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Pe ea Rojas Ma cos, F.; Pue o Albandoz, J.; Fe nández Ga cía, FR. (2012). A oiding
un ai ness o Owen alloca ions in linea p oduc ion p ocesses. Eu opean Jou nal o
Ope a ional Resea ch. 220(1):125-131. doi:10.1016/j.ejo .2012.01.013.
A oiding un ai ness o Owen alloca ions in linea
p oduc ion p ocesses
Fede ico Pe ea1(Co esponding Au ho ),
Jus o Pue o2, F ancisco R. Fe n´andez2
email: p[email p o ec ed] .es,{pue o, e nande}@us.es,
elephone and ax: +34 963877490 / +34 963877499
1Uni e si a Poli `ecnica de Val`encia (Spain)
2Uni e sidad de Se illa (Spain)
Abs ac
This pape deals wi h coope a ion si ua ions in linea p oduc ion p oblems
in which a se o goods a e o be p oduced om a se o esou ces so ha
a ce ain bene i unc ion is maximized, assuming ha esou ces no used
in he p oduc ion plan ha e no alue by hemsel es. The Owen se is a
well-know solu ion ule o he class o linea p oduc ion p ocesses. Despi e
hei s abili y p ope ies, Owen alloca ions migh gi e null payo o playe s
ha a e necessa y o op imal p oduc ion plans. This pape shows ha ,
in gene al, he a o emen ioned d awback canno be a oided allowing only
alloca ions wi hin he co e o he coope a i e game associa ed o he o iginal
linea p oduc ion p ocess. In his pape a new solu ion se named EOwen is
in oduced. Fo any playe whose esou ces a e needed in a leas one op imal
p oduc ion plan, he EOwen se con ains a leas one alloca ion ha assigns
a s ic ly posi i e payo o such playe .
Keywo ds: Coope a i e games, Linea p oduc ion games, Alloca ions.
P ep in submi ed o EJOR Sep embe 23, 2011
1. In oduc ion
A bene i coope a i e game is a pai (N, ), whe e N={1,2, ..., n}is
he se o playe s and : 2N→Ris he cha ac e is ic unc ion assigning o
e e y coali ion S⊂N he maximum bene i ha he coope a ion be ween
playe s in Swould yield. Fo a comple e in oduc ion on coope a i e game
heo y see o ins ance Owen (1995) o Fo g´o e al. (1999). Assuming ha
he game is supe addi i e, ha is (S) + (T)≤ (S∪T),∀S, T ⊂N,
coope a ion among all playe s is bene icial and, he e o e, he g and coali ion
Nis o o m.
One o he main ques ions in coope a i e game heo y is how o dis-
ibu e he bene i ob ained by he g and coali ion Namong he playe s.
An alloca ion is a ec o α∈Rn, such ha αiis he payo o playe iand
Pn
i=1 αi= (N). One well-accep ed way o alloca ing (N) among he play-
e s is o ind alloca ions in he co e. The co e o a game (N, ), deno ed by
Co e(N, ), is he se o alloca ions sa is ying ha no coali ion o playe s can
ob ain a be e payo by ac ing sepa a ely om he es o playe s. Tha is,
Co e(N, ) = {α∈Rn: (S)≤α(S)∀S⊂N, (N) = α(N)},(1)
whe e α(S) = Pi∈Sαi,∀S⊂N. In p inciple, he co e has a leas wo
p oblems: he co e o a game migh be emp y, ha is, he e a e games o
which no co e alloca ions exis , and inding a co e alloca ion migh be a NP-
ha d p oblem. Along he yea s, many o he alloca ion ules ha e appea ed
in he li e a u e. One o he mos used alloca ion ules is he Shapley alue,
2
which has a ac ed a lo o in e es o i s many applica ions, see Mo e i
and Pa one (2008).
A linea p oduc ion p oblem is a si ua ion in which ce ain goods ha
