The RepeatedPrisoners DilemmawithPrivate Monitoring: a N-player case

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1 The RepeatedPrsoners DlemmawthPrvate Montorng: a N-player case Ichro Obara Department of Economcs Unversty of Pennsylvana obara@ssc.upenn.edu Frst verson: December, 1998 Ths verson: Aprl, 2000 Abstract Ths paper studes the repeated prsoners dlemma wth prvate montorng for arbtrary number of players. It s shown that a mxture of a grm trgger strategy and permanent defecton can acheve an almost e cent outcome for some range of dscount factors f prvate montorng s almost perfect and symmetrc, and f the number of players slarge. Thsapproxmate e cecny result also holds when the number of players s two for any prsoners dlemma as long as montorng s almost perfect. A detaled characterzaton of these sequental equlbra s provded. 1. Introducton Ths paper examnes the repeated prsoners dlemma wth arbtrary number of players, where players only observe prvate andmperfectsgnals about the other players actons. Ths game belongs to the class of repeated games wth prvate montorng. Whle repeatedgames wthpublc montorng have beenextensvely analyzedn, for example, Abreu, Pearce and Stachett [1] or Fudenberg, Levn, and Maskn [8], few thngs are known about repeated games wth prvate montorng. It s shown n Compte [5] and Kandor and Matsushma [10] that a folk theorem stll holds n ths class of game wth communcaton between players, but t s d cult to analyze t wthout communcaton because the smple recursve structure s lost. The two player prsoner s dlemma was already examned n Sekguch [16], whch s the rst paper to show that the e cent outcome can be achevedn some repeated prsoner s dlemmas wth almost perfect prvate montorng. Ths paper s an extenson of Sekguch [16] n the sense that (1): a smlar grm trgger strategy s employed, I am grateful to my advsers George Malath and Andrew Postlewate for ther encouragement and valuable suggestons, and Johannes Horner, Stephen Morrs, Ncolas Persco, Tadash Sekguch for ther helpful comments. All remanng errors are mne.

2 (2): the e cent outcome s obtaned for any prsoner s dlemma wth two players, (3): ths e cency result for the two player case s extended to the case of arbtrary number of players wth some symmetrc condton on the montorng structure, and (4): the sequental equlbrum correspondng to ths grm trgger Nashequlbrum s explctly constructed. In Sekguch [16], the crtcal step of the arguments s to obtan the unque optmal acton wth respect to a player s subjectve belef aboutthe other player s contnuaton strategy. Snce players randomze between a grm trgger strategy and a permanent defectonn the rstperod, eachplayer s contnuatonstrategy s always one of these two strateges after any hstory. Ths means that a player s subjectve belef about the other player s strategy can be summarzed n one parameter: a subjectve probablty of the permanent defecton beng played by the other player. In ths paper, the clear cut characterzaton of the optmal acton s provded as a functon of ths belef. In Sekguch [16], t s shown that a player should start defectng f she s very con dent that the other player has started defectng, and a player should cooperate f she s really con dent that the other player s stll cooperatng. However, t s not clear what a player should do f the belef s somewhere n the mddle. Ths paper lls n that gap. Through a detaled examnaton of players ncentve, we also nd that the payo restrcton mposed n Sekguch [16] s not really necessary, that s, the approxmate e cency result s vald for any prsoner s dlemma game. Although the same knd of clear characterzaton of the optmal acton s possble wth many players, t s not straghtforward to extend ths e cency result to the n player case. The dynamcs of belef s rcherwthmore thantwo players. Inpartcular, t s possble to have a belef that some player started defectng but the other players are stll cooperatng. In such a case, a player mght thnk that t s better to contnue cooperatng because t mght keep cooperatve players from startng defecton. So, t s no longer clear when players should pull the trgger. Under the assumpton that the probablty of any sgnal pro le only depends on the number of the total errors t contans, t s shown that the e cency outcome can be supported wth the mxture of the permanent defecton and a certan knd of grm trgger strategy, where players start defectng f they observe even one sgnal of devaton by any other player. Ths strategy generates an extreme belef dynamcs under the assumptonon the sgnal dstrbuton, whch nturn ratonalzes the use of ths strategy. As soon as a player observes any bad sgnal from any other player, the player expects that some other players also got some bad sgnals wth hgh probablty. Then, she becomes pessmstc enough to start defectng for herself because defecton should preval among all players usng the same strategy at least n the next perod. A sequence of papers have re ned the result of Sekguch [16] for the two player case. Pccone [15] also acheves the e centoutcome forany prsoners dlemma wth two players and almost perfect prvate montorng. Moreover, he establshes a folk theorem for a class of prsoners dlemma usng a strategy whch allows players to randomze between cooperaton and defecton after every hstory. The strategy used n hs paper can be represented as an automaton wth countably n nte states. Ely and Välmäk [7] prove a folk theorem usng a smlar strategy, but ther strategy 2

3 s smple n the sense that t s a two states automaton. Bhaskar [3] s closest to ths paper n terms of results and strateges employed n the two player case. He essentally shows (2) and (4), and also proves a folk theorem for a class of prsoner s dlemma through a d erent lne of attack from Pccone [15] or Ely and Välmäk [7] usng a publc randomzaton devce. Malath and Morrs [11] s the rst paper to deal wth the n player case n the prvate montorng framework. They show that a subgame perfect equlbrum wth publc montorng s robust to the ntroducton of prvate montorng f players contnuatonstrateges are approxmately commonknowledge after every hstory. A folk theorem can be obtaned when nformaton s almost publc and almost perfect. Although the stage game n ths paper has a more spec c structure, the nformaton structure allowed n ths paper s not nested n ther nformaton structure. Especally, prvate sgnals can be ndependent over players n ths paper. Ths paper s organzed as follows. In Secton 2, the model s descrbed. In Secton 3, the assumptons on the nformaton structure are presented. Secton 4 dscusses the optmal acton wth respect to player s belefs and the belef dynamcs generated by the equlbrum strategy proposed n ths paper. A sequental equlbrum s constructed n Secton 5. Secton 6 gves a detaled characterzaton of the sequental equlbrum constructed n Secton 5. Secton 6 concludes wth a comparson to other related papers. 2. The Model Let N = f1;2;:::;ng be the set of players and g be the stage game played by those players. The stage game g s as follows. Player chooses an acton a from the acton set A = fc;dg : Actons are not observable to the other players and taken smultaneously. A n-tuple acton pro le s denoted by a 2 A = n =1 A : A pro le of all player s actons but player 0 s s a 2 j6= A j : Each player receves a prvate sgnal about all the other players actons wthn that perod. Let! = (! ;1 ;:::;! ; 1 ;! ;+1; :::;! ;n ) 2 fc;dg n 1 = be a generc sgnal receved by player ; where! ;j stands for the sgnal whch player receves about player j 0 s acton: A generc sgnal pro le s denoted by! = (! 1 ;:::;! n ) 2. All players have the same payo functon u. Player s payo u(a ;! ) depends on her own acton a and prvate sgnal!. Other players actons a ect a player s payo only through the dstrbuton over the sgnal whch player receves. The dstrbuton condtonal on a s denoted by p(!ja). It s assumed that p(!ja) are full support dstrbutons, that s, p(!ja) > 0 8a8!: The space of a system of full support dstrbutons fp(!ja)g a2a s denoted by P: Snce we are nterested n the stuaton where nformaton s almost perfect, we restrct attenton to a subset of P where nformaton s almost perfect. Informaton s almost perfect when every person s sgnal pro le s equal to the actual acton pro le n that perod wth probablty larger than 1 " for some small number ": To sum up, the space of the nformaton structure we manly deal wth n ths 3

