Power in plurality voting games

an den B ink, René; Dimi o , Dinko; Rusinowska, Agnieszka
A icle — Published Ve sion
Powe in plu ali y o ing games
Theo y and Decision
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: an den B ink, René; Dimi o , Dinko; Rusinowska, Agnieszka (2025) : Powe in
plu ali y o ing games, Theo y and Decision, ISSN 1573-7187, Sp inge US, New Yo k, NY, Vol. 99,
Iss. 1-2, pp. 359-375,
h ps://doi.o g/10.1007/s11238-025-10053-z
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/330384
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Theo y and Decision (2025) 99:359–375
h ps://doi.o g/10.1007/s11238-025-10053-z
Powe inplu ali y o ing games
René andenB ink1· DinkoDimi o 2 · AgnieszkaRusinowska3
Accep ed: 2 June 2025 / Published online: 13 June 2025
© The Au ho (s) 2025
Abs ac
Simple games in pa i ion unc ion o m a e used o model o ing si ua ions whe e
a coali ion being winning o losing migh depend on he way playe s ou side ha
coali ion o ganize hemsel es. Such a game is called a plu ali y o ing game i in
e e y pa i ion he e is a leas one winning coali ion. In he p esen pape , we in o-
duce an equal impac powe index o his class o o ing games and p o ide an
axioma ic cha ac e iza ion. This powe index is based on equal weigh o e e y pa -
i ion, equal weigh o e e y winning coali ion in a pa i ion, and equal weigh o
each playe in a winning coali ion. Since some o he axioms we de elop a e con-
di ioned on he powe impac o losing coali ions becoming winning in a pa i ion,
ou cha ac e iza ion hea ily depends on a new esul showing he exis ence o such
elemen a y ansi ions be ween plu ali y o ing games in e ms o single embedded
winning coali ions. The axioms es ic hen he impac o such elemen a y ansi-
ions on he powe o di e en ypes o playe s.
Keywo ds Axioma iza ion· Powe index· Plu ali y o ing game· Winning coali ion
We a e g a e ul o wo anonymous e e ees o hei help ul commen s and sugges ions.
Co esponding au ho .
* Dinko Dimi o
dink[email p o ec ed]land.de
René anden B ink
j. [email p o ec ed]
Agnieszka Rusinowska
agnieszka. usinow[email p o ec ed]
1 Depa men o Economics andTinbe gen Ins i u e, VU Uni e si y, De Boelelaan 1105,
1081HVAms e dam, TheNe he lands
2 Chai o Economic Theo y, Saa land Uni e si y, Campus C3 1, 66123Saa b ücken, Ge many
3 Uni e si é Pa is 1 Pan héon So bonne, CNRS, Cen e d’Economie de la So bonne, Pa is School
o Economics, 106-112 Bd de l’Hôpi al, 75647Pa isCedex13, F ance
360
R. an den B ink e al.
1 In oduc ion
In a simple game in pa i ion unc ion o m, a wo h is assigned o e e y so-called
embedded coali ion being a pai consis ing o a coali ion and a pa i ion ha con-
ains his coali ion. The wo h is one ( espec i ely ze o) i he co esponding coa-
li ion is winning ( espec i ely losing) in he pa i ion. Such a game is called a
plu ali y o ing game (c . an den B ink e al. 2021) i in e e y pa i ion, he e is
a leas one coali ion ha wins in ha pa i ion. So, in his case, winning does no
necessa ily mean ha he coali ion has a majo i y and can pass a bill, bu simply
ha i conside s i sel as leading in a gi en coali ional con igu a ion as ep e-
sen ed by he pa i ion.
The need o modeling pa liamen a y si ua ions as plu ali y o ing games
migh a ise when, a e an elec ion, no only he bigges pa y decla es i sel
he winne bu also a majo i y consis ing o ideological opponen s. Mo eo e ,
whe he he bigges pa y is a winne o no migh depend on whe he he ideo-
logical opposing pa ies o m a coali ion o no .
