A Note on Reputation E ects with Finite Memory

Transcription

1 A Note on Reputaton E ets wt Fnte Memory Andrea Wlson and Memet Ekmek y Deember 5, 2006 Abstrat Ts paper studes reputaton e ets n a 2-player repeated moral azard game. A long-lved player, Player, would bene t f e ould ommt to playng a partular aton w s strtly domnated n te stage game. Hs opponent, wo may be eter longlved or myop, beleves tere s a small probablty tat player s a ommtment type, and ea perod observes only a nosy sgnal about player s aton. We depart from te standard lterature by assumng tat player 2 as nte memory: e s restrted to use a nte automaton, bot to arry out s own strategy, and to update s belefs about player s strategy. We sow tat ts nte memory restrton enables player to permanently mantan a reputaton as a ommtment type (n ontrast to Crpps, Malat, Samuelson s result for unbounded players, w sowed tat under mperfet montorng, reputaton e ets are only temporary). However ts reles on player 2 avng a su ently large memory, and tere are also equlbra n w player does not buld a reputaton. Extremely prelmnary and nomplete, please do not rulate Correspondene address: Harvard Unversty, Eonoms Department; Cambrdge, MA Emal: wlson3@fas.arvard.edu. y Correspondene address: Nortwestern MEDS; emal m-ekmek@kellogg.nortwestern.edu

2 Introduton Ts paper studes reputaton games wt mperfet montorng, wen te unnformed player as a bounded memory. We study a repeated moral azard game, as n te followng example: Player 2 Player L R G (; ) ( ; 0) B (2; ) (0; 0) Here G s a strtly domnated aton for player, but bot players would bene t f e ould ommt to playng G wt probablty at least 2 : We wll also assume mperfet montorng: ea perod, player 2 observes only a nosy publ sgnal of player s aton. Fudenberg and Levne s (989) orgnal reputaton result looked at te perturbaton of ts game, n w tere s a small probablty tat player s a ommtment type wo always plays G. Tey sowed tat f player s su ently patent, ten s expeted payo aganst a myop opponent must be arbtrarly lose to (te ommtment payo ) n any NE of te repeated game wt nomplete nformaton. In 992, tey sowed tat ts result s robust to te addton of nose, n an ex ante sense: for xed but su ently g < ; player s average payo as alulated at te begnnng of te game s arbtrarly lose to, even wen s atons are mperfetly observed by player 2. More reently, Crpps, Malat, and Samuelson (2004) sowed tat ts reputaton result s neverteless a sort-run penomenon: wen montorng s mperfet, player 2 eventually learns player s true type wt probablty, and ene play eventually onverges almost surely to an equlbrum of te game wt nomplete nformaton. Te ntuton s tat one player suessfully bulds a reputaton as a ommtment type, player 2 s optmal strategy must beome almost unresponsve to new sgnals about player s atons. Ts destroys player s nentves to play G, so e wll ave an nentve to devate from te ommtment strategy. Ts means tat n te long run, te dstrbuton of sgnals wll statstally dentfy player s type: ene, player 2 eventually learns tat e s fang a normal type, at w pont reputaton e ets ollapse. Ekmek (2006) sowed ow t s possble to restore reputaton e ets by restrtng te nformaton observed by player 2. He studed ratng systems : rater tan seeng te entre story of sgnals, te sort-run players are nformed about te average frequeny wt w 2

