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^} ΠY
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.
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