A refinement of the concept of equilibrium in multiple objective continuous games

Re .R.Acad. Cienc.Exac .Fis.Na . (Esp) Vol. 93, N.'' 4, pp 457-462, 1999 Monog á ico: P oblemas complejos de decisión. II A REFINEMENT OF THE CONCEPT OF EQUILIBRIUM IN MULTIPLE OBJECTIVE CONTINUOUS GAMES* (Game heo y/mul iple objec i e games/ ec o p og amming/pe ec equilib ia) J. PUERTO, R. INFANTE AND F. R. FERNÁNDEZ Facul ad de Ma emá icas. Depa amen o de Es adís ica e In es igación Ope a i a. Uni e sidad de Se illa. 41012 Se illa. Spain. ABSTRACT This pape conside s a pe ec ion e inemen o he concep o equihb ium o mul iple objec i e non-ze o sum games. Based on he ideas o an Damme (1991) on pe u bed games and s abili y he concep o pe ec equilib ium is ex ended o a class o con inuous games wi h mul iple obje i es. Exis ence is shown and se e al ela ionships ha exis wi h he co esponding concep o scala games a e s a ed. 1. INTRODUCTION Al hough nea ly 40 yea s ha e passed he publica ion o Blackwell's pape (see Blackwell (1956)) which is he i s known e e ence on Mul iple Obje i e Games (MOG) only a ew pape s ha e been de oed o his pa - icula ield among he wide li e a u e o Game Theo y. Howe e , in ecen yea s he e has been some inc easing in e es in s udying games wi h ec o payo . One o he easons is ha his app oach ep esen s be e eal-wo ld si ua ions o game heo y. In ac , each compe i i e si u- a ion ha can be moedeled as a scala game has i s coun e pa as a mul iple objec i e game when mo e han one scena io has o be compa ed simul aneously (see Fe nández and Pue o (1996)). Hash's (1951) concep o equilib ium is p obably he mos impo an solu ion concep in non-coope a i e game heo y. The no ion o equilib ium in MOG was in- oduced and i s exis ence p o ed by Shapley (1959) un- de es ic i e hypo heses. The ounda ion behind his concep is ha i one playe does no speci y an equilib- ium as his s a egy, hen some playe could gain by changing his s a egy o some hing o he han wha was speci ied o him. Hence, no eason exis s o playe s o play s a egies ha a e no Nash's equilib ium. Howe e , i is also well-known ha any pa icula equilib ium does no ha e o be a easonable p edic ion o easonable be- ha io . We only can a gue ha any ou come ha is no an equilib ium would necessa ily be un easonable as a de- sc ip ion o how a playe should beha e. This ac leads se e al au ho s o conside o e inemen s o Nash's equi- lib ium concep . In ecen yea s, some esea ch has been de o ed o s udy solu ion s uc u es and algo i hms o mul ic i e ia games (see e.g. Be gs esse and Yu (1977) o Bo m e al. (1988)). Howe e , li le a en ion has been ocussed on he undamen al p oblem o exis ence o solu ions. Wang (1993) deal wi h his p oblem. In ha pape , ixed-poin heo ems and o he echniques a e used o de- i e condi ions o he exis ence o equilib ia in games wi h ec o payo s. No e heless, hese equilib ia a e s ill no s able agains small pe u ba ions o all playe s' s a egies. He e we a e in e es ed in a class o n-pe son non- coope a i e MOG wi h uncoun able se o s a egies. We p opose a e inemen o he concep o equilib ium o hose MOG based on he no ion o pe ec ness. This e- inemen has been al eady p oposed by Mendez-Naya e al. (1995) o