Fallahnejad, Reza; Moza a i, Mohammad Reza; Wanke, Pe e ; Tan, Yong
A icle
Nash Ba gaining Game enhanced Global Malmquis
P oduc i i y Index o C oss-P oduc i i y Index
Games
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Sugges ed Ci a ion: Fallahnejad, Reza; Moza a i, Mohammad Reza; Wanke, Pe e ; Tan, Yong (2024) :
Nash Ba gaining Game enhanced Global Malmquis P oduc i i y Index o C oss-P oduc i i y Index,
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Ci a ion: Fallahnejad, R.; Moza a i,
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Ba gaining Game Enhanced Global
Malmquis P oduc i i y Index o
C oss-P oduc i i y Index. Games
2024,15, 3. h ps://doi.o g/10.3390/
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games
A icle
Nash Ba gaining Game Enhanced Global Malmquis
P oduc i i y Index o C oss-P oduc i i y Index
Reza Fallahnejad 1,*, Mohammad Reza Moza a i 2, Pe e Fe nandes Wanke 3,* and Yong Tan 4
1
Depa men o Ma hema ics, Kho amabad B anch, Islamic Azad Uni e si y, Kho amabad 6817816645, I an
2Depa men o Ma hema ics, Shi az B anch, Islamic Azad Uni e si y, Shi az 71993-1, I an;
m [email p o ec ed]
3COPPEAD G adua e Business School, Fede al Uni e si y o Rio de Janei o, Rua Paschoal Lemme, 355,
Rio de Janei o 21941-901, B azil
4School o Managemen , Uni e si y o B ad o d, B ad o d BD7 1DP, UK; y. an9@b ad o d.ac.uk
*Co espondence: [email p o ec ed] (R.F.); [email p o ec ed] (P.F.W.)
Abs ac :
The Global Malmquis P oduc i i y Index (GMPI) s ands as an e olu ion o he Malmquis
P oduc i i y Index (MPI), emphasizing global echnology o inco po a e all- ime e sions o Decision-
Making Uni s (DMUs). This pape in oduces a no el app oach, in eg a ing he Nash Ba gaining
Game model wi h GMPI o es ablish a C oss-P oduc i i y Index. Ou p ima y objec i e is o de elop
a comp ehensi e amewo k u ilizing he Nash Ba gaining Game model o de i e equi able common
weigh s o di e en ime e sions o DMUs. These weigh s se e as a undamen al componen o
c oss-e alua ion based on GMPI, acili a ing a holis ic assessmen o DMU pe o mance o e a ying
ime pe iods. The p oposed index is designed wi h essen ial p ope ies: easibili y, non-a bi a iness
conce ning he base ime pe iod, echnological consis ency ac oss pe iods, and weigh uni o mi y o
GMPI calcula ions be ween wo- ime e sions o a uni . This esea ch amalgama es c oss-e alua ion
and global echnology while employing geome ic a e ages o de i e a conclusi e c oss-p oduc i i y
index. The co e mo i a ion behind his me hodology is o es ablish a eliable and ai means o
e alua ing DMU pe o mance, in eg a ing insigh s om Nash Ba gaining Game p inciples and GMPI.
This pape elucida es he a ionale behind me ging he Nash Ba gaining Game model wi h GMPI
and ou lines he objec i es o p o ide a comp ehensi e C oss-P oduc i i y Index, aiming o enhance
he obus ness and eliabili y o p oduc i i y assessmen s ac oss a ied ime ames.
Keywo ds:
da a en elopmen analysis; global Malmquis p oduc i i y index; common weigh s;
c oss-e alua ion; Nash ba gaining
1. In oduc ion
Da a En elopmen Analysis (DEA) has been in oduced as a ool o e alua ing he
pe o mance o homogeneous and simila Decision-Making Uni s (DMUs). Since hen,
nume ous heo e ical and p ac ical de elopmen s ha e been p esen ed o his ma hema -
ical p og amming-based echnique, and a ious opics ha e been discussed in i s scope.
One o hese opics in he con ex o e alua ing he pe o mance o DMUs is he de el-
opmen o pe o mance app aisal indexes o e ime, such as he Malmquis , Laspey es,
Paasche, Fishe , To nq is , and Hicks-Moo s een [
1
]. Among hese indexes, he Malmquis
P oduc i i y Index (MPI) has ecei ed conside able a en ion and has been he subjec o
nume ous s udies in his a ea.
The MPI, p oposed by Ca es e al. [
2
], is based on he i s -pe iod echnology o
pe o mance app aisal. By using he ixed pe iod as he undamen al echnology, he MPI
main ains ce ain desi able p ope ies such as consis ency in calcula ing he MPI due o
he same basis, ansi i i y, and ci cula i y. Howe e , he choice o he base ime pe iod is
a bi a y, and o he ime pe iods a e no aken in o conside a ion. To add ess his issue,
Games 2024,15, 3. h ps://doi.o g/10.3390/g15010003 h ps://www.mdpi.com/jou nal/games
Games 2024,15, 3 2 o 21
a ious sugges ions ha e been p oposed, such as using he geome ic mean o he MPIs
ob ained by conside ing bo h ime pe iods as he basis [3], and using global echnology.
