M. Álvarez-Mozos a, F. Ferreira b, J.M. Alonso-Meijide c & A.A. Pinto d a Department of Statistics and Operations Research, Faculty of

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1 Ths artcle was downloaded by: [b-on: Bbloteca do conhecmento onlne UP] On: 29 May 205, At: 02:46 Publsher: Taylor & Francs Informa Ltd Regstered n England and Wales Regstered Number: Regstered offce: Mortmer House, 37-4 Mortmer Street, London WT 3JH, UK Clck for updates Optmzaton: A Journal of Mathematcal Programmng and Operatons Research Publcaton detals, ncludng nstructons for authors and subscrpton nformaton: Characterzatons of power ndces based on null player free wnnng coaltons M. Álvarez-Mozos a, F. Ferrera b, J.M. Alonso-Mejde c & A.A. Pnto d a Department of Statstcs and Operatons Research, Faculty of Mathematcs, Unversty of Santago de Compostela, La Coruña, Span. b ESEIG, Polytechnc Insttute of Porto, Porto, Portugal. c Department of Statstcs and Operatons Research, Faculty of Scences of Lugo, Unversty of Santago de Compostela, La Coruña, Span. d LIAAD-INESC TEC, Departament of Mathematcs, Faculty of Scences, Unversty of Porto, Porto, Portugal. Publshed onlne: 22 Feb 203. To cte ths artcle: M. Álvarez-Mozos, F. Ferrera, J.M. Alonso-Mejde & A.A. Pnto (205) Characterzatons of power ndces based on null player free wnnng coaltons, Optmzaton: A Journal of Mathematcal Programmng and Operatons Research, 64:3, , DOI: 0.080/ To lnk to ths artcle: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francs makes every effort to ensure the accuracy of all the nformaton (the Content ) contaned n the publcatons on our platform. However, Taylor & Francs, our agents, and our lcensors make no representatons or warrantes whatsoever as to the accuracy, completeness, or sutablty for any purpose of the Content. Any opnons and vews expressed n ths publcaton are the opnons and vews of the authors, and are not the vews of or endorsed by Taylor & Francs. The accuracy of the Content should not be reled upon and should be ndependently verfed wth prmary sources of nformaton. Taylor and Francs shall not be lable for any losses, actons, clams, proceedngs, demands, costs, expenses, damages, and other labltes whatsoever or

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3 Optmzaton, 205 Vol. 64, No. 3, , Characterzatons of power ndces based on null player free wnnng coaltons M. Álvarez-Mozos a, F. Ferrera b, J.M. Alonso-Mejde c and A.A. Pnto d Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May 205 a Department of Statstcs and Operatons Research, Faculty of Mathematcs, Unversty of Santago de Compostela, La Coruña, Span; b ESEIG, Polytechnc Insttute of Porto, Porto, Portugal; c Department of Statstcs and Operatons Research, Faculty of Scences of Lugo, Unversty of Santago de Compostela, La Coruña, Span; d LIAAD-INESC TEC, Departament of Mathematcs, Faculty of Scences, Unversty of Porto, Porto, Portugal (Receved 9 May 202; fnal verson receved 3 December 202) In ths paper, we characterze two power ndces ntroduced n [] usng two dfferent modfcatons of the monotoncty property frst stated by [2]. The sets of propertes are easly comparable among them and wth prevous characterzatons of other power ndces. Keywords: smple game; power ndex; characterzaton. Introducton In the last decades, the measurement of power n decson-makng bodes such as the European Unon Councl of Mnsters or the Internatonal Monetary Fund has been a man topc n poltcal scences and many work has been done n order to attan an approprate measure. However, there s stll a debate even on the defnton of power. Most of the tmes, the power s understood as the ablty of an agent to nfluence the outcome. But, even when the defnton of power s agreed, the choce of an appropate rule to represent t s stll an open queston. Among the most studed power ndces n the lterature, one can fnd the Shapley Shubk ndex [3], the Banzhaf ndex [4], the Deegan Packel ndex [5] and the Publc Good ndex [6]. All the above power ndces are evaluatons of an agent s relatve sgnfcance to each of the coaltons that mght be formed. In ths work, we wll frst of all revew some of the man results related to these four power ndces. Some of the aforementoned power ndces restrct ther attenton to some knds of coaltons that are partcularly mportant. Indeed, the Deegan Packel and Publc Good ndces only take nto account the so-called mnmal wnnng coaltons. A wnnng coalton, that s, a group of agents that can pass a bll on ther own, s a mnmal wnnng coalton when the removal of any of ts members would prevent the coalton from passng the bll. More recently, other nterestng power ndces have been ntroduced. The Publc help ndex [7] s based on the set of all wnnng coaltons, more precsely, the power of each agent s proportonal to the number of wnnng coaltons n whch he partcpates. The Shft power ndex [8] consders only a subset of the Correspondng author. Emal: mkel.alvarez@ub.edu 203 Taylor & Francs

