Common Belief Foundations of Global Games

Transcription

1 Common Belef Foundatons of Global Games Stephen Morrs Prnceton Unversty Hyun Song Shn Prnceton Unversty October 2007 Abstract We provde a characterzaton of when an acton s ratonalzable n a bnary acton coordnaton game n terms of belefs and hgher order belefs. The characterzaton sheds lght on when a global game yelds a unque outcome. In partcular, we can separate those features of the nosy nformaton approach to global games that are mportant for unqueness from those that are merely ncdental. We derve two su cent condtons for unqueness that do not make any reference to the relatve precson of publc and prvate sgnals. Prelmnary verson. We acknowledge support from the NSF under the grant

2 1 Introducton Games often have many equlbra. Even when they have a sngle equlbrum, they often have many actons that are ratonalzable, and are therefore consstent wth common knowledge of ratonalty. Yet a pathbreakng paper by Carlsson and van Damme (1993) suggested a natural perturbaton of complete nformaton that gves rse to a unque ratonalzable equlbrum for each player. They ntroduced the dea of global games - where any payo s of the game are possble and each player observes the true payo s of the game wth a small amount of nose. They showed - for the case of two player two acton games - that as the nose about payo s become small, there s a unque equlbrum; the equlbrum strateges played also consttute the unque ratonalzable strateges. Ths result has snce been generalzed n a number of drectons and used n a number of applcatons. 1 When the global game approach can be appled to more general games, t can be used to derve unque predctons n settngs where the underlyng complete nformaton game has multple equlbra, makng t possble to carry out comparatve statc and polcy analyss. However, a number of recent papers have rased questons both on the basc theoretcal ratonale for global games and the applcablty of the framework for the analyss of real world problems. Three strands of the argument from the lterature are partcularly worthy of note. 1. In most economc envronments where coordnaton s mportant, nteractons endogenously generate nformatve publc nformaton that mght be used as a coordnaton devce. An especally mportant source 1 Morrs and Shn (1998) analyzed a global game wth a contnuum of players makng bnary choces, and ths case has been studed n a number of later applcatons. Morrs and Shn (2003) survey some theory and applcatons of global games. 2

3 of endogenous publc nformaton are market prces (see Atkeson (2001), Tarashev (2003), Hellwg, Mukherj and Tsyvnsk (2006) and Angeletos and Wernng (2006)) 2. When prces convey nformaton, ncreased precson of prvate nformaton wll feed ncreased accuracy of (endogenous) publc sgnals. sgnals are su cently accurate. Thus unqueness condtons wll fal f prvate 2. Whle asymmetrc nformaton may exst n a large varety of economc settngs, t does not always conform to the global game noton of nosy sgnals. Global game results turn on the relatve precson of prvate and publc sgnals, but f we do not know what these nosy sgnals are n real lfe, debates about relatve precsons have no conceptual bass (see, for example, Kurz (2006), Sms (2005a, 2005b), Svensson (2006), Woodford (2005)). 3. Whle common knowledge of payo s s relaxed n global games, there s stll assumed to be common knowledge of the nformaton structure, whch s surely a no more realstc assumpton. A recent paper by Wensten andyldz (2007) shows that the exact form of the perturbaton away from common knowledge of payo s s crucal n determnng the ratonalzable outcome. The global game predcton s not the only possble perturbaton that yelds unque ratonalzable outcomes. What clam does the global game approach have for beng a natural or reasonable perturbaton? The objectve of our paper s to evaluate these arguments and questons concernng the global game methodology, and to provde a framework that 2 Angeletos, Hellwg and Pavan (2006a, 2006b) note (nter ala) how other sources of endogenous publc nformaton may lead to multplcty n such coordnaton games. 3

4 can both deepen our understandng of the theoretcal bass for global games and to provde gudance for appled researchers on the scope (and lmtatons) of the global game approach. The canoncal nformaton structure assocated wth the global game approach s one where players observe the underlyng fundamental varable wth some nose. Ths s for the hstorcal reason that the early papers (Carlsson and van Damme (1993), Morrs and Shn (1998)) adopted ths formalsm. The nose s a convenent way to relax common knowledge of the fundamentals, but n subsequent applcatons of global games the nosy nformaton structure has been taken more lterally - as players falng (lterally) to observe the true fundamentals perfectly. Many of the crtcsms of the global game approach presumes such a lteral nterpretaton of the global games approach. However, there are ptfalls n takng the nosy nformaton structure too lterally, as the underlyng logc of the argument becomes dent ed wth a partcular formalsm, and the general scope of the approach becomes obscured by debates surroundng the merts or otherwse of the partcular formalsm. The logc underlyng the global game approach turns out to be more robust, and s not ted to takng nosy sgnals lterally. In ths paper, we revst the belef foundatons of global games. We know already that the falure of common knowledge of the fundamentals s a necessary condton for generatng the global game outcome, but the more demandng task s to show precsely how belefs depart from the complete nformaton benchmark. We have two objectves n ths paper. Frst, we lnk the global game analyss wth the earler lterature on common knowledge and nteractve epstemology - to the framework popularzed by Aumann (1976) and Monderer and Samet (1989). We provde a frame- 4

5 work that can encompass global games (especally ther countable state analogues) wthn a framework of nteractve belefs. We de ne an operator on the type space that has a strong resemblance to the p-belef operator of Monderer and Samet (1989), and show how ratonalzablty corresponds to common belef n ths generalzed belef operator. The perspectve s that of an outsde observer who observes only whether a player chooses one acton or the other. The fact that a partcular acton has been chosen reveals much about the player s belefs - both about the fundamentals of the envronment, but also about the belefs and hgher order belefs of other players. The belef operators that we dentfy correspond wth to the revealed strength of belefs that a player holds about the envronment and the other players. In ths sense, we take the vewpont of an outsde observer (such as an emprcal economst) who attempts to pece together the belefs from the acton chosen. In ths way, we can characterze the hgher order belefs that underpn play n global games, thereby answerng the queston of how the departure from common knowledge s acheved n global games. Second, the revealed belefs approach yelds nsghts on the queston of when there s a unque ratonalzable outcome n the global game. By usng the framework of the generalzed belef operators, we dentfy two sets of su cent condtons on the common belefs of the types that ensure unque ratonalzablty. Essentally, the property that matters s the statonarty of belefs wth respect to the orderng of types. Global game arguments work because the belefs that player types have over ther neghborng types do not change abruptly as we consder types along the orderng. A specal case of such nsenstvty of belefs along the type space s the case when each type beleves he s typcal. We show that unqueness n the nosy nformaton approach to global games wth publc and prvate nformaton uses precsely 5

