CHAPTR 5 SOM MINIMA AND SADDL POINT THORMS 5. INTRODUCTION Fied poit theorems provide importat tools i game theory which are used to prove the equilibrium ad eistece theorems. For istace, the fied poit theorems by Brouwer, Kautai ad Fa have bee amog the most used tools i ecoomics ad game theory, see [4], [3], [6]. These theorems are further applied to prove the miima theorems ad the eistece of saddle poits. The first miima theorem was give by vo Neuma [57] i 98. He established the followig. Theorem 5.. [88]. Let A be a m matri ad ad y be the sets of oegative row ad colum vectors with uit sum. The mi ma y Y T T Ay = ma mi Ay. (5..) y Y After ie years of this result vo Neuma [86], showed that the biliear character of Theorem 5.. was ot eeded. He eteded his result by usig Brouwer s fied poit theorem as follows. 9
Theorem 5.. [86]. Let ad Y be oempty compact, cove subsets of uclidea spaces ad f : Y R be oitly cotiuous. Suppose that f is quasicocave o ad quasicove o Y. The mi ma f ( = ma mi f (. y Y y Y Thereafter, may papers appeared i the literature verifyig the Neuma iequality. It is oticed that maority of miima theorems are proved by applyig fied poit theorems uder differet coditios o the spaces or o the mappigs. Sio [59] geeralized vo Neuma s result o the basis of Kaster, Kuratowsi ad Mazuriewicz theorem [6], which was give i the year 99 o the -dimesioal simple i the followig way. Theorem 5..3 [6]. Let D be the set of vertices of a -simple ad a multivalued map G : D be a KKM map (that is, co( A) G( A) for each A D ) with closed [resp., ope] values. The I G( z) φ. z D This theorem is a equivalet formulatio of the famous Brouwer fied-poit theorem. At the begiig, the results of the KKM theory were established for cove subsets of topological vector spaces maily by Ky Fa [6]-[64]. A umber of itersectio theorems ad applicatios were followed i the cove subsets of topological vector spaces. I 96, Ky Fa geeralized KKM theorem to ifiite dimesioal topological vector space, which is ow as Fa Kaster-Kuratowsi-Mazuriewicz theorem (i.e. Fa KKM theorem or KKMF theorem). Fa established followig miima iequality from the KKMF theorem. Theorem 5..4 [6]. Let be a oempty compact cove set i a topological vector space. If a fuctio ϕ : (, satisfies the followig coditios (i) for each, (ii) for each, ( ϕ is lower semicotiuous fuctio of y o, y ( ϕ is quasicocave fuctio of o. The the miima iequality mi supϕ( supϕ( ) holds. 3
Lassode [84] eteded the KKM theory to cove spaces by provig several variats of KKM theorems for cove spaces ad proposed a systematic developmet of the method based o the KKM theorem. The KKM theorem was further eteded to pseudocove spaces, cotractible spaces ad spaces with certai cotractible subsets or H-spaces by Horvath [66]-[67]. I these papers, most of the Fa s results i the KKM theory are eteded to H-spaces by replacig the coveity coditio by cotractibility. With these termiology, Par [68] established ew versios of KKM theorems, miima iequalities, ad others o H-spaces. Par ad Kim [69]-[73], Par [74]-[78] ad others eteded these results to more geeral spaces such as G-cove abstract cove ad KKM spaces. The purpose of this chapter is to preset a geeralized versio of the KKM theorem by usig the cocept of Chag ad Zhag [8]. As a applicatio of it, a geeralized miima iequality is obtaied ad a eistece result for the saddle-poit problem uder geeral settigs is derived. Cosequetly, a saddle poit theorem for two perso zero sum parametric game is also proved. Several eistig well ow results are obtaied as special cases. 