Blend-in fairness and equal split

Dimi o , Dinko; Sun, Ching-jen
A icle — Published Ve sion
Blend-in ai ness and equal spli
In e na ional Jou nal o Game Theo y
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Dimi o , Dinko; Sun, Ching-jen (2025) : Blend-in ai ness and equal spli ,
In e na ional Jou nal o Game Theo y, ISSN 1432-1270, Sp inge , Be lin, Heidelbe g, Vol. 54, Iss. 2,
h ps://doi.o g/10.1007/s00182-025-00949-z
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ORIGINAL PAPER
In e na ional Jou nal o Game Theo y (2025) 54:27
h ps://doi.o g/10.1007/s00182-025-00949-z
Abs ac
Blending in wi h o he s is a possible sel -se ing mo i a ion when people pa ici-
pa e in coope a i e si ua ions. We use his mo i a ion o o mula e a co esponding
ai ness p inciple, combine i wi h a he weak s anda d axioms om coope a i e
game heo y, and show ha i leads o equal spli o coali ional gains. The same
no ma i e p inciples cha ac e ize his solu ion when only cohesi e games (whe e i
is op imal o he coali ion o all playe s o o m) a e conside ed.
Keywo ds Blend-in ai ness · Cohesi e games · Coope a i e games · Equal
di ision solu ion
JEL Classi ica ion: A13 · C71 · D63 · D91
1 In oduc ion
A key ques ion in coali ional game heo y is how he o al payo a coali ion can
achie e h ough coope a ion should be di ided among i s membe s. One o he mos
in luen ial answe s o his ques ion wi hin he con ex o coope a i e ans e able u il-
i y games (TU-games) has been p o ided by he Shapley alue (c . Shapley 1953).
Acco ding o his single- alued solu ion, each playe should be assigned a sha e o
Accep ed: 10 June 2025
© The Au ho (s) 2025
Blend-in ai ness and equal spli
DinkoDimi o 1· Ching-jenSun2
We a e g a e ul o Syl ain Béal, Rene an den B ink, Youngsub Chun, Y es Sp umon , wo anonymous
e e ees, and a ious semina audiences o aluable sugges ions.
Dinko Dimi o
[email p o ec ed]
Ching-jen Sun
[email p o ec ed]
1 Chai o Economic Theo y, Saa land Uni e si y, Saa b ücken, Ge many
2 Deakin Business School, Deakin Uni e si y, Melbou ne, Aus alia
1 3
D. Dimi o , C.-j. Sun
he payo ha is p opo ional o his ma ginal con ibu ions in he co esponding
game. F om an axioma ic pe spec i e, he Shapley alue is he unique solu ion which
is e icien ( he en i e gain when all playe s coope a e (i.e., he wo h o he ‘g and’
coali ion) is dis ibu ed among he playe s), symme ic (playe s ecei e equal payo s
i hey a e exchangeable in gene a ing coali ional gains), sa is ies he null playe
p ope y (no con ibu ion o any coali ion esul s in ze o payo ), and is addi i e o e
games.
The expe imen al alidi y o he abo e p inciples was ecen ly es ed in de Clippel
and Rozen (2022). The da a analysis o hese au ho s p o ides s ong e idence o
he symme y and addi i i y axioms wi h he e iciency axiom being i ially sa is ied
due o he expe imen al design. Howe e , no such e idence was ound o he null
playe p ope y; ha is, e en null playe s we e assigned o payo s which a e signi i-
can ly di e en om ze o.
The lack o expe imen al e idence o he null playe p ope y na u ally d aws
one’s a en ion o ano he amous single- alued solu ion o TU-games, he equal
di ision solu ion. Fo each game, he la e dis ibu es he wo h o he g and coali-
ion equally among all playe s and i iola es he null playe p ope y while sa is ying
he o he h ee men ioned p inciples. As o iginally shown in an den B ink (2007),
he equal di ision solu ion can be cha ac e ized by eplacing, in he abo e axioma ic
sys em, he null playe p ope y by a nulli ying playe equi emen (each playe is
assigned ze o payo i he wo h o each coali ion con aining ha playe is ze o). We
e e he eade o Alonso-Meijide e al. (2019) o an excellen and de ailed su ey
o he co esponding s and o he li e a u e, o He nandez-Lamoneda e al. (2008)
o a cha ac e iza ion o he class o all addi i e, symme ic, e icien and con inuous
solu ions, as well as o Chap e 4 in B anzei e al. (2008) o an o e iew o o he
egali a ianism-based solu ion concep s.
In his pape we ollow a no ma i e pe spec i e on coope a i e games (c . Moulin
2003) and p o ide an axioma ic suppo o he idea ha he equal di ision solu ion
sp ings ou o a ai ness p inciple which is a he “... s ongly shaped by cul u al
alues” (c . Young 1994, p. xii). Tha is, ou wo k i s in o he s and o li e a u e
emphasizing he con ex -dependence o ai ness and equi y “... no because o he
lack o gene al p inciples o jus ice, bu due o i s e ec on he in e p e a ion o hose
p inciples” (c . Konow 2001, p. 139). Fo ins ance, Alesina and Angele os (2005)
s udy he ex en o which he di e ence in social pe cep ions ega ding he ai ness
o ma ke ou comes and he unde lying sou ces o income inequali y (Ame icans
s Eu opeans) a e consis en wi h equilib ium beha io . Almås e al. (2020) p o-
ide expe imen al suppo o he di e ences be ween Ame icans and Scandina ians
wi h espec o wha kind o inequali ies hey conside ai and o he impo ance
assigned o ai ness ela i e o e iciency. Cappelen e al. (2007) conside a dic a o
game in which he dis ibu ion phase is p eceded by a p oduc ion phase and show
how one may simul aneously es ima e he p e alence o di e en ai ness ideals and
he weigh people a ach o ai ness conside a ions. Finally, Gel and e al. (2002)
s ess he di e ences when i comes o sel -se ing mo i a ions in indi idualis ic o
in collec i is ic cul u es. These au ho s a gue ha , in he o me , he sel is se ed by
enhancing one’s posi i e a ibu es o “s and ou ” and be be e han o he s, while in
1 3
27 Page 2 o 21
Blend-in ai ness and equal spli
collec i is ic cul u es he ocus is a he on how indi iduals “blend in” and main ain
in e dependence wi h o he s.
