Probabilistic Owen-Shapley spatial power indices

Games and Economic Beha io 136 (2022) 524–541
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P obabilis ic Owen-Shapley spa ial powe indices
M.J. Albizu i∗, A. Goikoe xea
Uni e si y o he Basque Coun y, Facul y o Economics and Business, Depa men o Quan i a i e Me hods, Bilbao, Spain
a i c l e i n o a b s a c
A icle his o y:
Recei ed 21 Ap il 2021
A ailable online 26 Oc obe 2022
JEL classifica ion:
C71
Keywo ds:
Powe index
The spa ial Owen-Shapley index
P obabilis ic spa ial powe indices
In his pape we s udy p obabilis ic Owen-Shapley spa ial powe indices, which a e
gene aliza ions o he Owen-Shapley spa ial powe index (1977). We p o ide an explici
o mula o calcula ing hese spa ial indices o unanimi y games and gi e an axioma ic
cha ac e iza ion o he amily o p obabilis ic Owen-Shapley spa ial powe indices. We
employ an equal powe change p ope y, a spa ial dummy p ope y, anonymi y, a posi ional
in a iance p ope y, and a posi ional con inui y p ope y. Some examples a e also gi en.
©2022 The Au ho (s). Published by Else ie Inc. This is an open access a icle unde he
CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
1. In oduc ion
Coope a i e game heo y has been applied success ully o measu e he powe o agen s in o ing si ua ions, which a e
ep esen ed by simple games. A winning coali ion is assigned a wo h o one, and a losing one a wo h o ze o. The Shapley-
Shubik index (1954) and he Banzha index (1965) can be seen as he bes known indices o measu ing powe . They bo h
ake in o accoun whe he he p esence o an agen changes a losing coali ion in o a winning one, i.e. whe he an agen is
pi o al. In defining he Shapley-Shubik index, all possible o de ings o agen s a e conside ed. Each o de ing has a pi o al
agen associa ed wi h i : he one whose addi ion o he coali ion o med by he p e ious agen s changes ha coali ion om
losing o winning. When all o de ings a e equally p obable, he p obabili y o an agen being pi o al is by defini ion his o
he Shapley-Shubik index. O de ings a e no conside ed in defining he Banzha index. The index o an agen is he numbe
o coali ions in which he o she is pi o al.
A pape by Owen (1971)inspi ed Shapley (1977) o p opose a spa ial powe index: he Owen-Shapley powe index (see
also Owen and Shapley, 1989). In his new model, ideological di e ences be ween he agen s can be aken in o accoun .
I is o malized by means o a spa ial game, which is a simple game oge he wi h a cons ella ion o agen p ofiles, i.e., a
se o ec o s in he Euclidean space Rm ha ep esen s he ideological loca ions o o e s. The di e en dimensions can
be seen as ideological conside a ions o c i e ia, so each posi ion ep esen s he “ideal poin ” (o highes p e e ence) in he
space. Shapley (1977)w i es ha he use o he Euclidean space Rm“seems o lea e us ample scope o cap u ing many
kinds o poli ical and ideological pa ame e s wi hou an excess o abs ac ion and gene ali y”.
An issue is o malized by Shapley (1977) h ough a ec o ∈Rm. A playe in posi ion xis mo e in a o o han a
playe in posi ion yi he scala p oduc ·xis less han o equal o ·y. The e o e, playe s can be o de ed om he mos
o he leas en husias ic wi h espec o an issue, which implies ha one o hem is pi o al in ha o de ing. When all issues
a e equally likely, he p obabili y o a playe being pi o al is his o he Owen-Shapley spa ial powe index.
*Co esponding au ho .
E-mail add esses: [email p o ec ed] (M.J. Albizu i), goikoe x[email p o ec ed] (A. Goikoe xea).
h ps://doi.o g/10.1016/j.geb.2022.10.004
0899-8256/©2022 The Au ho (s). Published by Else ie Inc. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
Pe e s and Za zuelo (2017)s udy he Owen-Shapley spa ial powe index when he e a e wo dimensions. They p o ide a
o mula o calcula ing he index o unanimi y games and gi e an axioma ic cha ac e iza ion by means o a ans e axiom,
a dummy axiom, anonymi y and wo in a ian posi ional axioms ( eflec ion in a iance and posi ional in a iance).
A na u al a ia ion o he Owen-Shapley spa ial powe index is o conside ha no all issues a e equally p obable. Fo
example, hink abou a egional pa liamen o a coun y, in which local issues a e mo e ele an han he s a e ones (see
Sec ion 6). Ba and Passa elli (2009) conside ha he e is a p obabili y dis ibu ion o e issues, defined by a con inuous
densi y unc ion. They analyze he dis ibu ion o powe in he Council o he EU wi h wo dimensions. The s ances owa d
he EU on in e na ional issues and domes ic issues a e he wo dimensions ha a e aken in o accoun .
In his pape we also conside he a ia ion o he Owen-Shapley spa ial powe index wi h wo dimensions when he e is
a(ny) p obabili y dis ibu ion o e issues. We call hese spa ial powe indices p obabilis ic Owen-Shapley spa ial powe indices.
We gi e a o mula o calcula ing he indices o unanimi y games. The e o e, he index o any spa ial game can be easily
calcula ed by means o linea combina ions o indices o unanimi y games. We conduc an axioma ic s udy and p o e ha