5. PRLIMINARIS First we give basic defiitios used i our results. We follow [6], [8], [8],[85], [93], [95], [34] ad [35] for otatios ad prelimiaries. Defiitio 5.. [34]. A two-perso, zero-sum game G is defied by G = A, A, f, ) where, for i =,, we have (i) Ai is a fiite set of player i s actios, (ii) f i : A R, where A = A A, is player i s payoff fuctio, the player s payoff fuctios satisfy f a) + f ( a), for all a A. ( = ( f If we cosider two perso zero sum game geerated by fuctio f : Y R. This meas that the first player selects a poit from ad the secod player selects a poit y from Y. Because of this choice, the secod player pays the first oe the amout f (. Defiitio 5.. [35]. A poit * * (, y ) Y is said to be saddle poit of f i Y if 3
f ( *, * * * f (, y ) f ( y ) for all (, Y. * * A poit (, y ) Y is said to be a solutio of the game if ad oly if it is a saddle poit of f i Y. Defiitio 5..3 [93]. A two-perso zero-sum parametric game ( GP ) is defied by the followig form (, Y, f, g,, F ), where,. is a subset of a topological space, which is called the strategy set of player,. Y is a subset of a topological space, which is called the strategy set of player, 3. f : Y R ad g Y R, where R = (,, : + 4. is a real umber, which is called a parameter of the game, 5. F = f g : Y R, that is, for all (, Y, F ( = f ( g(, is a loss fuctio of player ad ( is a loss fuctio of player. F I geeral, F = if sup F ( is called the miimal worst loss of player ad F y Y = sup if F ( is called the maimal worst gai of player. y Y + We illustrate the above cocept of two perso zero sum parametric game by followig eample. ample 5... Cosider the sets =,,, }, Y = y, y, y, } ad parameter =. 3 4 3 y4 A fuctio f : Y R defied by the followig payoff matri y y y 3 y4 3 4 7 8 3-4 7 4 8-3 4 3 6 5-8 8 3
ad g : Y R + by y y y 3 y4 3 4 3 3 Whe =, the F : Y R becomes y y y 3 y4 3 4 3 4-4 5 4-5 -3 4 ad (, y3) is the saddle poit of the fuctio (. F KKM mappig i Hausdorff topological vector space is defied as follows. Defiitio 5..4 [6]. Let be a Hausdorff topological vector space ad be a oempty subset of. A multivalued mappig G :, that is, mappig with the values G( ), for each i, is called a KKM mappig if co,,..., } U G( ) for each fiite subset,,..., }. i i=,,..., }, where co,,..., } deotes the cove hull of the set Defiitio 5..5 [8]. Let be a oempty subset of a topological vector space. A multivalued mappig G : is called a geeralized KKM mappig, if for ay fiite set 33
,..., }, there eists a fiite subset y,..., y} such that for ay subset y,..., yi } y,..., y }, i co y,..., y } U G ( ). i i i =, we have We eted this defiitio for two maps i the followig maer. Defiitio 5..6. Let be a oempty subset of a topological vector space ad F, G :. The G is said to be a geeralized F-KKM mappig if for ay fiite set,..., } ad each i,..., i},,..., }, we have co U F ( )} U G ( ). i = = i It is remared that defiitio 5..6 becomes defiitio 5..5 whe F : ad F ( i ) = yi for each i,,..., }. I the followig eample we have show that cocept of F-KKM mappig is geeral tha KKM mappig. ample 5... Let = (,, = [, ]. Let F, G : be defied as F( ) = +,+ 3 3 ad G ( ) = +, + 5 5. Sice U G ( ) = [ 9/5,9/5], G ( ), [, 9/5) U (9/5, ]. So co,..., } U G( ) ad G is ot a KKM map. But it is a F-KKM map, sice i= i 5 U F( ) =, 3 3 ad 5 co U F( i )}, G( ) G( ). i = 3 5 = I U = Defiitio 5..7 [8]. Let be a topological vector space ad be a oempty cove subset of. A fuctio φ : (, is said to be γ geeralized quasicove i y for some γ (, if, for ay fiite subset y, y,..., y}, there is a fiite subset 34
