be drven by the assumpton made when the model s closed. Consder a stuaton where players have ncomplete nformaton about some payoff-relevant parameter. Each player has a probablty dstrbuton about the parameter, whch represents hs frst-order belefs, a probablty dstrbuton about other players frst-order belefs, whch represents hs second-order belefs, and so on. Imagne a researcher who has computed an equlbrum of ths game, where a type of a player s an nfnte-herarchy of hs belefs, and would le to mae predcton about the acton of a player accordng to ths equlbrum. Fx a type of player as hs actual type, and wrte A 1 ( ) for the set of all actons that are played by some type of whose frst-order belefs agree wth. Ths set s the set of actons that the researcher cannot rule out f he only nows the frst-order belefs and assumes that the player plays accordng to the equlbrum. Smlarly, wrte A ( ) for the set of actons that the researcher cannot rule out by loong at the frst orders of belefs. Wrte A ( ) for the lmt of these (decreasng) sets as approaches nfnty,.e., the set of all actons that cannot be ruled out by the researcher by loong at (arbtrarly many) fnte orders of belefs. Ths defnton can be put another way. Consder two researchers who agree on the equlbrum played. One researcher s certan that player s of type. The other (slghtly suspcous) researcher s wllng to agree wth ths assessment for the frst orders of belefs but does not have any further assumpton. The set A ( ) s precsely the set of actons that wll not be ruled out by the second researcher. In a model that s closed at order, all hgher-order belefs are determned by the frst orders of belefs and the assumpton that s made when the model 1 That s, n games where there are no tes and when the other players have more than one strategy, a player s payoffs from a strategy cannot be obtaned from a mxture of other strateges. Clearly the complement of ths set s of lower dmenson, and hence has Lebesgue measure zero.

3 FINITE-ORDER IMPLICATIONS 3 s closed. We wsh to emphasze the senstvty to the closng assumpton. In a gven equlbrum, the model predcts a unque acton for each possble belefs at orders 1 through, namely the equlbrum acton for the complete type mpled by ths belefs and the closng assumpton. But n the general model, every other acton n A ( ) s played by a type whose frst orders of belefs wll be exactly as ths type (but wll fal the closng assumpton.) Therefore, we cannot rule out any acton n A ( ) wthout resortng to the assumpton that s made at closng the model at order. In order to avod a techncal ssue 2, n ths verson we wll focus analyzng A ( ) for the case n whch, accordng to, the players frst-order belefs happen to be common nowledge. Of course, ths means that we wll be consderng devatons from common nowledge at orders hgher than. Our man result gves a lower bound for A ( ). Assume that our fxed equlbrum has full range,.e., every acton s played by some type. 3 For countableacton games, we show that A ( ) ncludes all actons whch survve the frst teratons of elmnatng all acton whch are never a strct best reply under. In partcular, A ( ) ncludes all actons that survve terated elmnaton of actons that cannot be a strct best reply. On the other hand, A ( ) s a subset of actons that survve the frst teratons of elmnatng strctly domnated actons, and hence A ( ) s a subset of ratonalzable actons. When there are no tes, these elmnaton procedures lead to the same outcome, and therefore A ( ) s precsely equal to the set of ratonalzable outcomes. We also extend ths characterzaton to games n whch the acton spaces are one-dmensonal compact ntervals and preferences are sngle-peaed wth contnuous best-reply functons. Of course, these games nclude many classcal economc models, such as Cournot olgopoly, provson of publc goods, etc. To llustrate the man argument n the proof of the lower bound, we now explan why A 1 ( ) ncludes all actons that survve the frst round of elmnaton process. Let vary over the set of types that agree wth at frst order (.e., concernng the underlyng parameter) but may have any belefs at hgher orders (.e., concernng the other players type profle.) Our full range assumpton mples that there are types wth any belefs whatsoever about other players equlbrum acton profle. Gven any acton a of that s a strct best reply to hs fxed belef about the parameter and some belef about the other players actons, there s a type who has these belefs n equlbrum, 2 There s a proof that would wor for arbtrary games and types but suffers from possble non-measurablty of certan functon. 3 Ths assumpton wll hold f the parameter space representng the underlyng uncertanty s rch enough.

