Jou nal o Indus ial Enginee ing and Managemen
JIEM, 2018 – 11(3): 528-534 – Online ISSN: 2013-0953 – P in ISSN: 2013-8423
h ps://doi.o g/10.3926/jiem.2430
A Reply o Pon e e al (2016) Supply Chain Collabo a ion:
Some Commen s on he Nucleolus o he Bee Game
Da id Muelle
B andenbu g Uni e si y o Technology, Co bus-Sen enbe g (Ge many)
da id.muelle @b- u.de
Recei ed: Sep embe 2017
Accep ed: Ma ch 2018
Abs ac :
Pu pose:
The aim o he pape is o pick up he esul o a p e iously published pape in o de o deepen
he discussion. We analyze he solu ion agains he backg ound o some well-known concep s and we
in oduce a newe one. In doing so we would like o inspi e he u he discussion o supply chain
collabo a ion.
Design/me hodology/app oach:
Based on game heo e ical knowledge we p esen and compa e se en
p ope ies o ai p o i sha ing.
Findings:
We show ha he nucleolus is a co e-solu ion, which does no ul il agg ega e mono onici y. In
con as he Shapley alue is an agg ega e mono onic solu ion bu does no belong o he co e o e e y
coope a i e game. Mo eo e , we p esen he Lo enz dominance as an addi ional ai ness c i e ia.
O iginali y/ alue:
We discuss he e y in ol ed p ocedu e o es ablishing lexicog aphic o de s o excess
ec o s o games wi h many playe s.
Keywo ds:
bee game, coope a i e game heo y, p o i alloca ion, Shapley alue, nucleolus, co e-selec ion,
agg ega e mono onici y, Lo enz se
1. In oduc ion
Pon e, Fe nández, Rosillo, Pa eño and Ga cía (2016) sugges ed in his jou nal he use o coope a i e game heo y
wi hin supply chain managemen and demons a ed he applica ion wi h he amous bee game. Emb acing his
inco po a ion o coope a i e game heo y in p inciple, some commen s and enhancemen s a e necessa y.
2. Aims and Solu ion Concep s o Coope a i e Game Theo y
2.1. Fundamen als o Game Theo y
Coope a i e game heo y is based on a ange o assump ions. Fo a de ailed discussion we e e o he app op ia e
li e a u e (Maschle , Solan & Zami , 2013, p. 659-662). A coope a i e game Γ is he pai (N, ), whe e N = {1,2, …,
n} deno es he se o playe s. No only is he amoun o all he playe s N impo an he e, bu also all he subse s o
N. Such a subse S  N is e e ed o as coali ion S, whe eby N i sel is desc ibed as a g and coali ion. Each
coali ion is ma ked by a alue unc ion (S). The unc ion assigns a alue o each subse S, which ep esen s he
economic pe o mance o his coali ion.
-528-
Jou nal o Indus ial Enginee ing and Managemen – h ps://doi.o g/10.3926/jiem.2430
2.2. P ope ies o a Game
To analyze a solu ion we ha e o in oduce some classes o games. Coope a ion may be success ul o no . To
conc e ize he e m “success'', some desi able p ope ies o games may be de ined. One goal is he gene a ion o a
esul which is no wo se han he esul s o isola ed ac ions. This is e e ed o as supe addi i i y. A game (N, ) is
supe addi i e i (R  S) ≥ (R) + (S) o all S, R  N wi h R  S = .
In he ollowing we concen a e on si ua ions in which o a leas one coali ion yields (R  S) > (R) + (S). In
consequence, he g and coali ion gene a es a be e esul han he sum o all s and-alone coali ions.
A game (N, ) is essen ial i (N) > ∑iN ({i}). By EN we deno e he se o all essen ial games wi h he se o
playe s N. In he ollowing essen ial games only a e analyzed. Supe addi i i y desc ibes he ela ionship o
coali ions o disjoin elemen s. A simila e ec may be claimed o coali ions o conjoin elemen s. This is called
con exi y. A game (N, ) is con ex (Maschle e al., 2013, p. 718) i (S {i}) – (S) ≤ (R  {i}) – (R) o all S
 R  N {i}. We will deno e he se o con ex games by CN.