6 ths strong verson of nsenstvty of belefs to the order. The rest of the paper s structured as follows. We begn n secton 2 wth a leadng example that llustrates many of the features that wll make an appearance n the general argument. We then characterze the hgher order belefs that are necessary and su cent for ratonalzablty, and revst some famlar examples of global games from the appled lterature, and llustrate our result. Secton 5 then bulds on earler results to shed lght on unqueness. We dscuss two su cent condtons for unqueness that do not make reference to nosy sgnals, or relatve precsons of prvate and publc sgnals. 2 Example There are I players who choose from fnvest, not nvestg. There s a cost of nvestng, p 2 (0; 1). The payo to nvestng depends on the fundamental state. There are domnance thresholds and wth < such that not nvest s domnant when falls below the lower threshold and nvest s domnant when s above the upper threshold. When <, the gross return to nvestng s zero rrespectve of the actons of the other players, so that nvestng yelds a sure payo of p. When >, the gross return to nvestng s 1 rrespectve of the actons of the other players so that nvestng yelds a sure payo of 1 p. When, the gross return to nvestng s 1 f and only f proporton q or more of the players (ncludng oneself) nvest, where 0 < q < 1. payo matrx s at least q nvest less than q nvest nvest 1 p p not nvest 0 0 The 6

7 2.1 Reconstructng the Belef Herarchy For an outsde observer (an emprcal economst, say), the observable features of the problem are qute coarse. The outsde observer sees only whether a player nvests or not. But when combned wth the knowledge of the payo s and the players ratonalty, the decson to nvest reveals much about that player s belefs - both about the fundamentals, but also about the belefs of other players. Suppose player s seen to nvest. Then, ether has a domnant acton to nvest, or he p-beleves all of the followng proporton q or more ether have a domnant acton to nvest or p- beleve that 3. proporton q or more ether have a domnant acton to nvest or p- beleve that [proporton q or more ether have a domnant acton to nvest or p-beleve that ] 4. and so on... p-belef of statement 1 s a necessary condton for nvestng, snce otherwse the expected payo to nvestng s negatve rrespectve of the actons of the other players. But then, other players wll have reached a smlar concluson. So, player must also p-beleve statement 2, snce otherwse there s probablty less than p that proporton q or more players consder nvest as beng rst-order undomnated. Then, fewer than q wll nvest. In general, the falue to p-beleve statement n + 1 s a reason not to nvest, because there s probablty less than p that proporton q or more players consder nvest as beng n-th order undomnated. 7

8 In ths way, unless nds t domnant to nvest, p-belef of all the statements n the lst s necessary for nvest to be chosen. Conversely, when a player p-beleves all of the statements n the lst, ths s also su cent for nvest to survve the terated deleton of domnated strateges. There s an exactly analogous herarchy of belefs that are revealed by a player who chooses not to nvest. Player j who does not nvest reveals that ether he nds t domnant not to nvest, of he (1 p)-beleves of all of the followng statements proporton 1 q or more ether have a domnant acton not to nvest or (1 p)-beleve that 3. proporton 1 q or more ether have a domnant acton not to nvest or (1 p)-beleve that [proporton 1 q or more ether have a domnant acton not to nvest or (1 p)-beleve that ] 4. and so on... These statements are ndvdually necessary and jontly su cent for not nvest to survve the terated deleton of nterm domnated strateges. 2.2 Informaton Structure To explore when one or other acton may be supported as an teratvely undomnated acton, we ntroduce an nformaton structure. Suppose takes realzatons n the set of ntegers Z = f ; 2 1; 0; 1; 2; 3; g and there s a pror densty over Z. There are I = 2n + 1 players who play the nvestment game. 8

9 The players receve nosy sgnals concernng. Let s be player s sgnal realzaton. s takes values n Z. Condtonal on, player s equally lkely to observe any sgnal between n to + n, but we depart from the famlar global game assumpton that players sgnals are ndependent condtonal on. The purpose of ths departure s to construct an nformaton structure that s as close as possble n sprt to the contnuum player global game, as we wll elaborate below. Condtonal on, each sgnal realzaton between n and + n s observed by precsely one player. No two players observe the same sgnal, and each possble realzaton between observed by some player. n and + n s One way n whch our nformaton structure could be generated s through the followng procedure. receve the hghest sgnal (namely + n). Condtonal on, a player s selected randomly to Each player has equal chance of beng selected. Next, the second hghest sgnal realzaton, + n 1 s gven to a player chosen from the remanng pool of players, where each player has equal chance of beng selected, and so on. Once the rankng has been chosen (unknown to the players themselves), each player observes hs sgnal, and makes nferences based on ths sgnal. The nformaton structure arrved at n ths way has the followng two features. Any two players can be strctly ranked accordng to ther sgnal realzatons. Condtonal on, player has equal chance of observng any sgnal realzaton between n and + n. Condtonal on observng sgnal realzaton s, player s posteror densty has support over the nterval [s n; s + n], and ( j s ) ( 0 j s ) = () ( 0 ) 9

10 for, 0 n the support. Among other thngs, ths means that the posteror denstes can be ranked by rst-degree stochastc domnance. We can trace a player s belefs about hs rank n the populaton, as measured by the realzaton of hs sgnal relatve to those of others. Player wth sgnal s has the hghest sgnal realzaton when = s n. So, player beleves he has the hghest sgnal wth probablty (s n j s ). In general, player wth sgnal s beleves that he has the k + 1-th hghest sgnal n the populaton wth probablty (s n + k j s ). Let k (s ) be the probablty that player assgns to there beng k 1 players wth sgnals lower than hmself, condtonal on sgnal s. Then k (s ) = (s + n k + 1 j s ). Let (s ) ( 1 (s ) ; 2 (s ) ; ; I (s )) be the pro le of s belefs over hs rank order, condtonal on s. 2.3 Evdent Events For the next step, see gure 1. Fx ^, and let ^s be the hghest sgnal realzaton such that proporton q or more of players have sgnal realzatons that are ^s or hgher at ^. Denote by ^p the probablty that ^ condtonal on ^s. We then have: 1. When ^, proporton q or more players receve sgnal ^s or hgher. Ths follows from the rst-degree stochastc domnance of sgnal realzatons as ncreases. 2. When s ^s, player ^p-beleves that ^. Ths follows from the rst-degree stochastc domnance of the posteror densty over as s ncreases. 10