3 player ose te good aton. Tere are only a nte number of possble ratngs, w are updated and publsed by an external ageny. In ts paper, we study weter t s possble for player to develop a permanent reputaton wen s opponent as nte memory. Followng Wlson (2005), we model ts by restrtng player 2 to use an optmal nte automaton strategy. Ea perod, e does observe te sgnal about player ; owever, e annot reall te entre observed story, and must nstead use s automaton to optmally keep trak of nformaton. As n Ekmek (2006), ts mples a nte number of possble ratngs ; te d erene s tat player 2 desgns te ratng grd mself, and optmally updates t as e observes new nformaton. More presely: ea state n te automaton an be dent ed wt a belef about player s strategy, and about te story to date. A strategy for player 2 spe es an aton for ea state n te automaton, togeter wt a transton rule, w spe es ow e updates te ratng n response to a new sgnal about player s aton. We study te long-run, steady-state equlbra of repeated moral azard games n w ()player s a smple ommtment type wt probablty > 0; ()player s atons are mperfetly observed; ()player 2 as nte memory. Our man result s tat f player 2 s memory s nte but su ently large, ten t s possble for player to permanently mantan a reputaton: tere s an equlbrum n w player expets to earn s maxmal ommtment payo after any story, and player 2 s automaton strategy s a best response ndependently of s dsount fator 2 : Unfortunately ts s only a possblty result, tere are also many equlbra n w player s expeted payo s sgn antly lower. 2 Model We onsder a smple reputaton game n w two players nterat repeatedly to play te followng moral azard stage game: Player Player 2 L R G (a ; ) ( ; 0) B (a; ) (0; 0) were a > 0: Tus te unque Nas equlbrum of te stage game s (B; R); as B s a domnant strategy for P (playng te good aton G nurs a ost regardless of P2 s aton), wle player 2 ooses aton L f and only f e expets G to be osen wt probablty at least 2 : However, te assumpton a > 0 mples tat bot players would bene t f P 3

4 ould ommt to playng G wt probablty at least 2 : It wll be assumed trougout te paper tat P s a long-run patent player, wle player 2 dsounts te future at rate 2 2 [0; ]: To allow for reputaton e ets, we make te followng assumpton: Assumpton : Wt probablty > 0; P s a smple ommtment type wo plays G wt probablty after every story. Ts s te reputaton game studed n Fudenberg and Levne (989), wo establsed tat for 2 near 0, and below but su ently lose to, Player must earn arbtrarly lose to s ommtment payo (a ) n any Nas equlbrum of te repeated game. 2. Imperfet Montorng Ea perod, bot players observe Player 2 s aton, but only a nosy publ sgnal of player s aton n te stage game. We restrt to a smple bnary symmetr nformaton struture: lettng Y fgl; gr; bl; brg denote te set of sgnal realzatons, te sgnal y s ondtonally d aordng to PrfgLjGLg = PrfgRjGRg = PrfbLjBLg = PrfBRjBRg = 2 PrfbLjGLg = PrfbRjGRg = PrfgLjBLg = PrfgRjBRg = 2 ; Tus player s aton s observed orretly (g f e plays G; b f e plays B) wt probablty 2 2 ; : Note tat ts struture sats es te typal dent aton assumpton: wt su ently many observatons, Player 2 would be able to dentfy any xed stage game strategy of Player. By Abreu-Peare-Staett (990): f Player 2 s myop, ts nformaton struture redues Player s average equlbrum payo n te omplete-nformaton repeated game to at most (a ) : 2 Fudenberg-Levne (992) sowed tat for any > 0; ter 989 reputaton result s robust to te addton of nose, n te followng sense: f s below, ten n any equlbrum of te repeated game, P s ex ante expeted payo must be arbtrarly lose to (a ) for su ently lose to. More reently, Crpps-Malat-Samuelson (2004) argued tat reputaton s neverteless a sort-run penomenon: n any Nas equlbrum of te game wt > 0 and mperfet montorng, Player 2 eventually learns Player s true type wt probablty, and ene 4

