Generalized Additive Games

Transcription

1 Noname manuscrpt No. (wll be nserted by the edtor) Generalzed Addtve Games Gula Cesar Roberto Lucchett Stefano Morett the date of recept and acceptance should be nserted later Abstract A Transferable Utlty (TU) game wth n players specfes a vector of 2 n 1 real numbers,.e. a number for each non-empty coalton, and ths can be dffcult to handle for large n. Therefore, several models from the lterature focus on nteracton stuatons whch are characterzed by a compact representaton of a TU-game, and such that the worth of each coalton can be easly computed. Sometmes, the worth of each coalton s computed from the values of sngle players by means of a mechansm descrbng how the ndvdual abltes nteract wthn groups of players. In ths paper we ntroduce the class of Generalzed Addtve Games (GAGs), where the worth of a coalton S N s evaluated by means of an nteracton flter, that s a map M whch returns the valuable players nvolved n the cooperaton among players n S. Moreover, we nvestgate the subclass of basc GAGs, where the flter M selects, for each coalton S, those players that have frends but not enemes n S. We show that well-known classes of TU-games can be represented n terms of such basc GAGs, and we nvestgate the problem of computng the core and the semvalues for specfc famles of GAGs. Keywords TU-games core semvalues arport games peer games argumentaton games. Gula Cesar Dpartmento d Matematca, Poltecnco d Mlano, Mlano, Italy and Unversté Pars- Dauphne, Pars, France E-mal: gula.cesar@polm.t Roberto Lucchett Dpartmento d Matematca, Poltecnco d Mlano, Mlano, Italy Stefano Morett CNRS UMR7243, PSL, Unversté Pars-Dauphne, Pars, France

2 2 Gula Cesar et al. 1 Introducton Snce the number of coaltons grows exponentally wth respect to the number of players, t s computatonally very nterestng to sngle out classes of games that can be descrbed n a concse way. In the lterature on coaltonal games there exst several approaches for defnng classes of games whose concse representaton s derved by an addtve pattern among coaltons. In some contexts, due to an underlyng structure among the players, such as a network, an order, or a permsson structure, the value of a coalton S N can be derved addtvely from a collecton of subcoaltons {T 1, T k }, T S {1,, k}. Such stuatons are modeled, for example, by the graph-restrcted games, ntroduced by Myerson [26] and further studed by Owen [27], the component addtve games [11] and the restrcted component addtve games [12]. Sometmes, the worth of each coalton s computed from the values that sngle players can guarantee themselves by means of a mechansm descrbng the nteractons of ndvduals wthn groups of players. As an example, consder a cost game where n players want to buy onlne n dfferent objects and the value of a sngle player n the game s defned as the prce of the object he buys. However, such a model may fal to reflect the mportance of a subset of players n contrbutng to the value of the coalton they belong to. In the prevous example, t s often the case that, by makng a collectve purchase, when a certan threshold prce s reached, some of the objects wll be sold for free and therefore the prce that a coalton S should pay wll depend only on the prce of a subset of purchased objects. In fact, n several cases the procedure used to assess the worth of a coalton S N s strongly related to the sum of the ndvdual values over another subset T N, not necessarly ncluded n S. Many examples from the lterature fall nto ths category, among them the well-known glove game, the arport games ([21],[22]), the connectvty game and ts extensons ([2],[20]), the argumentaton games [6] and classes of operaton research games, such as the peer games [8] and the mountan stuatons [24]: some of them wll be descrbed n Secton 4. In all the aforementoned models, the value of a coalton S of players s calculated as the sum of the sngle values of players n a subset of S. On the other hand, n some cases the worth of a coalton mght be affected by external nfluences and players outsde the coalton mght contrbute, ether n a postve or negatve way, to the worth of the coalton tself. Ths s the case, for example, of the bankruptcy games [3] and the mantenance problems ([19], [7]). In ths paper we ntroduce a general class of addtve TU-games where the worth of a coalton S N s evaluated by means of an nteracton flter, that s a map M whch returns the valuable players nvolved n the cooperaton among players n S.