Nash's equilib ia in con inuous scala games. Mo eo e , Van Megen e al (1999) and Pue o and Fe nández (1995,1999) conside simila e inemen s o Nash equilib ia o ini e mul iobjec i e games. By de- eloping a solu ion heo y ha is based on such e ine- men we equi e he e i ica ion o some p ope ies ha would be heo e ically desi able and, wha is equally im- po an , ha he e ined concep selec a nonemp y se o equilib ia o any con inuous MOG. Acknowledgemen s: The esea ch o he au ho s is pa ially suppo ed by Spanish DG-ICYT g an numbe PB97-0707. 458 J. Pue o e al. Re .RAcad.Cienc.Exac .Fis.Na . (Esp), 1999; 93 The pape is o ganized as ollows. Sec ion 2 is de o ed o s a e he gene al se ing whe e we o mula e he MOG and in oduces he concep o equilib ium o hose games. Sec ion 3 p oposes he e inemen o he concep o equilib ium. Sec ion 4 p esen s some conclusions and he pape ends wi h an appendix whe e echnical ools used in he pape a e desc ibed. 2. THE CONCEPT OF EQUILIBRIUM IN MULTIPLE OBJETIVE GAMES A con inuous mul iple objec i e game (MOG) in no - mal o m is de ined as a iple T = {N, Y', w'} whe e A^= {1,..., n] is he se o playe s. Fo each / e A^, F = [0, 1] is he se o pu e s a egies o playe /. The playe i's ec o payo u' is a con inuous unc ion de ined om u': F = n ^ 1 -^ R"'^'^ whe e m(i) is he numbe o objec- i es o his playe . I is clea he MOG di e s om single c i e ion (scal- a ) games only in he payo unc ions. In MOG, each playe has a ec o payo s o op imize, while in classical games hey ha e scala payo s. In pa icula , i m(l) = ••• = m(n) = 1 ou game T becomes a n-pe son game in no mal o m. 00 Now, le us in oduce he solu ion concep o a MOG p oblem. Le P(N) be he amily o all non emp y subse o A^, i.e. he di e en coali ions o playe s in A^. Fo any y = {y ..., /} G F and w = {u ..., u"} e IR'"^'^ whe e y e F and u' = (w'„ ..., w^.^) e W^'^'K le j^ = [Y -i^C} be he s a egies o coali ion C and y_c = y^/^ he s a- egies o playe s no in C. In he same way, le u^ = {u' : / G C] and u_c = {u' : ^ C} be he p ojec ions o u in o U^ and IR"^ espec i ely. Fo each playe / i s se o mixed s a egies S' is he se o all he Bo el-p obabili y measu es on [0, 1]. This is a subse o M he locally con ex linea space o all he signed measu ed on [0, 1]. M is he dual o C[0, 1] he space o all he con inuous unc ions om [0, 1] in o U. 5' is a weakly* compac subse o M, hence compac in i s weak* opology. As i is usual one can iden i y each measu e jneM wi h he con inuous linea unc ional </!,/> = J/d|i V/G C[0, 1] (see he Appendix o mo e de ails on he weak* opology.) A mixed s a egy o he MOG F is a combina ion s = (s ..., ^'') G 5 = (5^ X ••• X 5"). In he usual way, we can conside Y' imbedded in S' o all / because each y' G Y' co esponds o he mixed s a egy which assigns p ob- abili y 1 o y' and 0 anywhe e else. In he same way, any game wi h a ini e numbe o s a egies is also included in his amewo k. Once we ha e de ined he mixed s a - egy space, we can ex end he payo unc ions. uKs ..., ^'0 = {u (s ..., s-l ..., <,,(^', ..., n) (1) whe e u'j(s ..., s") = J u'jds^ ... ds'' o y = 1, ... m{í) is he in eg al wi h espec o he p oduc measu e gene a ed by s ..., s". I is wo h no ing ha u' is a weakly* con inuous unc ion because he payo unc ions u' a e con inuous. Le us in oduce he concep o equilib ium poin o he MOG F. We will use he no a ion x = (y^, y_c) o e e y coali ion C e P(N). De ini ion 1. A s a egy s = (s ...,s") e S is an equi- lib ium poin o = (N, Y', u'), i o each i e N, s' is a weakly e icien solu ion o he ec o maximum p oblem VM.