S a is ics & Ope a ions Resea ch T ansac ions
SORT 27 (2) July-Decembe 2003, 139-152
S a is ics &
Ope a ions Resea ch
T ansac ions
Pa ial coope a ion and con ex se s
J. En ique Rome o Ga c´
ıa∗, Jo ge J. L´
opez V´
azquez
Uni e sidad de Se illa, Spain
Abs ac
We conside games o ans e able u ili y, hose ha deal wi h pa ial coope a ion si ua ions, made
up o coali ion sys ems, in which e e y uni coali ion is easible and e e y coali ion o playe s can
be exp essed as a disjoin union o maximal easible coali ions. These sys ems a e named
pa i ion
sys ems
and cause es ic ed games. To sum up, we s udy easible coali ion sys ems de ined by a
pa ial o de designed o a se o playe s and we analyze he cha ac e is ics o a easible coali ion
sys em de eloped om a amily o con ex se s.
MSC:
90D12
Keywo ds:
coope a i e games, pa ial coope a ion, con ex se s
1 Pa ial coope a ion
A sys em o easible coope a ions is de ined by (N,F),F ⊆ 2N, ha p o es he ollowing
axiom:
(P1) ∅ ∈ F , and he g oup {i} ∈ F ∀ i∈N.
Conside ing he gi en explana ion, i esul s ha any coali ion S⊆Ncan be
exp essed by a disjoin union o easible coali ions, as
S=[
a∈S
{a}.
Howe e , his pa i ion o S o easible coali ions should no be unique. In gene al,
we will deno e PF(S) he se made up o pa i ions o S⊆Nin nonemp y easible
∗Add ess o co espondence: Jos´
e En ique Rome o Ga c´
ıa. [email p o ec ed]. Facul ad de Ciencias Econ´
omicas
y Emp esa iales. Uni e sidad de Se illa. A da. Ram´
on y Cajal, no1 41018-Se illa. Spain.
Recei ed: No embe 2001
Accep ed: Oc obe 2003
140
Pa ial coope a ion and con ex se s
coali ions. Ob iously PF(∅)={∅}. The p e ious easoning gi es sense o and makes
consis en he idea o a es ic ed coope a ion game:De ine he iple (N,F, ), in which
(N,F)is a easible coali ion sys em and (N, )a ans e able u ili y game. Then he
couple (N, F)in which
F: 2NR, F(S)=max 





X
i
(Ti)| {Ti} ∈ PF(S)






