Infinitely Split Nash Equilibrium Problems in Repeated Games

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1 Infntely Splt ash Equlbrum Problems n Repeated Games Jnlu L Department of Mathematcs Shawnee State Unversty Portsmouth, Oho 4566 USA Abstract In ths paper, we ntroduce the concept of nfntely splt ash equlbrum n repeated games n whch the profle sets are chan-complete posets. Then by usng a fxed pont theorem on posets n [8], we prove an exstence theorem. As an applcaton, we study the repeated extended Bertrant duopoly model of prce competton. Keywords: Repeated game; nfntely splt ash equlbrum; ash equlbrum; fxed pont theorem on posets. 00 Mathematcs Subject Classfcaton: 49J40; 49J5; 9A0; 9A5; 9A80. Introducton and Prelmnares.. Defntons and notatons n game theory In begn of ths secton, we revew some concepts and notatons n game theory that are used by many authors. The readers are referred to [], [67], [04] for more detals. Let n be a postve nteger greater than. An n-person noncooperatve strategc game, smply called an n- person game, conssts of the followng elements:. the set of n players denoted by wth = n;. the set of profles S = S, where S s the pure strategy set for player ; 3. the utlty vector mappng f = f: S R n, where f s the utlty (payoff functon of player, for. Ths game s denoted by G(, S, f. Throughout ths paper, we always assume that, n the products S, f and \{} S, the players appear n the same sequental orders. As usual, for every, we often denote a profle of pure strateges for player s opponents by x = (x, x,, x, x +, xn. The set of profles of pure strateges for player s opponents s then denoted by Hence we may wrte x S as S = \{} S.

2 Moreover, for every x S, we denote x = (x, x wth x S, for. ( f(s, x = {f(z, x: z S}. From f = f n the game G(, S, f, for any x S, we have f(x = f(x. One of the most mportant topcs n game theory s the study of ash equlbrum problems. It has been wdely studed by many authors and has been extensvely appled to economc theory, busness and related ndustres (see [], [7], [04]. We recall the defnton of ash equlbrum n n-person noncooperatve strategc games below. Let G(, S, f be an n-person game. A profle of pure strateges x = ( x, x,, x n S s a ash equlbrum of ths game f and only f, t satsfes f(z, x f( x, x, for every and for every z S, ( It can be rewrtten as f(z, x f( x, x, for every and for every z S. In an n-person game G(, S, f, we defne a mappng F: SS R n by F(z, x = f(z, x, for any x, z S. (3 F(z, x s called the utlty vector at profle x S assocated to z S. It s clearly to see F(x, x = f(x, any x S. (4 Let n be the component-wse partal order on R n satsfyng that, for x, z S, F(z, x n F(x, x = f(x, f and only f, f(z, x f(x, x, for all. (5 From (5, the ash equlbrum can be rewrtten by: a profle x S s a ash equlbrum of G(, S, f f and only f,.. n-person dual games F(z, x n F( x, x = f( x, for all z S. (6 An n-person game G = (, S, f s statc. Some games n the real world may not be statc. That s, t may not be one-shot nature. It s more realstc for ths game to be repeatedly played. The dynamc model of game based on an n-person game G = (, S, f s formulzed by the process that ths statc game s repeated nfnte perods (tmes. It s called an n-person repeated game, n whch there s a dscount factor nvolved for the utltes (see [0]. The dynamc model for n- person repeated games wll be studed n secton 3 In ths paper, we frst consder a specal model: n-person dual game. An n-person dual game based on an n-person game G(, S, f s modeled as follows: At frst, the players play the game