2.3. P ope ies o a Fai Solu ion
Looking a a game, he ques ion a ises o how o sha e he join ly gene a ed esul be ween he pa ne s,
which is equi alen o an alloca ion o he esul . A unc ion ( ) which assigns o a game (N, ) a, possible
emp y, subse ( ) o is called a solu ion concep . The unc ion dis ibu es (N ) and gene a es a payo ec o
x = (x1, x2, x3, … xn) wi h x . Such a unc ion is e e ed o as alloca ion scheme.
De ini ion 1: A solu ion is a single- alued solu ion i | ( )| = 1 o e e y . In his case, ( ) is ep esen ed by
an elemen o , i.e. ( ) = x.
Wi h he alloca ion o he join ly gene a ed esul , he p oblem o ai ness a ises. Se e al p ope ies o a ai
solu ion ha e been iden i ied in coope a i e game heo y in he las decades. The mos c ucial p ope ies a e
(González-Díaz, Ga cía-Ju ado & Fies as-Janei o, 2010, p. 226; Calleja, Ra els & Tijs, 2012; Muelle , 2018, p. 406-
408):
•E iciency: A single- alued solu ion is e icien i ∑iN i( ) = (N).
•Indi idual a ionali y: A single- alued solu ion is indi idual a ional i i( ) ≥ ({i}) iN.
•Equal- ea men -p ope y: A single- alued solu ion sa is ies equal- ea men p ope y i o he playe s i
and j, o which holds: (S  {i}) = (S  { j }) S  N wi h i, j  S yields i( ) = j( ).
•Dummy-playe -p ope y: A playe i is called a dummy playe i (S  {i}) = (S) + ({i}) o all S  N
wi h i  S. A single- alued solu ion sa is ies he dummy-playe -p ope y i o a dummy-playe i yields:
i( ) = ({i}).
•Addi i i y: A single- alued solu ion sa is ies addi i i y i o any wo games , w ollows
( + w) = ( ) + (w).
•Agg ega e mono onici y: A single- alued solu ion sa is ies agg ega e mono onici y i o all games , w
wi h (N) > w(N) and (S) = w(S) o all S N ollows: i( ) ≥ i(w) iN.
The e a e se e al o he p ope ies (A in & Ka se , 2018, p. 305) which a e no o in e es o u he discussion.
The i s wo p ope ies a e summe ized by de ining an impu a ion.
De ini ion 2: The se o impu a ions I( ) o a game N( ) is de ined by
Only hose impu a ions a e o in e es ha a e no domina ed by ano he impu a ion. The se o non-domina ed
impu a ions o ms he co e.
-529-
Jou nal o Indus ial Enginee ing and Managemen – h ps://doi.o g/10.3926/jiem.2430
De ini ion 3: The Co e( ) o a game N( ) is de ined by
The co e o a game con ains all solu ions which a e jus i ied as ai and, he e o e, a e s able. I may be small, e y
la ge, o emp y. Wi hou de ining he p ope y o balancedness in de ail we poin ou , ha he co e o a balanced
game is ne e emp y (Muelle , 2018, p. 405).
By BN we deno e he se o all balanced games wi h playe se N. Wi h his class a hand we in oduce he
co e-selec ion-p ope y. A solu ion sa is ies co e selec ion i i selec s a co e elemen o any game wi h a
non-emp y co e.
Co e selec ion: A single- alued solu ion sa is ies co e selec ion i ( )  Co e( ) o all  BN.
2.4. Cha ac e iza ion o he Shapley-Value and he Nucleolus
In o de o de e mine a ai sha e o playe i, he ollowing hough is wo h no ing: each playe ecei es a pa
depending on ha playe 's con ibu ions o he heo e ically possible, hus imaginable, coali ions. The con ibu ion
o he playe consis s in he inc ease in alue caused by ha playe 's pa icipa ion in he coali ion. The ques ion ha
has o be answe ed is which alue he coali ion has wi h playe i and which i would ha e wi hou playe i. This
di e ence is called he ma ginal con ibu ion.
De ini ion 4: The ma ginal con ibu ion mci o playe i o a coali ion S is de ined by: mci = (S  {i}) – (S).