5 play eventually onverges almost surely to an equlbrum of te repeated game wt omplete nformaton ( = 0): In partular, ts mples tat n te long run, player s expeted ontnuaton payo falls to at most (a ) 2 : In ts paper, we study te sustanablty of reputaton e ets wen Player 2 s memory s nte. We follow Wlson (2005) n de nng a nte-memory strategy as one w an be mplemented by a nte-state, non-determnst automaton. Te man result s tat, as n Ekmek (2006), ts nte memory restrton allows for sustanable reputaton e ets: tere are equlbra n w Player permanently mantans a reputaton for playng a strategy w would not be redble n te omplete-nformaton game. Unfortunately ts s only a possblty result, tere are also equlbra n w Player s average payo s sgn antly below s preferred ommtment payo. 2.2 Strateges and Equlbrum A beavor strategy for player s a map : [ t=0 Ht! (fg; Bg); were Ht s te set of t-perod prvate stores for player : H t = f(a 0 ; y 0 ); (a ; y ); :::; (a t ; y t )g were a t s player s realzed aton oe n perod t; and yt s te sgnal realzaton n perod t (w nludes player 2 s aton). We model Player 2 as an N-state, statonary, non-determnst automaton. A strategy for Player 2 s a trplet 2 = ( 0 ; ; d); were: 0 s te ntal memory state : N Y! (N ) s te transton rule, spefyng ow te memory N = f; 2; :::; Ng s updated after a new pee of nformaton y 2 Y: For ; j 2 N and y 2 Y; let y ;j (; y)(j) denote te probablty of a transton! j after sgnal realzaton y 2 Y: d : N! fl; Rg spe es an aton oe, as a funton of te urrent memory state 2 N : Note tat player 2 s automaton strategy s requred to be statonary: every tme e s n state 2 N ; e uses te same aton and transton rule. Te nterpretaton s tat player 2 s memory state 2 N represents all of te nformaton avalable to m; e an use ts nformaton, and understandng of te rule ; to make nferenes about te story, but annot 5

6 reall exatly w story e as observed. automata, and expltly fous on equlbrum steady states. We wll also restrt attenton to rreduble More presely: let fn; g denote te two possble types for Player, were n s te normal type wo plays strategy ; and s te ommtment type wo plays G wt probablty every perod. Let f ; f n denote te steady-state probabltes tat player 2 s n memory state ; ondtonal on P beng type ; n (respetvely). It s stragtforward to determne te dstrbuton ondtonal on type : namely, f te soluton to te followng N N system of equatons: 8j 2 N : f j = X fj2n jd(j)=rg f C gr ;j + ( )br ;j + X fj2n jd(j)=lg f C j gl ;j + ( Te dstrbuton ondtonal on type n s te soluton to a smlar system of equatons: f j = X f2n jd()=rg f C p gr ;j + ( p );j br + X f2n jd()=lg f C p gl ;j + ( p );j bl )bl ;j were p s te probablty (long-run frequeny) of a g-sgnal wen Player 2 s n state ; ondtonal on type n: Ten n memory state ; Player 2 beleves tat e s fang a ommtment type wt probablty f f + ( )f n and e expets to observe a g-sgnal wt probablty + ( )p : Player s problem s standard: de ne " E ( ; 2 ) ( ) # X t u (a t ; a t 2) j H t as s expeted ontnuaton payo ondtonal on H t ; were fht } t= t= s te ltraton on (A Y ) ndued by prvate stores for player, u () s player s stage game payo funton, and expetatons are taken wt respet to te probablty dstrbuton over (A Y ) ndued by ( ; 2 ); note tat player does not dretly observe player 2 s memory state. Say tat s a best response to 2 f for all beavor strateges 0 : " # " # X X ( ) t u (a t ; a t 2) E (0 ; 2) ( ) t u (a t ; a t 2) E ( ; 2 ) t= t= 6