4 paper s the followng subset of P: ( P " = fp(!ja)g a2a 2 < n (n 1) 2n ++ p(!ja) > 1 " f! = a for all ; and 8a, P p(!ja) = 1! ) and p " s a generc element of P " : We also ntroduce the perfectly nformatve sgnal dstrbuton P 0 = fp 0 (!ja)g a2a, where, for any a 2 A; p 0 (!ja) = 1 f! = a for all. The whole space of the nformaton structure P S P 0 s endowed wth the Eucldean norm. The stage game payo only depends on the number of sgnals C and D a player receves. Let d(! ) be the number of D s contaned n! :Then, u(a ;! 0 ) = u(a ;! 00 ) f d(! 0 ) = d(! 00 ) for any a : Let u a ;D k be the payo of player when d(! ) = k: The devaton gan when k defectons are observed s M (k) = u D;D k u C;D k > 0: The payo s P u(c;! ) p(!j(c;:::;c)) and P u(d;! )p(!j(d;:::;d)) are normalzed to 1 and 0 respectvely for all : It s assumed that (1;::;1) s an e cent stage!! game payo. The largest devatongan andthe smallest devaton gan are M and M respectvely, where M = max M (k) and M = mn M (k) 1 k n 1 1 k n 1 The stage game g s repeated n ntely many tmes by n players, who dscount ther payo s wth a common dscount factor ± 2 (0;1): Tme s dscrete and denoted by t = 1;2;::::Player s prvate hstory s h t = a 1 ;!1 ;:::; a t 1 ;! t 1 for t = 2 and h 1 = ;: Let H t be the set of all such hstory h t S and H = 1 H: t Player s strategy s a sequence of mappngs ¾ = (¾ ;1 ;¾ ;2 ;::::); each ¾ ;t beng a mappng from H t to probablty measures on A : Dscounted average payo s V (¾ : p;±) = P (1 ±) 1 ± t 1 E [u((a t ;!t ))j¾;p]; where the probablty measure on Ht s generated t=1 by (¾;p): The least upper bound and largest lower bound of the dscounted average payo are denoted by V and V respectvely. For ths n player repeated prsoner s dlemma, the grm trgger ¾ C and the permanent defecton ¾ D are de ned as follows: ½ C f h ¾ C (h t ) = t = ((C;C);:::;(C;C)) or t=1 D otherwse ¾ D (h t ) = D for all ht 2 H We also use ¾ C or ¾ D for any contnuaton strategy whch s dentcal to ¾ C or ¾ D, that s, any contnuaton strategy from the perod t+1 such that ¾ h t+k = ¾a h k for k = 1;2;::: and a = C or D: Moreover, any contnuaton strategy whch s realzaton equvalent to ¾ C or ¾ D s also denoted by ¾ C or ¾ D respectvely 1. Ths grm trgger strategy s the harshest one among all the varatons of the grm trgger strateges n the n player case. Players usng ¾ C swtch to ¾ D as soon as they observe any sgnal pro le whch s not fully cooperatve. When player s mxng ¾ C and ¾ D wth probablty (q ;1 q ); ths strategy s denoted by q ¾ C + (1 q )¾ D :. 1 A strategy s realzaton equvalent to another strategy f the former generates the same outcome dstrbuton as the latter ndependent of the other players strateges. t=1 4

5 Suppose that ¾ C or ¾ D s chosen by all players. Let µ 2 = f0;1;:::;n 1g be the number of players usng ¾ D as a contnuaton strategy among n players. Then a probablty measure q (h t ;p) 2 on the space = f0;1;:::;n 1g s derved condtonal on the realzaton of the prvate hstory h t : Clearly, ths measure also depends on the ntal level of mxture between ¾ C and ¾ D by every player, but ths dependence s not shown explctly as t s obvous. In the two player case, a player s strategy can be represented as a functon of belef, usng the fact that the other player s always playng ether ¾ C or ¾ D on and o the equlbrum path. Here the space of the other player s types s much larger. However, there s a convenent way of summarzng nformaton further. We classfy nto the two sets: f0g and f1;:::;n 1g; that s, the state no one have ever swtched to ¾ D and the state where there s at least one player who has already swtched to ¾ D. Player 0 s condtonal subjectve probablty that no player has started usng ¾ D s denoted by Á(q ) = q (h t ;p)(0): The reason why we just focus on ths number s because the exact number of players who are playng permanent defecton does not make much d erence n what wll happen n the future gven everyone s strategy. As soon as someone starts playng ¾ D, every other player starts playng ¾ D wth very hgh probablty from the very next perod on by the assumpton of almost perfect montorng. What s mportant s not how many players have swtched to ¾ D ; but whether anyone has swtched to ¾ D or not. Let V (¾ ;µ : p;±) be player s dscounted average payo when µ other players are playng ¾ D and n µ 1 other players are playng ¾ C : Ths notaton s just ed under the assumpton of the symmetry dstrbuton, whch s ntroduced n the next secton. Fnally, we need the followng notatons: V (¾ ;q : p;±) = M (q ;p) = n 1 X µ=0 V (¾ ;µ : p;±)q (µ) n 1 X U D;D µ : p U C;D µ : p ª q (µ) µ=0 3. Informaton Structure In ths secton, varous assumptons on the nformaton structure are proposed and dscussed. In the followng sectons, a sequental equlbrum s constructed wth a mxture ofgrmtrggerstrategy andpermanentdefecton, whchacheves anapproxmately e cent outcome for some range of dscount factors. As s the case wth any equlbrum based on smple grm trgger strateges, ths equlbrum sats es the followng property; players stck to the grm trgger strategy as long as they have an optmstc belef about the others, and they swtch to the permanent defecton once 2 Here, q s a mappng from the space of prvate hstory (and the montorng technology p) to a space of probablty measures on : However, we also use q as a pont n a n 1 dmensonal ½ smplex fp l g n 1 l=0 2< +j n 1 ¾ P p l =1 ; where p l means the probablty of the eventµ=l: l=0 5

