Learning from Private Information in Noisy Repeated Games

Transcription

1 Learnng from Prvate Informaton n Nosy Repeated Games Drew Fudenberg and Yuch Yamamoto Frst verson: February 18, 2009 Ths verson: November 3, 2010 Abstract We study the perfect type-contngently publc ex-post equlbrum (PTXE) of repeated games wth ncomplete nformaton where players observe mperfect publc sgnals of the actons and the map from actons to sgnal dstrbutons s tself unknown. The PTXE payoffs when players are patent are determned by the ntersecton of the maxmal half spaces n varous drectons; we focus on the cross-state drectons that consder payoffs n two or more states. We develop condtons under whch the maxmal half spaces n these drectons mpose no constrants on the equlbrum set, so that equlbrum play can be as f the players have learned the state. We use these condtons to provde a suffcent condton for the folk theorem, and a characterzaton of the PTXE payoffs n games wth a known montorng structure. Journal of Economc Lterature Classfcaton Numbers: C72, C73. Keywords: repeated game, publc montorng, ncomplete nformaton, perfect publc equlbrum, folk theorem, belef-free equlbrum, ex-post equlbrum. We thank NSF grant for fnancal support. Department of Economcs, Harvard Unversty, dfudenberg@harvard.edu Department of Economcs, Harvard Unversty, yamamot@fas.harvard.edu

2 1 Introducton The fact that repeated nteractons can allow new and more effcent equlbrum outcomes s one of game theory s most mportant nsghts. It has been shown to apply n a range of settngs, ncludng games wth mperfect publc nformaton about opponents actons, and games where the montorng structure- the map from actons to sgnal dstrbutons- s tself unknown. 1 It has also been shown n games wth prvate nformaton about the payoff functons. 2 Ths paper studes games wth the combnaton of these features: the montorng structure s unknown and there s also prvate nformaton about the montorng structure and/or the payoff functons. Ths descrbes, for example, a repeated partnershp game where players observe group output but do not observe each other s effort, and each player has prvate nformaton about the effect of her effort on the probablty dstrbuton of output. Our man goal n ths paper s to understand how the nformaton structure of the game- meanng the combnaton of the montorng structure and the ntal prvate nformaton- determnes the extent to whch the player s ntal prvate nformaton can be revealed n equlbrum. We address ths queston ndrectly, by computng the lmt of the equlbrum payoffs when players are patent. More specfcally, we restrct attenton to the perfect type-contngently publc expost equlbrum or PTXE (Fudenberg and Yamamoto (2010)). These are ex-post equlbra where each player s strategy depends only on the realzed publc outcomes and hs ntal prvate nformaton (hence type-contngent ) but not on the player s prvate nformaton about hs own past actons. PTXE generalzes several soluton concepts: It reduces to the PPXE of Fudenberg and Yamamoto (2010) f players have no prvate nformaton, the belef-free equlbra of Hörner and Lovo (2009) and Hörner, Lovo, and Tomala (2010) when 1 For repeated games wth publc montorng, see Green and Porter (1984), Radner (1985), Abreu, Pearce, and Stachett (1986), Abreu, Pearce, and Stachett (1990), Fudenberg and Levne (1994), Fudenberg, Levne, and Maskn (1994), Athey and Bagwell (2001), and Fudenberg, Levne, and Takahash (2007). Fudenberg and Yamamoto (2010) consder games where the montorng structure s unknown. 2 See Kohlberg (1975), Forges (1984), Sorn (1984), Hart (1985), Sorn (1985), Aumann and Maschler (1995), Crpps and Thomas (2003), Gossner and Velle (2003), Renault and Tomala (2004), Wseman (2005), Hörner and Lovo (2009), Wseman (2010), and Hörner, Lovo, and Tomala (2010) for games wth prvate nformaton. 1

3 actons are observed, 3 and the perfect publc equlbrum (PPE) of Fudenberg, Levne, and Maskn (1994, hereafter FLM) n complete-nformaton games wth a known montorng structure. As wth ex-post equlbra more generally, these equlbra are robust to the specfcaton of the players pror belefs: a PTXE for a gven pror dstrbuton s a PTXE for an arbtrary pror. 4 Any PPXE of the symmetrc nformaton game (where no player has ntal prvate nformaton about the state) nduces a PPXE of the game where some players do have prvate nformaton: these PPXE correspond to poolng equlbra of the ncomplete-nformaton game. Thus the folk theorems of Fudenberg and Yamamoto (2010) apply to games wth prvate nformaton. However, those theorems requre that the dstrbuton of sgnals vary wth the state n a suffcently rch way (essentally so that the state can be learned from the sgnals generated by some fxed acton profle), and ths s more restrctve than necessary when some players have prvate nformaton. For example, f one player knows the state, he may be able to communcate t to the others usng a strategy that condtons on the player s prvate nformaton. Ths paper takes the possblty of such mplct communcaton nto account, and so generates a larger set of equlbrum payoffs. In some cases, such as the partnershp games we defne n Secton 3, n whch a player s productvty s prvate nformaton, there often exst asymptotcally effcent equlbra, whle equlbrum payoffs are bounded away from effcency f the players gnore ther prvate nformaton. Moreover, we can characterze the lmt payoffs of PTXE wth lnear programmng technques. Specfcally, the set of lmt equlbrum payoffs s the ntersecton of maxmal half spaces n varous drectons, where the drecton vectors λ assgn weghts on each player s payoff n each state, the maxmal half space n drecton λ s all vectors v wth λ v no greater than the maxmum score for λ, and ths score s the hghest weghted sum of payoffs that can be obtaned wth contnuaton payoffs that satsfy the ncentve constrants and whose weghted sum s no hgher than the sum they are supportng. Roughly speakng, there are PTXE where players learn the state f the score 3 These equlbra are dfferent than the belef-free equlbra of repeated games wth prvate montorng (Pccone (2002), Ely and Välmäk (2002), Ely, Hörner, and Olszewsk (2005), Yamamoto (2007), Yamamoto (2009), and Kandor (2010)), whch requre that players be ndfferent. 4 See Bergemann and Morrs (2007) for a dscusson of varous defntons of ex-post equlbrum. 2