3 Generalzed Addtve Games 3 Our objectve s to provde a general framework for descrbng several classes of games studed n the lterature on coaltonal games and to gve a knd of taxonomy of coaltonal games that are ascrbable to ths noton of addtvty over ndvdual values. The general defnton of the map M allows varous and wde classes of games to be embraced. Moreover, by makng further hypothess on M, our approach enables to classfy exstng games based on the propertes of M. In partcular, we ntroduce the class of basc GAGs, whch s characterzed by the fact that the valuable players n a coalton S are selected on the bass of the presence, among the players n S, of ther frends and enemes, see Defnton 4. Several of the aforementoned classes of games can be descrbed as basc GAGs, as well as games dervng from real-world stuatons. As an example, ths model turns out to be sutable for representng an onlne socal network, where frends and enemes of the web users are determned by ther socal profles, as we shall see n Secton 4. The nterest of ths classfcaton s not only taxonomcal, snce t also allows to study the propertes of solutons for classes of games known from the lterature. We ndeed provde results on classcal soluton concepts for basc GAGs and we address the problem of how to guarantee that a basc GAG has a nonempty core. The outlne of the paper s as follows. Secton 2 provdes the basc defntons and notaton regardng coaltonal games. We ntroduce the model n Secton 3 and the class of basc GAGs s dscussed and characterzed n Secton 4. Next, Sectons 5 and 6 present some results on the core and the semvalues of the GAGs. Fnally, Secton 7 concludes the paper. 2 Prelmnares In ths secton we ntroduce some prelmnary notaton and defntons on coaltonal games. A TU-game, also referred to as coaltonal game, s a par (N, v), where N denotes the set of players and v : 2 N R s the characterstc functon, wth v( ) = 0. A group of players S N s called coalton and v(s) s called the value or worth of the coalton S. If the set N of players s fxed, we dentfy a coaltonal game (N, v) wth ts characterstc functon v. We shall assume that N = {1,, n} and for a coalton S, we shall denote by s ts cardnalty S. A partcular class of games s that of smple games, where the characterstc functon v can only assume values n {0, 1}. A game (N, v) s sad to be monotonc f t holds that v(s) v(t ) for all S, T N such that S T and t s sad to be superaddtve f t holds that v(s T ) v(s) + v(t )

4 4 Gula Cesar et al. for all S, T N such that S T =. Moreover, a game (N, v) s sad to be convex or supermodular f t holds that for all S, T N. v(s T ) + v(s T ) v(s) + v(t ) Gven a game v, an mputaton s a vector x R n such that N x = v(n) and x v({}) for all N. An mportant subset of the set of the mputatons s the core, whch represents a classcal soluton concept for TU-games. The core of v s defned as C(v) = {x R n : N x = v(n), S x v(s) S N} 1. Another class of soluton concepts for coaltonal games s the class of semvalues. A semvalue π p s defned for all N as: π p (v) = S N\{} p s [ v(s ) v(s) ], (1) where p = {p 0,, p n 1 } s such that p s 0 and n 1 ( n 1 ) s=0 s ps = 1: the non-negatve number p s represents the probablty that a coalton of sze s+1 wll form. A semvalue s sad to be regular f p s > 0 for all s {0,, n 1}. If p s = [ n ( )] n 1 1, s the correspondng semvalue defned va relaton (1) s the Shapley value of v and s shortly denoted by σ(v), whle f p s = 1 2, the n 1 correspondng semvalue s the Banzhaf value of v, shortly denoted by β(v). For a general ntroducton on cooperatve games, see Maschler et al. (2013). 3 Generalzed Addtve Games (GAGs) In ths secton we defne the class of games that s the object of the paper, and we provde some examples and basc propertes. The basc ngredents of our defnton are the set N = {1,...,, n}, representng the set of players, a map v : N R, assgnng a real value to each player and a map M : 2 N 2 N, called the coaltonal map, whch assgns a coalton M(S) to each coalton S N of players. Defnton 1 We shall call Generalzed Addtve Stuaton (GAS) any trple N, v, M, where N s the set of the players, v : N R s a map that assgns to each player a real value and M : 2 N 2 N s a coaltonal map, whch assgns a (possbly empty) coalton M(S) to each coalton S N of players and such that M( ) =. Defnton 2 Gven the GAS N, v, M, the assocated Generalzed Addtve Game (GAG) s defned as the TU-game (N, v M ) assgnng to each coalton the value 1 Note that all the prevous defntons hold for TU-games where v represents a gan, whle the nequaltes should be replaced wth when v s a cost functon.