The assump ion ha in e e y pa i ion he e is a leas one winning coali ion
comes om he ollowing wo obse a ions. Fi s , a e e e y pa liamen a y elec-
ion, be o e coali ions a e o med and we hus conside he pa i ion in o single-
ons, he e is usually a leas one pa y ha claims o be winne . As men ioned in
he p e ious pa ag aph, his can be he pa y ha go he mos numbe o o es,
bu no necessa ily. I can also be he bigges pa y o an ideological majo i y, a
cu en go e nmen pa y ha expec ed o do much wo se, a new pa y ha go
mo e o es han expec ed, and so on. E en when all pa ies go he same numbe
o o es, he e usually a e pa ies ha expec ed o ge less o es and hus conside
hemsel es a winne . In heo y, i is possible ha all pa ies eel ha hey los he
elec ion, bu his is e y unlikely. In pa i ions wi h a leas one non-single on,
pa ies o m a block, usually wi h he in en ion o become a winning coali ion
e en when he indi idual pa ies in he coali ion a e losing.
We assume plu ali y o ing games o be mono onic, bo h wi h espec o a coa-
li ion as well as o a ce ain ype o ex e nali ies ega ding o he coali ions. Spe-
ci ically, we assume ha (i) a winning coali ion canno become losing when i
g ows, and (ii) he e a e nega i e ex e nali ies o o he coali ions g owing in he
sense ha bigge ou side coali ions gi e ‘mo e esis ance’ and hus such coali-
ions becoming bigge canno u n a losing coali ion in o a winning one. This
e lec s ha in a ine pa i ion, he e is ‘less esis ance’ agains he winning
coali ion.
Whe eas in an den B ink e al. (2021) he ocus is mainly on he ep esen -
abili y o hese games by pa y weigh s, he cu en pape s udies powe dis ibu-
ions in plu ali y o ing games. Speci ically, we in oduce and axioma ize a powe
index o his class o games. In he o mula ion o ou axioms, we u ilize he
powe impac o losing embedded coali ions becoming winning in a pa i ion;
ha is, ou axioma iza ion hea ily depends on a new esul desc ibing elemen a y
ansi ions be ween plu ali y o ing games in e ms o single embedded winning
coali ions. Speci ically, we show ha in e e y plu ali y o ing game wi h a leas
361
Powe inplu ali y o ing games
one losing embedded coali ion, he e is always a losing embedded coali ion ha
can be u ned in o a winning one wi hou a ec ing he mono onici y o he game.
The es o he pape is o ganized as ollows. In he nex sec ion we in oduce he
basic ing edien s o plu ali y o ing games as a special class o simple games in pa -
i ion unc ion o m. Sec ion3 s a s wi h he o mal de ini ion o a powe index and
p esen s he men ioned use ul esul (P oposi ion 1) conce ning ansi ions be ween
plu ali y o ing games in e ms o single winning embedded coali ions. These an-
si ions a e co espondingly used as o o mula e ou i e axioms uniquely cha ac e -
izing he p oposed powe index (Theo em1). We conclude wi h some inal ema ks
in Sec .4. The Appendix con ains all p oo s.
2 Se up
We conside a ini e se N o playe s. Each non-emp y subse is called a coali-
ion. A collec ion
𝜋
o coali ions is a coali ion s uc u e i
𝜋
is a pa i ion o N,
i.e., i all coali ions in
𝜋
a e non-emp y, pai -wise disjoin , and hei union is N.