3 as a statc n-person noncooperatve strategc game. After ths game s played frst tme and before ths game s played agan, every player always consders the reacton of ts compettors to ts strategy appled n the frst tme. To see the optmzaton of the player s utltes, the players may mae arrangements of strateges to use n the second play. Suppose that ths performance s represented by a mappng A on S. Hence, f x S s the profle used by the players n the frst tme, then Ax S wll be the profle used by the players n the second tme. Ths n-person dual game s denoted by G(, S, f, A. Then we as: Is there a ash equlbrum x S of the game G(, S, f (frst play such that A x S s also a ash equlbrum of ths game n second play wth respect to the translated profles? It rases the so called splt ash equlbrum problems for dual games. In [5] and [9], multtudnous of teratve algorthms have provded for the approxmatons of splt ash equlbra for two games. In all results about estmatng ash equlbra n the lsted papers, there s a common essental prerequste: The exstence of a ash equlbrum n the consdered problem s assumed. It s ndubtable that the exstence of solutons for splt ash equlbrum problems s always the crux of the matter for solvng these problems. In [6], the present author proved an exstence theorem of splt ash equlbrum problems for related games by usng the Fan-KKM Theorem. Snce the present author has studed the fxed pont theory on posets for several years and has made some applcatons to ash equlbrum problems, so, n ths paper, we wll apply some fxed pont theorems on posets to study the solvablty of splt ash equlbrum problems for dual games. To ths end, the profle sets of games must be equpped wth partal orders that may be nether lnear spaces, nor topologcal spaces. The postve aspect of ths research s that the utlty functons n the consdered games are unnecessary to be contnuous and the mappng A that defnes the splt ash equlbrum problems s unnecessary to be lnear. In secton 3, we extend the concept of splt ash equlbrum problems for dual games to nfntely splt ash equlbrum problems for repeated games and prove an exstence theorem. As applcatons, n secton 4, we study the exstence of nfntely splt ash equlbrum and ash equlbrum for the repeated extended Bertrant duopoly Model of prce competton that s a specal repeated game.. Splt ash equlbrum problems n dual games.. Defntons and notatons for splt ash equlbrum problems n dual games Let G(, S, f be an n-person game. Throughout ths paper, unless otherwse stated, we assume that, for every player, hs strategy set S s nonempty and s equpped wth a partal order. That s, for every, player s strategy set s assumed to be a poset (S,. As the product partally ordered set of (S, s, the profle set s also a poset (S, n whch the partal order s the component-wse partal order of s. That s, for x = (x, x,, xn and y = (y, y,, yn S, we have that y x, f and only f, y x, for all.

4 For every, (S, s smlarly defned to be the product poset of (Sj, j s, j, n whch s the correspondng component-wse partal order of j s, j. Defnton. Let G(, S, f, A be an n-person dual game. The splt ash equlbrum problem assocated wth ths dual game, denoted by SE(G(, S, f, A, s formalzed as: to fnd a profle x S satsfyng f(z, such that the profle A x S solves the followng f((az, ( Ax f( (A x ( Ax x f( x, x, for every and z S, (7, From (6, a profle x S satsfyng (78 can be rewrtten as:, for every and z S. (8 F(z, x n F( x, x = f( x, for all z S, (9 and F(Az, A x n F(A x,a x = f( x, for all z S. (0 Such a profle x n S s called a splt ash equlbrum of ths splt ash equlbrum problem SE(G(, S, f, A. The set of all splt ash equlbrums s denoted by (G(, S, f, A. When loong at the equlbrum problems (5 and (6 separately, the problem (5 s the classcal ash equlbrums problem of strategc games. When, consderng a specal case, A = I (A s unnecessary to be lnear, that s the dentty mappng on S, SE(G(, S, f, I reduces to the classcal ash equlbrum problem for the game G(, S, f. In ths vew, splt ash equlbrum problems for dual games can be consdered as the natural extensons of the classcal ash equlbrum problems. A fxed pont theorem on posets s proved n [8]. In ths theorem, the underlyng space s a chancomplete poset and the consdered mappng s just requred to satsfy order-ncreasng upward condton wthout any contnuty condton (As a matter of fact, the underlyng space s just equpped wth a partal order and t may not have any topologcal structure. The values of the consdered mappng are unversally nductve that s a relatvely broad concept. Some propertes and examples of unversally nductve posets have been provded n [8]. We recall ths theorem below that wll be used n the proof of the man theorems n ths paper. Fxed Pont Theorem A (Theorem 3. n [8]. Let (P, P be a chan-complete poset and let : P P \{} be a set-valued mappng satsfyng the followng three condtons: A. s P -ncreasng upward; A. ((x, P s unversally nductve, for every x P; A3. There s an element y* n P and v* (y* wth y* P v*. Let ( denote the set of fxed ponts of. Then