6 they become pessmstc and never come back. Ths property s sats ed n games wth perfect montorng, but not so easly sats ed n games wth mperfect prvate montorng. In order to acheve a certan level of coordnaton, whch s necessary for an equlbrum wth trgger strateges, we mpose some assumptons on p(!ja) n addton to the assumpton that t s almost perfect. The rst assumpton, whch s mantaned throughout ths paper, s Assumpton 1 p(!ja) = p! () (j) j a (1) ;:::;a () ;:::;a (N) for any permutaton : N! N: Ths assumpton mples the followngs. Frst, each player s expected payo s the same under the same stuaton combned wth the assumpton on u: Second, only the number of C and D played matters n terms of expected payo. Fnally, the dynamcs of belef for each player s the same wth the same sort of prvate hstory. Ths assumpton allows us to treat agents symmetrcally and just es all the notatons ntroduced n Secton 2. Although Assumpton 1 s strong enough to acheve an almost e cent outcome for two players, a stronger assumpton s called upon to acheve smlar results wth more than two players. Let #(!ja) stand for the number of errors n!: The followng assumptons s strong enough for that purpose: Assumpton 2. p(! 0 ja) = p(! 00 ja) f #(! 0 ja) = #(! 00 ja) for any! 0 ;! 00 2 ;a 2 A A couple of remarks on these assumptons are n order. Frst, Assumpton 1 s a relatvely weak assumpton about the symmetry of a sgnal dstrbuton and sats ed n most of the papers n reference whch analyze the repeated prsoner s dlemma wth almost perfect prvate montorng. Second, whle Assumpton 2 s much stronger than Assumpton 1 n general, t s very close to Assumpton 1 n the two player case. Consequently, ths assumpton s also sats ed n those papers as most of them concentrate on the two player case. Assumpton 2 means that the probablty of some sgnal pro le only depends on the number of errors contaned n that pro le. For example, gven that everyone s playng C; the probablty that a player receves two D sgnals whle the other players get correct sgnals s equal to the probablty that two players receve one D whle the rest of the players gets correct sgnals. 6

7 n: For example, the followng nformaton structure sats es Assumpton2 for general ² Example: Totally Decomposable Case p(!ja) = j 6=j p(! ;j ja j ) for all a 2 A and! 2 ; where p(! ;j ja j ) = 1 " f! ;j = a j : Gven the acton by player j; the probablty that player 6= j receves the rght sgnal or the wrong sgnal about player j s acton s the same across 6= j. Also note that players sgnals are condtonally ndependent over players. 4. Belef Dynamcs and Best Response We represent our equlbrum strategy as a mappng from belef to actons 4fC; Dg. More explctly, our strategy takes the followng form: 8 < C f q 2 Q C ½(q ) = q f q 2 Q I : D f q 2 Q D where Q C ;Q I ;Q D are mutually exclusve and exhaustve sets n the space of the probablty measures on ; and q means playng C wth probablty q and playng D wth probablty 1 q: The steps to acheve the approxmate e cent payo s as follows. Frst, we gve a almost complete characterzaton of the unque optmal acton as a functon of belef. As a next step, we analyze the dynamcs of belef when players are playng ether ¾ C or ¾ D : Snce the dynamcs of belef n the n player case has some partcular feature whch does not appear n the two player case, we analyze t n detal. The thrd step s to construct a sequental equlbrum. We rst have to nd ½(q ) whch assgns the optmal acton for any possble belef on and o the equlbrum path. For ths purpose, we strengthen the characterzatonresult n the rst step to get ths functon ½(q ): In order to prove that ½(q ) s a sequental equlbrum, we also have to check consstency, that s, to check f players are actually playng ether ¾ C or ¾ D on the equlbrum path by followng ½(q ). Snce ths sequental equlbrum s constructed for a dscount factor n the mddle range, as the nal step, we dvde the orgnal repeated game to component repeated games or use a publc randomzaton devce to mplement the same payo by reducng hgh ± e ectvely Belef and the Best Response Before analyzng the unque optmal acton, we rst extend one property whch obvously holds n the two player case to the n player case. In the two player case, the d erence n payo s by ¾ C and ¾ D s lnear and there s a unque q (±;p 0 ) where a player s nd erent between ¾ C and ¾ D wth perfect montorng f ± s hgh enough 7

8 as shown n Fgure 1. When prvate montorng s almost perfect, there s q (±;p " ) where a player s nd erent between ¾ C and ¾ D and q (±;p " )! q (±;p 0 ) as "! 0: Put Fgure 1 here. When the number of players s more than two; the correspondng object V (¾ C ;q : p 0 ;±) V (¾ D ;q : p 0 ;±) for the n player case s a slghtly more complex object. Even when players randomze ndependently and symmetrcally playng C wth probablty q and D wth probablty (1 q), that s, q (µ) = n 1 P µ=0 (1 q) µ q n 1 µ n 1 µ for µ = 0;:::;n 1, t s a n 1 degree polynomal n q 2 [0;1]: Potentally, ths equaton may have n 1 solutons between 0 and 1 as shown n gure 2. We de ne q (±;p " ) to be the soluton whch s closest to 0 for V (¾ C ;q : p 0 ;±) V (¾ D ;q : p 0 ;±) = 0 where q (µ) = n 1 P µ=0 (1 q) µ q n 1 µ n 1 µ : Put Fgure 2 here. When montorng s almost perfect, V (¾ C ;q : p " ;±) V (¾ D ;q : p " ;±) s very close to V (¾ C ;q : p 0 ;±) V (¾ D ;q : p 0 ;±): Actually, t s easy to con rm that the former converges to the latter unformly n q as "! 0: 3 If ± > M(0) 1+M(0) ; then V (¾ C ;0 : p 0 ;±) V (¾ D ;0 : p 0 ;±) > 0; whch mples that there exsts q (±;p 0 ) between 0 and 1. The followng lemma extends a useful property for the two player case to the n player case. Lemma 1. q (±;p 0 )! 1 as ± # M(0) 1+M(0) Proof. See Appendx Now we show that the optmal acton as a functon of q s stll smlar to the one wth perfect montorng f the dynamcs of q s very close to the dynamcs of q wth perfect montorng. As a rst step to show that, the followng lemma shows that ¾ D s stll optmal f a player knows that someone has swtched to the permanent defecton and " s small. Lemma 2. There exsts a b" > 0 such that V (¾ ;q : p " ;±) s maxmzed by ¾ D for any p b" ; f q (µ) = 1 for any µ 6= 0. 3 Also note that convergence of V (¾ ;q j :p ") to V (¾ ;q j :p 0 ) s ndependent of the choce of assocated sequencefp " g because of the de nton ofp ". 8