4 676 M. Álvarez-Mozos et al. Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May 205 mnmal wnnng coaltons, the so-called mnmal wnnng coaltons wthout any surplus, n a sense, these coaltons are the most effcent mnmal wnnng coaltons. In ths paper, we take up agan the two power ndces ntroduced n [], namely f np and g np, and study ther propertes. These power ndces are also based on a partcularly mportant set of coaltons, specfcally on the wnnng coaltons that do not contan null players. A null player s a player whose partcpaton n any coalton does not change the stuaton,.e. the coalton contnues beng ether wnnng or loosng. Such set of coaltons contans the set of mnmal wnnng coaltons and s contaned n the set of wnnng coaltons. A frst consequence of ths fact s that f np and g np do not consder mnmal wnnng coaltons as the only source of power. Ths s the case n many real stuatons for nstance, many tmes the adopted decsons are more stable the greater the wnnng coalton supportng t s. Hence, the nformaton set on whch the new power ndces are based s wder than the nformaton set on whch the Deegan Packel and Publc Good ndces are based. The modellng of decson-makng bodes and votng procedures has been tackled usng smple games. The axomatc characterzaton of power ndces s a man topc n the feld for at least two reasons. Frst, characterzng a rule by means of a set of propertes may be more appealng than just gvng ts explct defnton. Second, decdng on whether to use a rule or another n a partcular stuaton may be done more easly takng nto account the propertes that each rule satsfes. In fact, many power ndces have shown to have dfferent sets of propertes that determne them unquely. In ths document, we present parallel characterzatons of the two power ndces ntroduced n [] n lne wth the characterzaton of the Deegan Packel power ndex by [9] and the Publc Good ndex by [0]. In ths way, the comparson among these four power ndces s much easer snce the characterzatons only dffer n one property. Moreover, the property n each of the characterzatons s a type of monotoncty n the sense that t descrbes the way n whch the payoff of an agent changes when hs poston n the stuaton s mproved. The rest of the paper s organzed as follows. In Secton 2, we announce notaton and present some prelmnary defntons and results such as the defntons and characterzatons of the Shapley Shubk and Banzhaf power ndces. In Secton 3, the Deegan Packel and Publc Good power ndces are presented together wth a par of characterzatons of each one of them. In Secton 4, the man results of the paper are presented, that s, the new power ndces f np and g np are characterzed by means of smlar propertes to the ones used n the characterzatons presented n Secton 3. Fnally, Secton 5 dscusses some concludng remarks. 2. Prelmnares A cooperatve transferable utlty game (just game from now on) s a par (N,v), where N ={,...,n} s the (fnte) set of players and v : 2 N R s the characterstc functon of the game, whch satsfes v( ) = 0. In general, we nterpret v(s) as the beneft that S can obtan by ts own,.e. ndependent to the decsons of players n N \ S. To avod cumbersome notaton, braces wll be omtted whenever t does not lead to confuson; for example, we wll wrte v(s ) or v(s \ ) nstead of v(s {}) or v(s \{}). A player N s a null player n a game (N,v)when hs margnal contrbuton to every coalton s zero,.e. when for every S N \, v(s ) v(s) = 0. Two players, j N are symmetrc n a game (N,v)f ther margnal contrbutons to every coalton concde,.e.