7 For Player 2: say tat 2 s a best response to f:. Gven te aton rule d : ( 0 ; ) maxmzes player 2 s average expeted payo, X [f + ( )f n ( () + ( )( ))] f2n jd()=lg were s te long-run frequeny wt w type n plays G wen player 2 s n memory state (te orrespondng probablty of a g-sgnal s p = + (2 ) ). 2. In ea memory state : d() maxmzes player s expeted ontnuaton payo, gven te state- belefs about player s strategy, and usng te ontnuaton payo s ndued by ( ; 2 ): Ts s equvalent to sayng tat player 2 as no nentves for one-sot devatons: gven te belefs and ontnuaton payo s mpled by s strategy, tere s no memory state n w e wses to devate from s presrbed aton and transton rules. De nton: An equlbrum of te game wt nomplete nformaton s a par ( ; 2 ) su tat s a best response to ; for 2 f; 2g: 3 Sustanable Reputatons Wt nte memory and nosy sgnals, t s mpossble for player 2 to be onvned tat e s fang a partular type of player : for all 2 N ; and for any strategy par ( ; 2 ); PrfCjg s bounded away from bot 0 and. Te fat tat PrfCjg s bounded above 0 mples tat permanent reputatons may be possble: n ontrast to te Crpps-Malat-Samuelson result wt unbounded players, t s no longer true tat an n nte number of devatons by Player from te ommtment strategy wll lead Player 2 to statstally dentfy m as te normal type. Hene, provded tat ; N are not too low, t s possble to onstrut an automaton w s optmal for player 2, yet provdes player wt nentves to play G often enoug to mantan a reputaton. Te d ulty s te upper bound max max 2N PrfCjg on player 2 s belefs, w depends on bot te ex ante pror and on te number of memory states. Proposton says tat f ; N are too low, ten tere are no reputaton e ets: player 2 an never beome onvned enoug of te ommtment type for player to bene t from reputaton-buldng: 7

8 q N Proposton : If < ; ten tere are no reputaton e ets:. If 2 s su ently lose to zero, ten n any NE: lm! E ( ) P t= t u (a t ; at 2 ) (a ) 2. For any 2, tere s a NE of te game wt nomplete nformaton n w Player s expeted payo s 0. Reall tat n te game wt omplete nformaton ( = 0), te set of NE payo s for player s gven by [0; (a ) ] (wenever te upper bound exeeds 0). Tus Proposton 2 establses tat f s too low, relatve to N and te nformatveness of te sgnal, ten te unertanty about player s type as no equlbrum e et: part () says tat player s payo annot exeed te upper bound of te set of omplete-nformaton NE payo s, and part () says tat tere s an equlbrum n w player expets to earn te worst possble payo of te omplete-nformaton game, 0. Te proof s n te appendx. Note tat, ontrary to reputaton games wt unbounded players, te ex ante pror does matter. For te range gven n Proposton, t s mpossble for player 2 to beleve tat e s fang a ommtment type wt probablty greater tan 2 : te proof smply sows tat for any automaton, and any player strategy ; player 2 wll always beleve s opponent s most lkely te normal type. At te oter extreme, a very g ntal pror q > N would mply tat for any strategy par ( ; 2 ); player 2 always beleves tat s opponent s most lkely a ommtment type: at ts pont te unquely optmal strategy for player 2 s to play L after all stores, regardless of ; and so player earns expeted payo a: (He plays B every perod, but neverteless playelr 2 s onvned tat e s fang a ommtment type). For te range n between, tere are multple equlbra. Proposton 2 desrbes te upper bound on Player s payo : we sow tat f te sgnal s not too nosy, and f player 2 s memory s nte but su ently large, ten tere s a NE n w player earns te maxmum possble ommtment payo, a 2 : (ts s wat e would obtan by publly ommttng to play G wt probablty 2 every perod): Proposton 2: For any > 0, > a ; and " > 0; tere exsts,n su tat wenever and N N (but nte): tere s a NE ( ; 2 ) su tat:. Player s expeted payo sats es E ( ) P t= t u (a t ; at 2 ) a 2 ". Player 2 s strategy 2 s optmal for any 2 2 [0; ] 8