.
is e med a game wi h es ic ed coope a ion by he easible coali ion sys em (N,F).
The supplied explana ion o game o es ic ed coope a ion by a sys em o easible
coali ions is o e e y coali ion o playe s, an ex ension o he one by Faigle (1989)
conce ning games wi h es ic ed coope a ions and by Be gan i˜
nos, Ca e as and
Ga c´
ıa–Ju ado (1993) when using communica ion g aphs o show incompa ibili y
among some o he playe s. Indeed, i can be shown ha F(S)≥Pi∈S ({i}).De ined
his way, he game is always supe addi i e.
Le (N,F)be a sys em o easible coali ions. Le S ⊆N. I is said ha T is F–
componen o S i i is p o ed ha T ∈ F and T0∈ F does no exis , as T ⊂T0⊆S .
Tha is o say, he S⊆NF–componen s a e he maximal easible coali ions included
in Sand, o any S⊆N, he F–componen s o S a e a collec ion {Tk}k⊂2Ssuch ha
S=[
k
Tk
Bu , he F–componen s o S⊆Na e no necessa ily a pa i ion o Sas i s in e sec ion
can be nonemp y.
I can be p o ed ha i we conside (N,F, ), whe e (N,F)is a easible coali ion
sys em, (N, )a supe addi i e game and, o each coali ion S ⊆N, he F–componen s
o S a e a pa i ion o i sel , hen he es ic ed coope a ion game (N, F) e i ies
F(S)=X
k
(Tk),
whe e {Tk}k∈ PF, he S pa i ion o i s maximal easible coali ions (F–componen s o
S ).
The e o e, i he F–componen s o any coali ion a e a pa i ion o i sel , and he
game (N, ) is supe addi i e, hen he es ic ed game by he sys em o easible coali ions
is de e mined by
F(S)=X
k
(Tk),
in which {Tk}kis he Spa i ion o maximal easible coali ions. As he p e ious
exp ession equi es ha maximal easible coali ions mus be disjoin ed, a new de ini ion
o a conc e e easible coali ions sys em has o be looked o . I will be named a pa i ion
sys em .
J. En ique Rome o Ga c´
ıa, Jo ge J. L´
opez V´
azquez
141
A pa i ion sys em is he couple (N,F),F ⊆ 2N ha e i ies he ollowing axioms:
(P1) ∅ ∈ F ,{i} ∈ F ∀ i∈N.
(P2) ∀S⊆N, he S maximal subse s in F(F–componen s o S ) a e a pa i ion o S ,
deno ed by
CF(S)={S1,...,Sk}.
E iden ly, a pa i ion sys em is a easible coali ions sys em, so, he Felemen s will
no change hei name.
Example 1 Le N ={1,2,...,n}, a na u al numbe n, and conside ing he collec ion
Lnmade o all he se s such as [i,j]={i,i+1,..., j−1,j} o 1≤i≤j≤n. This
model ep esen s a one-dimensional poli ical elec ion si ua ion and he couple (N,Ln)
is a pa i ion sys em.
Example 2 A communica ion si ua ion is he iple (N,G, ), in which (N, )is a game
and G =(N,E(N)) is a g aph. This idea was i s de eloped by Mye son (1977), and
esea ched by Owen (1986) and Bo m, Nouweland and Tijs (1992, 1993). I is easy o
see ha he couple (N,F), in which
F={S⊆N|(S,E(S)) is a connec ed subg aph o G},
is a pa i ion sys em. We mus poin ou ha he opposi e is no always ue, because
e e y G g aph is a collec ion o pai s {i,j}, and as a esul , he e mus be easible
collec ions made up o wo elemen s, bu his migh no happen.
The p e ious de ini ions come om an ex ension o communica ion si ua ion and
communica ion g aph- es ic ed game, de eloped by Mye son (1977) and Owen (1986).
The ollowing heo em shows a cha ac e iza ion o he concep o pa i ion sys ems.
Theo em 1 A easible coali ions sys em (N,F),F ⊆ 2Nis a pa i ion sys em i and
only i
∀A∈ F ,B∈ F ,con A ∩B,∅=⇒A∪B∈ F .
P oo . (⇐) Conside ing ha he F-componen s o A⊆N o m a eco e , i is
only necessa y o p o e ha e e y pai o F–componen s o Aa e disjoin ed. Le Ti,
Tj(i,j) maximal easible coali ions o A. I Ti∩Tj,∅,i would mean, hypo hesizing,
Ti∪Tj∈ F being Ti∪Tj⊂A. This con adic s ha Tiand Tja e maximal easible
coali ions o A.
(⇒) Le A∈ F ,B∈ F wi h A∩B,∅. I A∪B<F, hen
A∪B=[
k
Tk,
whe e {Tk}is he pa i ion o A∪B o maximal se s. As Aand Ba e easible coali ions
con ained in A∪B, hus A⊆Tj,B⊆Tp o e e y jand p. I j,p, hen Tj∩Tp=∅
142
Pa ial coope a ion and con ex se s
and, so, A∩B=∅agains he hypo hesis; hen A∪B∈ F . I j=p hen A⊆Tj⊆A∪B
and B⊆Tj⊆A∪B, implies A∪B=Tj∈ F .¤
2 Pa ially o de ed se es ic ed games
The aim o his sec ion is o s udy a easible coali ion sys em de ined by a pa ial o de
o all playe s. F om his momen only pose s P=(N,≤) will be conside ed and he
easible coali ion sys em cha ac e is ics de eloped om he amily o con ex se s will
be analyzed.
Le P=(N,≤) a pose . I is said ha A⊆Nis con ex in Pi i is p o ed ha
a∈A,b∈Aand a≤b=⇒[a,b]⊆A.
I P=(N,≤) is a pose , we a e in e es ed in ob aining P∗=(N,≤), he dual o P,
wi h
x≤yen P∗⇐⇒ y≤xen P.
I can be p o ed ha Co(P)≃Co(P∗), ∀P(Bi ko and Benne , 1985).The amily o
con ex se s in Pwill be deno ed
Co(P)={S⊆N|Sis con ex in P}.
This cha ac e iza ion implies, ∀i∈N,{i} ∈ Co(P) so he couple (N,Co(P)) is a
easible coali ions sys em. Then, gi en a game (N, ), i he e is an o de ela ion among
he playe s, i makes sense o ake in o conside a ion he iple (N,Co(P), ) and he
app op ia e pa ial coope a ion game,
Co(P)|2N−→ R, Co(P)(S)=max 