5 Generalzed Addtve Games 5 v M M(S) v() f M(S) (S) = 0 otherwse. (2) Example 1 (smple games) Let w be a smple game. Then w can be descrbed by the GAG assocated to N, v, M wth v() = 1 for all and { {} S f S W M(S) = otherwse where W s the set of the wnnng coaltons n w. In case there s a veto player,.e. a player such that S W only f S, then the game can also be descrbed by v() = 1, v(j) = 0 j and { T f S W M(S) = R otherwse wth T, R N such that T and / R. From Example 1 t s clear that the descrpton of a game as GAG need not be unque. Example 2 (glove game) Let w be the glove game defned n the followng way. A partton {L, R} of N s assgned. Defne w(s) = mn{ S L, S R }. Then w can be descrbed as the GAG assocated to N, v, M wth v() = 1 for all and { S L f S L S R M(S) = S R otherwse. Example 3 (bankruptcy games) Consder the bankruptcy game (N, w) ntroduced by Aumann and Maschler [3], where the value of a coalton S N s gven by w(s) = max{e d, 0}. N\S Here E 0 represents the estate to be dvded and d R N + s a vector of clams satsfyng the condton N d > E. It s easy to show that a bankruptcy game s the dfference w = v1 M v2 M of two GAGs v1 M, v2 M arsng, respectvely, from N, v 1, M 1 and N, v 2, M 2 wth v 1 () = E and v 2 () = d for all, { M 1 {} S f S B (S) = otherwse and { M 2 N \ S f S B (S) = otherwse for each S 2 N \ { }, and where B = {S N : N\S d E}.

6 6 Gula Cesar et al. Example 4 (connectvty games) ([2], [20]) Let Γ = (N, E) be a graph, where N s a fnte set of vertces and E s a set of non-ordered pars of vertces,.e. the edges of the graph. Consder the (extended) connectvty game (N, v Γ ), where each node of the underlyng graph s assgned a weght w. The weghted connectvty game s defned as the game (N, w), where { w(s) = S w f S N s connected n Γ and S > 1 0 otherwse. Then w can be descrbed as the GAG assocated to N, v, M wth v() = w for all and { S f S N s connected n Γ M(S) = otherwse. Some natural propertes of the map M can be translated nto classcal propertes for the assocated GAG. Defnton 3 The map M s sad to be proper f M(S) S for each S N; t s sad to be monotonc f M(S) M(T ) for each S, T such that S T N. Note that a map M can be monotonc but not proper, or proper but not monotonc. An example of map M whch s not monotonc s the one relatve to the glove game. Maps that are not proper wll be seen later. The followng results are straghtforward. Proposton 1 Let N, v, M be a GAS wth v R N + Then the assocated GAG (N, v M ) s monotonc. and M monotonc. Proposton 2 Let N, v, M be a GAS wth v R N + and M proper and monotonc. Then the assocated GAG (N, v M ) s superaddtve. Proof Let S and T be two coaltons such that S T =. By properness t s M(S) M(T ) =. By monotoncty t s Thus, snce v R N +, v M (S T ) = M(S T ) M(S) M(T ) M(S T ). v() M(S) M(T ) v() = v M (S) + v M (T ). Observe that Propostons 1 and 2 provde only suffcent condtons, for nstance the glove game s monotonc and superaddtve but the assocated map M s not monotonc. The followng example shows that, f the map M s proper and monotonc, the correspondng GAG does not need be convex.