4 s suffcently large n cross-state drectons that gve non-zero weght to two or more states. For ths to be the case, nformed players must be wllng to reveal ther nformaton, and unnformed players must not jam the nformaton revelaton of ther nformed opponents. A key pont s that the relevant condtons depend on whether the nformed player s payoff n a gven state s gven postve or negatve weght. Wth a postve weght the nformed player wants to reveal the state, and our condtons mply that other players cannot prevent ths; wth a negatve weght the nformed player mght prefer to hde the state, but under our condtons ths s not possble. We use these results to prove a folk theorem. Whle the exact condtons are complcated to state, the key assumpton s that for each par of players and j (where possbly = j) and each par of states ω and ω ω, ether () there s a player l, j whose prvate nformaton dstngushes ω and ω, and player l can reveal ths nformaton regardless of the actons of and j by choosng dfferent actons n state ω and state ω, () player or j (or both) can dstngush ω and ω usng ntal prvate nformaton, and the nformed player s wllng to reveal ths nformaton whle the other one cannot nterfere, or () there s an acton profle α (ndependent of the prvate nformaton) that dstngushes (more formally, statewse dentfes ) ω from ω. Condtons () and () lead to a sort of endogenous learnng where players transmt ther prvate nformaton to the opponents, whle condton () s a sort of exogenous learnng based on the dstrbuton of sgnals at a fxed acton profle. Note that condton () does not requre that player l s wllng to reveal hs nformaton. Ths s because the condton can be used for drectons where player l s contnuaton payoff has zero weght and hence s unconstraned, and our ndvdual full rank assumpton ensures that there s some specfcaton of the contnuaton payoffs that nduces player l to play the specfed actons n the two states. In contrast, f no thrd player can dstngush the states, then the ncentves of the revealng player become relevant, as wthout addtonal condtons t may be that any contnuaton payoffs that nduce player to reveal hs nformaton must ncrease or decrease player j s contnuaton payoff n a way that lowers the score. We then consder a few cases wth addtonal structure that smplfes our characterzaton. We begn wth the case where the state space has one component that only nfluences payoffs and a second component that only nfluences the montor- 3

5 ng structure; here we show that when the full rank condtons are satsfed the lmt set can be determned for each payoff functon separately. Next we consder games wth a product structure, where there s a separate and ndependent sgnal assocated wth each player s acton, and moreover each player knows the effect of hs acton on the sgnal dstrbuton whle the others do not. For example, n a game of blateral producton and exchange, the publc sgnal mght be the qualty of a player s output, wth each player havng prvate nformaton about the probablty that she wll make a hgh-qualty good when she exerts hgh effort. Here we show that the scores for two classes of cross-state drectons are hgh enough to be compatble wth the folk theorem, but that the scores n the remanng class need not be. Fnally, as an llustraton of our charaterzaton, we examne n detal a repeated partnershp example where only group output s observed, and the state determnes the productvty of player 2. We show that f player 1 s prvate nformaton reveals player 2 s productvty whle 2 has no prvate nformaton (.e. 1 knows 2 s productvty ), then the folk theorem holds n general, whle f only player 2 knows player 2 s productvty, the folk theorem can fal, and moreover the lmt equlbrum payoffs can be bounded away from effcency. Intutvely, player 2 cannot be nduced to reveal the state when dong so would lower hs equlbrum payoff, and ths leads to a bound on the extent to whch equlbra can trade off player 2 s payoffs between the two states; n some cases ths bound s so strong that t rules out the effcent outcome. Fnally, we specalze to the case of a known montorng structure, where we show that the set of lmt equlbrum payoffs wth mperfectly observed actons s the same as n the observed-acton case studed by Hörner and Lovo (2009) and Hörner, Lovo, and Tomala (2010) provded that the montorng structure satsfes a full-rank condton. Hörner, Lovo, and Tomala (2010) provde an equvalent characterzaton (for observed actons) that has a much dfferent form; each characterzaton may be better suted for some applcatons. Our results show that ther conclusons about lmt payoffs extend to mperfectly observed actons; ther work s complementary and more nformatve because t also explctly constructs equlbrum strateges. The assumpton of a known montorng structure also lets us provde a suffcent condton for the folk theorem that s easer to verfy: the key s that for every par of states ω and ω, there be at least three players whose prvate nformaton dstngushes between ω and ω ; Hörner, Lovo, and Tomala 4

6 (2010) use ths same condton to show n games wth observed actons the set of ex-post perfect equlbra s non-empty. In the case of one-sded ncomplete nformaton, we are able to further extend and refne ther results; for example, we fnd a smpler suffcent condton for the exstence of PTXE. 2 Framework 2.1 Model Let I = {1,,I} be the set of players. At the begnnng of the game, Nature chooses the state of the world ω from a fnte set Ω = {ω 1,,ω O }. Then each player observes a prvate sgnal, whch gves (possble mperfect) nformaton about the true state ω. The set of player s prvate sgnals, Θ, s a partton of Ω, and gven the true state ω Ω, he observes a prvate sgnal θ Θ that contans ω. For notatonal convenence, let θ (ω) denote ths θ,.e., ω θ (ω), and let θ(ω) = (θ (ω)) I. Gven θ Θ, player forms a pror about the true state ω, whch s denoted by µ (θ ) θ. Each perod, players move smultaneously, and player I chooses an acton a from a fnte set A. 5 Gven an acton profle a = (a ) I A I A, players observe a publc sgnal y from a fnte set Y accordng to the probablty functon π ω (a) Y ; we call the functon π ω the montorng structure. Player s realzed payoff s u ω (a,y), so that her expected payoff condtonal on ω Ω and a A s g ω (a) = y Y π ω y (a)u ω (a,y); g ω (a) denotes the vector of expected payoffs assocated wth acton profle a. If there are ω ω such that θ (ω) = θ (ω ) and u ω(a,y) u ω (a,y) for some a A and y Y, then we assume that player does not observe the realzed value of u as the game s played. 6 If there are no such ω ω, t s mmateral whether or not u s observed, as player can compute t from a, y, and θ. 7 5 All of our results extend mmedately to the case where A depends on θ. 6 As we explan n the next secton, the equlbra we consder reman equlbra when players are provded wth addtonal channels of nformaton about the state. Thus the assumpton that players do not observe ther realzed payoffs has no role n the results; t allows us to generalze past work (such as most of the references n footnote 2) that dd not requre players observe ther realzed payoffs. 7 We call ths the case of known own payoffs; note that t does not mply that each player knows the ther stage-game payoff functon g ω as that payoff s an expected value wth respect to 5

