File: 642J 209001 . By:BV . Date:22:01:96 . Time:16:08 LOP8M. V8.0. Page 01:01 Codes: 3585 Signs: 2385 . Length: 50 pic 3 pts, 212 mm oeeulbi htaesbett eitosmysilpritin persist still may deviations to subject are that equilibria some inuasee:I lyr r ieyt eit rmsm equilibrium, some from deviate deviate to ques- they likely identify important are will refinements an players leave these what they If While deviations, unanswered: to in 17]). tion susceptible prominently [10, most are have that 7]), (see refinements equilibria [2, literature many (see deviation-based cheap-talk Yet them- literature signaling Such the are equilibrium. the equilibria in deviations. Nash developed Nash a been to some be susceptible that will propose selves play concepts solution this standard play, predictable upon ril no. theory article economic of journal 2090  Theory Economic of Journal okhp tBree,Uiest fIw,adteUiest fCiaofrhlflcom- support. helpful financial for for Chicago Foundation of Science National University the the thanks and also Iowa, author of Each University ments. Berkeley, at workshops In Park, eateto Economics of Department etakAdesBue noi arls di ee-aa,JeFrel In-Uck Farrell, Joe Dekel-Tabak, Eddie Cabrales, Antonio Blume, Andreas thank We * eateto Economics of Department ti ai ee fnncoeaiegm hoyta fpaessettle players if that theory game non-cooperative of tenet basic a is It eitos yais n qiiru Refinements Equilibrium and Dynamics, Deviations, neweetercretfruain onot. do exist- C73. guarantee C72, formulations and B49, current realistic cheap- Numbers: existing more Classification their to concepts framework where solution our equiv- those ence applying is that criterion make are intuitive but refinements concept, they (1987) talk set to solution Kreps's because the and their respect that Cho to run with show on alent long We recurrent based refinements. equilibria are the signaling recurrent equilibria existing in of some which to recur underlying explore impervious deviations to equilibria We the likely to. only not deviated equilibria considering repeatedly also for but framework a deviations, propose We tions. xsigeulbimrfnmnsrl u aheulbi ucpil odevia- to susceptible equilibria Nash out rule refinements equilibrium Existing g aae,toaoyosrfre,a soit dtr n atcpnso theory of participants and editor, associate an referees, anonymous two Zapater, igo ~ 0001 eevdMy1,19;rvsdOtbr2,1994 24, October revised 1993; 11, May Received , , nvriyo California of University nvriyo California of University to 68 efe hsqeto sipratbecause important is question this feel We ? 5(1996) 125 , ate Rabin Matthew 1 Introduction . olSobel Joel and 1  96Aaei rs,Inc. Press, Academic 1996   a Diego San Berkeley ora fEooi Literature Economic of Journal l ihso erdcini n omreserved. form any in reproduction of rights All Copyright , , Berkeley aJolla La  96b cdmcPes Inc. Press, Academic by 1996 , 0022-0531 , California California  96 94720 92093  18.00 * 2 RABIN AND SOBEL

Figure 1

the long run because they may be repeatedly deviated to. It seems inap- propriate to omit such recurrent equilibria from the solution concepts with which we make predictions in economic models. In this paper, we develop a framework to address this question and use this framework to critique existing solution concepts. Our framework involves two steps. First, based on intuitions from existing refinements for when and how deviations occur, we develop an algorithm for specifying explicitly the outcomes that might arise following a deviation from an equi- librium. Second, we then develop solution concepts based on which equi- libria recur infinitely often given a theory of deviations. We show that our framework gives a more complete dynamic justification for a prominent existing solution concept, but also that it alters and (we feel) improves upon the solution concepts resulting from other theories of deviations. Our first step, specifying which equilibria might follow deviations, addresses a concern commonly expressed regarding signaling refinements. Signaling refinements have been criticized because proposed deviations from equilibria often rely on an apparent inconsistency: Some players defect from an equilibrium while others continue to believe in that equi- librium. Our approach to deviations takes this general criticism, often called the Stiglitz critique, into account.1 The standard SenderReceiver signaling game in Fig. 1 illustrates this critique. (The first of each pair of payoffs is that of the informed Sender; the second is that of the uninformed Receiver.)

1 As we discuss in the next section, we do not, however, concur with the more specific arguments of the example in [7] in which the critique is most often formulated.