7 Generalzed Addtve Games 7 Example 5 Let N = {1, 2, 3, 4}, v() > 0 N, and let M be such that M({2}) =, M({2, 3}) = {3} and M(S) = S for all S {2, 3}. Then M s proper and monotonc but the correspondng GAG s not convex, snce t holds that v M (S T )+v M (S T ) < v M (S)+v M (T ) for S = {1, 2, 3}, T = {2, 3, 4}. The next proposton shows that t s possble to provde suffcent condtons for a monotonc map M to generate a convex GAG. Proposton 3 Let N, v, M be a GAS wth v R N + and M such that M(S) M(T ) = M(S T ), (3) for each S, T 2 N. Then the assocated GAG (N, v M ) s convex. Proof It s easy to show that condton (3) mples monotoncty of M. Indeed, gven S T N t holds that S T = S and therefore M(S) M(T ) = M(S), whch mples that M(S) M(T ). In order to prove the convexty of the assocated GAG (N, v M ), we observe that the followng expresson holds for every S, T 2 N : v M (S) + v M (T ) = v() + v() = M(S) M(T ) v() + M(S) M(T ) M(S) M(T ) v(). (4) By monotoncty, we have that M(S) M(T ) M(S T ). Thus, f M(S T ) = M(S) M(T ) for each S, T 2 N, then from relaton (4) t follows that v M (S) + v M (T ) M(S T ) v() + M(S T ) = v M (S T ) + v M (S T ), v() whch concludes the proof. The condton provded by relaton (3) can be useful to construct a monotonc map M such that the correspondng GAG s convex when v R N +. The most trval example s the dentty map M(S) = S for each S 2 N. Another example s a map M of a GAS N, v, M wth N = {1, 2, 3} and v R N + such that M({1, 2, 3}) = {1, 2, 3}, M({1, 2}) = {1, 2}, M({2, 3}) = {2}, M({2}) = {2} and M({1}) = M({3}) = M({1, 3}) =.

8 8 Gula Cesar et al. 4 Basc GAGs We now defne an nterestng subclass of GASs. Consder a collecton C = {C } N, where C = {F 1,..., F m, E } s a collecton of subsets of N such that F j E = for all N and for all j = 1,, m. Defnton 4 We denote by N, v, C the basc GAS assocated wth the coaltonal map M defned, for all S N, as: M(S) = { N : S F 1,..., S F m, S E = } (5) and by N, v C the assocated GAG, that we shall call basc GAG. For smplcty of exposton, we assume w.l.o.g. that m 1 = m 2 = = m n := m. We shall call each F k, for all N and all k = 1,..., m, the k-th set of frends of, whle E s the set of enemes of. The basc GAG v C assocated wth a basc GAS can be decomposed n the followng sense: defne the collecton of n games v C, = 1,..., n, as { v() f S E =, S F v C (S) = k, k = 1,..., m 0 otherwse. (6) Proposton 4 The basc GAG v C assocated wth the map defned n (5) verfes: n v C = v C. (7) =1 A partcularly smple case s when every player has a unque set of frends, that we shall denote by F. Example 6 (arport games) ([21], [22]): Let N be the set of players. We partton N nto groups N 1, N 2,..., N k such that to each N j, j = 1,..., k, s assocated a postve real number c j wth c 1 c 2 c k (representng costs). Consder the game w(s) = max{c : N S }. Ths type of game (and varants) can be descrbed by a basc GAS N, (C = {F, E }) N, v by settng for each N j and each j = 1,..., k: - the value v() = cj N j, - the set of frends F = N j, and the set of enemes E = N j+1... N k for each N j and each j = 1,..., k 1 and E l = for each l N k. By usng smlar arguments, t s possble to show that also the mantenance games ([19], [7]), whch generalze the arport games, can be represented as basc GAGs. Example 7 (top-k nodes problem) [32,1] Let (N, E) be a co-autorshp network, where nodes represent researchers and there exsts an edge between two nodes f the correspondng researchers have co-authored n a paper. Gven a value

