A membrane computing approach to the generalized Nash equilibrium

A memb ane compu ing app oach o he gene alized Nash
equilib ium
Alejand o Luque-Ce pa
1
•Miguel A
´. Gu ie
´ ez-Na anjo
2
Accep ed: 17 Ma ch 2025
The Au ho (s) 2025
Abs ac
Gene alized Nash Equilib ium is an ex ended e sion o he s anda d Nash Equilib ium wi h impo an implica ions in eal-
li e p oblems such as economics, wi eless communica ion, he elec ici y ma ke , o enginee ing among o he a eas. In his
pape , we p opose a i s app oach o compu ing Gene alized Nash Equilib ia using Memb ane Compu ing echniques. We
model an e icien P sys em ha , based on Eule ’s me hod, compu es app oxima ions o Gene alized Nash Equilib ia o
popula ion games unde B own– on Neumann–Nash dynamics, b idging bo h a eas and opening a doo o a low o
p oblems and solu ions in bo h di ec ions.
Keywo ds Memb ane compu ing Gene alized Nash equilib ium E olu iona y game heo y
1 In oduc ion
E olu iona y Game Theo y (EGT) s udies he e olu ion o
a popula ion o agen s ha in e ac wi h each o he and ge
a payo in each in e ac ion Ho baue and Sigmund (2000).
The ob ained payo depends on he chosen s a egies o
he agen s which pa icipa e in he in e ac ion. Each agen
selec s only one s a egy a a ime, bu his choice can be
modi ied o e ime. The d i ing p inciple in his si ua ion
is ha indi iduals end o be sel ish, choosing s a egies
ha esul in highe payo s o hemsel es. In his con ex ,
a Nash equilib ium is eached when no agen can inc ease
i s payo by changing i s s a egy while o he agen s
main ain hei cu en ones Nash (1951).
In a Nash equilib ium p oblem, all he agen s compe e
among hem o maximize hei payo s, and each agen can
eely choose i s s a egy. The gene alized Nash equilib-
ium p oblem (GNEP) is a a ian o he Nash p oblem
in oduced in 1952 by G. Deb eu Deb eu (1952). In a
GNEP, he s a egy se o each playe may also depend on
he o he playe s’ s a egies. This GNEP models a la ge
numbe o eal-li e si ua ions, such as powe alloca ion in a
elecommunica ion sys em, en i onmen al pollu ion con-
ol, o ene gy ma ke model ( o a de ailed su ey, see,
e.g., Facchinei and Kanzow (2007)).
In his pape , we p opose o s udy he GNEP in he
amewo k o Memb ane Compu ing Pa
˘un (2002); Pa
˘un
e al. (2010). Memb ane Compu ing is a well-known a ea
o Compu e Science ha akes inspi a ion om he bio-
chemical eac ions inside he esicles o li ing cells. P
sys ems Pa
˘un (2000), he so-called Memb ane Compu ing
de ices, ha e been success ully conside ed o model many
dynamic p ocesses in eal-li e p oblems Colome e al.
(2011,2010); Ga cı
´a-Quismondo e al. (2017). F om he
ini ial de ini ion o P sys ems, many a ian s ha e been
explo ed by adding new ea u es o he ini ial model (see,
e.g., Song (2021) o a ecen su ey). Recen ly, P oba-
bilis ic P sys ems Ca dona e al. (2011), a kind o P sys em
designed o deal wi h p obabili y dis ibu ions in he
applica ion o ules, was conside ed o model he sp ead o
beha io s in s uc u ed popula ions in he amewo k o
EGT Ga cı
´a-Vic o ia e al. (2022). In his pape , we s udy
he GNEP by conside ing ansi ion P sys ems wi h ac i e
memb anes Pa
˘un (2001), also called memb ane
pola iza ion.
&Alejand o Luque-Ce pa
[email p o ec ed]
Miguel A
´. Gu ie
´ ez-Na anjo
[email p o ec ed]
1
Depa men o Compu e Science and Enginee ing, Chalme s
Uni e si y o Technology, Go henbu g, Sweden
2
Depa men o Compu e Science and A i icial In elligence,
Uni e sidad de Se illa, Se ille, Spain
123
Na u al Compu ing
h ps://doi.o g/10.1007/s11047-025-10014-z(0123456789().,- olV)(0123456789().,- olV)
The pape is o ganized as ollows: Sec . 2es ablishes
some backg ound on P sys ems and speci ies he ype o P
sys em we use: ansi ion P sys ems wi h memb ane
pola iza ion. Sec ion 3in oduces popula ion games unde
he B own- on Neumann-Nash (BNN) dynamics, ha will
be used as he amewo k o de ine ou P sys em. In Sec . 4,
we desc ibe he design o ou P sys em and analyze i s ime
complexi y, showing ha i does no depend on he numbe
o playe s o s a egies in ol ed. We also p esen an
expe imen o illus a e i s unc ioning. Finally, some
conclusions and hin s o u u e wo k a e p esen ed.
2 T ansi ion P sys ems wi h memb ane
pola iza ion
In his sec ion, we de ine he a ian o P sys ems ha we
used o sol e ou p oblem: ansi ion P sys ems wi h
memb ane pola iza ion. Then, in sec ion 2.1, we p o ide an
example o such a P sys em.
A e he de elopmen o he i s model o P sys em by
Gh. Pa
ˇun in 1998 Pa
˘un (2000), many a ia ions ha e been
p esen ed. In his wo k, we use a combina ion o wo
p oposed a ian s. The P sys em designed is a ansi ion P
sys em Pa
˘un (2000) wi h ac i e memb anes Pa
˘un (2001)
wi hou di ision ules, i.e., a ansi ion P sys em wi h
memb ane pola iza ion. A ansi ion P sys em wi h mem-
b ane pola iza ion o deg ee q1 is a cons uc :
P¼hC;l;w1;...;wq;ðR1;q1Þ;...;ðRq;qqÞ;iou i
whe e:
1. Cis he alphabe o objec s;
2. lis a hie a chical ee-like memb ane s uc u e o q
memb anes ha ha e a pola iza ion among 0;þ;;
3. w1;...;wqa e mul ise s o objec s o e C;
4. R1;...;Rqa e ini e se s o e olu ion ules o he
o m:
•u½ a
h!u0½ 0b
hwhe e u;u0; ; 0a e mul ise s o e
C,h2 1;...;qg,his no he label o he oo
memb ane in l, and a;b2 0;þ;g.