5 ( ( ((, P s a nonempty nductve poset; (([y*, P s a nonempty nductve poset; and has an P -maxmal fxed pont x* wth x* P y*.. An exstence theorem for splt ash equlbrum n dual games We need the followng concept, order-postve, for mappngs from posets to posets. It s an mportant condton for the mappng A for the exstence of splt ash equlbrum n splt ash equlbrum problems. Defnton. Let (X, X, (Y, Y and (U, U be posets. Let C, D be nonempty subsets of X and Y, respectvely. A mappng g: XY U s sad to be order-postve from XY to U whenever, for x, y D, f x Y y, then g(z, x U g(t, x mples g(z, y U g(t, y, for any z, t X. ( In partcular, f (U, U = (R m, m, where m s a natural number, a mappng g s order-postve from XY to (R m, m whenever, for x, y C, f x X y, then g(z, x m g(t, x mples g(z, y m g(t, y, for any z, t X. ( Let G(, S, f, A be an n-person dual game. To prove an exstence theorem for splt ash equlbrum problem SE(G(, S, f, A S, we need to defne a mappng : S, for x S, by (x ={t S: F(z, x n F(t, x and F(Az, Ax n F(At, Ax, for all z S}. (3 can be equvalently wrtten, for x S, as (x ={t S: f(z, x f(t, x and f((az, (Ax f((at, (Ax, for every and for all z S}. Observaton. In Theorem gven below, t s assumed that, for every x S, (x. It means that, for any gven profle x S and for every player, when player s opponents tae x to play, there exsts a strategy t S such that player wll optmze hs utlty at the profle (t, x. Hence, the condton that (x s nonempty s a reasonable condton and t should not be too strong. ow we prove one of the man theorems of ths paper. Theorem. Let G(, S, f, A be an n-person dual game. Suppose that, for every, (S, s a nonempty chan-complete poset. Let (S, be the product poset of (S, s equpped wth the component-wse partal order. If f and A satsfy the followng condtons: a. For every, f s order-postve from (S, (S, to (R, ; b. For every x S, (x s a unversally nductve subset of S; c. The operator A: S S s -ncreasng; d. There are elements x S and u (x satsfyng x u,

6 then the dual game G(, S, f, A has a splt ash equlbrum. Moreover ( ( ((G(, S, f, A, s a nonempty nductve poset; ((G(, S, f, A [x, s a nonempty nductve poset. Proof. Snce, for every, (S, s a nonempty chan-complete poset, then the profle set, as a product space of chan-complete spaces (S, s, (S, s a nonempty chan-complete poset, S where s the component-wse partal order of s. Defne a set-valued mappng : S by (x = (x ={t S: F(z, x n F(t, x and F(Az, Ax n F(At, Ax, for all z S}, for x S. (4 From condton b n ths theorem, t mples that, for every x S, (x. Hence the mappng S : S \{} s a well-defned set-valued mappng wth unversally nductve values n S. ext we show that s -ncreasng upward. otce that the partal order on S s the component-wse partal order of s on S s, respectvely. It mples that, for any x, y S, x y s equvalent to x y and x y, for every, From condton a, for every, f s order-postve from (S, (S, to (R,. From (3, t mples that F s order-postve from (S, (S, to (R n, n. Then, for arbtrary x, y S wth x y, t mples x y, every. From condton a, we then have It follows that f(z, x f(t, x f(z, y f(t, y, for z, t S. F(z, x n F(t, x mples F(z, y n F(t, y, for any z, t S. (5 From condton c, the operator A: S S s -ncreasng. It mples that f x y, then, Ax Ay. From condton a agan, smlar to (5, we have F(Az, Ax n F(At, Ax mples F(Az, Ay n F(At, Ay, for any z, t S. (6 (5 and (6 together mply that f x y, then (x (y. Hence s -ncreasng upward. The elements x S and u (x gven n condton d n ths theorem satsfy that u (x such that x u. So satsfes all condtons n the Fxed Pont Theorem A. It follows that ( and t satsfes the propertes ( and ( n Theorem A. From (9 and (0, the defnton of (G(, S, f, A, and (4, the defnton of, we obtan (G(, S, f, A = (. (7 By Applyng Theorem A and (7, the proof of ths theorem s completed mmedately..3. Applcatons to partally ordered Banach spaces. In ths subsecton, we consder a specal case of n-person dual games n whch the strategy set for every player s a nonempty and compact subset of a partally ordered Banach space. Ths case