{sJ : max {u (s 5_,), ..., ^^.^ (s sj) (2) Hence, i is easy o unde s and ha s is an equilib ium poin i each playe / chooses s' as he bes esponse o he s a egy s¿. The abo e de ini ion coincides wi h he con en ional de ini ion o con inuous Nash equilib ium when m(i) = 1 o all /, i.e. in he pa icula case o scala games. The in e es ed eade can see he pape s o Bu ge (1959) and Pa hasa a y and Ragha an (1971) o u he de ails and exis ence esul s in he scala case and he pape o Wang (1993) o he mul iobjec i e case. Fo he pu pose o he in oduc ion o he idea o equi- lib ium in MOG any concep o e iciency would be alid. Howe e , weak-e iciency induces he s onges dominance ela ion and he e o e a playe would only ag ee on de ia ion i he payo in all he c i e ia in- c eases (see e.g. Van Megen e al (1999)). In addi ion, o ex end he concep s o s able equilib ia o MOG we will need some opological p ope ies as closedness o he whole se o equilib ia ha equi es he use o weak e i- ciency a he han o he e iciency concep . Ou i s esul s a es he exis ence o equilib ium poin s unde gene al hypo heses. Le A^^^-^ = {1 G IR'''^'^ : Àj ^ 0, i:;"i'] A^. = 1} / = 1,..., n be a se o weigh s. Le us conside hen n-pe son unic i e ion gam (/l), whe e X = (À ..., ), À' G A,„(,.) wi h he payo unc ions uV) = IJÍ] ÀjUj(s',. s_¡) o a s_¿ ixed and / = 1, ..., n. We s a e in he ollowing lemma an s aigh o wa d esul which is a consequence o he gene al heo y o ec o op imiza ion. Lemma 1. Le F be a n-pe son MOG wi h con inu- ous payo unc ions u' o all / = 1,..., n and s a egy se s S' i = 1, ..., n. Then s = (s ..., s") is an equilib ium s a - egy o F i he e exis s A = (A',..., A") G A,„(,) X • • • X A^ so ha o all i p oblem I, ..., n s' is an op imal solu ion o he J. Pue o e al. Re .RAcad.Cienc.Exac .Fis.Na . (Esp), 1999; 93 459 m(/) max y Xiu](s', s ,) s'eS' (3) ;=i P oo . Since each u' is a con inuous mul i-linea unc- ion i is also conca e in s' whene e s_¡ is ixed. Hence he esul ollows. D The ollowing heo em whose p oo is di ec om Lemma 1 cha ac e izes he se o equilib ium poin s in MOG. Theo em 1. The se o equilib ium poin s o a MOG in he condi ions abo e coincides wi h he se o all Nash equilib ia o games T{X) when X a ies in A^^j^^ x • • • No ice ha his esul is well-known and i al eady ap- pea s sugges ed in he pape o Shapley (1959)) al hough o ini e wo-pe son games. A mixed s a egy 5^ G 5 is an equilib ium in an //-pe u - bed game i o all / = 1, ..., n, 5' is a weakly e icien solu ion o P.