7 In the nfntely repeated game, players have a common dscount factor δ (0,1). Let (a τ,yτ ) be the realzed pure acton and observed sgnal n perod τ, and denote player s prvate hstory from perod one to perod perod t 1 by h t = (a τ,yτ ) t τ=1. Let h0 = /0, and for each t 0, let H t be the set of all h t. Lkewse, a publc hstory up to perod t 1 s denoted by h t = (y τ ) t τ=1, and Ht denotes the set of all h t. A strategy for player s defned to be a mappng s : Θ t=0 H t A. Let S be the set of all strateges for player, and let S = I S. We defne the feasble payoff set n a gven state ω to be V ω co{(g ω (a)) a A} = {g ω (η) η (A)}, where (A) s the set of all probablty dstrbutons over A, 8 and we defne the set of feasble payoffs of the overall game to be V ω Ω V ω. Note that a feasble payoff vector v V may be generated usng dfferent acton dstrbutons η ω n each state ω. If players observe ω at the start of the game and are very patent, then any payoff n V can be obtaned by state-contngent strategy of the nfntely repeated game. 2.2 Prelmnares Player s strategy s S s type-contngently publc f t depends only on θ Θ and h t H t, that s, f s (θ,h t ) = s (θ, h t ) whenever ht and h t correspond to the same publc hstory. A strategy profle s S s type-contngently publc f s s type-contngently publc for each I. Gven a type-contngently publc strategy profle s S, let s (θ,h t ) denote player s contnuaton strategy when hs type s θ and the past publc hstory s h t, and let s (θ,h t ) = (s (θ,h t )) I. 9 Ths paper studes a specal class of Nash equlbra called perfect type-contngently publc ex-post equlbra or PTXE. Defnton 1. A strategy profle s S s a perfect type-contngently publc expost equlbrum (PTXE) f s s type-contngently publc, and f for any ω Ω the possbly unknown dstrbuton π ω. 8 As n the standard case of a game wth a known montorng structure, the feasble set V ω s both the set of feasble average dscounted payoffs n the nfnte-horzon game when players are suffcently patent and the set of expected payoffs of the stage game that can be obtaned when players use of a publc randomzng devce to mplement dstrbuton η over the acton profles. 9 Here, the word contnuaton strategy s an abuse of language, because s (θ,h t ) s not a strategy for the entre game; t specfes a play for a gven type θ but not for θ θ. 6

8 and h t H t, s (θ(ω),h t ) s a Nash equlbrum of the nfntely repeated game wth Ω = {ω}. Remark 1. PTXE s an ex-post equlbrum concept n the sense that t requres each player s strategy s a best response rrespectve of the true value of the state. For ths reason, the set of PTXE s ndependent of the players belefs about the state, whch makes the analyss of equlbra much smpler. The ex-post property also mples that a PTXE for a gven partton Θ s also a PTXE for any fner partton; n partcular a PTXE for the trval partton (where players have no prvate nformaton) remans a PTXE when Θ s nformatve. The PPXE we consdered n Fudenberg and Yamamoto (2010) are the same as the PTXE wth the trval partton; the pont of ths paper s that fner parttons on Θ can support a larger set of PTXE, as n the examples of Secton 6.3, where there are effcent lmt PTXE but the PPXE are bounded away from effcency unformly n δ. Remark 2. A second consequence of the ex-post nature of PTXE s that a PTXE of the game where players do not observe ther realzed stage game payoffs remans a PTXE f players do observe these realzed payoffs and the payoffs reveal nformaton about ω. That sad, addtonal equlbrum outcomes could arse here under a relaxed equlbrum defnton that allowed players to condton on ths addtonal prvate nformaton. We do not nvestgate that possblty n ths paper. Gven a dscount factor δ (0,1), let E(δ) denote the set of PTXE payoffs,.e., E(δ) s the set of all vectors v = (v ω ) (,ω) I Ω R I Ω such that there s a PTXE s satsfyng (1 δ)e [ t=1 δ t 1 g ω(at ) ] s,ω = v ω for all and ω. Note that v E(δ) specfes the equlbrum payoff for all players and all states. Let α = (α θ ) θ Θ where α θ A for each θ Θ, and let α = ( α ) I. Thus α s an acton profle contngent on prvate nformaton; t specfes a mxed acton α for each prvate sgnal θ of each player. Let g( α) = (g ω (αθ(ω) )) (,ω) denote the payoff vector of type-contngent profle α. If the acton profle α s used ndependently of prvate nformaton, we denote ts payoff vector by g(α) = (g ω (α)) (,ω). By defnton, any contnuaton strategy s h t = (s θ(ω),h t) ω Ω of a PTXE s also a PTXE. Thus any PTXE specfes PTXE contnuaton play after each sgnal y, where the contnuaton payoffs w(y) = (w ω (y)) (,ω) I Ω correspondng to ths 7

9 sgnal specfy the payoffs for every player and every state. We wll wrte π ω (α) w ω for the the expected contnuaton payoff at state ω under acton profle α. In Fudenberg and Yamamoto (2010), we showed that the lmt of the equlbrum payoffs as δ 1 s determned by the solutons k ( α,λ,δ) to the followng famly of lnear programmng problems; for each type-contngent acton profle α, drecton λ R I Ω \ {0}, and δ (0,1), k ( α,λ,δ) = max λ v v R I Ω w:y R I Ω subject to () v ω = (1 δ)g ω (α θ(ω) ) + δπ ω (α θ(ω) ) w ω (1) for all, ω, () v ω (1 δ)g ω (a,α θ (ω) ) + δπ ω (a,α θ (ω) ) w ω (2) for all, ω, and a A, () λ v λ w(y) for all y Y. If there s no (v,w) satsfyng the constrants, we set k ( α,λ,δ) = ; f for every K > 0 there s (v,w) satsfyng all the constrants and λ v > K, then let k ( α,λ,δ) =. Here condton () s the addng-up condton, condton () s ex-post ncentve compatblty, and condton () requres that the contnuaton payoffs le n half-space correspondng to drecton vector λ and payoff vector v. Note that when λ ω 0 and λ j ω 0 for some ω ω, condton () allows utlty transfer across states. As argued n Fudenberg and Yamamoto (2010), the score k ( α,λ,δ) s ndependent of δ, so we denote t by k ( α,λ). Let k (λ) = sup α k( α,λ) be the hghest score attanable n drecton λ for any choce of α. For each λ R I Ω \ {0} and k R, let H(λ,k) = {v R I Ω λ v k}, wth H(λ,k) = R I Ω for k = or λ = 0, and H(λ,k) = /0 for k = and λ 0. Now let H (λ) = H(λ,k (λ)) be the maxmal half-space n drecton λ, and let Q = λ R I Ω H (λ). 8

10 The followng proposton establshes that the ntersecton Q of the maxmal halfspaces s equal to the lmt set of PTXE payoffs as δ 1. The proof s omtted, as t s smlar to Fudenberg and Yamamoto (2010), whch bulds on the technques of Fudenberg and Levne (1994). Proposton 1. If dmq = I Ω, then lm δ 1 E(δ) = Q. Our goal n ths paper s to use ths characterzaton to compute lm δ 1 E(δ) n some cases of nterest. To do ths we provde condtons under whch the maxmal half spaces n the varous drectons are large. 3 Examples Before developng our general results, we provde a few examples of PTXE to llustrate the ways that players can learn the state n equlbrum. Example 1. Let I = {1,2} and Ω = {ω 1,ω 2 }, Θ 1 = {(ω 1 ),(ω 2 )} and Θ 2 = {(ω 1,ω 2 )}. Player 1 chooses ether U or D, and player 2 chooses ether L or R. The payoffs for state ω 1 are n the left panel, and those for state ω 2 are n the rght. L R U 2, 2 0, 1 D 1, 0 1, 1 L R U 1, 1 0, 1 D 1, 0 2, 2 In ths example, both (U,L) and (D,R) are statc ex-post equlbra. Assume that Y = A and πy ω (a) = ε f y a. Note that the sgnal dstrbuton does not depend on the state here, so that players cannot learn the state from state-ndependent actons. Instead, the effcent outcome ((2, 2),(2, 2)) can be approxmated f player 1 reveals hs prvate nformaton to player 2 through hs actons. Specfcally, consder the followng three-phase automaton. Phase 1. Player 1 chooses U f θ 1 = (ω 1 ), and D f θ 1 = (ω 2 ). Player 2 chooses L. If the observed sgnal s y = (U,L) or y = (D,R), then go to Phase 2. If y = (D,L), then go to Phase 3. If y = (U,R), stay. Phase 2. Players choose (U,L) n the rest of the game. 9