11 ŝ θˆ q s ŝ θˆ pˆ θ Fgure 1: Evdent events So, when ^, proporton q or more of the players ^p-beleve that n ^. We say that the event j ^ o s (q; ^p)-evdent when ths holds. Our de- nton generalzes Monderer and Samet s (1989) noton of p-evdent events, where we keep track of the proporton q of players who p-beleve an event. Suppose now that ^ n, and that player ^p-beleves j ^ o. Then he ^p-beleves all of the followng: proporton q or more ^p-beleve that 3. proporton q or more ^p-beleve that [proporton q or more ^p-beleve that 4. ] Say that there s common (q; p)-belef that when ths lst holds. From the lst of statements n secton 2.1, nvest s ratonalzable for f ^p p. We also have the reverse mplcaton. Invest s ratonalzable only 11

12 q 1 q s θ s θ s s θ pˆ rˆ s θ θ Fgure 2: Case when ^p p and ^r 1 p f ^p p. Ths s because when player has a domnant acton to nvest, he ^p-beleves that ^. So, the ether-or clause concernng domnant n acton types s subsumed under the ^p-belef of j ^ o. Snce our space of sgnals and fundamentals s countable, common (q; p)-belef mples the exstence of a (q; p)-evdent event, snce otherwse the countable ntersecton of events satsfyng each clause yelds the empty event. Followng an exactly analogous lne of reasonng for when not nvest s ratonalzable, we have: Clam 1 Invest s ratonalzable for f and only f p-beleves some (q; p)- evdent subset of fj g. Not nvest s ratonalzable for f and only f (1 p)-beleves some (1 q; 1 p)-evdent subset of j. Fgure 2 llustrates a case when both actons may be ratonalzable. When ^p p and ^r 1 p, the event fj g s (q; p)-evdent, and j s (1 q; 1 p)-evdent. Unque ratonalzablty rests on rulng out such cases. 2.4 Unqueness Consder the rank pro les (s ) and (s 0 ) at two d erent sgnal realzatons s and s 0. Wrte (s 0 ) D (s ) when (s 0 ) weakly domnates (s ) n the 12

13 µ ( θ ) θ θ θ Fgure 3: Decreasng rank belefs sense of rst degree stochastc domnance. Say that rank belefs are weakly ncreasng n sgnals when s 0 s mples (s 0 ) D (s ). Let s and s be sgnal realzatons llustrated n gure 2. s s the hghest sgnal such that at, proporton q or more have sgnal s or hgher. s s the lowest sgnal such that at, 1 q or more have sgnal s or lower. We then have the followng su cent condton for unqueness. Clam 2 If rank belefs are weakly decreasng n sgnals n fs j s s sg, then there s a unque ratonalzable outcome n the nvestment game, except possbly at one value of. When rank belefs are weakly decreasng n sgnals, a player beleves that hs rank s low when he receves a hgh sgnal. Suppose that a student nds out that hs test score s hgh, and what matters s just hs relatve rankng n the class. Is the hgh score good news or bad news? When rank belefs are decreasng n sgnals, ths s bad news. The fact that he has receved a hgh score ndcates that the test must have been very easy, so that others have receved even hgher scores. Such belefs correspond to prors that are locally U-shaped, such as that llustrated n gure 3. 13

14 Let ^p () be the largest probablty h wth whch f 0 j 0 g s common (q; h)-belef. When rank belefs are weakly decreasng, ^p () s ncreasng n. If ^p () < p for all n the undomnated regon, then there s no (q; p)-evdent subevent of fj g. Thus, suppose ^p () p above some threshold. Then, fj g s common (q; p)-belef at all, but fj g s not (1 q; 1 p)-belef at all <. Below the threshold, fj g s not common (q; p)-belef, but fj g s (1 q; 1 p)-belef. At tself, both actons may be ratonalzable, but ths s due to the probablty atom on arsng from the fact that s drawn from a dscrete space. Otherwse, there s a unque ratonalzable outcome. We note the followng corollares, bearng n mnd that the results hold except possbly at one value of. Corollary 3 If () s a constant functon over fs j s s sg, then there s a unque ratonalzable outcome n the nvestment game. For nstance, () would be constant over fs j s s sg f the pror s a geometrc densty over the relevant nterval, so that () = ( + 1) = ( + j) = ( + j + 1). Also, although we have conducted the dscusson wth a common pror, our argument could easly be extended for the case where players hold d erent prors over. Izmalkov and Yldz (2006) examne an nformaton structure where some players are systematcally more optmstc than others. Our framework could accommodate such nformaton structures. An even more restrctve specal case s when s not only constant over sgnals, but ts cross-secton s unform over the possble rank orders, n the sense that (s ) = 1; 1; ; 1 (1) I I I 14

15 If (1) holds, player beleves he has equal probablty of beng ranked anywhere n the populaton. Player beleves that he s typcal n qute a strong sense. The unqueness result for contnuum acton global games wth Gaussan fundamentals and sgnals rests of approachng the analogue of (1). When (s ) s unform, we can characterze the unque ratonalzable outcome crsply. Corollary 4 Suppose (s ) = 1; 1; ; 1 I I I whenever s s s. Then, nvest s the unque ratonalzable acton n the rst-order undomnated regon when p + q < 1. Not nvest s the unque ratonalzable acton n the rst-order undomnated regon when p + q > 1. The corollary follows from the fact that when unform, ^p = 1 q. Invest s ratonalzable when ^p > p. That s, when p + q < 1. Not Invest s ratonalzable when 1 ^p > 1 p. That s, when p + q > Comparson to Gaussan Informaton Structures Gven the mportance of rank order belefs, let us retrace what the analogous rank order belefs are n the famlar Gaussan nformaton structure that s commonly used n contnuum player global games. Player s prvate sgnal s gven by x = + " where s a Gaussan random varable wth mean y and varance 1=, and " s Gaussan wth mean zero and varance 1=. The random varables f" g are mutually ndependent, and ndependent of. Denote by (x) the proporton of players whose sgnal s x or less. The stands for lower. Then, (x) s a random varable wth realzatons n the unt nterval, and whch s a functon of the random varables f; " g 2[0;1] 15

