AN AXIOMATIC CHARACTERIZATION
OF THE POTENTIAL DECISIVENESS INDEX
Josep F eixas and Mon se a Pons∗
No embe 6, 2013
Abs ac
Le us conside ha somebody is ex emely in e es ed in inc easing he p ob-
abili y o a p oposal o be app o ed by a ce ain commi ee and le us assume
ha o achie ing his goal he/she is p epa ed o pay off one membe o he
commi ee. In a si ua ion like his one, and assuming ha o e-buying is al-
lowed and ee o s igma, which o e should be offe ed a b ibe? The po en ial
decisi eness index o simple games, which measu es he effec ha ensu ing
one posi i e o e p oduces in he p obabili y o pass he issue a hand, is a
good ool o ge he answe . An axioma ic cha ac e iza ion o his index is
gi en in his pape , and i s ela ion o o he classical powe indices is showed.
Key wo ds: Game heo y, Po en ial decisi eness index; a measu e o b ibes;
axioma iza ion; s anda d powe indices; ela ionship among se e al measu es.
Ma h. Subj. Class. (2000): 91A12, 91A40, 91A80, 91B12.
JEL Classi ica ion Numbe s: C71, D71.
∗“Depa amen de Ma em`a ica Aplicada III i Escola Poli `ecnica Supe io d’Enginye ia de Man-
esa (Uni e si a Poli `ecnica de Ca alunya).” Spain. Resea ch pa ially suppo ed by “Minis e io
de Econom´ıa y Compe i i idad p oyec o MTM2012-34426/FEDER” and “Go e n de la Gene ali-
a de Ca alunya SGR 2009–1029”.
E-mails: josep. [email protected], mon se a [email protected]
1 In oduc ion
Assume ha a p oposal has o be submi ed o a fini e se o o e s, ha each o e
has an independen a p io i p obabili y o o ing in a o o he p oposal and ha
some o ing ules a e es ablished o deciding i he p oposal will ei he be accep ed
o ejec ed a e he o es a e cas .
Suppose now ha an ex e nal influence is able o inc ease ill 1 he p obabili y
o a o e o accep ing he p oposal. O cou se, i his happens, and his o e
has e en a small influence in he final esul , he p obabili y o he p oposal being
app o ed will inc ease. The amoun o his inc easing effec is ob iously no he
same o all o he o e s. I depends on how c ucial is his/he o e and i also
depends on his/he ini ial p obabili y o o e o he p oposal.
A new index Ω o measu ing po en ial decisi eness o o e s in his con ex
was in oduced in [22], and i was p o ed ha ensu ing he a o able o e o he
o e wi h maximum Ω–measu e is he way o ob ain he g ea es inc emen in he
p obabili y o ge ing he p oposal app o ed.
Example 1.1 Assume ha a ju y has o ake a decision on a case. Fo pu poses
o he example, we will suppose he e a e 4 ju o s, one o whom is he p esiden o
he ju y. Each ju o will o e o ei he con ic ion o acqui al and he ou come o
he o e will be he majo i y decision o he ju y. Because ies a e possible, hese
will be esol ed by he cas ing o e o he p esiden , i.e., he p esiden plays he
ole o ieb eake in he ju y. Assume u he ha an ex e nal pe son, who is e y
in e es ed in he e dic o he ial, es ima es ha he p esiden will o e o acqui al
wi h p obabili y 1−p, while he o he 3 ju o s will o e o acqui al wi h an equal
p obabili y p. I he ou side conside s he possibili y o b ibing one o he ju o s o
ensu e his/he o e o acqui al wi h p obabili y 1, which one o he ju o s would
he/she a he selec o be offe ed he b ibe: he p esiden o any o he o he h ee
ju o s?
Some esul s in [23] allow us o selec a lis o o e s o be pe suaded, gi en
any pa icula anking o hei p edic ions, and his p ocedu e, which can be easily
implemen ed in a compu e , can be applied in an analogous way o selec a lis o
o e s o be b ibed, i.e. o e s wi h maximum alue o he Ω–measu e. We e e
o hese wo pape s o mo e examples abou he applicabili y o he Ω–measu e.