he amily o p obabilis ic Owen-Shapley spa ial powe indices can be ob ained by means o he axioms employed by Pe e s
and Za zuelo (2017), d opping eflec ion in a iance and adding con inui y. Reflec ion in a iance equi es he index no o
change when he cons ella ion o agen s is shi ed o o a ed. Howe e , as men ioned abo e, in a egional pa liamen , whe e
local issues a e mo e ele an han he s a e ones, all issues a e no ega ded equally likely. Conside also he possible
p esence o he agenda se e e ec , which influences he impo ance o likelihood o he di e en issues a s ake. Wi hin
he EU, he Commission would play he ole o agenda se e in he Council (see Passa elli and Ba , 2007). Thus, eflec ion
in a iance should no be imposed. On he con a y, posi ional in a iance changes ela i e posi ions o agen s and is sa isfied
by p obabilis ic Owen-Shapley spa ial powe indices. The e o e, a ans e axiom, a dummy axiom, posi ional in a iance and
con inui y gene a e he amily o p obabilis ic Owen-Shapley spa ial powe indices. We also gi e some illus a i e examples,
including an applica ion o he Basque Pa liamen .
O he spa ial powe indices ha e also been in oduced o e he yea s. Fo example Shenoy (1982)ex ends he Banzha
index o he spa ial se ing when o e s a e ep esen ed by poin s in Rm. Passa elli and Ba (2007)employ he mul ilinea
ex ension app oach o coope a i e games (Owen, 1972) o define a spa ial powe index when issues belong o Rm. Alonso-
Meijide e al. (2011)define a spa ial powe index aking in o accoun leng hs o pa hs connec ing playe s’ posi ions. Bena i
and Ma ze i (2013)ob ain a amily ha includes bo h he Shapley-Shubik index and he Owen-Shapley spa ial powe index
modeling o e s’ p opensi y o suppo an issue h ough a andom u ili y unc ion. Mul inomial alues a e in oduced by
Albina-Puen e and Ca e as (2015) o model di e en endencies o agen s. Blockmans and Gue y (2015)s udy he impac
o issue saliences and dis ance selec ion on he amily in oduced by Bena i and Ma ze i (2013). Ma in e al. (2017)
p opose a gene aliza ion o he Owen spa ial powe index (1971).
The e a e also o he s udies. Ál a ez-Mozos e al. (2017) add ess he p oblem o ex ending he Shapley-Shubik index
o he class o simple games wi h ex e nali ies. Ka os and Pe e s (2018)de elop a class o powe indices o e ec i i y
unc ions. An issue based powe index is also in oduced by Kong and Pe e s (2021)by means o o de ings o issues.
The s uc u e o he pape is as ollows. Sec ion 2gi es no a ion and p elimina ies. Sec ion 3p esen s p obabilis ic Owen-
Shapley spa ial powe indices. Sec ion 4calcula es hese spa ial powe indices o unanimi y games. Sec ion 5p esen s he
axioma ic cha ac e iza ion o he amily o p obabilis ic Owen-Shapley spa ial powe indices. We also show he indepen-
dence o he axioms employed. Some examples can be ound in Sec ion 6. Finally, Sec ion 7gi es some concluding ema ks
and poin e s o u u e wo k.
2. P elimina ies
2.1. No a ion
Gi en x, y ∈R2such ha x = y, he hal -line which s a s a xand passes h ough yis deno ed by [x, y, →), he
segmen wi h endpoin s xand yby [x, y], and we w i e (x,y)=[x, y] {x, y}. The pe pendicula bisec o line o [x, y]is
he line pe pendicula o he line h ough xand y ha passes h ough
1
2x +1
2y. Gi en x ∈R2and a line in R2such ha
x /∈, xdeno es he eflec ion o xwi h espec o . No ice ha is he pe pendicula bisec o line o [x, x]. I x ∈, hen
x=x.
The p ojec ion o x ∈R2on a line in R2, i.e.
1
2x +1
2x, is deno ed by ¯
x. Gi en x =(x1, x2)and y =(y1, y2) ∈R2,
x ·y =x1y1+x2y2. Gi en X⊆R2, co(X) e e s o he con ex hull o X.
2.2. Spa ial games and spa ial powe indices
Le Ube a se , he uni e se o playe s. A coali ion is a fini e nonemp y subse o U. A ans e able u ili y (TU) game is a
pai (N, )such ha Nis a coali ion and :2N→RN,
(∅)=0. A simple game is a TU game (N, )such ha (S) ∈{0, 1}
o all S∈2N, (N) =1 and (S) ≤ (T) o all S, T∈2Nwi h S⊆T( ha is, is mono onic). Le (N, )be a simple game.
A coali ion Sis winning in (N, )i (S) =1; o he wise Sis losing. A minimal winning coali ion is a winning coali ion wi h
no p ope winning subcoali ion. I (S) =1 and (S {i}) =0, playe iis said o be pi o al in S.
Deno e by GN he se o all TU games wi h se o playe s Nand by SN he subse o all simple games.
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M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
Fig. 1. Pola angle θ( ).
A cons ella ion o a se o playe s Nis a ec o p =(pi)i∈N∈(R2)Nsuch ha pi= pj o all i, j ∈Nwi h i = j. The se
o all cons ella ions o Nis deno ed by CN. Gi en p ∈CNand S⊆N, define pS∈(R2)Sby (pS)i=pi o all i ∈S. Wi h a
sligh abuse o no a ion, co(pS)deno es he con ex hull o he se {pi}i∈S.
Gi en a line in R2and p ∈CN, deno e by p he cons ella ion which is he eflec ion o pwi h espec o , i.e.
p=(p
i)i∈N.
A spa ial game wi h a se o playe s Nis a iple (N, , p)such ha (N, ) ∈SNand p ∈CN.
A spa ial powe index on A ⊆SNis a unc ion ϕ:A →RNsuch ha o all (N, , p) ∈Ai holds ha ϕi(N, , p) ≥0 o
all i ∈Nand i∈Nϕi(N, , p) =1.
The Owen-Shapley spa ial powe index is defined as ollows.