16 z x θˆ x j αy + βx α + β ( z ) G x x θˆ θ Fgure 4: Dervng G (zjx ) and the threshold x. We derve the densty functon of (x ) condtonal on x. Let G (zjx ) (2) be the cumulatve dstrbuton functon of (x ) condtonal on x, evaluated at z. In other words, G (zjx ) = Pr ( (x ) zjx ) (3) so that, G (zjx ) s the probablty that the proporton of players wth sgnal lower than x s z or less, condtonal on x. Fgure 4 llustrates the dervaton of G (zjx ). Gven, the proporton of players who have sgnal below x s p (x ) (4) where () s the cumulatve dstrbuton functon for the standard normal. Let ^ be the realzaton of at whch ths proporton s exactly z. In other 16

17 words, ^ = x 1 (z) p (5) The top panel of gure 4 llustrates ^. When ^, the proporton of players that have sgnal below x s z or less. In other words, (x ) z whenever ^. n Hence, G (zjx ) s the probablty of j ^ o condtonal on x. The bottom panel of gure 4 llustrates the argument. Condtonal on x, the densty over s normal wth mean y + x + (6) and precson +. densty to the rght of ^, namely Ths expresson gves G (zjx). The probablty that ^ s the area under ths 1 p + ^ Substtutng out ^ by usng (5) and rearrangng, we can re-wrte (7) to gve: y+x + (7) q G (zjx ) = p + + (y x ) + 1 (z) In the specal case when! 1, the prvate sgnal becomes n ntely precse. In ths lmt, so that G s the dentty functon. G (zjx)! 1 (z) = z (8) In other words, the c.d.f. of (x ) s the 45 degree lne, and hence the densty over (x ) s unform. Thus, n ths lmt, player beleves that he s typcal n qute a strong sense, n that he puts equal weght on every realzaton of (x ). In ths sense, the unform densty over s exactly analogous to the rank belef pro le (:) beng unform. 17

18 3 Common Belef n Global Games We now generalze our argument of the prevous secton. In so dong, we characterze the herarchy of belefs that underpn actons n global games. We wll also apply these nsghts n consderng the herarchy of belefs that ensure a unque ratonalzable outcome n the global game. 3.1 Settng There are I players, I = f1; 2; :::; Ig and a countable set of payo states,. A type space s a collecton T = (T ; ) I =1, where T s the set of types of player and : T! (T ). We consder bnary acton games, where each player wll choose a 2 f0; 1g. We wrte (Z; ) for the payo gan to player of choosng acton 1 over choosng acton 0 f Z I n fg s the set of hs opponents who choose acton 1 and the payo state s. In other words, f u (a; ) were player s payo f acton pro le a s chosen and state s, the functon s de ned as (fj 6= ja j = 1g ; ) = u (1; a ; ) u (0; a ; ). Thus a game s parameterzed by payo s =( 1 ; ::; I ). Throughout the paper, we wll consder supermodular games whch n ths context means: Assumpton. (Supermodularty) (Z; ) s ncreasng n Z,.e., Z Z 0 ) (Z; ) (Z 0 ; ) 3.2 Product Events The relevant state space for our problem s = T 1 T 2 ::: T I and an event would ordnarly be de ned as a subset of. However, we wll be nterested n a specal class of product events correspondng to each player 18

19 s type t belongng to a subset F T. Thus a product event s a vector F = (F 1 ; :::; F I ) 2 I 2 T. We wll be hghlghtng two nterpretatons of =1 product events. Frst, a product event F unquely de nes an equvalent ordnary event X F wth X F = f(t 1 ; :::; t I ; ) 2 jt 2 F for each = 1; :::; I g. Where no confuson arses, we wll dentfy a product event F wth ts equvalent ordnary event X F. In keepng wth ths nterpretaton, we wll wrte t 2 F f t 2 F for each = 1; :::; I and we wll de ne a natural partal order on product events by set ncluson, so F E f F E for each = 1; ::; I. Second, because we are focussng on bnary acton games, the set of product events s somorphc to the set of strategy pro les. Thus we can dentfy the product event F wth the strategy pro le where player chooses acton 1 f and only f t 2 F. Denote by S the class of product events. Now S s a complete lattce under the partal order and the jon E _ F and meet E ^ F of a par of events E and F are de ned as E _ F (E [ F ) I =1 E ^ F (E \ F ) I =1 We wrte 0 1 I tmes z } {? ::;? A and T = (T 1 ; :::; T I ) for the smallest and largest elements of S, respectvely. Notce that the meet operaton corresponds to ntersecton of the equvalent ordnary events,.e., X E^F = X E \ X F 19

20 and that the (set ncluson) orderng on product events generates the same orderng as set ncluson on ther equvalent ordnary events,.e., F E f and only f X F X E. There s also a natural de nton of the negaton of an event, :F, wth :F = : (F ) I =1 (v F ) I =1. Now the class of product events s closed under f_; ^; :g. The de ntons of jon can be extended to any countable collecton of smple events n the natural way, and we wll appeal to these de ntons later. Also, we note the followng propertes of these operatons. ::F = F; :; = T; :T = ; : (E _ F ) = :E \ :F 3.3 Generalzed Belef Operators We wll de ne player s -belef functon B : S! 2 T as follows. Let Z F; (t) be the set of players other than such that t j 2 F j ; thus Z F; : T! 2 I s de ned as Z F; (t 1 ; ; t I ) = fj 2 I j j 6= and t j 2 F j g. For any random varable f : T! R, wrte E t (f) for type t s expectaton of f, so E t (f) = X t ; [t ] (t ; ) f ((t ; t ) ; ). Now B (F ) = ft 2 F je t ( (Z F; ; )) 0g, 20

21 Thus B (F ) s player s best response to the strategy pro le F, snce t 2 B (F ) exactly f acton 1 s a best response for player f he thnks each opponent j chooses acton 1 only f t j 2 T j. We dub B because t 2 B a "belef functon" (F ) reveals that type t puts su cently hgh probablty on some or all of hs opponents havng types t j 2 T j. The more lkely s F, the greater s player s ncentve to play acton 1 hmself. Hence, hs takng acton 1 reveals that he places hgh weght on F. De ne B (F ) as the product set: B (F ) = B (F ) I =1 ; B (F ) dent es the set of type pro les for whom playng 1 s a best reply when other players play 1 on event F ; equvalently, t s the set of types wth hgh belefs that F s true. The generalzed belef operator B : S! S sats es the followng propertes: B1. F F 0 ) B (F ) B (F 0 ) B2. B (F ) F for all F B3. If F n s a decreasng sequence, then B (^nf n ) = ^nb (F n ). B4. B n (F ) s a decreasng sequence B1 states that B s an ncreasng operator on the lattce S; t s an mplcaton of supermodularty, and shows that our nterpretaton of revealed belefs s consstent wth the deductve closure of belefs. That s, f F mples F 0, then belef n F mples belef n F 0. B2 follows from the de nton. B3 s a contnuty axom: t s mpled by B1 f the type space s nte. In B4, B k denotes the k-fold applcaton of the B operator. B4 follows from B1 and B2. 21