can be sold in a ma ke a e o be p oduced om a se o a ailable dis inc
esou ces. An implici ea u e o he linea p oduc ion p oblems we deal
wi h in his pape is ha he esou ces no used in he p oduc ion plan
ha e no alue a all. Si ua ions like his may a ise when he esou ces a e
pe ishable and, i no used in he nex p oduc ion plan, hey a e was ed.
Ano he example o his si ua ion is ound in some indus ies in de eloped
coun ies ha gi e hei excesses o unde de eloped coun ies, o cha i y
o ganiza ions, o e en o o he companies wi hin he same a ea as long as
hey a e no compe ing ones. This is bene icial o bo h pa ies: he dono
pa y ge s id o excesses which, i no used, mus be elimina ed a ce ain
cos , and he ecei ing pa y only has o pay o he shipping cos s, which is
usually cheape han ha ing o buy he ma e ial.
In his pape we s udy a new se o alloca ions o linea p oduc ion
p ocesses (LP p ocesses o sho ), which a ise when a bunch o playe s N=
{1,...,n}wi h con lic ing objec i es con ol he esou ces o a LP p oblem.
A coope a i e game, called LP game, can be associa ed o each LP p ocess.
(No e ha di e en LP p ocesses may gene a e he same LP game.) An ea ly
e e ence o LP games can be ound in Owen (1975). LP games a e o ally
balanced games, so e e y subgame o a LP game has a non-emp y co e. By
sol ing he dual p oblem o he unde lying linea p oduc ion p oblem we
can ob ain a se o alloca ions o LP p ocesses known as Owen alloca ions
(see Owen (1975)), which has been well-s udied in he li e a u e. One o i s
3
main p ope ies is ha Owen alloca ions a e always co e alloca ions, and
a e easily compu ed. Mo e ecen ly, Van Gellekom e al. (2000) p o ided an
axioma ic cha ac e iza ion o his solu ion se . In his pape we show ha ,
despi e hei s abili y p ope ies, Owen alloca ions do no always yield a ai
dis ibu ion o he bene i ob ained. Fo ins ance, a playe whose esou ces
a e necessa y o any op imal plan may ecei e a null payo om Owen
alloca ions. Such d awback is discussed in his pape , and an al e na i e
alloca ion se is p oposed.
Since he pionee ing wo k by Owen, se e al gene aliza ions o LP games
ha e appea ed in he li e a u e. Dubey and Shapley (1984) s udy a game in
which playe s ha e pa ial con ol o e he cons ain s o a gene al ma hema -
ical p og amming p oblem. G ano (1986) in oduces ano he gene aliza ion
in which he esou ces owned by a coali ion a e no es ic ed o be he sum
o he esou ces o playe s in he coali ion. Cu iel e al. (1989) in oduce
LP games wi h commi ee con ol, ob aining esul s on he balancedness
o hese games, whose co e has been mo e ecen ly s udied by Molina and
Tejada (2004).
The goal o his pape is o in oduce a new se o alloca ions o linea
p oduc ion p ocesses ha a oid some o he a o emen ioned d awbacks o he
Owen se . To his end, he es o he pape is s uc u ed as ollows. Sec ion
2 gi es a sho in oduc ion o LP p ocesses and a mo i a ion o he s udied
p oblem. Some de ini ions and echnical esul s a e gi en in Sec ion 3. The
alloca ion se p oposed in his pape is in oduced and analyzed in Sec ion 4.
An axioma ic cha ac e iza ion and some o i s p ope ies a e gi en, as well
as a discussion abou he impossibili y o inding co e alloca ions ha a oid
4