•½ a
h!u0½ 0b
hwhe e u;u0; ; 0a e mul ise s o e C,
h2 1;...;qg,his he label o he oo memb ane
in l, and a;b2 0;þ;g.
The di e ence be ween he exp essions esides in ha
no objec s should be able o en e he skin memb ane
( he oo o l) om he en i onmen . The meaning o
hese ules can be easily unde s ood as combina ions o
he ollowing examples:
•½ua
h!½ a
h, also exp essed as ½u! a
h, is an objec
e olu ion ule, ha ans o ms he mul ise uin o
he mul ise .
•½ua
h! ½
b
his a send-ou communica ion ule, ha
ejec s he mul ise u, and ans o ms i in o he
mul ise .
•u½
a
h!½ b
his a send-in communica ion ule, ha
abso bs he mul ise u, and ans o ms i in o he
mul ise .
The gene al exp ession conside s combina ions o
hese cases, whe e some mul ise s can be abso bed in o
memb ane ha he same ime as o he s a e ans o med
o ejec ed.
5. q1;...;qqa e pa ial o de ela ions o e R1;...;Rq,
called p io i y ela ions. Gi en wo ules ; 0,we
ep esen ha has highe p io i y han 0by q [q 0.
P io i y indica es wha ule should be applied i bo h
a e applicable.
6. iou 2 0;1;...;qgis he ou pu egion, whe e 0
ep esen s he en i onmen .
A con igu a ion o Pis de ined by C ¼
ððw1; ;a1; Þ;...;ðw1; ;a1; Þ;w0; Þ o an ins an , whe e wh;
is he mul ise o objec s in memb ane ha ins an ,ah; is
he memb ane pola iza ion o memb ane h, and w0; is he
mul ise o objec s o he en i onmen . The ini ial con ig-
u a ion o Pis C0¼ððw1;0Þ;...;ðwq;0Þ;;Þ. We use he
no a ion C ¼l0 o deno e speci ic pa s om he con ig-
u a ion C whe e only he memb anes in he sub ee l0 om
la e conside ed.
Fo each con igu a ion, he ules a e applied in a pa allel
and maximal way. By maximal, we indica e ha no mo e
ules can be applied a he same ime. Fo mally, a mul ise
Uo ules is maximal i he e is no mul ise o applicable
ules U0such ha UU0. I wo applicable ules wi h he
same p io i y a e exclusi e, his is, igge ing one would
p e en he o he one om igge ing, hen only one o
hem is selec ed a andom and applied.
As in Ga cı
´a-Vic o ia e al. (2022), he seman ics o he
P sys em ollow he nex p inciples:
(I1) When an objec c osses a memb ane, i s pola iza-
ion may change. Rules can only be applied i he
pola iza ion is app op ia e.
(I2) I wo ules ha a ec he same memb ane can be
applied a he same ime, and one o he ules
changes he pola iza ion o he memb ane, bo h
ules a e applied. This means ha he change o
pola iza ion is pe o med a e all o he e olu ion
ules a e applied.
No ice ha p inciple (I2) ensu es ha ules a e applied in a
pa allel and maximal way. I his p inciple is no assumed,
and a ule can change he memb ane pola iza ion, hen he
o de o applica ion o he ules would be impo an du ing
a single ansi ion s ep. In ha case, mul ise s o ules ha
A. Luque-Ce pa, M. Á. Gu ié ez-Na anjo
123
can be applied would no be well-de ined, b eaking he
pa allelism and maximali y o he sys em.
No ice also ha , while he memb anes in Pha e labels
in 1;...;qg, we can always de ine a se Ho labels wi h
jHj¼qsuch ha he e is a bijec ion be ween he elemen s
o Hand he memb anes o l. The same applies o he
ules, ha we can exp ess as he uple ðR;qÞ. We use his
ac in Sec . 4.1 o p o ide a be e indexing.
2.1 Example o a P sys em
Le
P¼hC;l;w1;w2;ðR1; q 1[q 2gÞ;ðR2;;Þ;iou i
be a ansi ion P sys em wi h memb ane pola iza ion o
deg ee 2 whe e:
•C¼ a;b;cg;
•l¼½½
0
20
1;
•w1¼ k0g;
•w2¼ a3cg;
•R1¼ 1k0½
0
2!½kþ
2; 2½k0!k0
1g;
•R2¼ 3½a2!b0
2; 4½a!c0
2; 5½c!b0
2g;
•iou ¼2.
We ha e ha U1¼ 1; 3; 4; 5g,U2¼ 3g,U3¼
1; 3
4; 5ga e mul ise s o applicable ules. U2is no
maximal because U2U1.U1and U3a e he only maxi-
mal mul ise s o applicable ules, and hey could bo h be
applied in his con igu a ion because he e is no p io i y
be ween he ules in ol ed in each se . 2can no be pa o
a maximal mul ise o applicable ules because ule 1has
p io i y o e ule 2(q 1[q 2).
The mul ise U1would lead o he con igu a ion
C1¼½½b2ckþ
20
1, and he mul ise U3would lead o
C1¼½½bc3kþ
20
1. Because he pola iza ion o memb ane 2
changed o þin bo h con igu a ions, none o he ules in
R2can now be applied. Because he e a e no objec s k0in
memb ane 1, none o he ules in R1can be applied. The
compu a ion o he sys em is hen inished o bo h cases
a e one ansi ion s ep.