7 should be very useful n the applcatons. In [8], t was proved that every partally ordered compact Hausdorff topologcal space s both chan-complete and unversally nductve, as a consequence of Theorem, we have Corollary. Let G(, S, f, A be an n-person dual game. Suppose that, for every, S s a nonempty compact subset of a partally ordered Banach space (B,. Let B = B equpped wth the component-wse partal order of s. If f and A satsfy the followng condtons: a. For every, f s order-postve from (S, (S, to (R, ; b. For every x S, (x s a nonempty closed subset of S; c. The operator A: S S s -ncreasng; d. There are elements x S and u (x satsfyng x u, then the dual game G(, S, f, A has a splt ash equlbrum. Moreover, (G(, S, f, A has the propertes ( and ( lsted n Theorem. Remars. In Corollary, even though the profle set n the dual game G(, S, f, A s a subset of a Banach space, the operator A: S S s not requred to be lnear. It may be a nonlnear operator. 3. Infntely Splt ash equlbrum problems n repeated games 3.. Defntons and notatons of n-person repeated games Let G(, S, f be an n-person game. Recall that, for every, player s strategy set s assumed to be a poset (S,. (S, s the product poset, where s the component-wse partal order of s naturally equpped on S. For every natural number, after the players repeated play the game tmes and, for each tme, the game s played as a statc n-person smultaneous-move game, before they play ths statc game agan, every player always consders the reacton of ts compettors to ts strategy appled n the prevous tme. To optmze ther utltes, the players may mae arrangements of strateges to use n the next play (the ( + th play. Suppose that the profle of the arranged strateges s represented by the value of a mappng A: S S, for =,, 3, (Snce (S, s just a poset, t may not be equpped wth any algebrac structure. So the lnearty of A s may not be defned. To summarzng ths process, f x S s the profle used by the players n the frst tme, then Ax S wll be the profle used by the players n the second tme; AAx S wll be the profle used by the players n the thrd tme. Hence, for =,, 3,, A AAx S wll be the profle used by the players n the ( + th play. For smplcty, we wrte = A A AA0, for = 0,,,. (8 where A0 = I, that s the dentty operator on S. Then : S S s a sngle-valued mappng. In partcular, f A = A = = A = A, then we denote = A A0, for =,,. Suppose that the utltes of ths game are bounded. That s, there s a number M > 0 such that

8 (M f(x M, for every and for all x S. There s a dscount factor 0 < <. For every, player s dscounted value of utlty at a profle x S s h(x = 0 f ( x, (9 Player s dscounted value of utlty at the profle x assocated wth a profle z S s H(z, x = 0 f (( z,( x. (0 It mples H(x, x = h(x, for every and for all x S. The utlty vector wth dscounted values for ths repeated game at the profle x assocated wth a profle z S s H(z, x = H(z, x = ( 0 f (( z,( x. Under the boundedness condton (M, for every, both of h and H are well-defned real valued functons on S and S S, respectvely. Then t forms an n-person dynamc model based on an n-person game. It s called an n-person repeated game based on the n-person statc game G(, S, f and s denoted by Defnton 3. Let G S, f, A 0 G. S, f, A 0 be an n-person repeated game. A profle xˆ S s called a ash equlbrum of ths repeated game, f the followng nequaltes are satsfed H(z, xˆ H( xˆ, xˆ, for every and z S. ( Defnton 4. The nfntely splt ash equlbrum problem assocated wth the repeated game G, S, f,, denoted by SE( G, S, f,, s formalzed as: to fnd a profle x ( A 0 S satsfyng f((z, x ( ( A 0 f( ( x, x (, for every and z S, for = 0,,,. ( Such a profle x S s called an nfntely splt ash equlbrum. The set of all nfntely splt ash equlbrums of G, S, f, s denoted by ( G, S, f,. ( A 0 ( A 0 Proposton. Every nfntely splt ash equlbrum of an n-person repeated game s a ash equlbrum of ths repeated game. Proof. Suppose that, for every, for = 0,,,, the followng nequalty holds

9 f((z, x ( f( ( x, x (, for all z S. Snce 0 < <, t mples H(z, x = f (( z,( x 0 0 f (( x,( x = H( x, x. It completes the proof of ths proposton. Smlar to (3 for the defnton of the mappng, regardng to G S defne a mappng : S, for x S, by S, f, A 0, we need to (x ={t S: f((z, ( x f( ( t, ( x, for every, all z S, all = 0,, } can be rewrtten, for x S, as Theorem. Let (x ={t S: F(z, x n F(t, x, for = 0,,, and for all z S}. (3 G be an n-person repeated game. Suppose that, for every, S, f, A 0 (S, s a nonempty chan-complete poset. Let (S, be the product poset of (S, s equpped wth the component-wse partal order. Suppose that the followng condtons are satsfed: a. For every, f s order-postve from (S, (S, to (R, ; b. For every x S, (x s a nonempty unversally nductve subset of S; c. For every =,,, A: S S s an -ncreasng operator; d. There are elements x S and u (x satsfyng x u. Then the repeated game G S, f, A 0 has an nfntely splt ash equlbrum. Moreover ( ( (( G (( G S, f, A 0 S, f, A 0, s a nonempty nductve poset; [x, s a nonempty nductve poset. Proof. The proof of ths theorem s smlar to the proof of Theorem. As a product space of chan-complete posets (S, s, the profle set (S, s also a nonempty chan-complete poset, where s the component-wse partal orders s. By usng (3, we defne a set-valued S mappng : S, for x S, by (x = (x ={t S: F(z, x n F(t, x, for = 0,,, and for all z S}. (4 From (3 and (4, (x can be rewrtten as (x = (x ={t S: f((z, ( x f( ( t, ( x, for every, all z S, all = 0,, }