Since he Shapley alue a e ages playe s’ con ibu ions o e e y coali ion hey
join, i can be axioma ically oo ed ia a co esponding ma ginali y p inciple (c .
Young 1985) in a he indi idualis ic socie ies. In he p esen pape we ollow a pos-
sible sel -se ing mo i a ion o collec i is ic cul u es, in oduce a co esponding
ai ness p inciple (blend-in ai ness), and show ha i c ucially shapes he cha ac e -
iza ion o he equal di ision solu ion.
We s a in Sec . 2 by p o iding he o mal de ini ions o he men ioned s anda d
axioms and i s weaken e iciency and symme y o a symme ic e iciency p inciple.
The la e imposes on a single- alued solu ion he equal spli o he wo h o he
g and coali ion, p o ided ha all playe s a e symme ic in he co esponding game.
A weak null playe p ope y addi ionally pos ula es null playe s o be assigned ze o
payo s in games, whe e he o al gain when all playe s coope a e is ze o. These wo
p ope ies, oge he wi h addi i i y, a e clea ly sa is ied by bo h he Shapley alue
and he equal di ision solu ion.
In Sec . 3 we de elop a no ion o coope a ion equi alence o games o playe s.
To ix ideas, conside wo supe addi i e games ( ha is, games whe e he o ma ion o
la ge coali ions is wo hy) and a pa icula playe . Imagine now ha wha he playe
ealizes when compa ing hese wo games is ha he e is only a enaming o he o he
playe s bu no change in he co esponding coali ional wo hs. Tha is, in e ms o he
o e ed ‘blending-in wi h o he s’ possibili ies, he wo games a e coope a ion equi a-
len o ha pa icula playe . The blend-in ai ness p inciple s a ed in Sec . 4 hen
equi es a playe o be assigned he same payo in wo supe addi i e games which
a e coope a ion equi alen o him. This p inciple u ns ou o shape he cha ac e iza-
ion o he equal di ision solu ion bo h on he en i e se o games (Theo em 1) and
on he subse o cohesi e games, whe e i is op imal ha he g and coali ion o ms
(Theo em 2). We conclude in Sec . 5 wi h some inal ema ks.
The p oo s o ou cha ac e iza ion esul s a e elega ed o he Appendix, whe e
we i s in oduce he class o pa i ion games and show ha hey o m a basis o he
en i e se o games (Theo em 0). P oposi ions 1–5 a e hen c ucial o he p oo s o
Theo em 1 and Theo em 2 as hey explain how addi i i y, symme ic e iciency, he
weak null playe p ope y, and blend-in ai ness gene a e he equal di ision solu ion
on pa i ion games. The Appendix also con ains examples showing he independence
o he u ilized axioms.
2 No ma i e p inciples and solu ions
Le N be a se o n indi iduals. A coali ion is any subse o N. A coope a i e ans e -
able u ili y game (TU-game) is a pai
(N, )
, whe e is he cha ac e is ic unc ion
o he game assigning a wo h (S) o each coali ion
S⊆N
such ha
(∅)=0
. The
amoun (S) ep esen s how much he membe s o S can sha e should hey coope a e.
In wha ollows, we ix he playe se N and he e o e deno e he game
(N, )
by i s
cha ac e is ic unc ion . The no a ion
GN
s ands o he se o all TU-games on he
playe se N.
1 3
Page 3 o 21 27
D. Dimi o , C.-j. Sun
Two playe s
i, j ∈N
a e called symme ic in a game
∈G
N
, i
(S∪{i})= (S∪{j})
holds o e e y
S⊆N {i, j}
. A playe
i∈N
is a null
playe in
∈G
N
i
(S∪{i})= (S)
is alid o e e y
S⊆N {i}
. The sum o
wo games
,w ∈G
N
is de ined by
( +w)(S)= (S)+w(S)
o each
S⊆N
.
I is an implici assump ion in coope a i e game heo y models ha he g and
coali ion o ms (all playe s coope a e) and he basic ques ion is hen how should
he co esponding p oceeds be dis ibu ed among all indi iduals. Thus, a (single-
alued) solu ion
:GN→Rn
assigns a payo ec o o each game
∈G
N
. The
nex ou no ma i e p inciples ha could be imposed on a solu ion a e s anda d in
he li e a u e.
E iciency Fo all
∈G
N
:
Σi∈N i( )= (N)
.
Symme y Fo all
∈G
N
: I
i, j ∈N
a e symme ic in , hen
i( )= j( )
.
Null playe p ope y Fo all
∈G
N
: I
i∈N
is a null playe in , hen
i( )=0
.