We deno e by
P
he se o all pa i ions (coali ion s uc u es) o N and sligh ly
abuse no a ion by w i ing
(
T
1
,T
2
,…,T
k)
o he pa i ion
{
T
1
,T
2
,…,T
k}
. Fo
i∈N
and
𝜋∈P
, he no a ion
𝜋(i)
s ands o he coali ion in
𝜋
con aining playe
i. A pai
(S;𝜋)
consis ing o a non-emp y coali ion
S⊆N
and a pa i ion
𝜋∈P
wi h
S∈𝜋
is called an embedded coali ion. The se o all embedded coali ions is
E
=
{
(S;𝜋)∈
(
2
N
⧵
{
�
})
×P∣S∈𝜋
}
.
A simple game in pa i ion unc ion o m is a pai
(N, )
, whe e he pa i ion unc-
ion
∶E
→
{0, 1}
is such ha
(N;(N)) =1
. An embedded coali ion
(S;𝜋)∈E
is
called winning in he game
(N, )
i and only i
(S;𝜋)=1
. O he wise, i is called
losing. We some imes say ha coali ion S is winning in pa i ion
𝜋
when
(S;𝜋)
is
a winning embedded coali ion. The se o all winning embedded coali ions in he
game is deno ed by
EW( )
, while
E
𝜋
W
( )=
{
S∈𝜋∶(S;𝜋)∈E
W
( )
}
s ands o he
se o all coali ions which a e winning in
𝜋
.
This game o m allows o model ex e nali ies o coali ion o ma ion. Fo ins ance,
i can be ha a coali ion con ained in wo pa i ions
𝜋
and
𝜋′
is winning in
𝜋
bu los-
ing in
𝜋′
. Since he playe se N is ixed, we o en w i e a simple game in pa i ion
unc ion o m
(N, )
by i s pa i ion unc ion . We use he ollowing no ion o inclu-
sion, bo owed om Alonso-Meijide e al. (2017): Fo
(
S
�
;𝜋
�)
,(S;𝜋)∈E , we say
ha
(
S
′
;𝜋
′)
is weakly includedin
(S;𝜋)
, deno ed by
(
S
�
;𝜋
�)
⊆(S;𝜋
)
, i (i)
S′⊆S
, and
(ii) o each
T∈
𝜋
⧵{S}
, he e exis s
T�∈𝜋�
wi h
T⊆T′
. A game is hen de ined
as mono onic i
(
S
�
;𝜋
�)
,(S;𝜋)∈E wi h
(
S
�
;𝜋
�)
⊆(S;𝜋
)
implies
(
S
�
;𝜋
�)
≤ (S;𝜋
)
.
This mono onici y no ion e lec s (i) a nonnega i e e ec when a coali ion g ows,
and (ii) an idea o nega i e ex e nali ies when playe s ou side a coali ion o m la ge
coali ions. In pa icula , i implies ha when a coali ion is winning in a pa i ion,
hen i is winning in e e y ine pa i ion ha con ains his coali ion. In o he wo ds,
he idea exp essed he e is ha in a ine pa i ion he e is ‘less esis ance’ agains
he winning coali ion. Clea ly, a winning coali ion can become losing in a coa se
362
R. an den B ink e al.
pa i ion since o he playe s o ming coali ions migh gi e a ‘s onge esis ance’
agains he winning coali ion, o make he winning coali ion mo e likely o ‘b eak
down’.
We call a simple game in pa i ion unc ion o m a plu ali y o ing game i (i) i
is mono onic, and (ii) o each
𝜋∈P
we ha e
(S;
𝜋
)=1
o a leas one
S∈𝜋
. We
assume ha he playe se N is ixed and o size1
|N|=n≥3
, and iden i y a plu al-
i y o ing game by i s pa i ion unc ion. The se o all plu ali y o ing games on he
playe se N is deno ed by
GN
.
We close his sec ion by he ollowing example o a plu ali y o ing game.
Example 1 Le
N={1, 2, 3}
and
∈G{
1,2,3
}
be as de ined below whe e he unde -
lined coali ions a e winning in he co esponding pa i ion.