File: 642J 209002 . By:MC . Date:21:12:95 . Time:13:40 LOP8M. V8.0. Page 01:01 Codes: 2366 Signs: 1784 . Length: 45 pic 0 pts, 190 mm File: 642J 209003 . By:BV . Date:22:01:96 . Time:16:08 LOP8M. V8.0. Page 01:01 Codes: 3526 Signs: 2978 . Length: 45 pic 0 pts, 190 mm i rgnleulbimrsos to response equilibrium original his only that given response optimal the is which eit ysnigmessage sending by deviate a ie h qiiru epnet h message that, the argue They to criterion. response intuitive equilibrium Kreps's the and Cho given the fails outcome This eivsthat believes htteSilt rtqede nedhv ocrtoa lyr might continuing and players by equilibrium Cho equilibrium. forcerational feel this the we have from accept play sense, deviation indeed to we hypothesized some does the in While to critique unstable respond all. Stiglitz is after equilibrium the with this that deviate players that may to the argument they if Kreps's not equilibrium, Thus, this choose strategy. from deviation equilibrium reasonably a her anticipate to knowledge adhere common to Sender of h uuehaiy ota neeyrudpaesec r omaximize to try each players round every in that discount players so the that heavily, of assume implicitly that future we is observ- the period; considering game, each are a outcome that play we the equilibria repeatedly ing environment of players The set playtwo run. repeated a long ``non-strategic'' characterize the to in us persist allow may that dynamics of properties use may equilibrium. players the that from the strategies deviation of strategies a set of a following set specifies com- correspondence resulting with deviation the consistent a is label deviation We correspondence posited rationality. deviation a of expansion to knowledge This reactions that added. mon guarantees possible and be of above need set outlined strategies reasoning the more the respon- no parallels best formally until add process player iteratively we each deviation, by proposed ses a with Starting process: choose Sender epneto response a e os hnhreulbimpyf ydigs omte o she how matter inter- no therefore so should doing Receiver deviation The by the respond. payoff pret would equilibrium Receiver the her thought than worse get ale ruet n xmlso hnteSilt rtqede n osnthv oc,see force, have not does and does critique Stiglitz the when of [16]. examples and arguments earlier evbyws odvaeb edn h message the sending by deviate to wish ceivably hnteeulbimmessage, equilibrium the than iru sntstable. not is librium 1 2 ecmlt u rmwr nScin3b eciigsm broad some describing by 3 Section in framework our complete We expansion an defining by above arguments the formalize we 2, Section In then way, this in acts Receiver the If . the of types both which in game this of equilibrium sequential a is There efe hsaayi tp o on fteRcie elzsthat realizes Receiver the If soon. too stops analysis this feel We u eas hwta h tgizciiu osntawy eoe ntbeeulbi.For equilibria. unstable recover always not does critique Stiglitz the that show also we But t 2 m m EITOS YAIS N REFINEMENTS AND DYNAMICS, DEVIATIONS, a hrfr play therefore may 2 2 now , n h eevrchooses Receiver the and m 1 sbigsn by sent being as 2 o ie qiiru n hoyo deviations, of theory and equilibrium given a For . t 2 a ebte f sending off better be may m m 1 2 hnh ih epn to respond might he then , h n rp ocueta hsequi- this that conclude Kreps and Cho . m m 1 emyratto react may he , 1 hsi unmyla ohtypes both lead may turn in This . t 1 t 1 n hudrsodwt action with respond should and ol rfrt send to prefer would a t 2 m 2 given m ssending is 2 nytype only , 1 eas type because , m m 1 m 1 and fteReceiver the If . 1 m yplaying by 2 t ie such Given . r 1 m 1 ol con- could 2 given m t with t 2 1 1 would rather might m a r 2 2 2 3 , , . 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Kreps's to and susceptible not Cho concept our are to solution that basing equivalent equilibria proposed Despite only be our includes played. to equilibria, be out recurrent again turns on will concept equilibrium solution unstable no point m ept h tgizciiu,teeoe h equilibrium the therefore, a be critique, will Stiglitz there that the because probability occurs, Despite small deviation a a such is Once there to that deviation assume we period, that above argued itself. is correspondence deviation its stable, is both contains ilysprtn equilibrium separating tially h n rp ru sssetbet eito.Tedvaincorres- deviation The which deviation. a in to game, equilibrium susceptible this is from argue pondence Kreps and Cho hi n-htpayoff. one-shot their 4 rbblt 3 probability epc otenvrawa-etrsos rtro fadol fte r tbewt respect with stable are they if only criterion. and never-a-weak-best-response if with the criterion recurrent to never-a-weak-best-response are the propertyequilibria same to the respect has deviations. criterion to never-a-weak-best-response subject [15] are they if [29] ``unstable'' mutations. and as by and equilibrium, interrupted [14] tions in be be literature. periodically generally will equilibrium-refinement will play behavior the that equilibrium We guarantee that underlies quick. which but relatively models story is dynamic such equilibration explicitly and develop some rare relatively that are feel deviations previous which observe in who environments people of generation new a game. are the period of each players play the where once, only sueta lywill play that assume htaelkl ob lydrpael nteln u nsc nenviron- an such in run long the in repeatedly played as be define ment. to We likely equilibrium are correspondence. an strategies that to deviation play re-equilibrates may play the players and in equilibrium, correspondence, equi- deviation an certain the upset following in may periods deviations the that In but libria. behavior, equilibrium on down rtro lascnan neulbimta sstable. is that 3 equilibrium intuitive Section [7] an in Kreps's show contains and we Cho always on Indeed, based criterion 1. correspondence deviation Fig. the in that prediction long-run appropriate 1 6 5 4 3 hc fteetoeulbi ih cu nteln u?Wiewe While run? long the in occur might equilibria two these of Which iue1ilsrtsorapoc.Let approach. our illustrates 1 Figure n[6,w hwas,ta ysihl oiyn u rmwr,Khbr n Mertens' and Kohlberg framework, devia- our to modifying subject slightly by not that are also, they show if we ``stable'' [26], as In equilibria to refer we paper, the Throughout in sense makes outcomes game non-equilibrium the than plays rather person equilibrium each on focus which exclusive in Our situation a as interpreted be also can model Our iha qa itr of mixture equal an with 4  Y and 4 X n hspstta vnulyti eito iloccur. will deviation this eventually that posit thus and , t and 1 X ed h message the sends m 3 o' eesrl edvae wyfo naygiven any in from away deviated be necessarily won't Y 2 eventually easm htteei ednyfrpa osettle to play for tendency a is there that assume We ycnrs,bcueCoadKesagethat argue Kreps and Cho because contrast, By . ihpoaiiy1 probability with AI N SOBEL AND RABIN a Y 1 hnordvaincrepnec for correspondence deviation our Then . and X eit oasal qiiru,a which at equilibrium, stable a to deviate ilicueteohreulbimi this in equilibrium other the include will a 2 Y n to and m X 1 ssal,i ilb lydforever. played be will it stable, is  , ,adteRcie epnsto responds Receiver the and 4, etepoigeulbimthat equilibrium pooling the be t 2 m ed h message the sends 2 recurrent with r 2 ecl hspar- this call We . 5 hs equilibria those Y etherefore We 6 em the seems m 1 with Y X File: 642J 209005 . 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Length: 45 pic 0 pts, 190 mm eito orsodne o w-lyrgms lyr n have 2 and 1 Players sets games. pure-strategy two-player finite for correspondences deviation hncnld h ae ydsusn oeieso o oetn our extend to how on by behavior. ideas non-equilibrium some taken for discussing that allow by to to paper approach framework the conclude alternative then an discuss Bernheim we even First, existence, not. approach. guarantee do stable always concepts than we solution that rather current 3 where equilibria Section strong situations recurrent in in making on to show approach we focus practical equilibria, we Grossman a frame- as Because provides such our therefore predictions. concepts Applying other [11] inap- and Perry equilibria equilibria. concepts concepts and which solution solution deviation-prone cheap-talk these for all to because intuition work ``too out rather wrong are but rule the deviations, problems propriately have to that non-existence prone they feel are with We because strong. concepts not concepts has solution strong'' approach solution our their existing that like illustrates many who cheap-talk 4 those common Section for attractions equilibria, other many too where eliminate games of class not. guarantees do mop a existing refinements recurrent upon that in show improves also mop communication We recurrent concepts. where solution examples cheap-talk some provide and S aieapoc omkn rdcin nsc ae.Bsdo the on called Based concept games. solution such a develop in we predictions alter- [17], an making provides of framework to arguments our that approach show and native situation a such illustrates include therefore would equilibria long-run C plausible of notion Our equilibrium equilibrium to either attention limit us have A would literature equilibria, refinement standard three has of game theories a other that for hold instance, and not for need and Suppose, however, deviations. framework, our of feature in,and tions, # hl ete ssal,ec slkl oapa vradoe again. over and over appear to likely is each stable, is neither while ; eas ti h nysal qiiru.Ti em inappropriateif seems This equilibrium. stable only the is it because , nti eto,w eeo h is tpo u rmwr,constructing framework, our of step first the develop we section, this In nScin5 edsustoptnilfrhrapiain four of applications further potential two discuss we 5, Section In may refinements signaling existing that indicates framework our While nScin4 epoiea xeddeapei ha-akgmsthat games cheap-talk in example extended an provide we 4, Section In general a not is equilibria stable and recurrent between equivalence The S C 1 ups ht(codn osm theory) some to (according that Suppose . _ S 2 tal et n aoffntos( functions payoff and , B 2 eito orsodne n Expansions and Correspondences Deviation . and EITOS YAIS N REFINEMENTS AND DYNAMICS, DEVIATIONS, 4 o hnigaot`caiinpof'eulbi.We equilibria. ``coalition-proof'' about thinking for [4] . B C or edt eit oec te,but other, each to deviate to tend C cu,te equilibrium then occur, S 1 and u S 1 2 , ihtesto taeyprofiles strategy of set the with , u 2 .Frayfnt set finite any For ). A A sntsbett devia- to subject not is ilnvrocragain. occur never will not eurn mop recurrent Z edenote we , to A B The . A and , B 5 , , File: 642J 209006 . By:BV . Date:22:01:96 . Time:16:08 LOP8M. V8.0. Page 01:01 Codes: 3282 Signs: 2686 . 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( u Z m Z i j , ( i 2 Q h e fbs epne in responses best of set the ) h e fpoaiiydsrbtoson distributions probability of set the ) s ete osrc h eito orsodne hc saset a is which correspondence, deviation the construct then We . a # i *, ( a 1 .I i.1 h edrstp seither is type Sender's the 1, Fig. In ). m =( S 2 , i 1 2 Q p )_ z . sa lmn in element an is fh bevsmessage observes he if j # 2 ) j 1 ,where ), # , 2 # Z u ( [7 m 2 S j i n e fpsil eitosfo hsequilibrium, this from deviations possible of set a and ) ( 1 2 uhthat such s 1 otiigtoesrtge htrtoa lyr might players rational that strategies those containing ) i ( or , n p ), Q j 7 o all for ) m 7 A m i i (0) 2 lyr aeuiiyfunctions utility have Players . 2 for strategies mixed of set empty) (possibly a is ( M fe hc h eevrcosshsaction. his chooses Receiver the which after , scoe,adte hoe naction an chooses then and chosen, is n AI N SOBEL AND RABIN # u 2 )) lyr2i nuifre eevr who Receiver, uninformed an is 2 Player . i ( ] ( { S s h no fteest ilb h devia- the be will sets these of union The . s # Q i *, i i .W xed( extend We ). i # i z S # j S 7 7 ) yieaieyadn taeisthat strategies adding iteratively by , i m ] i i 1 . 0 ssml player simply is (0) if if to 0 and (0) 1 u n between and , nutv criterion intuitive i ( Z Q Q s p j i j htis, That . i i , { = vrtesrtge in strategies the over z t Z j # 7 o all for ) SBR < < u Q Z ota ie strategy mixed a that so , 1 T 2 j . , (0): 7 efr nincreasing an form we ? t u message a 1 i .Hrsto pure of set Her ). } ( i 2 0 ossso these of consists (0) 2 ( omxdstrategies mixed to ) Z or ( BR S j ) s j # i r ,w eoeby denote we ), t i 1 # nScin4, Section in ; 2 ( u Q n h can she and , Z i [s S sequilibrium 's 1 and ( ( j i i *# ) # ] t m # , sinput as ) Similarly, . m hc is which , [s S a r , Z 2 which , i a i *# there | and ) j fhe if such S T i | File: 642J 209007 . 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AI N SOBEL AND RABIN 1 k r a .Te let Then ). 2 2 1 ,bcueprevn eito ilmean will deviation a perceiving because ), 7 r ( 2 S 2 ahrta ( than rather ) 0)t eol rlmnr hypothesis preliminary a only be to (0)) 1 f( of ), 7 7 2 & # i i # ( ( hnec lyrogtrationally ought player each then , ( S , n m n Q m 7 2 trtvl ysetting by iteratively ) &1){ )ta lyr eiv possible believe players that )) 1 ( i i *= 1 m # # m ) denoted )), oalblesoe h new the over beliefs all to # 1 2 .I un hsmasthat means this turn, In ). # 7 ahrta ( than rather ) , ( expansion Q # i # eito correspondence deviation a ( 1 & n ( 1 # # *). i r 2 ) yieaieyadd- iteratively By )). 1 n eito sets deviation and ) n by and .Cneunl,we Consequently, ). i n hnw continue we then , uhta o all for that such * 7 m Exp : eas each Because 7 2 m 1 7 = ( m 7 2 1 m # ,( ), (0), , 2 i 7 1 i ( Q m ethen we , m n 1 a 7 ( 0,but (0), 2 )= 2 1 # ,then ), 7 i ,and ), r ) be )), ( 1 n 2 # )is )) (0)) ) i # S if i DEVIATIONS, DYNAMICS, AND REFINEMENTS 9