,..., } such that, for ay subset,..., },..., } ad ay co i,..., }, i we have γ maφ(, ). yi i i i Defiitio 5..8. Let be a topological vector space ad be a oempty cove subset of. Y Let γ (, ad F :, where, Y. The the fuctio φ : (, is said to be F γ geeralized quasicove i y if for ay fiite subset y, y,..., y}, each i,..., i},,..., }, ad ay co U F( i )} Y = ad ay y i F( i ), we have γ ma φ(, y ). i i Notice that the fuctio φ becomes γ geeralized quasi cove i y whe F : Y ad F ( i ) = y i for each i,,..., }. Defiitio 5..9 [95]. Let ad Y be two topological spaces. A multivalued mappig F Y : is said to be trasfer closed valued o if, I F( ) = I F( ). Defiitio 5.. [85]. Let ad Y be two topological spaces. A multivalued mappig F Y : is said to be itersectioally closed valued o Y if, I F( ) = I F( ). Defiitio 5.. [8]. Let, Y be two topological spaces. The a fuctio f : Y R = R ± } is said to be (i) quasicocave i if for each y Y ad λ R, the set : f ( λ} is cove (ii) quasicove i y if for each ad λ R, the set y Y : f ( λ} is cove 35
(iii) (iv) upper semicotiuous (resp. geerally upper semicotiuous) i if for each y Y ad λ R, the set : f ( λ} is closed (resp. itersectioally closed), lower semicotiuous (resp. geerally lower semicotiuous) i y if for each ad λ R, the set y Y : f ( λ} is closed (resp. itersectioally closed). Defiitio 5.. [8]. Let, Y be two topological spaces. The a fuctio f : Y R = R ± } is said to be (i) trasfer upper semicotiuous i if for each λ R, ad all, y Y with f ( < λ there eists a eighbourhood V() of ad a poit y Y such that f ( z, y ) < λ for all z V (). (ii) trasfer lower semicotiuous i y if for each λ R, ad all, y Y with f ( > λ there eists a eighbourhood V( of y ad a poit such that f (, u) > λ for all u V (. Defiitio 5..3 [35]. Let ad Y be two topological spaces. A fuctio φ : Y (, is said to be (i) (ii) γ trasfer lower semicotiuous fuctio i for some γ (, if, for all ad y Y with φ (, > γ, there eists some y Y ad some eighborhood N() of such that φ ( z, y ) > γ for all z N(). γ trasfer upper semicotiuous fuctio i y for some γ (, if, for all ad y Y with φ (, < γ, there eists some Y ad some eighborhood N( of y such that φ (, u) < γ for all u N(. Defiitio 5..4. Let, Y be two oempty sets. The mappig = = surective if F( ) U F( ) Y. F Y : is said to be 36
5.3 GNRALIZD KKM THORMS AND THIR APPLICATIONS First we prove the mai result of this sectio which will be used i the sequel. Theorem 5.3.. Let be a oempty cove subset of a Hausdorff topological vector space. Let G : be a multivalued mappig such that for each, G() is fiitely closed. The the family of sets G( ) : } has a fiite itersectio property if ad oly if the mappig G is a geeralized F-KKM mappig for some mappig F :. Proof. Let G( )} has fiite itersectio property. The for ay fiite subset,,..., }, I G( ). Taig I G( ), ad defie i = i φ * i = i F : by F ) = } for each. The for each i,..., i},,..., }, we have ( * co U F( i ) = co * } = * } G( U i = = ). This implies