PARAMETRIC STUDY ON PARETO, NASH MIN- MAX DIFFERENTIAL GAME

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1 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN ARAETRIC STUDY ON ARETO, NASH IN- AX DIFFERENTIAL GAE.S.Oma, o. th o Ramada Uvety, Egypt N.A. El-Kholy, D. Tata Uvety, Faculty o Scece, Egypt E.I.Solma, hd Studet AASTT, College o Egeeg ad Tech., Alexada, Egypt Abtact I paametc aaly, whch ote eee to paametc optmzato o paametc pogammg, a petubato paamete toducto to the optmzato poblem, whch mea that the coecet the obectve ucto o the poblem ad o the othe had o the cotat ae petubed. I th pape, we peet the qualtatve ad quattatve aaly o adapted appoach o -ax deetal game o xed duato wth geeal paamete the cot ucto ad cota betwee multple playe playg depedetly (the aeto cocept) ad othe depedetly (the Nah cocept), the olvablty et ad the tablty et o the t ad the ecod kd ae deed ad algothm o detemg thee et ae peeted. Keywod: aametc Aaly, Dyamc game, Stablty et o the t kd, Stablty et o the ecod kd. Itoducto Ude the m-max (ecuty) oluto cocept (Vo Neuma ad ogete,944) a paametc dyamc game tudy toduced betwee coopeatve collato o playe (aeto,896) ad othe playe act depedetly (Nah,95) (Sta ad Ho,967,969) whee the paamete peeted the obectve ucto ad cotat, qualtatve ad quattatve aaly o ome bac oto, uch a the et o eable paamete, the olvablty et ad the tablty et o the t ad the ecod kd, ae deed ad aalyzed qualtatvely ad quattatvely o ome clae o paametc optmzato poblem(.s. Oma,977)(.S. Oma, A. El-Baua ad E. Youe,986) The ame oto ae edeed 3

2 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN th pape ad appled to aeto,nah m-max deetal game o xed duato. oblem Fomulato k Code the deetal game ( γ, ) wth geeal paamete γ R k the obectve ucto ad geeal paamete R the cotat betwee o playe that agee to om a coalto ad coopeate a a gle playe I to mmze the collectve cot playg a m - max deetal game agat the othe N playe outde the coalto whch act depedetly to mmze the ow cot.let l u ( t) R, =,,,., l = l be the compote cotol o the playe amog the coalto, whle = m v ( t) R, =,,, N., m= m be the compote cotol o the playe outde the coalto, whee the compote cotol ( uv, ) R a elemet o the cota et Ω( )={( uv, ) R xt ( )= ( txuv,,,, ), xt ( )= x, htxuv (,,,, ) }, whee xt ( ) = ( txt, ( ), ut ( ), vt ( ), ), epeet the ytem o olea deetal equato that gove the game moto, the poblem ca be omulated a the ollowg poblem: ( γ, ): mj(, tuv ˆ,, γ)= wj(, tuv ˆ,, γ) ubect to u = t t = φ( xt ( )) (,, ˆ + I txuv,, γ) dt Ω( )={( uv ˆ, ) R xt ( )= ( txuv,, ˆ,, ), xt ( )= x, htxuv (,, ˆ,, ) }, ( γ, ): mmax (,,, γ)= (,,, γ) u v = t = φ( xt ( )) + I( txu,,, v, γ) dt t Jtu v wj tu v N = 33

3 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN ubect to Ω( )={( u, v) R xt ()= (, txu,, v, ), xt ( )= x, htxu (,,, v, ) } whee t t J( ut ( ), vt ( ), γ) = φ ( xt ( )) + I( txt, ( ), ut ( ), vt ( ), γ) dt, =,,, ae the cot ucto o the coopeatve playe φ( xt ( )) = wφ ( xt ( )) ad = w W, W = { w R w, w = }, ( γ, ): 3 = t + t I( txuv,,,, γ)= wi ( txuv,,,, γ) o each mj ( tuv,,, v, γ )= φ ( xt ( )) I( txuv,,,, v, γ ) dt=,,, N. v ubect to Ω( )={( uv, ˆ) R xt ( )= ( txuv,,,, v, ), xt ( )= x, htxuv (,,,, v, ) } = ad ( γ, ): 4 t mmaxj ( tuv,,, v, γ) = φ ( xt ( )) + I( txuv,,,, v, γ) dt, =,,, N. v u t ubect to Ω( )={( uv, ) R xt ()= (, txuv,,, v, ), xt ( )= x, htxuv (,,,, v, ) } whee t t J ( tuv,,, γ)= φ ( xt ( )) + I( txuv,,,, γ) dt, =,,, N. ae the cot ucto o the N o-coopeatve playe.hee, [ t, t ] deote the xed pecbed duato o the game, x : the tal tate kow by all playe, xt () R the tate vecto o the game, h(.) :[ t, t ] R R R R k q k, = l+ m. (.) :[ t, t ] R R R R k I (.) :[ t, t ] R R R R, =,,, N. k I (.) :[ t, t ] R R R R (.) : R R, =,,,. φ, =,,, N. φ, =,,,. (.) : R R ae cotuou deetable ucto. I the ollowg, we gve the deto o the olvablty ad tablty 34