5 Optmzaton 677 f for every S N \{, j}, v(s ) = v(s j). A game, (N,v), s called monotone f for every S, T 2 N wth S T, v(s) v(t ). Defnton 2. A smple game s a monotone game such that the worth of every coalton s ether 0 or and the worth of the grand coalton s. Formally, (N,v)s a smple game f and only f: Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May 205 (N,v)s monotone, for every S 2 N, v(s) {0, }, and v(n) =. The class of all smple games s denoted by SG. In a smple game (N,v) SG, a coalton S N s wnnng f v(s) =, and losng f v(s) = 0. W (v) denotes the set of wnnng coaltons of the smple game (N,v) and, gven N, W (v) denotes the subset of W (v) formed by the coaltons contanng player,.e. W (v) ={S W (v) : S}. Gven a smple game (N,v) SG, a swng for a player N s a coalton S 2 N such that S, S s a wnnng coalton and S \ s a losng coalton. The set of all swngs for player N s denoted by η (v). Any smple game (N,v) SG may be descrbed by ts set of wnnng coaltons W (v). Gven a player set N and an arbtrary famly of coaltons W 2 N, the par (N, W ) determnes a smple game f: / W, N W and for every S T N,fS W, then T W. A wnnng coalton S W (v) s a mnmal wnnng coalton f every proper subset of S s a losng coalton; that s, S s a mnmal wnnng coalton n (N,v) f v(s) = and v(t ) = 0 for any T S. W m (v) denotes the set of mnmal wnnng coaltons of the game (N,v)and W m (v) the subset of W m (v) formed by coaltons contanng player,.e. W m (v) ={S W m (v) : S}. Smlar to the case of wnnng coaltons, a smple game may also be defned by ts set of mnmal wnnng coaltons W m (v). Gven a player set N and an arbtrary famly of coaltons W m 2 N, the par (N, W m ) determnes a smple game f: / W m, W m = and for every S, T W m, S T and T S. It s easy to obtan the set of mnmal wnnng coaltons from the set of wnnng coaltons and vce versa,.e. W m (v) ={S W (v) : T S, T / W (v)}, W (v) ={S 2 N : T S, T W m (v)}. By a power ndex we mean a map f that assgns a vector f(n,v) R N to every smple game (N,v) SG. In the defntons below, two of the most popular power ndces are presented.

6 678 M. Álvarez-Mozos et al. Defnton 2.2 The Shapley Shubk power ndex [3], SS, s the power ndex defned for every (N,v) SG and N by SS (N,v)= s!(n s )!, n! where n = N and s = S. S η (v) Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May 205 Defnton 2.3 The Penrose Banzhaf Coleman power ndex [4,,2], PBC, s the power ndex defned for every (N,v) SG and N by PBC (N,v)= η (v) 2 n. In order to present characterzatons of SS and PBC formally, some propertes need to be presented. effa power ndex f satsfes effcency f for every (N,v) SG, f (N,v)=. N nppa power ndex f satsfes the null player property f for every (N,v) SG and each null player N n (N,v), f (N,v)= 0. sym A power ndex f satsfes symmetry f for every (N,v) SG and each par of symmetrc players, j N n (N, v), f (N,v)= f j (N,v). trp A power ndex f satsfes the transfer property f for every par of smple games (N,v),(N,w) SG, f(n,v)+ f(n,w)= f(n,v w) + f(n,v w), where (N,v w), (N,v w) SG are defned for all S N by (v w)(s) = max{v(s), w(s)} and (v w)(s) = mn{v(s), w(s)}. tpp A power ndex f satsfes the total power property f for every (N,v) SG, N f (N,v)= η (v) 2 n. N Next, n lne wth the characterzatons of the Shapley and Banzhaf values by [3], parallel characterzatons of SS and PBC are presented. Theorem 2.4 [4] The Shapley Shubk power ndex, SS, s the unque power ndex satsfyng eff, sym, npp and trp. Theorem 2.5 [5] The Penrose Banzhaf Coleman power ndex, PBC, s the unque power ndex satsfyng tpp, sym, npp and trp.