Assuming all o de s o o ming a coali ion o ha e he same p obabili y esul s in he weigh ed a e age o he
ma ginal con ibu ions o a playe , which is commonly desc ibed as he Shapley alue (Shapley, 1953, p. 311).
De ini ion 5: The Shapley alue o a playe φi in a game (N, ) is de ined by
.
The Shapley- alue (Muelle , 2018, p. 412):
•sa is ies e iciency, dummy-playe -p ope y, equal- ea men -p ope y, addi i i y and agg ega e
mono onici y o all coope a i e games,
•is indi idual a ional o each  EN,
•bu ul ils co e-selec ion only o each  C N.
The nucleolus was in oduced by Schmeidle (1969) and sea ches o a ai dis ibu ion by minimizing he maximal
dissa is ac ion o e e y playe . To achie e his, he dissa is ac ion o a coali ion wi h a conc e e payo ec o is
named in his connec ion as excess. I is necessa y o calcula e how unhappy a coali ion would be wi h a payo
ec o .
De ini ion 6: The excess (unhappiness) ex(S, x) o a coali ion S wi h a payo ec o x is de i ed by
ex(S, x) = (S) – ∑iS xi.
To de i e he nucleolus, he payo ec o s wi h he highes unhappiness o e e y playe a e sough in he nex
s ep. To do so, hese excess alues a e so ed in non-inc easingly o de (González-Díaz e al., 2010, p. 233; Muelle ,
2016, p. 205). The excess o a coali ion Si wi h espec o a payo ec o x is deno ed by ex(Si, x) = θi(x).
De ini ion 7: The ec o o non-inc easingly o de ed excess alues Θ is de ined by Θ(x) = θ1(x); θ2(x); θ3(x); …;
θ2n(x) wi h θi(x) ≥ θj(x) o 1 ≤ i ≤ j ≤ 2n.
-530-
Jou nal o Indus ial Enginee ing and Managemen – h ps://doi.o g/10.3926/jiem.2430
To compa e wo payo s, hei ec o s o non-inc easingly o de ed excess alues a e compa ed based on he
lexicog aphic o de . The ec o which is lexicog aphically smalle han he o he one is chosen as his ec o o e s
he minimum o he maximal dissa is ac ion o all playe s esul ing om he wo payo s.
De ini ion 8: I wo impu a ions x and y a e compa ed, hen x is conside ed o be lexicog aphically smalle (LEX)
han y i he e exis s an index m, wi h which esul s θk(x) = θk(y)  1 ≤ k ≤ m and θm(x) < θm(y).
Wi h hese explana ions, he nucleolus o a game can be de ined as ollows (González-Díaz e al., 2010, p. 232):
De ini ion 9: In a game (N, ) wi h I( ) ≠  he nucleolus nuc( ) is de ined by: nuc( ) = {x  I( )|Θ(x)LEXΘ(y)
 y  I( )}.
We summa ize ha he nucleolus (Muelle , 2018, p. 414):
•sa is ies e iciency, indi idual a ionali y, dummy-playe -p ope y, equal- ea men -p ope y and co e
selec ion o each  EN,
•does no ul il addi i i y o each  EN,
•is nei he o each  EN no o each  C|N|≥4 agg ega e mono onic.
2.5. Analyzing he Bee Game
Table 1 con ains he cha ac e is ic unc ion o he bee game.
S
(
S
)
S
(
S
)
S
(
S
)
S
(
S
)
{}0 {4} 400 {2,3} 550 {1,2,4} 850
{1} 100 {1,2} 400 {2,4} 650 {1,3,4} 1,250
{2} 200 {1,3} 450 {3,4} 750 {2,3,4} 1,050
{3} 300 {1,4} 600 {1,2,3} 800 {1,2,3,4} 1,500
Table 1. Cha ac e is ic unc ion o he bee game. Pon e e al. (2016, p. 1027).
Using he no a ion {1} = S, {1,4} = R, and {i}= 2 shows ha he game is no con ex as we ge :
400 – 100 850 – 600. To guide he ollowing p ocedu e, we ha e o check i he game has a non-emp y co e a all.
Es ablishing a sys em o inequali ies based on he cha ac e is ic unc ion shows ha his ques ion can be answe ed
in he a i ma i e. So, he co e is no emp y and he e exis s a ai solu ion.