X
i
(Ti)| {Ti} ∈ PCo(P)(S)






,
whe e PCo(P)(S) is he amily o pa i ions om he coali ion Sin con ex se s in P.
I is easy o p o e ha A,B∈Co(P), ha A∩B∈Co(P), impliying (N,Co(P)) a
closu e space. Also, Edelman and Jamison (1985), Bi ko and Benne (1985) hink ha
(N,Co(P)) p o es he Minkowski–K ein–Milman condi ion, and, he e o e an a omic
con ex geome y, named o de con ex in N.
As (N,Co(P)) is a easible coali ion sys em, e e y subse in Ncan be exp essed
as a union o i maximal con ex se s. In his pa icula case, he maximal con ex
de ini ion o S⊆Nin Pis equi alen o he one by Tijs (1993), which is due o he
wo (N,Co(P)) being a con ex geome y: Le (N,Co(P)) be a easible coali ion sys em
and le S ⊆N. I T ∈Co(P)and T ⊆N, hen T is maximal con ex S in P i and only i ,
∀i∈S T,T∪ {i}<Co(P).
No ice ha his cha ac e iza ion o maximal con ex is ce ain in all con ex
geome y, and, in gene al, he easible coali ion sys em (N,Co(P) is no a pa i ion
sys em.
J. En ique Rome o Ga c´
ıa, Jo ge J. L´
opez V´
azquez
143
Example 2 Le (N,≤)be a pose , whose Hasse diag am is shown in Figu e 1,
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Figu e 2:(Co(P),⊆)≃(24,⊆)

144
Pa ial coope a ion and con ex se s
The couple (N,Co(P)) is no a pa i ion sys em, applying Theo em 1, because {1,3} ∈
Co(P), {3,4} ∈ Co(P), he in esec ion is no emp y, howe e , {1,3} ∪ {3,4}<Co(P) due
o 1 ≤4 y [1,4] *{1,3,4}.
Le P=(N,≤) be a pose whose ange o leng h l(P) migh equal 1 o be less han 1.
Tha is o say:
l(P)=max{l(C)|Cis a chain in Pand l(C)=|C| − 1} ≤ 1.
Then (N,Co(P)) is a pa i ion con ex geome y. As e e y subse in Nis con ex, ei he
due o being an a om o a chain o wo elemen s om N, i implies ha Co(P)≃2N. Fo
example, in Figu e 2, Co(P)≃24. I l(P)≤1 and i i is conside ed a pa i ion sys em
(o pa i ion con ex geome y) es ic ed Co(P)–game linked o he h ee (N,Co(P), ),
i e i ies ha Co(P)(S)= (S),∀S∈2Nand, he e o e es ic ed game and o iginal
game a e he same.
I has been p o ed ha i l(P)≥2, he a omic con ex geome y (N,Co(P)) is no
necessa ily a pa i ion sys em . This is he eason why only pa ially o de ed se s wi h
l(P)≥2 a e aken in o conside a ion, and we sea ch o condi ions o se (N,Co(P)) as
a pa i ion sys em. We will in oduce he concep o comple ely cohe en o de ed se s as
gi en by Bi ko and Benne (1985).
A pose P=(N,≤) is cohe en i i is connec ed and no maximal elemen om P
co e s any minimal elemen om P.
Fo example, he pose in example 3 (Figu e 1) is cohe en . O he possible si ua ions
a e conside ed below:
• • • •
• • • • • •
• ••••
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 
(non connec ed)
 
(maximals a e abou minimals)
 