9 Generalzed Addtve Games 9 k, the top-k nodes problem conssts n the search for a set of k researchers who have co-authored wth the maxmum number of other researchers. The problem, ntroduced n [32], s formalzed as follows. For any S N, we defne the functon g(s) as the number of nodes that are adjacent to nodes n the set S. Gven a value k, the problem of fndng a set S of cardnalty k such that g(s) attans maxmum value s NP-hard [32]. Therefore, n [32] and later n [1] a slghtly dfferent problem s studed through a game-theoretcal approach, by usng the Shapley value of a properly defned cooperatve game as a measure of the mportance of nodes n the network. In the correspondng game (N, w), the worth of a coalton S, for each S N, S, s equal to the number of nodes that are connected to nodes n S va, at most, one edge. Formally, w(s) = S S N (E), where N (E) s the set of neghbours of N n the network (N, E). It s easy to check that game w can be descrbed as a basc GAS N, v, (C = {F, E }) N, where v() = 1, F = {} N (E) and E = N. We now provde further examples of classes of coaltonal games whch can be represented as basc GAGs, where n general each player can have several sets of frends (see the related lterature for a more detaled descrpton and analyss of such classes of games). Example 8 (argumentaton games) ([6]): Consder a drected graph N, R, where the set of nodes N s a fnte set of arguments and the set of arcs R N N s a bnary defeat (or attack) relaton (see Dung 1995). For each argument we defne the set of attackers of n N, R as the set P () = {j N : (j, ) R}. The meanng s the followng: N s a set of arguments, f j P () ths means that argument j attacks argument. The value of a coalton S s the number of arguments n the opnon S whch are not attacked by another argument of S. Ths type of game (and varants) can be descrbed as a basc GAS N, v, {F, E } by settng v() = 1, the set of frends F = {} and the set of enemes E = P (). Ths example stll falls n the settng of basc GAGs where each player has only one set of frends. However, there are also dfferent, and natural as well, types of characterstc functons that can be consdered. For nstance, t s nterestng to consder the game (N, v M ) such that for each S N, v M (S) s the sum of v() over the elements of the set D(S) = { N : P () S = and j P (), P (j) S } of arguments that are not nternally attacked by S and at the same tme are defended by S from external attacks: v M (S) = v(). (8) D(S) It s clear that such a stuaton cannot be descrbed by a basc GAG where each player has a unque set of frends. The game n (8) can however be descrbed as a basc GAG N, v C, where, gven a bjecton k : P () {1,, P () }, C = {F 1 P (),, F, E } s such that F k(j) = P (j) \ P () for all j P (), and E = P () for all N.