The mono onici y o he game equi es, o example, (i) playe 1 o be winning
in he pa i ion in o single ons since he is winning agains he coali ion consis ing
o playe s 2 and 3, and (ii) he coali ion consis ing o playe s 1 and 3 o be winning
agains playe 2 since playe 1 is winning in he pa i ion in o single ons. Obse e
addi ionally ha , al hough bo h playe s 1 and 2 a e winning in he la e pa i ion,
only one o hem is winning agains a wo-playe coali ion.
3 Axioms onpowe indices andcha ac e iza ion esul
A powe index o plu ali y o ing games is a mapping ∶G
N
→ℝn
+
sa is ying
Σi∈N i( )=1
o each
∈GN
. We in e p e he eal numbe
i( )∈[0, 1]
as he
powe o playe i in he game .
The axioms we in oduce in his sec ion conce n implica ions when he only di -
e ence be ween wo plu ali y o ing games is a single winning embedded coali ion.
Hence, he i s ques ion we ha e o answe is i o e e y plu ali y o ing game ha
has a leas one losing embedded coali ion, he e is a losing embedded coali ion such
ha u ning i in o a winning one, we s ill ha e a plu ali y o ing game; speci ically,
mono onici y equi es ha he new winning coali ion is also winning agains ‘less
esis ance’. Ou i s esul whose p oo is elega ed o he Appendix, answe s his
ques ion in he posi i e.
123
12
,3
13
,2
23
,1
1
,2
,3
1 No ice ha , when
n=2
, whe he an embedded coali ion is winning o losing in a pa i ion is in a
i ial way independen o how he es o he playe s a e o ganized. This is he eason o conside ing
only plu ali y games wi h a leas h ee playe s.

363
Powe inplu ali y o ing games
P oposi ion 1 Le
∈GN
be such ha
|
|
E
W
( )
|
|
<
|
E
|
. Then he e exis
(S;
𝜋
)∈E⧵EW( )
and
�
∈GN
such ha
EW( �)=EW( )∪{(S;
𝜋
)}
In o he wo ds, s a ing om any plu ali y o ing game in which no all embed-
ded coali ions a e winning, he e is always a pa h o games leading o he unique
game in which all embedded coali ions a e winning; along such a pa h, each nex
game di e s om i s di ec p edecesso only by one winning embedded coali ion.
Example 2 Conside ing Example 1, one can make playe 2 winning agains he coa-
li ion consis ing o playe s 1 and 3, as well as playe 3 in he pa i ion in o single-
ons. Howe e , playe 3 canno be made winning agains he coali ion o 1 and 2, as
mono onici y would equi e his playe o be winning in he pa i ion in o single ons
as well.
Le us now in oduce he equi emen s we impose on a powe index by using
he ollowing addi ional no a ion. Fo
(S;
𝜋
)∈E
, any wo games
, �∈GN
wi h
EW( �)=EW( )∪{(S
;𝜋
)}
, and any
T⊆N
, we se
△
i
( , �)=
i
( �)−
i
(
)
and
△
T
( , �)=Σ
i∈T
△
i
( , �
)
. Tha is,
△
i
( , �
)
displays he change in he powe o
playe
i∈N
(as measu ed by ) when a single embedded coali ion which is losing
in becomes winning in
′
. Co espondingly,
△
T
( , �
)
s ands o he change in he
powe o coali ion T. Fo
S,T∈𝜋
, we inally se
△
ST
( , �)=△
S
( , �)−△
T
( , �
)
saying how a apa a e he powe change in S and he powe change in T when S
becomes he new winning coali ion in he pa i ion
𝜋
.
We a e eady now o p esen ou axioms.
Unanimi y (U): Fo all
∈
G
N∶EW( )=E
implies
i( )= j( )
o all
i,j∈N
.
In e nal Impac (II): Fo all
,
�∈
G
N∶EW( �)=EW( )∪{(S
;𝜋
)}
implies
△
i
( , �)=△
j
( , �
)
o all
i
,j
∈
T
∈
E
𝜋
W(
�)
.