2
The usual model o a o ing scena io like he one desc ibed is a simple game,
ha is o say, a pai (N, W), whe e N={1,2, . . . , n}deno es he se o o e s, and
Wis he se o winning coali ions, i.e., se s o o e s whose a o able o e ensu es
he accep a ion o he p oposal. Subse s o N ha a e no in Wa e called losing
coali ions, and i is assumed ha : 1) ∅is losing; 2) subse s o losing coali ions a e
again losing (mono onici y). I is assumed ha W =∅, so ha Nis always a
winning coali ion. A winning coali ion is minimal i each p ope subse is a losing
coali ion. The se o minimal winning coali ions is usually deno ed by Wm, and,
because o mono onici y, i comple ely de e mines he game. Gi en S⊆N,S=∅,
he S–unanimi y game (N, US) is he game which has Sas he unique minimal
winning coali ion. I S=N he game (N, UN) is jus called he unanimi y game. A
o e i∈Nis null in (N, W) i idoes no belong o any minimal winning coali ion,
and i is a e oe i i belongs o all o hem. I is clea ha in he S–unanimi y
game (N, US) all o e s in N Sa e null and all o e s in Sa e e oe s.
Classically, he only elemen s which a e aken in o accoun o define he powe
o a pa icula o e ia e he se No all o e s and he o ing ule, defined by he
se Wo winning coali ions. The defini ions o powe indices y o eflec diffe en
aspec s o powe . Mos o hem ely on he idea o measu ing decisi eness (see [33]
[29], [34], [5], [14] o [15] among o he s), bu o he aspec s like success ha e also been
used ([31], [18], [9], [7], [8], [35], [24]). The e exis ano he app oach, which we call
he con ex ual app oach, ha akes in o accoun , o measu e he powe o a o e i,
no only he elemen s Nand Wbu also a p obabili y dis ibu ion po e he o e
configu a ions ha can eme ge. This con ex ual amewo k was in oduced, as a
as we know, by La uelle and Valenciano ([26]), al hough some au ho s had al eady
conside ed his kind o powe indices be o e, wi h diffe en p obabili y dis ibu ions
([18], [35]). When independence o o e ’s o es is assumed hen he p obabili y
dis ibu ion po e he o e configu a ions is comple ely de e mined by he p oba-
bili ies ec o p= (p1, . . . , pn)∈[0,1]n, whe e piis he a p io i p obabili y o each
o e i o o ing in a o o he p oposal.
Ei he in he classical app oach o in he con ex ual one, a powe index can
also be defined by a se o p ope ies which uniquely cha ac e ize i . This has been
done o mos o he classical indices, in pa icula he fi s axioma iza ion o he
Shapley–Shubik index on simple games was gi en in [17] and he fi s one o he
Banzha index in [18] (diffe en al e na i e axioma iza ions ha e been p oposed, see
3
o example [32], [19]). In he con ex ual app oach, a decisi eness index, which
ex ends he Banzha index, was p oposed in [18] and axioma ized in [10], diffe en
success indices we e in oduced in [26] and axioma ized in [2], and he po en ial
decisi eness index was in oduced in [22] and an axioma iza ion o i is p esen ed
in his wo k.
The pape is o ganized as ollows. In Sec ion 2 he defini ion and he mo i a ion
o he po en ial decisi eness a e ecalled. An axioma ic cha ac e iza ion o his
measu e is es ablished in Sec ion 3, and he independence o he axioms is p o ed.
Sec ion 4 is de o ed o ela e his index wi h he classical Banzha and Shapley–
Shubik indices, and Sec ion 5 summa izes he con en s o he pape and poin s ou
some u u e ques ions o wo k on.
2 The Ωmeasu e o po en ial decisi eness
Le (N, W) be a simple game, whe e N={1,2, . . . , n}deno es he se o o e s (we
assume ha n≥2) and Wis he se o winning coali ions. Assume ha each o e ’s
o e is independen o he o he s’ and le pibe he a p io i p obabili y o o e i
o o ing in a o o he p oposal. Ou con ex ual model is a iple (N, W,p),
whe e (N, W) is he simple game and p= (p1, . . . , pn)∈[0,1]nis he p obabili ies
ec o . In [10] and [11], his iple is called assessed simple game and we also use
his nomencla u e in his pape . The se o all assessed simple games is deno ed by
ASG.