{^[¿, 's) : max {u {s is.,), ..., u'^^^{s ~s_^) (5) Wi h hese p éliminai es we a e in posi ion o in o- duce he concep o pe ec equilib ium in MOG. De ini ion 2. We say ha s = (s ..., s'') G S is a pe - ec s a egy combina ion o he MOG T i he e exis s a sequence {pj} o pe u ba ion ec o s and a sequence {S]^} o s a egies wi h S ^ = (si, ..., s^) such ha : 1. Pk ^ 0; 2. Sj^ is an equilib ium in an pj^-pe u bed game o all k: 3. A REFINEMENT OF THE CONCEPT OF EQUILIBRIUM In o de o in oduce he e inemen o he concep o equilib ia o con inuous MOG we use he echnique o pe u bing he se o admissible s a egies and de ine he e ined equilib ia as limi s o sequences o equilib ia in he modi ied games. I should be no ed ha hese ideas we e i s ly applied by Sel en (1975) when he de ined he concep o pe ec equilib ia in scala games. The goal o his sec ion is o ex end he abo e men ioned concep o pe ec ness o con inuous MOG. In an equilib ium, each playe 's equilib ium s a egy is an e icien esponse o he o he playe s' equilib ium s a egies. In a pe ec equilib ium, he e mus also be a bi a ily small pe u ba ions o all playe s' s a egies such ha e e y pu e s a egy ge s s ic ly posi i e p ob- abili y and each playe 's equilib ium is s ill an e icien esponse o he o he playe s' pe u bed s a egies. Thus, as in Sel en's pe ec equilib ium we disc imina e sol- u ions which a e no s able agains any a bi a ily sligh pe u ba ion o he game s a egies. A pe u ba ion o e o ec o is a measu e p = (p ..., p'') E W;^ I 5", sa is ying o all /: 1. p'{(a, b]) > 0 o any in e al (a, b] ^ [0, 1] wi h a< b. 2. A^m 1])<1. The embling-hand s a egies associa ed o a pe - u ba ion ec o p a e he p obabili y measu es in X p^)x ••• xZVO whe e 3. 4 ^ s V/= 1, ..., n. I should be no iced ha when m(i) = 1 o all / = 1,..., n ou pe ec equilib ium educes o he concep o pe - ec equilib ium o con inuous games as in oduced in Mendez-Naya e al (1995). Fi s o all, we p o e ha e e y pe ec equilib ium is also an equilib ium. Then, we show ha he con e se is no ue. To p o e he inclusion we need some echnical esul s. Le YXp', s_¡) be he se o all he alues o he payo unc ion u'(s, s_¡) whene e s' e X p'). This is {p s_¡) = {j G R'"^'^ :}; = u s, 5_.), s e X'(p)}. This amily o se s induces a poin - o-se map: p^ - np s J (6) This applica ion is con inuous in he sense o Hogan (1973) (see he Appendix o de ails). Con inui y o F is used o p o e he exis ence o pe ec equilib ia as i will be shown in he ollowing. To p o e his p ope y we need o p o e p e iously he con inui y o he map X de- ined by (4). Lemma 2. The poin - o-se map X de ined by X:M ^ 2^ p ^ X(p) (7) X'ip) = {F" G S' : F'(I) ^ PXI) V/ = (a, è] ç [0, 1]} (4) is con inuous. 460 J. Pue o e al. Re .RAcad.Cienc.Exac .Fis.Na . (Esp), 1999; 93 P oo . Fi s , we p o e ha S is uppe semicon inuous (u.s.c). Indeed, le {/i^} ^ M be a sequence which con- e ges in he weak* opology o //, his is ju ,^^ ju, and le F^ e X{ i^) o all k sa is ying F^ ^ F. Conside he weak* con inuous unc ion g de ined by: g:M X M > M il^, F) > p-F (8) Then, g{p ^, F,) ^ g{p, F) = p- F. Mo eo e , since F^ e X(PJ) o all k hen (p^ - Fj)(I) ^ 0 o all / = (a, b] ^ [0, 1]. Hence (p-F)U) < O o all/= (a, b] ^ [0, 1] andFe X(p) which p o es ha X is u.s.c. Now, we p o e ha X is lowe semicon inuous (l.s.c). We ha e o p o e ha gi en {p/} ^ M a sequence which sa is ies p j^^ p, and F e X(p) hen he e exis s F^ G X(pj) excep o a ini e numbe o k, sa is ying F^ ^ F. We dis inguish wo cases. 