11 Phase 3. Players choose (D,R) n the rest of the game. We clam that the strategy profle wth ntal state Phase 1 s a PTXE f δ s close to one and ε s close to zero. Frst, players do not want to devate n Phase 2 or Phase 3, as (U,L) and (D,R) are statc ex-post equlbra. Also, player 1 wth θ 1 = (ω 1 ) does not want to devate n Phase 1. Indeed, f he devates to D, then players are lkely to go to Phase 3 and play (D,R) forever, whle f he does not devate, then players are lkely to go to Phase 2 so that (U,L) s played thereafter. Lkewse, we can check that player 1 wth θ 1 = (ω 2 ) does not want to devate n Phase 1. Player 2 s prescrbed play s always a statc best response, and snce 2 s play has no effect on the transtons between stages 2 does not want to devate ether. Note that the payoffs of ths equlbrum converge to ((2 ε,2 ε),(2 ε,2 ε)). Example 2. The next example s a two-player partnershp game wth two actons {C,D } per player, three possble outcomes H, M, L, and two states. 10 The realzed payoff functons are ndependent of ω and gven by u (C,y) = r (y) e and u (D,y) = r (y) for each I, ω Ω, and y Y. We assume that the state only nfluences the productvty of player 2 s effort: If player 1 chooses C 1 nstead of D 1 then the probabltes of H and M ncrease by p H and p M, ndependent of the state. In contrast, f player 2 chooses C 2 nstead of D 2 then the probabltes of H and M ncrease by q H and q M n state ω 1, but they ncrease only by βq H and βq M n state ω 2. If β < 1, the states have dfferent outcome dstrbutons, so can be dentfed by repeated observaton. We mpose restrctons on the realzed payoffs so that the stage game payoffs n each state correspond to a prsoner s dlemma: D s a domnant strategy, so (D 1,D 2 ) s a statc ex-post equlbrum, (C 1,C 2 ) s effcent, and V has a non-empty nteror If there were only two outcomes as n Radner, Myerson, and Maskn (1986), then payoffs are bounded away from effcency even f the state s known, whle wth three outcomes the folk theorem holds for generc sgnal dstrbutons as FLM shows. 11 Specfcally we assume r (H) > r (M) > r (L); e 1 > p H (r 1 (H) r 1 (L)) + p M (r 1 (M) r 1 (L)); e 2 > q H (r 2 (H) r 2 (L)) + q M (r 2 (M) r 2 (L)); e 1 < p H (r 1 (H) + r 2 (H) r 1 (L) r 2 (L)) + p M (r 1 (M) + r 2 (M) r 1 (L) r 2 (L)); and e 2 < βq H (r 1 (H) + r 2 (H) r 1 (L) r 2 (L)) + βq M (r 1 (M) + r 2 (M) r 1 (L) r 2 (L)). 10

12 Usng our results, we wll show that a folk theorem holds f player 1 knows the state and player 2 does not, but that PTXE payoffs are bounded away from effcency for some parameters f player 2 knows the state and player 1 does not. The key s that player 2 can learn whether the true state s ω 1 or ω 2 by playng C 2 no matter what player 1 does, snce player 2 s margnal productvty s dependent on the state but not on player 1 s acton. Thus for the case n whch player only 1 knows the state, even f player 1 tres to hde hs prvate nformaton, player 2 can learn the true state from the sgnal dstrbuton. On the other hand, f only player 2 knows the state and he tres to hde t then player 1 cannot learn the true state. That s, f player 2 chooses D 2, then for gven any player 1 s acton, the sgnal dstrbuton s the same for both states, and hence player 1 cannot learn from the observed sgnals. See Secton 6.3 for detals. 4 Suffcent Condtons for Effcent State Learnng In ths secton we develop dstngushablty condtons that are suffcent for lmt equlbra n whch payoffs are as f players have learned the true state. In Secton 4.2, we relate these condtons to the ncentves and nformaton of the players; roughly speakng, the dstngushablty condtons are equvalent to assumng that f nformed players are wllng to reveal the state then unnformed players cannot prevent them from dong so. When the dstngushablty condtons are satsfed, the maxmal half spaces n cross-state drectons (those that gve non-zero weghts to payoffs n two or more states) are the whole space, so the cross-state drectons mpose no constrants on the lmt equlbrum payoffs. The maxmal half spaces n drectons that gve non-zero weghts to a sngle state are the same as n the known-state case consdered by Fudenberg, Levne, and Maskn (1994), so combnng FLM s assumptons, our dstngushablty assumptons, and Proposton 1 establshes the exstence of lmt equlbra wth the desred propertes. 4.1 Statewse Full Rank and Statewse Dstngushablty We begn wth the statewse full rank condton, whch s suffcent for the maxmal score to be nfnty for all cross-state drectons. For each (,ω) I Ω and 11

13 each type-contngent acton profle α I θ Θ A, let Π (,ω) ( α) be a matrx wth rows (πy ω (a,α θ (ω) )) y Y for all a A. Let Π (,ω)( j,ω )( α) be a matrx constructed by stackng two matrces, Π (,ω) ( α) and Π ( j,ω )( α). Defnton 2. For each (,ω) and ( j,ω ) satsfyng ω ω, profle α has statewse full rank for (,ω) and ( j,ω ) f Π (,ω)( j,ω )( α) has rank A + A j. Statewse full rank mples that players can dstngush ω and ω even f player at state ω or player j at state ω devates. For each par (,ω) and ( j,ω ), there s more lkely to be a type-contngent profle α that has statewse full rank as the parttons Θ l for player l become fner. The ntuton s that f player l has more nformaton, then t s easer for the players to learn the true state through nferences based on player l s actons. Note that PPXE does not allow players to condton ther play on ther types, so t rules out ths nformaton channel. We say more about ths ssue n Secton 4.2. The next lemma shows that statewse full rank s suffcent for the maxmal score to be nfnty for all cross-state drectons. We say that a type-contngent profle α s ex-post enforceable f there are v R I Ω, δ (0,1), and w = (w ω ) ω Ω : Y R I Ω such that (1) holds for all and ω, and (2) holds for all, ω, and a. The proof of the lemma s omtted, as t s very smlar to Lemma 6 of Fudenberg and Yamamoto (2010). Lemma 1. Suppose profle α s ex-post enforceable and has statewse full rank for (,ω) and ( j,ω ) satsfyng ω ω. Then, k ( α,λ) = for λ such that λ ω 0 and λ ω j 0. Whle the statewse full rank condton s suffcent for effcent learnng, t requres at least A + A j sgnals. The followng condton, statewse dstngushablty, can be satsfed wth fewer sgnals and s suffcent for the maxmal score to be nfnty for all cross-state drectons that have at least one postve component. We wll soon relax ths condton even further, but ths defnton s a useful expostonal tool for explanng the more complcated defntons to come. Defnton 3. Profle α statewse dstngushes (,ω) from ( j,ω ) f there s ξ = (ξ (y)) y Y R Y such that () π ω (α θ(ω) ) ξ > π ω (α θ(ω ) ) ξ, 12