4 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN et o poblem. The Solvablty Set Deto. The Solvablty et B o poblem ( γ, ) deed a : = (, ) k + k B γ R the oluto o ( γ, ) ext o ome w W () { } The Stablty Set o The Ft Kd Deto. Suppoe that ( γ, ) B wth a coepodg et U o aeto,nah -ax oluto ( uv, ),( u, v),( uv, ) ad ( u, v ) o poblem ( γ, ), ( γ, ), 3( γ, ) ad (, ) 4 γ epectvely, the the tablty et o t kd o poblem ( γ, ) coepodg tou deoted by SU ( ) deed by ( )={(, ) k + k S U γ R U olve ( γ, ) o ome w W} ( Lemma.3 I o each =,,,. ad =,,, N. the cot ucto J (, tuvγ,, ) ad J (, tuvγ,, ) ae lea γ, ( txt, ( ), ut ( ), vt ( ), ) ad htxt (, ( ), ut ( ), vt ( ), ) ae lea ucto, the the et SU ( ) covex. oo. Suppoe that ( γ, ),( γ, SU ( ), the t ollow that o ( γ, ) J( uv,, γ) J( uv,, γ) xt ()= (, txt (), uv,, ) ht (, xt ( ), uv,, ) ( γ, ) J( u, v, γ) J( u, v, γ) J( u, v, γ) J( u, v, γ) xt ()= (, txt (), u, v, ) ht (, xt ( ), u, v, ) 3( γ, ) J (, tuv,, γ) J (, tuv,, γ) xt ()= (, txt (), uv,, ) ht (, xt ( ), uv,, ) 35

5 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN ( γ, ) J (, tu, v, γ ) J (, tu, v, γ ) ˆ ˆ J(, tu, v, γ ) J(, tuv,, γ ) xt txt u v ()= (, (),,, ) (, ( ),,, ) ht xt u v mlaly o ( γ, J( uv,, γ J( uv,, γ xt ()= (, txt (), uv,, ht (, xt ( ), uv,, ) ( γ, J( u, v, γ ) J( u, v, γ ) 3( γ, J( u, v, γ ) J( u, v, γ ) ()= (, (),,, (, ( ),,, xt txt u v ht xt u v J(, tuv,, γ ) J(, tuv,, γ ) xt ()= (, txt (), uv,, ht (, xt ( ), uv,, ) 4( γ, J (, tu, v, γ ) J (, tu, v, γ ) ˆ ˆ J(, tu, v, γ ) J(, tuv,, γ ) xt txt u v ()= (, (),,, (, ( ),,, ht xt u v (4) ow multplyg both de o the et o equato (3) by α ad both de o the et o equato (4) by ( α) ad addg coepodg equece o ame poblem togethe ug the leaty popety γ ad,we get (3) 36

6 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN J( uv,, αγ+ ( α) γ J( uv,, αγ+ ( α) γ xt ( ) = ( txt, ( ), uvα,, + ( α) ht (, xt ( ), uvα,, + ( α) ) J u v J u v (,, αγ+ ( α) γ (,, αγ+ ( α) γ J( u, v, αγ + ( α) γ ) J( u, v, αγ + ( α) γ ) ( ) = (, ( ),,, + ( ) (, ( ),,, + ( ) xt txt u vα α ht xt u vα α J( tuv,,, αγ + ( α) γ ) J( tuv,,, αγ + ( α) γ ) xt ( ) = ( txt, ( ), uvα,, + ( α) ht (, xt ( ), uvα,, + ( α) ) J tu v J tu v ˆ(,,, αγ+ ( α) γ ˆ(,,, αγ+ ( α) γ J( tu,, v, αγ + ( α) γ ) J( tuv,,, αγ + ( α) γ ) xt txt u v α + α ( ) = (, ( ),,, ( ) (, ( ),,, ( ) ht xt u v α + α (5) et o equato (5) yeld that ( αγ + ( α) γ, α + ( α) ) SU ( ) ad hece SU ( ) covex. Lemma.4 I o each =,,,. ad =,,, N. the cot ucto J (, tuvγ,, ) ad J(, tuvγ,, ) ae cotuou o ( txt, ( ), ut ( ), vt ( ), ) ad htxt (, ( ), ut ( ), vt ( ), ) ae cotuou o the et SU ( ) cloed. k R k R,, the oo. Let { γ, } be a equece SU ( ) uch that { γ, } covege to ( γ, ) a o all ( uv, ) Ω ( ) the o (, ) γ J( uv,, γ) J( uv,, γ) xt ()= (, txt (), uv,, ) 37