Addi i i y Fo all
,w ∈G
N
:
( +w)= ( )+ (w)
.
The Shapley alue is he unique solu ion which is e icien , symme ic, addi i e,
and sa is ies he null playe p ope y. This solu ion concep was in oduced in Shap-
ley (1953) and i assigns o each playe he a e age o his ma ginal con ibu ions in
he co esponding game. Gi en a game
∈G
N
, he ma ginal con ibu ion o playe
i o a coali ion
S⊆N {i}
is jus he di e ence
∆i(S)= (S∪{i})− (S)
. The
Shapley alue Sh is hen de ined by he condi ion
Sh
i( )=
1
|
N
|
!
∑
R∈R
∆i(Si(R)) o each i∈
N,
whe e
R
is he se o all
|N|!
o de ings o N and
Si(R)
is he se o playe s p eceding
i in he o de ing R.
F om he abo e ou axioms, i is only he null playe p ope y ha is iola ed by
ano he amous p ocedu e o sha ing coali ional gains, he equal di ision solu ion.
Fo each game
∈G
N
i is de ined by
ED
i( )=
(
N
)
n
o each i
∈N.
Tha is, his solu ion dis ibu es he wo h o he g and coali ion equally among all
playe s.
In he nex sec ion we p esen ou axioma ic cha ac e iza ion o he equal di i-
sion solu ion whe e all u ilized axioms bu one a e also sa is ied by he Shapley
alue. Besides addi i i y, hese include he weak null playe p ope y and symme ic
e iciency.
Weak null playe p ope y Fo all
∈G
N
: I
i∈N
is a null playe in and
(N)=0
, hen
i( )=0
.
Symme ic e iciency Fo all
∈G
N
: I all playe s a e symme ic in , hen
i(
)=
(
N
)
n
o each
i∈N
.
The weak null playe p ope y is clea ly implied by he null playe p ope y and i
is s onge han he null playe in a null en i onmen p ope y in oduced in Casajus
1 3
27 Page 4 o 21

Blend-in ai ness and equal spli
and Hue ne (2013); he la e imposes ha a null playe ge s a non-nega i e payo
in a game in which he g and coali ion has ze o wo h. On he o he hand, he com-
bina ion o e iciency and symme y (bu none o hese p ope ies aken sepa a ely)
implies he symme ic e iciency axiom. Symme ic e iciency is s onge han he
i iali y axiom in oduced in Chun (1989) and he weak symme y axiom used in
an den B ink (2007).
3 Coope a ion equi alence o games
Imagine a playe who e alua es his pa icipa ion in wo di e en coope a i e si ua-
ions and claims equal payo s in he co espondingly gene a ed games. As al eady
a gued abo e, one possible way o a ionalizing such a claim is o look a he ways
in which he pa icula playe main ains in e dependence wi h o he s and blends
in when wo king wi h hem in he espec i e si ua ions. The no ion o coope a ion
equi alence o games we in oduce below o malizes he idea ha , om he iew-
poin o a gi en playe , wo games o e he same possibili ies when i comes o
i ing in wi h o he playe s. We use hen his no ion o o mally s a e in Sec . 4 a
co esponding blend-in ai ness p inciple.
In o de o equip he eade wi h a sui able in ui ion o he no ion o coope a ion
equi alence, le us conside wo games in which h ee playe s coope a e on hei
in es men decisions.1 Playe s 1 and 2 a e domes ic in es o s lacking any expe i-
ence and sui able in es men echnologies, while playe 3 is an expe ienced o eign
in es o who possesses 100 uni s o capi al and is able o double any ac ion o i
he in es s. The e is no go e nmen es ic ion on such in es men s when made in
coope a ion wi h a leas one domes ic in es o . Howe e , he o eign in es o when
ac ing on his own is allowed o in es only 40 uni s (in he game ) o 30 uni s (in
he game w).
{1}{2}{3}{1
,
2}{1
,
3}{2
,
3}
N
10 20 2 ×40 + 60 30 220 240 260
w
20 10 2 ×30 + 70 30 240 220 260
In he game , playe s 1 and 2 would like o in es 10 and 20 uni s o capi al, espec-
i ely. Howe e , due o hei lack o in es men echnology, one has
({1}) = 10
,
({2}) = 20
, and
({1,2}) = 30
. In a coali ion con aining playe 3, he o eign in es-
o is allowed o use he en i e 100 uni s o his capi al and each domes ic coali ional
membe has access o he co esponding in es men echnology doubling he sum
o indi idual capi als. The in e p e a ion o he coali ional wo hs in he game w is
analogous.
Le us now compa e he wo games and w om he iewpoin o he o eign
in es o . In bo h games he in ends o in es 100 uni s o capi al and his is exac ly
he amoun he pu s on he able when joining any non-emp y coali ion o domes ic
1 Each o hese games can be seen as an app op ia e modi ica ion o he landlo d game in Shapley and
Shubik (1967).
1 3
Page 5 o 21 27
D. Dimi o , C.-j. Sun
playe s. Mo eo e , when looking a he domes ic in es o s in he wo games, he eal-
izes ha he e is only a enaming o hese playe s (playe 1 becomes playe 2 and
ice e sa) bu no change in he wo h o any coali ion con aining some o hese
playe s. No ice addi ionally ha each o hese games is supe addi i e as, exempli-
ied wi h espec o he game , he inequali y
(S∪T)≥ (S)+ (T)
holds o all
S, T ⊆N
wi h
S∩T=∅
. In o he wo ds, he domes ic in es o s and he o eign
in es o a e “incen i ized” in bo h games o o m la ge coali ions; hence, i seems
na u al o a playe o ocus a he on la ge coali ions when e alua ing his pa icipa-
ion in such si ua ions. Co espondingly, we decla e he wo supe addi i e games as
being coope a ion equi alen o he o eign in es o .