Le (N, , p)be a spa ial game and conside ∈R2such ha || || =1, whe e || || deno es he Euclidean leng h o . Each
ep esen s a possible issue o be ea ed by he agen s in N. A each he agen i ∈Nwho is pi o al in {j ∈N| ·pj≤ ·pi}
is calcula ed and iis said o be pi o al a . No ice ha iis unique excep o a fini e numbe o issues . We assume ha
all issues a e equally likely. The Owen-Shapley spa ial powe index o i, deno ed by i(N, , p), is he p obabili y o ibeing
pi o al a an issue .
We ake in o accoun he ollowing geome ical conside a ion. Gi en an issue , le be he line wi h di ec ion ec o
, and o each j ∈N he p ojec ion ¯
p
jon . We say ha j ∈Np ecedes k ∈Non i ¯
p
jp ecedes ¯
p
kin he di ec ion o
. Thus, i ∈Nis pi o al a i and only i iis pi o al in he se o agen s who p ecede him o he (including himsel o
he sel ) on .
2.3. Dummy playe in spa ial games
Playe i ∈Nis a dummy (Pe e s and Za zuelo, 2017) in a spa ial game (N, , p)i pi∈co
pS {i} o e e y coali ion Sin
which iis pi o al. Obse e ha i a playe is no pi o al in any coali ion, hen he o she is a dummy. And i a dummy playe
is pi o al in a coali ion S, hen he o she is su ounded, acco ding o p, by playe s in S, and i can he e o e be assumed
ha he dummy playe can no ake ad an age o his/he pi o al powe in S.
Obse e also ha i iis a dummy, hen iis ne e pi o al a any issue , and he e o e, i(N, , p) =0. The e can be wo
si ua ions. I iis no pi o al in any coali ion, iclea ly canno be pi o al a any issue . I iis pi o al in a coali ion S, gi en
ha iis a dummy, pi∈co
pS {i}. As a consequence, gi en any issue and a line wi h di ec ion ec o , he e is always
a playe jin Ssuch ha ¯
p
ip ecedes ¯
p
j. And hence, icanno be pi o al in he se o he agen s who p ecede him o he
(including i). The e o e, no is ipi o al a .
3. P obabilis ic Owen-Shapley spa ial powe indices
Le Bbe he σ-field gene a ed by subin e als in  =(0,2π]. The Owen-Shapley spa ial powe index a ises when he
Lebesgue p obabili y measu e λis conside ed on B. Indeed, he e is a bijec ion om he se o issues,  ∈R2:|| ||=1,
in o (0,2π] ha associa es each issue wi h i s pola angle θ( )∈(0,2π], as depic ed in Fig. 1. In defining he Owen-
Shapley spa ial powe index, i α, β∈(0,2π], α<β, he p obabili y o he issues sa is ying α<θ( )≤βis gi en by
λ
(α,β]=(β−α)/2π.
Bu no all he a eas o he issues migh be equally p obable when hei (Lebesgue) measu e is he same. All possible
si ua ions a e ep esen ed by p obabili y measu es Pon B. Mo eo e , he p obabili y o single issues is equi ed o be
ze o, i.e. Pmus be non-a omic, as happens o he Lebesgue p obabilis ic measu e.
A spa ial powe index ϕis said o be associa ed wi h a p obabili y P on Bi o each spa ial game (N, , p)and i ∈N,
ϕi(N, , p)is he p obabili y o ibeing pi o al a an issue , when he p obabili y o he issues , which a e ep esen ed
by θ( )∈(0,2π], is gi en by P. We w i e ϕ=Pand we say ha ϕis a p obabilis ic Owen-Shapley spa ial powe index.
As poin ed ou abo e, his is es ic ed o non-a omic p obabili y measu es.
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M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
Fig. 2. Di e en posi ions o −−→
pipj.
4. P obabilis ic Owen-Shapley spa ial powe indices o unanimi y games
Fi s no e ha i ideno es he union o he angle o angles o med by issues a which iis pi o al, hen
P
i(N, ,p)=P(i).
Le =uS, ∅ = S⊆N, be he unanimi y game on S, i.e.,
uS(T)=1i T⊇S,
0o he wise.
Obse e ha i i /∈S, P
i(N, , p) =0, since iis no pi o al a any issue .
I |S|=1, i is clea ha P
i(N, , p) =1 when i ∈S.
I |S|>1, le
S(p)=i∈S:pi/∈co pS {i},(1)
ha is, he se o non dummy playe s in (N, uS, p). I i ∈S S(p), hen icanno be pi o al a any issue , and he e o e
P
i(N, , p) =0. Fo i ∈S(p) ake in o accoun he ollowing. Le i, j ∈Sand an issue , which can be assumed o s a
om pi(we can conside as a ee ec o ). Le be he line h ough piand pj, ⊥be he pe pendicula line o  h ough
piand be a line in he di ec ion o (Fig. 2). No ice ha ¯
p
jp ecedes ¯
p
iin he di ec ion o i and only i is poin ing
in o he hal plane ou side pjwi h con ou ⊥.
Two cases a e dis inguished. I |S(p)|=2 and i ∈S(p), ihas h ee exp essions acco ding o di e en posi ions o −−→
pipj,
as depic ed in Fig. 2, and acco dingly,
P
i(N, ,p)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
Pθ−−→
pipj+π
2,θ−−→
pipj+3π
2i θ−−→
pipj≤π
2,
P0,θ−−→
pipj−π
2+Pθ−−→
pipj+π
2,2πi π
2<θ−−→
pipj≤3π
2,
Pθ−−→
pipj−3π
2,θ−−→
pipj−π
2i 3π
2<θ−−→
pipj.
I |S(p)|>2, conside co
pS(p), which is, by defini ion, he smalles con ex se ha con ains all poin s pisuch ha
i ∈S(p). By defini ion o S(p), he bounda y o co
pS(p)is he polygon whose e ices a e all poin s pisuch ha i ∈S(p).
Thus, o i ∈S(p), he e a e wo playe s j, k ∈S(p)
{i}such ha pjand pka e adjacen e ices o piin co
pS(p)(Fig. 3).
Conside ( esp. 
) o be he line h ough piand pj( esp. pk); ⊥( esp. 
⊥) he line pe pendicula o ( esp. 
) h ough
pi, and  o be a line in a di ec ion o an issue . I u ns ou ha ¯
p
kp ecedes ¯
p
iin he di ec ion o o all k∈Si and
only i is poin ing ou wa ds om co
pS(p)be ween ⊥and