7 Optmzaton 679 The man dfference between the Shapley value and the Banzhaf value s that the former s effcent whle the later satsfes the total power property. The characterzatons above show that ths dfference s transferred when smple games are consdered. Hence, the man dfference between SS and PBC s that the former s effcent whle the latter satsfes the total power property. Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May Power ndces based on mnmal wnnng coaltons The Deegan Packel power ndex [5] s based on the dea that when t comes to measure the power of an agent, t should only be consdered hs partcpaton n mnmal wnnng coaltons. Moreover, t assumes the followng three facts: Only mnmal wnnng coaltons wll emerge vctorous. Each mnmal wnnng coalton has an equal probablty of formng. Players n mnmal wnnng coaltons dvde the spols equally. The condtons above seem reasonable n many stuatons. The frst condton s a consequence of havng ratonal players n the sense that they seek for maxmzng ther power and hence, they wll only partcpate n mnmal wnnng coaltons. In other words, f a wnnng coalton s not a mnmal wnnng coalton, t means that there are players whose partcpaton n the coalton s not needed. Hence, the remanng players wll prefer to form the mnmal coalton contaned on the wnnng coalton snce there wll be less people to share the spols wth. The second condton states that all mnmal wnnng coaltons are equally lkely, whch s very reasonable once the frst condton s accepted. The last condton s a soldarty or equal treatment property. The requstes above lead to the followng defnton. Defnton 3. Gven a smple game (N,v) SG,the Deegan Packel power ndex [5], DP, s a vector n R N where each coordnate ( N) s defned as follows: DP (N,v)= W m (v) S W m (v) S. The DP power ndex s ntroduced n [5] together wth a characterzaton by means of four propertes. The characterzaton of SS presented before shares three of them, namely, eff, sym and npp. Indeed, DP concdes wth SSn the class of unanmty games. However, the DP power ndex does not satsfy trp. Instead, t satsfes the so-called DP-mergeablty property that s ntroduced next. Two smple games (N,v)and (N,w)are mergeable f for every par of mnmal wnnng coaltons S W m (v) and T W m (w), t holds that S T and T S. If two games (N,v) and (N,w) are mergeable, the mnmal wnnng coaltons n the maxmum game (N,v w) are precsely the unon of the mnmal wnnng coaltons n the two orgnal games (N,v)and (N,w). Hence, the mergeablty condton guarantees that W m (v w) = W m (v) + W m (w). Recall the defnton of a merged or maxmum game, (N,v w), gven for every S N by, (v w)(s) = max{v(s), w(s)}.

8 680 M. Álvarez-Mozos et al. Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May 205 dp-mera power ndex f satsfes DP-mergeablty f for every par of mergeable smple games (N,v),(N,w) SG, f(n,v w) = W m (v) f(n,v)+ W m (w) f(n,w) W m. (v w) The property above states that the power n a merged game s a weghted mean of the power n the two component games. The weghts come from the number of mnmal wnnng coaltons n each component game, dvded by the number of mnmal wnnng coaltons n the merged game. Hence, t concdes wth trp n the sense that t assesses the power n a merged game n terms of the power n the two component games. At ths pont, the propertes needed to present the frst characterzaton of the Deegan Packel ndex have been ntroduced. Theorem 3.2 [5] The Deegan Packel ndex, DP, s the unque power ndex satsfyng eff, sym, npp and dp-mer. More recently, [9] proposed a dfferent characterzaton of the Deegan Packel ndex. Ths characterzaton s based on the so-called DP-mnmal monotoncty property whch s nspred by the strong monotoncty property ntroduced n [2] to characterze the Shapley value. The property s formally ntroduced next. dp-mm A power ndex f satsfes DP-mnmal monotoncty f for every par of smple games (N,v),(N,w) SG and every player N such that W m (v) W m (w), f (N,v) W m (v) f (N,w) W m (w). The dp-mm property states that f the mnmal wnnng coaltons that contan a player N are mnmal wnnng coaltons n another game, then the power of player n the former game tmes ts number of mnmal wnnng coaltons s never bgger than the power of the player n the latter game tmes ts number of mnmal wnnng coaltons. Hence, dp-mm descrbes the way n whch the power of an agent changes when hs poston n the smple game s mproved. The characterzaton of DP power ndex proposed n [9] replaces the dp-mer property by the dp-mm property. Theorem 3.3 [9] The Deegan Packel ndex, DP, s the unque power ndex satsfyng eff, sym, npp and dp-mm. In the scentfc lterature concernng power ndces, one can fnd another power ndex that takes only mnmal wnnng coaltons nto account. The so-called Publc Good ndex proposed n [6] consders that each player s power s proportonal to the amount of mnmal wnnng coaltons n whch he partcpates. Defnton 3.4 Gven a smple game (N,v) SG,thePublc Good ndex [6], PG, sa vector n R N where each coordnate ( N) s defned as follows: W m (v) PG (N,v)= j N W m j (v). The frst characterzaton of ths power ndex by means of a set of propertes s proposed n [6]. The characterzaton follows the sprt of the characterzaton of the DP ndex