3 Popula ion games unde BNN dynamics
The pu pose o his sec ion is o in oduce popula ion
games. Speci ically, we in oduce popula ion games unde
BNN dynamics, which a e cen al o his pape . In sec ion
3.1, we gi e an example o such a game: he Ene gy
Ma ke Game, whe e playe s decide when o buy ene gy
and modi y hei s a egies depending on he decisions o
he es un il an equilib ium is eached. We use he Ene gy
Ma ke Game as a amewo k o de ine ou P sys em, and
we explain how he P sys em can be modi ied o adap i o
o he popula ion games unde BNN dynamics.
In a popula ion game, we ha e a socie y o decision-
making agen s di ided in o disjoin popula ions ha
ecei e di e en payo s depending bo h on he decisions
hey make and he decisions he es o he agen s make.
The goal o each popula ion is o maximize he payo
ecei ed. The decisions ha agen s can make depend on he
popula ion hey o m pa o . Each agen is endowed wi h a
e ision p o ocol, which p o ides condi ional swi ch a es
be ween s a egies acco ding o hei associa ed payo s
Sandholm (2010). These a es allow he agen s o change
hei s a egies o e ime. When he numbe o agen s is
la ge enough, his p ocess can be desc ibed by di e en ial
equa ions, e e ed o as he e olu iona y dynamics model
(EDM). In EDMs, he agen s can be modeled as eal
numbe s, he mass o agen s, ins ead o being modeled as
disc e e independen en i ies. The e a e mul iple EDMs,
bu we ocus on a speci ic EDM known as BNN dynamics
B own and on Neumann (1951), which a e desc ibed nex
(see Ma inez-Piazuelo e al. (2022) o de ails).
Le us conside a socie y o agen s di ided in o N2
Z1disjoin popula ions indexed by P¼ 1;2; :::; Ng.
Each popula ion k2Pis comp ised o a cons an mass o
decision-making agen s mk2R[0. The se o s a egies o
each agen in popula ion k2Pis SkZ1wi h
2nk¼jSkj 1. The amoun o agen s selec ing s a egy
i2Ska popula ion kis deno ed as xk
i2R0. No ice ha
agen s om di e en popula ions k1and k2can selec he
same s a egy ii i2Sk1and i2Sk2. Simila ly, he p o-
po ion o agen s selec ing s a egy i2Ska popula ion kis
deno ed as zk
i¼xk
i=mk. Fu he mo e, xk¼ðxk
i1; :::; xk
inkÞ>
and zk¼ðzk
i1; :::; zk
inkÞ>deno e he s a egic dis ibu ions o
popula ion k,x¼ðx1>;x2>; :::; xN>Þ>, and
z¼ðz1>;z2>; :::; zN>Þ>.Le 2Z0be he disc e e- ime
index; x( ) he alue o xa ime ;z( ) he alue o za ime
and pk
ið Þ2R he payo ecei ed by he agen s selec ing
s a egy i2Ska popula ion k2P.
Following he e ision p o ocol in oduced in Ma inez-
Piazuelo e al. (2022), he equa ions ha de ine he EDM
desc ibing he e olu ion o x( ) o e ime a e:
_
xk
ið Þ¼mk½^
pk
ið Þþxk
ið ÞX
j2Sk½^
pk
jð Þþð1Þ
^
pk
jð Þ¼pk
jð Þ 1
mkX
l2Sk
xk
lð Þpk
lð Þð2Þ
whe e ½þ¼maxð;0Þ, and _
xdeno es he de i a i e o
x. This EDM is known as he BNN dynamics B own and
on Neumann (1951).
A memb ane compu ing app oach o he gene alized Nash equilib ium
123
In ui i ely, Eq. (2) compu es he bene i o ha ing mo e
agen s ollowing s a egy jin popula ion k. The eason is
ha Equa ion (2) compu es he di e ence be ween he
payo ob ained by agen s xk
jand he a e age payo
ob ained a ime s ep . A posi i e alue o ^
pk
jð Þwould
indica e ha i would be be e o popula ion k o ha e
agen s swi ch o s a egy j. Equa ion (1) can hen be used o
decide how many agen s should swi ch o o he s a egies
a he nex ime s ep, gi en by he de i a i es _
xk
ið Þ.
A payo dynamics model (PDM) ha desc ibes he
e olu ion o p( ) is also in oduced, de ined by:
_
lð Þ¼Axð Þbð3Þ
pð Þ¼ ðxð ÞÞA>lð Þð4Þ
whe e is a i ness unc ion ha p o ides he payo o he
s a egies chosen a a gi en popula ion s a e, l ep esen s
some cons ain s o e such decisions, and such cons ain s
a e gi en by a ma ix Aand a ec o b ha depends on he
speci ic p oblem. In his con ex , Aand bjus in oduce
penaliza ions o he payo , ins ead o in oducing ha d
cons ain s.
Since he impo ance o his sys em lies in upda ing he
payo signal p( ) and ha ing a closed-loop con igu a ion
be ween p( ) and x( ), a simpli ied e sion o his sys em,
whe e we emo e he cons ain s o e he s a egies cho-
sen, is conside ed:
pð Þ¼ ðxð ÞÞ ð5Þ
An EDM whose payo unc ion ollows a PDM is hen
called an EDM-PDM.
I is no ha d o modi y he P sys em p oposed la e in
Sec . 4.1 o compu e he e ec o he cons ain s in o-
duced by Aand bin Eq. (5). Because hey a e linea , he
ex a compu a ion ime equi ed is cons an pe i e a ion
. Howe e , because ou goal is o show how can P sys ems
be used o compu ing Gene alized Nash Equilib ia (GNE),
we limi ou sel es o he case wi hou cons ain s.
3.1 Ene gy ma ke game
In he p e ious sec ion, we exp essed a popula ion game
unde BNN dynamics h ough Eqs. (1)(2) and (5). To sol e
a speci ic popula ion game, we need o de ine he payo
unc ion (Eq. 5) conside ing a speci ic unc ion ha is
di e en o each game. Taking his in o accoun , a speci ic
EDM mus be selec ed as a amewo k o de ine ou P
sys em. Because o his, we conside an example o he
Ene gy Ma ke Game Ma inez-Piazuelo e al. (2022)as
he amewo k.