Unde he assump ion o independence o o e ’s o es, he p obabili y o a
p oposal being accep ed in (N, W,p) is gi en by
(N, W,p) = ∑
S∈W ∏
i∈S
pi∏
i/∈S
(1 −pi).(1)
The unc ion is he mul ilinea ex ension (MLE) o he simple game (N, W)
which was in oduced by Owen in [28] in he gene al con ex o coope a i e games.
The MLE o a simple game is a polynomial unc ion. Thus, i is con inuous in i s
domain [0,1]nand diffe en iable in (0,1)n. I e ifies wo ypes o mono onici y
p ope ies:
• (N, W,p)≤ (N, W′,p) i W ⊆ W′,
4
• (N, W,p)≤ (N, W,p′) i p≤p′(componen wise).
We will use (p) ins ead o (N, W,p) whene e he e is no possible misunde -
s anding.
The inc emen on he p obabili y (p) due o an inc emen ∆pion piis:
∆i (p) = (p+ ∆i(p)) − (p) = i(p)∆pi(2)
whe e ∆i(p) = (0, . . . , 0,∆pi,0, . . . , 0), and is ands o he pa ial de i a i e o
wi h espec o he componen i, which is non-nega i e.
No e ha ∆i (p) depends on i(p) bu also on he alues ∆pi ha is possible o
achie e. Indeed, i is ob ious ha i pi= 1 no inc ease o his p obabili y is possible,
while i pi= 0 we can hink o an inc ease ∆pi= 1. So he po en ial decisi eness
impo ance o a o e idepends on wo ac o s: he a e o change i(p) and he a
p io i p obabili y pi. This is he mo i a ion gi en in [22] o defining he index Ω in
he ollowing way:
De ini ion 2.1 The po en ial decisi eness index Ω is he map ha assigns o e e y
(N, W,p)∈ASG a ec o Ω(N, W,p)∈[0,1]ndefined by:
Ωi(N, W,p) = (1 −pi) i(p).
The unc ion Ω is, o any fixed game (N, W), a con inuous unc ion on [0,1]n, di -
e en iable o any o de in i s in e io (0,1)n. F om (2) i is clea ha Ωi(N, W,p) =
(1i,p)− (p), whe e (1i,p) deno es he alue o on he ec o (1i,p) ob ained
om pby eplacing piwi h 1. Thus, his index gi es p ecisely he inc emen o
(p) ob ained by only changing he i–componen o p om pi o 1. Co olla y 3.3
in [22] shows ha 0 ≤Ωi(N, W,p)≤1, whe e 0 is only achie ed o null o e s o
o any o he o e wi h pi= 1 (i.e., pu e yes– o e s), whe eas 1 is only achie ed
o a dic a o being a pu e no- o e (i.e., Wm={{i}} and pi= 0).
As he diffe ence (1i,p)− (p) o , equi alen ly, Ωi(N, W,p) equals he inc ease
o p obabili y o he issue a hand o be passed when only o e ichanges his/he
o e om pi o 1, Ω is he mos na u al measu e, om he p obabilis ic poin o iew,
o b ibes o o e buying in he con ex o assessed simple games, when he alleged
b ibe is in e es ed in app o ing he p oposal. Once s a ed ha his obse a ion in
e ms o p obabili y is he main suppo o his measu e, we addi ionally p opose
in his pape a fi s axioma ic cha ac e iza ion o i . Thus, om he esul s o his
5

pape , he index Ω has suppo om bo h app oaches, p obabilis ic and axioma ic.
Needless o say ha finding o he axioma iza ions o Ω is an open issue.
This wo old cha ac e iza ion is a na u al p ocedu e o he jus ifica ion o well
known powe indices in simple games. Fo ins ance, ei he he Banzha o he
wo Coleman’s powe indices admi se e al axioma ic cha ac e iza ions bu also a
p obabilis ic in e p e a ion, see e.g. [27]. We also e e he in e es ed eade o [22]
and [23] o addi ional heo e ical in o ma ion abou Ω, which, as a as we know,
is he only ool exp essly in oduced o measu e he po en ial decisi eness o o e s
in he con ex o assessed simple games.