Le us deno e by
GN
sa
,
GN
sa ⊂G
N
, he se o all supe addi i e games on he playe se
N. Conside wo games
,w ∈G
N
sa
, and ix a playe
i∈N
whose iewpoin we would
like o desc ibe. Suppose u he ha he e exis s a bijec ion
σ:N→N
wi h
σ(i)=i
such ha
(S)=w(σ(S))
holds o all
S⊆N
wi h
j∈S
o some
j∈N {i}
. In
o he wo ds, when e alua ing he wo supe addi i e games, playe i ocusses on he
coali ions each o he o he playe s migh belong o and ealizes only a pe mu a ion
o he playe s’ names bu no changes a all in he co esponding coali ional wo hs.
We call he games and w coope a ion equi alen o playe i.
4 Blend-in ai ness leads o equal spli
The blend-in ai ness p inciple we in oduce below elies on he no ion o coope a-
ion equi alence o games o playe s. Mo e p ecisely, i equi es om a solu ion
o assign he same payo o a playe in supe addi i e games which a e coope a ion
equi alen o him.
Blend-in ai ness Fo all
,w ∈G
N
sa
and
i∈N
: I and w a e coope a ion equi a-
len o playe i, hen
i( )= i(w)
.
We would like o men ion he ac ha (1) i is a speci ic bijec ion
σ
( sa is ying
σ(i)=i
o he co esponding playe
i∈N
) we use in he o mula ion o coope a-
ion equi alence o games and (2) he implica ion e e s only o he payo o playe
i in he wo games. The eade migh hen wonde whe he he e is a logical ela ion
(on he class o supe addi i e games) be ween he abo e axiom and he anonym-
i y p ope y o single- alued solu ions equi ing ha a solu ion should no disc imi-
na e be ween he playe s solely on he basis o hei “names”. No ice ha , in he
de ini ion o blend-in ai ness, i migh happen ha
({i})=w({σ(i)})
o playe
i whose iewpoin he bijec ion
σ
desc ibes. Clea ly hen, he Shapley alue iola es
he newly in oduced equi emen , while sa is ying anonymi y. On he o he hand,
one can de ine a dic a o ship solu ion
dk:GN
sa →Rn
wi h espec o a p e-speci ied
playe
k∈N
, whe e o
∈G
N
sa
one has
d
k
j( )=
{
(
N
)i
j
=
k
,
0o he wise.
As i can be easily seen,
dk
sa is ies blend-in ai ness and iola es anonymi y. Hence,
hese wo equi emen s a e independen .
1 3
27 Page 6 o 21
Blend-in ai ness and equal spli
Ou main cha ac e iza ion esul is s a ed in Theo em 1 below and i basically says
ha , along wi h addi i i y and he weak e sions o s anda d equi emen s, i is he
blend-in ai ness ha leads o equal spli o coali ional gains.
Theo em 1 A solu ion
:GN→Rn
sa is ies addi i i y, he weak null playe p op-
e y, symme ic e iciency, and blend-in ai ness i and only i i is he equal di ision
solu ion.
I is wo h men ioning ha he equal di ision solu ion also sa is ies a s onge
e sion o he blend-in ai ness axiom equi ing ha o all pai s o (no necessa ily
supe addi i e) coope a ion equi alen games o some pa icula playe , his playe
ecei es he same payo in bo h games. As we show in he p oo o Theo em 1, he e
is no need o impose his s onge e sion on a solu ion since i s weake e sion s a ed
only wi h espec o supe addi i e games su ices o p o iding he cha ac e iza ion
o he equal di ision solu ion on he en i e se o games.
Obse e addi ionally ha wo o he o he axioms we u ilize, symme ic e iciency
and he weak null playe p ope y, a he indica e he o ma ion o he g and coali-
ion, he eade migh ask whe he a co esponding cha ac e iza ion could be eached
when only cohesi e games a e conside ed. In a cohesi e game
∈G
N
i is op i-
mal o he g and coali ion o o m since by de ini ion
(N)≥ΣK
k=1 (Sk)
holds
o e e y pa i ion
{S1,...,S
K}
o N (c . Osbo ne and Rubins ein (1994), p. 258).
Theo em 2 shows ha he e a e he same ou no ma i e p inciples2 which cha ac e -
ize he equal di ision solu ion on he men ioned subclass (deno ed by
GN
coh
) o games.
Theo em 2 A solu ion
:GN
coh →Rn
sa is ies addi i i y, he weak null playe p op-
e y, symme ic e iciency, and blend-in ai ness i and only i i is he equal di ision
solu ion.
As we show in he Appendix, he p oo o Theo em 2 elies on he ac ha all
games used in he p oo o Theo em 1 a e cohesi e (wi h he pa i ion games se ing
as a basis o he en i e se o games being e en supe addi i e (c . Rema k 0 in Sec .
6.1).