he un ai ness p oblem o he Owen alloca ions we add ess in his pape .
2. Linea P oduc ion p ocesses
A LP p oblem is a si ua ion in which he e is a ini e se o esou ces
R={1,2,..., }and om hose esou ces a se P={1,2,...,p}o con-
sump ion goods can be p oduced. The p oduc ion echnologies a e gi en by
a ma ix A∈R ×p, whe e Aij ≥0 deno es he amoun o esou ce inec-
essa y o p oduce one uni o p oduc j,∀i= 1, . . . , , j = 1,...,p. I is
also assumed ha he demand o e e y p oduc is la ge enough o sell all
p oduced p oduc s, he uni a y ma ke p ice o p oduc jbeing cj≥0. The
objec i e o a LP p oblem is o decide how much o each p oduc should be
p oduced so ha he gene al bene i is maximized.
Assume now ha a g oup o playe s N={1,...,n}con ol he esou ces
R={1,2,..., }, ha is, playe kowns Bik ≥0 uni s o esou ce i,k=
1, . . . , n, i = 1,..., . The e o e, le B= (Bik) ×nbe he esou ce-playe
ma ix. Le b∈R be he esou ce ec o , ha is b=BeN, whe e eS∈Rn
sa is ying (eS)k= 1 i k∈S, and ze o o he wise o all S⊆N. In o he
wo ds, biis he o al amoun o esou ce iowned by he g and coali ion, ha
is, bi=Pn
k=1 Bik ∀i∈R. Thus, he maximum p o i ha can be made by
he coope a ion o all playe s is he alue o p oblem PN:
max cx
s. . Ax ≤b
x≥0
(PN),
min yb
s. . yA ≥c
y≥0
(DN),(2)
whe e DNis he dual p oblem o PN(see Baza aa e al. (1990) o a de-
sc ip ion o duali y heo y in linea p og amming). I is easy o check ha ,
5
al hough playe s can y o p oduce sepa a ely, i is always mo e p o i able
o join hei esou ces since he bene i hey ob ain his way is a leas as
high as he sum o he possible coali ions’ p o i s sepa a ely. Fo a coali ion
S⊂N, we de ine i s cha ac e is ic unc ion, (S), ia he op imal alue o
p oblem PS:
max cx
s. . Ax ≤BeS
x≥0
(PS),
min yBeS
s. . yA ≥c
y≥0
(DS),(3)
whe e DSis he dual o PS.
P oblem PSis easible and bounded o all possible coali ions i BeS>0,
c≥0 and ∀j:cj>0 he e is a leas one esou ce i∈Rwi h Aij >0.
Each iple (A, B, c) sa is ying he condi ions abo e will be called in he
ollowing, acco ding o Van Gellekom e al. (2000), a linea p oduc ion p o-
cess. Le Ldeno e he class o LP p ocesses. F om he de ini ion o he
cha ac e is ic unc ion one can associa e o each LP p ocess a coope a i e
game (N, ). The eade may no e ha he same LP game can o igina e
om di e en LP p ocesses.
Now a na u al ques ion a ises: how o di ide he p o i made by he
g and coali ion among he playe s. Le us in oduce some no a ion ha will
be use ul in he es o he pape .
Le (A, B, c)∈ L. The easible egions o p oblems PNand DN, see (2),
a e deno ed by
Fmax(A, B, c) := {x∈Rp
+:Ax ≤b},
Fmin(A, B, c) := {y∈Rn
+:yA ≥c},
(4)
6
espec i ely. The op imal alues o p oblems PNand DNa e deno ed by
max(A, B, c) := max{cx :x∈Fmax(A, B, c)},
min(A, B, c) := min{yb :y∈Fmin(A, B, c)},
(5)
espec i ely, and he se o op imal solu ions o PNand DNby
Omax(A, B, c) := {x∈Fmax(A, B, c) : cx = max(A, B, c)},
Omin(A, B, c) := {y∈Fmin(A, B, c) : yb = min(A, B, c)}.
(6)
Asolu ion ule ϕon Lis a map assigning o e e y LP p ocess (A, B, c)∈
La se Γ ⊂Rnsuch ha Pi∈Nγi= max(A, B, c) o all γ∈Γ. Each membe
o his se is an alloca ion. A well-known solu ion ule o coope a i e games
is he co e, see (1). One well-accep ed solu ion ule speci ic o LP p ocesses
is he Owen se , de ined om op imal solu ions o he dual p oblem DN.
De ini ion 1. Le (A, B, c)∈ L. The Owen se o (A, B, c)is
Owen(A, B, c) := {yB :y∈Omin(A, B, c)}.(7)
Owen (1975) p o ed ha Owen(A, B, c)⊆Co e(A, B, c) o e e y (A, B, c)∈
L. Tha is, Owen alloca ions a e s able in he sense ha no g oup o playe s
can ob ain a be e payo by ac ing sepa a ely. Despi e hese good p op-
e ies, hey should no be conside ed as ideal alloca ions. See he ollowing
example.
Example 1. Conside he 3-playe game (A, B, c)∈ L whe e
A=