10 10 Gula Cesar et al. Example 9 (peer games) [8] Let N = {1,..., n} be the set of players and T = (N, A) a drected rooted tree descrbng the herarchy between the players, wth N as the set of nodes, 1 as the root (representng the leader of the entre group) and A N N s a set of arcs. Each agent has an ndvdual potental a whch represents the gan that player can generate f all players at an upper level n the herarchy cooperate wth hm. For every N, we denote by S() the set of all agents n the unque drected path connectng 1 to,.e. the set of superors of. Gven a peer group stuaton (N, T, a) as descrbed above, a peer game s defned as the game (N, v P ) such that for each non-empty coalton S N v P (S) = a. N:S() S A peer game (N, v P ) can be represented as the GAG assocated to the basc GAS on N where v() = a and where M s defned by relaton (5) wth collectons C = {F 1,..., F n, E } such that: F j = { {j} f j S() {} otherwse and E = for all N. Besdes the aforementoned classes of games from the lterature, ths model turns out to be sutable for representng socal network stuatons, as further dscussed n Secton 6. Wth the am of provdng a tool for the analyss of a wde range of coaltonal games, n Secton 5 we address the problem of how to guarantee that a basc GAG has a non-empty core, whle n Secton 6 we analyse classcal solutons from coaltonal game theory, lke the well known Shapley value [30], the Banzhaf value [4] and other semvalues [15]. 4.1 A characterzaton of basc GASs As t has been shown n the prevous sectons, a varety of classes of games that have been wdely nvestgated n the lterature can be descrbed usng the formalsm provded by basc GASs. Moreover, as we shall see n the next sectons, t s possble to produce, for basc GAGs, results concernng mportant soluton concepts, lke the core and the semvalues. It s therefore nterestng to study under whch condtons a GAS can be descrbed as a basc one. To ths purpose, the followng theorem provdes a necessary and suffcent condton when the set of enemes of each player s empty. Theorem 1 Let N, v, M be a GAS. The map M can be obtaned by relaton (5) va collectons C = {F 1,..., F m, E = }, for each N, f and only f M s monotonc.

11 Generalzed Addtve Games 11 Proof It s obvous that every map M obtaned by relaton (5) over a collectons C = {F 1,..., F m, E = }, for each N, s monotonc. Now, consder a monotonc map M and, for each N, defne the set M 1 = {S N : M(S)} of all coaltons whose mage n M contans. Let S M, be the collecton of mnmal (wth respect to set ncluson) coaltons n M 1, formally: S M, = {S M 1 : t does not exst T M 1 wth T S}. For each N, consder the collecton C = {F 1,..., F m, E = } such that {F 1,..., F m } = {T N : T S = 1 for each S S M, and T S M, }, (9) where each set of frends F k, k {1,..., m }, contans precsely one element n common wth each coalton n S S M, and no more than S M, elements. Denote by M the map obtaned by relaton (5) over such collectons C, N. We need to prove that M(S) = M (S) for each S 2 N, S. Frst note that for each N and for every coalton S M 1, we have M (S). Let us prove now that / M (S) for each S / M 1. Suppose, by contradcton, that there exsts T N wth T / M 1 such that F k T, for each k {1,..., m }. Consequently, by the defnton of S M,, we have that for every S S M,, S \ T. Defne a coalton U N of no more than S M, elements and such that U contans precsely one element of S \ T for each S S M,,.e. U (S \ T ) = 1 for each S S M, and U S M,. By relaton (9), U must be a set of frends n the collecton {F 1,..., F m }, whch yelds a contradcton wth the fact that U T =. It follows that for each N, M (S) f and only f M(S) for each S N, whch concludes the proof. Based on the arguments provded n the proof of Theorem 1, the followng example shows a procedure to represent a GAS wth a monotonc map M as a basc GAS. Example 10 Consder a GAS N, v, M wth N = {1, 2, 3, 4} and M such that M({1, 2, 3}) = {3}, M({3, 4}) = {2, 3}, M({2, 3, 4}) = {2, 3}, M({1, 3, 4}) = {2, 3, 4}, M(N) = {2, 3, 4}, and M(S) = for all other coaltons. The sets of mnmal coaltons are as follows: S M,1 =, S M,2 = {{3, 4}}, S M,3 = {{1, 2, 3}, {3, 4}}, S M,4 = {{1, 3, 4}}. Such a map can be represented va relaton (5) wth the collectons: F 1 1 =, {F 1 2, F 2 2 } = {{3}, {4}}, {F 1 3,..., F 4 3 } = {{1, 4}, {2, 4}, {3}, {3, 4}}, {F 1 4,..., F 3 4 } = {{1}, {3}, {4}}, where such collectons of frends are obtaned va relaton (9). The followng proposton characterzes monotonc basc GAGs. Proposton 5 Let N, v, C be a basc GAS wth v R N + and C = {C } N. Then the assocated GAG (N, v C ) s monotonc f and only f E = N.