⊥. These issues o m an a c i ha has se e al exp essions
depending on he posi ions o ec o s −−→
pipjand −−→
pipk. Fo he sake o simplici y we w i e θ=θ−−→
pipjand θ=θ−−→
pipk.
Wi hou loss o gene ali y we assume ha θ<θ
. Fi s ly, no e ha θ−θ=π, since, as w i en abo e, pi, pjand pka e
e ices o a polygon. Two main cases a e dis inguished.
1) θ−θ>π. Thus,
P
i(N, ,p)=Pθ+π
2,θ−π
2.
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M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
Fig. 3. iwhen |S(p)|>2.
2) θ−θ<π. Thus,
P
i(N, ,p)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
Pθ+π
2,θ +3π
2i θ≤π/2,
P0,θ −π
2+Pθ+π
2,2πi θ>π/2andθ<3π/2,
Pθ−3π
2,θ −π
2i θ>π/2andθ≥3π/2.
No e ha he pic u e in Fig. 3is case 2) when θ>π/2 and θ≥3π/2.
5. Axioma ic cha ac e iza ion o he amily o p obabilis ic Owen-Shapley spa ial powe indices
Pe e s and Za zuelo (2017) cha ac e ize he Owen-Shapley spa ial powe index by means o fi e axioms: Equal Powe
Change, Dummy P ope y, Anonymi y, Posi ional In a iance and Reflec ion In a iance. P obabilis ic Owen-Shapley spa ial
powe indices sa is y all bu he las o hem, which equi es symme y o he issues. The whole amily is cha ac e ized by
adding one axiom: a con inui y axiom.
The fi s is equi alen o he ans e axiom o Dubey (1975), as ema ked in Dubey e al. (2005).
Equal Powe Change (EPC) Fo all se o playe s N, all p ∈CN, and all , , w, w∈SN, i − =w −w≥0, hen
ϕ(N, ,p)−ϕ(N, ,p)=ϕ(N,w,p)−ϕ(N,w,p).
Acco ding o his axiom, i he same winning coali ions a e added when going om o as when going om w o
w, hen he change in powe o he playe s when going om o is also he same as when going om w o w.
In he second axiom, i (N, ) ∈SNis such ha |N|≥2 and i ∈N, hen (N {i}, −i) ∈SN {i}is defined by
−i(S)= (S∪{i})i ∅=S⊆N {i},
0i S=∅.
Game (N {i}, −i)can be seen as he esul ing game when playe ilea es and gi es consen o o he s, since any winning
coali ion in (N, )con aining icon inues being winning in (N {i}, −i)when playe iis no longe p esen .
Dummy P ope y (DP) Fo e e y spa ial game (N, , p)and e e y dummy iin (N, , p),
ϕj(N, ,p)=ϕj(N {i}, −i,pN {i})
o all j ∈N {i}.
The e o e, he p esence o a dummy playe does no a ec he powe o he o he playe s; and he powe o a dummy
playe is ze o.
Anonymi y is equi ed, i.e. powe is independen o he names o he playe s. Gi en a spa ial game (N, , p)and an
injec i e unc ion σ:N→U, define he spa ial game (σ(N), σ , σp)by σ (σ(S)) = (S) o all S⊆Nand (σp)σ(i)=pi
o all i ∈N.
Anonymi y (AN) Fo e e y spa ial game (N, , p)and e e y injec i e unc ion σ:N→U,
ϕσ(i)(σ(N), σ ,σp)=ϕi(N, ,p),
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M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
o all i ∈N.
The ollowing axiom, which is also sa isfied by he Owen-Shapley spa ial powe index, is a spa ial in a ian axiom. I he
ela i e posi ions o he agen s do no change wi h espec o a gi en agen , he index o ha agen does no change.
Posi ional In a iance (PI) Fo all playe se s Nand i ∈N, i p, p∈CNsa is y pi=p
iand pj∈[pi, pj, →) o all
j ∈N {i}, hen
ϕi(N, ,p)=ϕi(N, ,p).
The las axiom is a con inui y axiom wi h espec o cons ella ions, which is in oduced by Pe e s and Za zuelo (2017).