In he Ene gy Ma ke Game, N2Z1playe s compe e
o pu chase ene gy o e a ime ho izon o T2Z1 ime
slo s. Playe s who y o pu chase ene gy in he same ime
slo s will end up paying mo e o he ene gy, and he base
p ice o ene gy is highe o some ime slo s han o
o he s. The goal is o buy ene gy as cheaply as possible,
conside ing ha o he playe s ha e he same goal. Fo his
p oblem, we can conside each playe a popula ion k, and
he agen s xk
i ep esen he amoun o ene gy pu chased by
playe kin he ime slo i.
Following he no a ion desc ibed a he beginning o
Sec . 3, le Ck2RTnkbe a ma ix such ha each column
o Ckhas exac ly one elemen equal o 1 and he es equal
o 0, each ow o Ckhas a mos one elemen equal o 1,
and he j- h elemen o he i- h column o Ckis 1 i playe
kcompe es in ime slo jT. Le C¼½C1;C2;...;CN2
RTnbe he conca ena ion o he Ckma ices o all playe s,
whe e n¼Pk2Pnk. Then Cx co esponds o he collec i e
ene gy demand o all ime slo s. Le J:Rn!RTbe he
p icing unc ion gi en by JðxÞ¼DCx þ
J, whe e D2
RTT
0is diagonal and 
J2RT
0, and le Qk:Rnk
0!Rbe
he indi idual cos o each playe k2P, gi en by
QkðxkÞ¼P
i2Skðak
i=2Þðxk
iÞ2þbk
ixk
i, whe e ak
i2R0and
bk
i2R0.
Following he esul s om Ma inez-Piazuelo e al.
(2022), he payo unc ion pð Þ¼ ðxð ÞÞ o he Ene gy
Ma ke Game can be exp essed by ðxð ÞÞ ¼ Sxð Þ
C>
Jaxð Þbwhe e
•M¼diagðm1In1;m2In2;...;mNInNÞ,
•S¼diagðC1>DC1;...;CN>DCNÞþR>R, and
•R¼½ ffiffiffiffi
D
pC1;ffiffiffiffi
D
pC2;...;ffiffiffiffi
D
pCN.
•diag is he ope a ion ha cons uc s a ma ix using he
inpu elemen s as he diagonal, and whe e he es o he
elemen s a e null.
To de ine he payo o e zk
ið Þ, he ollowing ans o ma-
ion is pe o med o e Eq. (5):
pð Þ¼ ðxð ÞÞ ¼ ðMzð ÞÞ
¼SMzð ÞC>
JaðMzð ÞÞbð6Þ
Equa ions (1) and (2) also change o zk
ið Þ:
_zk
ið Þ¼_
xk
ið Þ
mk¼½^
pk
ið Þþzk
ið ÞX
j2Sk½^
pk
jð Þþð7Þ
^
pk
jð Þ¼pk
jð ÞX
l2Sk
zk
lð Þpk
lð Þð8Þ
A. Luque-Ce pa, M. Á. Gu ié ez-Na anjo
123
4 Design and unc ioning o he P sys em
In his sec ion, we in oduce i s he gene al idea behind
he design o a P sys em p oposed o compu e app oxi-
ma ions o GNE o popula ion games unde BNN
dynamics. Then, we de ine he P sys em in Sec . 4.1. A e
ha , we pe o m a compu a ion analysis in Sec . 4.2, whe e
we indica e he e olu ion ules de ined in Sec . 4.1 ha a e
applied o he con igu a ions. Finally, we include ou
expe imen al esul s in Sec . 4.3.
Le us conside he EDM-PDM sys em in oduced in
Sec . 3.1 by Eqs. (6), (7), and (8). In his sec ion, a P
sys em ha compu es app oxima ions o GNE unde he
BNN dynamics o his sys em is desc ibed. The compu-
a ion can be summa ized in a loop o i e s ages, ep e-
sen ed in Algo i hm 1. S ages 1 and 2 a e used o compu e
^
pð Þusing Eq. (8), S ages 3 and 4 a e used o compu e _zð Þ
using Eq. (7), and S age 5 is used o upda e he alue o
z( ). To sol e any o he EDM-PDM sys em, only he i s
s age o he P sys em has o be modi ied, while he es
emains unchanged.
The undamen al idea behind he sys em is o compu e
app oxima ions and disc e ize he alues in ol ed in he
EDM-PDM sys em by ounding o ndecimals and mul i-
plying by 10n. To show he unc ioning o ou P sys em, we
ix n¼2 om now on. Howe e , he sys em can easily be
modi ied o o he alues o n, p o iding be e app oxi-
ma ions wi h he cos o a longe un ime. A e dis-
c e izing, a P sys em can e ol e objec s ep esen ing z( ) o
compu e GNE. Fo n¼2, a single objec ha ep esen s
zk
ið Þ, ep esen s 1% o he agen s o popula ion k ha
ollow he s a egy i. Fo example, i we ha e 16 objec s
ha ep esen zk
ið Þ, hen zk
ið Þ¼0:16. Fo mally, o
app oxima e and disc e ize, we pe o m
oundðx;nÞ¼b10nxc. To ob ain he nex alue o he
a iables in he nex ins an þ s ep using he alues o
ins an , we use Eule ’s me hod Bu che (2016), his is,
zð þ s epÞ¼zð Þþ_zð Þ s ep.
In Algo i hm 1, o he s op condi ions can be easily
de ined, o example, compa ing he z( ) alues o one
i e a ion wi h hose o he p e ious one (in cons an ime)
and s opping i no di e ence is ound, bu mo e ules
would be necessa y. Fo he sake o simplici y, ou s op
condi ion is o limi he numbe o i e a ions in he loop.