9 3. Sket of proof: Consder te followng strategy 2 for player 2: In any state 6= ; N : Player 2 plays L w.p., moves up! + w.p. after a g-sgnal, and down! w.p. after a b-sgnal. In state N : Player 2 plays L w.p. a a ; stays n state N after a b-sgnal, jumps down to state N 2 after a g-sgnal. In state : Player 2 plays L w.p. a a 2 ( ) 2 ; stays n state after a b-sgnal, and moves to state 3 after a g-sgnal. (Note: ts probablty s postve for > a ): Note tat all transtons n ts automaton are determnst, so as long as tere are no devatons, player knows player 2 s memory state. Next, de ne V as player s expeted ontnuaton payo, ondtonal on (orretly belevng tat) player 2 beng n state 2 N : Now onsder te followng statonary strategy for player : As long as no devatons deteted by player 2: play G wt probablty wen e s beleved to be n state 2 N f any devaton s deteted (player 2 plays R wen expeted to play L, or ve versa), swt permanently to te strategy of playng B wt probablty after any story. We wll sow ere tat f player 2 follows te above strategy 2 ; ten ()player strtly prefers to play B wen player 2 s n state N, but s nd erent between playng G,B n all oter states; ()any su strategy ( ) 2N wt N = 0 earns expeted payo a 2: Ts mples tat te spe ed strategy for player s ndeed a best response to 2 ; and earns te maxmal ommtment payo a 2 : We sow n te appendx tat t s also possble to oose te s su tat 2 s a best response for player 2, for any 2 2 [0; ]: So, to prove optmalty for player : f player 2 follows 2 ; ten player s expeted ontnuaton payo s satsfy: lm! lm!! VN VN = V + V V lm! 3 V = a = ( p ) p lm! lm! V + N lm! V N () p N V V (a lm! V ) (3) p a 2 ( ) 2 (3) p 9

10 (For example, te rst equaton s obtaned by rearrangng, and takng lmts as! ; te followng equaton: V N = ( ) a N + p N V N + ( p N )V N Te term n square brakets s te expeted urrent-perod payo, and ten after a g-sgnal (probablty p N ) player 2 moves to state N ; after a b-sgnal e stays n state N): If we set N = 0; ten solvng tese equatons yelds, for all 2 N : lm! lm V n = a! 2 V = 2 2 V + Te rst lne mples tat player ndeed earns s maxmal ommtment payo a 2 wt ts strategy pro le. To verfy tat player s strategy s optmal: t s lear tat e must play B wt probablty n state N; sne playng G s bot ostly now, and yelds a worse expeted ontnuaton payo (t nreases te probablty of a g-sgnal, after w player 2 moves from state N to state N ; were VN < V N ): Te rest of s strategy s optmal f we an sow tat e s nd erent between playng G,B n all states 6= N: So onsder rst any state 6= ; N : ere, player 2 moves to state + after a g-sgnal, and to state after a b-sgnal. Player s ten nd erent between playng G,B f te expeted gan n s ontnuaton payo from playng G, (2 ) V + V ; exeeds te ost ( ); ts olds n te lmt as! ; as we ave above tat V+ V lm! = 2 Smlarly, e s nd erent between playng G,B n state, sne player 2 stays n after a b-sgnal, moves to 3 after a g-sgnal, and our automaton yelds lm! V 3 V = 2 : So, any strategy for player w sets N = 0 s a best response to 2 ; and earns te maxmal ommtment payo a 2 : It remans to prove tat we an oose (p ) 6=N su tat 2 s a best response for player 2; ts s sown n te appendx. 4 Unwrtten Unappealng Results Stll workng on lower bounds wen bot players are patent, tat s ; 2 bot near. In te semnar, I wll sket proofs of te followng possble teorems, lookng for opnons on w are wort wrtng up: 0