Ex e nal Impac (EI): Fo all
,
�
∈G
N
∶
EW
(
�
)=
EW
(
)∪{(
S
;𝜋
)}
implies
△
Q
( , �)=△
R
( , �
)
o all
Q,R∈E𝜋
W( )
.
Null Impac (NI): Fo all
,
�
∈G
N
∶E
W
(
�
)=E
W
( )∪{(S;𝜋
)}
implies
△
i
( , �)=
0
o all
i∈H∈
𝜋
⧵E𝜋
W( �)
.
Powe Di e ence (PD): Fo all
,
�
∈G
N
∶E
W
(
�
)=E
W
( )∪{(S;𝜋
)}
implies
Σ
T∈E𝜋
W
( )△
ST ( , �)=
1
|P|
.
Unanimi y equi es equal powe in case all embedded coali ions a e winning in
he co esponding game and hus, i can be seen as a weak symme y axiom.
In e nal Impac equi es ha a losing embedded coali ion becoming winning (i)
has he same impac on he powe s o he playe s in ha winning coali ion and (ii)
o each o he winning coali ion in he co esponding pa i ion (i.e., he pa i ion
ha con ains his new winning coali ion) i also has he same impac on he powe
o he playe s inside each such winning coali ion. Pa (i) can be seen as some kind
o Mye son (1977a) ai ness applied o his game model.2 Pa (ii) o his axiom
2 Mye son (1977a)’s ai ness is in oduced o communica ion g aph games and equi es ha b eaking
a link be ween wo playe s in a communica ion g aph has he same impac on he payo o hese wo
364
R. an den B ink e al.
ex ends his idea also o playe s in o he winning coali ions in he co esponding
pa i ion since also om he pe spec i e o each o hem he si ua ion changed in a
‘symme ic’ way.
Ex e nal Impac equi es ha he impac o a losing embedded coali ion becom-
ing winning is he same o each o he winning coali ion in he co esponding coa-
li ion in he sense ha he sum o he powe s o all playe s in each such winning
coali ion changes by he same amoun . This can also be seen as a kind o ai ness as
abo e, bu hen applied on he coali ion le el.
Null Impac is a a he s ong axiom ha equi es ha a losing embedded coali-
ion becoming winning has no e ec on he powe s o he playe s in losing coali ions
in he co esponding pa i ion. Al hough using a simila a gumen as he ai ness
c i e ia men ioned abo e i seems easonable ha wi hin each such losing coali ion
he changes in powe a e he same, equi ing he e ec o be ze o is an ex eme case.
Howe e , i we conside he pa i ion as a coali ion s uc u e (c . Aumann and D èze
1974 and Owen 1977) isola ed om he es , hen a ’null’ agen is powe less wha -
e e is he con igu a ion o winning coali ions in he coali ion s uc u e.
Finally, Powe Di e ence is a balance axiom in he s yle o he collusion neu al-
i y axioms o TU-games in Halle (1994) and Malawski (2002). They speak abou a
pai wise powe di e ence axiom, whe e a ce ain change in he game (collusion o
playe s) does no change he sum o he payo s o he wo playe s in ol ed. We con-
side changes in he ‘game’ in he sense o losing embedded coali ions becoming
winning in hei pa i ions. A simila equi emen as ha o collusion neu ali y
would equi e ha he sum o he powe s o he new winning coali ion and each
o he winning coali ion in ha pa i ion would be ze o. In ou con ex his is, how-
e e , a e y s ong equi emen . The e o e, we modi y his equi emen in wo ways:
i s , we weaken his idea by equi ing he sum o hese powe di e ences o be con-
s an and second, his cons an is no ze o, bu
1
|P|
. This e lec s an equal impo ance
o he pa i ions in he ollowing sense. Since a powe index is non-nega i e and he
indi idual powe s add up o one, i seems easonable ha he maximal powe swi ch
ha can be ob ained by any change in he game is 1. Then, making a minimal change
(i.e., changing he s a us o only one coali ion) in one pa i ion being equal o
1
|
P
|
could indeed be seen as e lec ing equal impo ance o he pa i ions. (This is an
ex eme case, and we ge back o his in he Conclusion).