14 () π ω (α θ(ω) ) ξ = π ω (a,α θ (ω) ) ξ π ω (a,αθ (ω) ) ξ for all a suppα θ (ω) and a A, () π ω (α θ(ω ) ) ξ = π ω (a j,α θ j(ω ) j ) ξ for all a j A j. To nterpret ths condton, wthout loss of generalty we assume π ω (α θ(ω ) ) ξ = 0. Clause () of ths condton assures that the sgnals generated by α statstcally dstngush ω from ω, and moreover pcks out a drecton ξ where the dfference has a partcular sgn. Clause () says that changng player s contnuaton payoff functon n state ω from w ω(y) to wω (y) + ξ (y) preserves ncentve compatblty for player, and clause () says that the change n player s contnuaton payoff (of w ω (y) ξ (y)) can be offset to preserve the feasblty constrant (λ ω w ω ω (y)+λ j w j ω (y) = 0) wthout changng player j s expected contnuaton payoff to any acton. Snce clause () mples π ω (α θ(ω) ) ξ > 0, ths change n the contnuaton payoffs ncreases player s expected contnuaton payoff at state ω, whch mples an ncreases n the score for λ such that λ ω > 0. Note that ths defnton s not symmetrc between and j because condton () s an nequalty and condton () s an equalty. When ths condton s satsfed, scalng up the vector ξ can generate arbtrarly large scores for all cross-state drectons λ that have at least one postve component. Our next step s to replace statewse dstngushablty wth an ensemble of three weaker condtons- ths ensemble s weaker because t wll allow dfferent acton profles to be used n dfferent drectons. Defnton 4. Profle α m-statewse dstngushes (,ω) from ( j,ω ) f there s ξ = (ξ (y)) y Y R Y such that () π ω (α θ(ω) ) ξ > π ω (α θ(ω ) ) ξ, () π ω (α θ(ω) ) ξ = π ω (a,α θ (ω) ) ξ π ω (a,αθ (ω) ) ξ for all a suppα θ (ω) and a A, () π ω (α θ(ω ) ) ξ = π ω (a j,α θ j(ω ) j ) ξ π ω (a j,αθ j(ω ) j ) ξ for all a j suppα θ j(ω ) j and a j A j. Note that ths condton relaxes statewse dstngushablty by replacng the last equalty n () wth an nequalty. Lemma 4(a) below shows that a profle that 13

15 m-statewse dstngushes (,ω) from ( j,ω ) can be used to generate an nfnte score for all λ such that λ ω > 0 and λ j ω < 0; the m refers to the fact that postve and negatve components are mxed n these drectons. Defnton 5. Profle α p-statewse dstngushes (,ω) from ( j,ω ) f there s ξ = (ξ (y)) y Y R Y such that () π ω (α θ(ω) ) ξ > π ω (α θ(ω ) ) ξ, () π ω (α θ(ω) ) ξ = π ω (a,α θ (ω) ) ξ π ω (a,αθ (ω) ) ξ for all a suppα θ (ω) and a A, () π ω (α θ(ω ) ) ξ = π ω (a j,α θ j(ω ) j ) ξ π ω (a j,αθ j(ω ) j ) ξ for all a j suppα θ j(ω ) j and a j A j. Lemma 4(b) below shows that a profle that p-statewse dstngushes (, ω) from ( j,ω ) can be used to generate an nfnte score for all postve drectons λ such that λ ω > 0 and λ j ω > 0. As ths suggests, ths condton s symmetrc: Lemma 2. Suppose α p-statewse dstngushes (,ω) from ( j,ω ). Then α p- statewse dstngushes ( j,ω ) from (,ω). Proof. Let ξ be a vector utlzed to p-statewse dstngush (,ω) from ( j,ω ). Then the vector ξ satsfes all the condtons of p-statewse dstngushablty of ( j,ω ) from (,ω). Q.E.D. Note that f α statewse dstngushes (,ω) from ( j,ω ), then t m-statewse dstngushes ths par and p-statewse dstngushes ths par. As we wll explan later, the combnaton of m- and p-statewse dstngushablty s suffcent for a statc-threat folk theorem. However, t s not suffcent for a perfect folk theorem, because the maxmal score mght not be hgh enough n cross-state drectons where all the non-zero components are negatve. The followng condton s suffcent for the score to be nfntely large for these drectons. Defnton 6. Profle α n-statewse dstngushes (,ω) from ( j,ω ) f there s ξ = (ξ (y)) y Y R Y such that 14

16 () π ω (α θ(ω) ) ξ > π ω (α θ(ω ) ) ξ, () π ω (α θ(ω) ) ξ = π ω (a,α θ (ω) ) ξ π ω (a,αθ (ω) ) ξ for all a suppα θ (ω) and a A, () π ω (α θ(ω ) ) ξ = π ω (a j,α θ j(ω ) j ) ξ π ω (a j,αθ j(ω ) j ) ξ for all a j suppα θ j(ω ) j and a j A j. Lemma 4(c) below shows that a profle that n-statewse dstngushes (, ω) from ( j,ω ) can be used to generate an nfnte score for all negatve drectons λ such that λ ω < 0 and λ j ω < 0. Also, n-statewse dstngushablty s symmetrc, as the next lemma shows. We omt the proof, snce t s very smlar to that of Lemma 2. Lemma 3. Suppose α n-statewse dstngushes (,ω) from ( j,ω ). Then α n- statewse dstngushes ( j,ω ) from (,ω). Now we state the man result of ths secton, whch shows that the score for cross-state drectons can be nfnty f the correspondng statewse condton s satsfed. The proof can be found n the appendx. Lemma 4. (a) Suppose α s ex-post enforceable and m-statewse dstngushes (, ω) from ( j,ω ). Then k (α,λ) = for λ such that λ ω > 0 and λ j ω < 0. (b) Suppose α s ex-post enforceable and p-statewse dstngushes (,ω) from ( j,ω ). Then k (α,λ) = for λ such that λ ω > 0 and λ j ω > 0. (c) Suppose α s ex-post enforceable and n-statewse dstngushes (, ω) from ( j,ω ). Then k (α,λ) = for λ such that λ ω < 0 and λ j ω < Suffcent Condtons for Statewse Dstngushablty In games wth ncomplete nformaton, players have three possble sources of nformaton about the state: () nference based on the publc sgnals at a statendependent acton profle; () the nformaton contaned n ther own types; and () nferences based on the correlaton between the opponents actons and the 15