4 4 JONATHAN WEINSTEIN AND MUHAMET YILDIZ and therefore must play the strct best reply, a, n equlbrum. Ths argument wll be formalzed as part of an nductve proof of the man result. For generc games, the above procedures are equvalent, and we have a characterzaton. Nevertheless, usually, when we fx a game tree, there wll be tes n the normal-form representaton. In that case, these procedures may dffer sgnfcantly. Whle the set of ratonalzable strateges remans large, our elmnaton procedure wll lead to a subset of the result of terated admssblty,.e., terated elmnaton of wealy domnated strateges. In perfect nformaton games, terated admssblty yelds bacwards nducton, and hence our bounds wll not be powerful. In such games, usng our technques one can fnd a characterzaton for sequental equlbra n terms of terated admssblty. Of course, such a characterzaton wll be a postve result. We extend our result beyond the full-range assumpton. To do ths, gven any subset of the range of equlbrum, we consder a new teratve process that starts wth ths subset. At each teraton we consder the set of all strct best responses aganst belefs wth supports n the prevous set. 4 Now A ( ) wll ncludeallactonsavalable atth teraton. Notce that ths s not an elmnaton procedure and the sets may easly grow, yeldng a large lower bound. As an example, we consder Cournot olgopoly wth a general nverse-demand functon wth usual regularty condtons and suffcently many frms wth dentcal constant (postve) margnal cost. We assume that demand depends on a realvalued parameter θ such that a frm s best reply s contnuous and ncreasng wth respect to θ. Consder agan the confdent researcher and hs slghtly septcal frend. The former s confdent that t s common nowledge that θ = θ, whle the latter s only wllng to concede that t s common nowledge that θ θ ε and agrees wth the th-order mutual nowledge of θ = θ. Hesan arbtrarly generous septc; he s wllng to concede the above for arbtrarly small ε > 0 and arbtrarly large fnte. We show that the septc nonetheless cannot rule out any output level that s not strctly domnated. Our results have two mplcatons. Frstly, whle there are epstemc foundatons for teratve admssblty (Brandenburger and Kesler (2002)) and ratonalzablty (Bernhem (1985), Pearce (1985)), the epstemc foundatons for Nash equlbrum n general games are very wea (Aumann and Brandenburger (1995)). Despte ths, equlbrum analyss s frequently used because t offers much sharper predctons. Our result shows that ths predctve power s deceptve because t requres the full power of the (common-nowledge) assumpton made n closng the model; f we weaen that assumpton, however slghtly, equlbrum s no more powerful than terated admssblty. 4 These sets are analyzed also by Mlgrom and Roberts (1991).

5 FINITE-ORDER IMPLICATIONS 5 Secondly, our result ponts to a close ln between the equlbrum mpact of hgher-order uncertanty and hgher-order reasonng. For generc games, assumng th-order mutual nowledge of payoffs andthatafxed equlbrum (wth full support) s played s equvalent to assumng th-order mutual nowledge of ratonalty and common nowledge of payoffs. 5 Ths mples that when the equlbrum mpact of hgh-order uncertanty s large, the mpact of hgh-order falures of ratonalty s also large, suggestng that predctons wll be hghly unrelable wthout a very accurate nowledge of players reasonng capacty. 2. Basc Defntons and Prelmnary Results Notaton 1. Gven any lst X 1,X 2,... of sets, wrte X = Q j6= X j, x = (x 1,...,x 1,x +1,...) X,and(x,x )=(x 1,...,x 1,x,x +1,...). Lewse, for any famly of functons f j : X j Y j,wedefne f : X Y by f (x )=(f j (x j )) j6=. Gven any metrc space (X, d), wrte (X) for the space of probablty dstrbutons on X, suppressng the fxed σ-algebra on X whch at least contans all open sets and sngletons; when we use product spaces, we wll always use the product σ-algebra. We wll wrte δ x for the probablty dstrbuton that puts probablty 1 on {x}. We consder a game wth fnte set of players N = {1, 2,...,n}. Thesource of underlyng uncertanty s a payoff-relevant parameter θ Θ where (Θ,d) s a compact, complete and separable metrc space, where d s a metrc on set Θ. Each player has acton space A and utlty functon u : Θ A R where A = Q A. We endow the game wth the unversal type space of Brandenburger and Deel (1993), a varant of an earler constructon by Mertens and Zamr (1985). Types are defned usng the auxlary sequence {X } of sets defned nductvely by X 0 = Θ and X =[ (X 1 )] n X 1 for each > 0. We endow each X wth the wea topology and the σ-algebra generated by ths topology. A player s frstorderbelefs(about theunderlyng uncertanty θ) are represented by a probablty dstrbuton 1 on X 0, second order belefs (about all players frst order belefs and the underlyng uncertanty) are represented by a probablty dstrbuton 2 on X 1,etc. Therefore,atype of a player s a member of Q =1 (X 1). Snce a player s th-order belefs contan nformaton about hs lower-order belefs, we need the usual coherence requrements. We wrte T = Q N T for the subset of ( Q =1 (X 1)) n n whch t s common nowledge that the players belefs are coherent,.e., the players now ther own 5 Of course, there s a close relatonshp between assumptons about ratonalty and payoff uncertanty. But ths relatonshp s not straghtforward. In non-generc games, A dffers from ratonalzablty, and the ratonalty assumpton for teratve admssblty cannot be common nowledge (Brandenburger and Kesler (2002)), whle payoffs can be common nowledge n our model.