that G : is a geeralized F-KKM mappig. Now cosider G : to be a geeralized F-KKM mappig, F : ad suppose the family G( ) : } does ot have the fiite itersectio property, i.e., there eists some fiite subset,,..., } such that I G( ) =. Sice G is a geeralized F-KKM i = i φ mappig, the for each i,..., i },,..., }, we have co U F( i ) G( ). U i = particular, co ( )} G( ) F for each i,,..., }. i i = I Let S = co y, y,..., y}, L = spa y, y,..., y}. The S L. Sice G is fiitely closed, G( i ) I L is a closed set. Let d be the uclidea metric o L. It is easy to see that d L I G( )) > L I G( ) (5.3.) ( i i Now we defie a fuctio f : S [, as f ( c) = d( c, L I G( i )) y i i= It follows from (5.3.) ad I G( = φ that for each c S, f ( c) >. Let i = i ) 37
R( c) = d( c, L I G( i )) y i. (5.3.) i= f ( c) Thus R is a cotiuous fuctio from S ito S. By the Brouwer fied poit theorem, there eists a elemet Deote c * S such that c* = F( c* ) = d( c*, L I G( i )) y i (5.3.3) f ( c) i= I = i,,..., }: d( c*, G( i ) I L > ). (5.3.4) The for each i I, c G( i ) I. Sice c, c G( ), i I, so we have * i i I * L * L * i c U G( ) (5.3.5) From (5.3.3) ad (5.3.4), we have c = d( c, LI G( i)) yi co( yi : i I}. * * i I f( c* ) However, sice G is a geeralized F-KKM mappig from ito, which follows that co U F( i ) U G( ). i Therefore we have = = c co( y : i I} U G( ). (5.3.6) * i i i I This cotradicts (5.3.5). Hece G( ) : } has the fiite itersectio property. This completes the proof. It is remared that if F : is defied as a sigle valued map with F ( i ) = yi for each i,,..., }, the above result implies the followig results of Chag ad Zhag [8, Theorem 3.]. Corollary 5.3. [8]. Let be a oempty cove subset of a Hausdorff topological vector space. Let G : be a multivalued mappig such that for each, G() is fiitely closed. The the family of sets G( ) : } has the fiite itersectio property if ad oly if the mappig G is a geeralized KKM mappig. Followig is a geeralized versio of the KKM mappig theorem. 38
Theorem 5.3.. Let Y be a oempty cove subset of a Hausdorff topological vector space, φ Y ad maps F, G : with G a itersectioally closed valued map of Y. Furthermore, assume that there eists a oempty subset, cotaied i some precompact cove subset Y of Y, such that G( ) is compact subset of Y for at least oe ad let : be a geeralized F-KKM mappig. The I G( ) φ. G Proof. For ay fiite subset,,..., } of, let = U,,..., }. Sice Y is a precompact cove subset of Y, the cove hull of Y U,,..., } is also a compact cove subset of Y, ad deote it by K. For each y, let T ( = G( I K. Sice G( is itersectioally closed valued map of Y ad K is a compact subset of Y, each T( is also compact. Further, sice : is defied by ( ) G( ) G G = for each ad G is a geeralized F-KKM mappig with closed values. So, we ca easily show that T is also a F- KKM map. Therefore, by Theorem 5.3., we have I T ( φ. Hece we have φ I T ( ) = = I G( ) I K I I G( G( ) I G( ) I... I G( ) I G( ) I... I G( Let C deotes the compact set I G ). The we have I G( ) I C φ for every fiite ( subset,,..., }. Sice each G() is a itersectioally closed valued map of Y ad ) ). i= i C is a compact subset of Y, each G( ) I C is also a compact subset of Y. Sice the family G( ) I C } has the fiite itersectio property, we have I G( ) I C = I G( ) φ. This completes the proof. Followig Corollary is the special case of Theorem 5.3.. Corollary 5.3.. Let be a oempty cove subset of a Hausdorff topological vector space ad maps F, G : with G a itersectioally closed valued map such that 39