Because pe o ming mul iplica ions using P sys ems is
no i ial, we de ine a P sys em ha eplica es he Russian
peasan mul iplica ion algo i hm Came on (1994)in
Appendix A. The eason o using his speci ic algo i hm is
ha he numbe o ime s eps equi ed o compu e a mul-
iplica ion is uppe bound by a cons an o all mul ipli-
ca ions o ou P sys em. We use his mul iplica ion P
sys em as a module o ou P sys em.
Algo i hm 1 Gene al o e iew o he P sys em compu a ion
4.1 De ini ion o he P sys em
The P sys em o compu e app oxima ions o GNE unde
he BNN dynamics is de ined as he cons uc :
P¼hC;H;l;ðwhÞh2H;ðR;qÞ;iou i
whe e he alphabe o objec s is gi en by:
C¼ hP od;k;i;li;hk;i;lijk2P;i2Sk;l¼X
j kjSjjþig
[ hP od2;k;i;li;ha;k;i;lijk2P;i2Sk;l¼X
j kjSjjþig
[ c; em;mul 0;mul 1;p od;p od0;e;e0;pos;q;s0;s1;zneg;z a p;z a ng
[ a;b;d;m; ;y0;p;n;compg[ plj1lX
k2PjSkjg
[ mi;ki;ai;bi; i;yij1i6g[ o e ;p1;e ; g
[ y0;0;y0;1;y0;2;y2;0;y2;1;y2;2;y3;0;y4;0;y4;1;y4;2;y4;3;y5;0g
[ y7;kjk2Pg[ hAUX;ni;hAUX1;ni;hCLK;nijn0g
[ y3;j;i;mul zi;0;mul zi;1jk2P;i2Sk;1j7g
[ Ck;hq;ii;di;qi;negi;posijk2P;i2Sk;1j7g
[ y0;k;y0;i;y5;j;y5;3;i;y5;11;ijk2P;i2Sk;1j10g
[ y5;12;k;wi;compwi;zi;jk2P;i2Sk;1j10g
[ EXITi;hEXIT;k;i;l;nijk2P;i2Sk;1j10;
n1g[ hINIT;k;i;lijk2P;i2Sk;1j10;n1g;
he se o memb ane labels is gi en by
H¼ 0g[P[ Si;kg8i2Sk8k2P[ RESi;kg8i2Sk8k2P
[ MULTi;kg8i2Sk8k2P[ M1;M2g
[ MULT2i;kg8i2Sk8k2P[ UPDi;kg8i2Sk8k2P
[ ACUMkg8k2P;
he memb ane s uc u e lis ep esen ed in Fig. 1, and is
de ined as ollows:
•Memb ane skin wi h label 0, inside o which we ind:
1. One memb ane wi h label P.
2. Nmemb anes wi h labels P. Inside o each
memb ane k2P:
2:1. Skmemb anes wi h labels Si;k8i2Sk. Inside
each Si;k:
A memb ane compu ing app oach o he gene alized Nash equilib ium
123

– One memb ane wi h label RESi;k
2:2. Skmemb anes wi h labels MULTi;k8i2Sk.
Inside each memb ane MULTi;k:
– One memb ane wi h label M1
– One memb ane wi h label M2
2:3. Skmemb anes wi h labels MULT2i;k8i2Sk.
Inside each memb ane MULT2i;k:
– One memb ane wi h label M10
– One memb ane wi h label M20
2:4. Skmemb anes wi h labels UPDi;k8i2Sk
2:5. One memb ane wi h label ACUMk
he ou pu egion iou is he skin (label 0);
he ini ial mul ise s a e wP¼y0,wSi;k¼hk;i;lizk
i8i2Sk
8k2Pwi h zk
i:¼b
100
nkc o 1 i max Skgand
zk
max Skg:¼100 ðjSkj1Þb100
nkc,wRESi;k¼hAUX;0i8i2
Sk8k2P, and o any o he memb ane m, he ini ial
mul ise is wm¼;;
and he se o ules Ris gi en by he ollowing ules,
sepa a ed by he co esponding s age o Algo i hm 1,
whe e k ep esen s he emp y mul ise , he ule RSm;n ep-
esen s he n- h ule o he m- h s age, he ules a e de ined
8k2P,8i2Sk, and l¼P
j kjSjjþi, and qm;n ep esen s
he p io i y o he ule RSm;n:
S age 1 (Compu a e payo p( ))
To use (z( )) in he P sys em, le j:¼b100 ðC>
J
bÞcbe he cons an pa o Eq. (6), and le aj;l:¼bðSMÞjlc
and bl:¼bðSMÞll þðaMÞlc o 1 j;lPk2PjSkjbe
he coe icien s ha will mul iply zk
iin Eq. 6. No ice ha
because aj;land bla e used o compu e p oduc s by mul-
iplying hem by zk
i, and zk
iis al eady mul iplied by 100,
nei he aj;lno bla e mul iplied by 100 when ounded.
The in ui ion behind he ules o his s age is ha objec s
hk;i;li ep esen zk
ið Þ, and hey will be used o compu e
plð Þby gene a ing objec s pl(see Eq. (6)).
Rules RS1;1,RS1;3, and RS1;4mo e objec s hk;i;li, which
ep esen zk
ið Þ, un il hey a e in memb ane P. A he same
ime, hey p oduce objec s ha will be used in la e s ages
(c,hP od;k;i;li), and objec s ha will be used o
Fig. 1 Memb ane s uc u e o
ou P sys em. All memb anes
s a wi h pola iza ion 0
A. Luque-Ce pa, M. Á. Gu ié ez-Na anjo
123
coo dina ion (Ck). Rule RS1;2gene a es objec s pl ep e-
sen ing he cons an pa o Eq. 6.
RS1;1½hk;i;li0
Si;k!½c0
Si;khk;i;li
RS1;2½y0!pj1
1pj2
2...pjn
n0
P
RS1;3½hk;i;li0
k!½hP od;k;i;li0
khk;i;liCk
RS1;4hk;i;li½ 0
P!½hk;i;li0
P
Rules RS1;5and RS1;6a e used o coo dina e he sys em.