22 De nton 5 Event F s -evdent f t s a xed pont of B,.e., F = B (F ) By B2, ths s equvalent to the requrement that F B (F ). Note that event F s -evdent f and only f the strategy pro le F s an equlbrum of the ncomplete nformaton game (where nd erent types choose acton 1). De nton 6 Event C (F ) s the largest -evdent contaned n F, so (by B1) C (F ) ^ k1 B k (F ). If t 2 C (F ), we say that there s common -belef at t. At t, everyone -beleves F, everyone -beleves that everyone -beleves F, and so on. These de ntons parallel de ntons n the formal economcs lterature on common belefs, and we can use them to report a xed pont characterzaton of common -belef n the manner of Aumann (1976) and Monderer and Samet (1989): Proposton 7 Event F s common -belef at t f and only f there exsts a -evdent event F 0 such that t 2 F 0 F ; Proof. For the f drecton, note that snce F 0 s -evdent, we have F 0 B (F 0 ) B B (F 0 ). From property B1, we then have F 0 F B (F ) B B (F ). Hence, F s -evdent at t. For the only f drecton, f F s common -belef at t, then C (F ) = B that C (F ) s -evdent. C (F ), so Lemma 8 C (T ) s the largest -evdent event,.e., f F 0 s -evdent then F 0 T. 22

23 Ths lemma shows that C (T ) s the equlbrum of the ncomplete nformaton game where acton 1 s played the most. It s therefore a very specal case of the observaton of Vves (1990) that the largest equlbrum of a supermodular game can be found lookng at the lmt of best response dynamcs startng at the largest strategy pro le. 3.4 Characterzng Ratonalzablty We now characterze ratonalzable strategy pro les n terms of our generalzed belef operators, n the analogous way that we characterzed ratonalzable strateges n our leadng example of the nvestment game. We rst de ne ratonalzable actons as follows. De nton 9 Acton a s ratonalzable for type t f a 2 R (; t ), where R 0 (; t ) = f0; 1g 8 >< R k+1 (; t ) = a 2 R k (; t ) >: R (; t ) = \ k1 R k (; t ) there exsts 2 (T f0; 1g) such that (1) (t ; ; a ) > 0 ) a j 2 R k j (; t j ) for all j 6= (2) X a (t ; ; a ) = (t ; jt ) (3) a 2 arg max a 0 X t ;;a (t ; ; a ) u ((a 0 ; a ) ; ) 9 >= >; Ths corresponds to the de nton of nterm correlated ratonalzablty n Dekel, Fudenberg and Morrs [DFM] (2007), who gave a formal epstemc argument that the nterm correlated ratonalzable actons are exactly those that are consstent wth common knowledge of ratonalty and a type s hgher order belefs about. They also show that there s the standard equvalence between (correlated) ratonalzablty and terated domnance. An acton s nterm correlated ratonalzable f and only f t survves terated deleton 23

24 of strctly nterm domnated strateges (clam 1). The "correlaton" n the de nton arses because a player s type s allowed to have any - perhaps correlated - belefs over others actons, types and payo states as long as he puts probablty 1 on others actons beng ratonalzable for ther types (part (1) of the de nton) and hs belefs are consstent wth that type s belefs about others types and payo state. The alternatve "nterm ndependent ratonalzablty" soluton concept dscussed n DFM puts condtonal ndependence restrctons on those belefs. However, there wll not be a d erence between the ex and nterm soluton concepts n ths envronment because supermodularty wll ensure that the crtcal condtonal belefs over opponents actons wll be pont belefs. Now we have our characterzaton of ratonalzable actons. Proposton 10 Acton 1 s ratonalzable for type t f and only f t 2 C (T ). Recall that a product event F can be understood as a strategy pro le, where F s the set of types of player. The operator B s then the best response map on strategy pro les. Now T corresponds to the largest strategy pro le and C (T ) s the strategy pro le that arses n the lmt when we teratvely apply the best response functon. Thus the above proposton re ects the well known fact that best response dynamcs startng wth the largest strategy pro le converges to the largest equlbrum n an ncomplete nformaton game wth supermodular payo s (Vves (1990)) and the largest equlbrum also correspondence to the largest ratonalzable strategy pro le (Mlgrom and Roberts (1991)). As noted above, the d erence between ex ante and nterm ratonalzablty wll not matter n ths settng. For completeness, we wll report a drect argument for the proposton whch hgh- 24

25 lghts the "nfecton argument" logc from the hgher order belefs lterature and ntroduces some technques we wll appear to later. Proof. In provng ths result, t s nsghtful to ntroduce a dual operator to the B operator. To nterpret S (F ), note that B For any product event F, de ne S (F ) as S (F ) :B (:F ) (9) I S (F ) = v B (v F ) I =1 =1 (v F ) I =1 s the set of player s types for whom acton 1 s a best reply when, for all j 6=, player j plays acton 0 on F j. Then v B I =1 v F s the set of player s types who strctly prefer to play acton 0 when player j plays acton 0 on F j, for all j 6=. Thus, S (F ) s the set of type pro les who strctly prefer to play acton 0 when acton zero s played on F. that S (F ) s a smple event, when F s a smple event. Note In partcular, when F = ;, the event S (;) conssts of the type pro les for whom playng acton 0 s strctly domnant. Ths s so, snce the these types strctly prefer to play acton 0 even f no other types play acton 0. The event S S (;) conssts of type pro les who strctly prefer to play acton 0 when all type pro les n S (;) play acton 0. In other words, S S (;) s the set of type pro les who strctly prefer acton 0 when faced wth types who do not use rst-order domnated actons. Iteratng the S operator, the event S k+1 (;) 25

