When is Reputation Bad? 1

Transcription

1 When is Repuaion Bad? Jeffrey Ely Drew Fudenberg David K. Levine 2 Firs Version: April 22, 2002 This Version: Ocober 23, 2002 Absrac: In radiional repuaion heory, repuaion is good for he longrun player. In "Bad Repuaion," Ely and Valimaki give an example in which repuaion is unambiguously bad. This paper characerizes a more general class of games in which ha insigh holds, and presens some examples o illusrae when he bad repuaion effec does and does no play a role. The key properies are ha paricipaion is opional for he shor-run players, and ha every acion of he long-run player ha makes he shor-run players wan o paricipae has a chance of being inerpreed as a signal ha he long-run player is bad. We also broaden he se of commimen ypes, allowing many ypes, including he Sackelberg ype used o prove posiive resuls on repuaion. Alhough repuaion need no be bad if he probabiliy of he Sackelberg ype is oo high, he relaive probabiliy of he Sackelberg ype can be high when all commimen ypes are unlikely. We are graeful o Adam Szeidl for careful proofreading, o Juuso Valimaki for helpful conversaions, and o Naional Science Foundaion Grans SES-97308, SES , SES , and SES Deparmens of Economics, Norhwesern Universiy, Harvard Universiy and UCLA.

2 . Inroducion A long-run player playing agains a sequence of shor-lived opponens can build a repuaion for playing in a specific way and so obain he benefis of commimen power. To model hese repuaion effecs, he lieraure following Kreps and Wilson [982] and Milgrom and Robers [982] has supposed ha here is posiive prior probabiliy ha he long-run player is a commimen ype who always plays a specific sraegy. 3 In Bad Repuaion, Ely and Valimaki [200] (henceforh EV) consruc an example in which inroducing a paricular commimen ype hurs he long-run player. When he game is played only once and here are no commimen ypes, he unique sequenial equilibrium is good for he long-run player. This remains an equilibrium when he game is repeaed wihou commimen ypes, regardless of he player s discoun facor. However, when a paricular bad commimen ype is inroduced, he only Nash equilibria are bad for a paien longrun player. 4 Wha is no clear from EV is when repuaion is bad. This paper exends he ideas in EV o a more general class of games in an effor o find he demarcaion beween bad and good repuaion. In addiion, we ry o relae he EV conclusions o pas work on repuaion effecs. Repuaion effecs are mos powerful when he long-run player is very paien, and Fudenberg and Levine [992] (FL) provided upper and lower bounds on he limiing values of he equilibrium payoff of he longrun player as ha player s discoun facor ends o. The upper bound 3 See Sorin [999] for a recen survey of he repuaion effecs lieraure, and is relaionship o he lieraure on merging of opinions. 4 I is obvious ha incomplee informaion abou he long-run player s ype can be harmful when he long-run player is impaien, since incomplee informaion can be harmful in one-sho games. Fudenberg-Kreps [987] argue ha a beer measure of he power of repuaion effecs is o hold fixed he prior disribuion over he repuaionbuilder s ypes, and compare he repuaion-building scenario o one in which he repuaion builder s opponens do no observe how he repuaion builder has played agains oher opponens. They discuss why repuaion effecs migh be derimenal in he somewha differen seing of a large long-run player facing many simulaneous small bu long run opponens.

3 2 corresponds o he usual noion of he Sackelberg payoff. The lower bound, called he generalized Sackelberg payoff, weakens his noion o allow he shor-run players o have incorrec beliefs abou he long-run player s sraegy, so long as he beliefs are no disconfirmed by he informaion ha he shor-run players ge o observe. When he sage game is a one-sho simulaneous-move game, acions are observed, payoffs are generic, and commimen ypes have full suppor, hese wo bounds coincide, so ha he limi of he Nash equilibrium payoffs as he long-run player s discoun facor ends o one is he single poin corresponding o he Sackelberg payoff. For exensive-move sage games, wih public oucomes corresponding o erminal nodes, he bounds can differ. However, alhough FL provided examples in which he lower bound is aained, in hose examples he upper bound was aained as well, and we are no aware of pas work ha deermines he range of possible limiing values for a fairly general class of games. Here we examine he upper bound more closely for a specific class of games designed o capure he insigh of EV. Specifically, we define a class of bad repuaion games, in which he long-run player can do no beer han if he shor-run players choose no o paricipae. This exends he EV example in a number of ways. We allow a broad class of sage games in which paricipaion by he shor-run players is opional; allowing for many acions, many signals, many shor-run players, and a wide variey of payoffs. Especially imporan, we allow for a broad range of ypes, including ypes ha are commied o good acions, as well as ypes ha are commied o bad acions. Earlier research suggess ha o aain he upper bound on he long-run player s payoff, i can be imporan o include he Sackelberg ype ha is commied o he sage-game acion he long-run player would choose in a Sackelberg equilibrium. 5 We 5 EV consider wo specificaions for he bad ype, eiher commied (o playing he bad acion) or sraegic (willing o play a differen acion occasionally o increase enry and he fuure payoff from playing bad. ) In a relaed model, Mailah and Samuelson [998] argue ha bad ypes and specifically sraegic bad ypes are more plausible han Sackelberg ypes. We are sympaheic o he argumen ha sraegic bad ypes may be

4 3 find ha he EV resul fails if he probabiliy of his Sackelberg ype is oo high, bu exends o he case where he probabiliy of he Sackelberg ype is sufficienly low, bu nonzero. This shows ha i is no essenial o rule ou he ypes ha suppor good repuaion effecs in order o derive he bad repuaion resul. Moreover, he relaive probabiliy of he Sackelberg ype can be high when all commimen ypes are unlikely; in his sense our condiions hold for almos all sufficienly small commimen-ype perurbaions of he complee informaion limi. By exending he EV example o a broad class of sage games we are able o more clearly idenify he ypes of assumpions key o a bad repuaion. There are several such properies, noably ha he shor-run players can eiher individually or collecively choose no o paricipae. However, mos of he assumpions on he srucure of he game seem o involve lile loss of economic applicabiliy. The key subsanive assumpion seems o be ha every acion of he long-run player ha makes he shor-run players wan o paricipae in he game has a chance of being misconsrued as a signal of a bad repuaion. EV moivae heir example by considering an auomobile mechanic who has specialized knowledge of he work ha needs o be done o repair he car. We hink ha we have idenified a broader class of bad repuaion games ha can be inerpreed as exper advice. This includes consuling a docor or sockbroker, or in he macroeconomics conex, can be he decision wheher or no o urn o he IMF for assisance. In EV, he shorrun players observe only he advice, bu no he consequences of he advice. Here we explicily consider wha happens when he shor-run players observe he consequences as well. We also show ha here are oher disinc classes of games wih raher differen observaion srucures ha are bad repuaion games, such as our eaching evaluaion game, where advice is no an issue because he long-run player does no more likely han commimen ypes, bu his does no imply ha he probabiliy of commimen ypes should be zero. Insead, we would argue ha i is preferable for models o allow for a wide range of ypes, especially hose wih fairly simple behavior rules.