Once he pola iza ion o he memb ane Pchanges o þ,
ule RS1;7is applied, and he objec s plgene a ed a e mixed
wi h he objec s pl ha we e al eady in memb ane P
because o ule RS1;2, compu ing indeed p( ) as exp essed
in Eq. 6.
RS1;5½C100
1C100
2...C100
N!y10
0
RS1;6y1½
0
P!½y2þ
P
RS1;7½hk;i;li!pa1;l
1pa2;l
2...pal1;l
l1pbl
lpalþ1;l
lþ1...pan;l
nþ
P
Rules RS1;8,RS1;9, and RS1;11 a e used o coo dina ing.
The coo dina ion is achie ed by changing he pola iza ion
o memb ane P. Rule RS1;10 ex ac s om Pobjec s pl, ha
now ep esen plð Þ¼pk
ið Þ, as objec s ha;k;i;li.
RS1;8½y2!y3þ
P
RS1;9½y3þ
P!½y4
P em
RS1;10 ½pl
P!½

Pha;k;i;li
RS1;11 ½y4!y5
P
Rule RS1;12 mo es each objec ha;k;i;li o i s co e-
sponding memb ane k. These objec s will la e be used in
S age 2. Rules RS1;13 o RS1;15 a e used o coo dina e he
beginning o S age 2. Rule RS1;16 is a cleaning ule used o
emo e objec s em ha a e now useless.
RS1;12 ha;k;i;li½ 0
k!½ha;k;i;li0
k
RS1;13 ½y5!y6
P
RS1;14 ½y6
P!½
0
Py7;1y7;2:::y7;N
RS1;15 y7;k½
0
k!½mul jSkj
0
k,8k2P.
RS1;16 ½ em !ks
m,8m2H,8s2 þ;;0g(Clean-
ing ule).
S age 2 (Compu e sums P
j2Sk
zk
jð Þpk
jð Þ)
To compu e he p oduc s o zk
jð Þand pk
jð Þ, we use he
memb anes MULTi;k, which wo k as mul iplica ion mod-
ules. These modules compu e he p oduc o wo numbe s
and e u n he esul in a maximum o 43 ansi ion s eps.
Each memb ane MULTi;kcon ains wo memb anes, M1
and M2. When objec s aa e placed in M1, objec s ba e
placed in MULTi;k, an objec k1is placed in M1, and he
pola iza ion o he h ee memb anes is 0, he mul iplica ion
module will compu e he p oduc o he numbe s ep e-
sen ed by objec s aand b. Memb anes MULTi;kexpel
objec s d, ep esen ing he p oduc , and an objec ep e-
sen ing ha he mul iplica ion p ocess is inished. Fo he
sake o simplici y, we le he de ails abou he mul ipli-
ca ion p ocess in Appendix A, including a compu a ion
analysis.
The in ui ion behind he ules o his s age is ha objec s
hP od;k;i;liwe e p oduced in S age 1 o ep esen zk
ið Þ.
These objec s a e ans o med in o objec s p od. Toge he
wi h objec s ha;k;i;li, ha ep esen pk
ið Þ, and he mul i-
plica ion modules MULTi;k, we ob ain he esul o mul i-
plying zk
ið Þby pk
ið Þ. The sum Pj2Skzk
jð Þpk
jð Þis ob ained
by g ouping all he esul ing objec s in he memb anes
ACUMi;k. The es o he ules a e o coo dina ion,
ounding, o p oducing objec s necessa y o la e s ages.
Rules RS2;1 o RS2;9a e used o coo dina e he gene a-
ion o objec s a,b, and k1in memb anes M1, MULTi;k, and
M1, espec i ely. Once his is done, he mul iplica ion
p ocess will begin, and he p oduc o zk
ið Þ( ep esen ed by
objec s a) and pk
ið Þ( ep esen ed by objec s b) will be
compu ed. Objec s hP od2;k;i;liand posia e also gene -
a ed o la e s ages.
RS2;1½hP od;k;i;li!p od hP od2;k;i;li
k
RS2;2½ha;k;i;li!e posi
k
RS2;3mul 0½
0
MULTi;k!½mul 0þ
MULTi;k
RS2;4p od½
þ
MULTi;k!½p odþ
MULTi;k
RS2;5e½
þ
MULTi;k!½eþ
MULTi;k
RS2;6½mul 0þ
MULTi;k!½k00
MULTi;k em
RS2;7p od !½a0
M1
RS2;8½e!b0
MULTi;k
RS2;9k0½
0
M1!½k10
M1
Rule RS2;10 is used o send objec s posi o memb anes
UPDi;k, which will be used in la e s ages.
RS2;10 posi½
0
UPDi;k!½posi0
UPDi;k
Rule RS2;12 akes he mul iplica ion esul s, ep esen ed
by objec s d, and sends hem o memb ane ACUMk ans-
o med in o objec s neg. When all mul iplica ions a e in-
ished, he sum Pj2Skzk
jð Þpk
jð Þis gi en inside ACUMk,
ep esen ed by objec s neg. Then, ule RS2;11 changes he
pola iza ion o ACUMk o indica e ha he sum is com-
pu ed. Because zk
jð Þand pk
jð Þwe e ounded by mul iplying
by 100, i is necessa y o use ules RS2;13,RS2;14, and RS2;15
o ound he p oduc again and ejec hem om memb ane
ACUMk.
A memb ane compu ing app oach o he gene alized Nash equilib ium
123
RS2;11  jSkj½
0
ACUMk!½y2;0þ
ACUMk
RS2;12 d½
0
ACUMk!½neg0
ACUMk
RS2;13 ½neg100þ
ACUMk!½
þ
ACUMkneg
RS2;14 ½neg51þ
ACUMk!½
þ
ACUMkneg, wi h q2;13 [q2;14.