The concep o global echnology as he benchma k echnology o e alua ing he
pe o mance o DMUs o e ime was i s added o he MPI concep s a DEA by [
4
]. The
use o his echnology as a se o all ime e sions o all DMUs led o he in oduc ion o he
Global Malmquis P oduc i i y Index (GMPI). This index has ea u es such as easibili y, in-
dependence o esul s om di e en echnologies, ci cula i y, and consis ency. Consis ency
means ha he basis o calcula ing he index is he same o di e en DMUs, which makes
he DMUs compa able based on he sco es. Bu he consis ency o [
4
]’s GMPI was only
linked o he echnology used. The au ho s o [
5
] poin ed ou a kind o inconsis ency in his
index, which is he possibili y o using di e en weigh s o calcula e he GMPIs o di e en
DMUs. Geome ically, no only di e en DMUs, bu e en di e en ime e sions o a DMU,
can use a di e en hype plane o he global echnology as he benchma k. To add ess
such inconsis encies, e . [
5
] p oposed a me hod in ol ing a Common Weigh (CW) o all
DMUs ac oss all imes, aiming o mi iga e dispa i ies a ising om a ied echnologies and
suppo hype planes. Howe e , his me hod showed limi a ions, lacking pee e alua ion
and being hea ily elian on weigh selec ion.
C oss-e alua ion can also be chosen as an in e media y solu ion o dynamic e alua-
ion o DMUs: be ween ull weigh eedom in [
4
], and ull weigh limi a ion o all DMUs
o a CW in [
5
]. Ding e al. [
6
] used he concep o c oss-e alua ion in calcula ing MPI.
Howe e , his applies exclusi ely o global echnology, and c oss-e alua ion is u ilized
solely o iden i ying e iciencies, no p oduc i i y indexes. In he con en ional MPI o -
mula, only C oss-E iciency (CE) sco es eplace he e iciency sco es o con en ional DEA
models. Addi ionally, he weigh s used in e alua ions a e no common, in oducing a
challenge o inconsis ency. In addi ion, he unc ion used o agg ega e pe o mance is o an
a i hme ic ype ha is no compa ible wi h MPI s udies in he use o geome ic unc ions.
Homayoni e al. [
7
] p oposed a c oss-p oduc i i y index o add ess he issues iden i ied
in [
6
], whe e he weigh s om sel -e alua ion and pee e alua ion we e used di ec ly in
he p oduc i i y index calcula ion. Howe e , he p oblem o inconsis ency s ill exis s, as
he e iciencies in hei MPI a io a e calcula ed based on di e en weigh s.
In his pape , we in oduce a me hodology o calcula ing GMPIs ha add esses he
limi a ions o p e ious app oaches. We conside he global echnology se , comp ised o
all DMUs in wo di e en ime pe iods, as he benchma k. To ensu e ai e alua ion and
accoun o DMUs’ p e e ed weigh s, we sugges using se e al CWs o sel and pee
e alua ion, ins ead o elying on jus one CW. Fo bo h ime e sions o a DMU, we ob ain
a CW and use i o calcula e he GMPI o ha DMU and o he s.
Due o he compe i i e na u e o he DEA e alua ion p ocess, we employ he Nash
ba gaining game model wi h sui able b eakdown poin s o es ablish ai CWs. This co-
ope a i e model is o mula ed o simul aneously maximize he u ili y o all pa icipan s.
Addi ionally, i esul s in a Pa e o-op imal solu ion, which mo i a es playe s o accep i .
We c ea e a c oss-e alua ion ma ix con aining alues o sel and pee p oduc i i y and
agg ega e he GMPIs using he geome ic mean o main ain he mul iplica i e s uc u e a
he agg ega e le el. This app oach o e comes he inconsis ency issue o p e ious me hods
and p o ides a mo e accep able e alua ion o DMUs. Fu he mo e, ou me hod accoun s
o DMUs’ p e e ences and ensu es ai ness in he e alua ion p ocess.
A e examining he li e a u e on CW and MPI in DEA, i is appa en ha no a emp
has been made o ob ain a CW o he ime e sions o a DMU. Fu he mo e, no s udy has
ye calcula ed a GMPI ma ix. A e iew o he li e a u e also sugges s ha game heo y has
no been used o gene a e p oduc i i y indexes in DEA.
The p oposed in eg a ed amewo k o GMPI, CW, Nash ba gaining, and c oss-
e alua ion allows he cons uc ed GMPI o possess ce ain p ope ies. The use o a CW in
calcula ing he e iciencies o ime e sions o a DMU ensu es ha he GMPI ac ion is well
de ined and consis en ac oss all DMUs. The use o global echnology ensu es ha he mod-
els used o es ima e e iciencies a e always easible [
4
]. Addi ionally, bo h sel -e alua ion
Games 2024,15, 3 3 o 21
and pee e alua ion a e conside ed, esul ing in an agg ega ed GMPI ha inhe i s he
desi ed c oss-e alua ion ea u es. This includes educing he dependence o esul s on a
pa icula weigh , inc easing s abili y and eali y, a oiding o e es ima ion, being ai due
o he use o desi ed weigh s o each uni , and e lec ing eali y mo e accu a ely.
The pape is s uc u ed as ollows. The second sec ion o e s an o e iew o essen ial
backg ound in o ma ion. In he hi d sec ion, we in oduce he p oposed me hod along
wi h i s p ope ies, and he ou h sec ion p esen s an examina ion o he me hod h ough a
nume ical example. Finally, he las sec ion p esen s ou conclusions and sugges ions o
u u e esea ch.
2. P elimina ies
In his sec ion, we p o ide a b ie o e iew o some undamen al concep s and p inci-
ples o DEA and game heo y ha a e necessa y o unde s anding he p oposed me hod.
Fo ease o e e ence, all he symbols and abb e ia ions used h oughou he pape a e
compiled in Table 1.
Table 1. The abb e ia ions and nomencla u es used in his pape .
Nomencla u es
j = 1,. . ., n Se o Obse ed DMUs o The index o he unde e alua ion DMUo.
i = 1,. . ., m Se o inpu s Eoo CCR E iciency o DMUo
=1,. . ., s Se o ou pu s Ejo C oss-e iciency alues o DMUo u ilizing weigh s o DMUj
XjInpu ec o o DMUj Ek,G
oE iciency o DMUo in ime k ela i e o he global on ie
Yjou pu ec o o DMUj uk,G
j
Ou pu weigh ec o s o DMUj a ime k in ela ion o he
global on ie
U Vec o s o ou pu weigh s k,G
j
Inpu weigh ec o s o DMUj a ime k in ela ion o he
global on ie
V Vec o s o inpu weigh s Ek,G
j,o
C oss-e iciency o DMUo a ime k using op imal weigh s o
DMUj a ime k wi h espec o he global on ie .