9 Proof. Take ¾ D and any strategy whch starts wth C. The least devaton gan s (1 ±) 4: The largest loss caused by the d erence n contnuaton payo s wth ¾ D and the latter strategy s ±"V : Settng b" small enough guarantees that (1 ±)4 > ±"V for any " 2 (0;b"): Then, D must be the optmal acton for any such ": Snce players are usng permanent defecton, q (µ) = 1 for some µ 6= 0 n the next perod. Ths mples that D s the optmal acton n all the followng perods. Usng p( j ) and gven the fact that all players are playng ether ¾ C or ¾ D ; we ntroduce a notaton for the transton probablty of the number of players who have swtched to ¾ D : Let ¼ (ljm) be a probablty that l players wll play ¾ D from the next perod when m players are playng ¾ D now. In other words, ths ¼ (ljm) s a probablty that l m players playng C receve the sgnal D when n m players play C and m players play D: Of course, ¼ (ljm) > 0 f l = m and ¼ (ljm) = 0 f l < m. The followng lemma, whch s a sort of Maxmum theorem, provdes varous nformatve anduseful bounds onthe varatons ofdscountedaverage payo s caused by ntroducng small mperfectness n prvate montorng. Lemma nf V (¾ C ;0 : p " ;±) = (1 ±)+±"V p 1 ±(1 ") "2P " 2. Gven ± 2 ³ M(0) 1+M(0) ;1 sup V (¾ ;0 : p " ;±) 5 1 ±+±"V 1 ±(1 ") ¾ ;p "2P " Proof. (1): For any " 2 (0;1) and p " 2 P " ; So, ; There exsts a " > 0 such that for any " 2 [0;"]; V (¾ C ;0 : p " ;±) = (1 ±) + ±¼ (0j0)V (¾ C ;0 : p " ;±) + ± (1 ¼ (0j0))V (2): Gven ± 2 V (¾ C ;0 : p " ;±) = ³ M(0) 1+M(0) ;1 (1 ±) + ± (1 ¼ (0j0))V 1 ±¼ (0j0) = (1 ±) + ±"V 1 ± (1 ") ; t s easy to check that V (¾ C ;0 : p 0 ;±) > V (¾ D ;0 : p 0 ;±) : Pck " small enough such that () V (¾ C ;0 : p " ;±) > V (¾ D ;0 : p " ;±) for any p " and () " < b": Let ¾ 0 be the optmal strategy gven that everyone s usng ¾ C : 4 Suppose that ¾ 0 assgns D for the rst perod. Then for any " 2 [0;"]; V (¾ 0 ;0 : p ") 5 (1 ±)U D;D 0 ½ : p " + ± ¼ (1j1) V (¾ 0 ;0 : p P ";±)+ n µ=2 ¾ ¼ (µj1)v (¾ D ;µ 1 : p " ;±) 4 Ths ¾ exsts because the strategy space s a compact space n product topology, on whch dscounted average payo functons are contnuous. Of course, ths¾ depends on the choce of p ": 9

10 In ths nequalty, the second component represents what player could get f she knew the true contnuaton strateges of her opponents at each possble state. To see that ths addtonal nformaton s valuable, suppose that the contnuaton strategy of ¾ 0 leads to a hgher expected payo than V (¾ 0;0 : p " ;±) or V (¾ D ;µ 1 : p " ;±) at the correspondng states, then ths contradcts the optmalty of ¾ 0 or ¾ D by Lemma 2. So ths nequalty holds. Then, for any " 2 [0;"]; V (¾ 0;0 : p " ;±) 5 P (1 ±)U (D;D 0 : p " ) + ± n ¼ (kj1)v (¾ D ;µ 1 : p " ;±) = V (¾ D ;0 : p " ;±) < V (¾ C ;0 : p " ;±) µ=2 1 ±¼ (1j1) Snce ths contradcts the optmalty of ¾ 0; ¾ 0 has to assgn C for the rst perod. Now, So, V (¾ 0;0 : p " ;±) 5 (1 ±) + ±¼ (0j0)V (¾ 0;0 : p " ;±) + ± (1 ¼ (0j0))V V (¾ 0;0 : p " ;±) 5 (1 ±) + ± (1 ¼ (0j0))V 1 ±¼ (0j0) 5 (1 ±) + ±"V 1 ± (1 ") Ths mples that sup V (¾ ;0 : p " ;±) 5 1 ±+±"V 1 ±(1 ") for any " 2 [0;"]: ¾ ;p " 2P " (1) means that a small departure from the perfect montorng does not reduce the payo of ¾ C much when all the other players are usng a grm trgger strategy. (2) means that there s not much to be exploted by usng other strateges than ¾ C wth a small mperfecton n the prvate sgnal as long as all the other players are usng a grm trgger strategy. The followng result shows that the unque optmal acton s almost completely characterzed as a functon of q except for an arbtrary small neghborhood and equvalent to the optmal acton wth perfect montorng. Proposton 1. Gven ±; for any > 0; there exsts a " > 0 such that for any p " ; ² t s not optmal to play C for player f q sats es Á(q ) 5 1 ± ± M (q ;p 0 ) ² t s not optmal to play D for player f q sats es Á(q ) = 1 ± ± M (q ;p 0 )+ Proof: (1): It s not optmal to play C f 10