G ) = K ( is compact for at least oe ad let mappig. The I G( ) φ. Proof. Sice : is defied by ( ) G( ) G G : be a geeralized F-KKM G = for each, we have that G is a geeralized F-KKM mappig with closed values. By Theorem 5.3. the family of sets G( ) : } has the fiite itersectio property. Sice G ) is compact, we have I G( ) φ. Sice G is itersectioally closed valued mappig I G( ) = I G( ) φ. ( Notice that if F : be a sigle valued map ad F ( i ) = yi for each i,,..., }, the Corollary 5.3. cotais the result of Asari et al [79, Th..]. Corollary 5.3.3. [79]. Let be a oempty cove subset of a Hausdorff topological vector space ad map : with G a trasfer closed valued map such that G ) = K G ( is compact for at least oe I G( ) φ. ad let G : be a geeralized KKM mappig. The The relatio betwee geeralized F-KKM mappigs ad (quasi-cocavit is the followig. S γ geeralized quasi-coveity Theorem 5.3.3. Let be a oempty cove subset of a topological vector space. Let φ : (,, γ (, ad F :. The the followig are equivalet (i) The mappig G : defied by G( = : φ( z) γ,for all z F( )} [ resp. G( = : φ( z) γ,for all z F( )} for each y is a geeralized F-KKM mappig. (ii) φ( is S γ geeralized quasi-cocave, [resp. S γ quasicove] i y. geeralized 4
Proof. For the sae of simplicity, we prove the coclusio oly for the first case i (i) ad (ii). The other case ca be proved similarly. ( i) ( ii). Sice G : is a geeralized F-KKM mappig, for ay fiite set,,..., }, ad each i,..., i},,..., }, we have co U F( i ) G( U = = i ) ad hece for ay y co U F( i ), y G( i ). U So there = eists some m,,..., }, such that G( ), z F ). Therefore, we have im ( i m y i m mi(, zi ) γ for all z F ). im = y so we have ϕ( y, ) γ ( i m z im for all i.e. ϕ is F-γ geeralized quasi-cocave i y. ( ii) ( i). Sice ϕ is F- γ geeralized quasi-cocave i y, for ay fiite set,,..., }, each i,..., i},,..., }, ad ay y co U F( i ), = miϕ ( y, zi ) γ for all z F ). im Hece there eists some m : m, such that ϕ y, ) γ for all z F ). This shows that y G( ) ad hece y UG( ). i m i = ( i m ( z i m im ( im By the arbitrariess of y co U F( ) i, we have = co U F( i ) G( U = = i ). This implies that G : is a geeralized F-KKM mappig. Let = be a topological vector space, G : defied by G ( = : φ( γ} ( resp. G( = : φ( γ}) for each y be a geeralized KKM mappig, ψ ( is a γ geeralized quasicocave (resp., cove) fuctio i y ad F : be a sigle valued map, F ( i ) = yi for each i,,..., }, the Theorem 5.3. implies Propositio 3. of Asari et al [79]. 4
Corollary 5.3.4 [79]. Let be a oempty cove subset of a topological vector space. Let φ : (, ad γ (,. The the followig are equivalet. (ii) The mappig G : defied by G ( = : φ( γ} [ resp. G( = : φ( γ}] is a geeralized KKM mappig. (iii) φ( is γ geeralized quasi-cocave, [resp. γ geeralized quasicove] i y. Some Applicatios As applicatios of the above results we preset miima ad saddle poit results. First we state ad prove the followig geeral miima iequality. Theorem 5.3.4. Let be a oempty closed cove subset of a Hausdorff topological vector space ad F :. Let γ (, be give umber ad maps φ, ψ : (, satisfy the followig coditios. (i) For ay fied, y φ( is a γ geerally lower semicotiuous fuctio i. (ii) For ay fied, ψ ( is a F-γ geeralized