17 opponents types. The frst nformaton source s studed by Fudenberg and Yamamoto (2010). The second nformaton source s suffcent for perfect learnng f every player can dstngush ω and ω. (Note that ths corresponds to assumpton () of condton (SFR) n Secton 5.) Here we nvestgate the thrd nformaton source: nferences based on the correlaton between the opponents actons and the opponents types. For ths nformaton to generate large scores n cross-state drectons, the nformed player must be wllng to reveal hs nformaton, and unnformed players must not jam the nformaton revelaton of ther nformed opponents. We address these ssues by provdng smple suffcent condtons under whch a type-contngent acton profle satsfes the varous dstngushablty condtons. Defnton 7. Player can reveal whether ω or ω f there are a A and a A such that π ω (a) π ω (a,a ). Ths says that player can generate dfferent sgnal dstrbutons at ω and ω, usng a type-contngent acton. Note that ths s necessary for players to learn the state from the correlaton between player s actons and hs types. As the next lemma shows, ths condton s suffcent for p-statewse dstngushablty for (,ω) and (,ω ). Lemma 5. Suppose θ (ω) θ (ω ) and player can reveal whether ω or ω. Then there s α that p-statewse dstngushes (,ω) from (,ω ). Proof. Let a A and a A be such that π ω (a) π ω (a,a ). Then there s ξ R Y such that π ω (a) ξ > π ω (a,a ) ξ. Let a argmax a π ω (a,a ) ξ and a argmn a π ω (a,a ) ξ. Let α be a type-contngent acton profle such that players play (a,a ) at state ω and (a,a ) at state ω. Then ths α p-statewse dstngushes (,ω) from (,ω ). Indeed, clause () follows from π ω (a,a ) ξ π ω (a) ξ > π ω (a,a ) ξ π ω (a,a ) ξ. Also, clauses () and () hold, by defnton of a and a. Q.E.D. To get the ntuton, recall that p-statewse dstngushablty s relevant to drectons λ that put postve weghts on payoff for (,ω) and (,ω ). In these drectons, player s payoffs at ω and ω are both maxmzed, so she s wllng to reveal her nformaton at both states. 16

18 In contrast, even f player can reveal whether ω or ω there mght be no profle that m-statewse dstngushes (,ω) from (,ω ). The reason s that f λ puts negatve weght on payoffs at (,ω ), player s payoffs at ω s mnmzed n the correspondng LP problem so that he mght not want to reveal the true state. However, the followng condton s suffcent for m-statewse dstngushablty; the dea s that player at state ω cannot conceal hs prvate nformaton f he cannot generate the same sgnal dstrbuton as n state ω. Defnton 8. Player at ω cannot hde state ω f there s a A such that π ω (a) s not n the convex hull of {π ω (a,a )} a A. Lemma 6. Suppose θ (ω) θ (ω ) and player at ω cannot hde state ω. Then there s α that m-statewse dstngushes (,ω) from (,ω ). Proof. Let a A be such that π ω (a) s not n the convex hull of {π ω (a,a )} a A. Then from the separatng hyperplane theorem, there s ξ such that π ω (a) ξ > π ω (a,a ) ξ for all a A. Let a argmax a πω (a,a ) ξ and a argmax a π ω (a,a ) ξ. Let α be a type-contngent acton profle such that players play (a,a ) at state ω and (a,a ) at state ω. We clam that ths α m-statewse dstngushes (,ω) from (,ω ). Clause () follows from π ω (a,a ) ξ π ω (a) ξ > π ω (a,a ) ξ. Also, clauses () and () hold, by defnton of a and a. Q.E.D. A smlar dea apples to n-statewse condton; here a relevant drecton λ puts negatve weghts on payoffs at (,ω) and (,ω ), so we need to take nto account player s ncentve for nformaton revelaton at both states. Defnton 9. Player cannot shuffle states ω and ω f there s a A such that the convex hull of {π ω (a,a )} a A and the convex hull of {π ω (a,a )} a A do not ntersect. Lemma 7. Suppose θ (ω) θ (ω ) and player cannot shuffle states ω and ω. Then there s α that n-statewse dstngushes (,ω) from (,ω ). Proof. Let a A be such that the convex hull of {π ω (a,a )} a A and the convex hull of {π ω (a,a )} a A do not ntersect. Then from the separatng hyperplane theorem, there s ξ such that π ω (a,a ) ξ > π ω (a,a ) ξ for all a A and a A. Let a argmn a π ω (a,a ) ξ and a argmax a π ω (a,a ) ξ. Let 17

19 α be a type-contngent acton profle such that players play (a,a ) at state ω and (a,a ) at state ω. Then as n the proof of the last lemma, we can show that ths α n-statewse dstngushes (,ω) from (,ω ). Q.E.D. Next we consder statewse dstngushablty for (,ω) and ( j,ω ) where j and only player knows the state;.e., θ (ω) θ (ω ) and θ j (ω) = θ j (ω ). Defnton 10. Player j at state ω s rrelevant for (,ω) f there are a A and a A such that π ω (a) s not a lnear combnaton of {π ω (a,a j,a j)} a j A j. Ths says that there s an acton profle a such that f player wants to reveal whether the true state s ω or ω by choosng a at ω and a at ω, the unnformed player j cannot nterfere, n the sense that a change n player j s acton at state ω cannot result n the overall dstrbuton when plays a beng the same as the dstrbuton n ω under a. For an example where ths condton fals, suppose that there are two players wth A = {a,a } for each, and that π ω (a 1,a 2 ) = πω (a 1,a 2 ) = πω (a 1,a 2 ) = πω (a 1,a 2 ) and πω (a 1,a 2 ) = π ω (a 1,a 2 ) = πω (a 1,a 2 ) = πω (a 1,a 2 ). Here player j at state ω s not rrelevant for (,ω). On the other hand ths condton can be satsfed even f player j has an opton to jam player s nformaton revelaton: Suppose that there are two states, ω 1 and ω 2, and that player 1 knows the state whle other players do not. Let A 1 = {U,D} and A 2 = {J,NJ}. Suppose that player 1 s acton s observable f player 2 chooses NJ, whle t s unobservable f player 2 chooses J. Suppose that player l s actons are always observable for each l 1. Let a be an acton profle such that a 1 = U and a 2 = NJ, and let a 1 = D. Then πω 1(a) s not a lnear combnaton of {π ω 2(a 1,a 2,a 12)} a 2 A j, so that player 2 at state ω 2 s rrelevant for (1,ω 1 ). The followng lemma shows that ths rrelevance condton s suffcent for statewse dstngushablty (and hence suffcent for p- and m-statewse dstngushablty). Lemma 8. Suppose θ (ω) θ (ω ) and player j at ω s rrelevant for (,ω). Then there s α that statewse dstngushes (,ω) from ( j,ω ). Proof. Let a A and a A be such that π ω (a) s not a lnear combnaton of {π ω (a,a j,a j)} a j A j. Then there s ξ such that π ω (a) > 0 and π ω (a,a j,a j) 18