Posi ional Con inui y (PC) Le (N, , p)be a spa ial game and {pm}be a sequence o cons ella ions pm∈CNsuch ha
pm→p. Then,
lim
m→∞ϕ(N, ,pm)=ϕ(N, ,p).
Theo em 1. A spa ial powe index ϕsa isfies EPC, DP, AN, PI and PC i and only i he e exis s a non-a omic p obabili y measu e Pon
Bsuch ha ϕ=P.
I is wo h no ing ha Pe e s and Za zuelo (2017)ga e a second cha ac e iza ion adding PC o he fi e a o emen ioned
axioms, weakening DP. The weak DP axiom equi es dummy playe s o ha e ze o powe . In he case o he p obabilis ic
Owen-Shapley spa ial powe indices, DP can no be weakened, as shown by his coun e example. Le 
Pbe a p obabili y
measu e on Bsuch ha 
Pπ
4,3π
4=1. Conside he spa ial powe index ha sa isfies EPC and coincides wi h  o
unanimi y games uSsuch ha |S|=2 and wi h 
Pwhen |S|>2. This spa ial powe index sa isfies EPC, AN, PI, PC and he
weak DP, and i is no a p obabilis ic Owen-Shapley spa ial powe index.
We p o e Theo em 1in h ee s eps (P oposi ions 1, 2and 3). In he fi s s ep we employ his lemma, which is used by
Pe e s and Za zuelo (2017). I ollows om Lemma 2.3 in Einy (1987), see also Einy and Haimanko (2011).
Lemma 1. Le ϕbe a spa ial powe index ha sa isfies EPC and (N, , p)be a spa ial game such ha S1, ..., Ska e he minimal
winning coali ions o (N, ). Then,
ϕ(N, ,p)=
∅=I⊆{1,...,k}
(−1)|I|+1ϕ(N,uk∈ISk,p).
P oposi ion 1. I a spa ial powe index ϕsa isfies EPC, DP, AN, PI and PC, hen, o e e y x ∈R2 he e exis s a non-a omic p obabili y
measu e Pxon Bsuch ha
ϕi(N, ,p)=Px
i(N, ,p)
i pi=x.
P oo . Le ϕbe a spa ial powe index ha sa isfies EPC, DP, AN, PI and PC, and x ∈R2. A p obabili y measu e Pϕ
xon B
mus be ound such ha
ϕi(N, ,p)=Px
i(N, ,p)
i pi=x. Since Pϕ
xis a p obabili y measu e, Pϕ
x(∅)=0. Conside now subin e als in (0,2π]. I α, β∈(0,2π]sa is y α<β
and β−α<π, le i, j, k ∈Uand p ∈C{i,j,k}such ha pi=x, he pola angle θ−−→
pipjis smalle han θ−−→
pipk,
θ−−→
pipj=β+π
2i β+π
2≤2π,
β−3π
2i β+π
2>2π,
and
θ−−→
pipk=α+3π
2i α+3π
2≤2π,
α−π
2i α+3π
2>2π.
Fig. 4shows he case in which β+π
2≤2πand α+3π
2≤2π. Define
Pϕ
x(α,β]=ϕi({i,j,k},u{i,j,k},p).
529
M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
Fig. 4. The case in which β+π
2≤2πand α+3π
2≤2π.
Fig. 5. G aphic o i,iand i.
This is well defined. Indeed, by PI, jand kcan be loca ed nea e o o u he om iin a s aigh line. And by AN, his
index does no depend on he names o he playe s.
I α, β∈(0,2π]sa is y α<β and β−α≥π, hen he e exis s γ∈(0,2π]such ha α<γ<β, γ−α<πand
β−γ<π, and define
Pϕ
x(α,β]=Pϕ
x(α,γ]+Pϕ
x(γ,β].
To p o e ha i is also well defined ake γ∈(0,2π]such ha α<γ<β, γ−α<πand β−γ<π, and p o e ha
Pϕ
xα,γ+Pϕ
xγ,β=Pϕ
x(α,γ]+Pϕ
x(γ,β].(2)
Assume, wi hou loss o gene ali y, ha γ<γ; hen he abo e equali y becomes
Pϕ
xα,γ=Pϕ
x(α,γ]+Pϕ
xγ,γ.(3)
By defini ion o Pϕ
x, he e exis i, j, k ∈U, and p ∈C{i,j,k}such ha pi=xand
Pϕ
xα,γ=ϕi({i,j,k},u{i,j,k},p).
Le nbe he line ha passes h ough piand has a di ec ion ec o wi h pola angle γ(Fig. 5). The e exis y ∈pi,pj
and z∈(pi,pk)such ha he line ha passes h ough yand zis pe pendicula o n. Applying ii) o Lemma 2in he
Appendix,
ϕi({i,j,k},u{i,j,k},p)=ϕi0({i0,i1,j},u{i0,i1,j},
q)+ϕi1({i0,i1,k},u{i0,i1,k},
q),
whe e