Abb e ia ions
DEA Da a En elopmen Analysis CEM C oss-E iciency Ma ix
DMU Decision Making Uni GMPI Global Malmquis P oduc i i y Index
CE C oss-e iciency SGMPI
Sel -E alua ion Nash CW Global Malmquis P oduc i i y Index
MPI
Malmquis P oduc i i y Index
PGMPI Pee E alua ion Nash CW Global Malmquis
P oduc i i y Index
CW Common Weigh CGMPI C oss E alua ion Nash Common Weigh Global Malmquis
P oduc i i y Index
2.1. C oss-E iciency
The c oss-e alua ion me hod, in oduced by [
8
,
9
], u ilizes pee e alua ion in addi ion
o sel -e alua ion, es ablishing a ious se s o inpu and ou pu weigh s o each DMU
and compu ing i s e iciencies based on all hese weigh combina ions. Consequen ly,
each DMU will ob ain mul iple dis inc e iciency sco es, and he a e age po ays he
comp ehensi e pe o mance o he DMU. DMUs can hen be compa ed and anked based
on a e age c oss-e iciencies, ha exhibi obus disc imina ion capabili ies and adhe e o
easonable logic. The DEA li e a u e has ex ensi ely s udied c oss-e iciency, wi h ecen
s udies including [10–17].
Employing he con en ional no a ion in DEA, we conside a se o n DMUs o be
e alua ed based on m inpu s and s ou pu s. Each DMU is deno ed as DMUj = (X
j
, Y
j
)
(j = 1,
. . .
, n), whe e X
j
= (x
1j
,
. . .
, x
mj
) and Y
j
= (y
1j
,
. . .
, y
sj
) ep esen he inpu and ou pu
ec o s o DMUj, espec i ely.
The e iciency a ing o any gi en DMUo can be compu ed using he ollowing a io
o m o he CCR model [18]:
Eo=Maxuo·yo
o·xo
S. .
Games 2024,15, 3 4 o 21
uo·yj
o·xj
≤1, j =1, . . . , n,
o≥0,uo≥0 (1)
u
o
and
o
ep esen he ec o s o ou pu and inpu weigh s o DMU
o
, espec i ely. This
p oblem is sol ed by sequen ially e alua ing he e iciency o each DMU, al e ing i as he
uni unde e alua ion, esul ing in n e alua ions o de e mine he ela i e e iciency o all
DMUs. The e iciency sco e anges om 0 o 1, wi h DMUs sco ing 1 conside ed e icien
and hose sco ing less han 1 conside ed ine icien .
When he mos p e e ed weigh s ob ained by model (1) o a gi en DMUj ( a ing
DMU) a e employed o compa e he e iciency sco e o ano he DMUo ( a ed), he so-called
c oss-e iciency alue is:
Ejo =uo·yj
o·xj
(2)
The esul ing c oss alues be ween each DMU can be collec ed and a anged in a
C oss-E iciency Ma ix (CEM), as shown in Table 2. The en ies in column ‘o’, which
co espond o he a ed DMUo, ep esen he e alua ions ob ained by using he p e e ed
weigh s o all n DMUs. These e alua ions include he sel -e alua ion (based on i s own
weigh s) and he e alua ions based on he weigh s o n
−
1 o he DMUs (pee -e alua ion).
On he o he hand, he en ies in ow j ep esen he e alua ions o all DMUs based on he
p e e ed weigh s o a ing DMUj. The e iciency alues e alua ed o each DMU wi h
espec o i sel a e posi ioned on he main diagonal o he CEM.
Table 2. C oss-E iciency Ma ix.
Ra ing DMU Ra ed DMUs
1. . . o. . . n
1 E11 . . . E1o . . . E1n
. . . . . . . . . . . .
j Ej1 . . . Ejo . . . Ejn
. . . . . . . . . . . .
n En1 . . . Eno . . . Enn
The c oss-e iciency sco e o DMU0 is ypically compu ed as he a e age o i s c oss-
e iciencies using (3):
CEo=1
n
n
∑
j=1
Ejo (3)
2.2. Malmquis P oduc i i y Index
The Malmquis P oduc i i y Index (MPI) is a widely used ool in p oduc i i y analysis,
pa icula ly in DEA. The MPI e alua es and compa es p oduc i i y changes ac oss DMUs
o e ime. Essen ially, he MPI gauges p oduc i i y changes by examining shi s in he
e iciency on ie o bes p ac ice echnology ac oss di e en ime pe iods. I allows
analys s o measu e changes in p oduc i i y by conside ing bo h echnological p og ess
and e iciency imp o emen s. This me hod compa es he e iciency o DMUs be ween
di e en ime pe iods, e lec ing al e a ions in echnology o e iciency enhancemen s
wi hin DMUs.
Conside he pe o mance e alua ion and p oduc i i y change o n DMUs a wo
imes, and + 1. Le
E ,
o
and
E , +1
o
ep esen he e iciencies o uni o a ime ela i e o
he echnology on ie o ime and + 1, espec i ely. Simila ly,
E +1,
o
and
E +1, +1
o
deno e
Games 2024,15, 3 5 o 21
he e iciencies o his uni a ime + 1 ela i e o he echnological bounda ies o ime
and + 1. The e iciencies a e compu ed using model (1). The MPI is de ined as ollows:
MPIo="E +1,
o
E ,
o
E +1, +1
o
E , +1
o#1
2
(4)
Less, mo e, and equal MPI alues o 1 indica e eg ess, p og ess, and ze o p oduc i i y
change o e ime and + 1, espec i ely.