9 Optmzaton 68 Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May 205 presented n Theorem 3.2. Indeed, the property of dp-mer s replaced by the pg-mer whch s formally ntroduced below. pg-mer A power ndex f satsfes PG-mergeablty f for every par of mergeable smple games (N,v),(N,w) SG and every N, f (N,v w) = f (N,v) j N W m j (v) +f (N,w) j N W m j (w) j N W m. j (v w) Hence, pg-mer descrbes the power n the merged game as a weghted mean of the powers n the two component games as the dp-mer does. However, the weghts used dffer from the ones used n dp-mer. Next, the counterpart of Theorem 3.2 for the PG ndex s presented. Theorem 3.5 [6] The Publc Good ndex, PG, s the unque power ndex satsfyng eff, sym, npp and pg-mer. More recently, [0] proposed a dfferent characterzaton of the Publc Good ndex. Ths characterzaton s parallel to the one for the DP ndex presented n Theorem 3.3. It s based on the so-called PG-mnmal monotoncty property whch s smlar to the DP-mnmal monotoncty property stated above. The property s formally ntroduced next. pg-mm A power ndex f satsfes PG-mnmal monotoncty f for every par of smple games (N,v),(N,w) SG and every player N such that W m (v) W m (w), f (N,v) W m j (v) f (N,w) W m j (w). j N The pg-mm property keeps a close relaton wth the pg-mm property. Both propertes descrbe the relaton between the power of an agent n two dfferent smple games when the mnmal wnnng coaltons that contan the player n one game are mnmal wnnng coaltons n the other game. The dfference les on the scalars that multply the power n each of the smple games. Hence, usng pg-mm property, a counterpart of Theorem 3.3 s obtaned for the PG ndex. Theorem 3.6 [0] The Publc Good ndex, PG, s the unque power ndex satsfyng eff, sym, npp and pg-mm. The four characterzaton results presented n ths secton are summarzed n Table. 4. Two new power ndces based on null player free wnnng coaltons In Secton 2, thess and PBC power ndces are ntroduced. These ndces are based on the swngs of each player, that s, on the wnnng coaltons contanng the player that become loosng when the player leaves them. In Secton 3, power ndces based on mnmal wnnng coaltons are ntroduced, namely DP and PG. In ths secton, two new power ndces are ntroduced followng []. A wnnng coalton S W (v) s sad to be a null player free wnnng coalton f no null player s contaned on t, that s, f for every S there s T W (v) such that T \ / W (v). The set of null player free wnnng coaltons s denoted by W np (v). As before, for every j N

