Journal of Mathematical Economics

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1 Journal of Mathematcal Economcs 46 (2010) Contents lsts avalable at ScenceDrect Journal of Mathematcal Economcs journal homepage: A note on the evaluaton of nformaton n zero-sum repeated games Ehud Lehrer a,b, Dnah Rosenberg c,d, a School of Mathematcal Scences, Tel Avv Unversty, Tel Avv 69978, Israel b INSEAD, Bd. de Constance, Fontanebleau Cedex, France c HEC department of economcs and decson scences, 1, rue de la lbératon Jouy en Josas, France d Laboratore d Econométre de l Ecole Polytechnque, Pars, France artcle nfo abstract Artcle hstory: Receved 9 March 2006 Receved n revsed form 19 August 2007 Accepted 21 February 2010 Avalable onlne 12 March 2010 Keywords: Repeated games Zero-sum games Value-of-nformaton Two players play a zero-sum repeated game wth ncomplete nformaton. Before the game starts one player receves a prvate sgnal that depends on the realzed state of nature. The rules that govern the choce of the sgnal are determned by the nformaton structure of the game. Dfferent nformaton structures nduce dfferent values. The value-of-nformaton functon of a game assocates every nformaton structure wth the value t nduces. We characterze those functons that are value-of-nformaton functons for some zero-sum repeated game wth ncomplete nformaton Elsever B.V. All rghts reserved. 1. Introducton In Bayesan games wth nformaton structures players have a pror dstrbuton of and receve a partal nformaton on the game actually played. Ths nformaton affects the players posteror dstrbuton and thereby the equlbrum payoffs of the game. An outsde observer, who does not observe the pror dstrbuton nor the players actons, collects data about the outcomes assocated wth dfferent nformaton structures. The queston arses as to whch observatons can refute the model of ratonal players playng a Bayesan game. Ths ssue was frst addressed by Glboa and Lehrer (1991) who dealt wth a decson maker who knows the cell of a partton contanng the realzed state (as n Aumann, 1976). We consder a smlar queston n the framework of repeated zero-sum games wth ncomplete nformaton that were ntroduced and studed by Aumann and Maschler (1995). The frst advantage of zero-sum games s that they have a unque equlbrum payoff, the value. The second advantage s that n these games recevng more nformaton can never be harmful. The latter property s partcular to zero-sum games, as shown frst by Hrshlefer (1971) and later by Kamen et al. (1990) and Bassan et al. (1999). Gossner and Mertens (2001) compared dfferent nformaton structures n zero-sum repeated games wth ncomplete nformaton. Pror to startng the game one player s nformed of the cell of a partton that contans the realzed state. Dfferent parttons nduce dfferent repeated games and typcally result n dfferent values. We called a value-of-nformaton functon the one that assocates parttons wth the correspondng values n the nduced Bayesan games. We prove that any monotonc functon over parttons s a value-of-nformaton functon of some repeated game wth ncomplete nformaton Ths research was supported by a grant from the Mnstry of Scence and Technology, Israel, and the Mnstry of Research, France. The frst author acknowledges the support of the Israel Scence Foundaton (Grant #762/045). Correspondng author at: HEC department of economcs and decson scences, 1, rue de la lbératon Jouy en Josas, France. E-mal addresses: rosenberg@hec.fr, dnah@math.unv-pars13.fr (D. Rosenberg). URL: lehrer (E. Lehrer) /$ see front matter 2010 Elsever B.V. All rghts reserved. do: /j.jmateco