Figu e 3
J. En ique Rome o Ga c´
ıa, Jo ge J. L´
opez V´
azquez
145
A pose P, wi h l(P)≥2, is comple ely cohe en i any subpose in e ed by P, P0
wi h l(P0)≥2, is cohe en . The ollowing igu es illus a e his concep . Figu e 4 shows
diag ams o cohe en pose s ha a e no comple ely cohe en . On he o he hand, Figu e
5, shows examples o comple ely cohe en pose s.
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676
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1 2 31 2
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Figu e 4
146
Pa ial coope a ion and con ex se s
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Figu e 5
No ice ha comple ely cohe en pose s in Figu e 5, excep he i s o hem, e i y
ha P ex(P) is a chain. This p ope y will be e y impo an o p o e ha he cou-
ple (N,Co(P)) is a pa i ion sys em.
Theo em 2 Le P =(N,≤)be a comple ely cohe en ini e pose , as P ex(P)is a chain C.
Then, e e y maximal elemen om P co e s he maximum in chain C and he minimal
elemen om C co e s e e y minimal elemen om P.
P oo . I Pis cohe en , i is connec ed and i s maximal elemen s do no co e any
minimal. The e o e, i xis maximal, i ollows ha y∈Pis such ha xÂyin which
y<ex(P) because se ex(P) is he union o maximal and minimal elemen s. Then, y∈C
/y≤maxCexis s.
• •
•
.
.
.
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y
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x0
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I y,maxC, as maxCis no maximal in P, he e is
x0ÂmaxC. The induced subpose P0, made up o he
elemen s {y,maxC,x,x0} e i ies ha l(P0)=2 and is no
cohe en , in opposi ion o he hypo hesis. Consequen ly,
y=maxC.
The easoning o minimal elemen s is equi alen o he one abo e. ¤
J. En ique Rome o Ga c´
ıa, Jo ge J. L´
opez V´
azquez
147
The ollowing heo em is he main esul om his esea ch. I es ablishes al e na i e
cha ac e iza ion o he wo (N,Co(P)) o be a pa i ion sys em.
Theo em 3 Le P =(N,≤)be a ini e pose . The couple (N,Co(P)) is a pa i ion sys em
i and only i P is comple ely cohe en and P ex(P)=C is a chain.
P oo . (⇒) Conside ha (N,Co(P)) is a pa i ion sys em. We mus p o e ha Pis
comple ely cohe en and P ex(P)=C.
I P ex(P),C, he e a e a,b∈P ex(P) so ha {a,b}is an an ichain. As
{a,b}*ex(P), conside he se s
m(a)={m∈P|m≺a},M(a)={m0∈P|a≺m0},
and, analagously, m(b) and M(b). Ob iously, hese a e no emp y se s, and i is easy o
no ice ha m(a)∩M(b)=m(b)∩M(a)=∅. Howe e , m(a)∩m(b) and M(a)∩M(b) ,
hese in e sec ions canno be emp y. So, hese a e he al e na i es:
(1) m(a)∩m(b),∅
(2) M(a)∩M(b),∅
(3) m(a)∩m(b)=M(a)∩M(b)=∅
Using he duali y Co(P)≃Co(P∗), we only need o pay a en ion o (1) and (3).
(1) Le m∈m(a)∩m(b), m0∈M(a). I b£m0(Figu e 6), he se {m,b,m0}<Co(P)
and hei maximal con exes {{b,m0},{m,b}} a e no i s pa i ion. I b≤m0(Figu e 7),
{m,a,m0}<Co(P) and hei maximal con exes {{a,m0},{m,a}} a e also no i s pa i ion.
•
•
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Figu e 6 Figu e 7
(3) Suppose ha m(a)∩m(b)=M(a)∩M(b)=∅and le m∈m(a) and m0∈M(a). I
he e is no connec ion be ween band elemen s m,m0, hen {m,b,m0}<Co(P) and hei
maximal con exes{{b,m0},{m,b}} a e no i s pa i ion (Figu e 8). I he e was connec ion
i would be because, m≤b,b≤m0, one o bo h o hem. In e e y si ua ion, m<m(b)
and m0<M(b) such ha m(a)∩m(b)=M(a)∩M(b)=∅. In all si ua ions, we canno
ind con ex se s in which hei maximal con exes a e no a pa i ion. Indeed, i m≤b
he e is a b1such ha m≤b1≤b(Figu e 9) and, o {m,a,b}<Co(P) hei maximal
con exes {{m,a},{a,b}} a e no i s pa i ion. I b≤m0 he easoning is equi alen .