1) 2) F(T) = p(I) o any / = (a, è] ^ [0, 1]. In his case, we can ake F^ = pj^ and he esul ollows. F / ju. Since /i is a ini e measu e,^he e exis s a measu e F e M such ha F(/) > F(/) > p(I) i F(/) > p(I) and F(/) > p(I) i F{I) = p{I). Conside he sequence F^ = (1 - Xj)F + XJ^ wi h A^ G [0, 1]. I is clea ha F^ ^ F i A^ -^ 0. On he o he hand, F^ G X{pj) i F^ - /i^ ^ 0 which is equi alen o 4(F - F) + F - ^^ ^ 0. Now, o any / = {a, b] ^ [0, 1] i F(7) > p(I) hen (F, - p,) (I) ^ 0 i 4 is small enough. O he wise, his is i F(/) = p(I) hen (F^ - pj) (I) ^ 0 o any A^. Hence, i A^ ^ 0 hen: and he p oo is comple e. D Lemma 3. The poin - o-se map F de ined by (6) is con inuous a p o any i = 1, ..., n. P oo . Le {p,} a M, pj^ ^ p and y^ e Y' p,^, s_¿) o all k, y^ -^ y. F om he de ini ion o Y'(p ^, s_- he e exis s {Fj} cz X' such ha F^ G X'(JU^) and y^ = w(F^, 5_.). Since S' is weak* compac he sequence [Fj^ (o some sub- sequence) con e ges in he weak* opology o some F. Then, F belongs o X p) because X' is^u.s.c... Now, since u is weak* con inuous w(F^, 5_-) -^ u(F, 5_-). This implies ha y = M(F, 5_,) G F'(/Î, ^_.). Hence F is u.s.c. Le {/i^} cz M,ji¡^ ^ p and y G F(/î, 5_,). By de ini ion o F' he e exis s F G X'(/Î) such ha y = u{p, 5_.). Now, since X' is l.s.c. he e exis s a sequence {F^} wi h F^^ X pj^) excep o a ini e numbe o k such ha Fj. ^ F. Take y^ = u(p,^, s), hen y^ G Y'ip,^, s¡). Mo eo e , he con- inui y o u implies ha y^ -^ y. This p o es ha F is l.s.c. a p. D Le N'ip, s_¡) be he se o weakly e icien solu ions o P-(p, s) on he image space. This is NXP, S J = {ye Tip, s J : ^y' e Tip,, s J wi h y' > y componen wise}. Now, we conside he poin - o-se map de ined by he bo e in oduced amily o se s. p - N p s_¡) Lemma 4. The poin - o-se map N' is u.s.c. a p. (9) P oo . Apply Theo em 4.2.1 in Sawa agy e al. (1985) aking in o accoun ha in ou case he domina- ion s uc u e D = R'^^'^ is cons an and con ex and ha by Lemma 3 F is a con inuous poin - o-se map. D Le us conside he poin - o-se map M' de ined as M' p, 5_,) = [s^ G {p) : u s sj G N p, 5_,)} (10) We s a e he ollowing lemma which is used in he nex heo em. This is a consequence o he p e ious e- sul s. Howe e , o he sake o comple eness a p oo is gi en. Lemma 5. The poin - o-se map M' de ined in (10) is an uppe semicon inuous map a p, o any i = 1, ..., n. P oo . Le {p,} ^ S, p,^ p, F, G M'(p,. ~s_.), and F, ^ F Since X' is u.s.c. 3. p, F e X p). F om he de ini ion o M{p, 5=_-), w(F^, ^_.) G N pj^, s_¡). F om he weak* con i- nui y o u: u(F,^, s_.) > u{F, ~s_¡). Hence, w(F, s_^^ N p, s_¡), since he map A^' is u.s.c. a p. The e o e, F G M p, s_¡), and so he map M is u.s.c. a p. D Le us assume he hypo hesis o Lemma 5 hen we ha e he ollowing esul which s a es he ela ion be- ween he se o equilib ia and pe ec equilib ia o a MOG. Theo em 2. Fo any con inuous MOG in no mal o m, he se o pe ec s a egy combina ions is a subse o he se o equilib ia. J. Pue o e al. Re .RAcad.CiencExac .Fis.Na (Esp), 1999; 93 461 P oo . Le us conside a pe ec equilib ium s and le {5^} and [jij] be wo sequences such ha ^^ is an equilib- ium in an /i^-pe u bed game o all k and e i ying ha {sj^, ii) ^ (s, 0). Hence, we ha e o all / = 1, ..., n ha s[ e M i[ (s_.),^) hen he u.s.c. o M' impUes ha s' G M'(0, s_¡) o all / = 1, ..., n. Tha is, s' is a weakly e icien solu ion o he p oblem