20 ξ = 0 for all a j A j. Let a argmax a A πω (a,a ) ξ, and let α be a typecontngent acton profle such that players play (a,a ) at state ω and (a,a ) at state ω. We clam that ths α statewse dstngushes (,ω) from ( j,ω ). Clause () of statewse dstngushablty follows from a argmax a A πω (a,a ) ξ. Also, snce π ω (a,a ) π ω (a) > 0 and π ω (a,a j,a j) ξ = 0 for all a j A j, clause () and () hold. Q.E.D. The ntuton s as follows. Recall that statewse dstngushablty s a combnaton of p- and m-statewse dstngushablty, so the correspondng drecton λ gves postve weght to player s payoff at state ω. Therefore, player at state ω s wllng to reveal hs prvate nformaton. Also, player j at state ω cannot nterfere wth ths nformaton revelaton. For n-statewse dstngushablty, we need a stronger assumpton, snce the correspondng λ puts negatve weght on player s payoff at state ω, so that he mght want to hde hs prvate nformaton. Defnton 11. Player j at state ω s strongly rrelevant for (,ω) f there are (a,a ) A such that any convex combnaton of {π ω (a,a )} a A s not a lnear combnaton of {π ω (a,a j,a j)} a j A j. Ths condton s a combnaton of cannot hde and rrelevant condtons: here player at state ω cannot conceal hs prvate nformaton and player j at state ω s rrelevant to player s nformaton revelaton. Lemma 9. Suppose θ (ω) θ (ω ) and player j at ω s strongly rrelevant for (,ω). Then there s α that n-statewse dstngushes (,ω) from ( j,ω ). Proof. Let a A and a A be such that any convex combnaton of {π ω (a,a )} a A s not a lnear combnaton of {π ω (a,a j,a j)} a j A j. Then there s ξ such that π ω (a,a ) > 0 for all a A and π ω (a,a j,a j) ξ = 0 for all a j A j. Let a argmn a A πω (a,a ) ξ, and let α be a type-contngent acton profle such that players play (a,a ) at state ω and (a,a ) at state ω. Then as n Lemma 8, we can show that ths α n-statewse dstngushes (,ω) from ( j,ω ). Q.E.D. When player j knows the state and player does not (.e., θ j (ω) θ j (ω ) and θ (ω) = θ (ω )), the statewse condtons are satsfed under the same condtons 19

21 as the case where player knows the state. Recall that p- and n-statewse condtons are symmetrc (Lemmas 2 and 3), so the suffcent condtons for the prevous case apply. For the m-statewse condton, we obtan the followng lemma. Lemma 10. Suppose θ j (ω) θ j (ω ) and player j at ω s strongly rrelevant for ( j,ω ). Then there s α that m-statewse dstngushes (,ω) from ( j,ω ). Proof. Let a A and a j A j be such that any convex combnaton of {π ω (a j,a j)} a j A j s not a lnear combnaton of {π ω (a,a j,a j)} a A. Then there s ξ such that π ω (a j,a j) < 0 for all a j A j and π ω (a,a j,a j) ξ = 0 for all a A. Let a j argmax a j A j πω (a,a j,a j) ξ, and let α be a type-contngent acton profle such that players play (a j,a j) at state ω and (a j,a j) at state ω. Then we can show that ths α n-statewse dstngushes (,ω) from ( j,ω ). Q.E.D. Fnally we consder pars (,ω) and ( j,ω ) where there s a player l, j who knows the state (here possbly = j). If ether player or j can dstngush ω from ω, then the prevous lemmas stll apply. Thus the nterestng case s when both player and j do not know the state. Defnton 12. Both player at ω and player j at ω are rrelevant for nformaton revelaton by player l, j f there are a A and a l A l such that any lnear combnaton of {π ω (a,a l,a l )} a A s not a lnear combnaton of {π ω (a j,a l,a jl)} a j A j. Ths says that f player l wants to reveal hs prvate nformaton then nether player at ω nor player j at ω can nterfere. The next lemma shows that ths condton s suffcent for p-, m-, and n-statewse dstngushablty. Lemma 11. Suppose there s I, j I, l, j such that θ l (ω) θ l (ω ) and that both player at ω and player j at ω are rrelevant for nformaton revelaton by player l. Then there s an α that p-, m-, and n-statewse dstngushes (,ω) from ( j,ω ). Proof. Let a A and a l A l be such that any lnear combnaton of {π ω (a,a l,a l )} a A s not a lnear combnaton of {π ω (a j,a l,a jl)} a j A j. Then there are ξ and κ > 0 such that π ω (a,a,a l ) = κ for all a A and π ω (a j,a l,a jl) ξ = 0 for all a j A j. Let α be a type-contngent acton profle such that players play a at state ω and (a l,a l) at state ω. Then t s easy to check that ths α satsfes all the condtons of p-, m-, and n-statewse dstngushablty. Q.E.D. 20

22 5 Ex-Post Folk Theorems In ths secton we provde two sorts of folk theorem n PTXE: The frst shows that all feasble ndvdually ratonal payoffs can be approxmated by payoffs of PTXE, and the second uses weaker condtons to obtan a statc-threats verson. In both cases, the key s fndng the approprate condtons on the combnaton of ntal prvate nformaton and the nformaton revealed by the publc outcomes. Recall that Π (,ω) ( α) s a matrx wth rows (π ω y (a,α θ (ω) )) y Y for all a A, and that Π (,ω)( j,ω )( α) s a matrx constructed by stackng two matrces, Π (,ω) ( α) and Π ( j,ω )( α). Defnton 13. Profle α has ndvdual full rank for (,ω) f Π (,ω) ( α) has rank A. Profle α has ndvdual full rank f t has ndvdual full rank for all players and all states. Ths condton mples that at each state, every possble devaton of any one player leads to a statstcally dfferent dstrbuton on outcomes. Defnton 14. For each (,ω) and ( j,ω) satsfyng j, profle α has parwse full rank for (,ω) and ( j,ω) f Π (,ω)( j,ω) ( α) has rank A + A j 1. Note that parwse full rank mples ndvdual full rank; t mples that devatons by one player can be dstngushed from devatons by another. Condton IFR. Every pure acton profle α has ndvdual full rank. Condton PFR. For each (,ω) and ( j,ω) satsfyng j, there s a profle α that has parwse full rank for (,ω) and ( j,ω). Condton SFR. For each par of states (ω,ω ) satsfyng ω ω, at least one of the followng two condtons holds: () For each and j (possbly = j), there s a profle α that has statewse full rank for (,ω) and ( j,ω ). () θ l (ω) θ l (ω ) for all l I. (SFR) requres that for each par of states ω and ω ω, ether () for every (, j) there s a profle that lets players dstngush state ω from state ω, regardless of whether player devates n state ω or player j devates n state ω, or () players can dstngush these ω and ω usng ther prvate nformaton θ. 21