No e ha i he alleged b ibe was in e es ed in de ea ing he p oposal (ins ead
o app o ing i ) hen he diffe ence (p)− (0i,p) would be he app op ia e measu e
because i gi es he inc ease o p obabili y, in absolu e alue, o he issue a hand
o be de ea ed when only o e ichanges his/he o e om pi o 0. In his las
exp ession, (0i,p) deno es he alue o on he ec o (0i,p) ob ained om pby
eplacing piwi h 0. This measu e o assessed simple games is somehow analogous o
Ω because, by applying (2) wi h ∆i(p) = (0, . . . , 0,−pi,0, . . . , 0), we ob ain (p)−
(0i,p) = pi i(p).
Be o e con inuing wi h he axioma ic cha ac e iza ion le us e u n o Exam-
ple 1.1.
Example 2.2 (Example 1.1 e isi ed)
Fo he o ing sys em in Example 1.1, we ha e N={1,2,3,4}, whe e 1deno es
he p esiden ,
W={{1,2},{1,3},{1,4},{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}}.1
Fo his game, exp ession (1) gi es:
(p) = p1p2+p1p3+p1p4−p1p2p3−p1p2p4−p1p3p4+p2p3p4.
Thus, he pa ial de i a i es a e:
1(p) = p2+p3+p4−p2p3−p2p4−p3p4, 2(p) = p1−p1p3−p1p4+p3p4,
3(p) = p1−p1p2−p1p4+p2p4, 4(p) = p1−p1p2−p1p3+p2p3.
Le us conside now he pa icula alue o p= (1 −p, p, p, p) o some 0<p<1.
Fo his p obabili y ec o we ha e: 1(p) = 3p(1 −p)and 2(p) = 3(p) = 4(p) =
1This game can also be ep esen ed by he weigh ed game wi h ep esen a ion [3; 2,1,1,1].
6
1−3p+3p2.Thus, we can compa e he po en ial decisi eness index o he p esiden ,
i.e., playe 1, wi h he po en ial decisi eness index o any o he ju o . Wi hou loss
o gene ali y we ake playe 4:
Ω1(N, W,p) = (1 −p1) 1(p)=3p2(1 −p),
Ω4(N, W,p) = (1 −p4) 4(p)=1−4p+ 6p2−3p3.
Thus, Ω1(N, W,p)−Ω4(N, W,p) = −3p2+ 4p−1and
Ω1(N, W,p)−Ω4(N, W,p)>0⇔p∈(1/3,1),
Ω1(N, W,p)−Ω4(N, W,p)<0⇔p∈(0,1/3).
Hence, acco ding o he po en ial decisi eness index, he p esiden o he ju o is he
bes candida e o be b ibed in (N, W,p)i p > 1/3, while o p < 1/3any o he ju o
should be chosen as a candida e o be b ibed. No e also ha he maximum diffe ence
in he in e al (1/3,1) is achie ed o p= 2/3.
We ema k ha compu ing he MLE o a simple game is a complex ask when he
numbe o a iables in ol ed is high. Some bounds a e ob ained in [20], and a ious
compu a ion me hods can be ound, in ano he con ex , in [6] and [25].
F om now on we es ic ou wo k in p o ing some p ope ies o he Ω measu e,
and in gi ing an axioma ic cha ac e iza ion o i .
3 Axioma ic cha ac e iza ion o he Ωmeasu e
In his sec ion we es ablish some ma hema ical p ope ies o he Ω measu e and use
hem o gi e an axioma ic cha ac e iza ion o i . These p ope ies a e consequence
o some cha ac e is ics o he MLE o a simple game ha we collec in he ollowing
lemma. The fi s pa will be used in he axioma iza ion o his index, while he
second pa is basic o es ablishing he ela ionship o he Ω measu e wi h he
Shapley-Shubik index.
Lemma 3.1 Le (N, W,p)be an assessed simple game and (N, W,p)i s MLE as
defined in (1).
(a) I (N,
W,p)is ano he assessed simple game, hen
(N, W ∪
W,p) + (N, W ∩
W,p) = (N, W,p) + (N,
W,p).
7
(b) I π:N→Nis a pe mu a ion on N, hen (N, π(W),p) = (N, W, π(p)),
whe e π(W) = {π(S)|S∈ W} and, π(p) = (pπ(1), . . . , pπ(n)).