10 S From condton b n ths theorem, the mappng : S \{} s a well-defned set-valued mappng wth unversally nductve values n S. ext we show that s -ncreasng upward. From condton a, for every, f s order-postve from (S, (S, to (R,. From condton c, t mples that, for every = 0,,, : S S s -ncreasng. Then, for arbtrary x, y S wth x y, smlarly to (5 and (6, we can show that F(z, x n F(t, x F(z, y n F(t, y, for any z, t S, = 0,,. (5 (5 mples that f x y, then (x (y. Hence s -ncreasng upward. The elements x S and u (x gven n condton d n ths theorem mples that u (x wth x u. So satsfes all condtons of Fxed Pont Theorem A. Rest of the proof s the same to the proof of Theorem. Smlar to Corollary, as an applcaton of Theorem to partally ordered Banach spaces, we have Corollary. Let G be an n-person repeated game. Suppose that, for every, S, f, A 0 S s a nonempty compact subset of a partally ordered Banach space (B,. Let (B, be the product partally ordered Banach space of (B, s, where s the component-wse partal order of s. Suppose that the followng condtons are satsfed: a. For every, f s order-postve from (S, (S, to (R, ; b. For every x S, (x s a nonempty closed subset of S; c. For every =,,, A: S S s an -ncreasng operator, d. There are elements x S and u (x satsfyng x u. Then the repeated game G S, f, A 0 has an nfntely splt ash equlbrum. Moreover ( G, S, f, has the propertes ( and ( gven n Theorem. ( A 0 By usng Proposton, as applcatons of Theorem, or n Corollary, we obtan the followng exstence results about ash equlbrum of n-person repeated games. Corollary 3. Let G S, f, A 0 be an n-person repeated game as gven n Theorem (or n Corollary. If condtons a-d lsted n Theorem (or n Corollary are satsfed, then ths repeated game has a ash equlbrum. Remars. Theorems, and Corollary,, 3 provde some condtons for the exstence of nfntely splt ash equlbrum or ash equlbrum n repeated games. otce that these condtons are just necessary condtons and are not suffcent condtons. Hence, f the condtons of these exstence results do not hold for some repeated games, there stll may exst an nfntely splt ash equlbrum. It only means that t cannot be assured that there s one, f these condtons are not satsfed. 4. Applcatons to repeated extended Bertrant duopoly model of prce competton

11 In [6], the present author generalzed the Bertrant duopoly model of prce competton wth two frms from the same prce model (see [0] to the model wth possbly dfferent prces. Then the dual extended Bertrant model s ntroduced and an exstence theorem of splt ash equlbrum for the Marov dual extended Bertrant duopoly model of prce competton s proved n [6]. We revew ths duopoly model below. The extended Bertrant duopoly model of prce competton s a model of olgopolstc competton that deals wth two proft-maxmzng frms, named by and, n a maret. In ths model, t s assumed that the two frms have constant returns to scale technologes wth costs c > 0 and c > 0, per unt produced, respectvely, where the costs c and c are possbly dfferent. Wthout loss of the generalty, we assume c c, (6 The nequalty (6 means that the qualtes of the products by these two frms may be dfferent. More precsely, the qualty of the products n frm may not be as good as the qualty of the products n frm. Let pj be the prce of the products by frm j, for j =,. Let (p, p be the demand functon n ths duopoly maret. Let j(p, p be the sale functon for frm j, for j =,. f and j are assumed to be contnuous functons of two varables and strctly decreasng wth respect to every gven varable. Suppose that there are postve numbers p, for j =,, such that, for all p j j(pj, p 0, for all pj [0, p j and j(pj, p = 0, for all pj p j. (7 Suppose that the socally optmal (compettve output level n ths maret s strctly postve and fnte for every frm 0 < (c, c <. For gven prces p, p, set by frms and, respectvely, the maret s assumed to be clear. That s, (p, p = (p, p + (p, p. Let = c / c, that defnes the rato of the qualtes of the products by frm to frm. From the assumpton (7, we have (0, ]. Consdered as a noncooperatve strategc game, the competton taes place as follows: The two frms smultaneously name ther prces p, p, respectvely. The sales (p, p and (p, p are then satsfed ( p, p ( p, p = 0, c c c, f, f p p f p p, p p (8 and