Fo
i∈N
and
∈GN
, le
P
i
=
{
𝜋∈P∶𝜋(i)∈E
𝜋
W
( )
}
be he se o all pa i-
ions
𝜋
, whe e playe i belongs o a winning coali ion in
𝜋
. We de ine he equal
impac powe index
∗
such ha , o each
∈GN
and
i∈N
,
(1)
∗
i( )=
1
|
P
|∑
𝜋∈P
i
1
|
|
|
E𝜋
W( )
|
|
|
⋅
|
𝜋(i)
|
.
Foo no e 2 (con inued)
playe s. No ice ha we apply ai ness o a coali ion ha migh con ain mo e han wo playe s, and in his
espec ou axiom looks mo e simila o he ai ness in Algaba e al. (2001) applied wi h espec o union
s able sys ems whe e a link (called suppo ) migh con ain mo e han wo playe s.
365
Powe inplu ali y o ing games
This index can be seen as an applica ion o he p inciple o insu icien eason in he
sense ha he alloca ion o powe is based on assigning equal sha es. Speci ically,
∗
ollows a ‘ h ee-s ep’ p ocedu e: (i) he assump ion o equal weigh /impo ance o
e e y pa i ion esul s in he alloca ion o
1
|P|
o e he playe s in e e y pa i ion; (ii)
in any pa i ion, his numbe is equally alloca ed o e he winning coali ions in he
pa i ion ( ecall ha in a plu ali y o ing game he e is a leas one winning coali ion
in e e y pa i ion); and (iii) he powe assigned o a winning coali ion in a pa i ion
is equally alloca ed o e he playe s in ha winning coali ion.
Ano he way o measu ing powe in simple pa i ion unc ion o m games can be
gene a ed by applying he p inciple o insu icien eason only wi h espec o mini-
mal winning coali ions. As shown in Alonso-Meijide e al. (2017) in he con ex whe e
he e can be pa i ions wi hou any winning coali ion, such an applica ion esul s in a
‘ wo-s ep’ p ocedu e o i s sha ing he ull powe equally among all minimal winning
coali ions and second alloca ing he powe assigned o each minimal winning coali ion
equally o e he playe s in ha coali ion.
In o de o show ha
∗
is a powe index, ix
∈GN
and obse e ha , due o
E{N}
W
( )={N
}
, he lowes alue
∗
i( )
can ake o
i∈N
is when playe i belongs o a
winning coali ion only in he pa i ion
(N)
; hence, in such a case, we ha e
∗
i( )=
1
n
⋅
|
P
|
>
0
.
On he o he hand,
and hus,
∗
is indeed a powe index.
We ha e he ollowing cha ac e iza ion esul .
Theo em1 A powe index sa is ies U, II, EI, NI, and PD i and only i
= ∗
The p oo o Theo em1 is elega ed o he Appendix.
∑
i
∈N
∗
i( )=
∑
i∈N
1
|P|
∑
𝜋∈P
i
1
|
|
|E𝜋
W( )|
|
|
⋅|𝜋(i)|
=1
|P|∑
𝜋∈P
∑
i∈N,𝜋∈P
i
1
|
|
|E𝜋
W( )|
|
|
⋅|𝜋(i)
|
=1
|P|∑
𝜋∈P
∑
S∈E𝜋
W( )