7 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN ht (, xt ( ), uv,, ) (, ) γ J( u, v, γ ) J( u, v, γ ) (,, γ) (,, γ) ()= (, (),,, ) J u v J u v xt txt u v ht (, xt ( ), u, v, ) (, ) 3 γ J(, tuv,, γ ) J(, tuv,, γ ) xt ()= (, txt (), uv,, ) ht (, xt ( ), uv,, ) (, ) 4 γ J (, tu, v, γ ) J (, tu, v, γ ) ˆ (,,, ) (,,, ) ˆ J tu v γ J tuv γ xt ()= (, txt (), u, v, ) ht (, xt ( ), u, v, ) (6) Now takg the lmt o both de o the et o equato (6) a t yeld that 3 J( uv,, γ) J( uv,, γ) xt ()= (, txt (), uv,, ) ht (, xt ( ), uv,, ) J u v J u v (,, γ) (,, γ) J( u, v, γ ) J( u, v, γ ) ()= (, (),,, ) (, ( ),,, ) xt txt u v ht xt u v J(, tuv,, γ ) J(, tuv,, γ ) xt ()= (, txt (), uv,, ) 38

8 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN ht (, xt ( ), uv,, ) ˆ(,,, γ) ˆ(,,, γ) J tu v J tu v J(, tu, v, γ ) J(, tuv,, γ ) xt txt u v ()= (, (),,, ) (, ( ),,, ) ht xt u v (7) om equato (7) we deduce that ( γ, ) SU ( ) ad SU ( ) cloed. Theoem I ( uv, ) Ω R aeto oluto o the playe de the coalto wth tate taectoy x coepodg to poblem ( γ, ),the thee ext cotuou cotate ucto p:[ t, t ] R, q δ R uch that the ollowg elato ae ated x ()= t (, t x, u, v, ), x ( t)= x (8) Htx (,, uvw,,,, w, p, δγ,, ) pt ()= xt () (9) φ x t pt ( )=, = w ( x( t )) () ( ( )) φ φ xt ( ) = Htx (,, uvw,,,, w, p, δγ,, ) = () δht (, x, uv,, )= ( ht (, x, uv,, ) (3) δ (4) whee T Htxuvw (,,,,,, w, p, δγ,, )= wi( txuv,,,, γ) + p ( txuv,,,, ) = δ T ht (, xuv,,, ) (5) the Hamltoa ucto o the playe de the coalto ad t patal devatve.futhemoe, I ( u, v) Ω R -ax pot o the coopeatve playe wth tate taectoy x coepodg to poblem ( γ, ),the thee ext cotuou cotate ucto p:[ t, t ] R, q q ς R ad η R uch that the ollowg elato ae ated x ()= t (, t x, u, v, ), x ( t )= x (6) 39

9 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN Htx (,, u, vw,,, w, p, ςη,, γ, ) pt ()= (7) xt () φ x t pt ( )=, = w ( x( t )) (8) ( ( )) φ φ xt ( ) = Htx u vw w pςη γ Htx (,, u, vw,,, w, p, ςη,, γ, ) = v (,,,,,,,,,,, ) = (9) ( ςht (, x, u, v, )= () η ht (, x, u, v, )= ( ht (, x, u, v, ) (3) ς (4) η, =,,, N. (5) whee T Htxuvw (,,,,,, w, p, ςηγ,,, )= wi( txuv,,,, γ) + p ( txuv,,,, ) = ς T ht (, xuv,,, ) (6) the Hamltoa ucto o the playe de the coalto ad t patal devatve evaluated ug the two et o multple ς ad η. Fo the playe outde the coalto, I ( uv, ) Ω R equlbum pot o the N playe coepodg to poblem 3 ( γ, ), q the thee ext cotuou cotate ucto q :[ t, t ] R, δ R uch that the ollowg elato ae ated x ()= t (, t x, u, v, ), x ( t)= x (7) H ( tx,, uvq,,, δ, γ, ) q k ()= t xk () t (8) φ ( x ( t )) qk( t ) =, k =,,, xk( t ), =,,, N (9) H ( tx,, uvq,,, δ, γ, ) =, v =,,, N (3) δ ht (,,,, ) =, =,,, (3) ht (, x, uv,, ) (3 3