10 682 M. Álvarez-Mozos et al. Table. Parallel characterzatons of DP and PG. DP PG dp-mer pg-mer eff eff [5] sym sym [6] npp npp Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May 205 dp-mm pg-mm eff eff [9] sym sym [0] npp npp player N, W np (v) denotes the set of null player free wnnng coaltons contanng player,.e. W np (v) ={S W np (v) : S}. Note that for every (N,v) SG,the followng relaton holds, W m (v) W np (v) W (v). Thus, the set of null player free wnnng coaltons can be seen ether as a refnement of the set of wnnng coaltons or as an extenson of the set of mnmal wnnng coaltons. Note that a smple game s determned by ts set of null player free wnnng coaltons, W np (v). The clam before holds snce the set of wnnng coaltons can be easly obtaned from W np (v),.e. W (v) ={T 2 N : there s S W np (v) such that S T }. It s also easy to obtan the set of mnmal wnnng coaltons gven the set of null player free wnnng coaltons and vce versa, as follows, W m (v) ={T W np (v) : for every S T, S / W np (v)}. () W np (v) ={S U : there s T W m (v) such that T S} (2) U W m (v) In [] two new power ndces based on null player free wnnng coaltons are ntroduced. In the paper, the new power ndces are denoted by f and g. However, the notaton s slghtly modfed here for the sake of clarty. The new power ndces consder that only null player free wnnng coaltons should be taken nto account when t comes to measurng the power. In other words, these power ndces are based on the nformaton contaned on the set W np. In ths way, null players, whch by defnton do not partcpate n coaltons of W np,are assgned no power. The formal defntons are ntroduced next. Defnton 4. The f np power ndex s defned for every (N,v) SG and N by, f np (N,v)= W np (v) S W np (v) S.

11 Optmzaton 683 Defnton 4.2 The g np power ndex s defned for every (N,v) SG and N by, g np (N,v)= j N W np (v) W np j (v). Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May 205 The dea behnd the power ndces defned above s n lne wth the defntons of the Deegan Packel and the Publc Good ndces (see Defntons 3. and 3.4). The only dfference s that f np and g np consder all wnnng coaltons that do not contan null players nstead of only consderng mnmal wnnng coaltons. A drect consequence of ths fact s that non-null players whch do not partcpate n many mnmal wnnng coaltons are allocated more power. Consequently, f np consders that all null player free wnnng coaltons are equally lkely and that the players n a null player free wnnng coalton dvde the spols equally. g np assumes that the power of each player s proportonal to the number of null player free wnnng coaltons n whch he partcpates. The Shft power ndex ntroduced n [8] s smlar to the Publc Good ndex and the g np power ndex; however, t s based on a set of coaltons whch s contaned on the set of mnmal wnnng coaltons. A set of few ndependent propertes s a convenent tool to descrbe a power ndex and eases the comparson among dfferent power ndces. In order to characterze f np and g np, the followng monotoncty propertes are ntroduced. f np -mm A power ndex f satsfes f np -mnmal monotoncty f for every par of smple games (N,v),(N,w) SG and every player N such that W m (v) W m (w), f (N,v) W np (v) f (N,w) W np (w). g np -mm A power ndex f, satsfes g np -mnmal monotoncty f for every par of smple games (N,v),(N,w) SG and every player N such that W m (v) W m (w), f (N,v) j N W np j (v) f (N,w) j N W np j (w). The f np -mm and g np -mm propertes are based on the strong monotoncty property used n [2] to characterze the Shapley value. Indeed, both descrbe the behavour of a value n two smple games, (N,v)and (N,w), n whch there s a player N such that W (v) W (w), n other words, v(s ) v(s) w(s ) w(s) for every S N \. The dfference les on the relaton between the power of player n both games. The strong monotoncty property states that player s power n (N,w) s at least as bg as hs power n (N,v). Instead, f np -mm and g np -mm propertes state that the relaton holds after multplyng the payoffs by the denomnator of the defntons of f np and g np respectvely. The followng results show that f np and g np are characterzed wth a close set of propertes of the ones used n Theorems 3.3 and 3.6 to characterze DP and PG, respectvely. Theorem 4.3 and f np -mm. The power ndex f np s the unque power ndex that satsfes eff, npp, sym Proof () Exstence. From Defnton 4., t straghtforward to check that f np satsfes eff, npp and sym. Forf np -mm property, note that by Equaton (2), W m (v) W m (w)