11 (1 ±) M (q ;p " ) ½ ¾ > ± Á(q ) (1 ") sup V (¾ ;0 : p " ;±) + "V ¾ ; + (1 Á(q ))"V By Lemma 3.2., ths nequalty s sats ed for any " 2 [0;"] and any p " f (1 ±) M (q ;p " ) ½ > ± Á(q ) (1 ") 1 ± + ±"V ¾ 1 ± (1 ") + "V + (1 Á(q ))"V LHS converges to (1 ±) M (q ;p 0 ) and RHS converges to ±Á(q ) as "! 0: So, f q sats es Á(q ) 5 1 ± ± M (q ;p 0 ) for any > 0; then there exsts a " 0 (±; ;q ) 2 (0;") and a neghborhood B (q ) of q such that C s not optmal for any p "0 (±; ;q ) and any q 0 2 B (q ): Ths " 0 (±; ;q ) > 0 can be set ndependent of q by the standard arguments because q s n a compact space: (2): It s not optmal to play D f (1 ±) M (q ;p " ) < ± Á(q )f(1 ")V (¾ C ;0 : p " ;±) + "V g + (1 Á(q ))"V "V ths nequalty s sats ed for " 2 (0;1) and any p " f < ± (1 ±) M (q ;p " ) Á(q ) ½ (1 ") 1 ± + ±"V 1 ± (1 ") + "V ¾ + (1 Á(q ))"V "V Ths nequalty converges to Á(q ) = 1 ± ± M (q ;p 0 ) as "! 0: So, f q sats es Á(q ) = 1 ± ± M (q ;p 0 ) + for any > 0; there exsts a " 00 (±; ;q ) such that D s not optmal for any p " 2 P " 00 (±; ;q ) and any q 0 around q : Agan, " 00 (±; ;q ) can be set ndependent of q : Fnally, settng "(±; ) = mnf" 0 (±; );" 00 (±; )g completes the proof. Ths proposton mples that the optmal acton can be completely characterzed except for an arbtrary small neghborhood of the manfold satsfyng Á(q ) = 1 ± ± M (q ;p 0 ) n a n 1 dmensonal smplex; where player s nd erent between ¾ C and ¾ D wth perfect montorng: Note that ths unque optmal acton s equvalent to the one wth perfect montorng. Snce the contnuaton payo s wth prvate montorng converge to the contnuaton payo wth perfect montorng as montorng gets accurate, a player prefers C(D) to D (C) n prvate montorng f and only f she prefers 11

12 C (D) to D (C) n perfect montorng except for the regon where she s almost ndfferent between C and D wth perfect montorng. A slght nose stll matters n ths regon, but we also characterze the best response for ths regon later. Although ths argument s essentally the path domnant argument used n Sekguch [16] for n = 2, t extends that argument to the n player case and provdes a sharper characterzaton even for n = 2: In fact, ths proposton s the reason why an almost e cent outcome s acheved for any prsoner s dlemma n the two player case. An mmedate corollary of ths proposton s that C s the unque optmal acton gven that Á s su cently close to 1, ± > M(0) 1+M(0) ; and " s small: Corollary 1. Gven ± > M(0) 1+M(0) ; there exsts Á > 0 and " > 0 such that for any p "; t s not optmal for player to play D f Á 2 Á;1 : 4.2. Belef Dynamcs Snce the optmal acton s almost characterzed as a functon of q ; we have to analyze the dynamcs of q assocated wth grm trgger strateges. Gven the optmal acton shown above, all we have to make sure for the consstency of grm trgger strateges s that q stays n the C area descrbed by Proposton 1 as long as player has observed fully cooperatve sgnals from the begnnng and q stays n the D area once player receved a bad sgnal or started playng defecton for herself. Assumpton 2 on the sgnal dstrbuton s requred here for the rst tme as the followng arguments show. Frst, consder a hstory where player observes some defecton for the rst tme. If ths s the rst perod, player nterprets ths as a sgnal of ¾ D rather than as an error f " s small 5. Suppose next that ths knd of hstory s reached after the rst perod. Also suppose that the number of players s three for smplcty and player 1 observes 1 defecton by player 2. Wth Assumpton 2, player 1 can nterpret ths as a 1-error event and stll beleve that everyone s cooperatve. On the other hand, t s equally lkely that player 2 s observaton contaned 1 error n the last perod and the current sgnal s correct. Note that there are two such events. The player for whom player 2 observed \D" last perod can be player 1 or player 3. Snce someone should have already defected after all other possble hstores, the probablty that no one has swtched to ¾ D s at most 1 3. Obvously, ths exblty of nterpretaton ncreases as the number of players ncreases, whch makes t easer to move Á closer to 0 after ths sort of hstory. Note that ths upper bound of Á does not depend on the level of ": Second, consder a hstory where player has already started defecton. Suppose that all players but player have been cooperatve untl the present. Also suppose agan that the number of players s three and = 1 for the sake of smple exposton: 5 We need to let players to randomze between¾ C and¾ D n the ntal perod for ths argument to work n the ntal perod. If players start wth, say, ¾ C wth probablty 1, no learnng occurs after the ntal perod. Ths s rst observed by Matsushma [12]. Note that ths s the only reason why the ntal randomzaton s needed. The rest of arguments does not depend on ths ntal randomzaton. 12

13 For everyone to be stll cooperatve after the current perod, all players but player 1 should observe the wrong sgnal C about player 1 and the correct sgnal C about the other players n the current perod. Agan, there are other events wth the same probablty, where some player swtches to ¾ D : For example, player 2 may observe the correct sgnal D about player 1 and the wrong sgnal D about player 3. Ths event contans the same number of errors. Snce there are 5 such events, the probablty that at least one player has swtched to ¾ D s at least 5 6 even though t s assumed that all players butplayerhave beencooperatve untl the current perod. Wthpostve probablty that someone has already started defecton, the posteror Á s strctly less than 1 6 : Ths argument s agan ndependent of the level of ":6 The above arguments strongly suggest that once players start playng ¾ D ; then they contnue to do so. The followng proposton summarzes the above arguments and also provde another useful property to support ¾ C : Let W k = q já(q ) = 1 ± ± M (q ;p 0 ) k ª ; where k > 0 and 0 < 1 ± ± M (n 1) k: When k s small, W k just covers the regon where C s the unque optmal acton and the unque optmal acton s ndetermnate. We show that Á t > Á f a player plays C, observes C; and Á t 1 2 W k. Proposton 2. Suppose that every player plays q¾ C + (1 q) ¾ D wth q 2 (0;1) n the rst perod, and () : Assumpton 1 s sats ed and n = 2, or () : Assumpton 2 s sats ed. Then for all and t = 2;3;::: ² For any Á 0 > 0;k > 0; there exsts e" such that for any " 2 (0;e") Á q h t+n = Á 0 for h t+n = h t 1 ;(C;C);:::;(C;C) when Á q h t 1 2 W k ² Á(q (h t )) 5 1 n after hstores such as ½ h t = h t 1 ; C;! t 1 for t = 3 h t = C;! t 1 for t = 2 wth! t 1 6= C or ½ h t = h t 1 ; D;! t 1 for t = 3 h t = D;! t 1 for t = 2 Proof. See appendx. 6 Ths last argument s spec c to the n = 3 player case. The two player case has to be treated separately. See Sekguch [16] for that case. 13