2 394 E. Lehrer, D. Rosenberg / Journal of Mathematcal Economcs 46 (2010) on one sde. Hence, essentally no observed outcomes wth respect to nformaton can dsprove the asserton that agents are Bayesan ratonal. Ths result stands n sharp contrast to one-player decson problems where an addtonal condton beyond monotoncty s needed. A companon paper (Lehrer and Rosenberg, 2006) deals wth value-of-nformaton functons of one-shot zero-sum games. Whle n one-shot games t s always optmal to use all avalable nformaton, n repeated games a full use of the nformaton can be sub-optmal. One mght therefore expect that recevng more nformaton n repeated games s less benefcal than n one-shot games and that the mpact of nformaton on the value n one-shot games and n repeated games s dfferent. It turns out that ths ntuton s msleadng and the sets of the value-of-nformaton functons for one-shot and repeated games concde. Whle the problems treated here and n Lehrer and Rosenberg (2006) are smlar, the proofs n the two papers use totally dfferent technques. The proof uses a few propertes of the value functon of one-shot zero-sum Bayesan games n whch players know only the pror. 1 As a by-product we characterze these functons n the case of two states and leave the general problem open. Ths paper s organzed as follows. The next secton presents the model of nformaton structures n repeated games wth one-sded nformaton. Secton 3 descrbes the man result. A sketch of the proof s gven n Secton 4, and the paper ends wth fnal comments and open problems. 2. The model 2.1. Informaton structures We consder two-player games wth ncomplete nformaton. A state k s randomly selected from a fnte state space K accordng to a common pror p. One-sded parttonal nformaton structure (or smply nformaton structure) s represented by a partton Q of the set K. Denote by Q(k) the cell of Q that contans k. When k s realzed, player 1 gets to know Q(k) and player 2 knows nothng about k beyond the pror p. Note that any game has fntely many parttonal nformaton structures The repeated game The n-stage game wth one-sded nformaton, denoted by Ɣ n (p, Q), s defned by an nteger n, an nformaton structure Q, a probablty p over K, a fnte set of actons for each player {1, 2}, A, and a payoff functon g from K A 1 A 2 to the reals. The payoff assocated wth (k, a 1,a 2 ) s denoted by g k (a 1,a 2 ). The game s played as follows: At stage 0, nature chooses an element k of K wth probablty p, player 1 s then nformed of Q(k). The game s played n n stages. At stage m = 1,...,n, players 1 and 2 smultaneously choose actons accordng to probablty dstrbutons that may depend on the hstory of prevous actons and sgnals. If the realzed state s k and the par of chosen actons s (a m 1,am 2 ) the payoff at stage m s g k(a m 1,am ). The par of chosen actons (and not the payoff) s then 2 announced to both players and the game proceeds to the next stage. The payoff n Ɣ n (p, Q) s the expected average of the n-stage payoffs receved durng the game. A behavor strategy of player 1 s a sequence 1 = ( 1 1,2 1,...,m 1,...), where m s a functon from hs nformaton at stage 1 m, Q (A 1 A 2 ) m 1 to the set 2 (A 1 ) of probablty dstrbutons over hs set of actons. A behavor strategy of player 2 s a sequence 2 = ( 1 2,2 2,...,m 2,...), where m 2 s a functon from hs nformaton at stage m, (A 1 A 2 ) m 1 to the set (A 2 ). When appled to the game Ɣ n (p, Q), all m, m>n,are payoff rrelevant. The probablty dstrbuton p, the partton Q and the par of strateges 1, 2 nduce a probablty over the set of hstores of length n, H n = K (A 1 A 2 ) n. The expectaton wth respect to ths probablty wll be denoted by E p,q 1 2 or smply by E when no confuson can arse. If the players use the strateges 1, 2, the assocated payoff n the n-stage game s n Q ( 1, 2,p) = E p,q n 1 2 [1/n m=1 g k(a m 1,am )]. Ths s the expected average payoff receved along the n stages of the game. 2 The game Ɣ n (p, Q) s a fnte game and therefore, by the mnmax theorem, has a value denoted by v Q n (p, (g k ) k K ). The followng proposton whose proof s omtted states that ths sequence converges. Proposton 1. The sequence v Q n (p, (g k ) k K ) has a lmt denoted by v Q (p, (g k ) k K ). The functon v Q (p, (g k ) k K ) wll be referred to as the long-run value of the game. Ths s an approxmaton of the equlbrum payoff n long games. A long duraton of the games mples that player 1 mght want to lmt hs use of avalable nformaton because t may gve a relatve nformatonal advantage to player 2. Excessve use of nformaton can ncrease player 1 s payoffs n the short-run but mght be harmful to hm n the long run. 1 See Aumann and Maschler (1995) and Mertens and Zamr (1971). 2 Throughout ths paper (X) denotes the set of probablty dstrbutons over a set X.