11 Assumng tat publ randomzaton s not allowed: f exeeds te bound n Proposton, ten n any NE of te game wt nomplete nformaton, lm V! 2 (reasonable bound for nosy sgnals, useless as! ) If player s also an automaton, wt te same # states as player 2: s mnmum payo lmbs farly qukly from te above bound. If player unbounded (but stll no publ randomzaton), an onstrut equlbra wt te above payo for any If we allow publ randomzaton: no lower bound, get a folk teorem 5 Conluson to be wrtten... A Appendx A. Proof of Proposton : Frst, to sow tat tere s a NE of te game wt nomplete nformaton n w player s expeted payo s 0: Let be te beavor strategy aordng to w P plays B wt probablty after every story. Ten Player 2 s problem s to oose a transton rule X f2n jd()=lg [f ( )f n ] togeter wt a deson rule satsfyng d() = L PrfCjg PrfNjg: Ts s dental to q N te problem studed n Wlson (2005), w establsed tat f <, ten te upper bound on Player 2 s expeted payo s 0, attaned by an automaton w plays R n all memory states. Gven ts automaton, t s ndeed a best response for player to play B after any story. Next, to sow tat 2 near 0 mples tat player s average dsounted payo annot exeed (a ) 2 (te maxmum NE payo n te game wt omplete nformaton), we rst alulate bounds on player 2 s belefs about te type of player. Fx strateges ( ; 2 ); f and order te states n N su tat te ndued belefs f n are weakly nreasng n : De ne as te total probablty of an! j transton ondtonal on type s 2 f; ng : eg for a state s ;j

12 wt d() = L; ;j = gl ;j + ( fj s j; s ; we ave for all 2 N : P j P j f j j; f n j n j; )bl ;j : Rearrangng te steady-state equatons f s = f P j6= ;j f n P j6= ;j n P j+ f j j; P j+ f j n j; n = P j Note also tat ;j n ;j tat te LHS above s at most f N fn fn n fn 2 fn n 2 fn fn n f n N for all ; j: For = N; our orderng of te states mples ; wle te RHS s at least f N f n N : ene, we ave : Substtutng ts nto te equaton for N : te LHS above s at most ; wle te RHS s at least f N fn n ; ene, we ave f N fn n f N 2 fn n 2 ts argument: fn fn n 2 fn f n N ::: f f n 2(N ) : Iteratng Moreover, te orderng of te states mples f f n : (It s not possble tat all states n N are reaed more frequently ondtonal on tan n): Hene, for any strategy par ( ; 2 ); we ave: Prfjg max 2N Prfnjg = fn fn n If te bound n te proposton olds, ten ts s below. 2(N ) Fnally, let 2 N be te memory state n w player s ontnuaton payo s gest. De ne V ( ) E[( ) P t= t u (a t ; at 2 ) j ] as player s expeted ontnuaton payo, ondtonal on knowng : If d( ) = R ten we are done, as ts mples tat player s expeted payo s at most 0. So, assume tat d( ) = L: Sne we sowed above tat player 2 always beleves e s more lkely to be fang a normal type, d( ) = L an only be optmal for a myop player 2 f e expets te normal type to play G wt su ently g probablty n : For player to want to play G n ; we need (2 ) g V ( ) max j6= V (j) ; b ; () To sow tat ts s mpossble wen player s payo s too g: let tat player plays G n ; and p = + (2 ) denote te probablty te mpled probablty of a g-sgnal n state : Ten we ave, V ( ) ( )(a )+V ( ) p ( g ; ) + ( p )( b ; ) V ( ) max V (j) j6= w rearranges to V ( ) lm! max j6= V (j) a lm! V ( ) p ( g ; ) + ( p )( b ; ) 2