14 5. Sequental Equlbrum wth the Grm Trgger Strategy As n secton 2, the equlbrum strategy s represented as a mappng from the space of belef Q to 4fC;Dg : The followng notatns are useful: Q C " Q I " = fq jv (¾ C ;q : p " ;±) > V (¾ D ;q : p " ;±)g = fq jv (¾ C ;q : p " ;±) = V (¾ D ;q : p " ;±)g Q D " = fq jv (¾ C ;q : p " ;±) < V (¾ D ;q : p " ;±)g ½ Q n = q já(q ) 5 1 ¾ n These are subsets of the n 1 dmensonal smplex on : The subset Q n s a set contanng the absorbng set of the dynamcs of q under the grm trgger. Q I " s a manfold where player s nd erent between ¾ C and ¾ D : In partcular, q (±;p ") 2 Q I " by de nton. Note that Q C " converges to Q C 0 = q já(q ) > 1 ± ± 4(q ;p 0 ) ª and Q D " converges to Q D 0 = q já(q ) < 1 ± ± 4(q ;p 0 ) ª as "! 0: Now de ne ½ as a mappng from q 2 Q to 4fC;Dg n the followng way: 8 < C f q 2 Q C ½ " " (q ) = q (±;p " ) f q 2 Q I " : D f q 2 Q D " We know from Proposton 1 that ths functon assgns the best response acton almost everywhere except for a neghborhood of Q I " when " s small. Thanks to the last proposton about the dynamcs of belef, now we can verfy that ½ " (q ) actually assgns the unque optmal acton for any belef q =2 Q I " wth one weak assumpton. Proposton 3. Gven ±; f Q n 2 Q D 0 and " s small enough, then ² C s the unque optmal acton f and only f q 2 Q C " [ Q I " ² D s the unque optmal acton f and only f q 2 Q D " [ Q I " Proof: Fx > 0 n Proposton 1 and set k > 0 slghtly larger than > 0: Now take any q such that Á(q ) > 1 ± ± 4(q ;p 0 ) k and q 2 Q D " : We prove that D s the unque optmal acton n ths regon. If a player plays C, then Proposton 2 mples that the contnuaton strategy s ¾ C f " s su cently small: Snce ¾ C s domnated by ¾ D n ths regon, the unque optmal acton should be D: Smlarly, take any Á such that 1 ± ± 4(q ;p 0 ) + > Á(q ) and q 2 Q C " : If D s played, then Proposton 2 and the assumpton Q n 2 Q D 0 mples that the contnuaton strategy s ¾ D as long as " s small enough. Snce ¾ D s domnated by ¾ C n ths regon, the unque optmal acton should be C: Snce any other q 2 Q D " sats es Á(q ) 5 1 ± ± 4(q ;p 0 ) k and any other q 2 Q D " sats es Á(q ) = 1 ± ± 4(q ;p 0 ) + sats es q 2 Q C " for small " > 0; the proof s complete. 14

15 The last thng we have to make sure s that players are actually playng ether ¾ C or ¾ D on the equlbrum path after they ntally randomze between these strateges. It s almost obvous from the arguments we used to prove the propostons. Proposton 4. Suppose that () : Assumpton 1 s sats ed and n = 2, or () : Assumpton 2 s sats ed. Gven ± 2 ³ M(0) 1+M(0) ;1 ; f Q n ½ Q D 0 ; then there s a " > 0 such that for any p " ; The followng proposton shows that ½ " (q ) s a symmetrc sequental equlbrum, whch generates the same outcome dstrbuton as (:::;q (±;p " )¾ C + (1 q (±;p " ))¾ D ;:::): Proof. ³ ;1 Snce ± 2 M(0) 1+M(0) ; q (±;p " ) exsts n (0;1) f " s small enough. Q n ½ Q D " s also sats ed for small ": Suppose that player chooses ¾ C as her strategy n the rst perod. Set " > 0 small enough for Corollary 1 and Proposton 2 to hold. As long as (C;C) s the outcome, Á > Á and C s the unque optmal acton by Corollary 1 and Proposton 2. Consder a hstory where player observed some D for the rst tme or a hstory where player started playng D; After ths sort of hstory, Á t = Á(q (h t )) s gong to stay n Q n forever by Proposton 2 as long as she contnues playng D: Snce Q n ½ Q D ",ths contnuaton strategy s clearly ¾ D : We also know that ½ " (q ) assgns the best response after any o the equlbrum hstory. Moreover, consstency s obvous. Takng " small such that all the above arguments go through, we con rm that ½ " (q ) s a symmetrc sequental equlbrum, whch generates the same outcome dstrbuton as (:::;q (±;p " )¾ C + (1 q (±;p " ))¾ D ;:::) by constructon: Snce the probablty that everyone chooses ¾ C n ths sequental equlbrum; q (±;p " ) n 1 ; gets closer to 1 as ± gets closer to by Lemma 1,an outcome M(0) 1+M(0) arbtrary close to the e cent outcome can be acheved for ± arbtrary close to M(0) 1+M(0). For hgh ±; we can use a publc randomzaton devce agan to reduce ± e ectvely. It s also possble to use Ellson s trck as n Ellson [6] to acheve an almost e cent outcome although the strategy s more complex and no longer a grm trgger.. Here s the corollary of Proposton 3 regardng to an approxmately e cent outcome. Corollary 2. Suppose that () : Assumpton 1 s sats ed and n = 2, or () : Assumpton 2 s sats ed. For any k > 0; f Q n ½ Q D 0 ; then there s a " > 0 such that for any p " ; there exsts a sequental equlbrum whose symmetrc equlbrum payo s more than 1 k: Fnally, we have to check f the assumpton Q n ½ Q D 0 used above to prove propostons s not so strong. Frst of all, ths s always sats ed when n = 2 for a range of ± wth q (±;p 0 ) ;1 ; that s, when ± s between M(0) M(0)+M(1) 1+M(0) and 1+M(0)+M(1) : The followng proposton provdes su cent condtons for Q n ½ Q D 0 to hold for general n. 15