The MPI has unde gone nume ous heo e ical ad ancemen s since i s incep ion, all
gea ed owa d e ining i s applica ion and augmen ing i s analy ical p owess wi hin he
ealm o p oduc i i y analysis. Table 3encapsula es a selec ion o hese heo e ical ex en-
sions o he MPI.
Table 3. Theo e ical ex ensions o MPI.
Topic Re e ences
1 Cos ype MPI [19–24]
2 Global/biennial/o e all MPI [4,5,20,25–28]
3 Ne wo k [29–32]
4 Combina ion o MPI wi h o he Indexes and Indica o s [33–38]
5 Dis ance unc ion/Di ec ional p oduc i i y [39–43]
6Agg ega ion/disagg ega ion/ decomposi ion/Cen alized
scena io/me ge s/me a- on ie [44–58]
7 O he s [59–72]
2.3. Game Theo y
Game heo y is a ma hema ical amewo k ha helps analyze decision making and
s a egic in e ac ions be ween di e en indi iduals o g oups. Coope a i e game heo y, a
subse o game heo y, cen e s on scena ios in which playe s can uni e and collabo a e o
a ain hei goals. In coope a i e games, playe s can ag ee on how o dis ibu e he payo
among hemsel es, which is he essence o he ba gaining me hod. The ba gaining me hod
is a coope a i e game heo y solu ion concep ha p o ides a way o alloca e he gains
om coope a ion among he playe s.
The ba gaining me hod assumes ha playe s nego ia e a coope a i e solu ion ha
maximizes he sum o hei payo s. The solu ion mus be easible, meaning ha i
should lie wi hin he playe s’ nego ia ion ange. The nego ia ion ange is de e mined
by he playe s’ ese a ion alues, which a e he minimum payo s hey a e willing o
accep . The ba gaining me hod p o ides di e en solu ions depending on he nego ia ion
p ocess, such as he Nash ba gaining solu ion, he Kalai-Smo odinsky solu ion, and he
egali a ian solu ion.
Assuming n playe s, deno ed as {1,
. . .
, n}, engage in a ba gaining scena io, whe e he
payo ec o is ep esen ed as an elemen in R
n
. He e, R
n
signi ies he easible se and se es
as he b eakdown poin . Acco ding o [
73
], a easonable solu ion in ba gaining should
adhe e o undamen al p ope ies in ba gaining heo y. These p ope ies encompass: Pa e o
e iciency, ensu ing ha no pa icipan can be made be e o wi hou making ano he
pa icipan wo se o ; in a iance conce ning a ine ans o ma ions, implying he solu ion
emains unchanged despi e linea ans o ma ions; independence o i ele an al e na i es,
asse ing ha he chosen solu ion emains consis en i espec i e o i ele an op ions;
and symme y, signi ying he equi able ea men o all pa icipan s. The au ho s o [
73
]
demons a ed he exis ence o a unique solu ion sa is ying hese ou p ope ies, e e ed o
as he Nash solu ion. The Nash solu ion can be de i ed by maximizing
∏n
i=1(ui−bi)
whe e
u
i
and b
i
a e he elemen s o u and b, espec i ely. This in ol es selec ing a ec o om he
easible solu ions o he game ha maximizes he p oduc o he di e ences be ween each
playe ’s desi ed u ili y and hei espec i e b eakdown poin s, wi h he less desi able poin
se ing as he b eakdown.
Games 2024,15, 3 6 o 21
The combina ion o game heo y and DEA has been explo ed in a ious s udies, which
aim o e alua e he e iciency o decision-making uni s. The use o game heo y models
can help cap u e he s a egic in e ac ions be ween he uni s, while DEA can measu e hei
e iciency. Table 4summa izes he indings o some o hese s udies and hei con ibu ions
o he ield. O e all, he combina ion o game heo y and DEA can p o ide a comp ehensi e
amewo k o analyze decision-making p ocesses in complex and dynamic en i onmen s.
Table 4. A li e a u e e iew on DEA-Game.
Subjec A ea o DEA Game Re e ences
1
Connec ing e iciency games o a ious e sions o DEA models
[74–77]
2
B eaking down he o e all sys em e iciency in a mul is age and
pa allel-p ocesses ne wo k o calcula ing he e iciency o
p ocesses in a ne wo k
[78–85]
3E alua ing e iciency when mul iple g oups o inpu s o
di e en pe spec i es a e aken in o conside a ion [86–88]
4
Reaching consensus among indi iduals o o ganiza ions
employing mul iple c i e ia o pe o mance e alua ion, and
add essing he Egois ’s dilemma
[89–96]
5 Technology sha ing and esou ce pooling [97–101]
6Es ablishing a uni ied se o weigh s o e alua ing he
e iciency o DMUs [102–105]
3. C oss Common Weigh s Global Malmquis P oduc i i y Index Based on
Ba gaining Games
3.1. Mo i a ion h ough a Nume ical Example
In his sec ion, ou ocus is on delinea ing he mo i a ion behind he p oposed me hod
by u ilizing a hypo he ical example. Ou aim is o explica e he me hodology o compu ing
he GMPI h ough he lens o a s aigh o wa d nume ical illus a ion. To emba k on his,
we will del e in o a da ase encompassing 8 DMUs, each cha ac e ized by wo inpu s and
one ou pu , assessed a wo dis inc ime poin s: and + 1. A ime , hese DMUs a e
deno ed as A h ough G, while a ime + 1, hey a e ep esen ed as A’ h ough G’. The
pe inen da a conce ning hese DMUs is delinea ed in Table 5, laying he ounda ion o
ou illus a i e example.
Table 5. Da a o nume ical example.