6 6 JONATHAN WEINSTEIN AND MUHAMET YILDIZ belefs and ther margnals from dfferent orders agree. We wll use the varables, T as generc types of any player and, T asgenerctypeprofles. For every T, there exsts a probablty dstrbuton κ on Θ T such that (2.1) = δ 1 marg Θ [ (X 2 )] N\{}κ, ( ) and 1 =marg Θ κ, where marg denotes the margnal dstrbuton. Conversely, gven any dstrbuton κ on Θ T,wecandefne T va (2.1). A strategy of a player s any measurable functon s : T A.Gvenanytype and any profle s of strateges, we wrte π (,s ) (Θ A ) for the jont dstrbuton of the underlyng uncertanty and the other players actons nduced by and s. Formally, π (,s )=κ β 1 where β :(θ, ) 7 (θ,s ( )). Defnton 1 (Best Reply). For each N and for each belef π (Θ A ), we wrte BR (π) forthesetofmaxmzersofe π [u (θ,a,a )] over a A, where E π s the expectaton operator wth respect to π. WhenBR (π) s sngleton, wth slght abuse of notaton, we wrte BR (π) for ts unque member. We sometmes wrte BR θ, σ for BR (δ θ σ ). Astrategyprofle s =(s 1,s 2,...) s a Bayesan Nash equlbrum ff at each, s ( ) BR π,s. An equlbrum s s sad to have full range ff ((FR)) s (T )=A. The followng assumpton mples that every equlbrum s has full range. Assumpton 1 (Rchness of Θ). Gven any N, anyµ (A ),andany a,thereexstsν (Θ), suchthat BR (ν µ) ={a }. Lemma 1. Under Assumpton 1, every equlbrum s has full range. Proof. Theproofsthatareomttednthetextarentheappendx. Notaton 2. For some arbtrary θ 1,...,θ n Θ N,wrte for the type of a player who s certan that t s common nowledge that each player j s certan that θ = θ j. 6 For notatonal convenence we wll suppress the dependence on θ 1,...,θ n. 6 The assumpton that the players frst-order belefs are of ths degenerate form s only for the sae of smplcty. As long as the players frst order belefs are common nowledge, our results reman vald.