q∈C{i0,i1,j}and

q∈C{i0,i1,k}sa is y

qi0=pi,
qj=pj,
qi1=pi+z−y,
and
530
M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
Fig. 6. G aphic o x,y,z∈R2, he pola angle αand he sequences αmand pm.

qi1=pi,
qk=pk,
qi0=pi+y−z.
And aking in o accoun he defini ion o Pϕ
x, equali y (3)is p o ed. Subs i u ing in (2)gi es
Pϕ
xγ,γ+Pϕ
xγ,β=Pϕ
x(γ,β],
which is ue because β−γ<πand can be p o ed wi h he same easoning as abo e.
I can be p o ed simila ly ha i α, β∈(0,2π]sa is y α<β, hen
Pϕ
x(α,β]=
m

k=1
Pϕ
x(αk,β
k],(4)
when
(α,β]=
m

k=1
(αk,β
k]
and he in e als (αk,β
k]a e pai wise disjoin .
Lemma 4in he Appendix p o es ha Pϕ
xhas a unique ex ension (w i en also Pϕ
x) on B ha is a p obabili y measu e.
We now p o e ha Pϕ
xis non-a omic, ha is, Pϕ
x(α)=0 o all α∈(0,2π].
Take x, y, z∈R2such ha x ∈(y,z)and le αbe he pola angle o a di ec ion ec o o he line ha passes h ough x
and is pe pendicula o he line ha passes h ough yand z(as depic ed in Fig. 6). Le αmbe a sequence in (0,2π]such
ha {αm}→α, i, j, k ∈Uand pm∈C{i,j,k}be a sequence such ha
pm
i=x,pm
j=y,pm
k→z
and
Pϕ
x(αm,α]=ϕi({i,j,k},u{i,j,k},pm), (5)
whe e αmis he pola angle o a di ec ion ec o o he line ha passes h ough xand is pe pendicula o he line ha
passes h ough xand pm
k. Thus,
lim
m→∞ϕi({i,j,k},u{i,j,k},pm)=ϕi({i,j,k},u{i,j,k},(x,y,z))=0,
whe e he fi s equali y holds by PC and he second by DP.
The e o e, by (5),
lim
m→∞ Pϕ
x(αm,α]=0.
Since Pϕ
xis a p obabili y measu e,
lim
m→∞ Pϕ
x(αm,α]=Pϕ
x(α),
and hence, Pϕ
x(α)=0.
Finally, we p o e ha
ϕi(N, ,p)=Px
i(N, ,p)
531
M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
i pi=x. Since ϕsa isfies EPC, by Lemma 1i is sufficien o conside he equali y o unanimi y games. By cons uc ion,
he equali y holds o (N, uS, p)i |S|>2 and he e a e a leas h ee non-dummy playe s in S. I ob iously holds when
|S|=1. The e o e, by DP, i is sufficien o conside (S, uS, p)whe e |S|=2, i.e., S={i,j}⊆N.
I mus now be p o ed ha ϕi({i,j}, u{i,j}, p)coincides wi h Pϕ
x(i)(see Fig. 2). Le y ∈pi,pjand {ym}⊆R2be a
sequence such ha {ym}→y. Le k ∈N {i,j}and pm∈C{i,j,k}be a sequence such ha pm
{i,j}=p{i,j}and pm
k=ym. Thus,
PC and DP gi e
ϕi({i,j},u{i,j},p)=lim
m→∞ϕi({i,j,k},u{i,j,k},pm). (6)
By defini ion o Pϕ
x, i ollows ha ϕi({i,j,k}, u{i,j,k}, pm) =Pϕ
xm
i, whe e m
iis a sequence o in e als o unions o wo
disjoin in e als in (0,2π]whose limi is i. Since Pϕ
xis a p obabili y measu e,
lim
m→∞ Pϕ
xm
i=Pϕ
x(i),
which, oge he wi h (6), implies he equi ed esul . 
P oposi ion 2. I a spa ial powe index ϕsa isfies DP, AN, PI and PC, hen, o e e y x, y ∈R2,
Pϕ
x=Pϕ
y.
P oo . i) Le α, β∈(0,2π]. We p o e ha Pϕ
x()=Pϕ
y()when  =(α,β]⊆(0,2π], whe e β−α=π, o  =(α,2π]∪
(0,β]⊆(0,2π], whe e 2π−α+β=π.
Le i, j ∈Uand p ∈C{i,j}such ha pi=x, pj=yand  =i, whe e iis he se o med by he pola angles o he
ec o s ha s a a piand poin o he hal -plane wi h con ou ⊥ ha does no con ain pj(see Fig. 2). Since ϕis a spa ial
powe index,
ϕi({i,j},u{i,j},p)+ϕj({i,j},u{i,j},p)=1.
By P oposi ion 1,
ϕi({i,j},u{i,j},p)=Pϕ
x(i)
and
ϕj({i,j},u{i,j},p)=Pϕ
yj,
whe e j=(0,2π] i. Subs i u ing he wo equali ies in he fi s one and aking in o accoun ha Pϕ