1 0
1 1
0 1
1 2








, B =








101
040
100
050








, c =
1
2

.
7
The co esponding dual p oblem D(N)is
min 2y1+ 4y2+y3+5y4
s. . y1+y2+y4≥1
y2+y3+2y4≥2
y1, y2, y3, y4≥0.
(8)
The cha ac e is ic unc ion o he associa ed game is ({i}) = ({1,3}) =
0∀i= 1,2,3, ({1,2}) = 3, ({2,3}) = 1, ({1,2,3}) = 4. I can be
checked ha Omin(A, B, c) = {(1,0,2,0)}and, he e o e, Owen(A, B, c) =
{(1,0,2,0)B}={(3,0,1)}.
This alloca ion is in he co e o he game bu , is i a “ ai ” alloca ion?
No e ha playe 2 ecei es no hing bu , wi hou his esou ces, he op imal
p oduc ion plan canno be achie ed. So, he Owen alloca ion gi es a null
payo o a playe whose esou ces a e necessa y o he op imal p oduc ion
plan.
Wha happened in Example 1 is a gene al d awback o he Owen se in LP
p ocesses. This is a consequence o he complemen a y slackness heo em,
see Baza aa e al. (1990), which says ha i he e is some su plus o esou ce
iin an op imal solu ion x∗∈Omax(A, B, c) (meaning (Ax∗)i< bi), hen
y∗
i= 0 ∀y∗∈Omin(A, B, c). This condi ion o op imali y con eys a simple
economic p inciple: i he e is a posi i e slack in a cons ained p imal e-
sou ce, i.e. he e a e le o e s, hen he addi ional quan i ies o ha esou ce
mus ha e no alue (shadow p ices a e ze o). This means ha only playe s
owning esou ces ha gene a e no su plus ha e he chance o ecei ing a
s ic ly posi i e payo om Owen alloca ions, which a e based on shadow
p ices. This ac could make playe s ge id o hei su pluses so ha he
8
P oposi ion 1. Le b
B∈[Bx∗, B] o some educed ma ix Bx∗associa ed o
an op imal p oduc ion plan x∗. Then we ha e ha :
Owen(A, B, c)⊆Owen(A, b
B, c)⊆Owen(A, Bx∗, c).
P oo . Le α∈Owen(A, B, c)⇒ ∃ y∈Omin(A, B, c) : α=yB. Le us
see ha y∈Omin(A, b
B, c).
min yb
s. . yA ≥c
y≥0
(D),
min yb
b
s. . yA ≥c
y≥0
(b
D).
Since y∈Omin(A, B, c), by de ini ion one has ha yis op imal o p oblem
D. The e o e, yis easible o p oblem b
D. Besides, as we p o ed in Lemma 1,
pa 4, yb =yb
b. Thus y∈Omin(A, b
B, c). Applying again he complemen a y
slackness heo em, one has ha :
αk=
n
X
i=1
yiBik =X
i:yi6=0
yiBik =X
i:yi6=0
yib
Bik ⇒α=yb
B.
The e o e we ha e p o en ha α∈Owen(A, b
B, c), and as a consequence
Owen(A, B, c)⊆Owen(A, b
B, c). Owen(A, b
B, c)⊆Owen(A, Bx∗, c) can be
p o en analogously. 
An immedia e co olla y o he p e ious esul s a es ha he name Ex ended
Owen se is meaning ul, as EOwen con ains he Owen se .
Co olla y 1. Owen(A, B, c)⊆EOwen(A, B, c) o all (A, B, c)∈ L.
The ollowing example p o es ha he inclusions in P oposi ion 1 and Co ol-
la y 1 may be s ic .
15