5 Concluding ema ks
The no ion o blend-in ai ness in oduced in his pape elies on he e y basic idea
ha indi iduals in coope a i e si ua ions a e usually cons ued as being undamen-
ally connec ed o o he s. Such a undamen al connec ion in supe addi i e games is
mi o ed by he o ma ion o la ge coali ions; in o he wo ds, playe s’ “incen i es”
o look a indi idual wo hs when compa ing wo such games a e a he mode a e.
The la e in e p e a ion na u ally leads o he de eloped no ion o coope a ion equi -
2 Addi i i y, he weak null playe p ope y, and symme ic e iciency should hen be s a ed wi h espec o
cohesi e games. The e is no need o change he o mula ion o blend-in ai ness since i is anyway s a ed
wi h espec o supe addi i e games and supe addi i e games a e also cohesi e.
1 3
Page 7 o 21 27
D. Dimi o , C.-j. Sun
alence o games wi h blend-in ai ness equi ing a playe o be assigned he same
payo in wo supe addi i e games ha a e coope a ion equi alen o ha playe .
As al eady elabo a ed in Sec . 4, he s anda d anonymi y axiom (sa is ied by he
Shapley alue) and blend-in ai ness ( iola ed by he Shapley alue) a e independen
equi emen s. Ne e heless, blend-in ai ness can al e na i ely be seen as a local e -
sion o anonymi y in he sense ha (1) anonymi y in he co esponding supe addi i e
games holds wi h he possible excep ion o he playe unde conside a ion and (2)
he payo equi alence is equi ed only o ha pa icula playe (whe eas i holds o
all playe s in he classical axiom o anonymi y). In o he wo ds, an axiom in he spi i
o anonymi y (blend-in ai ness) combined wi h (small modi ica ions o ) classical
axioms sa is ied by he Shapley alue u ns ou o ha e a he d ama ic consequences.
I seems he e o e easonable o u he s udy and axioma ically cha ac e ize solu-
ions ha sa is y he blend-in ai ness axiom and a e as close as possible o he o mal
de ini ion o he Shapley alue. No ice ha he Shapley alue iola es blend-in ai -
ness due o he ac ha i pa icula ly accoun s o playe s’ ma ginal con ibu ions
o he emp y se and o single on coali ions. Hence, any solu ion espec ing only he
ma ginal con ibu ions o coali ions o size a leas wo would sa is y ou ai ness
axiom. One pa icula example o a symme ically e icien and addi i e solu ion
sa is ying blend-in ai ness is p o ided in Sec . 6.3.
Appendix
In Sec . 6.1 we in oduce he collec ion o pa i ion games and show ha hey o m
a basis o he space o all games. This ac is hen used in Sec . 6.2 o show how
blend-in ai ness shapes he cha ac e iza ion o he equal di ision solu ion (Theo em
1 and Theo em 2). Sec ion 6.3 con ains examples o he independence o he u ilized
axioms.
Pa i ion games
Le
T⊆N
be a nonemp y coali ion. The pa i ion game o e T is he game
uT
de ined as ollows.
(1) I
T=N
, hen o each coali ion
S⊆N
,
u
T(S) :=
{0i (1)
S∈ {∅,T,N}
o (2)
|S|>|T|
wi h (
S, T
)pa i ioning
N
,
−
1o he wise.
(2) I
T=N
, hen o each coali ion
S⊆N
,
u
N(S) :=
{0i
S
=
N
,
1o he wise.
1 3
27 Page 8 o 21
Blend-in ai ness and equal spli
Lemma 3 Fix
T⊂N
,
k∈T
wi h
|T|≥2
, and le
:GN→Rn
sa is y addi i i y,
he weak null playe p ope y, and symme ic e iciency. Then,
(1) [
|T|=|N|/2
and
k(uT)=0
]
⇒
k
(
u
T {k})=0
;
(2) [
|T|=|N|/2
and
k
(
uT
)=
k
(
u
N T)=0
]
⇒
k
(
u
T {k})=0
.
P oo Fix
T⊂N
wi h
|T|≥2
and
k∈T
. Clea ly hen, he coali ions T and
T {k}
a e nonemp y. Recall u he ha
u
T
(
N
)=
u
N T(
N
)=0
holds.
(1) No ice ha
|T|=|N|/2
implies ei he (
|T|>|N|/2
and
|T {k}| ≥ |N|/2
) o
(
|T|<|N|/2
and
|T {k}| <|N|/2
). We conside hese wo possibili ies sepa a ely.
(1.1) (
|T|>|N|/2
and
|T {k}| ≥ |N|/2
): Fo he wo pa i ion games
uT
and
uT {k}
we ha e:
uT(S)=0
o
S∈ {∅,T,N}
and
uT(S)=−1
, o he wise;
uT {k}(
S
)=0
o
S∈ {∅,T {k},N}
and
uT {k}(
S
)=−1
, o he wise. De ine he
games
cT
and
cT {k}
by
c
T(S)=
{
−
1i
S
=
T
,
0o he wise,
and
c
T {k}(S)=
{
−
1i
S
=
T {k}
,
0o he wise,
and p oceed as ollows.