12 12 Gula Cesar et al. Proof The suffcent condton s obvous. Moreover, suppose E for some and let j E. Consder S = F 1 F m. Then M(S), whle / M(S j). By Proposton 1, Theorem 1 and Proposton 5 we have the followng corollary. Corollary 1 Let N, v, M be a basc GAS wth v R N +. Then the assocated basc GAG (N, v C ) s monotonc f and only f M s monotonc. 5 The Core of GAGs In ths secton we present some results concernng the core of a GAG. The frst one s qute smple, and relates to the core of general GAGs. Moreover, we provde some suffcent condtons for the nonemptyness of the core of basc GAGs and we show by an example how they mght be useful to derve several allocatons n the core for partcular subclasses of games. Proposton 6 Let N, v, M be a GAS such that v R N + and M s proper and such that M(N) = N. Then, the core of the assocated (reward) GAG (N, v M ) s non-empty. Proof Let x R N be the allocaton wth x = v() for each N. Consder the game w defned as: w(s) = S v(). Notce that x C(w). Moreover, t holds that v M (S) w(s) S N, by properness of M, and v M (N) = w(n). Thus x C(v M ). We now turn our attenton to basc GAGs. To start wth, we observe that n general M s not proper. However suffcent condtons n order to apply Proposton 6 are easly seen: 1. for every there s j such that F j = {} 2. E = for all. Condton 1 mples that the map M s proper, whle condton 2 s equvalent to sayng that M(N) = N. Next, let us observe that C(v C ) f C(v C ) for all. Thus let us see now condtons under whch C(v C ). Denote by I = {j N : F k C s.t. F k = {j}} the set of players that appear n collecton C as sngletons. Note that I may be empty. Otherwse, players n I are veto players n the assocated game v C. From the above consderatons the followng Proposton holds, whch characterzes the core of the game v C. Proposton 7 Consder the game v C, where M s defned by relaton (5) wth collectons C = {F 1,..., F m, E = }. Then C(v C ) f and only f I. Moreover, f I, then t holds that: C(v C ) = {x R N + : j I x j = v()} (10)

13 Generalzed Addtve Games 13 Proof Note that I s the set of veto players n v C. Therefore, C(v C ) f and only f I. Moreover, f I, relaton (10) smply follows from the fact that v C (N) = v(). It follows that f I N, then C(v C ). However, when games v C such that I = are combned wth games v Cj such that I j, the resultng GAG can have a nonempty core, as shown n the followng example. Example 11 Consder a two-person basc GAS N, v, C wth N = {1, 2} such that v(1) = α, v(2) = 2 and C 1 = {{1}, {2}}, E 1 = }, C 2 = {{1, 2}, E 2 = }. By Proposton 7, we have that C(v C1 ), and C(v C2 ) =. The core of the resultng GAG v C = v C1 + v C2 s non-empty when α 2. The stuaton descrbed n the prevous example can be generalzed as follows. We frst defne the set I = { N : I } as the set of players that have at least one sngleton among ther sets of frends. The followng proposton provdes a necessary and suffcent condton for the non-emptness of the core of a specal class of basc GAGs of the type ntroduced n Example 11. Proposton 8 Consder the GAG v C correspondng to a basc GAS N, v, C wth v() 0 and C = {F 1,..., F m, E = } for each N. Suppose there exsts a coalton S N, S, satsfyng the followng two condtons: ) S I for each I; ) for each N \ I, there exsts k {1,..., m} such that F k = S. Defne the equal splt allocaton among players n S as the vector y such that y = e S vc (N), s where s s the cardnalty of S and where e S {0, 1} N s such that e S k = 1, f k S and e S k = 0, otherwse. The allocaton y s n the core of the game vc ff v C (N) s N\I v(). (11) Proof Condton () means that the players n S appear as sngletons n the set of frends of every player that has at least one sngleton among hs sets of frends, whle condton () means that, for those players that do not have sngletons among ther sets of frends, there exsts a set of frends concdng wth S. Frst, we prove that condton (11) s necessary. Suppose that (11) does not hold,.e. v C (N) < s N\I v(). Consder a coalton T N, such that T 2, T S = {t} and T F k N \ I, k = 1,, m. It holds that: j T y j = y t = vc (N) s < N\I v() = v C (T )