Example 2. Take he LP p ocess om Example 1. One can see ha Omax(A, B, c) =
{x∗= (2,1)}. The e o e, Bx∗(in his case unique) and a choice o b
Ba e:
Bx∗=








1 0 1
0 3 0
1 0 0
0 4 0








,b
B=








1 0 1
0 3 0
1 0 0
0 5 0








.
Then, we ob ain ha p oblem DN(Bx∗)and ha co esponding o b
B,DN(b
B),
a e, espec i ely:
min 2y1+ 3y2+y3+4y4
s. . y1+y2+y4≥1
y2+y3+2y4≥2
y1, y2, y3, y4≥0
DN(Bx∗),
min 2y1+ 3y2+y3+5y4
s. . y1+y2+y4≥1
y2+y3+2y4≥2
y1, y2, y3, y4≥0
DN(b
B).
(16)
F om he solu ions o hese p oblems, which a e he con ex hulls o {(1,0,2,0),
(0,1,1,0),(0,0,0,1)}and {(1,0,2,0),(0,1,1,0)}, espec i ely, one can see
ha Owen(A, Bx∗, c)(which in his example coincides wi h EOwen(A, B, c))
is he con ex hull o {(3,0,1),(1,3,0),(0,4,0)}and Owen(A, b
B, c)is he
con ex hull o {(3,0,1),(1,3,0)}. This way we p o e ha he ela ions de-
sc ibed in P oposi ion 1 a e s ic . Besides, no e ha he e a e alloca ions in
EOwen(A, B, c) ha gi e playe 2 a s ic ly posi i e payo , unlike Owen(A, B, c).
No e as well ha EOwen(A, B, c)⊂Co e(A, B, c).
16
F om hei de ini ions, i is easy o p o e ha o educed LP p ocesses,
as in oduced in De ini ion 2, he EOwen se and he Owen se coincide.
Analogously, one can s a e ha in LP p ocesses whe e all he esou ces a e
comple ely used, he EOwen se coincides wi h he Owen se .
The ollowing p oposi ion p o es ha alloca ions in he EOwen se dis-
ibu e exac ly (N) among he playe s.
P oposi ion 2. Le (A, B, c)∈ L and le γ∈EOwen(A, B, c). Then γis
e icien .
P oo . Le γ∈EOwen(A, B, c). Then he e exis s x∗∈Omax(A, B, c),
Bx∗∈ B(A, B, x∗) and by∈Omin(A, Bx∗, c) such ha γk=P
i=1 byiBx∗
ik ∀k=
1,...,n. The e o e
γ(N) =
n
X
k=1
γk=
n
X
k=1
X
i=1 byiBx∗
ik =
X
i=1 byi
n
X
k=1
Bx∗
ik =
X
i=1 byibx∗
i=bybx∗.(17)
Since by∈Omin(A, Bx∗, c) and x∗∈Omax(A, B, c), we know ha bybx∗=cx∗=
(N). This concludes ha γ(N) = (N). 
Ano he in e es ing p ope y s a es ha , o all playe s whose esou ces a e
necessa y o p oduce he maximum bene i (N) in some op imal p oduc ion
plan, he e exis s an alloca ion in EOwen ha assigns hem a s ic ly posi i e
payo .
Theo em 1. Le (A, B, c)∈ L and x∗∈Omax(A, B, c), and le k∈Nbe a
playe such ha some o he esou ces ha he owns a e needed o he op imal
p oduc ion plan x∗ o be de eloped. Then he e exis s α∈EOwen(A, B, c)
such ha αk>0.
17
P oo . Le x∗∈Omax(A, B, c), and Bx∗∈ B(A, B, x∗). By he s ic
complemen a y slackness heo em, i he slack in he i h cons ain o p oblem
PN(Bx∗) is ze o, hen he e exis s a solu ion y o DN(Bx∗) such ha yi>0
(see Theo em 10.7 in Vande bei (1997)). Assuming ha he uni s o he i h
esou ce owned by playe ka e needed o he op imal p oduc ion plan x∗,
i is easy o see ha Bx∗
ik >0. Then, he payo o playe k om he EOwen
alloca ion αBx∗=yBx∗is, a leas , yiBx∗
ik >0.