Fo he game
uT+cT
we ha e
(uT+cT)(S)=0
o
S∈ {∅,N}
and
(uT+cT)(S)=−1
, o he wise. Tha is, all playe s a e symme ic in
uT+cT
and
hus, by symme ic e iciency,
k(uT+cT)=0
in pa icula holds. By assump ion,
k(uT)=0
holds as well and hus, by addi i i y,
k(
c
T)=0
(8)
ollows. Conside ing now he game
c
T
+cT {k}
wi h
(
cT
+
c
T {k})(
S
)=−1
o
S∈{T,T {k}}
and
(
c
T+
c
T {k})(
S
)=0
, o he wise, we ha e ha playe k
is a null playe in he game. By
(
cT
+
c
T {k})(
N
)=0
and he weak null playe
p ope y,
k(
c
T+
c
T {k})=0
(9)
ollows. We ha e hen om (8), (9), and addi i i y ha
k
(
c
T {k})=0
(10)
should hold. No ice inally ha o he game
uT {k}+cT {k}
we ha e
(
u
T {k}+
c
T {k})(
S
)=0
o
S∈ {∅,N}
and
(
u
T {k}+
c
T {k})(
S
)=−1
, o h-
e wise. In o he wo ds, all playe s a e symme ic in
uT {k}+cT {k}
and hus, by
1 3
Page 15 o 21 27

D. Dimi o , C.-j. Sun
symme ic e iciency,
k(
u
T {k}+
c
T {k})=0
in pa icula holds. By (10) and
addi i i y,
k
(
uT
{
k
})=0
ollows.
(1.2) (
|T|<|N|/2
and
|T {k}| <|N|/2
): Recall ha he coali ions T and
T {k}
a e nonemp y and obse e u he ha he coali ions
N T
and
N (T {k})
a e nonemp y as well. Fo he wo pa i ion games
uT
and
uT {k}
we ha e:
uT(S)=0
o
S∈ {∅,T,N T,N}
and
uT(S)=−1
, o he wise;
uT {k}(
S
)=0
o
S∈ {∅,T {k},N (T {k}),N}
and
uT {k}(
S
)=−1
, o he wise. De ine he
games
cT
and
cT {k}
by
c
T(S)=
{
−
1i
S∈{T,N T}
,
0o he wise,
and
c
T {k}(S)=
{
−
1i
S∈{T {k},N
(
T {k}
)
}
,
0o he wise,
and p oceed as ollows.
Fo he game
uT+cT
we ha e
(uT+cT)(S)=0
o
S∈ {∅,N}
and
(uT+cT)(S)=−1
, o he wise. Tha is, all playe s a e symme ic in
uT+cT
and
hus, by symme ic e iciency,
k(uT+cT)=0
in pa icula holds. By assump ion,
k(uT)=0
holds as well and hus, by addi i i y,
k(
c
T)=0
(11)
ollows. Conside now he game
c
T
+cT {k}
wi h
(
cT
+
c
T {k})(
S
)=−1
o
S∈{T {k},T,N T,N (T {k})}
and
(
c
T+
c
T {k})(
S
)=0
, o he wise.
Clea ly, playe k is a null playe in his game. By
(
cT
+
c
T {k})(
N
)=0
and he
weak null playe p ope y,
k(
c
T+
c
T {k})=0
(12)
ollows. We ha e hen om (11), (12), and addi i i y ha
k
(
c
T {k})=0
(13)
should hold. No ice inally ha o he game
uT {k}+cT {k}
we ha e
(
u
T {k}+
c
T {k})(
S
)=0
o
S∈ {∅,N}
and
(
u
T {k}+
c
T {k})(
S
)=−1
, o h-
e wise. In o he wo ds, all playe s a e symme ic in
uT {k}+cT {k}
and hus, by
symme ic e iciency,
k
(
u
T {k}+
c
T {k})=0
in pa icula holds. By (13) and
addi i i y,
k(
u
T {k})=0
ollows.
(2) No ice ha
|T|=|N|/2
implies
|T|=|N T|=|N|/2
. Conside hen
he pa i ion games
uT
,
uN T
, and
uT {k}
ecalling ha hey a e de ined as ol-
lows:
uT(S)=0
o
S∈ {∅,T,N}
and
uT(S)=−1
, o he wise;
uN T(
S
)=0
o
S∈ {∅,N T,N}
and
uT(S)=−1
, o he wise;
uT {k}(
S
)=0
o
1 3
27 Page 16 o 21
Blend-in ai ness and equal spli
S∈ {∅,T {k},N (T {k}),N}
and
uT {k}(
S
)=−1
, o he wise. De ine he
games
cT
,
cN T
, and
cT {k}
by
c
T(S)=
{
−
1i
S
=
T
,
0o he wise,
c
N T(S)=
{
−
1i
S
=
N T
,
0o he wise,
and
c
T {k}(S)=
{
−
1i
S∈{T {k},N
(
T {k}
)
}
,
0o he wise,
and p oceed as ollows.
Fo he game
uT+cT
we ha e
(uT+cT)(S)=0
o
S∈ {∅,N}
and
(uT+cT)(S)=−1
, o he wise. Tha is, all playe s a e symme ic in
uT+cT
and
hus, by symme ic e iciency,
k(uT+cT)=0
in pa icula holds. By assump ion,
k(uT)=0
holds as well and hus, by addi i i y,
k(
c
T)=0
(14)
ollows. Applying he same a gumen wi h espec o he game
uN T+cN T
and
ecalling
k
(
u
N T)=0
holds by assump ion, we ge
k(
c
N T)=0.