23 Note that (SFR) fals for (,ω) and (,ω ) f π ω s ndependent of ω (so that the montorng structure s known) and θ j (ω) = θ j (ω ) for all j (so no player s prvate nformaton dstngushes between ω and ω ). We say more about the case of a known montorng structure n Secton 7. The next proposton establshes a general folk theorem n PTXE. Let V {v V I ω Ω v ω v ω } where vω = mn α max a g ω (a,α ). A subset W of R I Ω s smooth f t s closed and convex; t has a non-empty nteror; and there s a unque unt normal for each pont on bdw. 12 Proposton 2. Suppose (IFR), (PFR), and (SFR) hold. Then for any smooth strct subset W of V, there s δ (0,1) such that W E(δ) for all δ (δ,1). To prove ths proposton, we compute the maxmal scores for each drecton. The key pont s that (SFR) mples the maxmal score for cross-state drectons can be made large enough to establsh the folk theorem. When the frst condton n (SFR) holds, that concluson comes from Lemma 1. When the second condton holds, the followng lemma apples: Lemma 12. Suppose (PFR) and (IFR) hold. Let λ be such that θ (ω) θ (ω ) for all I, ω Ω and ω ω satsfyng (λ j ω ) j I 0 and (λ j ω ) j I 0. Then k (λ) max v V λ v. Ths lemma shows that the maxmal score n cross-state drectons doesn t exclude any feasble payoffs f all players know the state. The ntuton behnd the lemma s smple. If each player can dstngush ω and ω usng prvate nformaton θ, players can choose dfferent acton profles contngent on whether the true state s ω or ω. Therefore we expect that the score on state ω wll not constran the score on state ω so that the maxmal score for drectons vectors that only weght these two states wll be hgh enough to acheve the folk theorem. The formal proof s delegated to the appendx. Combnng ths lemma and Lemma 1 shows that the maxmum score n all cross-state drectons s at least max v V λ v. Ths mples that the set Q s determned by λ that has non-zero components only for a sngle state. The followng lemmas show that (IFR) and (PFR) mply that the maxmal score for such 12 A suffcent condton for each pont on bdw to have a unque unt normal s that bdw s a C 2 -submanfold of R I Ω. 22

24 drectons s max v V λ v. The proofs are omtted, as they are straghtforward generalzatons of FLM. Lemma 13. Suppose (PFR) holds. Then k (λ) = max v V λ v for all λ such that () (λ ω ) I 0 for some ω and (λ ω ) I = 0 for all ω ω, and () (λ ω ) I has at least two non-zero components or at least one postve component. Lemma 14. Suppose (IFR) holds. Then k (λ) = max v V λ v for all λ such that λ ω < 0 for some (,ω) and λ j ω = 0 for all ( j,ω ) (,ω). From these lemmas, we obtan Q = V and hence Proposton 2 follows. Thus the folk theorem obtans f (IFR), (PFR), and (SFR) hold and f V s full dmensonal. As we have seen n Secton 4.1, statewse full rank s stronger than needed for effcent learnng, and can be replaced wth statewse dstngushablty. Condton Pontwse-SD. For each ω and ω satsfyng ω ω, at least one of the followng condtons holds: () For each and j (possbly = j), there s an ex-post enforceable acton profle α that m-statewse dstngushes (, ω) from ( j,ω ), there s an ex-post enforceable acton profle α that p-statewse dstngushes (,ω) from ( j,ω ), and there s an ex-post enforceable acton profle α that n-statewse dstngushes (,ω) from ( j,ω ). () θ l (ω) θ(ω ) for all l I. Ths says that for each par of states ω and ω ω, ether () for every (, j) there s a profle that lets players dstngush state ω from state ω, regardless of whether player devates n state ω or player j devates n state ω, or () players can dstngush these ω and ω usng ther prvate nformaton θ. Note that (Pontwse-SD) s weaker than (SFR), snce f α has statewse full rank, then t satsfes the m-, p-, and n-statewse dstngushablty condtons, but the converse s false, as the pontwse condton allows dfferent profles to be used for dfferent drectons. On the other hand, (Pontwse-SD) s a strong form of statewse dstngushablty as t requres the n-statewse condton. Proposton 3. Suppose (IFR), (PFR), and (Pontwse-SD) hold. Then for any smooth strct subset W of V, there s δ (0,1) such that W E(δ) for all δ (δ,1). 23

25 The proof of ths proposton parallels to that of Proposton 2, wth the dfference that Lemma 4 s used nstead of Lemma 1 for the concluson that the maxmum scores n cross-state drectons s nfnte. An even weaker condton s suffcent for a statc-threat folk theorem: For that result t s suffcent that the m- and p-statewse condtons can each be satsfed for some profle. Condton Pontwse-WeakSD. For each ω and ω satsfyng ω ω, at least one of the followng condtons holds: () For each and j (possbly = j), there s an ex-post enforceable acton profle α that m-statewse dstngushes (, ω) from ( j,ω ) and there s an ex-post enforceable acton profle α that p-statewse dstngushes (,ω) from ( j,ω ). () θ l (ω) θ(ω ) for all l I. Proposton 4. Suppose (PFR) and (Pontwse-WeakSD) hold. Assume that there s an ex-post equlbrum α 0,.e., α such that α θ (ω) argmax α g ω(α,α θ (ω) ) for all I and ω Ω. Let V 0 {v V I, ω Ω v ω g ω(α0 )}. Then, for any smooth strct subset W of V 0, there s δ (0,1) such that W E(δ) for all δ (δ,1). The proof of ths proposton s smlar to that of Proposton 3, wth the followng dfferences. In ths proposton, we do not assume (IFR) or (Pontwse- SD), so that Lemma 4(c) and Lemma 14 may not apply. Therefore, t mght be that k (λ) < max v V λ v for each λ 0 such that λ ω 0 for all (,ω). For these drectons, we apply the next lemma to show that k (λ) max v V 0 λ v. The proof s straghtforward and hence omtted. Lemma 15. Suppose there s a statc ex-post equlbrum α 0. Then, for any drecton λ, k ( α 0,λ) λ g( α 0 ). Also, snce Proposton 4 does not assume (IFR), Lemma 12 does not apply, so t mght be that k (λ) < max v V λ v for some cross-state drectons λ. For these drectons, we use the followng lemma to show that k (λ) max v V 0 λ v. The proof of the lemma can be found n the appendx. Lemma 16. Suppose (PFR) holds. Let λ be such that θ (ω) θ (ω ) for all I, ω Ω and ω ω satsfyng (λ j ω ) j I 0 and (λ j ω ) j I 0. Then, k (λ) max v V 0 λ v. 24