5 4 privaely observe anyhing ha is payoff-relevan for he shor-run player. Finally, we illusrae he boundaries of bad repuaion by giving a number of examples and classes of paricipaion games ha are no bad repuaion games. 2. The Model 2.. The Dynamic Game There are N + players, a long run-player, and N shor-run players 2 N +. The game begins a ime = and is infiniely repeaed. Each period, each player i chooses from a finie acion space i A. We denoe individual acions a i, and acion profiles by a. We also use i a - o denoe he play of all players excep player i and denoe he play of all players excep players i and j. i j a -- The long-run player discouns he fuure wih discoun facor d. Each shor-run player plays only in one period, and is replaced by an idenical shor-run player in he nex period. There is a se Q of ypes of long-run player. There are wo sors of ypes: ype 0 ÎQ is called he raional ype, and is he focus of our ineres, wih uiliy described below. For each pure acion a, ype q ( a ) is a commied ype, ha is consrained o play a. These are he only possible ypes in Q. Noe ha we do no require ha every pure acion commimen ype has posiive i probabiliy. The sage game uiliy funcions are ua (), where u () a corresponds o he long-run player of ype q = 0. The common prior disribuion over long-run player ypes is denoed µ (0). There is a finie public signal space Y wih signal probabiliies r ( y a). All players observe he hisory of he public signals. Shor-run players observe only he hisory of he public signals, and in paricular observe neiher he pas acions of he long-run player, nor of previous shor-run players. We do no assume ha he payoffs depend on he acions only hrough he signals, so he shor-run players a dae need no o

6 5 know he realized payoffs of he previous generaions of shor-run players. 6 We le h = ( y, y2, y) denoe he public hisory hrough he end of period. We denoe he null hisory by 0. We le h denoe he privae hisory known only o he long-run player. This includes his own acions, and may or may no include he acions of he shor-run players he has faced in he pas. A sraegy for he long-run player is a sequence of maps s ( h, h,) q Î conhulla º A ; a sraegy profile for he shor-run j j j players is a sequence of maps s ( h) Î conhulla º A. (Noe ha - j A denoes he produc of he A s, no he convex hull of he a - is a Nash response o i i -- i i i -- i produc.) A shor-run profile a if u ( a, a, a ) ³ u ( a, a, a ) for all a i Î A i. We denoe he se of shor-run Nash responses o a by B( a ). Given sraegy profiles s, he prior disribuion over ypes m (0) and a public hisory h ha has posiive probabiliy under s, we can calculae from s he condiional probabiliy of long-run player acions a ( h ) given he public hisory. A Nash Equilibrium is a sraegy profile s such ha for each posiive probabiliy hisory - s ( h) Î B( a ( h)) [shor-run players opimize] ) 2) ( h, h,( a )) a s q = [commied ypes play accordingly] 3) s (,,0) is a bes-response o s - [raional ype opimizes]. 2.2 The Ely-Valimaki Example We will use he EV example o illusrae our assumpions and definiions. In EV, he long-run player is a mechanic, her acion is a map from he privaely observed sae of he cusomer's car ω { ET, } o 6 Fudenberg and Levine [992] assumed ha a player s payoff was deermined by his own acion and he realized signal, bu ha assumpion was no used in he analysis. The assumpion is used in models wih more han one long-run player o jusify he resricion o public equilibria, bu i is no needed here.

7 6 announcemens { e,}, where E means he car needs a new engine, T means i needs a une-up, and he announcemens, which are wha he mechanic says he car needs, deermine wha is acually done o he car. Thus he long-run player s acion space is a map from privaely observed saes o announcemens, A = { ee, e, e, }, where he firs componen indicaes he announcemen in response o he signal E and he second o T. There is one shor-run player each period who chooses an elemen of A 2 = { InOu, }. The public signal akes on he values Y = {,, eou}. If he shor-run player chooses Ou he signal is Ou, ha is r ( Ou a, Ou) = ; oherwise he signal is he announcemen of he long-run player. The wo saes of he car are assumed o be i.i.d. and equally likely, so r( ( e e, In)) = r( ( e e, In)) = /2, r ( ( e ee, In)) =, and r ( (, e In)) = 0. If he shor-run player chooses Ou, each player ges uiliy 0. If he plays In and he long-run player s announcemen is ruhful (ha is, maches he sae), he shor-run player receives u; if i is unruhful, i is - w where w > u > 0. The raional ype of long-run player has exacly he same sage-game payoff funcion as he shor run players. Thus when he long-run player is cerain o be he raional ype, he sraegic form of he sage game is ee In Ou ( u -w )/2,( u - w )/2 0,0 e uu, 0,0 e -w,- w 0,0 ( u -w )/2,( u - w )/2 0,0 Figure When he raional ype is he only ype in he model, here is an equilibrium where he chooses he acion ha maches he sae, all shorrun players ener, and he raional ype's payoff is u. However, EV show ha when here is also a probabiliy ha he long-run player is a bad