13 So for () to old, we need g ; b ; a lm V ( )! 4 = 2 p ( g ; ) + ( p )( 3 b ; ) 2 " ( )( g ; ) + ( b ; ) # + b 2 ; g ; 5 So, b ; a lm V ( )! 2 > ( g;) a lm V ( ) +! 2 Te RHS s non-negatve (te payo annot possbly exeed a); so ts requres lm V ( ) a = (a )! 2 2 as desred. (Ts proof s nomplete beause te last part of te argument reles on player knowng wen player 2 s n state : ) A.2 Proof of Proposton 2 A.2. Steady-State Probabltes It wll be useful to alulate te steady-state dstrbuton over N ; ondtonal on player s type and on ( ; 2 ): De ne p s as te probablty of a g-sgnal wen player 2 s n state 2 N ; ondtonal on player s type s 2 f; ng: De ne f s as te steady-state probablty of 2 N ondtonal on s: If player 2 follows te transton rule spe ed by 2 ; ten f s s te soluton to te followng system of equatons: = N : fnp s s N = fn s p s N 4 N : f s = f s p s + f+( s p s +) = 3 : f3 s = fp s s + f2p s s 2 + f4( s p s 4) = 2 : f2 s = f3( s p s 3) = : fp s s = f2( s p s 2) (For example: ea perod, player s n state f eter e was already ere te prevous perod and observed a b-sgnal, or f e was n state 2 and observed a b-sgnal: ene, f s = 3

14 f s( ps ) + f 2 s( ps 2 ): Solvng ts system reurvsely yelds: 3 N : f s = NY 2 j= p s j+ p s j p s N p s fn s N f s 2 = ( p s 3)f s 3 f s = ( ps 2 ) p s ( p s 3)f3 s Wrte all of tese n terms of f s N and use te fat tat probabltes sum to, to solve for f s : Under te ommtment strategy, p = 8 2 N ; ts mples: fn C = 4 N 2 5 ; f C = N N () For te normal type: reall tat as long as tere are no devatons, player always knows player 2 s state 2 N ; e must play B n state N; so p n N = ( ); but s wllng to oose any probabltes p n n te remanng states. For future referene: f e sets p n = 2 8 6= ; N ; N; ten we obtan f n N = f N = + p n N + p n N + 2( p n N )(N 4) ( ) p n N 2p n p n N p n + 2( p n N )(N 4) p n ; (2) A.2.2 Optmalty for Player 2 We need to oose (p n ) 2N su tat: PrfjNg = 2 : ts mples tat player 2 s nd erent between playng L; R (ene wllng to randomze) n state N; gven tat e expets te normal type to play B ere Player 2 s nd erent between L; R ondtonal on state (to be wllng to randomze n state ) Player 2 s nd erent between L; R ondtonal on observng a b-sgnal n state 2 (Ts s requred for optmalty of te 2! transton, 2 b = : note tat transtons out of states,2 are dental (go to 3 after a g-sgnal, after a b-sgnal), so movng from 2 to only a ets te probablty of playng L n te subsequent perod) 4

15 If tese ondtons are sats ed, and p n 2 8 6= ; N; ten player 2 s strategy s optmal for any 2 : To see ts: n all states 6= ; N; e s supposed to play L : ts s a myop best response to (sne normal P plays G ere wt probablty at least 2 ); and a strt myop best response to te ommtment type s strategy. Te above ondtons guarantee tat player 2 s aton oe s also a myop best response at all oter nformaton sets (states,n, and wen rst movng nto state ). Terefore, any one-sot devaton n te aton an only redue te urrent-perod payo, and may trgger te permanent punsment pase by Player. A one-sot devaton n te transton only matters f t anges te sgnal sequenes tat take player 2 to states,n: and n ts ase, agan te result s tat e wll play R (wt postve probablty) wen supposed to play L wt probablty, trggerng a permanent swt by player to te strategy of always playng B. So, tere are no nentves for one-sot devatons. It s also stragtforward to sow tat player 2 s expeted payo n ts equlbrum exeeds s payo from optmzng aganst te belef tat type n always plays B (e, ount on trggerng a devaton and desgn te orrespondng optmal automaton). To sow tat t s possble to satsfy te above ondtons for N su ently g: te rst ondton, aton nd erene n state N, requres PrfjN g PrfnjNg fn fn n = (3) For te seond and trd ondtons (aton nd erene n, and after observng a b-sgnal n state 2) to old, player must play G wt a slgtly lower probablty after te rst b-sgnal n state 2, tan after two or more onseutve b-sgnals startng n state 2. (Te probablty of a ommtment type s lower n te latter ase, so we need to nrease te probablty tat te normal type plays G to keep player 2 nd erent). More presely: after every story n w player (orretly) beleves tat player 2 s n state 2, let m play G wt probablty 0 after te rst b-sgnal, and wt probablty after ea subsequent onseutve b-sgnal. Also de ne p 0 ; p as te probabltes of a g-sgnal ndued by 0 ; (e 0 = +(2 )0 ): Ten te long-run frequeny wt w player s type n and plays G wen player 2 s n state s gven by: ( )f N ( )f N 2 ( p 2 ) 0 + ( p 0 ) + ( p 0 )( p ) + ::: = f N 2 ( p 2) p And te probablty tat player 2 s n state ondtonal on type n s: ( ) 0 + f n (f n 2 ( p 2 ) + ( p 0 ) + ( p ) + ( p ) 2 + ::: = f n 2 ( p 2) p + (2 ) 0 5