3 E. Lehrer, D. Rosenberg / Journal of Mathematcal Economcs 46 (2010) Value-of-nformaton functons We focus on the long-run value of the game vewed as a functon of the nformaton structure. In the one-sded nformaton case, let (g k ) k K be a payoff functon and let p be a dstrbuton over K. When the nformaton s gven through Q the value of the nduced game s denoted v Q (p, (g k ) k K ). Defnton 1. A functon f defned over all parttons of K s the value-of-nformaton of an ncomplete nformaton game wth one-sded nformaton and parttonal nformaton structure f there exst () a dstrbuton p over K and () payoff functons (g k ) k K such that for any partton Q over K, f (Q) = v Q (p, (g k ) k K ). 3. The result Descrbng the propertes that value-of-nformaton functons must owe n repeated zero-sum games s the man theme of ths note. Snce n zero-sum games t s always benefcal for a player to have more nformaton, the followng monotoncty condton s clearly necessary for a value-of-nformaton functon. Defnton 2. A functon v from the set of parttons of a fnte set K to the real numbers s sad to be monotonc f for any two parttons Q and Q, the fact that Q refnes Q (.e., for any T Q there s an T Q such that T T ) mples v(q) v(q ). The followng theorem states that ths condton s also suffcent. Theorem 1. Let V be a functon from the set of parttons of K to the real numbers. The functon V s a value-of-nformaton functon of a repeated game wth state space K, one-sded nformaton, and parttonal sgnalng f and only f t s monotonc. The sketch of the proof of ths theorem s gven n Secton 4 and a detaled proof s postponed to the Appendx A. The man mplcaton of ths theorem s that even when one takes nto account the strategc use of nformaton the Bayesan paradgm mposes no restrcton on value-of-nformaton functons beyond monotoncty. It mples, for nstance, that there s no restrcton on the frst dervatve of the value, whch could mean that the effect of a fxed addton of nformaton on the value may go down or up wth the nformaton that already exsts. Lehrer and Rosenberg (2006) proved that the same characterzaton holds for one-shot games. Therefore, repetton does not affect the set of value-of-nformaton functons, although t mght affect the value of a partcular game. 4. The proof In ths secton, we provde a sketch of Theorem 1 s proof. An elaborate proof of ths theorem s deferred to Appendx A. The proof employs a few tools that wll be descrbed n the followng subsectons The one-shot game The frst tool we need s the value of the one-shot Bayesan game. Suppose that p = (p k ) (K). The value of the one-shot Bayesan game wth null nformaton (.e., no player obtans addtonal nformaton about the realzed state beyond the pror p) s denoted u(p). Formally, Notaton 1. u(p) s the value of the game defned by the acton sets A 1,A 2 and the payoff k K p(k)g k(a 1,a 2 ), when the par of actons (a 1,a 2 ) s played. Ths functon also plays a central role n the theory of repeated games wth ncomplete nformaton (see Aumann and Maschler, 1995). Defnton 3. A functon u defned on (K) srealzable f there are games G k, k K, wth the same acton sets and payoff functons g k, such that the value of Ḡ(p) = k K p(k)g k s u(p) for every p n (K), where Ḡ(p) s the game wth the same acton sets and payoff functon k K p(k)g k. The proof of Theorem 1 reles on the fact that the set of realzable functons s large. Mertens and Zamr (1971) proved that the set of realzable functons s dense n the set of contnuous functons. Ths s not suffcent for our purposes snce we need a precse realzaton and not merely an approxmaton. The followng proposton s what we need and s specfc to repeated games n whch the long-run value s characterzed by the value of the game wth no nformaton. 3 3 It s worth emphaszng that the proof of the man result hnges on Proposton 2 whch bears no relaton to the proof of a smlar result for one-shot games that bulds on Glboa and Lehrer (1991).