No e ha his p oposi ion allows us o s a e ha he EOwen se always
o e comes he un ai ness p oblem illus a ed in Example 1.
Le us now in oduce he p ope y o uppe limi inclusion (ULI), which
will be use ul o ou cha ac e iza ion o EOwen.
P ope y 1 (ULI). A solu ion ule ϕsa is ies ULI i o e e y (A, B, c)∈
L, e e y x∗∈Omax(A, B, c), and e e y ma ix B′such ha B′∈[Bx∗, B] o
e e y Bx∗∈ B(A, B, x∗), we ha e ha ϕ(A, B′, c)⊆ϕ(A, B, c).
The ollowing esul p o es ha EOwen sa is ies his p ope y.
P oposi ion 3. EOwen sa is ies ULI.
P oo . Le (A, B, c)∈ L. Conside x∗∈Omax(A, B, c), and le B′∈
[Bx∗, B] o e e y Bx∗ educed ma ix associa ed o x∗. Simila ly as we
p o ed in P oposi ion 1, i can be seen ha Omax(A, B′, c)⊆Omax(A, B, c).
Besides, since we p o ed in Lemma 2 ha B(A, B, x∗)⊇ B(A, B′, x∗), we
18
ha e
EOwen(A, B, c) = Sx∗∈Omax(A,B,c)SBx∗∈B(A,B,x∗)Owen(A, Bx∗, c)
⊇Sx∗∈Omax(A,B′,c)SBx∗∈B(A,B,x∗)Owen(A, Bx∗, c)
⊇Sx∗∈Omax(A,B′,c)SBx∗∈B(A,B′,x∗)Owen(A, Bx∗, c)
=EOwen(A, B′, c).

Now we a e eady o gi e a cha ac e iza ion o he EOwen solu ion ule
o linea p oduc ion p ocesses.
Theo em 2. Le ϕbe a solu ion ule o e L.ϕsa is ies ULI, coincides wi h
he Owen se o e LP p ocesses wi hou le o e s and is minimal i and only
i ϕ≡EOwen.
P oo .
•Clea ly EOwen coincides wi h he Owen se in LP p ocesses wi hou
le o e s, and as p o en in P oposi ion 3, EOwen sa is ies ULI. Le
us see ha EOwen is minimal. Fo his pu pose, le ϕbe a solu ion
se ha coincides wi h he Owen se in LP p ocesses wi hou le o e s
and sa is ies ULI, and le (A, B, c)∈ L. The e o e, Owen(A, Bx∗, c) =
ϕ(A, Bx∗, c)⊆ϕ(A, B, c) o all x∗∈Omax(A, B, c) and all Bx∗∈
B(A, B, x∗). Hence
EOwen(A, B, c) = Sx∗∈Omax(A,B,c)SBx∗∈B(A,B,x∗)Owen(A, Bx∗, c)
=Sx∗∈Omax(A,B,c)SBx∗∈B(A,B,x∗)ϕ(A, Bx∗, c)
⊆ϕ(A, B, c)
which p o es ha EOwen is minimal.
19
•Le ϕbe a solu ion ule o e Lsa is ying he hypo heses o he heo em.
The e o e ϕ(A, Bx∗, c) = Owen(A, Bx∗, c)∀x∗∈Omax(A, B, c) and o
e e y Bx∗∈ B(A, B, x∗), because (A, Bx∗, c) has no le o e s. Now,
since ϕsa is ies ULI we ha e ha ϕ(A, Bx∗, c)⊆ϕ(A, B, c), he e o e
EOwen(A, B, c) = [
x∗∈Omax(A,B,c)[
Bx∗∈B(A,B,x∗)
ϕ(A, Bx∗, c)⊆ϕ(A, B, c).
F om he minimali y o ϕ, and since EOwen sa is ies he hypo heses o
he heo em, ϕ(A, B, c)⊆EOwen(A, B, c). Thus, EOwen(A, B, c) =
ϕ(A, B, c).