10 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN δ, =,,, N (33) whee T H ( txuvq,,,,, δ, γ, )= I( txuv,,,, γ) + q ( txuv,,,, ) δ T ht (, xuv,,, ), =,,, N (34) ae the Hamltoa ucto o each o the N playe ad t patal devatve.futhemoe, (, ) ˆ u v D Ω R a -ax pot o playe o the Nah playe coepodg to poblem (, ) 4 γ, the thee q ext cotuou cotate ucto q :[ t, t ] R, ς R ad η uch that the ollowg elato ae ated x ()= t (, t x, u, v, ), x ( t )= x (35) (,,,,, ς, η, γ, ) H tx u v q q k ()= t (36) x () t ( x ( t )) φ qk( t ) =, k =,,,, =,,, N (37) x ( t ) H tx u v q k ˆ (,,,,, ς, η, γ, ) v ˆ k q R =, =,,, N., ˆ =,,, N (38) H ( tx,, u, v, q, ς, η, γ, ) =, =,,, N (39) ˆ ς h( t, x, u, v, ) =, =,,, N ˆ =,,, N (4) η ht (, x, u, v, ) =, =,,, N (4) ht (, x, u, v, ) (4 ς, =,,, N (43) η, =,,, N (44) whee T H ( txuvq,,,,, ς, η, γ, )= I( txuv,,,, γ) + q ( txuv,,,, ) ς T ht (, xuv,,, ), =,,, N (45) ae the Hamltoa ucto o the Nah playe ad t patal devatve evaluated ug the two et o multple ς ad η. Detemato o The Stablty Set o The Ft Kd I th ecto we toduce method to detemg the tablty et o t kd SU ( ) o aeto,nah -ax cotuou deetal game that ca be ummazed the ollowg tep:. Stat wth ( γ, ) B coepodg to aeto,nah -ax oluto 3

11 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN U. Subttutg the ytem o equato gve by theoem ( we obta q I (,,,, ) (,,,, ) (,, tx uvγ tx uv h tx uv,, ) w + p δ = = = = δht (, x, uv,, )= I( tx,, u, v, ) ( tx,, u, v, ) h( tx,, u, v, ) = q γ w + p ς = = = I( tx,, u, v, ) ( tx,, u, v, ) h( tx,, u, v, ) = q γ w + p η = v = v = v ςht x u (,,, v, )= η ht (, x, u, v, ) =, =,,, N. q I (,,,, ) tx uv γ (, tx, uv,, ) h (, tx, uv,, ) q + δ = v v v = = δ ht x uv N (,,,, ) = =,,,. I (, tx, u, v, γ ) (, tx, u, v, ) h (, tx, u, v, ) = ˆ ˆ ˆ q + q ς vˆ = vˆ = vˆ ˆ ˆ ς h( t, x, u, v, ) =, =,,, N., =,,, N. (,,,, γ ) (,,,, ) (,, q,, ) + q η = = = ht (, x, u, v, ) =, =,,, N. I tx u v tx u v h tx u v η (46) Th ytem o equato (46) deoted by Fwδςη (,,,, δ, ς, η, γ, ), epeet = + N( m+ l) + q+ Nq(3 + N) equato o the + k+ k + q(+ N) ukow, whch ae olea k k q γ R, R ad lea w, δςη,,, δ, ς ad η ( R ). I = + k+ k + q(+ N) the we may obta the ukow explctly.i = k+ k ad γ Fw (, δςη,,, δ, ς, η, γ, ), Fw (, δςη,,, δ, ς, η, γ, ) ext ad cotuou o evey ( w, δ, ς, η, δ, ς, η, γ, ) D ( w, δ, ς, η, δ, ς, η, γ, ), whee D α the eghbohood o the pot o oluto α (,,,,,,,, ) w δ ς η δ ς η γ o 3