4 396 E. Lehrer, D. Rosenberg / Journal of Mathematcal Economcs 46 (2010) Proposton 2. () Gven a fnte number of pars (x l,y l ) (K) IR, l = 1,...,L, there s a realzable functon u such that u(x l ) = y l, l = 1,...,L. () If C 1 s fnte, C 2 s a unon of closed polygons, C 1 C 2 = and c 1 and c 2 are two numbers such that c 1 >c 2, then there s a realzable functon u that satsfes u(x) c 2 when x C 2, u(x) = c 1 when x C 1, and u(x) c 1 otherwse. () Let C 1 and C 2 be two dsjont closed sem-algebrac sets, and f 1 and f 2 two realzable functons. Then, there s a realzable functon u that satsfes u(x) = f 1 (x) when x C 1, and u(x) = f 2 (x) when x C The long-run value Let f be a real-valued functon defned on (K) and cav(f ) the mnmal concave functon whch s greater than or equal to f. For a partton Q and B Q let p(b) = k B p(k). Denote by p( B) the condtonal probablty over K, gven B K (.e., p (k B) = p(k)/p(b), k B). Notaton 2. For a partton Q denote by M(Q) the set of matrces M = (m B ) D wth D lnes (D can be any postve nteger) B Q and Q columns such that for any, B, m B 0, and for any B, D m B = p(b). For such a matrx we denote by m the quantty B Q m B, and by p (M) the probablty dstrbuton over K defned by B Q p (M)(k) = m B p (k B) m. The followng proposton extends Aumann and Maschler (1995) that states that when player 1 s fully nformed of k the long-run value s cav(u)(p). Proposton 3. The value of the game wth one-sded nformaton s { } v Q (p, (g k ) k K ) = max m u(p (M)) M M(Q). (1) D The concavfcaton of u can also be defned as cav(u)(p) = max{ D m u(p ) m 0, m = 1, m p = p}. Proposton 3 states that the long-run value s the maxmum over a smaller class of possble (m,p ), D. In other words, v Q (p, (g k ) k K ) s a knd of a local concavfcaton of the functon u A sketch of the proof of Theorem 1 Let V be a monotonc functon of parttons. Our goal s to construct a game for whch the long-run value assocated wth the partton Q s equal to V(Q). To ths end we construct a realzable functon u such that V(Q) = v Q (p, (g k ) k K ) as defned by Eq. (1). Let p be the unform probablty (1/ K,...,1/ K ). Note that n order to prove that V s a value-of-nformaton functon we need the exstence of a probablty p, whch can therefore be chosen to be the unform one. A probablty dstrbuton over the state space s represented by a pont n the smplex. A subset B of K s represented by the pont n the smplex assocated wth the condtonal probablty ( B). A partton s represented by a lnear subspace, denoted by H(Q), spanned by ( B), where B runs over all the cells of the partton. Note that H(Q) H(Q ) f and only f Q refnes Q. Note, moreover, that for every partton Q, p H(Q) (see the fgure below). To satsfy Eq. (1) t s suffcent to fnd u that satsfes the followng two condtons: () If Q refnes Q, then u s less than or equal to V(Q )onh(q); and () for each partton Q there exst ponts p 1,...,p Q n H(Q), such that u(p l ) = V(Q) and p can be wrtten as a convex combnaton of p 1,...,p Q. Such a realzable functon exsts by Proposton 2. The followng fgure llustrates the case of K ={1, 2, 3}. The partton Q ={1, 2, 3} s represented by the 0-dmensonal space H({1, 2, 3}), whch s the center of the trangle, denoted n the fgure as {1, 2, 3}; the parttons of K nto two sets are represented by lnes and H({{1}, {2}, {3}}) s the whole smplex. The ponts marked are on H({{1}, {2, 3}}) and the center of the trangle s a convex combnaton of them.