Le us now in oduce he ollowing inc easing mono onici y (IM) p ope y
ha will lead us o ano he cha ac e iza ion o EOwen se :
P ope y 2 (IM). A solu ion ule ϕsa is ies IM i o e e y (A, B, c)∈ L,
e e y x∗∈Omax(A, B, c), and e e y B1, B2such ha B1∈[Bx∗, B] o each
Bx∗∈ B(A, B, x∗)and B2∈[B1, B], we ha e ha ϕ(A, B1, c)⊆ϕ(A, B2, c).
The ollowing esul s a es ha inc easing mono onici y is equi alen o up-
pe limi inclusion.
Lemma 4. Le ϕbe a solu ion ule o e L. Then ϕsa is ies ULI i and only
i ϕsa is ies IM.
P oo . T i ially, i ϕsa is ies IM hen i sa is ies ULI oo. Con e sely,
le (A, B, c)∈ L, and x∗∈Omax(A, B, c). Mo eo e , le B1∈[Bx∗, B]
o each Bx∗∈ B(A, B, x∗) and B2∈[B1, B]. I is s aigh o wa d ha
20

x∗∈Omax(A, B2, c). Then, since B1∈[Bx∗, B2] and ϕsa is ies IM, we ha e
ha ϕ(A, B1, c)⊆ϕ(A, B2, c), which concludes he p oo . 
F om he equi alency be ween inc easing mono onici y and uppe limi in-
clusion, he ollowing al e na i e cha ac e iza ion o EOwen i ially ollows
om Theo em 2.
Co olla y 2. Le ϕbe a solu ion ule o e L.ϕsa is ies IM, coincides wi h
he Owen se o e LP p ocesses wi hou le o e s and is minimal i and only
i ϕ≡EOwen.
We inish his sec ion by s udying he ela ion be ween EOwen, he Owen
se , and he co e o he o iginal game. I is ob ious ha Co e(A, B, c)⊆
Co e(A, Bx∗, c) o e e y op imal p oduc ion plan x∗and e e y educed ma-
ix Bx∗associa ed o i , since Bx∗
(S)≤ (S) and Bx∗
(N) = (N), whe e
Bx∗
deno es he cha ac e is ic unc ion o he co esponding educed game.
The ollowing example shows ha his ela ion may be s ic .
Example 3. Conside he LP p ocess (A, B, c)whe e
A=





1 0
1 1
0 1




, B =





1 0
1 3
1 0




, c =
2
1

.
I can be seen ha Omax(A, B, c) = {x∗= (1,1)}, and he cha ac e is ic
unc ion o he associa ed game is ({1}) = 2, ({2}) = 0, ({1,2}) = 3.
The e is a su plus o esou ce 2. One educed ma ix (in which bo h playe s
21
d op hal o he uni s o esou ce 2 hey had) is
Bx∗=





1 0
0.5 1.5
1 0




,
and x∗({1}) = 1, x∗({2}) = 0, ({1,2}) = 3. The e o e, Co e(A, B, c)
Co e(A, Bx∗, c).
Un o una ely, no all EOwen alloca ions a e co e alloca ions. Howe e ,
i is no always possible o ind alloca ions ha a oid he un ai ness d awback
o he Owen se men ioned in his pape and emain in he co e o he o iginal
game a he same ime. The e o e, one has o look o some comp omise
be ween null-payo o absolu ely necessa y playe s and un-s abili y. The
ollowing example illus a es he abo e s a emen s.
Example 4. Conside he LP p ocess (A, B, c)wi h he ollowing da a,
A=