Figure 2

rationality.8 An important feature of the expansion process is that it will always contain at least one Nash equilibrium. This fact follows once we note that the hypothetical game defined by permitting players to use only mixtures of strategies in D(#) (and in which they receive the payoffs of the real game) has a Nash equilibrium. That Nash equilibrium must also be a Nash equilibrium of the real game; otherwise the expansion process would not have stopped.9 Because the intuitive-criterion deviation correspondence constructed from the pooling equilibrium X in Fig. 1 contains X itself, the Stiglitz criti- que applies. Figure 2 demonstrates, however, that the Stiglitz critique does not always redeem unintuitive equilibria.

The pooling equilibrium ((m2 m2), (a2r2)) fails the intuitive criterion in

this example because t1 (and only t1) could gain by deviating. Because this is the only deviation designated by the intuitive criterion, 71(0)=Q1=

[(m1m2)] and 72(0)=Q2=[(a1r2)]. In the iteration process, (a1 r3) will be included in 72(1), but nothing more is added into either player's

8 Because deviation correspondences are sets of strategies consistent with common knowledge of rationality without imposing the equilibrium assumption, our approach here is similar to that taken in [21, 24], where an approach to combining behavioral assumptions with rationalizability is developed. Game-theoretic applications using similar ideas include [6], [22, 23, 25], [28], and [30]. 9 The sets (71(0), 72(0)) need not contain an equilibrium (they don't in our example for Fig. 1). It is necessary to expand the set of strategies in the deviation correspondence beyond

(71(0), 72(0)) in order to develop a theory which guarantees that it is possible for players to play an equilibrium following a deviation. This expansion process also guarantees that our theory of deviations is consistent with common knowledge of rationality.

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best-response set. The deviation correspondence will therefore be (71*, 7*),2

where 7*=1 [(m1m2)] and 7*=2 [(a1r2), (a1 r3]). These sets contain only the separating equilibrium to this game, ((m1 m2), (a1 r3)), and not the original equilibrium. The unintuitive equilibrium does not therefore become plausible even if we assume common knowledge of the posited deviation, so the outcome here is not stable in any sense: Even when we take the Stiglitz critique fully into account, the pooling equilibrium does not survive a deviation. We have not described the only possible way to formalize the Stiglitz critique. In fact, if one applies our approach to the BeerQuiche example in which Cho and Kreps [7] discuss the Stiglitz critique, one finds that the bad equilibrium is not included in its deviation correspondence. Figure 3, which is closely related toil the BeerQuiche example, clarifies the contrast between our approach and the original formulation of the Stiglitz critique.

Here, as in Fig. 1, there is a pooling sequential equilibrium, (m2 m2 , a2 r1);

as in Fig. 1, this equilibrium is subject to a deviation in which t1 plays m1 , and the Receiver responds to m1 with a1 . It is easy to confirm that the deviation correspondence based on this deviation does not add back the equilibrium strategy for either player, because once the players realize that

t1 might deviate and play m1 , the Receiver would prefer to respond to m1

with the strategy a1 no matter what else he believed; the expansion process does not add back the original equilibrium. Our formulation captures what the players might rationally choose in response if they come to believe the proposed deviation, and only the proposed deviation, is likely. The specific argument by Joseph Stiglitz