5 E. Lehrer, D. Rosenberg / Journal of Mathematcal Economcs 46 (2010) The detaled proof, provded n the appendx, shows that f V s monotonc, then there s a realzable functon u such that the value u attans at the center s less than or equal to V({{1}, {2, 3}}), whle the values u attans at the ponts marked are equal to t. 5. Comments and open problems 5.1. General nformaton structures The man theorem dscusses parttonal structures. It would be nterestng to study value-of-nformaton functons defned over more general sgnalng structures Repeated games wth lack of nformaton on both sdes The model studed here can be extended to stuatons where player 1 s nformed through a partton P and player 2 through a partton Q. Mertens and Zamr (1971) proved the exstence of the long-run value of the game (denoted by v P,Q (p, g)) n ths case (defned as the lmt of the values of n-stage games as n goes to nfnty, or of -dscounted games as goes to 0). Characterzng the value-of-nformaton functons n ths case s an open problem The general null nformaton case Null nformaton s the case where player 2 gets no nformaton at all on the state. One can fully characterze the value of games wth null nformaton and only two states. Indeed, along the same lnes as the one provded n the appendx, one can prove that when there are two states, f f s a real polynomal that does not vansh, then 1/f s realzable. Ths n turn mples that a functon s realzable f and only f t s pecewse ratonal. In the case of more than two states we could prove that every polynomal s realzable, and that, for every contnuous and pecewse ratonal functon q and every ε>0, there s a realzable functon u that concdes wth q over a set that occupes (n the sense of Lesbegue measure) 1 ε of (K). We conjecture that every contnuous and pecewse ratonal functon over (K), as n the two-state case, s realzable. Appendx A. The proof of Proposton 2 s broken nto three parts, one for each tem. Proof of Proposton 2 () Lemma A.1. If u and v are realzable, then so s mn(u, v). If the game G 1,...,G K realzes u and G 1,...,G realzes v then the followng two stage game realzes mn(u, v): = player K 2 frst chooses acton L or R, player 1 sees the choce and then the game s G 1,...,G K f the choce was L and G 1,...,G K f t was R. Lemma A.2. If u s realzable, then so s u 2. Proof. If u s realzable by the game G 1,...,G K, then the game whose -th matrx game s ( ) G 0 realzes u 2. 2G 1 G Lemma A.3. Any polynomal s realzable. Proof. Mertens and Zamr (1971) showed ( that f u and v are realzable then so s u + v. ByLemma A.2, (u + v) 2 s also realzable as are u 2 and v 2. Therefore, uv = 1 2 (u + v) 2 u 2 v 2) s also realzable.

6 398 E. Lehrer, D. Rosenberg / Journal of Mathematcal Economcs 46 (2010) Moreover, any constant functon s realzable. The game G 1,...,G K all of whose matrces are dentcally 0, except for the -th one whch s dentcally 1, realzes the polynomal f (p) = p. Any polynomal s therefore realzable by teratvely addng and multplyng constants and the polynomals f. Proposton 2 () follows from the fact that for every (x l,y l ) (K) IR, l = 1,...,L, there s a polynomal u such that u(x l ) = y l,l= 1,...,L. Proof of Proposton 2 () Lemma A.4. Let C be a closed sem-algebrac set. Then there s a realzable functon u that satsfes u(x) 0 when x C, and u(x) > 0, otherwse. Proof. Let D be a set of the form {x R k ; f 1 (x) = f 2 (x) =...