26 s the set of type pro les who strctly prefer acton 0 when faced wth types who do not use kth order domnated actons. Then, the jon de ned as W k1 S k (;) (10) s the smple event consstng of type pro les who strctly prefer to play acton 0 after the terated deleton of strctly domnated strateges. Thus, acton 1 s ratonalzable for player f only f acton 1 s a best reply when other types play acton 1 n the negaton of (10). That s, acton 1 s ratonalzable for type t player f and only f t 2 B Ths proves the proposton. = B = B = B : W k1 S k (;) T k1 : S k (;) T k1 B k (T ) C (T ) Naturally, we can carry out an exactly analogous analyss for acton 0. De ne e be Then we have e (Z; ) = (I n (Z [ fg)). Proposton 11 Acton 0 s ratonalzable for type t f and only f t 2 C e (T ). and t. Say that domnance solvablty holds f R ( ; t ) = f0g or f1g for all Corollary 12 There s a unque ratonalzable acton for each type f and only f C (T ) = :C e (T ). 26

27 4 Characterzng Belef Herarches We are now n a poston to utlze our result on ratonalzablty to characterze the belef herarches of players n a global game. We take the pont of vew of an outsde observer. We have just observed a player takng acton 1. What can we nfer from the acton about the belefs of the player? We llustrate the scope of the generalzed belef operator by lstng a number of examples of global games, some of whch have receved attenton n the appled lterature n nancal economcs and macroeconomcs. We start wth our leadng example, dscussed n an earler secton. Investment Game Revsted When, then successful coordnaton s possble only f proporton q or more nvest. The cost of nvestng s p 2 (0; 1), and the gross return to nvestng s 1. The payo to not nvestng s 0. In ths case, we have 8 < 1 p f > (Z; ) = 1 p, f #Z+1 q and : I p, otherwse From our proposton on ratonalzablty, nvest s ratonalzable for a player f and only f the player p-beleves all of the followng or proporton at least q p-beleve that 0 3. or proporton at least q p-beleve that [ or proporton at least q p-beleves that 0] 4. and so on... 27

28 Regme Change Game There s a cost of nvestng of p 2 (0; 1). The gross return to nvestng s 1 f proporton nvestng s at least f (), and t s 0 otherwse. The payo to not nvestng s 0. These are the payo s n Morrs and Shn s (1998) paper on currency attacks. The functon that correspondng to these payo s s gven by 1 p, f #Z+1 f () (Z; ) = I p, otherwse Coordnaton s successful only f the proporton nvestng s least f (), where f s a non-ncreasng functon of the fundamentals. Assume that f () > 1 f and only f < 0. In ths case, Invest s a ratonalzable acton for a player f and only f he p-beleves all of the followng the proporton of players who p-beleve that 0 s at least f () 3. the proporton of players who p-beleve that [the proporton of players who p-beleve that 0 at least f ()] s at least f () 4. and so on... Lnear Regme Change Game Ths s the specal case of the regme change game where f () = 1. Thus, the gross return to nvestng s 1 f proporton nvestng s at least 1, and t s 0 otherwse. The payo to not nvestng s 0. The functon 28

29 correspondng to these payo s s 1 p, f #Z+1 1 (Z; ) = I p, otherwse These payo s have become the canoncal global game payo structure n recent papers, such as Dasgupta (2001), Metz (2002), Angeletos, Hellwg and Pavan (2006, 2007), and others. 3 For the lnear regme change game, nvest s a ratonalzable acton for a player f and only f he p-beleves all of the followng the proporton of players who p-beleve that 0 s at least 1 3. the proporton of players who p-beleve [the proporton of players who p-beleve that 0 at least 1 ] s at least 1 4. and so on... Lnear Payo Game Payo to nvest s Payo to not nvest s 0. l, where l s the proporton of opponents not nvestng. (Z; ) = 1 + #Z I 1 These payo s were examned by Morrs and Shn (2001, 2003), and has gured n appled papers such as Plantn, Sapra and Shn (2005). ratonalzable for player 1 only f all of the followng hold. 1. player 1 s expectaton of s at least 0,.e., E 1 () 0 Invest s 3 Morrs and Shn ntroduced these payo s n ther 1999 nvted lecture at the Econometrc Socety European meetngs n Santago de Compostella, eventually publshed as Morrs and Shn (2004). 29

30 2. player 1 s expectaton of s at least one mnus player 1 s expectaton of the proporton of others wth expectaton of at least 0,.e., E 1 () 1 Pr 1 (E 2 () 0) 3. player 1 s expecaton of s at least one mnus player 1 s expectatons of the proporton of others wth expectaton of at least one mnus others expectaton of the proporton of others wth expectaton of at least 0 4. and so on... The two person verson of ths game has an especally smple structure. The payo functon s Invest Not Invest Invest ; 1; 0 Not Invest 0; 1 0; 0 Then nvest s ratonalzable for player 1 f and only f all of the followng hold. 1. player 1 s expectaton of s at least 0,.e., E 1 () 0 2. player 1 s expectaton of s at least one mnus player 1 s probablty that player 2 s expectaton of s at least 0,.e., E 1 () 1 Pr 1 (E 2 () 0) 3. player 1 s expecaton of s at least one mnus player 1 s probablty that player 2 s probablty that player 1 s expectaton of s at least 0,.e., E 1 () 1 Pr 1 (E 2 () 1 Pr 2 (E 1 () 0)) 4. and so on... 30

31 Contrbuton Game The publc good contrbuton game s a prvate values verson of a global game. Let R I. The cost of nvestng s. The return to nvestng s 0 f proporton at least nvest, (Z; ) = 1 otherwse., f #Z I 1 1, otherwse In ths context, nvest s ratonalzable for player 1 only f all of the followng hold. 1. player 1 s expectaton of 1 s at least 0, e.g., E 1 ( 1 ) 0 2. player 1 s expectaton of 1 s at least one mnus player 1 s probablty that the proporton of others wth expectaton of at least 0 s at least. 3. and so on... 5 Unqueness We now turn our attenton to su cent condtons for domnance solvablty. The perspectve of common belef gves us new nsghts nto the propertes belef herarches that yeld unqueness. We report on two su cent condtons for unqueness. We begn wth the common certanty of rank belefs. 5.1 Common Certanty of Rank Belefs Common certanty of rank belefs reles on a large degree of symmetry n the game, and has consderable a nty wth many uses of global games seen n the appled lterature. The argument for unqueness s a generalzaton 31