12 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN the ytem F, the by the mplct ucto theoem,γ ad ca be expeed a ucto o w, δςη,,, δ, ς, η. The Stablty Set o The Secod Kd Deto.5 Suppoe that ( γ, ) B wth a coepodg et o aeto,nah -ax oluto ( uv, ),( u, v),( uv, ) ad ( u, v ) o poblem ( γ, ), ( γ, ), ( γ, ) ad ( γ, ) epectvely,ad 3 4 σγ (,, I)={( uv, ) R xt ()= (, txuv,,, ), xt ( )= x, h( txuv,,,, ) =, I {,,, q}, h( txuv,,,, ), I} (47) whch deote ethe the uque de o σ o t( σ ) whch cota U,the the tablty et o ecod kd o poblem ( γ, ) coepodg to σγ (,, I) deoted by Q( σγ (,, I)) deed by Q( σγ (,, I))= {( γ, ) B Ω ( γ, ) σγ (,, I) φ} (48) whee Ω ( γ, )={( u, v ) Ω( ) ( u, v ) olve ( γ, ) o ome w W} (49) Lemma.6 I o each =,,,. U the cot ucto J(, tuvγ,, ) l m tctly covex wth epect to u R ad cocave wth epect to v R,ad o each =,,, N. the cot ucto J(, tuvγ,, ) tctly covex m l wth epect to v R ad cocave wth epect to u R ad let σγ (,, I), σγ (,, I ae two dtct de o Ω ( ), the Q( σγ (,, I)) Q( σγ (,, I) = φ oo. Code that ( γ, ) Q( σγ (,, I)) Q( σγ (,, I), the we have Ω ( γ, ) σγ (,, I) φ, Ω ( γ, ) σγ (,, I φ whch cotadct that σγ (,, I), σγ (,, I ae two dtct de. Detemato o the Stablty Set o the Secod Kd To deteme the tablty et o ecod kd Q( σγ (,, I)) o aeto,nah -ax cotuou deetal game that ca be ummazed the ollowg tep:. Stat wth ( γ, ) B coepodg to aeto,nah -ax oluto U o poblem ( γ, ). 33

13 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN Detemg the de σγ (,, I) by ubttutg the cotat to obta the actve et o cotat. 3. Fom the deto o the tablty Set o The Secod Kd by takg I = {,,, d} {,,, q} the we have the ytem o equato: (,,,, ) d I (,,,, ) (,, tx uvγ tx uv h tx uv,, ) w + p δ = = = = h( tx,, uv,, ) = =,,, d. I( tx,, u, v, ) ( tx,, u, v, ) h( tx,, u, v, ) = d γ w + p ς = = = I( tx,, u, v, ) ( tx,, u, v, ) h( tx,, u, v, ) = d γ w + p η = v = v = v h ( tx,, u, v, ) = =,,, d. I (,,,, ) d tx uv γ (, tx, uv,, ) h (, tx, uv,, ) q + δ = v v v = = h tx uv d (,,,, ) = =,,,. I (, tx, u, v, γ ) (, tx, u, v, ) h (, tx, u, v, ) = ˆ ˆ d ˆ + q ς vˆ = vˆ = vˆ (,,,, γ ) (,, d,, ) (,,,, ) + q η = = = (,,,, ) = =,,,. I tx u v tx u v h tx u v h tx u v d (5) Th ytem o equato (5) epeet + Nm ( + l+ d) + 3d equato o the + k+ k + d(+ N) + ukow, whch ae olea k γ R, k R, u l R, v m R ad lea,,,,, w δςη δ ς ad q η ( R ), om whch we ca obta γ ad a ucto o w, δςη,,, δ, ς, η, u ad v. Reeece: Nah J.F.,'No-coopeatve game', Aal o athematc,vol. 54, No., (95)..S. Oma, Qualtatve aaly o bac oto paametc covex pogammg I (aametea the Cotat), Apl. at. CSSR Akad. Ved. mgue (977). 34

14 Euopea Scetc Joual Jauay 5 edto vol., No.3 ISSN: (t) e - ISSN S.Omau, A. El-Baua ad E. Youeee, O a geeal cla o paametc covex pogammg, ASE e I5 (l), (986). aeto,v., Cou d'ecoome oltque,rouge, Lauae,Swtzelad,( 896). Sta, A.W., ad Ho,Y. C.; Nozeo-um deetal game. Joual o Optmzato Theoy ad Applcato, Vol. 3, No. 3, (967) Sta, A.W., ad Ho,Y. C.; Futhe opete o Nozeo-um deetal game. Joual o Optmzato Theoy ad Applcato, Vol. 3, No. 4, (969). Vo Neuma,J. ad ogete, O. Theoy o game ad ecoomc behavo, ceto Uvety e, ceto. N J, (944). 35