= f l (x) = 0,r 1 (x) 0,r 2 (x) 0,...,r m (x) 0}, where f 1,...,f l,r 1,...,r m are polynomals. By Proposton 3, f 1,...,f l,r 1,...,r m are realzable and by Mertens and Zamr (1971), u D = mn{f 1,...,f l, f 1,..., f l,r 1,...,r m } s realzable. Clearly, u D (x) 0 when x C and u D (x) < 0, otherwse. Snce any closed sem-algebrac set s a fnte unon of such D s (see, Bochnak et al., 1988, p. 46), the desred u s u = max {u D } D, whch s also realzable. A unon of closed polygons s a closed sem-algebrac set. By Lemma A.4, there s a realzable functon u whch s less than or equal to 0 on C 2 and greater than 0 otherwse. Let c be the mnmum of u over the set C 1. By multplyng u wth the constant c 1 c 2 /c and addng c 2 one obtans a realzable functon u that satsfes u (x) c 2 when x C 2 and u (x) c 1 when x C 1. Takng the mnmum of u and c 1 would yeld the realzable functon needed for Proposton 2 (). Proof of Proposton 2 () By Lemma A.4, for = 1, 2, there s u whch s at least 0 on C and less than 0 otherwse. Consder u = max{mn{u + 1, 1}, 0}. u s realzable, bounded between 0 and 1, and equal to 1 on C, and less than 1 otherwse. For any nteger l, u l s also realzable. It s bounded between 0 and 1, and s equal to 1 on C and less than 1 otherwse. By addng a large postve number, say M, we may assume that the functons f 1 and f 2 are postve. There are suffcently large l s such that f j >f u l, /= j,onc j. Thus, max{f 1 u l 1 1,f 2u l 2 } s realzable and (subtractng M, f necessary) satsfes Proposton 2 2 (). Proof of Proposton 3. We frst defne an auxlary game wth one-sded nformaton wth Q beng the set of states, player 1 s fully nformed of the state, the sets of the players actons reman unchanged, and the payoff assocated wth a par of actons a 1,a 2 and state B s ḡ B (a 1,a 2 ) = k K p(k B)g k (a 1,a 2 ). The probablty of the state B s p(b). Denote ths probablty over Q by r. ByAumann and Maschler (1995) the long-run value of ths game s cav(ū)(r), where ū s the value of the auxlary game played once wth player 1 s nformaton beng trval. For any q n (Q), let p q (K) satsfy p q (k) = B Q q(b) p(k B). Note that ū(q) = u(p q ). Hence, m 0, m = 1 cav(ū)(r) = max m ū(q ) q (Q), m q = r. For any q n (Q), denote m B = m q (B) and defne the matrx M = (m B ). Snce, {,B m B = m = 1 and } m B = r(b), we conclude that M M(Q). Moreover, ū(q ) = u(p (M)) and cav(ū)(r) max D m u(p (M)) M M(Q). On the other hand, for any M M(Q), m u(p (M)) = m ū(q ), wth q (B) = m B /m. Therefore, u(p (M)) = ū(q ) mplyng D m u(p (M)) cav(ū)(r). The last two nequaltes yeld the desred result. A detaled proof of Theorem 1. Let V be a monotonc functon over parttons. Let p be (1/ K,...,1/ K ). Order all parttons of K: P 1,...,P l, so that f j<then V(P j ) V(P ), whch mples that P j does not refne P. Notaton 3. Let P be a partton. Denote by H(P) the space n R K spanned by {1l T ; T P}, where 1l T denotes the ndcator vector of the set T. Note that the partton P refnes P f and only f H(P ) s a subspace of H(P). Ths mples that f >j, then 4 dm [ H(P j ) H(P ) ] < dmh(p ). We prove that for any l the followng property, denoted by E, holds: there s a realzable functon u such that for any j and every p H(P j ), u(p) V(P j ). Furthermore, for any j there s a D j P j matrx 5 M j M(P j ) such that for any r D j, u(p r (M j )) = V(P j ). 4 dm(h) denotes the dmenson of H. 5 Recall Notaton 2. P j n M(P j ) stands for the nformaton structure that corresponds to the partton P j.