12 684 M. Álvarez-Mozos et al. mples W np and hence, (v) W np (w). Then, f np (N,w)= = W np (w) W np (w) S W np (w) S W np (v) S S + W np (w) S W np (w)\w np (v) S, Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May 205 f np (N,w) W np (w) = S W np (v) S + S W np (w)\w np (v) S S W np (v) = f np S (N,v) W np (v). (2) Unqueness. The unqueness s proved by nducton on the number of mnmal wnnng coaltons. If W m (v) =, then v = u S where W m (v) ={S}. If a power ndex, f satsfes eff, npp and sym, we have, f (N,v)= { S f S 0 f / S. Hence, the unqueness holds when W m (v) =. Next, assume that a power ndex satsfyng the propertes s unque for every (N,v) SG wth less than m > mnmal wnnng coaltons,.e., f s unque for every (N,v) SG such that W m (v) < m.let(n,v) SG wth W m (v) = {S,...,S m }. Take T = m k= S k. Then, for each / T let us defne (N,w) SG by W m (w) = W m (v). Then, snce W m (v) = W m (w), applyng f np -mm twce, f (N,v) W m (v) =f (N,w) W m (w). Fnally, note that W m (w) < m and hence, by nducton, the rght hand sde of the equalty above s unque. It remans to prove the unqueness for T.Bysym, there s a constant c R such that for every T, f (N,v) = c. Moreover, by eff and the unqueness for any / T, c s unque whch concludes the proof. Theorem 4.4 and g np -mm. The power ndex g np s the unque power ndex that satsfes eff, npp, sym, Proof The proof follows mmedately from a reasonng smlar to the one used n the Proof of Theorem 4.3. Hence, the two Theorems above show that the dfferences between f np and g np are restrcted to a monotoncty property. Moreover, the only dfference among SS, DP, PG, f np and g np s the type of monotoncty satsfed by each power ndex. Fnally, the parallel characterzatons of f np and g np are depcted n Table Concluson In a smple game, a member s consdered crtcal for a wnnng coalton when hs elmnaton from the coalton turns t nto a losng one. In a mnmal wnnng coalton, every

13 Optmzaton 685 Table 2. Parallel characterzatons of f np and g np. f np f np -mm eff sym npp g np g np -mm eff sym npp Downloaded by [b-on: Bbloteca do conhecmento onlne UP] at 02:46 29 May 205 member of the coalton s crtcal. The Deegan Packel and Publc Good ndces are based on mnmal wnnng coaltons. A player s null when t s not crtcal for any wnnng coalton. However, n most of the cases, a null player partcpates n many wnnng coaltons. For ths reason, the new ndces, f np and g np, are based on wnnng coaltons that do not contan null players. The characterzatons provded n ths paper hghlght the fact that the new power ndces share most of ther defnng features wth other well-known ndces. The project ntated wth ths paper s not closed and we have related future work n mnd. The two man lnes of ths future work are to extend these ndces to more complex models and to propose tools to compute these ndces. One of the models developed for representng decson-makng bodes more adequately s the one of smple games wth a coalton structure. In [7,9] extensons of the Deegan Packel and Publc Good ndces are proposed and characterzed for smple games wth a coalton structure. The ndces characterzed n ths paper can be extended to the model of smple games wth a coalton structure. Moreover, we also want to consder more complex models lke games wth graph-restrcted communcaton or games wth levels structure of cooperaton. Although the mathematcal expresson of the ndces characterzed n ths paper s smple, one of the dffcultes s ther computaton for games wth a bg number of players. Two tools have been used to compute power ndces n large games: the multlnear extensons [20,2] and [0] and the generatng functons [22]. One of the deas that we want to develop n the near future s to fnd methods whch allow to compute the proposed ndces by means of these two tools. Acknowledgements A. A. Pnto acknowledges the fnancal support of LIAAD INESC TEC through program PEst, USP-UP project, Faculty of Scences, Unversty of Porto, Calouste Gulbenkan Foundaton, FEDER, POCI 200 and COMPETE Programmes, PTDC/MAT/207/200 and Fundação para a Cênca e a Tecnologa (FCT). J. M. Alonso-Mejde and M. Álvarez-Mozos acknowledge the fnancal support of the Spansh Mnstry of Economy and Compettveness through Projects ECO C02-02/ECO and MTM C03-03 and of Xunta de Galca through Project INCITE PR. The authors would also lke to thank the comments and suggestons of two anonymous referees. References [] Alonso-Mejde JM, Ferrera F, Álvarez-Mozos M, Pnto AA. Two new power ndces based on wnnng coaltons. J. Dffer. Equ. Appl. 20;7:

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