Compu ing he Shapley alues o he playe s leads o he ollowing esul s: φ1 = 262.50, φ2 = 254 , φ3 = 445 and
φ4 = 537,50. As poin ed ou , he non-con exi y o he game may cause ha he Shapley- alue does no belong o
he co e. Analyzing coali ion {1,3,4} shows ha hese playe s ge a alue o 1,245 . This is less han he alue
which hey gene a e (1,250). Tha ’s why he Shapley alue is no a co e-alloca ion and we ha e o compu e he
nucleolus.
Conce ning he calcula ional e o o he nucleolus, he e y in ol ed p ocedu e o es ablishing lexicog aphic
o de s o excess ec o s o games wi h many playe s mus be men ioned. The e a e some mis akes in compu ing
he nucleolus caused by o e looking he possibili y ha a linea p og am can ha e mul iple solu ions (Guaja do &
Jø ns en, 2015). Ne e heless, he nucleolus has been co ec ly compu ed in se e al publica ions (e. g. F omen,
1997; Halle jo d, Helming & Jø ns en, 1995; Kimms & Çe ine , 2012).
The solu ion o he bee -game is x = (225, 225, 410.5, 639.5) (Pon e e al., 2016, p. 1029). Checking De ini ion 3
indica es ha his solu ion belongs o he non-emp y co e. Tha means i is a s able and ai solu ion. Un o una ely,
his esul is no he nucleolus. To p o e ha we compu e he esul ing ec o o non-inc easingly o de ed excess
alues Θ(x). We ge :
-531-
Jou nal o Indus ial Enginee ing and Managemen – h ps://doi.o g/10.3926/jiem.2430
Θ(x) = (0, 0, -25, –25, –50, –60.5, -85.5, –110.5, –125, -185.5, –214.5, -225, –239.5, 239.5, –264.5, –300).
We may educe he maximal dissa is ac ion by choosing he impu a ion y = (250, 225, 410.50, 614.5). This leads o
he ec o o non-inc easingly o de ed excess alues Θ(y), wi h:
Θ(y) = (0, 0, –25, –25, –75, –85.5, –85.5, –110,5. –150, –189.5, –200, –210.5, –214.5, –239.5, –264.5, –275).
Compa ing hese ec o s, we can conclude ha Θ(y) LEX Θ(x), wha indica es ha x is no he nucleolus.
Con inuing he p ocedu e, we ge he nucleolus wi h nuc( ) = (291 , 225, 441 , 541 ). This gene a es he ollowing
ec o o non-inc easingly o de ed excess alues:
Θ(nuc) = (0, 0, –25, –25, –116 , –116 , –116 , –141 , –141 , –158 , –158 , –191 , –208 , –233 , –233 , –283 ).
This esul di e s signi ican ly om he o iginal alue. Bu we ha e poin ed ou ha bo h alues belong o he non-
emp y co e. The p oblem is ha he co e o he discussed bee game is la ge and con ains a lo o possible
alloca ions. So he ques ion ises o a jus i ica ion o an alloca ion which is wide han he co e a gumen a ion.
Besides he p esen ed nucleolus we wan o p esen ano he possible a gumen – he Lo enz se .
2.6. Lo enz Dominance, he Lo enz Se , and he Lo enz Solu ion
Lo enz dominance was es ablished o mi o he concen a ion o weal h in a quan i a i e way. S a ing poin
o he a gumen a ion is a socie y o n indi iduals in which he o al income o I is dis ibu ed by he alloca ion
x (B ânzei, Dimi o & Tijs, 2008, p. 37). The ec o esul s om ea anging x acco ding o
= ( ).
The ec o Lo enz domina es he ec o
^y
o any x, y  wi h
i o all p  {1, …, n – 1} wi h a leas one s ic inequali y. In his case we deno e x LOR y.
Lo enz dominance implies an alloca ion wi h less inequali y. Some concep s o egali a ian solu ion ha e been
de eloped based on Lo enz dominance (c . A in, Kuipe s & Ve meulen, 2008: p. 569-571). As he bee game is no
con ex we p esen a newe solu ion concep – he Lo enz se (Hougaa d, Peleg & Tho lund-Pe e sen, 2001;
Hougaa d & Smilgins, 2016).