16 Proposton 5. : ³ ² If M (k) =M for k = 1;:::;n 1, then Q n ½ Q D 0 and Q I 0 6= ; for ± 2 M 1+M ; nm 1+nM ² Regardng n as a parameter, take a sequence of the stage game wth n = 2;3;:::::: If there exsts a lower bound M > 0 such that M (k) = M ndependent mn 1 k n 1 of n; then ³ there exsts n such that for all n = n; Q n ½ Q D 0 (n) and Q I 0 (n) 6= ; for ± 2 M(0) 1+M(0) ; nm 1+nM Proof: If the devaton gan s constant, Q I 0 = q já(q ) = 1 ± ± 4ª 1 So, n < 1 ± ± 4 Qn ½ Q D 0 (n): Combnng ths nequalty wth M 1+M < ± for QI 0 6= ;; ± 2 ³ s obtaned: M 1+M ; nm 1+nM If M s ndependent of n; Q n ½ Q D 1 ³ 0 (n) f n < 1 ± ± M: So, Qn ½ Q D 0 (n) and Q I 0 (n) 6= ; for ± 2 M(0) 1+M(0) ; nm M(0) 1+nM for all n = n f n s chosen such that 1+M(0) < nm 1+nM The assumpton of constant devaton gan s natural n the team producton, where a player s cost comes from dsutlty assocated wth the e ort level. 6. Concluson In ths paper, we clarfy the ncentve structure n a general repeated prsoner s dlemma wth prvate montorng when players are usng a mxture of a grm trgger strategy and permanent defecton, and provde the su cent condtons under whch the smple grm trgger strategy supports the e cent outcome as a sequental equlbrumfor some range of dscount factors. It s also shownthat the best response to a mxture of grm trgger strategy and permanent defecton can be characterzed almost unquely, whch makes t possble to provde the clear representaton of the sequental equlbrum supportng the e cent outcome. There have been two lnes of research explorng the sustanablty of the e cent outcome or the possblty of any folk theorem n repeated prsoners dlemma wth prvate montorng. One drecton of research s based on grm trgger strateges. Bhaskar [3] and Sekguch [16] belong to ths lterature as well as ths paper. The emphass of these papers are on coordnaton of players actons and belefs. As shown n Malath and Morrs [11], the almost common knowledge of contnuaton strateges s very mportantfor subgame perfectequlbra or perfect publc equlbra to be robust to the perturbaton wth respect to prvate nose because t enables players to keepcoordnatng wthothers wthoutpublc sgnals. In ths paper, nfact a player strongly beleves that the other players are playng ¾ C when she contnues playng C; and beleves that the other players are playng ¾ D once she started playng D: However, note that our strategy does not exhbt such a strong coordnaton as requred n Malath and Morrs [11]. Whle Malath and Morrs [11] requres that players contnuaton strateges are almost common knowledge after any hstory, a dscoordnaton of belef or contnuaton strategy can occur almost surely for our 16

17 mxng grm trgger strategy. When a player observes D for the rst tme, she s stll uncertan about the other player s contnuaton strateges. The other drecton of research s based on a complete mxed strateges whch makes the other player nd erent over many strateges. Ely and Välmäk [7] and Pccone [15] are among papers n ths drecton. 7 One advantage of the rst approach compared to the second approach s: 1. The equlbrum needs to use mxng only at the rst perod of the game, whle the latter approach uses a behavor strategy whch let players randomze at every perod after every hstory. Another advantage, whch s closely related to the rst one, s as follows: 2. Snce our strategy s almost pure, t s very easy to justfy the use of a mxed strategy. Especally, our strategy sats es the re nement crteron for repeated game equlbra wth respect to ncomplete nformaton on players payo s, whch s proposed by Bhaskar[4]. Ths crteron prohbts players to play dfferent contnuaton strateges after d erent hstores f players are nd erent between these contnuaton strateges after all these d erent hstores. Strateges n Ely and Välmäk [7] or Pccone [15] volate ths crteron at the second perod when prvate sgnals are ndependent. Whle pur caton s straghtforward for our strategy by ntroducng a small amount of uncertanty nto stage game payo s, payo uncertanty has to depend on a prvate hstory n a partcular way to purfy equlbra n Ely and Välmäk [7] or Pccone [15]. It s also easy to adopt Nash s populaton nterpretaton to purfy our equlbrum. What we have n our mnd s a pool of players who are matchng wth the other players to play repeated games, where most of the players use grm trgger and only a small porton of the players use permanent defecton. The case where prvate montorng s not almost perfect has not been systematcally nvestgated and t remans as a mportant topc for future research. 7 Obara [14] uses the same knd of strategy for repeated partenershp games wth mperfect publc montorng and constructs a sequental equlbrum whch cannot be supported by publc perfect equlbra. 17

18 Appendx. Proof of Lemma 1. When ± = 4(0) 1+4(0) ; q (±;p 0 ) = 1 s the soluton of the equaton n q: where q (µ) = n 1 P V (¾ C ;q : p 0 ;±) V (¾ D ;q : p 0 ;±) = 0 µ=0 (1 q) µ q n 1 µ n 1 µ for kµ = 0;:::;n 1:: < 0 usng the mplct functon theo- We just need to show (±;p j ±= 4(0) 1+4(0) rem. Snce V (¾ C ;q : p 0 ;±) V (¾ D ;q : p 0 ;±) n 1 X µ = (1 ±) (1 q) µ n 1 q n 1 µ 4 (µ;p 0 ) ±q n 1 µ (±;p 0 ) ±= 4(0) (¾ C ;q :p 0;±) V (¾ D ;q :p (¾ C;q :p0;±) V(¾D;q j ±= 4(0) 1+4(0) 1 + 4(0) = (1 ±)(n 1) 4 (1) + ± (n 1) j ±= 4(0) 1+4(0) = 1 1 n 1 4(0) + 4(1) > 0 Proof of Proposton2 case 1: Let h t = h t 1 ;(C;C) such that Á t 1 = Á q h t 1 2 Wk : Applyng Bayes rule 8, Á t = Á(q (h t )) = Á t 1 P(! t 1 =Cj h t 1 (1 Á t 1 )P(! t 1 =Cjµ t 16=0;h t 1 )+Á t 1 P(! t 1 =Cj h t 1 Ths functon s ncreasng n Á t 1 and crosses 45 ± lne once. Note that ths functon s bounded below by ' Á t 1 Á = t 1 (1 ") : Let Á b be the unque xed (1 Á t 1 )"+Á t 1 pont of ths mappng. Gven that Á t = Á q h t 1 2 Wk ; t s easy to see that ¾ D : 8 All the condtonal dstrbutons mplctly depend on the level of ntal mxture between ¾ C ; ) ) 18