7 FINITE-ORDER IMPLICATIONS 7 We are nterested n how equlbrum s robust aganst the falure of common nowledge assumpton n hgh orders. We now formalze our noton of robustness. Equlbrum predcton wth fnte-order nformaton of payoffs. Letus fx an equlbrum s and a type of a player, whobelevesthattscommon nowledge that each player j s certan that θ = θ j for some θ 1,...,θ n. Accordng to equlbrum, he wll play s. Now magne a researcher who only nows the frst th-order belefs of player and nows that equlbrum s s played. All the researcher can conclude from ths nformaton s that wll play one of the actons n A s, = ns ( ) l = l l o. Assumng plausbly that a researcher can verfy only fnte orders of a player s belefs, all a researcher can ever now s that player wll play one of the actons n A \ s, = =1 A s,. It turns out that the set A s, s closely related to the set of ratonalzable actons. To establsh ths close ln, we wll now defne ratonalzablty and a strong verson of teratve admssblty at nstances n whch the players perceptons of underlyng uncertanty s common nowledge. Ratonalzablty and Iterated Admssblty. Wewlldefne the ratonalzable actons for a type. Ths set wll be equal to the usual ratonalzable actons of player forthegamenwhchtscommonnowledgethateach player j s certan that θ = θ j. Defne sets Sj, j N, =0, 1,..., teratvely as follows. Set Sj 0 = Aj.Foreach>0, letσ j 1 be the set of all probablty dstrbutons on S j 1,.e., the set of all possble belefs of player j on other players allowable actons that are not elmnated n the frst 1 rounds. Wrte [ Sj = BR j θ j, σ j σ j Σ 1 j for the set of all all actons a j of j that are best reply aganst some of hs belefs n Σ j 1. The set of all ratonalzable actons for player (wth type )s \ S = S. =0 Next we defne the set of strateges that survve teratve elmnaton of strateges that are never strct best reply, denoted by W, smlarly. We set

8 8 JONATHAN WEINSTEIN AND MUHAMET YILDIZ W 0 j = Aj, W j = aj BR j θ j, σ j = {aj } for some σ j W 1 j ª, and \ W = W. =0 Notce that we elmnate a strategy f t s not a strct best-response to any belef on the remanng strateges of the other players. Clearly, ths yelds a smaller set than the result of teratve admssblty (.e., teratve elmnaton of wealy domnant strateges). 7 In some games, teratve admssblty may yeld strong predctons. For example, n fnte perfect nformaton games t leads to bacward nducton outcomes. Nevertheless, n generc normal-form games all these concepts are equvalent and usually have wea predctve power. Lemma 2. For fnte-acton games n normal form, f the payoffs aregenerc under,then W = S. Our next result states that, nowng fntely many orders of a player s belefs and the equlbrum, a researcher can predct that a ratonalzable strategy wll be played. Proposton 1. For any equlbrum s, any player, andany as n Notaton 2, n partcular, A s, S A s, S. 0; Proof. For =0, the proposton s true by defnton. Assume that t s true for some 1 0,.e.,forany, A 1 s, S 1.Nowtaeany wth l = l for all l. Under,player s certan that θ = θ and l = l for each l 1. Hence, by nducton hypothess, he s certan that s ( ) S 1. Thus, π,s = δθ µ for some µ A 1 s,. Therefore, showng that A s ( ) BR π,s S, s, S. 7 In partcular, f we use non-reduced normal-form of an extensve-form game, many strateges wll be outcome equvalent, n whch case our procedure wll elmnate all of these strateges. To avod such over-elmnaton, we can use reduced-form, by representng all outcomeequvalent strateges by only one strategy.

9 FINITE-ORDER IMPLICATIONS 9 Snce the set of ratonalzable strateges s typcally very large, t s not surprsng that, usng the nowledge of fnte orders of belefs and equlbrum, a researcher can predct that a ratonalzable acton wll be played. Nevertheless, we wll llustrate next that, typcally, ths s all a researcher can deduce. 3. Countable-acton games We wll now consder the countable-acton games and show that equlbrum predctons wth the nowledge of fnte-order belefs cannot be sharper than that of teratve elmnaton of strateges that are never strct best-response. Snce the latter s equvalent to ratonalzablty for generc games, ths wll yeld a characterzaton for these games. (A game s sad to be a countableacton game ff A s countable or fnte for each N.) Proposton 2. For any countable-acton game, any equlbrum s wth full range, any N, N, andany n partcular, W as n Notaton 2, A s, ; W A s,. Proof. We wll use mathematcal nducton. For =0, the statement s gven by the full-range assumpton. Assume that the proposton s true for some,.e., W j A j s, ( j) for some. Wewllfx arbtrary and show that W +1 A +1 s,. Tae any a W +1.Bydefnton, there exsts a probablty dstrbuton σ on W such that (3.1) BR θ, σ = {a }. By nducton hypothess, for each a W, there also exsts [a ] T such that (3.2) s ( [a ]) = a and (3.3) l [a ]= l l.