Time Time + 1
DMU X1X2YDMU X1X2Y
A 0.5 2.5 1 A’ 0.25 2.25 1
B 0.75 1.25 1 B’ 1 2 1
C 1.5 2 1 C’ 0.75 2.25 1
D 1.5 0.75 1 D’ 1.5 0.5 1
E 2 0.75 1 E’ 2 0.25 1
F 3.25 0.75 1 F’ 2.5 1 1
G 2.5 1.5 1 G’ 2.5 2 1
H 2 2.5 1 H’ 3 2.25 1
The echnology o he 8 DMUs in wo ime pe iods and wo-dimensional space is
p esen ed in Figu e 1. The cons an e u n o scale echnology is depic ed wi h he bold
(black) bo de , which ep esen s he - ime echnology on ie . DMUs A, B, D, E, and F a e
loca ed on his on ie . The ain (o ange) on ie wi h poin s A’, D’, and E’ on i ep esen s
he echnology bounda y o ime + 1. Di ec ional ec o s a e included in he igu e o
indica e he change in s a e o he DMUs om ime o + 1.
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Games 2024, 15, x FOR PEER REVIEW 7 o 23
B 0.75 1.25 1 B’ 1 2 1
C 1.5 2 1 C’ 0.75 2.25 1
D 1.5 0.75 1 D’ 1.5 0.5 1
E 2 0.75 1 E’ 2 0.25 1
F 3.25 0.75 1 F’ 2.5 1 1
G 2.5 1.5 1 G’ 2.5 2 1
H 2 2.5 1 H’ 3 2.25 1
The echnology o he 8 DMUs in wo ime pe iods and wo-dimensional space is
p esen ed in Figu e 1. The cons an e u n o scale echnology is depic ed wi h he bold
(black) bo de , which ep esen s he - ime echnology on ie . DMUs A, B, D, E, and F
a e loca ed on his on ie . The ain (o ange) on ie wi h poin s A’, D’, and E’ on i
ep esen s he echnology bounda y o ime + 1. Di ec ional ec o s a e included in he
igu e o indica e he change in s a e o he DMUs om ime o + 1.
To gauge he pe o mance change o uni C, adi ionally, analys s employ he geo-
me ic mean o efficiency be ween C’ and C a bo h ime poin s, and + 1, wi hin he
amewo k o he MPI. The MPI se es as a cus oma y measu e o assessing p oduc i i y
al e a ions o e ime. In his con ex , efficiencies a e compu ed by measu ing he adial
dis ance be ween he unde -e alua ed DMU and i s adial p ojec ion on o he on ie s, o
he o igin. In Figu e 1, he c oss signs ma k he p ojec ions o poin C and C’ on he on-
ie s o imes and + 1. The p ojec ion poin s on he ime on ie lie on he line segmen s
AB and BD, while he p ojec ion poin s on he ime + 1 on ie lie on he segmen A’D’.
The e a e ou p ojec ion poin s and h ee diffe en aces, which means ha he pe o -
mance o uni C is calcula ed based on he dis ance o diffe en bases o diffe en suppo -
ing hype -planes o he wo- ime echnologies. This leads o he use o diffe en no mal
ec o s and diffe en inpu and ou pu weigh s o calcula ing he co esponding pe o -
mance o a DMU, esul ing in inconsis ency in e alua ing he pe o mance o he unde -
e alua ed uni .
Figu e 1. Two- ime echnology compa ison o nume ical example DMUs.
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5
A
A'
G
F
E
D
H
C
E'
F'
D'
B'
H'
G'
C'
B
X2
X1
Figu e 1. Two- ime echnology compa ison o nume ical example DMUs.
To gauge he pe o mance change o uni C, adi ionally, analys s employ he geo-
me ic mean o e iciency be ween C’ and C a bo h ime poin s, and + 1, wi hin he
amewo k o he MPI. The MPI se es as a cus oma y measu e o assessing p oduc i i y
al e a ions o e ime. In his con ex , e iciencies a e compu ed by measu ing he adial
dis ance be ween he unde -e alua ed DMU and i s adial p ojec ion on o he on ie s,
o he o igin. In Figu e 1, he c oss signs ma k he p ojec ions o poin C and C’ on he
on ie s o imes and + 1. The p ojec ion poin s on he ime on ie lie on he line
segmen s AB and BD, while he p ojec ion poin s on he ime + 1 on ie lie on he
segmen A’D’. The e a e ou p ojec ion poin s and h ee di e en aces, which means ha
he pe o mance o uni C is calcula ed based on he dis ance o di e en bases o di e en
suppo ing hype -planes o he wo- ime echnologies. This leads o he use o di e en
no mal ec o s and di e en inpu and ou pu weigh s o calcula ing he co esponding
pe o mance o a DMU, esul ing in inconsis ency in e alua ing he pe o mance o he
unde -e alua ed uni .
To o e come inconsis encies esul ing om he use o di e en echnologies and ime
bases, Pas o and Lo ell [
4
] p oposed he use o a global echnology ha includes all DMUs
om bo h ime pe iods. Figu e 2displays he global echnology and i s co esponding
global on ie , whe e e icien DMUs A’, D’, and E’ a e o ime + 1, and e icien uni B
is o ime . To calcula e he e iciencies used in MPI, we now examine he p ojec ions o
poin s C and C’ on his global on ie , esul ing in only wo p ojec ion poin s: one on he
A’B line segmen , and he o he on he BD line segmen . Al hough he numbe o bases has
been educed, he poin o conce n is ha he e iciencies o C and C’ a e s ill calcula ed
ela i e o di e en line segmen s and, he e o e, di e en bases.