8 7 ee, he longamoun ha converges o 0 as he discoun facor goes o. The inuiion for his resul has four seps. Firs of all, he raional ype mus e if is equilibrium p 0, because he shor- -run player is oo number K such ha ee. Second, from Bayes rule i follows ha here is some E in periods where he raional ype is playing honesly will make he poserior probabiliy of he bad ype so high ha all subsequen shor-run players play ou. Third, when here have been K - successive observaions of E, he raional ype of long run player is emped o play insead of e, even hough his lowers his shor-run payoff, o avoid driving ou he shor-run players wih anoher observaion of E. Thus, he long-run player is emped o ake an acion ha is worse for boh himself and he shor-run players in order o avoid being incorrecly agged as a bad ype; an inducion argumen shows ha hones play by he raional ype unravels Paricipaion Games and Bad Repuaion Games We consider paricipaion games in which he shor-run players may choose no o paricipae. The crucial aspec of non-paricipaion by he shor-run players is ha i conceals he acion aken by he long-run player from subsequen shor-run players; his is wha allows he lower bound on he long-run player's Nash equilibrium payoff in he EV example o be lower han Sackelberg payoff. We will hen define bad repuaion games as a subclass of paricipaion games ha have he addiional feaures needed for he bad repuaion resul; he following is a brief summary of he key feaures of hese games. Firs of all, here will be a se of friendly acions ha mus receive sufficienly high probabiliy o induce he shor-run players o paricipae, such as e in he EV example. Nex, here are bad signals ha are mos srongly linked o some unfriendly acions ha deer paricipaion, bu which also have

9 8 posiive probabiliy under friendly play; in EV he bad signal is E. Finally, here are some acions ha are no friendly, bu reduce he probabiliy of he bad signals, such as in EV; we call hese acions empaions. If here is a posiive prior probabiliy ha he long-run player is a bad ype ha is commied o one of he unfriendly acions, some hisories of play will induce he shor-run players o exi, so o avoid hese hisories he raional ype of long-run player may choose o play one of he empaions; foreseeing his, he shor-run players will chose no o ener. Our main resul shows ha his leads o a bad repuaion resul as long as he prior does no assign oo much probabiliy o ypes ha are commied o play friendly acions. To model he opion o no paricipae, we assume ha cerain public signals y e Y e are exi signals. Associaed wih hese exi signals are exi profiles, which are pure acion profiles shor run players. e For each such e, ρ ( e e y a, e ) ρ( y e ) e E A for he = for all a, and ry ( e - ) =. In oher words, if an exi profile is chosen, an exi signal mus occur, and he disribuion of exi signals is independen of he longrun player's acion. Moreover, if a Ï E hen r( y a, a - ) = e for e all a ÎA, y Î Y. We refer o A e E as he enry profiles. Noe ha an enry profile canno give rise o an exi signal. A paricipaion game is a game in which E - specializes o paricipaion games. ¹Æ. The remainder of he paper We begin by disinguishing acions by he long-run player ha cause he shor-run players o exi (unfriendly acions), and hose ha are needed o ge hem o ener (friendly acions). Definiion : A non-empy finie se of pure acions for he long-run player N is unfriendly if here is a number y < such ha a ( N ) ³ y implies B( α ) conhull E. Remark: This definiion says ha unfriendly acions induce exi, in he srong sense ha exi is he only bes response if he probabiliy of he

10 9 unfriendly acions is sufficienly high. There will ofen be many ses of unfriendly acions. In he EV example he se { ee,, e } is unfriendly, and so is any subse. Definiion 2: A non-empy finie se of mixed acions F for he long run player is friendly if here is a number g > 0 such ha B A implies a ³ gf for some f Î F. ( α ) conhull( E ) The number g is called he size of he friendly se. Remark: This definiion says ha he probabiliies given o every pure acion mus be bounded below by a scale facor imes some friendly mixure if he shor-run players are no o exi. Noe ha weigh on a friendly acion is necessary for enry, bu need no be sufficien for enry, and ha a friendly se mus be non-empy. There may also be many differen friendly ses. Suppose ha F is friendly of size ĝ, and le F º min{ fa () > 0 f Î F }. Then if f Î F i may be replaced by any mixure over he suppor of f, and he resuling se will be friendly of size g ˆF. Similarly, if we have a friendly se and we eliminae mixures f ÎF whose suppor conained in he suppor of some differen f Î F, we ge a new friendly se wih a smaller value of a. In he EV example, he acion e is friendly, wih w -u a =. w + u/2 Finally, consider a pure acion a such ha Ba ( ) A conhull( E ). Since a is pure, a ³ gf is possible only if f = a. In oher words, any pure acion ha permis shor-run enry (such as e in he EV example) mus be in every friendly se. Definiion 3: The suppor AF ( ) of a friendly se are played wih posiive probabiliy: F is he acions ha A( F ) º { a Î A f ( a ) > 0, f Î F }

11 We say ha a friendly se F is orhogonal o an unfriendly se N if N Ç AF ( ) =Æ. 0 Nex we consider wha signals may reveal abou acions. Definiion 4: We say ha a se of signals Y is unambiguous for a se of acions N - if for all - a Ï E, y ÎYn, ÎN, a ÏN we have r( y n, a - ) > r( y a, a - ). This is a srong condiion: every acion in probabiliy o each signal in Y han any acion no in N mus assign a higher N. A given se of acions may no have signals ha are unambiguous; in he case of he EV example, E is an unambiguous signal for he unfriendly se { ee }. Definiion 5: An acion a is vulnerable o empaion relaive o a se of signals Y i if here exis numbers rr, > 0 and an acion b such ha - - ) If a Ï E, y ÎY, hen ) ) ρ( y b, a ) ρ( y a, a ) ρ. 2) If a e Ï E and y ÏY ÈY hen ρ( yba, ) ( +% ρρ ) ( ya, a ) ) For all e Î E, u ( b, e ) ³ u ( a, e ). The acion b is called a empaion, and he parameers rr, are he empaion bounds. In oher words, an acion is vulnerable if i is possible o lower he probabiliy of all of he signals in Y by a leas r while increasing he probabiliy of each oher signal by a leas he muliple ( + ρ% ). Noice ha for an acion o be vulnerable o a empaion, i mus place a leas weigh r on each signal in Y. Noice also ha he definiion does no conrol he payoff o he vulnerable acion when he shor-run players paricipae he empaion here is no o increase shor-run payoff, bu raher o decrease he probabiliy of he signals in Y. In he EV example, he acion e is vulnerable relaive o { E }. The empaion b i is, which sends he