10 10 JONATHAN WEINSTEIN AND MUHAMET YILDIZ Now, let be the type of player who s certan that θ = θ and assgns probablty σ (a ) to each type [a ],.e., X κ = δ θ σ (a ) δ [a ]. a W [ ] Frstly, and agree up to the +1st order. To see ths, note frst that 1 = δ θ = 1.Moreover,foranyl wth 1 <l +1,wehave marg [ (Xl 2 )] N\{}κ = X σ (a ) marg [ (Xl 2 )] N\{}δ [a ] = X σ (a ) δ l 1 [a ] = X σ (a ) δ ( ) = δ l 1 ( ) l 1, where the summatons are taen over a W, and the thrd equalty s due to (3.3). Hence, by (2.1), l = δ θ δ l 1 δ ( ) = l l 1. But, under, s ( ) s dstrbuted wth σ,.e., the jont dstrbuton of θ and s ( ) s π,s = δθ X σ (a ) δ s ( [a ]) = δ θ X σ (a ) δ a, where the summatons are taen over a W and the second equalty s by (3.2). Therefore, s ( ) BR π,s = BR θ, σ = {a }. s,, and thus W +1 Ths shows that a = s ( ) A +1 A +1 For generc fnte-acton games, Propostons 1 and 2 yeld a characterzaton. Proposton 3. Under the notaton and the assumptons of Proposton 2, f the payoffs are generc under,then A s, = S. That s, n generc fnte-acton games, a researcher s predctons based on fnte orders of players belefs and equlbrum wll be equvalent to the predctons that follow from ratonalzablty. For non-generc games, we have weaer conclusons: W A s, S. s,.

11 FINITE-ORDER IMPLICATIONS Nce games We wll now consder a class of nce games, whch are wdely used n economc theory, and show that A s, = S for each whenever equlbrum s has full range. Defnton 2. Agamessadtobence ff for each, A =[0, 1] and u (θ,,a ) s a sngle-peaed functon maxmzed at some BR (θ,a ) where BR (θ,a ) s contnuous n a. 8 Our frst lemma presents smple but very useful propertes of ratonalzable strateges n nce games. Lemma 3. For any nce game and any as n Notaton 2, the followng are true. (1) Each S saclosedntervalna. (2) For each a S, there exsts a 1 S 1 a = {BR θ,a 1 }. such that Lemma 3.2 mples that a researcher s predctons about the equlbrum behavor wth only fnte-order nformaton of belefs wll be equvalent to predctons that follow from ratonalzablty as our next proposton establshes. Proposton 4. For any nce game, any equlbrum s wth full range, for any N, N, and as n Notaton 2, S n partcular, S = A s, ; = A s,. Proof. Proposton 1 establshes already that A s, S.Wewll now prove the reverse nequalty usng symmetrc arguments to the proof of Proposton 2. Assume that S 1 A 1 s,. Then, Lemma 3.2 states that each a S s a unque best reply to some a 1 = s ( ) where l = l for each l 1. Now consder the type that puts pont mass at θ,. Then, s ( ) BR θ,s ( ) ª = BR θ,a = a. But, by defnton of,wehave l = l for each l, showngthat a S,.e., S A s,. 8 Ths assumpton s satsfed f u s strctly quas-concave n a and contnuous n a.