Due o he inconsis encies highligh ed, [
5
] p oposed a me hod in ol ing a CW applied
uni o mly ac oss all DMUs and ime pe iods. This aimed o mi iga e dispa i ies a ising
om dispa a e echnologies and suppo ing hype planes. Howe e , his app oach lacked
he capaci y o pee e alua ion and showed a high eliance on he weigh selec ion,
posing limi a ions.
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Games 2024, 15, x FOR PEER REVIEW 8 o 23
To o e come inconsis encies esul ing om he use o diffe en echnologies and ime
bases, Pas o and Lo ell [4] p oposed he use o a global echnology ha includes all
DMUs om bo h ime pe iods. Figu e 2 displays he global echnology and i s co e-
sponding global on ie , whe e efficien DMUs A’, D’, and E’ a e o ime + 1, and effi-
cien uni B is o ime . To calcula e he efficiencies used in MPI, we now examine he
p ojec ions o poin s C and C’ on his global on ie , esul ing in only wo p ojec ion
poin s: one on he A’B line segmen , and he o he on he BD line segmen . Al hough he
numbe o bases has been educed, he poin o conce n is ha he efficiencies o C and C’
a e s ill calcula ed ela i e o diffe en line segmen s and, he e o e, diffe en bases.
Due o he inconsis encies highligh ed, [5] p oposed a me hod in ol ing a CW ap-
plied uni o mly ac oss all DMUs and ime pe iods. This aimed o mi iga e dispa i ies a is-
ing om dispa a e echnologies and suppo ing hype planes. Howe e , his app oach
lacked he capaci y o pee e alua ion and showed a high eliance on he weigh selec ion,
posing limi a ions.
In he upcoming sec ion, we p opose an al e na i e me hod cen e ed on employing
a global on ie , ensu ing consis ency in he unde lying echnology o efficiency calcu-
la ions. By using a sha ed weigh ac oss wo e sions o a DMU, ou app oach main ains
uni o mi y in efficiency calcula ions, le e aging he no mal ec o o a hype plane de i ed
om he global on ie . This ob ia es inconsis encies seen in p e ious me hods. Fu he -
mo e, he common weigh we in oduce adhe es o he Pa e o p ope y, acili a ing c oss-
compa isons among DMUs, hus enabling pee e alua ion—a ea u e lacking in he [5]
me hod. Unlike [5], ou app oach’s analy ical ou pu is no elian on a singula weigh ,
enhancing con idence in he e alua ion while emb acing addi ional a o able cha ac e is-
ics.
Figu e 2. Global echnology o nume ical example da a.
3.2. Me hod Desc ip ion
In his sec ion, we ou line he s eps o ou p oposed me hod o calcula e a C oss
Global Malmquis p oduc i i y index. We will conside a se o 10 DMUs, aiming o ana-
lyze how hei p oduc i i y changes ac oss diffe en ime pe iods.
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5
A
A'
G
F
E
D
H
C
E'
F'
D'
B'
H'
G'
C'
B
X2
X1
Figu e 2. Global echnology o nume ical example da a.
In he upcoming sec ion, we p opose an al e na i e me hod cen e ed on employing a
global on ie , ensu ing consis ency in he unde lying echnology o e iciency calcula ions.
By using a sha ed weigh ac oss wo e sions o a DMU, ou app oach main ains uni o mi y
in e iciency calcula ions, le e aging he no mal ec o o a hype plane de i ed om he
global on ie . This ob ia es inconsis encies seen in p e ious me hods. Fu he mo e, he
common weigh we in oduce adhe es o he Pa e o p ope y, acili a ing c oss-compa isons
among DMUs, hus enabling pee e alua ion—a ea u e lacking in he [
5
] me hod. Un-
like [
5
], ou app oach’s analy ical ou pu is no elian on a singula weigh , enhancing
con idence in he e alua ion while emb acing addi ional a o able cha ac e is ics.
3.2. Me hod Desc ip ion
In his sec ion, we ou line he s eps o ou p oposed me hod o calcula e a C oss Global
Malmquis p oduc i i y index. We will conside a se o 10 DMUs, aiming o analyze how
hei p oduc i i y changes ac oss di e en ime pe iods.
Fo he i s ime, he global echnology and i s bounda y, i.e., he global on ie , was
used by [
4
] o calcula e GMPI. Following [
4
], we conside a global echnology cons uc ed
om da a encompassing all DMUs o e wo dis inc ime pe iods (all pe iods in he case
in which we may ha e mo e pe iods). The e iciency o
DMUk
o=(xk
o, yk
o
o k = , + 1
ela i e o he global on ie , can be ob ained using model (5).
Ek,G
o=Maxuk,G
oyk
o
k,G
oxk
o
s. uk,G
oyk
j
k,G
oxk
j
≤1, j =1,. . . , n, k = , +1
uk,G
o, k,G
o≥0, k = , +1 (5)
Games 2024,15, 3 15 o 21
Now we conside he CWGMPI model. I we use his model o e alua e he 8 DMUs
o he nume ical example, only one common weigh ec o is gene a ed o all uni s, and
all he e iciencies used o calcula e he CWGMPI a e based on his one ec o . In Table 11,
GMPI alues based on his ec o o uni s a e gi en. As you can see, no only is his
index no based on pee e alua ion, bu i is no e en based on sel -e alua ion, and all
calcula ions and ankings depend on his common weigh .
Table 11. The op imal weigh s and alue o CWGMPI.
V1*V2*U*
CW 0.190853 0.187448 0.37745
DMUs CWGMPI DMUs CWGMPI
A,A’ 1.101632 E,E’ 1.218691
B,B’ 0.667168 F,F’ 1.14487
C,C’ 1.170434 G,G’ 0.889999
D,D’ 1.12332 H,H’ 0.855186
Now le us e iew he CGMPI alues ob ained by he p oposed me hod which can be
seen in Table 9. Fo a be e isual unde s anding, we use a ada cha (Figu e 3).