Figure 3

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Fig. equilibrium in equilibrium unituitive ti tagtowr oso hti inln ae ihtotpso edr w signals, two Sender, of types two with games signaling in that show to straightforward is It 10 hte n ple tgizsSilt rtqeo ai n Sobel's and Rabin or critique Stiglitz Stiglitz's applies one Whether n ol nayeetmdf u eea praht noprt the incorporate to approach general our modify event any in could One tgizsoiia rtqesest et rps htteei agrset larger a is there that propose to be to seems critique original Stiglitz's t ent htee tgizsoiia omlto osntageaanttrwn u the out throwing against argue not does formulation original Stiglitz's even that note We 1 t hnsetowudb epe ytedvain fteRcie for Receiver the if deviation; the by tempted be would too she then , a 1 1 slkl,bcuei type if because likely, is t hs h rgnleulbimmgtb played. be might equilibrium original the Thus, . 1 hnh ih esnbyrsodt h eito by deviation the to respond reasonably might he then , # Y nyif only EITOS YAIS N REFINEMENTS AND DYNAMICS, DEVIATIONS, novssm iig ti losrihfradt eosrt htthe that demonstrate to straightforward also is It mixing. some involves Q 1 , Q # 2  n pligorepninpoest these to process expansion our applying and ) D ( # .O ore whenever course, Of ). # susal nta lyr ih eit.In deviate. might players that in unstable is t 2 aet eiv htteRcie would Receiver the that believe to came t 2 D 10 a oelkl than likely more was ( # ){ [#] n might one , a 2 rather 11 File: 642J 209012 . By:BV . Date:22:01:96 . Time:16:08 LOP8M. V8.0. Page 01:01 Codes: 3015 Signs: 2318 . 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Definition other out rule always not not does framework Our # hsasmto ilb ai o eei inln ae se[,pg 9,Fc 2]). Fact 190, page [7, (see games signaling generic for valid be will assumption This )= [#] ,o hudb ul u ouincnet ae nequilibria on based concepts solution our build be should or ), 2 3 . . D ih not might } as (}) nequilibrium An neulbimis equilibrium An neither ND 3 Dynamics . us-tbeequilibrium quasi-stable edvain ie,where (i.e., deviations be .W sueta ahset each that assume We ). } ( AI N SOBEL AND RABIN # iwtksasfiinl yai iwof view dynamic sufficiently a takes view $# # ND is quasi-stable stable ( ( # otof Sort .Atiilcaso qiirathat equilibria of class trivial A ). if D Q ) ( if i # = : )= ND < # [#] ( # # o each for )= D . # ND ( # lyeventually play [#] ) ntenext the In ))? sfinite. is ) } ( . i Because . 11 File: 642J 209013 . By:BV . Date:22:01:96 . Time:16:08 LOP8M. V8.0. Page 01:01 Codes: 3798 Signs: 2975 . 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Fig. run? in long game the in occur out ruling outcomes, equilibrium inappropriate. is on equilibria focus quasi-stable exclusive abandon to unwilling al,w ruderirthat earlier argued we mally, epooet ueout rule B to propose we from deviation equilibrium the occurs, it if stable, otisn rpr o-mt ust htaeasrigst.A equi- An sets. absorbing are that subsets non-empty librium proper, no contains ND oteoiia qiiru hni etequilibrates. next it when equilibrium original the to euneo eitosfrom deviations of sequence ae n hoyo eitossy hti sntsal.Frsc neape e Farrell [13].) see (See example, assumption. an this such relax For can stable. We not is it that says deviations of theory [10]. a and game, a  stenme fprosapoce infinity approaches periods that follows of It probability. number positive with the arise will as equilibria, these only and rbblt nteln u r ecie ntefloigdefinition. following the in described are run long the in probability ml httepoaiiyo eigteequilibrium the seeing of probability the that imply cusi h eteulbae eidwt rbblt tleast at probability with period equilibrated next the in occurs equilibrium the occurs iru oocrfloigadvaini iie ute ups htthere that equi- suppose an further for finite; takes is it deviation periods some a of exists following number occur the to that librium Suppose play. repeated of # 1 ilb utfe fw aetefloigasmtosaottedynamics the about assumptions following the make we if justified be will 14 13 12 n oegnrly htcnw a bu h e feulbi htmight that equilibria of set the about say we can what generally, More ae nteasmto hta n lyo h aeteemgtb a be might there game the of play any at that assumption the on Based Definition return must play occurs, equilibrium quasi-stable a from deviation a If ne hs odtos h e feulbi htapa ihpositive with appear that equilibria of set the conditions, these Under ( # B ( ti eeta eueorasmto httenme feulbimotoe sfinite. is ideas. outcomes related equilibrium investigate of [13] number and the that [12], assumption [8], our use we that here in is equilibrium It unique a is there when are equilibria quasi-stable of examples clearest The # )= n Y ) ( # # ND ). # G ND srcreti ti otie nsm eurn set. recurrent some in contained is it if recurrent is B o all for ( *( Y # p ( ), X 0sc htwhenever that such >0 # lal ih cu eetdy Because repeatedly. occur might clearly 4 ,advainfrom deviation a ), EITOS YAIS N REFINEMENTS AND DYNAMICS, DEVIATIONS, ossso hs qiirata a rs olwn a following arise can that equilibria those of consists ) X B X . n # to ( ih cu eetdya el u eas ti lotrue also is it because But well. as repeatedly occur might # e fNs equilibria Nash of set A # )= G Y h set The . n htoc hr s lywl ee eunto return never will play is, there once that and , # [# X ilnvrocraan uigotteequilibrium the out Ruling again. occur never will "| # saln-u qiiru.Mr eeal,let generally, More equilibrium. long-run a as "# # .If ND G ND is ( # X ( # $# X Y recurrent )frsome for $) ilnvraanoccur. again never will )= # ND `ih' cu.Ad because And, occur. ``might'' cus n qiiru in equilibrium any occurs, [Y] ( # but ) fi sa bobn e and set absorbing an is it if # and G # aheulbimin equilibrium each # $# $# # san is B  ND # ND 12 n B X ovre ozero. to converges &1 seilyi eare we if Especially *( ( # ( X # ( # ND # bobn set absorbing and ) )= Y ) hnonce then $), ) X ] ssal with stable is , ( [X and X snt For- not. is p ,arguably ), . 13 , # Y] B  ecall We *( ND B B 14 The . *( *( # Y )= ( # # 13 X # # is $) ), if ) $ , File: 642J 209014 . By:BV . Date:22:01:96 . Time:16:08 LOP8M. V8.0. Page 01:01 Codes: 3737 Signs: 3101 . Length: 45 pic 0 pts, 190 mm eurn e,te oeulbimotieo h eurn e ilb obser- be will each set for recurrent Moreover, the future. of the outside in equilibrium ved know no then we set, set, recurrent absorbing an also exist. is sets sets recurrent absorbing that of intersection any since ess nteln u,bcuei sisl o outt eitos n ol ee be never would and deviations, to robust not in contained itself not is equilibrium any it Likewise, because in equilibrium. another not run, by is to long equilibrium deviated an the If in equilibrium. persist other some of correspondence pnec.I ihrw iht osdrtents-ogln u,o fw ontwn to want not equi- do recurrent we Let of if definition follows. or weaker as run, a For corre- iteratively if long probabilities, deviation game. defined run deviation a not-so-long be long on in the could very equilibria bound consider libria the all lower to in to a wish appear deviations such of we to assume probability either unlikely the very If on are spondence. bounds which lower but are there often, infinitely occur ``could'' ol edee eurn eas ti otie nisondvaincorrespondence. deviation own its in contained is it because recurrent deemed be would n that ,w s h olwn iperesult: Proposition simple prove to following set order the the In use (i.e., criterion). we intuitive correspondence that intuitive the 2, deviation pass those with intuitive the that with the equilibria equilibria coincide to of to recurrent correspondence according framework deviation stable the intuitive our are that the section. applying of demonstrates to next the by respect existence 2 in the section Proposition deviations guarantee of this criterion. theory not a conclude do such We that study deviations We equilibria. of stable theories those even recurrent is it if only and if probability nontrivial is dynamic te ad ih cu niieyotn ne hsdfnto,teequilibrium the definition, this Under often. infinitely occur might hand, other ue h e feulbi htwl ess nteln run. long the in persist cap- will fully that recurrence equilibria of of definition the set that the is conclude often tures we infinitely reason, played this be For will zero. equilibrium non-recurrent a that probability ee eunto return never eurne ic n us-tbeeulbimi rval eurn,Lemma recurrent, trivially is equilibrium quasi-stable any Since recurrence. # 14 ${ ilntpriti h ogrn n qiiru hc scnandin contained is which equilibrium Any run. long the in persist not will 15 ti la rmDfnto hti lystlso neulbimi a in equilibrium an on settles play if that 4 and Definition set, from absorbing clear an is itself It is game the of equilibria all of set the Since Proposition Lemma to respect with equilibria recurrent of existence guarantees 1 Proposition em olw ietyfo h eiiin fqaisaiiyand quasi-stability of definitions the from directly follows 1 Lemma sw ics ntetx,ordfnto frcrec oeie ue u qiirathat equilibria out rules sometimes recurrence of definition our text, the in discuss we As # B , *( hn#i o recurrent not is # then # n otisarcretstfreach for set recurrent a contains ) 1 0 let >0, . fteeeit us-tbeeulbim# equilibrium quasi-stable a exists there If 1 # . 