P oo :
(a) Fo any subse Ao 2Nwe define
(N, A,p) = ∑
S∈A ∏
i∈S
pi∏
i/∈S
(1 −pi).
I (N, A) is a simple game hen is i s MLE as defined in (1). I is also clea ha i
{W1,W2}is a pa i ion o W hen (N, W,p) = (N, W1,p) + (N, W2,p). Thus,
(N, W ∪
W,p) = (N, W
W,p) + (N,
W W,p) + (N, W ∩
W,p)
= (N, W,p)− (N, W ∩
W,p)+ (N,
W,p)− (N, W ∩
W,p)+ (N, W ∩
W,p)
= (N, W,p) + (N,
W,p)− (N, W ∩
W,p).
(b) F om (1) we can w i e
(N, π(W),p) = ∑
S∈π(W)∏
k∈S
pk∏
k/∈S
(1 −pk) = ∑
π−1(S)∈W ∏
k∈S
pk∏
k/∈S
(1 −pk)
=∑
S∈W ∏
k∈π(S)
pk∏
k/∈π(S)
(1 −pk) = ∑
S∈W ∏
π−1(k)∈S
pk∏
π−1(k)/∈S
(1 −pk)
= (N, W, π(p))

In he ollowing heo em, ou basic p ope ies o he po en ial decisi eness index
Ω a e es ablished. We will p o e la e ha hese axioms comple ely cha ac e ize his
index. In he ollowing defini ion we in oduce some new concep s needed o enounce
he heo em.
De ini ion 3.2
Le i∈Nand N−i=N {i}. The new game (N−i,W−i) is defined by
S∈ W−ii and only i S⊆N−iand S∪ {i} ∈ W
The game (N−i,W−i) is he educed game o (N, W) de e mined by N {i}as
defined in [36]. The no a ion we use is bo owed om [10].
8
Theo em 3.3 Le (N, W,p)∈ASG and Ωbe he po en ial decisi eness index.
(A1) Null o e p ope y: I jis null in (N, W) hen Ωj(N, W,p) = 0.
(A2) Ex e nal null o e p ope y. I jis null in (N, W) hen
Ωi(N, W,p) = Ωi(N−j,W−j,p−j)
o any i∈N(i=j), whe e he j h componen o phas been dele ed in p−j.
(A3) T ans e p ope y: I (N,
W)is ano he simple game, hen
Ω(N, W ∪
W,p) + Ω(N, W ∩
W,p) = Ω(N, W,p) + Ω(N,
W,p).
(A4) Unanimi y p ope y: I (N, UN) is he unanimi y game hen
Ωi(N, UN,p) = (1 −pi)∏
k∈N
k=i
pk
o all i∈N.
P oo :
We s a by p o ing ha i jis null in (N, W), hen (N, W,p) = (N−j,W−j,p−j).
I jis null in (N, W) hen i can no belong o any minimal winning coali ion, so
ha S∈ W and j∈Simplies ha S {j} ∈ W. Thus, we can w i e:
(N, W,p) = ∑
S∈W
j /∈S∏
k∈S
pk∏
k/∈S
(1 −pk) + ∑
S∈W
j∈S∏
k∈S
pk∏
k/∈S
(1 −pk)
= (1 −pj)∑
S∈W
j /∈S∏
k∈S
pk∏
k/∈S
k=j
(1 −pk) + pj∑
S∈W
j /∈S∏
k∈S
pk∏
k/∈S
k=j
(1 −pk)
=∑
S∈W
j /∈S∏
k∈S
pk∏
k/∈S
k=j
(1 −pk) = ∑
S∈W−j∏
k∈S
pk∏
k/∈S
(1 −pk)
= (N−j,W−j,p−j).
(3)
(A1) I jis null in (N, W) hen, om (3), i is clea ha pjdoes no appea in he
exp ession o (N, W,p) so ha i s co esponding pa ial de i a i e j(N, W,p) =
0. Thus Ωj(N, W,p) = 0.
(A2) I jis null in (N, W), hen, om (3), (N, W,p) = (N−j,W−j,p−j), and
he e o e hei espec i e pa ial de i a i es wi h espec o any componen i=j
coincide. Thus, Ωi(N, W,p) = Ωi(N−j,W−j,p−j) o any i=j.
9
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