Games 2024, 15, x FOR PEER REVIEW 17 o 23
Figu e 3. C oss-GMPI ma ix ada diag am o he p oposed me hod o he nume ical example.
In a ada cha , he diffe en le els, o concen ic polygons ( om inside o ou side),
ep esen he alues o scales o he dimensions being measu ed. The dis ance o a poin
om he cen e o he cha o a speci ic le el on each axis co esponds o he alue o he
a iable o ha pa icula en i y. The in e p e a ion o le els in he ada cha based on
he da a p o ided enables a quick assessmen o he ela i e pe o mance o DMUs (A o
H) ac oss mul iple weigh s (Common weigh s be ween DMUs A,A’ o H,H’ in Table 7).
DMUs A, B, G, and H seem o ha e nea ly iden ical pe o mance ac oss all weigh s. They
main ain consis en alues ac oss he ada cha , indica ing simila s eng hs and weak-
nesses in he measu ed weigh s. DMUs C, D, E, and F exhibi mo e a ia ion in hei pe -
o mance ac oss diffe en weigh s. The hi d numbe s in he columns o Table 8 a e he
e ices o he oc agons o Figu e 3. When one le el is inside ano he le el, i sugges s ha
he DMU co esponding o he ou e le el pe o ms be e compa ed o he DMU associ-
a ed wi h he inne le el. B ie ly, we no ice ha DMU B is a he inne mos le el o he
g aph, and his means ha his uni undoub edly has he leas change in pe o mance
compa ed o o he uni s. A e ha uni , G and H uni s a e loca ed. On he o he hand,
uni s E and C a e he la ges oc agons, i.e., polygons wi h he g ea es dis ance om he
cen e o he ada . These esul s a e consis en wi h he CGMPI alues ob ained by he
p oposed me hod in he las ow o Table 8. The e o e, i can be in e ed ha he conclu-
sion ega ding he s a e o change in he pe o mance o he uni s is based on an a e age
alue as ep esen ed by an oc agon. I is ob ious ha he in e ence and analysis o he
si ua ion based on such a p ocess is much mo e eliable han he analysis based on only
one o i s alues.
The p oposed in eg a ed amewo k o GMPI, CW, Nash ba gaining, and c oss-e al-
ua ion allows he cons uc ed GMPI o possess ce ain p ope ies. The use o a CW in
0
0.2
0.4
0.6
0.8
1
1.2
1.4
A,A'
B,B'
C,C'
D,D'
E,E'
F,F'
G,G'
H,H'
CGMPI
A
B
C
D
E
F
G
H
Figu e 3. C oss-GMPI ma ix ada diag am o he p oposed me hod o he nume ical example.
In a ada cha , he di e en le els, o concen ic polygons ( om inside o ou side),
ep esen he alues o scales o he dimensions being measu ed. The dis ance o a poin
om he cen e o he cha o a speci ic le el on each axis co esponds o he alue o he
a iable o ha pa icula en i y. The in e p e a ion o le els in he ada cha based on
Games 2024,15, 3 16 o 21
he da a p o ided enables a quick assessmen o he ela i e pe o mance o DMUs (A o H)
ac oss mul iple weigh s (Common weigh s be ween DMUs A,A’ o H,H’ in Table 7). DMUs
A, B, G, and H seem o ha e nea ly iden ical pe o mance ac oss all weigh s. They main ain
consis en alues ac oss he ada cha , indica ing simila s eng hs and weaknesses in
he measu ed weigh s. DMUs C, D, E, and F exhibi mo e a ia ion in hei pe o mance
ac oss di e en weigh s. The hi d numbe s in he columns o Table 8a e he e ices o
he oc agons o Figu e 3. When one le el is inside ano he le el, i sugges s ha he DMU
co esponding o he ou e le el pe o ms be e compa ed o he DMU associa ed wi h
he inne le el. B ie ly, we no ice ha DMU B is a he inne mos le el o he g aph, and
his means ha his uni undoub edly has he leas change in pe o mance compa ed o
o he uni s. A e ha uni , G and H uni s a e loca ed. On he o he hand, uni s E and C a e
he la ges oc agons, i.e., polygons wi h he g ea es dis ance om he cen e o he ada .
These esul s a e consis en wi h he CGMPI alues ob ained by he p oposed me hod in
he las ow o Table 8. The e o e, i can be in e ed ha he conclusion ega ding he s a e
o change in he pe o mance o he uni s is based on an a e age alue as ep esen ed by
an oc agon. I is ob ious ha he in e ence and analysis o he si ua ion based on such a
p ocess is much mo e eliable han he analysis based on only one o i s alues.
The p oposed in eg a ed amewo k o GMPI, CW, Nash ba gaining, and c oss-
e alua ion allows he cons uc ed GMPI o possess ce ain p ope ies. The use o a CW in
calcula ing he e iciencies o ime e sions o a DMU ensu es ha he GMPI ac ion is
well de ined and consis en ac oss all DMUs. This means ha he basis o compa ison is
he same o pe o mance app aisal. Tha is, no only is he e iciency used in calcula ing
he Malmquis index based on common bounda ies, bu also on common weigh s o com-
mon suppo ing hype planes o bounda ies [
5
]. Since a common weigh is employed in
he compu a ion o p oduc i i y indexes o all uni s each ime, he base o compa ison
emains consis en , ensu ing he consis ency and compa abili y o esul s.
The Malmquis index measu es p oduc i i y change by compa ing a uni ’s e iciency
a ime + 1 o ime . In DEA, hese e iciencies a e ela i e and elian on a on ie
compa ison. Malmquis indexes a e ca ego ized based on single o mul iple echnology
use: e e ence-based (using only one echnology, e.g., ime o global echnology) o
adjacen -based (employing mo e han one ime echnology). Adjacen -based indexes may
encoun e in easibili y issues due o some uni s’ supe -e iciency in one pe iod ela i e o
ano he . Employing global echnology ensu es easibili y in e iciency es ima ion models [
4
].