(15)
Conside now he game
c
T
+cN T+cT {k}
wi h
(
cT
+
c
N T+
c
T {k})(
S
)=−1
o
S∈{T {k},T,N T,N (T {k})}
and
(
c
T+
c
N T+
c
T {k})(
S
)=0
, o he -
wise. Clea ly, playe k is a null playe in his game. By
(
cT
+
c
N T+
c
T {k})(
N
)=0
and he weak null playe p ope y,
k(
c
T+
c
N T+
c
T {k})=0
(16)
ollows. We ha e hen om (14), (15), (16), and addi i i y ha
k
(
c
T {k})=0
(17)
should hold. No ice inally ha o he game
uT {k}+cT {k}
we ha e
(
u
T {k}+
c
T {k})(
S
)=0
o
S∈ {∅,N}
and
(
u
T {k}+
c
T {k})(
S
)=−1
, o h-
e wise. In o he wo ds, all playe s a e symme ic in
uT {k}+cT {k}
and hus, by
symme ic e iciency,
k(
u
T {k}+
c
T {k})=0
in pa icula holds. By (17) and
addi i i y,
k
(
u
T {k})=0
ollows.
1 3
Page 17 o 21 27
D. Dimi o , C.-j. Sun
Rema k 3 No ice ha he games
cT
,
cT {k}
, and
cN T
cons uc ed in he p oo o
Lemma 3 a e no supe addi i e bu cohesi e.
P oposi ion 4 I a solu ion
:GN→Rn
sa is ies addi i i y, he weak null playe
p ope y, symme ic e iciency, and blend-in ai ness, hen
i(uT)=0
o each
i∈N
and
T⊂N
.
P oo We p oceed by induc ion on he ca dinali y o
T⊂N
.
Ini ializa ion: Fo
|T|=|N|−1
,
i(uT)=0
o each
i∈N
ollows om P opo-
si ion 3.
Induc ion Hypo hesis: Suppose ha
i(uT)=0
holds o each
i∈N
and each
T⊂N
wi h
|T|>
.
Take
T⊂N
wi h
|T|=
and
i∈N T
. No ice ha i
=1
, hen he asse -
ion di ec ly ollows om P oposi ion 2. Suppose now ha
≥2
holds. Since
|T∪{i}| = +1>
, we ge
i
(
u
T∪{i})=0
by he induc ion hypo hesis. By
he same easoning and o he case when
+1=|N|/2
, we addi ionally ge
i
(
u
N (T∪{i}))=0
due o
|N (T∪{i})|= +1
. Applying Lemma 3 esul s
hen in
i(uT)=0
. Since playe
i∈N T
was a bi a y chosen, we conclude ha
i(uT)=0
holds o each
i∈N T
.
Hence, i emains o be shown ha
i(uT)=0
holds o each
i∈T
as well. Fo
his, ix
i∈T
, deno e by
T|T|
he se o all coali ions o size
|T|
and by
T|T|(
i
)
he se o all coali ions in
T|T|
con aining playe i. No ice ha we ha e
i/∈T′
o
each
T′∈T
|T| T|T|(
i
)
. Hence, as al eady shown abo e,
i(uT′)=0
holds o each
T′∈T
|T| T|T|(
i
)
.
Conside now he game
u=Σ
T
′∈T|T|u
T
′
and no ice ha , when
|T|≥|N|/2
holds, we ha e
u
(S)=



0i
S∈ {∅,N}
,
−

T|T|

+1 i |S|=|T|,
−T|
T
|
o he wise,
while, when
|T|<|N|/2
is he case, we ha e
u
(S)=



0i
S∈ {∅,N}
,
−

T|T|

+1 i |S| ∈ {|T|,|N T|} ,
−T|
T
|
o he wise.
Obse e ha , in ei he o hese cases, all playe s a e symme ic in u. By symme -
ic e iciency,
i(u)=0
. By
i(uT′)=0
o each
T′∈T
|T| T|T|(i)
and addi i i y,
T|T|(
i
)
x
i=0
should hold. We ha e hen
xi=0
and hus,
i(uT)=0
ollows.
Rema k 4 No ice ha he game u cons uc ed in he p oo o P oposi ion 4 is supe -
addi i e and hus, cohesi e.
1 3
27 Page 18 o 21
Blend-in ai ness and equal spli
P oposi ion 5 I a solu ion
:GN→Rn
sa is ies symme ic e iciency, hen
i
(u
N
)=1
n
o each
i∈N
.
P oo Recall ha
uN(N)=1
and
uN(S)=0
o each
S=N
holds. Clea ly hen,
all playe s a e symme ic in
uN(N)
and hus, by symme ic e iciency,
i
(
uN
)= 1
n
o each
i∈N
ollows.
□
P oposi ions 1–5 show ha , hanks o addi i i y, he weak null playe p ope y,
symme ic e iciency, and blend-in ai ness, one has
i(uT)=EDi(uT)
o each
pa i ion game
uT
(
T⊆N
,
T=∅
) and each
i∈N
.
P oo o Theo em 1 I can be easily e i ied ha he equal di ision solu ion sa is ies
he ou axioms. As o he e e se implica ion, le T be a non-emp y coali ion, b a
eal numbe , and de ine he game
ub
T
as ollows:
(1) I
T=N
, hen o each coali ion
S⊆N
,
u
b
T(S) :=
{0i (1)
S∈ {∅,T,N}
o (2)
|S|>|T|
wi h (
S, T
)pa i ioning
N
,
−
1o he wise.
(2) I
T=N
, hen o each coali ion
S⊆N
,
u
b
N(S) :=
{0i
S
=
N
,
bo he wise.
In iew o P oposi ions 1–5, we conclude o each
i∈N
ha
i(ub
T)=
{0i
T⊂N
,
b
n
i T=N.