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Length: 45 pic 0 pts, 190 mm rtro,tedfnto fdvainst eursta taeyin strategy a that requires sets sends deviation Sender of of type one definition which the criterion, u nadnmcfaeok tipista n qiiru htfails pro- that positive with equilibrium often any infinitely that played be implies not It bability. will framework. criterion intuitive dynamic the a in que 1. Lemma from follows then result nutv rtro ntehpteia ae When game. hypothetical the in criterion intuitive xaso of expansion ht ihrsett hshpteia ae hr exists there game, hypothetical this to respect with that, etrsos ota eito,ms eaalbei h hypothetical the # in available as be well must as deviation, criterion, that intuitive the to game. fail response to best equilibrium a an causes that Sender eiscnandin contained tegies nyi ti us-tbe ned hscs ple oteitiiedeviation intuitive and the if to applies recurrent case is this equilibrium Indeed, con- quasi-stable. an correspondence: is equilibrium then it every if from equilibrium, only correspondence quasi-stable deviation a the tains if that means 1 l etrsosst hssrtg utb in be must strategy this to responses best all opeetepofw ilso httebs epne to responses exists best the there the that Then show survives not. will in also we Suppose contained proof it game. the then actual complete message, game, the unsent hypothetical in an the criterion in intuitive criterion intuitive criterion. intuitive the passes r h aea h aebigeaie,bti hc nytemessages the only which in but examined, ( being in game actions the and as same the are cin lydwt oiiepoaiiy(no f h qiiru ah by path) equilibrium the off in or (on strategy probability some positive with played actions hr exists there hc ntr speieytesto qiirata assteintuitive the passes that equilibria of set the equilibria stable precisely of is set criterion turn the precisely in are which correspondence deviation intuitive o odfrsome for hold not if trivially holds result , rpsto eed h nutv rtro gis h tgizcriti- Stiglitz the against criterion intuitive the defends 2 Proposition h ro fPooiin2i iet eso htfrayequilibrium, any for that show We direct. is 2 Proposition of proof The Proposition ocmlt u ro,w li hti neulbimsrie the survives equilibrium an if that claim we proof, our complete To Proof D ( # otisa qiiru htpse h nutv rtro.The criterion. intuitive the passes that equilibrium an contains ) pligLma1 ene nyso ht o l equilibria all for that, show only need we 1, Lemma Applying . . # uvvn h nutv rtro uhthat such criterion intuitive the surviving $ # Q o osdrtehpteia aei hc h payoffs the which in game hypothetical the consider Now . 2. EITOS YAIS N REFINEMENTS AND DYNAMICS, DEVIATIONS, ,wihcnrdcsteasmto that assumption the contradicts which *, M 7 , h e feulbi htaercretwt epc othe to respect with recurrent are that equilibria of set The .W nwb h h n rp xsec theorem existence Kreps and Cho the by know We *. Q 2 # m Q )aeaalbet h lyr,where players, the to available are *) # ,o h otivkdi h nutv rtro.To criterion. intuitive the in invoked sort the of *, aln h nutv rtro.Lt( Let criterion. intuitive the failing utas ei ( in be also must uvvsteitiieciein ups hti does it that Suppose criterion. intuitive the survives m utb neeetof element an be must * 7 *, 1 7 Q ) n eito ythe by deviation any *), 2 2 ec,snealstra- all since Hence, . _ al h intuitive the fails _ 7 # #( Q $# Q _ *, 1 stestof set the is * 7 1 ND m uvvsthe survives Moreover, . 7 *, 1 utbe must * )b the be *) 2 7 ( # )that 2 *) .This ). 15 # , , File: 642J 209016 . By:BV . Date:22:01:96 . Time:16:08 LOP8M. V8.0. Page 01:01 Codes: 3547 Signs: 2760 . Length: 45 pic 0 pts, 190 mm neeyeulbim n o vr usto types of subset every for and equilibrium, every in 16 iru n ahohrgopsnsadfeetmsae hyrqiethat require They equi- message. the different with a sticks sends group type group each one other where each and groups, librium disjoint into are types that Sender's deviations on based credible correspondences weakly deviation construct shall We ru hnwhat than group aof ontdpn ietyo h esg etb h edr othat so functions Sender, utility the the by by sent message representable the are on payoffs directly the depend not do payoffs unstable some con- where the correspondence in recurrent. deviation present are a we equilibria games section, that cheap-talk this indicated In of 2 recurrent. text Proposition were on point; equilibria this based stable reflect correspondence not only deviation did the criterion intuitive Yet the recurrent. nonetheless are that noacutteseilntr fceptl ae n h deviation the and games cheap-talk of nature Matthews special that that the deviations account of game, class into the broader run. of the long play the consider given in to a occur us in may leads occur necessarily perspective deviations our which of Matthews theory find we While uso fti su.W aeafrhrrsrcinfrsmlct htde o substantively not does that dis- simplicity a for for restriction [22] Matthews and further While a [17] beliefs, results: some See make if our types. We deviation relevant change issue. a the this allow benefits We of Receiver prefer. the cussion Sender by of Matthews response types beliefs, optimal relevant and given the to that action response the optimal choose one than more has librium in,w nyalwfrpr-taeydeviations. pure-strategy for allow only we tions, oelbrlnto falwbedvain hnd Matthews do than deviations allowable of notion liberal more xssapriinof partition a exists 16 Definition ha-akgmsaesgaiggmswt h rpryta players' that property the with games signaling are games Cheap-talk equilibria unstable be may there that is framework our of point major A ems oiytedfnto fdvaincrepnecst take to correspondences deviation of definition the modify must We hl nprdb Matthews by inspired While pcfcly hnteeaemlil ttmnsta ih eblee,o fteReceiver the if or believed, be might that statements multiple are there when Specifically, (iii); (ii); (i); # alasubset a Call . t u u a esahge aoffo h eevrsotmlrsos to response optimal Receiver's the from payoff higher a gets r max arg # * 1 1 i ( ( t t 5 , , . a a ntesneo Matthews of sense the in tal et t i *)< k *)>max 4 o l # t all For ol e fiiae nte ru' esg.Formally: message. group's another imitated if get could ha akadRcretMops Recurrent and Talk Cheap . J emt eflo arl 1]i suigthat assuming in [10] Farrell follow We permit. . u into 1 *( a J # t tal et S o all for ) of [u 2 J AI N SOBEL AND RABIN  i T , *( 1 T tal et i t ' etitoscmeln fitrrtda a as interpreted if compelling restrictions .'s let , 1 ..., =1, # t a Ji ), rsn eiiinvldfrmxdsrtg devia- mixed-strategy for valid definition a present . tal et efsgaigfamily self-signaling u u u 2 1 ( 1 *( ( t t , 1] eiiin5icroae a incorporates 5 Definition [17], . , t j i a a n actions and , etepyfstype payoffs the be ) if ) i *) ? t ] ( tal et  t for ) J . 1] hypriinthe partition They [17]. . for i 1 ..., =1, tal et t a # uhthat: such i * u eaieto relative o sueta ewill he that assume not . J X 1 k ( t and j , ; a T and ) t hr exists there , esi equi- in gets i { # k , u fthere if tal et 2 ( t , a . t 16 's ). DEVIATIONS, DYNAMICS, AND REFINEMENTS 17