quasicocave fuctio i y. (iii) φ( ψ ( for all (,. (iv) The set : φ( y ) γ} is precompact for at least oe y. The there eists such that if sup φ( z) γ. z F ( ) Proof. Defie maps T, G : by T ( = : ψ ( z) γ for all z F( )} ad G( = : φ ( z) γ for all z F( )}. Coditio (i) implies that G is a itersectioally closed-valued mappig o. Ideed, if G(, the φ (, z) > γ. Sice φ( is γ geerally lower semicotiuous fuctio i there eists a y ad a eighborhood N() of such that φ ( r, y ) > γ, for all r N(). The G( y ) \ N( ). Hece G( y ). Thus G is itersectioally closed-valued. From coditio (ii) ad Theorem 5.3., T is a geeralized F-KKM mappig. From coditio (iii), we 4
have T ( G(, ad hece G is also geeralized F-KKM mappig. So G is also geeralized F-KKM mappig. By coditio (iv) G y ) is precompact. Hece, G y ) ( ( is compact. Now by Theorem 5.3.., it follows that I G( ) φ. Therefore, there eists such that φ( z) γ, for all z F( ). I particular, we have if sup φ( z) γ. This completes the proof. z F ( ) It is remared that If F :, F ( i ) = yi for each i,,..., } ad ψ ( is a γ geeralized quasicocave fuctio i y. The Theorem 5.3.3 implies Theorem 3. of Asari et al [79]. Corollary 5.3.5 [79]. Let be a oempty closed cove subset of a Hausdorff topological vector space. Let γ (, be give umber, ad maps φ, ψ : (, satisfy the followig coditios. (v) For ay fied y, φ( is a γ trasfer lower semicotiuous fuctio i. (vi) For ay fied, ψ ( is a γ geeralized quasicocave fuctio i y. (vii) φ( ψ (, for all (,. (viii) The set : φ( y ) γ} is precompact for at least oe y. The there eists such that if sup φ( γ, z F ( ) for all y. We ow preset the followig saddle poit results. First, we preset a saddle poit theorem for γ geerally lower semicotiuous (upper) fuctio i ad F- γ geeralized quasicocave (cove) fuctio i a Hausdorff topological vector space usig Theorem 5.3.4. Theorem 5.3.5. Let be a oempty closed cove subset of a Hausdorff topological vector space ad F :. Let γ (, be give umber, F ad let φ : (, satisfy the followig coditios. : a surective mappig 43
(i) φ( is a γ geerally lower semicotiuous fuctio i ad F- γ geeralized quasicocave fuctio i y. (ii) φ( is a γ geerally upper semicotiuous fuctio i y ad F- γ geeralized quasicove fuctio i. (iii) There eists, y such that the sets : φ( y) γ} ad y : φ(, γ} are precompact. The, there eists a saddle poit of φ ( ; that is, there eists (, such that φ( φ( φ(, for all y. Moreover, we also have if supφ ( = sup if φ( = γ. Proof. By Theorem 5.3.4 with φ = ψ, there eists such that Sice F is surective, the φ(, z) γ, for all z F(. sup if φ( z) = sup if φ( ad hece Z F ( φ(, γ, for all y. (5.3.7) Let f : (, be defied as f ( y, ) = φ (. By assumptio (ii) f ( is γ geerally lower semicotiuous fuctio i ad F γ geeralized quasicocave fuctio i y. Therefore, agai by Theorem 5.3.4, there eists y such that sup if φ ( z, z F ( ) = if z F ( ) sup f ( y, z) Note that F is surective. The if supφ( z, = if supφ( ( ) z F ad hece φ(, γ, for all. (5.3.8) Combiig (5.3.7) ad (5.3.8) we have φ ( = γ ad φ( φ( φ(, for all y. (5.3.9) Fially, agai by (5.3.7) ad (5.3.8), we get supif φ( if supφ( 44