RS2;15 ½neg !kþ
ACUMk, wi h q2;14 [q2;15.
Rules RS2;16,RS2;17, and RS2;18 coo dina e he beginning
o he nex s age.
RS2;16 ½y2;0!y2;1þ
ACUMk
RS2;17 ½y2;1þ
ACUMk!½
0
ACUMky2;2
RS2;18 ½y2;2
k!½y3;00
k em
S age 3 (Compu e ½^
pk
iþand P
j2Sk½^
pk
jþ)
The goal o his ules is, because o Eq. (8), o compu e
^
pk
iby compu ing he di e ence o wo numbe s: pk
ið Þand
Pl2Skzk
lð Þpk
lð ÞIn his s age, we compu e ½^
pk
iþand
Pj2Sk½^
pk
jþ.
F om now on, in his s age, le Sk¼ i1;...;ijSkjg.
Rules RS3;1and RS3;2a e used o coo dina ion.
RS3;1½y3;0!y3;1;i1... y3;1;ijSkj0
k
RS3;2y3;1;i½
0
UPDi;k!½ emþ
UPDi;ky3;2;i
Because we need o compu e ^
pk
ið Þ o each i, we use
ule RS3;3 o c ea e jSkjcopies o Pl2Skzk
lð Þpk
lð Þ, each
ep esen ed by negi. These copies a e hen sen in o
memb anes UPDi;kby using ule RS3;4.
RS3;3½neg !negi1... negijSkj0
k
RS3;4negi½
þ
UPDi;k!½negiþ
UPDi;k
Rules RS3;5and RS3;6a e used o coo dina e he com-
pu a ion o pk
ið ÞPl2Skzk
lð Þpk
lð Þby changing he
pola iza ion o memb anes UPDi;k o -.
RS3;5½y3;2;i!y3;3;i0
k
RS3;6½y3;3;i½
þ
UPDi;k!½y3;4;i
UPDi;k0
k
Rules RS3;7,RS3;8, and RS3;9compu e
pk
ið ÞPl2Skzk
lð Þpk
lð Þ. I he di e ence is posi i e,
objec s qi emain. O he wise, because we a e in e es ed in
using his di e ence o compu e ½^
pk
iþ, objec s negia e
elimina ed. ½^
pk
iþis hen ep esen ed by objec s qi.
RS3;7½negiposi!k
UPDi;k
RS3;8½negi!k
UPDi;k, wi h q3;7[q3;8.
RS3;9½posi!qi
UPDi;k, wi h q3;7[q3;9.
Rules RS3;10 and RS3;11 a e used o coo dina ion. Rule
RS3;12 gene a es objec s qand ejec s hem om memb anes
UPDi;k. By doing his, we compu e he sum Pj2Sk½^
pk
jþ,
while ½^
pk
iþis s ill ep esen ed by objec s qi. Rule RS3;13
coo dina es he beginning o he nex s age.
RS3;10 ½y3;4;i!y3;5;i
UPDi;k
RS3;11 ½y3;5;i
UPDi;k!½y3;6;i0
UPDi;k em
RS3;12 ½qi0
UPDi;k!½
0
UPDi;kqq
i
RS3;13 ½y3;6;i0
UPDi;k!½
0
UPDi;ky3;7;i
S age 4 (Compu e _zk
ið Þ)
Because o Eq. (7), o compu e _zk
ið Þwe need o compu e
i s zk
iPj2Sk½^
pk
jþ. F om S age 3, we ha e objec s q ha
ep esen Pj2Sk½^
pk
jþ, and om S age 2 we ha e objec s
hP od2;k;i;li ha ep esen zk
i. As in S age 2, we can use
memb anes MULT2i;kas mul iplica ion modules. These
modules wo k he same as MULTi;k om S age 2, com-
pu ing p oduc s in a maximum o 43 ansi ion s eps, and
wi h he only di e ence ha MULT2i;k e u ns objec s di
ins ead o dand objec s 1ins ead o (see Appendix A o
mo e de ails).
When he mul iplica ion p ocess is inished, we will
ha e objec s qi om S age 3 ha ep esen ½^
pk
iþ, and we
will ha e ob ained objec s di ha ep esen zk
iPj2Sk½^
pk
jþ.
We can hen compu e _zk
ið Þ(see Eq. (7)), esul ing in
objec s z a p i i is posi i e, o z a n i i is nega i e.
F om now on, in his s age, le Sk¼ i1;...;ijSkjg.
Rules om RS4;1 o RS4;12 coo dina e he beginning o
he mul iplica ion p ocess inside memb anes MULT2i;k,
mul iplying zk
i, ep esen ed by objec s hP od2;k;i;li, and
Pj2Sk½^
pk
jþ, ep esen ed by objec s q.
RS4;1½y3;7;i1...y3;7;ijSkj!mul zi1;0...mul zijSkj;00
k
RS4;2½mul zi;0!mul zi;10
k
RS4;3½q!hq;i1i...hq;ijSkji0
k
RS4;4hP od2;k;i;li½ 0
MULT2i;k!½p od0
MULT2i;k
RS4;5hq;ii½ 0
MULT2i;k!½e0
MULT2i;k
RS4;6mul zi;1½
0
MULT2i;k!½mul 0þ
MULT2i;k
RS4;7½p od !p od0þ
MULT2i;k
RS4;8½e!e0þ
MULT2i;k
RS4;9½mul 0þ
MULT2i;k!½mul 10
MULT2i;k em
RS4;10 p od0!½a0
M10
RS4;11 ½e0!b0
MULT2i;k
RS4;12 mul 1½
0
M10!½k10
M10
Rules RS4;13 and RS4;14 coo dina e he beginning o he
compu a ion o ½^
pk
ið Þþzk
ið ÞP
j2Sk½^
pk
jð Þþin memb anes
Si;k.