Figure 4

a ``neologism''an unused message m(X) that means ``I am some type t # X.'' We assume that there are a finite number of messages used in equi-

librium and a finite set of potential neologisms, [m(L)]L T. For a given

equilibrium #, Definition 6 can be used to construct sets (Q1 , Q2). For every self-signaling family of types J, partition [J1, ..., JN] and best respon-

ses a*toi Ji meeting the criteria of Definition 6, let _2 be the strategy that responds to the neologism m(Ji) by action ai* and to every other message

according-to the equilibrium strategy. Then let Q2 be the set of all such _2 and let Q1 be the set of optimal responses to beliefs over the set Q2 . Once the set Q has been specified, we create the deviation corre- spondence by iteratively adding strong best responses, where the Sender's strategies are restricted to sending those messages that are used with 17 positive probability in either the original equilibrium or in Q1 . We call the solution concept created by applying our framework to this deviation correspondence recurrent mop. The game in Fig. 4 illustrates some implications of recurrent mop, and how it differs from existing solu- tion concepts. Assuming that the four types of Sender are equally likely, there is a pool- ing equilibrium where all types of the Sender send the same messages and the Receiver always takes action G. The pooling outcome is not plausible

according to the arguments of Matthews et al.: types t1 and t2 could improve their payoffs by jointly deviating with the message ``I am either t1

or t2 ,'' leading to action C, and types t3 and t4 could jointly deviate by sending the message ``I am either t3 or t4 ,'' leading to action F. For that reason, the pooling equilibrium is not stable with respect to the mop

17 This restriction is necessitated by the structure of cheap-talk games, and is made consis- tent with rationality by assuming that the Receiver interprets any message as if it were one

of the messages sent in the original equilibrium or in the set Q1 .

File: 642J 209017 . By:MC . Date:21:12:95 . Time:13:42 LOP8M. V8.0. Page 01:01 Codes: 2724 Signs: 2056 . Length: 45 pic 0 pts, 190 mm File: 642J 209018 . By:BV . Date:22:01:96 . Time:16:08 LOP8M. V8.0. Page 01:01 Codes: 4172 Signs: 3569 . Length: 45 pic 0 pts, 190 mm n ntelgco epnigt hs eitossget httypes that suggests deviations these to responding noth- of but further, equilibrium logic themselves pooling [t the separate the to partially in to deviation trying the ing back types a players different from is, lead the Deviations involve never That will correspondence. equilibrium. equilibrium quasi-stable deviation pooling pooling is mop partially it the the because from mop, to recurrent respect a with is equilibrium pooling partially taeisweetypes where strategies s esgsta eentue ihpstv rbblt nete h rgnleulbimor equilibrium original the either in probability positive with used not Q were that messages use huh h esg ol esn ihpoaiiyzr.Smlry h eevrcould Receiver the Similarly, zero. probability with with messages sent unsent be previously would to respond message the thought ouincnetta atrsta at ycnrs,tenneitneof non-existence a the formulate contrast, By we fact. prediction-making, that for captures Matthews standard that concept the solution as recurrence using 1]peit rcsl h poiecnlsota nytepooling the that only note conclusionthat also opposite We the game. precisely this predicts in concept [10] outcomes solution possible the the about agnostic 18 action increpnec.I sssetbe o ntne otype to instance, for susceptible, is It correspondence. tion message n ec sneteeaen te qiiru ucms h atal oln equilibrium pooling partially the equilibrium, outcomes) pooling equilibrium fully the other add no not are quasi-stable. do is there We (since neologisms). (using the hence equilibrium or and pooling messages, partially original original the the either contains correspondence deviation the ticular, ust ftypes of subsets odn to ponding stmtdt efsga esl oidc h action the induce to herself self-signal to tempted is game. the of and will outcome equilibrium correspondence sequential deviation other only the the contain Moreover, correspondence. deviation ol send would ihactions with taeywhere strategy oteSne' taeistepsiiiythat possibility the strategies Sender's the to efsgaighref nuigteaction the inducing herself, self-signaling ti motn htw o nld `ek'bs epne uigieain on owould message so doing to iteration; respond during to responses Receiver best ``weak'' the include instance, not for we that allow, important is it m epnigt h message the to responding nw that knows 12 1 18 1 Matthews efe htrcretmpmkstergtpeito nti gamethe this in prediction right the makes mop recurrent that feel We hsohreulbimi atal oln ucm nwihtypes which in outcome pooling partially a is equilibrium other This )I h trtv rcs,teRcie ol o d nsrtge hr ersod to responds he where strategies in add now would Receiver the process, iterative the In .) , with omly osdrteprilypoigeulbimi hc types which in equilibrium pooling partially the consider Formally, t t 2 2 ] F oladidc h action the induce and pool m B ol iht olwt types with pool to wish would hseulbimi lontsal ihrsett h o devia- mop the to respect with stable not also is equilibrium This . 12 nta of instead t m m n types and C 1 12 ( tal et and t t and 3 tal et 1 .Ti,i un en htteRcie ih epn to respond might Receiver the that means turn, in This, ). and sends t F 3 ' rgnlslto ocp sepyi hsgm.Btthe But game. this in empty is concept solution original .'s .'s m [t r o edn hi qiiru esgs ie hs ewudadd would we this, Given messages. equilibrium their sending not are nta of instead C t t 34 1 m n to and , 3 1 and ( and , ihactions with noneetpofequilibrium announcement-proof t t 1 m ), 2 elgs-ro equilibrium neologism-proof ( ] t t t t 2 4 2 1 ihaction with ) and edtemessage the send edsm obnto fmessages of combination some send m sends B 34 AI N SOBEL AND RABIN and with [t m C E 12 3 hs h eito orsodnewl oss of consist will correspondence deviation the Thus, . and E , , C t t nta of instead 3 hl types while , 4 D t A ] 2 sends F hti h ed o lo h edrto Sender the allow not do we why is That . n message and , ol nfc edtemessage the send fact in would h set The . ilsprt rmec te.By other. each from separate will [t m A 3 34 m , ahrthan rather h o eito set deviation mop The . ( t F t 4 3 hsrsos sotmlbcuehe because optimal is response this ; ,and ), ] Q . 18 m 1 t ( ilteeoecnanol the only contain therefore will t 3 m 1 t ihaction with ) and 4 ( eeoe yFarrell by developed D t sends 3 ihaction with ) . evsteanalyst the leaves C m t ieie type Likewise, . 4 ( m t oladinduce and pool 3 34 and ) t t 1 1 Nt eewhy here (Note . m and eitn by deviating Q ( D t 2 1 m m and ) D ilinclude will t eas he because ( 34 2 t n res- and , 1 npar- In . edthe send and ) m ( t t t t 3 3 1 4 ) DEVIATIONS, DYNAMICS, AND REFINEMENTS 19