1 0
1 1
0 1




, B =





100
010
001




, c =
1
1

.
The cha ac e is ic unc ion o he associa ed game is ({1,2}) = ({2,3}) =
({1,2,3}) = 1, and ze o o any o he coali ion. The e o e, he co e o
his game consis s o he single on {(0,1,0)}. No e as well ha wi hou he
esou ces o playe s 1 and 3, playe 2 ge s no hing, bu i is no possible o
gi e a posi i e payo o playe s 1 and 3 wi h a co e alloca ions.
Le us calcula e he EOwen se o his example. One can see ha he
ex eme op imal p oduc ion plans a e x1= (1,0) and x2= (0,1), and ha
22
B(A, B, x1)and B(A, B, x2)consis only o one ma ix each (named B1and
B2, espec i ely). B1has a diagonal equal o (1,1,0) and B2has a diagonal
equal o (0,1,1). All non-diagonal en ies a e null o bo h ma ices. I is
easy o see ha Omin(A, B1, c)is he union o e 1≥1and 2≥0o he
con ex hulls o {(1,0, 1),(0,1, 2)}, and ha Omin(A, B2, c)is he union o e
1≥1and 2≥0o he con ex hulls o {( 1,0,1),( 2,1,0)}. The e o e,
Owen(A, B1, c)is he con ex hull o {(1,0,0),(0,1,0)}and Owen(A, B2, c)
is he con ex hull o {(0,0,1),(0,1,0)}. Fo e e y non-ex eme op imal solu-
ion xa= (a, 1−a) (a∈(0,1)),B(A, B, xa)consis s only o one ma ix Ba,
in which he diagonal is (a, 1,1−a), and he es is ze o. The co espond-
ing Omin(A, Ba, c)is he con ex hull o {(0,1,0),(1,0,1)}, and he e o e he
Owen se o he co esponding educed LP p ocess is {(0,1,0),(a, 0,1−a)}.
The e o e, EOwen(A, B, c)is he con ex hull o {(1,0,0),(0,1,0),(0,0,1)}
(all playe s can ob ain a posi i e payo om alloca ions in his se ).
To summa ize, he Owen se , he EOwen se and he co e o he o iginal
game ha e a ela ionship as shown in Figu e 1.
&%
'$
&%
'$
Owen
EOwen
Co e(A, B, c)
Figu e 1: Gene al ela ion be ween EOwen,Co e and Owen .
23
Conclusions
In his wo k we ha e in oduced he EOwen se , a new solu ion ule on
he class o linea p oduc ion p ocesses which o e comes ce ain d awbacks o
he well-known Owen se , in he sense ha one can always ind an alloca ion
ha gi es a s ic ly posi i e payo o playe s whose esou ces a e needed o
(a leas ) one op imal p oduc ion plan. Some examples in he pape show
ha Owen alloca ions do no sa is y his p ope y.
EOwen is de ined as he union, o e all possible op imal p oduc ion plans
and all possible educed ma ices, o he Owen se s o e he co esponding
educed LP p ocesses, in which playe s ge id o he le o e s in hei e-
sou ces acco ding o a educed ma ix. Se e al heo e ical p ope ies and an
axioma ic cha ac e iza ion o his new solu ion ule a e gi en. By means o
an example we also p o e ha , in gene al, i is no possible o ind alloca ions
ha gi e non-null payo s o playe s ha a e necessa y in o de o achie e
he op imal alue o (N) by es ic ing o he co e.
We no e ha wo ypes o playe s ha e been in ol ed in his pape : g oup
T1, consis ing o playe s such ha some o he esou ces hey own a e needed
o (a leas ) one op imal p oduc ion plan; and g oup T2, consis ing o playe s
whose esou ces a e ne e comple ely used in any o he op imal p oduc ion
plans. Theo em 1 ensu es ha playe s in T1 can always ind an alloca ion in
EOwen ha gi es hem a s ic ly posi i e payo . Examples ha e shown ha
playe s in T2 may ecei e only ze o payo s om Owen alloca ions. Playe s
in bo h T1 and T2 a e o special in e es , since hey ecei e s ic ly posi i e
payo s om EOwen alloca ions, and may only ecei e ze o payo s om
Owen alloca ions (see o ins ance examples 1 and 2).
24