32 of the argument we gave for the example of the nvestment game gven n secton 2. Payo s are separable-symmetrc f there exst a non-decreasng functon g : f0; 1; ::; I 1g! R and a functon h :! R such that (Z; ) = g (#Z) + h () for all = 1; ::; I, Z I=fg and 2. We wll mantan ths assumpton throughout ths secton. Wth separable-symmetrc payo s, a type t 2 T has a strctly domnant strategy to choose acton 1 f g (0) + X (t ) [t ; ] h () > 0; t ; and a type t 2 T has a strctly domnant strategy to choose acton 0 f g (I 1) + X t ; (t ) [t ; ] h () < 0. Lmt domnance s sats ed f there exsts at least one type of one player wth a strctly domnant strategy to choose acton 1 and at least one type of one player wth a strctly domnant strategy to choose acton 0. A type s sad to be strategc f nether acton s strctly domnant for that type. We ntroduce the followng complete order on the unon of all types, T [ = [ I =1T : t t j f X (t ) [t ; ] h () X j (t j ) [t j ; ] h (). t ; t j ; In other words, each type s ordered by hs belefs on the fundamentals. Hgh types are those that have hgh expectatons of fundamentals. Now let : T! (f0; :::; I 1g) be a player s belef about hs rank, so (t ) [k] = X (t ) [f(t ; ) j# fj 6= jt j t gg = k 1]. t ; 32

33 Now (t ) [k] s the probablty that player attaches to there beng exactly k 1 players havng a lower expectaton of. De ne (t ) ( (t ) [1] ; (t ) [2] ; ; (t ) [I]) as the mappng that assocates wth each type the densty over possble ranks for that player. Constant common rank belefs of strategc types s sats ed f there exsts r 2 (f0; :::; I 1g) such that for each player and each strategc type t 2 T, (t ) = r. Fnally, we wll use three "techncal" assumptons. We label them techncal assumptons because they sats ed for free n the standard contnuous sgnal global game envronment wth smooth denstes. One mert of our dscrete formulaton s that t forces us to make explct assumptons that are mplct n the standard formulaton. There s unform separaton f there exsts " > 0 such that for any and t ; t 0 2 T, X (t 0 ) [t ; ] h () (t ) [t ; ] h () 6= X t ; t ; X ) (t ) [t ; ] h () t ; X (t 0 ) [t ; ] h () " In other words, f one type of a player has a hgher expectaton of than another, the d erence exceeds some unform amount ". There are no rank tes f t t j or t j t for all 6= j. There are no common rank payo tes f XI 1 r (n + 1) g (n) + X (t ) [t ; ] h () 6= 0 t ; n=0 for all and t 2 T. t ; 33

34 Proposton 13 If separable-symmetrc payo s, lmt domnance, constant common rank belefs of strategc types, unform separaton, no rank tes and no common rank payo tes sats ed, then domnance solvablty holds. If r s the common rank belef held by all strategc types, acton 1 s the unque ratonalzable acton for type t of player f XI 1 r (n + 1) g (n) + X (t ) [t ; ] h () > 0; t ; n=0 and acton 0 s the unque ratonalzable acton of type t of player f XI 1 r (n + 1) g (n) + X (t ) [t ; ] h () < 0. t ; n=0 We can paraphrase our result as: Common certanty of common rank belefs for strategc types mples domnance solvablty, where "common certanty" denotes "common 1-belef," whch s often descrbed as common knowledge n the economcs lterature. Proof. Lmt domnance mples that there exsts a player j and type t j such that c = g (0) + X j t j [t j ; ] h () > 0. (11) t j ; Now for each, t 2 T t t j = 8 9 < : t 2 T g (0) + X = (t ) [t ; ] h () c ; (12) t ; S (?). Now we establsh the followng clam by nducton on k: for each and 34

35 k = 0; 1; ::: 8 8 >< XI 1 < t 2 T : n=0 >: S k (?), S S r (n + 1) g (n) + X t ; (t ) [t ; ] h () 9 9 = I 1 ; c + X >= r (n + 1) g (n) g (0) " k (13) n=0 >; > 0 where " s de ned by the unform separaton assumpton. Recall that S k (?) s the set of types of player such that hs unque kth level ratonalzable acton s to play 1. For k = 0, the clam follows from (11) and (12). Suppose that the clam holds for k 1 and that t 2 T sats es XI 1 r (n + 1) g (n)+ X XI 1 (t ) [t ; ] h () = c+ r (n + 1) g (n) g (0) " k > 0. t ; n=0 S n=0 (14) If t has a domnant strategy to play acton 1, then t 2 S (?) S k (?); (14) mples that t does not have a domnant strategy acton to play acton 0. If t does not have a domnant strategy, then common rank belefs mples (t ) = r. Type t s certan (by unform separaton and the nducton hypothess) that all hgher ranked players have types t j 2 S k 1 (?) and therefore have a unque (k 1) th ratonalzable acton S j j to play acton 1. and t 2 S So the expected payo to playng acton 1 s at least XI 1 r (n + 1) g (n) + X (t ) [t ; ] h () t ; n=0 S k (?). Ths establshes the nductve step. Now (13) mples 8 9 < : t XI 1 2 T r (n + 1) g (n) + X = (t ) [t ; ] h () > 0 ; [ k1s S k (?) t ; n=0 35 = ft 2 T jr (; t ) = f1gg.

36 A symmetrc argument mples 8 9 < : t XI 1 2 T r (n + 1) g (n) + X = (t ) [t ; ] h () < 0 ; [ k1s e t ; n=0 No payo tes mples 8 9 < : t XI 1 2 T r (n + 1) g (n) + X = (t ) [t ; ] h () < 0 ; =?. t ; n=0 h S e k (?) = ft 2 T jr (; t ) = f0gg. We bre y report two smple weakenngs of the common rank belefs under whch the result wll contnue to hold. Frst, consder the rst order stochastc domnance order on rank belefs, so that r D r 0 f, for each n = 1; :::; I, nx r () =1 nx r 0 () Say that there s decreasng common rank belefs f, for any t 2 T and t j 2 T j, =1 t t j ) (t ) E j (t j ) Now f we replaced the assumpton of common and constant rank belefs of strategc types wth common and decreasng rank belefs of strategc types, we would agan have domnance solvablty. In partcular, acton 1 (0) would be the unque ratonalzable acton f XI 1 n=0 (t ) [n + 1] + X t ; (t ) [t ; ] h () > ( < )0. Second, suppose that rank belefs were not constant but that they dd not change too fast relatve to the expectatons of fundamentals. = g (I 1) g (0) 36 Let