12 probabiliy of he signal E o zero. (Since here is one oher signal, condiion 2 of he definiion is immediae.) Noice ha if an acion a is vulnerable, i canno be he case ha - - Ï conhulle hen r( a, a ) = r( a ) ; he disribuion if a - - of signals mus be in some way dependen on he long-run player s acion if he shor-run players do no exi. This is relaed o he noion of an acion being idenified, as in Fudenberg, Levine and Maskin [994]. Here we allow he possibiliy ha here are sraegies such as e and e from he EV example ha are no idenified, bu do no allow complee lack of idenificaion unless he shor-run players play in E - one. Definiion 6: A mixed acion wih probabiliy a for he long run player is enforceable if here does no exis anoher acion ã such ha for all - - u ( a, a ) ³ u ( a, a ) and for all a - - Î E, a ÎA - E, - - u ( a, a ) > u ( a, a ) and r( a, a - ) = r( a, a - ). When no enforceable, we say ha he acion ã defeas a. a is If an acion is no enforceable hen here is necessarily lack of idenificaion, since a and ã induce exacly he same disribuion over signals. The key poin is ha if he shor-run players ener wih posiive probabiliy, he raional ype canno play an acion ha is no enforceable: by swiching o ã he would sricly increase his curren payoff, while mainaining he same disribuion over signals, and so he same fuure uiliy. Noe also ha a mixed acion ha assigns posiive probabiliy o unenforceable acions is no enforceable: if a assigns probabiliy p o unenforceable acion a, hen a is defeaed by he mixed acion â formed by replacing he probabiliy on a. a wih he acion Definiion 7: A paricipaion game has an exi minmax if ã ha defeas

13 2 - max - - max u ( a, a ) a ÎE Çrange( B) - min - max u ( a, a ) a Îrange( B) a a = In oher words, any exi sraegy forces he long-run player o he minmax payoff, where he relevan noion of minmax incorporaes he resricion ha he acion profile chosen by he shor-run players mus lie in he range of B. 7 I is convenien in his case o normalize he minmax payoff o 0. bad repuaion games. We are now in a posiion o define a class of games we call Definiion 8: A paricipaion game is a bad repuaion game if i has an exi minmax, here is an unfriendly se orhogonal o N, a friendly se F ha is N, and a se of signals Y ha are unambiguous forn, and such ha every enforceable f Î F is vulnerable o empaion relaive o Y. The signals Y are called he bad signals. In paricular, he EV game is a bad repuaion game. We ake he friendly se o be { e }, he unfriendly se o be { ee } and he unfriendly signals o be { E }. We have already observed ha { e } is a friendly se and { ee } unfriendly. The wo are obviously orhogonal, and { E } is unambiguous for { ee }. In a bad repuaion game, he relevan empaions are hose relaive o Y. For he remainder of he paper when we examine a bad repuaion game and refer o a empaion, we will always mean relaive o he se Y. 7 When here is a single shor-run player his resricion collapses o he consrain of no playing sricly dominaed sraegies, bu when here are muliple shor-run players i involves addiional resricions. I is clear ha no equilibrium could give he long-run player a lower payoff han he minmax level defined in definiion 7. Conversely, in complee-informaion games, any long-run player payoff above his level can be suppored by a perfec Bayesian equilibrium if acions are idenified and he public observaions have a produc srucure (Fudenberg and Levine [994]). This is rue in paricular when acions are publicly observed as shown in Fudenberg, Kreps and Maskin [990].

14 3 For any bad repuaion game, i is useful o define several consans describing he game. Recall ha y is he probabiliy in he definiion of an unfriendly se and ha g is he scale facor in he definiion of a friendly se. Since he friendly se is finie, we may define j > 0 o be he minimum, aken over elemens of he friendly se, of he values ρ in he definiion of empaion. Define r = min r( y n, a ). r( y a, a ) n ÎN, a ÏN, a Ïconhull( E ), yy Î - Since he friendly se is non-empy and orhogonal o he unfriendly se, he denominaor of his expression is well defined, and since Y is unambiguous for he unfriendly se, r >. Also define h =- log( gj)/logr which is posiive, and k 0 log( y) =- log ( ) r ( y + - y ). 3. The Theorem We now prove our main resul: In a bad repuaion game wih a sufficienly paien long-run player and likely enough unfriendly ypes, in any Nash equilibrium, he long-run player ges approximaely he exi payoff. The proof uses several Lemmas proven in he Appendix. We begin by describing wha i means for unfriendly ypes o be likely enough. Le Q ( F ) be he commimen ypes corresponding o acions in he suppor of F ; we will call hese he friendly commimen ypes. Le ˆQ be he unfriendly commimen ypes corresponding o he unfriendly se N. Definiion 9: A bad repuaion game has commimen size ef, if

15 4 where f > 0., æ m(0)[ Q ] ö m(0)[ Q ( F )] e m çè (0)[ Q( F )] ø f This noion of commimen size places a bound on he prior probabiliy of friendly commimen ypes ha depends on he prior probabiliy of he unfriendly ypes. Since f is posiive, he larger he prior probabiliy of Q, he larger he probabiliy of he friendly commimen ypes is allowed o be. The hypohesis ha he priors have commimen size ef, for sufficienly small e is a key assumpion driving our main resuls. Noe ha he assumpion of a given commimen size does no place any resricions on he relaive probabiliies of commimen ypes. In paricular, le m be a fixed prior disribuion over he commimen ypes, and consider priors of he form lm, where he remaining probabiliy is assigned o he raional ype. Then he righ-hand side of he inequaliy defining commimen size depends only on m, and no on l, while he lef-hand side has he form lm. Hence for sufficienly small l he assumpion of commimen size eh, is saisfied. Noe ha he EV example has commimen size 0,h since he only ypes are he raional ype and he commimen ype who plays ee. r = Define min min f Î F max a () min{0, } U = u a - u, and le r. Recall ha g is he scale facor in he definiion of a friendly se. F º min{ fa () > 0 f Î F } and ha Theorem : In a bad repuaion game of commimen size ( + h) ( gf /2), h le v be he supremum of he payoff of he raional ( ) ype in any Nash equilibrium. Then