VM¿(s_¿) o all z = 1, ..., n wha by De ini ion 1 implies ha 5" is an equilib ium poin . D In he ollowing, we p o ide an example which shows ha he concep o pe ec equilib ium is a s ic e ine- men o he concep o equilib ium in con inuous mul i- objec i e games. Example 1. Le us conside ha wo-pe sons wo ob- jec i es con inuous game F wi h ec o payo s gi en by: u x, y) = u x, y) = (xy, x^y) o all (x, y) e [0, 1] x [0, 1]. The game T induces wo con inuous single objec i e games T. wi h i = 1,2 wi h he payo unc ions u¡, u] de ined on [0, 1] x [0, 1] by: u {x, y) = u]{x, y) = xy ^li^i y) = ^ {^y y) = ^y- I is clea ha any equilib ium (F, G) in T is also an equilib ium in F- / = 1, 2. Indeed, i we assume ha (F, G) is no an equilib ium in F, hen i mus exis F* such ha : xy dF*(x)dG(y) < xy dF(x)dGCy). (11) Now, since o any (Xj, ji), (^2, ^2) ^ [0' 1] ^ [0' 1] we ha e ha Xjji implies ha : < x^2 i ^i^d only i Xjjj ^ x^2 hen (11) xVdF*(x)dGCy)< x'y àF{x)àG{y). (12) Thus, u F'',G) = (J xy dF*(x)dG(j), J x^y dF*(jc)dG(j)) < (J xy dF(x)dGCy), J ^y dF(x)dGCy)) = = w'(F, G). This las inequali y means ha (F, G) would no be an equilib ium in F because F* is a be e esponse o G han F This con adic ion p o es ha any equilib ium in F is also an equilib ium in he associa ed single objec i e games F,, F2. No ice ha he same a gumen can be ap- plied o he //^-pe u bed games. Le us deno e by è{x) he degene a e p obabili y measu e which assigns p obabili y 1 o JC and 0 e e y- whe e else. I is clea ha (¿(0), <5(0)) is a mul iobjec i e equilib ium o he game F. Assume ha i is pe ec . Then, he e exis s a sequence {(F^ G^)}^^ j o Nash equi- lib ia o he co esponding sequence o /¿^-pe u bed games, con e ging in he weak* opology o (¿(0), <5(0)). I is clea ha u è{l), G^) > w(F, G^) (componen wise) o all k^ I (p o ided ha F 7^ ô(l)). Then, by applica- ion o Theo em 3 in Mendez-Naya e al. (1995) o he single objec i e games F. / = 1, 2, i ollows ha F^ co- incides wi h jJ" on [0, 1). The e o e, since F^ is a p ob- abili y measu e, when pa goes o ze o F^ con e ges o (5(1). This con adic s ha (¿(0), ¿(0)) is he limi o he sequence {(F^ G^)}^^i. Hence (¿(0), ¿(0)) is an equilib- ium which is no pe ec . A Finally, he exis ence o such pe ec equilib ium poin s is s a ed in he ollowing heo em. Theo em 3. Fo any con inuous MOG in no mal o m F, he e exis s a leas one pe ec equilib ium. P oo . Since each XXiij) is a weak* compac , con ex se and he objec i e unc ions M'(-, S_¡) a e linea o any s_¿ ixed hen he e exis s 5*^ being an equilib ium o he ^^-pe u bed (1993)). game (see e.g. Co olla y 3.2 in Wang Now, {si}¡^^ 1 is a sequence in 5^ x ••• x 5^ which is a weak* compac se . Then, he e exis s a weak* con e - gen subsequence {s^ }^^ j included in {s,}¡^^ j. Hence, ap- plying he uppe semicon inui y o M' o all z = 1, ..., n he esul ollows. D 4. CONCLUDING REMARKS In his pape he concep o pe ec equilib ium in con- inuous MOG is in oduced as a e inemen o he con- cep o equilib ium in MOG. To his end, we ha e used he app oaches ollowed by an Damme (1991) and Mendez-Naya e al. (1995) based on p ubed games. Ou i s ema k is on he de elopemen o p ocedu es ha compu e hese kinds o equilib ia. A i s sigh , i seems o be a ha d