1
|
|
|
E𝜋
W( )|
|
|
=1
|
P
|∑
𝜋∈P
1=1
|
P
|
⋅
|
P
|
=1
366
R. an den B ink e al.
4 Conclusion
The s udy o plu ali y o ing games combines ideas om he analysis o simple
games (c . Shapley 1962) and insigh s om he gene al li e a u e on pa i ion unc-
ion o m games ini ia ed by Th all (1962) and Th all and Lucas (1963).3 Speci i-
cally, he p esen pape con ibu es o he s and o li e a u e de o ed o powe indi-
ces in simple games and o alues o games in pa i ion unc ion o m. Wi hin his
s and o li e a u e, he ocus is p edominan ly on ex ending he Shapley alue o
games wi h ex e nali ies (e.g., Mye son 1977b; Albizu i e al. 2005; Macho-S adle
e al. 2007; McQuillin 2009; Du a e al. 2010; G abisch and Funaki 2012). I is
hus no su p ising ha he p oposed powe indices in his con ex a e based on he
ma ginal con ibu ions o he playe s. Speci ically, he classes o games on which
powe indices ha e been de ined and axioma ically cha ac e ized include he class
o games whe e in e e y pa i ion he e can be a mos one winning coali ion (c .
Bolge 1986) o he class o games whe e he e migh be no winning coali ions in
some pa i ions (c . Alonso-Meijide e al. 2017; Ál a ez-Mozos e al. 2017). In con-
as o Bolge (1986) and Ál a ez-Mozos e al. (2017), bu simila o Alonso-Mei-
jide e al. (2017), he axioms we u ilize in he p esen pape a e no based on playe s’
mo emen s om one coali ion o ano he wi hin a pa i ion bu a he conside he
di ec impac on powe o losing embedded coali ions becoming winning in a pa i-
ion. The o mula ion o such axioms is based on a new esul showing ha in e e y
plu ali y o ing game wi h a leas one losing embedded coali ion, he e is always a
losing embedded coali ion ha can be u ned in o a winning one wi hou a ec ing
he mono onici y o he game.
We ema k ha ou axioms also can be applied o he class o simple games in
pa i ion unc ion o m (wi h possibly pa i ions wi h no winning coali ion) whe e
he equi emen o Powe Di e ence should be applied only when he o iginal game
(be o e u ning a losing coali ion in o a winning one) has a leas one winning coali-
ion. Rede ining a powe index such ha he sum o powe s always equals he ac-
ion o pa i ions ha ha e a leas one winning coali ion, hese axioms could be
used o cha ac e izing he powe index ha has he same o mula as in Equa ion (1).
An in e es ing ques ion o u u e esea ch, along wi h he s udy o he logical
independence o he axioms, is o p o ide axioma iza ions o powe indices o so-
called decisi e plu ali y o ing games, being games such ha e e y pa i ion con-
ains exac ly one winning coali ion. This is challenging since we canno jus u n
one losing coali ion in o a winning one wi hou des oying he decisi eness o he
game. So, we can look o axioms whe e he eplacemen o he winning coali ion in
one pa i ion by ano he coali ion in he same pa i ion will ha e enough bi e.
Ano he ques ion o u u e esea ch is o weaken some o he axioms and cha ac-
e ize he class o powe indices ha we ob ain in his way. Two candida es a e he
ollowing. Null Impac can be weakened by equi ing ha u ning one losing coali-
ion in o a winning coali ion in a pa i ion has he same impac on he powe o
3 We e e he eade o Lucas and Ma celli (1978) o a s udy o gene al p ope ies o pa i ion unc ion
o m games and o Koczy (2018) o a de ailed li e a u e su ey.
373
Powe inplu ali y o ing games
Hence,
holds and hus, NI is sa is ied.
Powe Di e ence In iew o (2) and (3), we ha e
as equi ed o he ul illmen o PD.
Suppose now ha sa is ies he abo e axioms. To show ha he powe index is
uniquely de e mined, conside
∈GN
and le us p oceed by induc ion on he ca -
dinali y o he se
EW( )
o winning embedded coali ions in .