16 5 k * æ ö æ ö v (- d) k ç + U èr ø çè r ø min * min where k* = k0 + log ( m(0)[ Q ]/og ) l ( y + (- y) ) d v. lim 0 r,. In paricular, To prove his we use a series of Lemmas proven in he Appendix. For he res of his secion, we fix an arbirary Nash equilibrium. Given his equilibrium, le v ( h ) denoe he expeced coninuaion value o he raional long-run player, and le m( h ), a - ( h ) be he poserior beliefs and sraegy of he shor-run players a hisory h. Noice ha he expeced presen value o he raional long-run player condiional on a posiive probabiliy public/privae hisory pair mus no depend on he privae hisory h, or he raional long-run ype would be failing o opimize. If has posiive probabiliy under a - ( h ), hen we define a ( h ), and a - d d r y a posiive probabiliy under v ( h,) a º (- ) u () a + å ( y av ) ( h,) y. pu weigh only on such posiive- When mixed acions a and probabiliy, way. a - a a -, i is convenien also o define v ( h, a ) in he naural Lemma : In a paricipaion game, if h is a posiive probabiliy hisory in which y ÎY occurs in period and m( h- )[ Q ( F )] gf /2 hen a0( h) ³ ( g/2) f for some friendly f. In oher words, when he prior on friendly ypes is sufficienly low, enry can occur only if he raional ype is playing a friendly sraegy wih appreciable probabiliy. This is a consequence of he definiion of friendly sraegies: enry requires ha he overall sraegy assigns some minimum probabiliy o a friendly sraegy, and if he friendly ypes are unlikely,

17 6 hen a non-negligible par of his probabiliy mus come from he play of he raional ype. Recall ha h =- log( gj)/logr. Lemma 2: In a bad repuaion game, if h is a posiive probabiliy hisory, and he signals in h all lie in e Y ÈY, hen a) A mos k log ( m(0)[ Q] ) = k0 + log ( ) r ( y + -y ) of he signals are in Y. b) If he commimen size is m( h )[ Q ( F )] ( g/2) F. + h (( g/2) F ), h hen Remark: The inuiion for par a is simple, and closely relaed o he argumen abou he deerminisic evoluion of beliefs in FL: The shor-run players exi if hey hink i is likely ha enry will lead o he observaion of a bad signal. Hence each observaion of a bad signal is a surprise ha increases he poserior probabiliy of he bad ype by (a leas) a fixed raio greaer han, so along a hisory ha consiss of only bad signals and exi signals, he poserior probabiliy of he bad ype evenually ges high enough ha all subsequen shor-run players exi. This argumen holds no maer wha oher ypes have posiive probabiliy, and i is he only par of his lemma ha would be needed when here are only wo ypes, one raional and one bad, as in EV. However, as we will show by example below, we canno expec he bad repuaion resul o hold when here is sufficienly high probabiliy of he Sackelberg ype. Par b of he lemma says ha if he

18 7 iniial probabiliy of he friendly ypes is sufficienly low compared o he prior probabiliy of he bad ypes, hen along any hisory on he pah of play which consiss only of exi oucomes and bad oucomes, he probabiliy of he Sackelberg ype remains low. The inuiion for his resul is ha because of he assumpion ha he unfriendly and friendly ses are orhogonal, r >, so each observaion of a bad signal no only increases he probabiliy of he bad ype, i increases he relaive probabiliy of his ype compared o any friendly commimen ype, and his bounds he rae of growh of he probabiliy of friendly ypes. 8 Define ìæ ö U y Y + Î u (, y r) r = ï ïèç í ø ïïïî 0 oherwise ìïd + y ÎY d (, y r ) = ï ír ïïïî d oherwise e and Yh ( ) = { y ÎY È Y r( y a ( h ), a - ( h)) > 0}. Lemma 3: In a paricipaion game if a - h Î E - ( ) conhull( ), or a - ( h) Ï conhulle - and a 0 ( h ) ³ cf for some c > 0 and vulnerable friendly acion f wih empaion bounds r, ρ% hen yyh Î ( ) - d r + d r v ( h) max ( ) u (, y ) (, y ) v ( h, y). Remark: This lemma says ha if he raional ype is playing a friendly sraegy, his payoff is bounded by a one-period gain and he coninuaion payoff condiional on a bad signal. This follows from he assumpion ha for every enry-inducing sraegy i is possible o lower he probabiliy of all of he signals in Y by a leas j while increasing he probabiliy of 8 If here were a ype wih a hisory-dependen sraegy, his par of he lemma would need o be modified.

19 8 each oher signal by a leas he muliple r min. The fac ha he raional ype chooses no o reduce he probabiliy of he bad signal means ha he coninuaion payoff afer he bad signal canno be much worse han he overall coninuaion payoff. Proof of Theorem : Given an equilibrium, we begin by consrucing a posiive probabiliy sequence of hisories beginning wih an iniial hisory a dae 0. Given h already consruced, we define h+ = ( hy, + ) where + Î yyh Î ( ) - d r + d r y argmax ( ) u (, y ) (, y ) v ( h, y). We know ha Y( h ) is no empy because eiher α ( h ) conhull α ( h ) conhull E E. This laer case implies ha a ( h ) gf, or ³ for some friendly f, and since only enforceable acions can be played in equilibrium, his f mus be vulnerable o empaion, ) ) so ρ( Y α ( h), α ( h)) γρ( Y f, α ( h)) > 0. Now apply Lemma 2 o conclude ha for each h a mos k * of he signals are in Y and m( h)[ Q ( F )] gf /2. Consider an h such ha know a - ( h) Ï conhulle - ha a ( h ) gf g. From he definiion of a friendly acion, we ³ for some friendly m( h )[ Q ( F )] F /2 and Lemma implies haa0 ( h ) ³ gf /2. Now apply Lemma 3 o conclude ha for each h - d + r min + d + r min + v ( h ) ( ) u ( y, ) ( y, ) v ( h ) Since v ( h ) U, i follows ha -d P = 2d rmin r = min. v (0) ( ) å ( y, ) u ( y, ). Since e u ( y, r min) = 0, and y f, so ÎY a mos k * imes, his gives he desired bound. Noice ha he fac ha u ( y, r ) = 0 follows from he assumpion of exi minmax: i is here ha we make use of he fac ha exi e min gives he long-run player no more han he minmax.

20 9 4. Examples We now consider a number of examples o illusrae he scope of Theorem, and also he exen o which he assumpions are necessary as well as sufficien. To begin, Example 4. illusraes wha happens when he prior pus oo much weigh on some commied ypes for he hypohesis of commimen size g F /2 o be saisfied. Example 4.2 shows ha he EV conclusion is no robus o he addiion of an observed acion ha makes he shor-run players wan o ener. Example 4.3 examines paricipaion games ha are no bad repuaion games, and example 4.4 illusraes he role of he exi-minmax assumpion. In all of he examples bu 4., we assume ha he hypohesis of commimen size g F /2 is saisfied, and invesigae wheher he game is a bad repuaion game. The following secion considers a class of bad-repuaion principalagen games. 4.: EV Wih Sackelberg Type We have relaxed he original assumpions of EV in a number of ways. One imporan exension is ha we allow for posiive probabiliies of all commimen ypes. In paricular, we allow a posiive probabiliy of a Sackelberg ype commied o he hones sraegy e, which is he opimal commimen. However, a hypohesis of he heorem is ha he prior saisfy he commimen size assumpion. Here we illusrae ha assumpion in he conex of he EV example. Suppose in paricular ha here are 3 ypes, raional, bad, and Sackelberg. The se of possible priors can be represened by he simplex in figure 2.