12 ( p, p ( p, p =, c c c 0, f, f f p p p p p p. (9 We assume that the frms produce to order and so they ncur producton costs only for an output level equal to ther actual sales. Therefore, for gven prces p, p, the frm j has profts uj(p, p = (pj cjj(p, p, for j =,. (30 In [6], an exstence theorem for ash equlbrum of the extended Bertrant duopoly model s proved. We recalled t below for easy reference. Theorem 6. n [6]. In the extended Bertrant duopoly Model, there s a unque ash equlbrum ( p ˆ, p ˆ. In ths equlbrum, both frms set ther prces equal to ther costs, respectvely: ˆp = c, ˆp = c. Ths extended Bertrant duopoly model of prce competton wth two frms s a -person statc game. It s denoted by G(, S, u, where = {, }, Sj [0, p ], u = (u, u, and uj s defned by (30, for j =,, respectvely. For every natural number, after the two frms repeated play the game tmes and, for each tme, the game s played as a statc -person smultaneous-move game and before they name ther prces to play agan, every frm always consders the reacton of ts compettor to ts strategy (prce appled n the prevous tme. Suppose that when ths game s played n the th tme, the two frms names ther prces as j p j, for j =,, respectvely. To optmze ther utltes, for example, frm could try to ncrease ts profts by ncreasng the prce from p to p (never excess p, even though decreasng ts sales. Meanwhle, frm could try to ncrease ts profts by decreasng the prce from p to p (never lower than p for ncreasng ts sales. Suppose that such performance s defned by a lnear transformaton A from p p to p, p. Here we assume 0 p = p, and p 0 = p, that are the prces set by the two frms n the very frst tme. So, for =,,, there s a matrx M:, M =, (3 where 0,, such that ( p, p = A (( p, p = ( p, p. (3 It mples 0 p p and 0 p p, for =,,. (33

13 Hence the process of repeatedly playng the statc game G(, S, u wth the sequence of lnear transformatons A } defned by (3 s a dynamc game, that s the repeated extended Bertrant { 0 duopoly model of prce competton. It s a specal repeated game denoted byg, S, u,. ( A 0 From Defnton 4, p = p, S s an nfntely splt ash equlbrums of the repeated game G S, u, A 0 u((p, ( p ( p, f t satsfes u( ( p, p (, for every =,, all p S, every = 0,,,. (34 Theorem 3. For the nfntely splt ash equlbrum problem of the repeated extended Bertrant duopoly model G, S, u,, we have ( ( A 0 If c = c = c, then, for any sequence of lnear transformatons { A } 0 defned n (3, p = p, = (c, c s the unque nfntely splt ash equlbrum; ( p ( If c < c, then there exsts a unque nfntely splt ash equlbrum p = ( p, p = (c, c, only f all lnear transformatons A s equal to the dentty, that s A = 0 0, for = 0,,,. (35 Proof. Part ( s an mmedate consequence of Theorem 6. n [6]. To prove part (, notce that every nfntely splt ash equlbrum of the repeated game G S, u, A 0 s a splt ash equlbrum of the dual game G S, u, A studed n [6]. From Theorem 6. n [6], p, = (c, c s the unque splt ash equlbrum of the dual game G S, u, A, only f ( p 0 A =. 0 It mples = A. Applyng Theorem 6. n [6] agan, t follows that ( p, p = (c, c s the 3 unque splt ash equlbrum of the trple game G S, u, I, A, only f A = 0 0. Then (35 s proved by nducton. Snce and are contnuous functons, from the condton (7 and the defnton (30 of the utlty functons, t mples that there exsts M > 0, such that (M u(p M, for =, and for all p S. Let be the dscount factor of ths dynamc game. By Proposton, as a consequence of Theorem 3, we have

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