By u ilizing global echnology, ou p oposed index a oids po en ial ine iciency conce ns
in e iciency calcula ions.
DEA’s no able unc ionali y lies in i s abili y o assis each DMU in selec ing he mos
ad an ageous weigh s o mul iplie s o inpu s and ou pu s du ing e iciency calcula ions.
Howe e , adi ional DEA models, pa icula ly in measu ing MPI, end o o e s a e e i-
ciency due o he lexibili y in weigh selec ion o inpu s and ou pu s. Consequen ly, his
o e es ima ion a ec s he a ionali y o de i ed MPI alues. Va ious s a egies add ess
his issue, no ably he common se o weigh s and he c oss-e alua ion p ocess. C oss-
e alua ion, a signi ican componen o DEA heo y, su passes he common se o weigh
me hod in applica ions [
111
]. I p esen s clea bene i s, including pe sonalized DMU o de -
ing and he a oidance o un ealis ic weigh schemes wi hou necessi a ing expe weigh
es ic ions [112].
U ilizing a sel -e alua ion and pee e alua ion me hodology akin o c oss-e alua ion
o compu e c oss GMPI, ou app oach yields ealis ic alues o judgmen . C oss GMPI,
de i ed om he mean o p oduc i i y sco es, ep esen s an a e age p oduc i i y change,
ensu ing s abili y and eliabili y in assessmen .
5. Conclusions
This s udy p esen s an inno a i e, in eg a ed app oach o compu e he MPI h ough a
combina ion o global echnology, Common Weigh s, Nash ba gaining, and c oss-e alua ion
echniques. Ou p oposi ion ad oca es o ob aining CWs o wo- ime e sions o a DMU
Games 2024,15, 3 17 o 21
based on global echnology, acili a ing e iciency calcula ions in andem wi h a singula
on ie , and ensu ing consis ency and compa abili y in esul s ac oss di e en e alua ions.
By employing he Nash ba gaining game model, hese CWs sa is y desi able p ope -
ies, ensu ing incen i e alignmen among DMUs and yielding Pa e o-e icien solu ions.
Addi ionally, he no el y o inco po a ing CWs o pee and c oss-sec ional e alua ions
s ands ou as a no el con ibu ion, enhancing he me hod’s obus ness and applicabili y.
This in eg a ed amewo k yields a p oposed index boas ing se e al desi able p ope -
ies: easibili y, non-a bi a iness in base ime pe iod selec ion, echnological and weigh
consis ency, esul s abili y and eliabili y, and equi able assessmen s. The compa ison
o ou C oss-GMPI me hod wi h exis ing models, namely he GMPI and CWGMPI, un-
de sco es i s supe io i y in o e ing a holis ic e alua ion while main aining ai ness and
eliabili y ac oss e alua ions.
The nume ical example showcases he e ec i eness o ou p oposed me hod. U ilizing
ada cha s o isual ep esen a ion allows o a swi ye comp ehensi e assessmen
o DMU pe o mance ac oss a ious weigh s. This isual depic ion highligh s consis en
pe o mances among ce ain DMUs and a ia ions in o he s, p o iding nuanced insigh s
in o hei ela i e s eng hs and weaknesses.
In essence, ou in eg a ed app oach no only esol es e iciency o e es ima ion con-
ce ns p e alen in adi ional DEA models, bu also ensu es a mo e equi able and eliable
assessmen . I s dis inc ad an ages, such as a oidance o un ealis ic weigh schemes, and
enhanced s abili y, unde line i s supe io i y in pe o mance e alua ion.
The p esen s udy could be imp o ed o expanded in se e al ways. This esea ch
discussed a me hodology o de e mining a CW ec o o DEA based on he Nash ba -
gaining solu ion. In gene al, he esul ing Nash ba gaining game model is non-linea ,
gi en he na u e o a io o ms o DEA e iciency. Con e ing he nonlinea model in o
a pa ame ic linea p og amming p oblem wi h one pa ame e whose lowe and uppe
bounds can be de e mined, and using a heu is ic sea ch on he single pa ame e employing
he Kalai-Smo odinsky solu ion (which is he unique alloca ion ule o wo-playe ba -
gaining p oblems wi h linea p og amming), o using Egois ’s dilemma o his pu pose,
could be conside ed o u u e esea ch. De eloping he p esen s udy o mo e han wo
ime pe iods using he Ex ended Nash ba gaining game, using mo e ad anced and newe
indexes in he li e a u e o calcula e c oss-sec ional MPI alues, and conside ing he e ec s
o e u n o scale and o in e nal s uc u es o DMUs, could also be guidelines o u u e
esea ch. I is ecommended ha s udying he e ec s o common inpu s, me ging and
decomposi ion o DMUs, and decision-making in a cen alized scena io and in unce ain y
condi ions, be conside ed o u u e esea ch.
Au ho Con ibu ions:
Concep ualiza ion, R.F., M.R.M., P.F.W. and Y.T.; Me hodology, R.F.; So wa e,
M.R.M.; Fo mal analysis, R.F., P.F.W. and Y.T.; Da a cu a ion, R.F.; W i ing—o iginal d a , R.F. and
M.R.M.; W i ing— e iew & edi ing, R.F., P.F.W. and Y.T.; P ojec adminis a ion, P.F.W. and Y.T. All
au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
Da a A ailabili y S a emen : The da a p esen ed in his s udy a e a ailable in he a icle.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Re e ences
1.
Walhee , B. Malmquis P oduc i i y Index o Mul i-Ou pu P oduce s: An Applica ion o Elec ici y Gene a ion Plan s. Socioecon.
Plann Sci. 2019,65, 76–88. [C ossRe ]
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The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
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