Theo em 0 u he implies he exis ence o eal numbe s
(a
T
){T⊆N,T =∅}
such ha ,
o
∈G
N
, one has
=Σ
T⊆N,T =∅
aTu
b
T
. By he addi i i y o ,
i(
)=Σ
T⊆N,T =∅
i(
a
T
u
b
T)
(18)
ollows. We u he make use o he ollowing esul . Claim
ΣT⊂N,T =∅
i(
a
T
u
b
T)=0
.
□
P oo o he Claim No ice i s ha P oposi ion 1–5 and he addi i i y o gi e us
i(aub
T)=0
o each
i∈N
, each non-emp y
T⊂N
, and any
a≥0
; in pa icu-
la , we ha e
i(aTub
T)=0
o each
i∈N
and each non-emp y
T⊂N
whene e
aT≥0
holds. Suppose now ha we ha e
aT<0
o some non-emp y
T⊂N
.
No ice hen ha , wi h z being he ze o game,
aTub
T+(−aT)ub
T=z
holds and hus,
due o he addi i i y o and
(z)=0
ollowing om symme ic e iciency, we ge
i(aTub
T)=0
o each
i∈N
. Hence, he asse ion ollows.
1 3
Page 19 o 21 27
D. Dimi o , C.-j. Sun
The combina ion o (18) wi h he abo e Claim gi es us
i
(
)=
i
(
aNub
N)=
i
(
ubaN
N)=
i
(
u
(
N
)
N)=
(
N
)
n
o each
i∈N
.
□
P oo o Theo em 2 Recall ha pa i ion games a e supe addi i e (Rema k 0) and hus,
cohesi e, and ha all games cons uc ed in he co esponding p oo s o P oposi ions
1–5 a e cohesi e as well (Rema ks 1–4). Since he ze o game is also supe addi i e,
he p oo o Theo em 2 is ully analogous o he p oo o Theo em 1.
□
Independence o he axioms
In wha ollows, we assume ha he playe se con ains a leas h ee playe s and cons uc
ou examples whe e he co espondingly de ined solu ion sa is ies all axioms bu he men-
ioned one. Obse e ha all solu ions u ilized o show axioms’ logical independence on
GN
can also be used o show he co esponding independence on
GN
coh
as well. This is due
o he ac ha each non-supe addi i e game cons uc ed in hese examples is cohesi e.
Addi i i y The solu ion
A
, gi en by
A( )=ED( )
i
∈G
N
sa
and
A( )=Sh( )
i
∈G
N GN
sa
, sa is ies all axioms bu addi i i y. As o see he la e
ac , le
N={1,2,3}
and conside he games and w de ined as ollows:
(S)=1
i
S∈ {{1,2},N}
, and
(S)=0
, o he wise;
w(S)=1
i
S∈ {{3},N}
, and
w(S)=0
,
o he wise. The game is supe addi i e bu w is no and hus, we ha e
A(
)=(1
3
,
1
3
,
1
3)
and
A(
w
)=(1
6
,
1
6
,
4
6)
. Fo he game
( +w)
we ha e
( +w)(N)=2
,
( +w)(S)=1
i
S∈ {{3},{1,2}}
, and
( +w)(S)=0
, o he wise. Since
( +w)
is no a supe addi i e game, we ge
A(
+
w
)=(4
6
,
4
6
,
4
6)=
A(
)+
A(
w
)
.
Weak null playe p ope y Deno e by
R3
i
he se o all pe mu a ions o he playe se
N a which playe
i∈N
is a leas he hi d playe in he co esponding o de . No ice ha
R3
i=R3
j
holds o all
i, j ∈N
and se :
=R3
i
. Fo
i∈N
, le
g
i( )=
1
∑
R
∈R
3
i
∆i(Si(R))
.
De ine hen he solu ion
WNP
by
WNP
i( )=gi( )+
(
N
)−Σ
i∈Ngi
(
)
n
o each i
∈N.
In o he wo ds, he payo
WNP
i( )
is jus he sum o he a e age o playe i’s ma ginal
con ibu ions o coali ions o size a leas wo (
gi( )
) and he equal di ision o he excess
(N)−Σi∈Ngi( )
. This solu ion clea ly sa is ies addi i i y and symme ic e iciency.
In o de o see ha i sa is ies blend-in ai ness as well, ecall ha
σ(i)=i
holds o he
pe mu a ion
σ
guiding he coope a ion equi alence o wo supe addi i e games and w
o playe
i∈N
. Gi en ha
(S)=w(σ(S))
holds o all
S⊆N
wi h
j∈S
o some
j∈N {i}
,
WNP
i( )= WNP
i(w)
ollows. No ice inally ha
WNP
iola es he
weak null playe p ope y. As o see i , ake
N={1,2,3}
and conside he supe addi-
i e game de ined by
(S)=0
i
S∈ {{1},{2,3},N}
, and
(S)=−1
, o he wise.
No ice ha playe 1 is a null playe in and
(N)=0
. We ge
g( ) = (0,1,1)
and
1 3
27 Page 20 o 21

Blend-in ai ness and equal spli
WNP(
)=(−2
3
,
1
3
,
1
3)
in iola ion o he weak null playe p ope y equi ing playe 1
o ge ze o payo in he game .
Symme ic e iciency The solu ion
SE
, de ined by
SE( )=0
o each
∈G
N
,
sa is ies all axioms bu symme ic e iciency.
Blend-in ai ness The Shapley alue iola es blend-in ai ness while sa is ying all
o he axioms.
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