Figure 5

equilibrium will occur. In our opinion, this is because Farrell is overly con- servative in determining what constitutes credible deviations from an equi- librium, but is overly liberal in rejecting the partially pooling equilibrium because it is not stable.19 So far, all our examples of recurrent equilibria have been quasi-stable. In our final example, we illustrate that non-quasi-stable equilibria can be recurrent. Consider the game in Fig. 5.20 There are three equilibrium outcomes that involve partial pooling (for

example, type t1 induces the action D, while the other two types induce A), a separating equilibrium, and a pooling equilibrium in which the Receiver takes action F. It is straightforward to check that all equilibria permit a mop deviation. Starting from the outcome in which the Receiver only plays actions A and D with positive probability, there is a deviation that induces C. The deviation theory must then include the action F (as the Receiver

allows the possibility that typed t1 and t2 will induce C and type t3 will send another message). Consequently, the deviation correspondence must

contain the outcome in which types t1 and t2 separate from the other type. In this way, one can verify that any minimal set for the game in Fig. 5 must

19 While our framework can be used to guarantee existence, Fig. 4 illustrates why we don't feel it should be applied only when there is non-existence using other solution concepts. There are instances where existing refinements clearly are over-selective permitting equilibria even when existence is not a problem. In Fig. 4, for instance, a recurrent version of Farrell's [10] solution concept would conclude that both equilibria are possible, whereas Farrell allows only the fully pooling equilibrium. While Farrell identifies a sense in which the partially pooling equilibrium is prone to deviations, the deviation that he proposes involves the players separat- ing even morenothing in the logic of his notion of stability suggests that the players would deviate back to the pooling equilibrium. Even applying his own theory of deviations in this game, we feel Farrell's solution concept, by including only the pooling equilibrium, is too selective. 20 This game is modeled after an example in [19], which is itself a variation of a game introduced by [18].