19 ' Á t 1 ³ and Á b can be made larger than Á 0 > 0 for any q unformly by choosng " small enough. If Á t 1 < Á; b then, as long as players contnue to observe C; ' n Á 1 s gong to ncrease monotoncally to Á: b On the other hand, snce Á t = ' Á t 1 and Á t+n = ' Á t+n 1 ; Á t+n 1 s larger than ' n Á t 1 for any n = 0;1;::: If Á t 1 = Á; b then Á t+n 1 = Á b > Á 0 for any n: Ths mples that Á t+n ª 1 n=0 s always above Á0 : ½ h t case 2: = h t 1 wth! t 1 h t = C;! t 1 =! 0 6= C ; C;! t 1 Suppose that t = 3: By Bayes Rule, for t = 3 for t = 2 Á t = Á(q (h t )) Á = t 2 P((! t 2 ;! t 1 ;!t 1 )=(C;C;! 0 )jµ t 2 =0) (1 Á t 2 )P((! t 2 ;! t 1 )=(C;! 0 )jµ t 2 6=0; h t 2 )+Á t 2 P((! t 2 ;! t 1 Ths s bounded above by )=(C;! 0 )jµ t 2 =0) 5 = P! t 2 ;! t 1 ;!t 1 P! t 2 ;! t 1 = (C;C;! 0 )jµ t 2 = 0 = (C;! 0 )jµ t 2 = 0 P (#(! 0 jc)) P (#(! 0 jc)) + (n jc) 1)#(!0 P (#(! 0 jc)) (n 1) #(!0jC) 5 1 n where P (#(! 0 jc)) s the probablty of the event that #(!0 jc) errors occur. So, once players observed a bad sgnal for the rst tme, the posteror Á t jumps down at least below 1 n ndependent of the pror Át 1 or q t 1 for t = 3: Ths argument s ndependent of the level of ": When t = 2; Á t decreases enough to be less than 1 n f " s very small. Ths s because players do not nterpret t as an error but as a sgnal of ¾ D at the rst perod. ½ h t case 3: = h t 1 ; D;! t 1 h t = D;! t 1 for t = 3 for t = 2 wth! t 1 =! 00 we have to treat () n = 3 and () n = 2 separately agan. (): n = 3 By Bayes Rule, 19

20 Á t = Á(q (h t )) = Ths s bounded above by 5 = Á t 1 P((! t 1 ;!t 1 )=(C;! 00 (1 Á t 1 )P(! t 1 =! 00 jµt 16=0)+Át 1 P(! t 1 =! 00 P! t 1 ;!t 1 = (C;! 00 )jµ t 1 = 0 P! t 1 =! 00 jµ t 1 = 0 P (#(! n 1 P m=1 jc) + n 1) + P (#(!00 1 (n 2)(n 1) n 1 m m )jµ t 1=0) P (#(! 00 jc) + n 1) n 1 P jc) + n 1) 5 1 (Ths holds wth equalty when n = 3) 6 Ths argument s ndependent of "; too. (): n = 2 Á q h t 1 ;(D;C) = Át 1 Á q h t 1 ;(D;D) = m=1 jµt 1=0) (1 2P(1) P(2))+(1 Á t 1 )(P(1)+P(2)) Á t 1 (1 P(1) p(2))+(1 Á t 1 )(P(1)+P(2)) Á t 1 P(1)+(1 Á t 1 )(1 P(1) P(2)) (P(1)+P(2))+(1 Á t 1 )(1 P(1) P(2)) Á t 1 (n 2)(n 1) m n 1 m where P (k) s a probablty that k errors occur. It can be shown that Á q t h t 1 ;(D;! ) when " s small. See Sekguch [16] for detal. Wth case 2 and case 3, I can conclude that Á(q (h t )) 5 1 n after any hstory such as ½ h t ² = h t 1 ; C;! t 1 for t = 3 h t = C;! t 1 for t = 2 wth! t 1 6= C or ½ h t ² = h t 1 ; D;! t 1 h t = D;! t 1 for t = 3 for t = 2 20

21 References [1] Abreu, D. Pearce, and E. Stachett, Toward a theory of dscounted repeated games wth mperfect montorng, Econometrca 58 (1990), [2] V. Bhaskar, Informatonal constrants and overlappng generatons model: folk and ant-folk theorem, Rev. of Econ. Stud 65, (1998), [3] V. Bhaskar, Sequental equlbra nthe repeatedprsoner s dlemma wthprvate montorng, mmeo, [4] V. Bhaskar, The robustness of repeated game equlbra to ncomplete payo nformaton, mmeo, [5] O. Compte, Communcaton n repeated games wth mperfect prvate montorng, Econometrca 66 (1998), [6] G. Ellson, Cooperaton n the prsoner s dlemma wth anonymous random matchng, J. Econ. Theory 61 (1994), [7] J. C. Ely and J. Välmäk, A robust folk theorem for the prsoner s dlemma, mmeo, [8] D. Fudenberg, D. Levne, and E. Maskn, The folk theorem wth mperfect publc nformaton, Econometrca 62 (1994), [9] J. Harsany, Games wth randomly dstrbuted payo s: a new ratonal for mxedstrategy equlbrum ponts, Int. J. Game Theory 2, (1973), [10] M. Kandor and H. Matsushma, Prvate observaton, communcaton, and colluson, Econometrca 66 (1998), [11] G. Malath and S. Morrs, Repeated games wth almost-publc montorng, mmeo, [12] H. Matsushma, On the theory of repeated game wth non-observable actons, part I: Ant-Folk Theorem wthout communcaton. Econ. Letters 35 (1990), [13] I. Obara, Repeatd prsoners dlemma wth prvate montorng: a N-player case, mmeo, 1999 [14] I. Obara, Prvate strategy and e cency: repeated partnershp games revsted, mmeo, [15] M. Pccone, The repeated prsoner s dlemma wthmperfectprvate montorng, mmeo, [16] T. Sekguch, E cency nrepeated prsoner s dlemma wthprvate montorng, J. Econ. Theory 76 (1997),