12 12 JONATHAN WEINSTEIN AND MUHAMET YILDIZ 5. Extensons We have so far assumed that the equlbrum at hand has full range, whch allowed us to consder large changes. A researcher may be certan that t s common nowledge that the set of parameters are restrcted to a small subset, or equvalently, the equlbrum consdered may not vary much as the belefs about the underlyng uncertanty change. We wll now present extensons of our result to such cases. Local Ratonalzablty. Taeany and any B A such that s B. WeDefne sets S B;, N, N, by settng S 0 B; = B, [ S B; = BR θ, σ. σ (S 1 [B; ]) Notce that ths s the same procedure as terated strct domnance, except that the ntal set s restrcted to a subset. Unle terated strct domnance, these sets can become larger as ncreases. Hence we defne the set of locally ratonalzable strateges by S \ B; = [ =0 l= Notce that the set S B; verson of W, smlarly, by settng W 0 B; W B; S l B;. may be much larger than B. We defne local = B, = a A BR θ, σ = {a } for some σ W 1 = T =0 and W B; n our elmnaton process. B; ª, S l= W l B;. Notce that we consder all actons Proposton 5. Let s be an equlbrum, N and be as n Notaton 2. Ifthegamehascountableactons,then W s (T ); A s, S s (T );. Ifthegamesnceoragenercfnte-acton game (under ), then for any B s (T ), S B; A s,. The last statement mples that, for nce and fnte-acton games, even the slght changes n very hgher-order belefs wll have substantal mpact on equlbrum behavor, unless the game s locally domnance-solvable,.e., S B; = s ª for some open neghborhood B of s. In our next secton, we wll show that many mportant games lac local domnance-solvablty, and

13 FINITE-ORDER IMPLICATIONS 13 hence anythng less than common nowledge of players percepton wll lead to substantally dfferent outcomes. 6. Applcatons 6.1. Cournot Olgopoly. Frst consder the lnear case. When we have only a duopoly, the game s domnant-solvable, and hence Proposton 1 mples that hgher-order belefs have neglgble mpact on equlbrum whenever the players frst-order belefs turn out to be common nowledge. Wensten and Yldz (2003) shows that ths s n fact the case for entre type space, a fact that s also mpled by a result of Nyaro (1996). On the other hand, when n 3, any producton level that s less than or equal to the monopoly producton s ratonalzable, and hence Proposton 4 mples that a researcher cannot rule out any such output level for a frm no matter how many orders of belefs he specfes. We wll now show a more dsturbng fact. For farly general olgopoly models we wll show that when n s suffcently large, any such outcome wll be n S B; for every neghborhood B of s. Therefore, by Proposton 5, even a slght doubt about the model n very hgh orders wll lead a researcher to fal to rule out any outcome that s less than monopoly outcome as a frm s equlbrum output General Cournot Model. Consder n frms wth dentcal constant margnal cost c>0. Smultaneously, each frm produces q [0, 1] at cost q c and sell ts output at prce P (Q; θ) where Q = P q s the total supply. (Here we mpose capacty constrants n order to be consstent wth our general model. We wll focus on the cases n whch these constrants are not bndng.) For some fxed θ, we assume that Θ s a closed nterval wth θ Θ 6= θª. We also assume that P 0; θ > 0, P ; θ s strctly decreasng when t s s postve, and lm Q P Q; θ =0. Therefore, there exsts a unque ˆQ such that ³ P ˆQ; θ = c. We assume that, on [0, ˆQ], P ; θ s contnuously twce-dfferentable and P 0 + QP 00 < 0. It s well nown that, under the assumptons of the model, () the proft functon, u q, Q; θ = q (P (q + Q) c), s strctly concave n own output q; () the unque best response q (Q ) to others aggregate producton Q s strctly decreasng on [0, ˆQ] wth slope bounded away from 0 (.e., q / Q λ for some λ < 0); () equlbrum outcome at θ,... θ, s θ,... θ,s unque and symmetrc (Ouguch and Suzumura (1971)).