16 Condtonal on beng n state, player 2 s ten nd erent between playng L; R te total probablty tat player plays G s 2 : f C + ( )f n f C + ( )f n = 2, f C ( )f2 n( p 2) = 0 + (2 ) Smlarly, ondtonal on beng n state 2 and observng a b-sgnal, e s nd erent between L; R f: Solvng, we need: f C + ( )f 2 n( p 2) 0 f C + ( )f 2 n( p 2) = 2, f C ( )f n 2 ( p 2) = ( 20 ) f C ( )f n 2 ( p 2) = ( 20 ) and = 2 So, for player 2 s strategy to be optmal, t su es to oose (p ) su tat p 2 8 6= ; N ; N; and equatons (3),(4) are sats ed. For example, f e sets p n = 2 (4) 8 6= ; N; ten substutng (),(2) nto (3),(4), we need: (2 ) + 2( ) + (N 4) p n N 2 = (3a) (2 ) N ( ) + (N 4) p n 7 4 ( ) N 2 5 = 2 0 (4a) 3 were p n s te average probablty of a g-sgnal ondtonal on state and type n : namely, te soluton to p n = f 2 n( p 2) f n ; w at = 2 s pn = : Te LHS of (3a) goes +2 2(2 ) 0 to n nty as N! ; wle te RHS of (4a) goes to 0; sne we an oose 0 arbtrarly, and are also free to nrease any p n for 6= N (note tat at p n = te ommtment and normal strateges are dental, so te LHS and RHS of (3a),(4a) would be very lose to te ex ante pror ); t s always possble to satsfy tese equaltes for N su ently large. For example: at = :95 and = :05; we need ; equaton (3a) needs N = 205 and 0 very lose to (slgtly below) 2 : Te mnmal N requred s strtly dereasng n bot and : Ts ompletes te proof, as optmalty for player was sown n te text. Referenes [] Abreu, Peare, and Staett (990), Toward a Teory of Dsounted Repeated Games wt Imperfet Montorng, Eonometra, 58,

17 [2] Celentan, Fudenberg, Levne, and Pesendorfer (996), Mantanng a Reputaton aganst a Long-Lved Opponent, Eonometra, 64, [3] Crpps, Malat, and Samuelson (2004), Imperfet Montorng and Impermanent Reputatons, Eonometra, 72, [4] Fudenberg and Levne (989), Reputaton and Equlbrum Seleton n Games wt a Patent Player, Eonometra, 57, [5] Fudenberg and Levne (992), Mantanng a Reputaton wen Strateges are Imperfetly Observed, Revew of Eonom Studes, 59, [6] Fudenberg, Levne, and Maskn (994), Te Folk Teorem wt Imperfet Publ Informaton, Eonometra, 62, [7] Ekmek (2006), Sustanable Reputatons wt Ratng Systems, workng paper [8] Wlson, A (2005), Bounded Memory and Bases n Informaton Proessng, stll under revson 7