37 measure the strategc senstvty of the game. Wrte (r; r 0 ) for the dstance between the rank belefs r and r 0 : (r; r 0 nx ) = max r () n=1;::;i =1 nx r 0 (). Say that there s near constant common rank belefs f, for any t 2 T and t j 2 T j, (t ) ; j (t ) X : (t ) [t ; ] h () t ; =1 X j tj 0 [t j ; ] h (). Now f we replaced the assumpton of common and constant rank belefs of strategc types wth near constant common rank belefs of strategc types, we would agan have domnance solvablty. t j ; Morrs and Shn (2005) descrbe a unqueness result usng ths dea (where the near constant rank belefs s delvered by "bounded margnals on d erences" property. Valentny (2006) also used a related dea. 5.2 Common Certanty of Belefs n D erences Mason and We now present a second set of su cent condtons for unqueness that allows for asymmetry across players. Payo s are separable f there exst ncreasng functons 1 : 2 I=fg! R and 2 :! R such that (Z; ) = 1 (Z) + 2 () The type space of each player s two-dmensonal. A type has two components. The rst component s completely ordered and we dentfy t wth the set of ntegers Z. The second component s any nte set. Thus, for each, we have a bjecton g : T! Z. 37

38 The rst component of a type can be nterpreted as a sgnal receved about the fundamentals, so that hgher rst components are assocated wth hgher belefs about. The second component s some other dmenson along whch players vary. However, note that the orderng apples only to the types of a sngle player, whereas the condton of common certanty of rank belefs appled the orderng to the unon of all types, and so we were rankng across players, also. We now ntroduce our assumptons. Denote by g 1 (t ) the rst component of g (t ). Assumpton (Unform Monotoncty): There exsts " > 0 such that g 1 (t ) > g 1 (t 0 ) ) X (t ) [t ; ] 2 () > X (t ) [t ; ] 2 () + " t ; t ; for all, t, t 0. Assumpton (Lmt Domnance): For each, there exst t and t such that 1 (I=fg) + X t ; (t ) [t ; ] 2 () < 0 and 1 (?) + X t ; t [t ; ] 2 () > 0. Assumpton (-D use Belefs): There exsts > 0 such that, for each and, for each j 6=, h j : j! Z, X (t ) [t ; ] < ft :g j1 (t j )=h j (g j2 (t j )) for some jg; The last assumpton and the unformty requrement n the rst assumpton can be thought of as techncal assumptons: they are requred 38

39 only because we are allowng for dscrete type spaces and are not requred (or are mplct) n the standard contnuous sgnals global games framework. Fnally, we come to our key de nton. De ne player s belefs about d erences : T! (Z j) j6= as follows: h (t ) j; j j6= ; hn = (t ) o g 1 j g 1 (t ) + j ; j. (15) j6= To grasp the expresson on the rght hand sde, note that g 1 j g 1 (t ) + j ; j s the type of player j whose rst component s g 1 (t )+ j, and whose second component s j. Thus, type t s belefs about d erences are t s belefs over other players types where player j s type s dstance j away along the rst component. Our su cent condton for unqueness rests on the belefs about d erences beng nsenstve to the rankng of a partcular player s type. In other words, the functon de ned n (15) s a constant functon wth respect to the rst component of a player s type. Proposton 14 Assume unform monotoncty and lmt domnance. Then there exsts > 0 such that, f and there are -d use belefs, then common certanty of belefs n d erences mples domnance solvablty. Proof. Fgure 5 llustrates the argument. For each k = 0; 1; :::, there exsts h k :! Z and non-ncreasng h k :! Z such that h k ( ) s non-decreasng n k for each, h k ( ) s non-ncreasng n k for each, 1 2 R k (; t ) f and only f g 1 (t ) h k (g 2 (t )) and 0 2 R k (; t ) f and only f g 1 (t ) h k (g 2 (t )) 39

40 c ψ * h * ( ψ * ) h * ( ψ * ) Fgure 5: Translaton Ths can be proved by standard monotone methods, see, e.g., van Zandt and Vves (2006). Thus there exst h ; h :! Z such that h ( ) h ( ) for all 2 and 1 2 R (; t ) f and only f g 1 (t ) h (g 2 (t )) (16) and 0 2 R (; t ) f and only f g 1 (t ) h (g 2 (t )) Now we prove unqueness by rst supposng that h 6= h (and then provng a contradcton). Let c be the smallest nteger such that h ( ) h ( )+c for all and 2. Observe that c > 0 and that there exsts and 2 satsfyng h ( ) = h ( ) + c. and thus 8 X >< >: t ; + X t ; Now observe that by (16), we know that 1 2 R ; g (h ( ) ; ) g (h ( ) ; ) [t ; ] 1 g (h ( ) ; ) [t ; ] 2 () j : gj1 (t j ) h j (g j2 (t j )) 9 >= 0 >; 40

41 Now suppose that player s type g 1 (h ( ) + c; ) and beleves ths hs opponents are choosng acton 1 f and only f g j1 (t j ) h j (g j2 (t j ))+c. Then by common knowledge of belefs n d erences X t ; = X t ; g (h ( ) + c; ) [t ; ] 1 g (h ( ) ; ) [t ; ] 1 j : gj1 (t j ) h j (g j2 (t j )) + c j : gj1 (t j ) h j (g j2 (t j )) These are the payo s from the strategc part of the payo functon that depends on the actons of others. On the other hand, the part of the payo functon that depends on the fundamentals can be ordered by the assumpton of monotoncty X t ; X t ; g (h ( ) + c; ) [t ; ] 2 () g (h ( ) ; ) [t ; ] 2 () + c" so, addng the two parts of the payo functon together, we have 8 X >< g (h ( ) + c; ) [t ; ] 1 j : gj1 (t j ) h j (g j2 (t j ) + c) 9 >= >: t ; + X t ; g (h ( ) + c; ) [t ; ] 2 () Now observe that j : g j1 (t j ) h j (g j2 (t j ) + c) unless g j1 (t j ) = h j (g j2 (t j )) for some j. X t ; X t ; Thus c" >; n o j : g j1 (t j ) > h j (g j2 (t j )) g (h ( ) + c; ) n o [t ; ] 1 j : g j1 (t j ) > h j (g j2 (t j )) g (h ( ) + c; ) [t ; ] 1 j : gj1 (t j ) h j (g j2 (t j ) + c) 41