yis a p obabili y
measu e on B, Pϕ
x(i)=Pϕ
y(i)is p o ed.
ii) Le α, β∈(0,2π]⊆(0,2π]. Le  =(α,β]⊆(0,2π], whe e β−α<π, o  =(α,2π]∪(0,β]⊆(0,2π], whe e
2π−α+β<π. Le (see Fig. 7) i, j, k, i0, i1∈U, p ∈C{i,j,k}such ha pi=x, y ∈pi,pj, z∈(pi,pk)and is he se o
pola angles o he ec o s ha s a a piand poin o he in e sec ion o he hal -plane wi h con ou ⊥ ha does no
con ain yand he hal -plane wi h con ou 
⊥ ha does no con ain z. By i) o Lemma 2,
ϕi({i,j,k},u{i,j,k},p)
=ϕi0({i0,i1,j,k},u{i0,i1,j,k},q)+ϕi1({i0,i1,j,k},u{i0,i1,j,k},q), (7)
whe e q ∈C{i0,i1,j,k}sa isfies q{j,k}=p{j,k}, qi0=yand qi1=z. Mo ing pkclose o yin a s aigh line, by PI, he fi s index
on he igh -hand side o (7)does no change. Mo eo e , when pk∈co
y,pj,z, playe kbecomes a dummy playe . Thus,
DP implies
ϕi0({i0,i1,j,k},u{i0,i1,j,k},q)=ϕi0({i0,i1,j},u{i0,i1,j},q{i0,i1,j}).
Simila ly, app oaching pjclose o zin a s aigh line, by PI, he second index on he igh -hand side o (7)does no change.
And playe jbecomes a dummy playe when pj∈co
({y,pk,z}). Again by DP,
ϕi1({i0,i1,j,k},u{i0,i1,j,k},q)=ϕi1({i0,i1,k},u{i0,i1,k},q{i0,i1,k}).
Then, (7) u ns in o
ϕi({i,j,k},u{i,j,k},p)
=ϕi0({i0,i1,j},u{i0,i1,j},q{i0,i1,j})+ϕi1({i0,i1,k},u{i0,i1,k},q{i0,i1,k}).
532
M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
By defini ion o Pϕ
x, and applying PI (app oaching ymclose o xin a s aigh line) and DP (i2becomes a dummy playe ),
ϕi0N,uN,pm=Pϕ
xπ,3π
2.
Applying PI (mo ing zmclose o xmin a s aigh line) and DP (i3becomes a dummy playe ),
ϕi1N,uN,pm=ϕi1{i0,i1,i2},u{i0,i1,i2},pm
{i0,i1,i2}.(16)
And by PI (mo ing pm
i0and pm
i2 u he om pm
i1in s aigh lines),
ϕi1{i0,i1,i2},u{i0,i1,i2},pm
{i0,i1,i2}=ϕi1{i0,i1,i2},u{i0,i1,i2},pi1m,(17)
whe e pi1m
i1=pm
i1, pi1m
i0=pm
i1−(1,0)and pi1m
i2=pm
i1+(0,1).
Since pm
i1→x, i ollows ha pi1m→pi1, whe e pi1∈C{i0,i1,i2}sa isfies pi1i1=x, pi1i0=x −(1,0)and pi1i2=
x +(0,1). And he e o e, PC implies
lim
m→∞ϕi1{i0,i1,i2},u{i0,i1,i2},pi1m=ϕi1{i0,i1,i2},u{i0,i1,i2},pi1,
which, oge he wi h (16) and (17), implies
lim
m→∞ϕi1N,uN,pm=ϕi1{i0,i1,i2},u{i0,i1,i2},pi1.
By defini ion o Pϕ
x, he igh -hand side o his equali y equals o Pϕ
x3π
2,2π, and hence,
lim
m→∞ϕi1N,uN,pm=Pϕ
x3π
2,2π.
Simila ly, we ha e
lim
m→∞ϕi2N,uN,pm=Pϕ
x0,π
2
and
lim
m→∞ϕi3N,uN,pm=Pϕ
xπ
2,π.
Consequen ly, aking limi s on bo h sides o (15),
Pϕ
xπ,3π
2+Pϕ
x3π
2,2π+Pϕ
x0,π
2+Pϕ
xπ
2,π=1.
Tha is, by (4), Pϕ
x(0,2π]=1. 
Lemma 4. I ϕis a spa ial powe index ha sa isfies DP, AN, PI and PC, hen, o e e y x ∈R2, Pϕ
xis a p obabili y measu e on B.
P oo . We p o e ha Pϕ
xis a p obabili y measu e on he field o fini e unions o in e als, deno ed by B0. The esul ollows
since any p obabili y measu e on a field has a unique ex ension ha is a p obabili y measu e on he associa ed σ-field.
Pϕ
xhas o sa is y se e al p ope ies in o de o be a p obabili y measu e on B0.
i)We p o e in he p e ious lemma ha Pϕ
x(0,2π]=1.
ii)Pϕ
xis coun ably addi i e on B0. We p o e his in wo s eps.
•Fi s , on he class o in e als. Le (αm,β
m]be a fini e o infini e sequence o pai wise disjoin in e als. I needs o be
p o ed ha i
(α,β]=∞