supφ( φ( if φ( by (5.3.9) supif φ(. Cosequetly, if supφ ( = sup if φ( = γ. This completes the proof. Let φ ( be a quasicocave fuctio i y ad γ geeralized quasicove fuctio i F :, F ( i ) = yi for each i,,..., }. The Theorem 5.3.4 implies Theorem 3. of Asari et al [79]. Corollary 5.3.6 [79]. Let be a oempty closed cove subset of a Hausdorff topological vector space. Let γ (, be give umber. Suppose that φ : (, satisfy the followig coditios. (i) φ( is a γ trasfer lower semicotiuous fuctio i ad γ geeralized quasicocave fuctio i y. (ii) φ( is a γ trasfer upper semicotiuous fuctio i y ad γ geeralized quasicove fuctio i. (iii) There eists, y such that the sets : φ( y) γ} ad y : φ(, γ} are precompact. The, there eists a saddle poit of φ ( ; that is, there eists (, such that φ( φ( φ(, for all y. Moreover, we also have if supφ ( = sup if φ( = γ. Now we preset the saddle poit theorem for parametric games i topological vector spaces. Theorem 5.3.6. Let ad F be two Hausdorff topological vector space ad let ad Y F be two oempty closed cove subsets. Let γ (, be give umber, F be a surective mappig ad let G = f g : Y Rsatisfy the followig coditios. 45
(i) f ( is a γ geerally lower semicotiuous fuctio i ad F- γ geeralized quasicove fuctio i y. (ii) f ( is a γ geerally upper semicotiuous fuctio i y ad F- γ geeralized quasicocave fuctio i. (iii) g ( is a γ geerally lower semicotiuous fuctio i y ad F- γ geeralized quasicove fuctio i. (iv) g( is a γ geerally upper semicotiuous fuctio i ad F- γ geeralized quasicocave fuctio i y. (v) There eists, y such that the sets : G ( y) γ} ad y : G (, γ} are precompact. The, there eists a saddle poit of G ( ; that is, there eists (, such that G ( G ( G (, for all y. Moreover, we also have if supg ( = sup if G ( = Proof. Sice, from (ii) ad (iii) i the theorem, it follows that the fuctio G ( is γ geerally upper semicotiuous fuctio i y ad F- γ geeralized quasicocave fuctio i. Similarly G ( is γ geeralized lower semicotiuous fuctio i ad F- γ geeralized quasicove fuctio i y. The from Theorem 5.4.5, ( has a saddle poit (, Y. γ. G The followig result is derived from Theorem 5.3.5. Corollary 5.3.7. Let ad F be two Hausdorff topological vector spaces ad let ad Y F be two oempty closed cove subsets. Let γ (, be give umber ad let G = f g : Y Rsatisfy the followig coditios. (i) f ( is a γ trasfer lower semicotiuous fuctio i ad γ quasicove fuctio i y. (ii) f ( is a γ trasfer upper semicotiuous fuctio i y ad γ quasicocave fuctio i. (iii) g ( is a γ trasfer lower semicotiuous fuctio i y ad γ quasicove fuctio i. geeralized geeralized geeralized 46
(iv) g( is a γ trasfer upper semicotiuous fuctio i ad γ geeralized quasicocave fuctio i y. (v) There eists, y such that the sets : G ( y) γ} ad y : G (, γ} are precompact. The, there eists a saddle poit of G ( ; that is, there eists (, such that G ( G ( G (, for all y. Moreover, we also have if supg ( = sup if G ( = γ. As a illustratio we give the followig eample. ample 5.3.. Cosider the followig payoff matri 3 F = 4 Let the strategies p ad q of the players ad be give respectively by p = ( ) ad q = ( y,, where, y [, ]. The we have 3 y F( p, q) = ( ) 4 y = 3y + ( + ( ) y + 4( )( = 4y 3 y + 4 = f (. Now we have fuctio f ( = 4y 3 y + 4, y [, ], whose graph is give lie this: 4 3.8..4.6.8.6.4. 47
This fuctio is cotiuous so both upper ad lower semicotiuous ad cove so both quasicove ad quasicocave. Now the slope of this fuctio is f (, f ( = (4y 3, 4 ), which is for y = / ad y = 3/ 4. Now f ( / + ε, 3/ 4 + δ ) = 5/ + 4εδ yields that the poit ( /, 3/ 4) is a saddle poit ad the value of the game is F ( p, q) = 5/. 48