21 20 Figure 2 When he prior falls ino he region A, he probabiliy of he bad ype is oo high, and he shor run players refuse o ener regardless of he behavior of he raional ype. Bad repuaion arises because he long-run player ries o preven he poserior from moving ino his region. In EV he prior assigned probabiliy zero o he Sackelberg ype. Thus he prior and all poseriors on he equilibrium pah belong o he lower boundary of he simplex. When here is a sufficienly high probabiliy of he Sackelberg ype, he shor-run players will ener regardless of he behavior of he raional ype; his is region B. Noe ha he boundaries of hese regions inersec on he righ edge of he simplex: his poin represens he mixure beween eeand ewhich makes he shor-run player indifferen beween enry and exi.

22 2 When he prior is in region A, here will be no enry and he longrun player obains he minmax payoff of zero. On he oher hand, when he prior is in region B, here is a Nash (and indeed sequenial) equilibrium in which he long run player receives he bes commimen payoff, which is u in he noaion of EV. The equilibrium is consruced as follows. Consider he game in which he poserior probabiliy of he bad ype is zero. In his game here exiss a sequenial equilibrium in which he long-run player ges u. Suppose ha we assume ha his is he coninuaion payoff in he original game in any subform in which he longrun player played a leas once in he pas. A sequenial equilibrium of his modified game is clearly a sequenial equilibrium of he original game, and by sandard argumens, his modified game has a sequenial equilibrium. How much does he raional long-run player ge in his sequenial equilibrium? One opion is o play in he firs period. Since he shor-run player is enering regardless, his means ha beginning in period 2 he raional ype ges u. In he firs period he ges ( u - w)/2. Hence in equilibrium he ges a leas (-d)( u - w)/2 + du, which converges o u as d. Our heorem is abou he se of equilibrium payoffs for priors ouside of hese wo regions. The heorem saes ha here is a curve, whose shape is represened in he figure, such ha when he prior falls below his curve (region C), he se of equilibrium payoffs for he long-run player is bounded above by a value ha approaches he minmax value as he discoun facor converges o. The diagram shows ha he lef boundary of he simplex is an asympoe for his curve as i approaches he complee informaion prior (i.e. µ (0)( θ 0) = ) in he lower lef corner. This illusraes he imporan aspec of he commimen size resricion: i is saisfied for almos all sufficienly small perurbaions of he complee informaion limi.

23 22 4.2: Adding an Observed Acion o EV We now modify he EV game by adding a new observable acion "g" for he long-run player called give away money. This acion induces he shor-run players o paricipae ( Bg ( ) conhull( E ) = ) so i mus be in every friendly se. Since he acion is observable, i is no vulnerable o empaion wih respec o any signals ha are unambiguous for he unfriendly acions, so his is no a bad repuaion game. Moreover, even wihou a Sackelberg ype he EV conclusion fails in his game: here is an equilibrium where he raional ype plays g in he firs period. This reveals ha he is he raional ype, and here is enry in all subsequen periods, while playing anyhing else reveals him o be he bad ype so ha all subsequen shor run players exi. Thus he assumpion ha every friendly acion is vulnerable o empaion is seen o be boh imporan and economically resricive Orhogonaliy Issues Suppose friendly acions send he bad signal by puing posiive weigh on unfriendly acions. An imporan class of games in which his is he case are hose in which, condiional on enry, he long-run players acions are observed. In his case he bad signals correspond o unfriendly acions, and bad signals can only have posiive probabiliy when he unfriendly acion is played wih posiive probabiliy. Moreover, in some games, he only friendly sraegies involve randomizing in his way. Proposiion : If for every friendly se and unfriendly se here is a friendly acion whose resricion o he complemen of he unfriendly se has probabiliy 0 of generaing a signal ha is unambiguous for he unfriendly se, he game is no a bad repuaion game. Proof: The assumpion ha he friendly and unfriendly ses are orhogonal is violaed.

24 23 To see ha his makes a difference, consider he following woperson game wih observed acions: L M R U 0,4,3 0,0 D 0,0,3 0,4 Figure 3 where L and R correspond o exi and M o enry. 9 In his case enry can be induced only by mixing wih probabiliy of U beween ¼ and ¾. We will firs explain why he bad repuaion heorem does no hold here, and hen show ha is conclusion fails as well. Boh U and D are unfriendly, and we need o choose eiher one or boh of hem o be in he unfriendly se. If we include boh acions in he unfriendly se, hen i is impossible o find an orhogonal friendly se. If we include only one of he acions in he unfriendly se, and chose a friendly se ha includes mixed acions, hen orhogonaliy is again violaed, while if we specify ha he friendly se is he singleon corresponding o he oher acion, hen he friendly acion is no vulnerable o empaion. The conclusion of he heorem fails here as well: suppose ha he only commimen ype wih posiive probabiliy is D, and ha he probabiliy of he bad ype is less han ¼. Consider he following sraegies: For any curren probabiliy m ( h )[ D] less han ¼ he raional ype mixes so ha he overall probabiliy of D is exacly ¾. (In paricular, his is rue when he long-run player has been revealed o be raional, so ha m ( h )[ D] = 0.) The shor-run player always eners. If U is observed, 9 In his example, he shor-run player has several exi acions, and his payoff depends on he long-run player s acion. This is a necessary feaure of wo-player games where he only friendly sraegies are mixed, bu i is no necessary in hree-player games hink of a game where player 3 has veo power, 3 only plays In if 2 plays M, and 2 s payoffs o M are as in he payoff marix of his example.