De ini ion 10: The Lo enz se L( ) o a game is de ined by:
L( ) = {x  Co e( )| y  Co e( ): y LOR x}.
The Lo enz se consis s o Lo enz undomina ed solu ions, which belong o he co e. We can s a e ha
L( )  Co e( ) and ha L( ) ≠  i Co e( ) ≠  (Hougaa d & Smilgins, 2016, p. 153).
The Lo enz se coincides wi h equal-dis ibu ion-solu ion i his solu ion belongs o he co e. I he equal
dis ibu ion does no belong o he non-emp y co e, hen he e exis s a unique alloca ion x  Co e( ), which
minimizes he Euclidian dis ance om he equal dis ibu ion o he co e and x  L( ). This is he Lo enz-solu ion
based on leas squa es solu ion (A in e al., 2008, p. 569). We de ine he Euclidean leng h o an alloca ion x
wi h: .
De ini ion 11: The Lo enz solu ion LS( ) o a balanced game is he alloca ion x  Co e( ) o which
o all y  Co e( ).
-532-

Jou nal o Indus ial Enginee ing and Managemen – h ps://doi.o g/10.3926/jiem.2430
To analyze he nucleolus and he solu ion o Pon e e al. we will ea ange he alues in non-dec easing o de wha
leads o = (225, 225, 410.5, 639.5) o he solu ion o Pon e e al. and = (225, 291 , 441 ,541 ) o he
nucleolus. I becomes appa en ha he nucleolus Lo enz domina es he o he solu ion. Bu he nucleolus is no a
membe o he Lo enz se . I we ha e a look a he impu a ion x3 = (416 , 250, 416 , 416 ), we ge he o de ed
ec o = (250, 416 , 416 , 416 ). This ec o Lo enz domina es all he o he ec o s and minimizes he
Euclidean dis ance o he equal dis ibu ion.
3. Summa y
Inco po a ing game heo e ic solu ion concep s in o supply chain collabo a ion is a welcome b oadening o he
managemen o his p ocess. In e p e ing such si ua ions as a coope a i e game may lead o some use ul insigh s.
Beside he well-es ablished concep s we ha e in oduced a newe concep o ai ness. The o iginal esul s we e
e lec ed agains he backg ound o he new solu ion. We ha e shown ha he nucleolus is a use ul concep o
balanced games which a e no con ex. The Lo enz solu ion en iches he discussion by in oducing he
Lo enzean unde s anding o ai ness. In doing so we would like o inspi e he u he discussion o supply chain
collabo a ion.
Decla a ion o Con lic ing In e es s
The au ho decla ed no po en ial con lic s o in e es wi h espec o he esea ch, au ho ship, and/o publica ion
o his a icle.
Funding
The au ho ecei ed no inancial suppo o he esea ch, au ho ship, and/o publica ion o his a icle.
Re e ences
A in, J., & Ka se , I. (2018). The SD-p enucleolus o TU-Games: coali ional mono onici y and co e s abili y. In
Muelle , D., & T os , R. (Eds.), Game heo y in managemen accoun ing - implemen ing incen i es and ai ness (301-321).
Be lin e al.: Sp inge .
A in, J., Kuipe s, J., & Ve meulen, D. (2008). An axioma ic app oach o egali a ianism in TU-games. In e na ional
Jou nal o Game Theo y, 37(4), 565-580. h ps://doi.o g/10.1007/s00182-008-0133-6
B ânzei, R., Dimi o , D., & Tijs, S.H. (2008). Models in Coope a i e Game Theo y (2nd ed.). Be lin.
Calleja, P., Ra els, C., & Tijs, S.H. (2012). Agg ega e mono onic s able single- alued solu ions o coope a i e games.
In e na ional Jou nal o Game Theo y, 41(4), 899-913. h ps://doi.o g/10.1007/s00182-012-0355-5
F omen, B. (1997). Reducing he numbe o linea p og ams needed o sol ing he nucleolus p oblem o n-pe son
game heo y. Eu opean Jou nal o Ope a ional Resea ch, 98(3), 626-636. h ps://doi.o g/10.1016/0377-2217(95)00341-X
González-Díaz, J., Ga cía-Ju ado, I. & Fies as-Janei o, M.G. (2010). An in oduc o y cou se on ma hema ical game heo y.