RS4;13  jSkj
1!yjSkj
4;0
A. Luque-Ce pa, M. Á. Gu ié ez-Na anjo
123
RS4;14 y4;0½
0
Si;k!½y4;1þ
Si;k
Rules RS4;15 o RS4;23 compu e such di e ence. I he
di e ence is posi i e (nega i e), hen objec s z a p (z a n)
a e p oduced.
RS4;15 qi½
þ
Si;k!½s0þ
Si;k
RS4;16 di½
þ
Si;k!½diþ
Si;k
RS4;17 ½s0!s1þ
Si;k
RS4;18 ½d100
i!znegþ
Si;k
RS4;19 ½d51
i!znegþ
Si;k, wi h q4;18 [q4;19.
RS4;20 ½di!kþ
Si;k, wi h q4;19 [q4;20.
RS4;21 ½s1zneg !kþ
Si;k
RS4;22 ½s1!z a pþ
Si;k, wi h q4;21 [q4;22.
RS4;23 ½zneg !z a nþ
Si;k, wi h q4;21 [q4;23.
Rules RS4;24,RS4;25, and RS4;26 a e jus o coo dina ion,
p epa ing he sys em o he nex and inal s age.
RS4;24 ½y4;1!y4;2þ
Si;k
RS4;25 ½y4;2!y4;3þ
Si;k
RS4;26 ½y4;3þ
Si;k!½y5;0
Si;k em
S age 5 (Upda e z( ) and ou pu esul s)
Since S age 1, we ha e objec s c ha ep esen zk
i. In his
s age, we combine hem wi h objec s z a p o z a n, ep-
esen ing _zð Þ, o upda e z( ) using Eule ’s me hod:
zð þ s epÞ¼zð Þþ s ep _zð Þ.
Rule RS5;1is used o coo dina ion. Because we ake
s ep ¼0:01, ules RS5;2 o RS5;10 a e used o compu e
zð Þþ s ep _zð Þ. Because no hing gua an ees ha he new
alues sa is y 0 zk
ið Þ100, he e a e some special cases
o conside . Mos o he ules om RS5;11 o RS5;38 a e used
o deal wi h hese cases, and he es a e o coo dina ion.
These cases a e u he de eloped in sec ion 4.2, S age 5.
RS5;1½y5;0!y5;1
Si;k
RS5;2½z a n100 c!k
Si;k
RS5;3½z a n51 c!k
Si;k, wi h q5;2[q5;3.
RS5;4½z a p100 !p
Si;k
RS5;5½c!p
Si;k,wi hq5;3[q5;5.
RS5;6½z a n100 !n
Si;k, wi h q5;3[q5;6.
RS5;7½z a n51 !n
Si;k, wi h q5;6[q5;7.
RS5;8½z a n !k
Si;k, wi h q5;7[q5;8.
RS5;9½z a p51 !p
Si;k, wi h q5;4[q5;9.
RS5;10 ½z a p !k
Si;k, wi h q5;9[q5;10.
RS5;11 ½y5;1!y5;2comp100
Si;k
RS5;12 ½p100
Si;k!½o e 
Si;kw100
i
RS5;13 ½y5;2
Si;k!½y5;30
Si;ky5;3;i
RS5;14 ½p0
Si;k!½
0
Si;kp
RS5;15 ½o e comp100 !k
Si;k
RS5;16 ½n0
Si;k!½
0
Si;kn
RS5;17 ½comp0
Si;k!½
0
Si;kcompwi
RS5;18 ½p comp !p1
Si;k, wi h q15 [q18.
RS5;19 ½p10
Si;k!½
0
Si;kwi
RS5;20 ½pn!k0
k
RS5;21 ½p compwi!wi0
k, wi h q20 [q21.
RS5;22 ½nw
i!compwi0
k, wi h q20 [q22.
RS5;23 ½p!e 0
k, wi h q21 [q23.
RS5;24 ½n!e 0
k, wi h q22 [q24.
RS5;25 ½y5;3!y5;40
Si;k
RS5;26 ½y5;4!y5;50
Si;k
RS5;27 ½y5;50
Si;k!½
þ
Si;ky5;6
RS5;28 ½y5;3;i1...y5;3;ijSkj!y5;40
k
RS5;29 ½y5;40
k!½y5;5 100þ
k em
RS5;30 ½wi !ziþ
k
RS5;31 ½wi!kþ
k, wi h q5;30 [q5;31.
RS5;32 ½compwi!kþ
k
RS5;33 ½ !ziþ
k, wi h q5;30 [q5;33.
RS5;34 ½y5;5þ
k!½
0
k em
RS5;35 zi½
þ
Si;k!½ziþ
Si;k
RS5;36 y5;6½
þ
Si;k!½y5;70
Si;k
RS5;37 y5;7½
0
RESi;k!½y5;8þ
RESi;k
RS5;38 zi½
þ
RESi;k!½EXITiþ
RESi;k
Rules om RS5;39 o RS5;47 a e used o coo dina ion and
o p epa ing he ou pu o he P sys em. Objec s hAUX;ni
a e used o coun how many i e a ions o Algo i hm 1 ha e
been comple ed. Objec s hEXIT;k;i;l;ni ep esen he
alues zk
ið Þa i e a ion n. Objec s hINIT;k;i;lia e used o
ese he P sys em, p epa ing S age 1 o a new i e a ion o
he algo i hm.
RS5;39 ½hAUX;ni!hCLK;nþ1i100hAUX1;nþ1iþ
RESi;k
RS5;40 ½y5;8þ
RESi;k!½y5;9
RESi;k em
RS5;41 ½EXITihCLK;ni
RESi;k!½

RESi;khEXIT;k;i;l;ni, wi h n1
RS5;42 ½hCLK;ni!k
RESi;k, wi h n1 and q5;41 [q5;42.
RS5;43 ½y5;9
RESi;k!½
0
RESi;ky5;10
RS5;44 ½hAUX1;ni!hAUX;ni0
RESi;k, wi h n1
A memb ane compu ing app oach o he gene alized Nash equilib ium
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