25 24 he ype is revealed o be raional. If D is observed, he probabiliy of he bad ype increases by a facor of 4/3, so when i firs exceeds ¼ i is a mos equal o /3. A his poin, he raional ype may reveal himself by playing U wih probabiliy, while preserving he incenive of he shorrun player o ener. I is easy o see ha his is a Nash equilibrium for any discoun facor of he long-run player, ye in his equilibrium, he long-run player's payoff is. We say ha an acion f * is sufficien for enry if, for some a <, a ³ af * implies ha here is a 2 Î B( a ) wih posiive probabiliy of enry. In he example above he friendly acion is sufficien for enry, and sends he bad signal only because of mixing ono he unfriendly acion. Tha is, he sufficien acion mixes beween a pure acion ha does no send he bad signal, and an unfriendly acion. If here is a friendly acion ha does no send he bad signal a all, hen he game is no a bad repuaion game since such an acion canno admi a empaion. More srongly, if an acion sufficien for enry does no send he bad signal a all, hen a paien raional player can do almos as well as in he absence of bad repuaion effecs. Proposiion 2: If here is an acion f * ha is sufficien for enry and does no send any bad signal, he only commimen ypes are unfriendly ypes, and he probabiliy of commimen ypes is sufficienly low, hen as d here are sequenial equilibrium payoffs for he raional ype ha approach he highes sequenial equilibrium payoff wihou commied ypes. Proof: Suppose ha he prior probabiliy of commied ypes is sufficienly low ha he shor-run players will ener when he raional ype plays f *. Then i is a sequenial equilibrium for he raional ype o play f * in he iniial period wih enry by he shor-run players. Subsequenly, if a bad signal was observed, he shor-run players say ou. If a bad signal

26 25 was no observed, he probabiliy of commied ypes is zero, and he coninuaion equilibrium is he bes possible wihou commied ypes. On he equilibrium pah, he raional ype payoff clearly approaches ha of he highes payoff wihou commied ypes, since he ges ha amoun beginning in period 2, and payoffs in period are bounded below. 4.4: Exi Minmax In paricipaion games, repuaion plays a role because he shor run players will guard agains unfriendly ypes by exiing. This is bad for he long-run player only if exi is worse han he payoff he oherwise would receive, and he exi minmax assumpion ensures ha his is he case. In paricipaion games wihou exi minmax, here are oucomes ha are even worse for he long-run player han obaining a bad repuaion. In his case i is possible ha here exis equilibria in which he long-run player is deerred from his empaion o avoid exi by he even sronger hrea of a minmaxing punishmen. For example consider he game in Figure 4, where he firs marix represens he payoffs, and he second represens he disribuion of signals condiional on enry. In Ou 2 Ou g b r F, 0,0-2,0 F ½ ½ 0 U,0 0, -2,0 U 0 0 T,0 0,0-2, T ½ 0 ½ Figure 4 This game is a paricipaion game wih exi acions Ou and Ou 2, unfriendly acion U and friendly acion F vulnerable o empaion T. There are only wo ypes, he raional ype and a bad ype ha plays U. Exi minmax fails because he maximum exi payoff exceeds he minmax payoff, and we claim ha here are good equilibria in his game because

27 26 he hrea of exiing wih Ou 2 is worse han he fear of obaining a repuaion for playing U which would only lead o exi wih Ou. To see his, consider he following sraegy profile. The raional ype plays F a every hisory unless he signal r has appeared a leas once; in ha case he raional ype plays T. The shor run player plays Ou 2 if a signal of r has ever appeared. Oherwise, he shor run player plays Ou if he poserior probabiliy of he bad ype exceeds ½ and In if his probabiliy is less han ½. Observaions of r are inerpreed as signals ha he long-run player is raional. Since ( TOu, 2) is a Nash equilibrium of he sage game, he coninuaion play afer a signal of r is a sequenial equilibrium. When r has no appeared, he long run player opimally plays F. Playing U gives no shor-run gain and hasens he onse of Ou, and playing T shifs probabiliy from he bad signal bo he signal r which is even worse. 0 The shor-run players are playing shor-run bes responses. In his equilibrium, he long run player does no give in o he empaion o play T. As a resul, wih posiive probabiliy, he shor-run players never become sufficienly pessimisic o begin exiing, and so he long run player achieves his bes payoff. In he above example here were wo exi acions. The nex proposiion saes ha when here is only one exi acion and he long-run player s exi payoff is independen of his own acion, he wors Nash equilibrium payoff for he long run player is (no much worse han) his exi payoff. Noe ha his condiion is saisfied in he principal-agen applicaions discussed in secion 5. The proposiion is a consequence of FL (992). Proposiion 3: Consider a paricipaion game wih a single shor-run player and a unique exi acion. If 0 Playing T gives probabiliy ½ of shifing o he absorbing sae where payoffs are 2. Playing he equilibrium acion of F has probabiliy a mos ½ of swiching o he sae where payoffs are 0.

28 27 (i) here exiss a pure acion â, such ha Ba ( ˆ ) { exi} =, (ii) he prior disribuion assigns posiive probabiliy o a ype ha is commied o â, and (iii) he long-run player s acion is idenified condiional on enry hen here is a lower bound on he payoffs o he raional ype which converges o he exi payoff, as he discoun facor approaches. Proof: FL (992) esablished 2 ha for any game here exiss a lower bound b( δ ) on he se of Nash equilibrium payoffs for he raional ype, and ha as δ, b( δ) converges o a limi ha is a leas where player o 2 max min 2 u ( α, α ) a C α % Ba ( ) % ( ) is he se of self-confirmed bes-responses o for he shor-run Ba a, and C is he se of acions corresponding o he suppor of he prior disribuion over commimen ypes. Because he long-run player s acion is idenified condiional on enry and Ba ( ˆ ) = { exi}, we have Ba % ( ˆ ) = { exi}, and because he ype ha plays â has posiive prior probabiliy, he FL (992) bound is a leas ˆ u ( a, exi ). For games saisfying he condiions of he proposiion, he exi minmax condiion is no necessary for bad repuaion. The wors The assumpion ha his is a pure acion is no necessary here; we sae he resul his way for consisency wih he res of he paper. 2 The saemen of he FL heorem requires ha commimen ypes including mixing ypes have full suppor, in which case he se C is he space of all (mixed) acions, bu he proof given here also shows ha he version of he lower bound given here is correc.

