An axiomatic characterization of the potential decisiveness index

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 [7] B. Ba y. Is i be e o be powe ull o lucky? pa i. Poli ical S udies, pages 183–194, 1980. [8] B. Ba y. Is i be e o be powe ull o lucky? pa ii. Poli ical S udies, pages 338–352, 1980. [9] S.J. B ams and M. Lake. Powe and sa is ac ion in a ep esen a i e democ acy. In P. O deshook, edi o , Game Theo y and Poli ical Science, pages 529–562. New Yo k Uni e si y P ess, 1978. [10] F. Ca e as. α-Decisi eness in simple games. Theo y and Decision, 56:77–91, 2004. [11] F. Ca e as. A decisi eness index o simple games. Eu opean Jou nal o Op- e a ional Resea ch, 163:370–387, 2005. [12] F. Ca e as and J. F eixas. Some heo e ical easons o using egula semi al- ues. In H. De Swa , edi o , Logic, Game Theo y and Social Choice. P oceedings o he In e na ional Con e ence, LGS, pages 140–154, Tilbu g, he Ne he lands, 1999. [13] F. Ca e as and J. F eixas. Semi alue e sa ili y and applica ions. Annals o Ope a ions Resea ch, 109:341–356, 2002. [14] J.S. Coleman. Con ol o collec i i ies and he powe o a collec i i y o ac . In B. Liebe man, edi o , Social Choice, pages 269–300. Go don and B each, New Yo k, USA, 1971. [15] J.S. Coleman. Combina o ics: Indi idual In e es s and Collec i e Ac ion: se- lec ed Essays. Camb idge Uni e si y P ess, 1986. [16] L. Diffo Lambo and J. Moulen. O dinal equi alence o powe no ions in o ing games. Theo y and Decision, 53:313–325, 2002. [17] P. Dubey. On he uniqueness o he Shapley alue. Ame ican Poli ical Science Re iew, 4:131–139, 1975. [18] P. Dubey and L.S. Shapley. Ma hema ical p ope ies o he Banzha powe index. Ma hema ics o Ope a ions Resea ch, 4:99–131, 1979. 16 [19] V. Fel kamp. Al e na i e axioma ic cha ac e iza ions o he Shapley and he Banzha alues. In e na ional Jou nal o Game Theo y, 24:179–186, 1995. [20] J. F eixas. Bounds o he owen’s mul ilinea ex ension. Jou nal o Applied P obabili y, 44:852–864, 2007. [21] J. F eixas. On o dinal equi alence o he Shapley and Banzha alues o coop- e a i e games. In e na ional Jou nal o Game Theo y, 39:513–527, 2010. [22] J. F eixas and M. Pons. Two measu es o ci cums an ial powe : influences and b ibes. Homo Oeconomicus, 22:569–588, 2005. [23] J. F eixas and M. Pons. Ci cums an ial powe : op imal pe suadable o e s. Eu opean Jou nal o Ope a ional Resea ch, 186:1114–1126, 2008. [24] T. K¨oning and T. B ¨auninge . The inclusi eness o Eu opean decision ules. Canadian Jou nal o Theo e ical Poli ics, 10:125–142, 1998. [25] W. Kuo and M.J. Zuo. Op imal Reliabili y Modeling: P inciples and Applica- ions. Wiley, New Yo k, 2003. [26] A. La uelle and F. Valenciano. Assessing success and decisi eness in o ing si ua ions. Social Choice and Wel a e, 24:171–197, 2005. [27] A. La uelle and F. Valenciano. Vo ing and Collec i e Decision–Making. Cam- b idge Uni e si y P ess, 2008. [28] G. Owen. Mul ilinea ex ensions o games. Managemen Science, 18:64–79, 1972. [29] L.S. Pen ose. The elemen a y s a is ics o majo i y o ing. Jou nal o he Royal S a is ical Socie y, 109:53–57, 1946. [30] R. Pongou, B. Tchan cho, and L. Diffo Lambo. Poli ical influence in mul i- choice ins i u ions: cyclici y, anonymi y and ansi i i y. Theo y and Decision, 70:157–178, 2011. [31] D. Rae. Decision ules and indi idual alues in cons i u ional choice. Ame ican Poli ical Science Re iew, 63:40–56, 1969. 17 [32] A.E. Ro h. In oduc ion o he Shapley alue. In A.E. Ro h, edi o , The Shap- ley alue. Essays in Hono o L.S. Shapley, pages 1–27. Camb idge Uni e si y P ess, 1988. [33] L.S. Shapley. A alue o n-pe son games. In A.W. Tucke and H.W. Kuhn, edi o s, Con ibu ions o he heo y o games II, pages 307–317. P ince on Uni- e si y P ess, P ince on, USA, 1953. [34] L.S. Shapley and M. Shubik. A me hod o e alua ing he dis ibu ion o powe in a commi ee sys em. Ame ican Poli ical Science Re iew, 48:787–792, 1954. [35] P.D. S affin, M.D. Da is, and S.J. B ams. Powe and sa is ac ion in an ideo- logically di ided o ing body. In M.J. Holle , edi o , Powe , o ing and o ing powe , pages 239–253. Physica Ve lag, W¨u zbu g–Wien, 1982. [36] A.D. Taylo and W.S. Zwicke . Simple games: desi abili y ela ions, ading, and pseudoweigh ings. P ince on Uni e si y P ess, New Je sey, USA, 1999. [37] B. Tchan cho, L. Diffo Lambo, R. Pongou, and B. Mbama Engoulou. Vo e s’ powe in o ing games wi h abs en ion: Influence ela ion and o dinal equi a- lence o powe heo ies. Games and Economic Beha io , 64:335–350, 2008. 18