P o idence: Ame ican Ma hema ical Socie y. h ps://doi.o g/10.1090/gsm/115
Guaja do, M., & Jø ns en, K. (2015). Common mis akes in compu ing he nucleolus. Eu opean Jou nal o Ope a ional
Resea ch, 241(3), 931-935. h ps://doi.o g/10.1016/j.ejo .2014.10.037
Halle jo d, Å., Helming, R., & Jø ns en, K. (1995). Compu ing he nucleolus when he cha ac e is ic unc ion is
gi en implici ly: a cons ain gene a ion app oach. In e na ional Jou nal o Game Theo y, 24(4), 357-372.
h ps://doi.o g/10.1007/BF01243038
Hougaa d, J.L., Peleg, B., & Tho lund-Pe e sen, L. (2001). On he se o Lo enz-maximal impu a ions in he co e
o a balanced game. In e na ional Jou nal o Game Theo y, 30(2), 147-165. h ps://doi.o g/10.1007/s001820100070
Hougaa d, J.L., & Smilgins, A. (2016). Risk capi al alloca ion wi h au onomous subuni s: The Lo enz se . Insu ance:
Ma hema ics and Economics, 67(3), 151-157. h ps://doi.o g/10.1016/j.insma heco.2015.12.002
-533-
Jou nal o Indus ial Enginee ing and Managemen – h ps://doi.o g/10.3926/jiem.2430
Kimms, A., & Çe ine , D. (2012). App oxima e nucleolus-based e enue sha ing in ai line alliances. Eu opean Jou nal
o Ope a ional Resea ch, 220(2), 510-521. h ps://doi.o g/10.1016/j.ejo .2012.01.057
Maschle , M., Solan, E., & Zami , S. (2013). Game heo y. Camb idge: Camb idge Uni e si y P ess.
h ps://doi.o g/10.1017/CBO9780511794216
Muelle , D. (2016). Wha ’s in i o me? - An analysis o en i onmen al in es men s h ough a game heo e ic lens.
In e na ional Jou nal o Inno a ion and Sus ainable De elopmen , 10(2), 198-218.
h ps://doi.o g/10.1504/IJISD.2016.075551
Muelle , D. (2018). The usabili y and sui abili y o alloca ion schemes o co po a e cos accoun ing. In Muelle , D.,
& T os , R. (Eds.), Game heo y in managemen accoun ing - implemen ing incen i es and ai ness (401-427) Be lin e al.:
Sp inge . h ps://doi.o g/10.1007/978-3-319-61603-2_19
Pon e, B., Fe nández, I., Rosillo, R., Pa eño, J., & Ga cía, N. (2016). Supply chain collabo a ion: a game- heo e ic
app oach o p o i alloca ion. Jou nal o Indus ial Enginee ing and Managemen , 9(5), 1020-1034.
h ps://doi.o g/10.3926/jiem.2084
Schmeidle , D. (1969). The nucleolus o a cha ac e is ic unc ion game. Jou nal o Applied Ma hema ics, 17(6),
1163-1170. h ps://doi.o g/10.1137/0117107
Shapley, L.S. (1953). A alue o n-pe son games. In Kuhn, H.W., & Tucke , A.W. (Eds.), Con ibu ions o he heo y o games
(II) (307-317). P ince on: P ince on Uni . P ess. h ps://doi.o g/10.1515/9781400881970-018
Jou nal o Indus ial Enginee ing and Managemen , 2018 (www.jiem.o g)
A icle’s con en s a e p o ided on an A ibu ion-Non Comme cial 4.0 C ea i e commons In e na ional License. Reade s a e
allowed o copy, dis ibu e and communica e a icle’s con en s, p o ided he au ho ’s and Jou nal o Indus ial Enginee ing and
Managemen ’s names a e included. I mus no be used o comme cial pu poses. To see he comple e license con en s, please
isi h ps://c ea i ecommons.o g/licenses/by-nc/4.0/.
-534-