29 28 equilibrium coninuaion value ha he shor-run players could inflic is arbirarily close o he exi payoff and hence a paien long run player could no be deerred from his empaion o avoid a bad repuaion. 5. Poor Repuaion Games and Srong Tempaions Recall ha an acion is vulnerable o a empaion if when he shor-run players paricipae, he empaion lowers he probabiliy of all bad signals, and increases he probabiliy of all ohers. In his case he bad repuaion resul requires he exi minmax condiion, as demonsraed by he example in Secion 4.4. Noice, however, ha in he example he relaive probabiliy of g and r is changed by he empaion. If he empaion saisfies he sronger propery ha he relaive probabiliy of he oher signals remains consan, hen we can weaken he assumpion of exi minmax. In his secion we develop his resul, and give an applicaion o games wih wo acions. Firs we give a formal definiion of a srong empaion: Definiion 0: An acion a is vulnerable o a srong empaion relaive o a se of signals Y if here exiss a number r > 0 and an acion b i such ha - - ) If a Ï E, y ÎY hen ) ) ρ( y b, a ) ρ( y a, a ) ρ - - 2) If a Ï E and yy, ' ÏY ÈY e hen ρ ρ ( y b, a ) ρ( y a, a ) ( y' b, a ) ( y' a, a ) =. ρ 3) For all e Î E, u ( b, e ) ³ u ( a, e ). The acion b i is called a srong empaion. The firs and hird pars of his definiion are he same as in he definiion of a empaion; he addiional srengh comes from par (2), which requires ha he empaion no merely increase he probabiliy of all of he good signals, bu leave heir relaive probabiliies unchanged.

30 29 Noe ha srong empaion is equivalen o empaion in games in which he se Y \( Y ÈY e ) has a single elemen, for example games in which here are only wo enry signals; in paricular applies when he game of Secion 4.4 is modified so ha he only signals when enry occurs are g and r. This condiion les us prove an analog of lemma 3: and d: Lemma 4: In a paricipaion game, if a - ( h ) ÎconhullE - or a - ( h) Ï conhulle - and a0 ( h ) ³ gf for some g > 0 and friendly acion f ha is vulnerable o a srong empaion size r, hen where yîyh ( ) d r d rd v ( h ) max (- ) u (, y ) + (, y ) v ( h,) y, ìæ ö U if y Y + Î u (, y r) = ï íçè Y r ø ï uˆ ïî oherwise ì æ ö d y Y and d (, y r) + Î = í ï èç Y r ø. ï ïî d oherwise The proof, which is in he Appendix, follows ha of lemma 3, bu akes advanage of he fac ha he long-run player s coninuaion expeced value, condiional on a friendly acion, a non-exi profile, and a signal no e in Y ÈY, is he same for he equilibrium acion and he srong empaion b. Define - = - - u a a a, a Îconhull( E ) Çimage( B) uˆ max (, ) This is a bound on he long-run player s payoff when he shor-run players play exi acions ha are a bes response o some (possibly incorrec) conjecures.

31 30 Definiion : A paricipaion game is a poor repuaion game if here is an unfriendly se N, a friendly se F ha is orhogonal o N, and a se of signals Y ha are unambiguous forn, and such ha every enforceable f Î F is vulnerable o srong empaion relaive o Y. The nex resul says ha poor repuaion games have much he same consequences as bad repuaion games. Theorem 2: In a poor repuaion game of commimen size gf /2, h, ˆ limd v u. Wih lemma 4 in hand, he proof of Theorem 2 is very close o ha of Theorem, and is omied. Noice ha i is possible for a game o be boh a bad repuaion game and a poor repuaion game, and, since srong and ordinary empaion are equivalen when Y \( Y ÈY e ) is a singleon, he wo are necessarily equivalen in his case. The original EV game is such an example. Noice also ha example 4.4 in which we consruc a non-bad equilibrium has hree signals raher han wo. Wih wo signals, he game would sill fail he exi minmax condiion and fail o be a bad repuaion game, bu i would sill be a poor repuaion game, and would no admi a good equilibrium. Finally, observe he proofs of boh Lemma 3 and 4 can be generalized, so ha he difference beween he bes equilibrium payoff (in he limi as d ) and he mos favorable oucome wih exi is bounded by a scale facor imes he he produc of wo erms, namely (i) he change in relaive probabiliies induced by a empaion and (ii) he excess of he bes resul given exi over he minmax. In paricular, he bound on he difference is coninuous in he each of hese erms, so ha if eiher is small he bes equilibrium payoff for a paien long-run player can only exceed he bes exi payoff by a small amoun. We urn now o he special case of wo-player paricipaion games where here is only one signal in Y and shor-run player payoffs depend only on he signal. We focus on he case where one signal in e Y \( Y ÈY ), so ha bad repuaion implies poor repuaion. We show

32 3 ha hese games are no poor repuaion games (and by implicaion no bad repuaion games eiher). Proposiion 4: In a wo-player paricipaion game suppose here are only wo enry signals (ha is wo elemens of Y Y e ), ha he shor-run player has only wo acions, and ha he shor-run player s realized payoff is deermined by he signal. Then he game is no a poor repuaion game. Proof: Noice ha since he shor-run player has only wo acions, hey correspond o enry and exi respecively. Consequenly, he shor-run player payoff condiional on enry depends only on he disribuion over signals induced by he long-run player acion. If we normalize he shorrun player s payoff funcion so ha his exi payoff is 0, and suppose ha boh he friendly and unfriendly ses are non-empy, hen one signal yields a negaive payoff and he oher signal s payoff is posiive; call hese he bad and good signals respecively. If he game has no non-empy unfriendly se, i is no a poor repuaion game; so we can suppose here is a leas one non-empy unfriendlly se. Any unfriendly se A consiss of acions wih a sufficienly high probabiliy of sending he bad signal, and he bad signal (as a singleon se) is he only se Y ha can be unambiguous for A. Le f be he friendly acion in he (finie) friendly se ha maximizes he shor-run player's payoff. The payoff o his acion, condiional on i no generaing he bad signal wih he negaive payoff, is posiive, and since any empaion relaive o Y mus reduce he probabiliy of he bad signal, a empaion mus give he shor-run player a higher payoff han his friendlies friendly acion. For his o be rue, here mus be a pure sraegy ˆb ha gives he shor-run player a leas his same uiliy. Clearly ˆb induces enry, and since i is a pure sraegy, i mus be in he friendly se. This conradics he fac ha f was assumed o maximize shor-run player uiliy in he friendly se.