7 E. Lehrer, D. Rosenberg / Journal of Mathematcal Economcs 46 (2010) We proceed by nducton over. Let = 1. The partton P 1 s the trval partton and u can be taken to be the constant functon V(P 1 ). Now assume that E 1 holds and denote by u 1 the correspondng realzable functon. Step 1: Defnton of a class of matrces. Let D = P and consder the square matrx M 1 wth D columns, all of whose off-dagonal entres are zero. The dagonal entry correspondng to T P s p(t). Note that for every row r of M 1, p r (M 1 ) /= p. Let M 2 be a matrx of the same dmenson whose entres n the column correspondng to the cell T P are all equal to p(t)/d. Obvously, M 1,M 2 M(P ). Defne M = M 1 + (1 )M 2.If s postve and suffcently small, then M M(P ) and for every row r of M, p r (M) /= p. Furthermore, all entres of M are strctly postve. Defne H(P ) 0 ={v; v = ε T p ( T), ε T P T = 0}. For every v H(P ) 0 let a rt = m r ε T p r (M) p, v, r D, T P. Consder the matrx M v whose (r, T)-th entry s m rt + a rt.iftheε T s that defne v are small enough, then the entres of M v are postve. We show now that M v s n M(P ). Ths s so snce r m rt + a rt = p(t) + r m rε T p r (M) p, v =p(t) + ε T r m rp r (M) p, v =p(t). The last equalty s due to the fact that r m rp r (M) = p. Step 2: There s v H(P ) 0 such that for any row r, p r (M v ) / j< H(P j ). Note that a T P rt = m T P r ε T p r (M) p, v =m r p r (M) p, v ε T P T = 0. Therefore, p r (M v ) = p r (M) + p r (M) p, v T ε T p ( T) = p r (M) + p r (M) p, v v. Assume by contradcton that the clam s false. Then there s a neghborhood of H(P ) 0 around the orgn, denoted W, such that for every v W there s j<and a row r such that p r (M v ) = p r (M) + p r (M) p, v v H(P j ). Defne the set F rj to be the set contanng v W such that p r (M) + p r (M) p, v v H(P j ), where W s the closure (the relatve one n H(P ) 0 )ofw. F rj s a closed set for every r and j. By assumpton, the unon of the closed sets F rj contans W. As a complete space, W s of category II. Thus, at least one of the F rj s contans an open set. Therefore, there are j and r so that p r (M) + p r (M) p, v v H(P j ) for v s n an open (n W) set. Note that for every v W, p r (M) + p r (M) p, v v H(P ) (K). Furthermore, snce p r (M) p /= 0, the map v p r (M) + p r (M) p, v v s an open map. Thus, H(P j ) (K) contans an open set of H(P ) (K). Snce both are ntersectons of lnear spaces whose spannng [ vectors ] are n (K) wth (K) tself, t mples that H(P j ) (K) contans H(P ) (K). However, when j<, dm H(P j ) H(P ) < dmh(p ). Ths mples that H(P ) (K) s not ncluded n H(P j ) (K). We therefore conclude that there exsts a matrx, denoted M,nM(P ) that satsfes p r (M ) / j< H(P j ) for every row r of M. Step 3: Concluson of the proof. By Proposton 2 (), there s a realzable functon f that satsfes the followng: For every row r of M, f attans ts maxmum, V(P ), on p r (M ); and f s smaller than or equal to mn 1 j 1 V(P j )on 1 j 1 H(P j ) (K). By takng the maxmum of f and the functon u 1 we get a realzable functon u that satsfes: (a) For any j, u s smaller than or equal to V(P j )onh(p j ) (K); and (b) for any j, there s a D j l matrx M j M(P j ) such that for any 1 r D j, u(p r (M j )) = V(P j ). Property E l s therefore proven by nducton. Let (g k ) k K be the payoff functons that realze u l. (a) Imples that v P O (p, (g k ) k K ) V(P ) for every l. Proposton 3 and (b) mply that v P O (p, (g k ) k K ) = V(P j ), whch completes the proof. References Aumann, R.J., Maschler, M., Repeated Games wth Incomplete Informaton. MIT Press. Bassan, B., Gossner, O., Scarsn, M., Zamr, S., Postve value of nformaton n games. Internatonal Journal of Game Theory 32, Bochnak, J., Coste, M., Roy, M-F., Real Algebrac Geometry. Sprnger, Berln, Germany. Glboa, I., Lehrer, E., The value of nformaton an axomatc approach. Journal of Mathematcal Economcs 20, Gossner, O., Mertens, J.-F., The Value of Informaton n Zero-Sum Games. Mmeo. Hrshlefer, J., The prvate and socal value of nformaton and the reward to nventve actvty. Amercan Economc Revew 61, Kamen, M., Tauman, Y., Zamr, S., Informaton Transmsson. In: Ichsh, T., Neyman, A., Tauman, Y. (Eds.), Game Theory and Applcatons. Academc Press, pp Lehrer, E., Rosenberg, D., An axomatc approach to the value of nformaton n games. Journal of Mathematcal Economcs 42, Mertens, J.-F., Zamr, S., The value of two-person zero-sum repeated games wth lack of nformaton on both sdes. Internatonal Journal of Game Theory 1,