14 14 JONATHAN WEINSTEIN AND MUHAMET YILDIZ Lemma 4. In the general Cournot model, for any equlbrum s and for under whch t s common nowledge that θ = θ for arbtrary θ, there exsts n < such that for any n> n and for any B = Q N s ², s + ² A wth ²>0, we have B; = 0,q M ( N), S where q M [0, 1] s the monopoly outcome under P ; θ. Proof. Let n be any nteger greater than 1+1/ λ, whereλ s as n (). Tae any n> n. By (), B =[q 0, q 0 ] n for some q 0, q 0 [0, 1] wth q 0 < q 0. By (), for any >0, S B; =[q, q ] n,where q = q (n 1) q 1 and q = q (n 1) q 1. Defne Q (n 1) q, Q (n 1) q,andq =(n 1) q,sothat Q = Q Q 1 and Q = Q Q 1. Snce the slope of Q, (n 1) λ, s strctly less than 1, Q decreases wth and becomes 0 at some fnte, and Q ncreases wth and taes value Q (0) = (n 1) q M at +1.Thats,S B; = 0,q Mn for each >. Therefore, S B; = 0,q Mn. Together wth Proposton 5, ths lemma yelds the followng. Proposton 6. In the general Cournot model, assume that Θ = θ ε, θ + ε for arbtrarly small ε > 0, and that the best-response functon q (Q ; θ) s a contnuous and strctly ncreasng functon of θ at Q, θ where Q = (n 1) s j s the others aggregate output n equlbrum, and s the type profle under whch t s common nowledge that θ = θ. Then, A s, = 0,q M ( N), where q M [0, 1] s the monopoly outcome under P ; θ. Proof. Frstly, by () above, we have a nce game. Moreover, by the assumpton n the hypothess, there exsts B s (T ) as n Lemma 4. Hence, Lemma 4 and Proposton 5 mply 0,q M = S B; A s, 0,q M, yeldng the desred equalty. In Proposton 6, the assumpton that q (Q ; θ) s responsve to θ guarantes that θ s a payoff-relevant parameter. Hence, our proposton can be spelled outasfollows. Assumethattsactuallycommonnowledgeamongthefrms that a payoff-relevant parameter θ taes value θ. Imagne a researcher who

15 FINITE-ORDER IMPLICATIONS 15 does not now ths but s certan that t s common nowledge that θ s n some arbtrarly gve neghborhood of θ. Hence the researcher has a very slght uncertanty about the complete nformaton case. Suppose n addton the researcher can learn the players belefs at arbtrary fnte orders. If there are suffcently many frms n the maret, no matter how many orders of belefs the researcher correctly specfes, he wll not be able to rule out any output level that s not strctly domnated. In that case, any predcton based on equlbrum that s not mpled by strct domnance wll be nvald whenever we slghtly devate from the dealzed complete nformaton model. Appendx A. Omtted proofs ProofofLemma1.Tae any and any a. Tae any γ (T ), and let µ = γ s 1 (A ).Letνbeas n Assumpton 1. Defne as the type such that κ = ν γ. Notcethatπ ³,s = κ β 1 =(ν γ) β 1 = ν γ s 1 = ν µ. Hence,s ( )=BR π,s = BR (ν µ) =a. ProofofLemma3.We wll use nducton on. For =0, part 1 s true by defnton. Assume that part 1 s true for some 1,.e.,Sj 1 s a closed subset of A j =[0, 1] for each j. That s, S 1 s a compact set. Hence, there exst a =mn a S 1 [ ] BR θ,a and ā =max a S 1 [ ] BR θ,a. Snce the preferences are sngle-peaed and all the acton profles outsde of S 1 are elmnated for j 6=, ā strctly domnates each a > ā,anda strctly domnates each a < a. Hence, S a, ā. Moreover, snce S 1 convex and BR θ ³,a s contnuous n a, BR θ,s 1 s connected. Thus, a, ā BR ³θ,S 1 ³ S. Therefore, S = BR θ,s 1 = a, ā, a closed nterval. Snce S = BR ³θ,S 1, by defnton, each a S s the unque best response to some a 1 S 1. References [1] Aumann and Brandenburger (1995): Epstemc Condtons for Nash Equlbrum, Econometrca, 63, [2] Bernhem, D. (1984): Ratonalzable strategc behavor, Econometrca, 52, [3] Brandenburger, A. and E. Deel (1993): Herarches of Belefs and Common Knowledge, Journal of Economc Theory, 59, [4] Brandenburger, A. and Kesler... [5] Carlsson, H. and E. van Damme (1993): Global Games and Equlbrum Selecton, Econometrca, 61, [6] Fenberg, Y. and A. Srzypacz (2002): Uncertanty about uncertanty and delay n barganng, mmeo.

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