Ini ializa ion: Suppose
|
|
E
W
( )
|
|
=
|
E
|
. By U and he de ini ion o a powe index,
i( )=
1
n
o each
i∈N
ollows.
Induc ion Hypo hesis: Suppose ha he powe index is uniquely de e mined o
each
∗∈GN
wi h
|
|
E
W
(
∗
)
|
|
>
|
|
E
W
( )
|
|
.
By P oposi ion 1, he e exis s
�∈GN
such ha
EW
(
�
)=E
W
( ) ∪ {(S;𝜋
)}
o
some
(S;
𝜋
)∈E⧵EW( )
. Obse e ha , by
|
|
E
W
(
�
)
|
|
>
|
|
E
W
( )
|
|
and he Induc ion
Hypo hesis, he powe ec o
( �)
is uniquely de e mined. In wha ollows, we
show ha ( ) is uniquely de e mined as well.
Fo his, and w.l.o.g., le
𝜋
=
(
S1,…,S
k−
1,S
k
,S
k+
1,…,S
K)
wi h
(
S𝓁;𝜋
)
=
1
o each
𝓁∈{1, …,k−1}
,
Sk=S
, and
(
S𝓁;𝜋
)
=
0
o each
𝓁∈{k+1, …,K}
.
No ice hen ha
i
(
�
)−
i
( )=
0
holds o each
i
∈∪
K
𝓁=k+1
S
𝓁
due o NI. Tha is,
he exac de e mina ion o any such
i( )
di ec ly ollows om he ac ha
i
(
�)
has al eady been ixed.
Fo each
T⊆N
, se
T( ) ∶= Σi∈T i( )
and
T
(
�
) ∶= Σ
i∈T
i
(
�)
. No ice u he
ha , by PD we ha e
∗
i( �)=
1
|P|
∑
𝜋∈P �
i
1
|
|
|E𝜋
W( �)|
|
|
⋅|𝜋(i)|
=
1
|P|
∑
𝜋∈P �
i⧵{𝜋∗}
1
|
|
|E𝜋
W( �)|
|
|
⋅|𝜋(i)
|
=1
|
P
|∑
𝜋∈P
i⧵{𝜋∗}
1
|
|
|
E𝜋
W( )
|
|
|
⋅
|
𝜋(i)
|
.
△ ∗
i
( ,
�
)=
∗
i
(
�
)−
∗
i
( )=
0
∑
T
∈E𝜋∗
W( )
△
∗
ST ( , �)=
|
||E𝜋∗
W( )
|
||⋅△
∗
S( , �)−
∑
T∈E𝜋∗
W( )
△
∗
T( , �)
=|S|⋅|||E𝜋∗
W( )|||
|P|⋅|S|⋅(|||E𝜋∗
W( )|||+1)+|||E𝜋∗
W( )|||
|P|⋅(|||E𝜋∗
W( )|||+1)⋅|||E𝜋∗
W( )
|||
=1
|
P
|
⋅
(|||
E𝜋∗
W( )
|||
+1
)
⋅
(|||
E𝜋∗
W( )
|||
+1
)
=1
|
P
|

374
R. an den B ink e al.
whe e he las equali y ollows om he de ini ion o a powe index. Thus, we can
exac ly de e mine
S( )
due o (1)
( �)
being al eady ixed by he Induc ion Hypo h-
esis, and (2)
S
𝓁( )
being also ixed o each
𝓁∈{k+1, …,K}
by NI as shown
abo e.
Obse e u he ha axiom EI equi es he di e ence
Q( �)− Q( )
o be he
same o each
Q
∈
{
S
1
,…,S
k−1}
and hus,
Q( )
is uniquely de e mined as well.
Finally, II equi es o each
Q
∈
{
S
1
,…,S
k−1}
he powe di e ences
i( �)− i( )
o
be he same o each
i∈Q
. Since
( �)
is known by he Induc ion Hypo hesis, we
conclude ha ( ) is uniquely de e mined.
◻
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