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A Ä ( 2 12 *( 22 21 t ( odfn uhgmsfral,w edt eeo oeiiildefini- initial some develop to need we formally, games such define To we but mop, recurrent for theorem characterization broad no have We Definition ; t , neapeo aeo ata omnitrssi xml f[2,wihRabin which [22], of 5 Example is interests common partial of game a of example An message the send types all where equilibrium, pooling the consider Formally, , L L B t A 1 a , , ) and ) D or ? # m ,or + ) 13 # F ( min t t ie uhsrtisfrteReceiver, the for stratgies such Given . , 3 C )| F t [ osend to , 2 n to and , r max arg B + [u tityprefers strictly and , t 6 1 1 . and sapoaiiydsrbto upre on supported distribution probability a is ( t hc r ae hr h edrhsacompelling a has Sender the where games are which , J m , ha-akgm a ata omnitrssi there if interests common partial has game cheap-talk A m A 1 a 13 t 22 ..., , epciey ie h eevrsrsoss type responses, Receiver's the Given respectively. , ): 23 2 and ,  ilsrcl rfrsending prefer strictly will with F a t h eitosfo hseulbimaefor are equilibrium this from deviations The . # J # L m j t A 2 A of 12 u and *( , and , 2 AI N SOBEL AND RABIN F ( T L ,or t , t ) 3 uhthat: such L a ] osend to t E ) while , 3 tti on ofrhrsrtge ol eadded be would strategies further no point this At . ? m tityprefers strictly eoetelws aofta type that payoff lowest the denote ( 21 12 t ) with ] t m h set The . a bani opeeypooling completely a in obtain can 23 A n o h eevrt epn othese to respond to Receiver the for and , C *( t m 1 L or L ih edeither send might 12 let , , E m osending to L ? ,to 23 eursteRcie to Receiver the requires ) A rmteprior the from ota h xaso includes expansion the that so , A *( m *( m 13 L 12 L otisalo the of all contains ) with with m ) # 123 B L [ . m C r max arg n let and , or t , 13 t 1 1 E and or D ,or tityprefers strictly ? n to and , lolet Also . m L] t m D 23 2 partial ,to  123 Given . osend to u and , and t 1 p # m m ( L t 13 23 ) t File: 642J 209021 . 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Length: 45 pic 0 pts, 190 mm nDfnto ,let 8, Definition in ihoeo hs cin n htfreach action for the that takes and Receiver actions the these which of one with ohpaesfrtpsi sets in types for players both iin htteei orcretmpcontaining mop recurrent no is that there demonstrates that 2 nitions Step while t oietf hmevsa ebr of members as themselves identify to A rfl.Ti odto olw rm()whenever (i) that strategy from a follows at condition arrives This population one profile. the use partition least We once at pool. the that partition to, reveals the show belongs of it to elements set condition different the this of of members member when a loses as type treated being by available rmtpsi other in types from odto i sasrn odto htgaate httpsin types that guarantees that condition strong a is (i) Condition ata omninterests common partial ati unalw st sals h olwn result: following the establish to us allows turn in fact irawt h aepoet.Fral,let Formally, property. same the with libria sapr-taeyNs qiiru and equilibrium Nash pure-strategy a is eso httedvaincrepnec htsat rma equilibrium different an from in starts types that correspondence which deviation in the that show we o all for rfrt dniyhrefa ebro h atto htcnan e type her would contains type that partition each pooled. the that of be states member than a (ii) rather as herself Condition identify partition. to the prefer of element other eaiet n oln equilibrium. pooling any to relative oei hc ye ntesame the in types which in come that 8 Definition xssan exists igfo oln qiiru utcnanaprilyrvaigequi- revealing sets partially different a in contain types begin- must which correspondence equilibrium in deviation pooling librium the a interests from common ning partial with game i *( 23 # eiiin6i en ocpueteitiinta ti nteitrs of interest the in is it that intuition the capture to meant is 6 Definition Proof Proposition J odtos()ad(i hrfr oehrgaatethat guarantee together therefore (ii) and (i) Conditions J ii if (iii) i i i)freach for (ii) , (i) and t ? i epoetepooiini w tp.Frtw hwta na in that show we First steps. two in proposition the prove We . for ) # i u J and Ä t 1 i k and ; ( L i t # 1 ..., =1, i & ; EITOS YAIS N REFINEMENTS AND DYNAMICS, DEVIATIONS, 3 t J J i . k J i # B )>max k for # L opoigeulbimi eurn o nagm with game a in mop recurrent a is equilibrium pooling No { i P snnepy(tms oti atal oln out- pooling partially a contain must (it non-empty is hr exists there , & j J J eapoigeulbimadlet and equilibrium pooling a be < pcf htteRcie epnst ahmessage each to responds Receiver the that Specify . i twl ee oet esifraiestrategy informative less a to move never will it , i 23 J { ;t upr h qiiru,slc actions select equilibrium, the support to ); . i o tlattwo least at for [u odto ii eursta,rltv owa is what to relative that, requires (iii) Condition uhthat such k] J 1 tp1dmntae that demonstrates 1 Step . 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Page 01:01 Codes: 3144 Signs: 2582 . Length: 45 pic 0 pts, 190 mm hrfr,i type if Therefore, olw rm(i)o eiiin6ta n efsgaigfml htexists that family self-signaling any in that equilibrium 6 an Definition to of relative (iii) from follows prpit qiiru strategies. equilibrium appropriate 22 ini tefssetbet eeoito.Te eieaslto concept, solution a coali- ap- define deviating it They the renegotiation. by because negotiated to equilibria, outcome susceptible the many itself whether is too to tion as out test rules no plies equilibrium Nash strong ucpil oaysc eitn olto.Bernheim coalition. deviating such any to equilibrium com- susceptible private Nash for strong opportunity is off, there worse if players likely other be some this munication. their leave may play with to will renegotiation continue to this problem players such as other While so a that strategies. behavior given their out equilibrium payoffs, renegotiating higher pointing players all of them in may coalition yield equilibrium a Nash to [1] Pareto-efficient susceptible A be follow games: multi-person They in equilibria hypothesis played. Pareto-efficient be only applied. be environments, will could communication-rich paper in this that, in outlined principles general the which to both are that concepts. existing framework than that our realistic demonstrates within more 3 and concepts Proposition powerful solution section, more construct this can of examples we the with Along from starting correspondence # ilntbbl nnomtvl oee.Blume forever. uninformatively babble not will the completes This partition. the of elements different proof. from types pool can model. rcs nyamt cin otie in contained expansion the actions that conjectures admits on only assumption full-support process the and (i) from m euti hi td feouinr tblt nceptl ae,btto concept. but solution games, cheap-talk other cheap-talk no in for stability holds result evolutionary the of knowledge, study our their in result # i 24 trmist hwthat show to remains It oda ihti su,Amn 1 rpsdteslto ocp of concept solution the proposed [1] Aumann issue, this with deal To research game-theoretic of Bernheim areas further two discussing by conclude We rpsto ttsta ngmswt ata omnitrss players interests, common partial with games in that states 3 Proposition iha lmn of element an with B aciio[7 ugssta eae eutmgtb osbei i yai learning dynamic his in possible be might result related a that suggests [27] Sanchirico hneeysrtg o h eevrta sin is that Receiver the for strategy every then , tal et 4 osdrtefeun upsto ngm theory game in supposition frequent the consider [4] . t i # J A i *( hc ue u l aheulbi htare that equilibria Nash all out rules which , ed message sends J ND i .I olw rm(i)ta oeeetof element no that (iii) from follows It ). AI N SOBEL AND RABIN # B 5 ( P Discussion . B utb upriinof subpartition a be must otisti outcome: this contains ) B tr ihayeulbimin equilibrium any with Start . m i ihpstv rbblt under probability positive with A *( tal et L for ) 5 banarelated a obtain [5] . D ( # Q L utrsodto respond must ) tal et 2 and [J J i i ru that argue . ] o some for tfollows It . Q 1 d the add ND B .It ( # 24 i ) . File: 642J 209023 . By:BV . Date:22:01:96 . Time:16:08 LOP8M. V8.0. Page 01:01 Codes: 3423 Signs: 2886 . Length: 45 pic 0 pts, 190 mm iru ssal codn otempdvaincrepnec.Tu,the Thus, correspondence. deviation mop the to according stable is outcomes. librium all equilibria not among their do selecting also we for but only if equilibria, not even all justification among run, (partial) long a the provide would in suggest occur would those will framework priori equilibria (even our stable intuitive then correspondence only equilibrium, the deviation that stable from every a constructed settle. that contains will outcomes) criterion out play non-equilibrium turned which from it upon generated outcomes if as fact, candidates In natural longer be no outcomes. still it equilibrium the would set, on maintains only recurrent definition focus a to this to automatically While converges us set. eventually leads play recurrent that some hypothesis in contained set a are that said and D outcomes, equilibrium outcomes as of well as analysis outcomes equilibrium equilibrium of theories. foundations non-equilibrium framework the our useful exploring how developing discussing in in by both conclude and (see, to useful wish years be We recent in could [20]). attack and under [3] come e.g., has itself hypothesis equilibrium the ocp hticroae nutv oin fwe qiiraaesubject are Bernheim equilibria and Aumann when ence. unlike of equilibrium but, notions an renegotiation obtain intuitive to con- could incorporates and we renegotiations that so, such Doing concept all implications. unlikely. allow long-run are instead their renegotiations would sider stable we such to framework, that lead our inference not In their need question equilibrium we Nash behavior, strong by allowed rather renegotiations ``stable,'' is mis- equilibrium ``recurrent.'' be an is strong whether may it on the whether equilibrium much on both Nash too focusing that coalition-proof in the suggests how to leading of and framework hard model equilibrium is our explicit it Nash Yet an because communicate. without mostly etc., but players knowledge, games, multi-player common for conceptualize it formalized from not free itself is that renegotiation. renegotiation coalitional coalitional further beneficial some exists there equilibrium Nash coalition-proof ( 25 sw hwdi eto ,hwvr hr r ae hr oequi- no where games are there however, 4, Section in showed we As they sets, recurrent themselves are equilibria stable because course, Of non- to respect with correspondences deviation defined we that Suppose paper, this throughout outcomes equilibrium on focused have we While hl Bernheim While have we because partly issue, this to apply directly not does model Our # he[]sget eae approach. related a suggests [8] Chwe ) 25 ou nyo qiira hs h n rp' hoyo deviations of theory Kreps's and Cho Thus, equilibria. on only focus O o all for O EITOS YAIS N REFINEMENTS AND DYNAMICS, DEVIATIONS, srcreti ti iia ihrsett h property the to respect with minimal is it if recurrent is # # tal et O eurn ucmswudb hs ucmsthat outcomes those be would outcomes Recurrent . a ecreti ugsigta ayo the of many that suggesting in correct be may . nwiheulbi r ue u nyif only out ruled are equilibria which in , tal et urnesexist- guarantees . 23 a File: 642J 209024 . By:BV . Date:22:01:96 . Time:16:08 LOP8M. V8.0. Page 01:01 Codes: 3646 Signs: 2939 . Length: 45 pic 0 pts, 190 mm ehv oei hsppr u lot netgt hte lywl con- non- will which often. play converge, infinitely necessarily whether not recur as investigate does might to run, play outcomes long also if equilibrium the but and, in paper, equilibrium occur to this will verge in equilibria done which have investigate partial we to is only not mop there used which recurrent in types. equilibrium, differing outcomes pooling the only of would partially separation include it nonetheless the although would would instance, predict rationalizability for It 3, uniquely Example outcomes. In not power. equilibrium include predictive as would some have rationalizability'' well mop as ``recurrent non-equilibrium concept solution implied 24 10. 11. 15. 14. 13. 12. 16. 18. 17. 6. 7. 5. 2. 1. 4. 3. 9. 8. h bv ojcue l ugs htavrato u rmwr a be can framework our of variant a that suggest all conjectures above The .K Cho, I.-K. .K-h n .M Kreps, M. D. and I.-K.-Cho Sobel, J. and Kim, Y.-G. Blume. A. Sobel, J. and Banks J. Aumann, R. .D enem .Plg n .Whinston, M. and Peleg, B. Bernheim, D. B. .D Bernheim, D. B. .Cafr n .Sobel, J. and Crawford V. .Farrell, J. Chwe, M. .Gosa n .Perry, M. and Grossman S. .Khbr n .F Mertens, F. J. and Kohlberg E. Rob, R. and Mailath, G. Kandori, M. .Kli .Pze,adD Schmeidler, Schmeidler, D. D. and Kalai and E. Pazner, A. Kalai, E. .Mailath, G. .Mui n .P Vial, J.-P. and Moulin H. .Mthw,M kn-uiaa n .Postlewaite, A. and Okuno-Fujiwara, M. Matthews, S. okn ae 217 nvriyo hcg,My1992. May Chicago, of University 92-127, Paper Working 1959. 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