m=1
(αm,β
m],
hen
Pϕ
x(α,β]=∞

m=1
Pϕ
x(αm,β
m].
539

M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
The fini e case is al eady p o en (exp ession (4)), so now he infini e case is conside ed. Since Pϕ
x(αm,β
m]≥0 o all m ∈N,
he e ms in his se ies can be ea anged so ha βm+1<β
m o all m ∈Nand β1=β(no e also ha {αm}→α). Hence,
aking in o accoun (4),
M

m=1
Pϕ
x(αm,β
m]=Pϕ
x(αM,β]
holds o all M∈N, so he equali y o be p o ed is
Pϕ
x(α,β]=lim
M→∞ Pϕ
x(αM,β].
We dis inguish wo cases.
I β−α<π, le i, j, k ∈Nand p ∈C{i,j,k}such ha pi=xand
Pϕ
x(α,β]=ϕi({i,j,k},u{i,j,k},p), (18)
whe e α( esp. β) is he pola angle o a di ec ion ec o o he line which is pe pendicula o he line ha passes h ough
piand pk( esp. pj) (Fig. 4). Le {ym}⊆R2be a sequence such ha {ym}→pkand αm( o big enough m ∈N) is he abo e
pola angle associa ed wi h piand ym. Le pm∈C{i,j,k}such ha pm
{i,j}=p{i,j}(obse e ha pm
i=x) and pm
k=ym. Thus, PC
implies
ϕi({i,j,k},u{i,j,k},p)=lim
m→∞ϕi({i,j,k},u{i,j,k},pm). (19)
By he defini ion o Pϕ
x,
ϕi({i,j,k},u{i,j,k},pm)=Pϕ
x(αm,β],
so his equali y, oge he wi h (18) and (19), implies he equi ed esul .
I β−α≥π, he e exis s γsuch ha α<γ<β, γ−α<π, (and by (4))
Pϕ
x(α,β]=Pϕ
x(α,γ]+Pϕ
x(γ,β],
and
Pϕ
x(αm,β]=Pϕ
x(αm,γ]+Pϕ
x(γ,β].
Hence, he equali y o be p o ed educes o
Pϕ
x(α,γ]=lim
M→∞ Pϕ
x(αM,γ],
which is ue because his is he fi s case again.
•Now we p o e ha Pϕ
xis coun ably addi i e on B0.
Fi s , i
A=
m

k=1
Ik∈B0,
whe e Ika e pai wise disjoin in e als, define
Pϕ
x(A)=
m

k=1
Pϕ
x(Ik).
This is well defined because i
A=
m

l=1
I
l,
hen
A=
m

k=1
m

l=1Ik∩I
l,
and he e o e,
540
M.J. Albizu i and A. Goikoe xea Games and Economic Beha io 136 (2022) 524–541
Pϕ
x(A)=
m

k=1
m

l=1
Pϕ
xIk∩I
l,
and by (4), his double addi ion coincides wi h bo h
m

k=1
Pϕ
x(Ik)and
m

l=1
Pϕ
xI
l.
To p o e ha Pϕ
xis coun ably addi i e on B0, le Akbe a sequence o pai wise disjoin elemen s in B0such ha
A=∞

k=1
Ak∈B0.
Since A ∈B0and Ak∈B0, i ollows ha
A=
m

l=1
Iland Ak=
mk

l=1
Ik
l,
whe e Iland Ik
la e pai wise disjoin in e als. Thus, since Pϕ
xis coun ably addi i e on he class o in e als,
Pϕ
x(A)=
m

l=1
Pϕ
x(Il)=
m

l=1
∞

k=1
mk

l=1
Pϕ
xIl∩Ik
l=∞

k=1
mk

l=1
Pϕ
xIk
l=∞

k=1
Pϕ
x(Ak).
iii)Taking in o accoun ha ϕis non-nega i e, Pϕ
x(A)≥0 o all A ∈B0and Pϕ
x(A)≤1 o all A ∈B0(since
Pϕ
x(0,2π]=1). 
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