ask because i is e en ha de han in he scala case. Thus, mo e esea ch should be di ec ed owa ds his pa icula poin . Secondly, we would poin ou ha he basic a ionale o conside ing pe u bed model wi h small p obabili y o e o s is ha hey gi e us a way o es he p inciple ha an equilib ium does no depend on he un easonable assump ion ha playe s igno e he pu e s a egies o he game ha ing ze o p obabili y in he equilib ium. Hence, when we use his app oach we a e only using pa o he basic a ionali y p inciples abou playe s' a ional beha - io in a mul iple objec i e game. Un o una ely, we hink 462 J. Pue o e al. Re .RAcad.Cienc.Exac .Fis.Na . (Esp), 1999; 93 ha as i happens in he scala case he e is no solu ion concep ha e i ies all he a ionali y p inciples o in el- ligen beha io in MOG. APPENDIX An in e es ig and impo an no ion ha a ises na u ally when we conside he mixed s a egies in he con inuous games is he weak* opology. S a ing wi h C[0, 1] he no med space o he con inuous unc ions in [0, 1] we o m i s opological dual M he se o all he Bo el signed measu es on [0, 1]. De ini ion 3. A sequence [s^ a M is said o be weak^ con e gen o an elemen s e M i o e e y e C[0, 1] </ s,^y -^ {x, s}. In his case we w i e s^^ ^ s. Based on his con e gence i is possible o conside a no ion o compac ness (less se e e han he usual one) bu s ill su icien o p o ide al e na i e explana ion o he exis ence o solu ions o op imiza ion p oblems. De ini ion 4. A se K ^ M is said o be weak^ com- pac i e e y in ini e sequence om K con ains a weak"^ con e gen subsequence. Sepa able no med linea spaces posses an impo an p ope yu cha ac e izing his kind o compac ness. Lemma 6. Le Xbe a sep able, no med linea ec o space and X^^ i s opological dual. E e y bounded se- quence inX^ con ains a weak^ con e gen subsequence. A well-known consequence o his esul is ha he closed bounded se s in X* a e weak* compac . Since C[0, 1] is a sepa able space hese esul s apply. De ini ion 5. A unc ional L de ined on M is said o be weak^ con inuous a s^e M i gi en e > 0 and a ini e collec ion {/i,/2, ..., ,} om C[0, 1] sduch ha L(s) - L(SQ) < 8 o all s e M such ha {s - SQ, } < ô o all i - 1, ..., n. In he ollowing, we ecall some esul s conce ning he s abili y o op imal solu ion se s o mul iobjec i e op i- miza ion p oblems ega ding pe u ba ions o easible solu ion se s. A poin - o-se map o mul imapping F om a se X in o a se F is a map ha associa es a subse o F wi h each poin o X. In wha ollows, we in oduce se e al concep s wi h ega ds o he con inui y o hese maps. We ollow he de ini ions o Hogan (1973). Fo addi ional de ails on his and o he subjec s we e e he in e es ed eade s o he book o Sawa agi e al. (1985). Le F be a poin - o-se map om a se X in o a se F. De ini ion 6. F is said o be. 1. lowe semicon inuous (l.s.c.) a poin x e X i {x^} c: X, x^ ^^ X, and y G F(x) all imply he exis - ence o an in ege m and a sequence {y^} Œ Y such ha y^ e Fix'') o k^m and j^ -^ y; 2. uppe semicon inuous (u.s.c.) a poin x e X i {x^} c: X, x^ ^ X, y^ G F(x^) and y'' -^ y all imply ha y e F(x); 3. con inuous a poin x e X i i is bo h l.s.c. and u.s.c. a X. REFERENCES 1. Blackwell, O. (1956). 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