14 14 Gula Cesar et al. and therefore y / C(v C ). Moreover, we prove that condton (11) s also suffcent. Clearly, y = v C (N) = v(). (12) N N y = S In order to prove that y C(v C ) we have to show that f relaton (11) holds then T y v C (T ) for each coalton T N. Frst consder a coalton T N such that S T. Note that N v() v C (T ), and then by relaton (12), T y S y = N v() vc (T ). Now consder a coalton T N such that S T =. By condton () and (), we have that v C (T ) = 0, whch s not greater than T y snce y 0 for each N. Fnally, consder a coalton T N such that S T and S T. Snce S I for each I, then no term v() wth I contrbutes to the worth of T. Ths means that v C (T ) = v(), (13) M(T ) wth M(T ) N \ I and where M s defned by relaton (5). Now consder a player S T. If condton (11) holds, then we have that y j y v() v C (T ), j T N\I where the last nequalty follows by relaton (13), whch concludes the proof. Observe that, even f the equal splt allocaton does not belong to the core, the core mght be non empty, as the followng example shows. Example 12 Consder a GAS G = N, v, M wth N = {1, 2, 3} and such that C 1 = {{1}, {2}, {3}, E 1 = }, C 2 = {{2, 3}, E 2 = } and C 3 = {{2, 3}, E 2 = }. The coalton {2, 3} satsfes the hypothess n Proposton 8 but relaton (11) s not satsfed. However, the allocaton (v(1), v(2), v(3)) belongs to the core of v C. The prevous Propostons provde suffcent condtons for the nonemptness of the core for basc GAGs. If a game can be represented n terms of a basc GAG, these condtons can be drectly verfed by consderng only the collectons of frends and enemes of each player, wthout havng to check any further property of the characterstc functon (for nstance, the balancedness property [5, 31]), that n general nvolves much more complex procedures. Consequently, the results provded n ths secton can be used to construct non-trval classes of games wth a non-empty core, or to easly derve core allocatons. For nstance, even f a game s not tself representable n terms of a basc GAG, Proposton 7 may be appled to generate allocatons n the core of games that can be descrbed as the sum of proper basc GAGs where no player has enemes, as next example shows.

15 Generalzed Addtve Games 15 Example 13 Consder the game ntroduced n [13], n whch the players are nodes of a graph wth weghts on the edges, and the value of a coalton s determned by the total weght of the edges contaned n t. Formally, an undrected graph G = (N, E) s gven, wth weght w,j on the edge {, j} 2, and the game v s defned, for every S N, as v(s) =,j S w,j. If all weghts are nonnegatve, the game s convex and therefore the core s non empty. However, fndng allocatons n the core s not straghtforward and n [13] necessary and suffcent condtons for the Shapley value to belong to t are provded. Indeed, game v can be descrbed as the sum of n basc GAGs, one for each player N, where each other player j contrbutes to the worth of a coalton S N wth half of the weght w,j f and only f and j belong to S, whle contrbutes to any coalton t belongs to wth the weght w,. Formally, v = N vc, where v C s a proper basc GAG assocated to collectons Cj = {F j 1 = {}, F j 2 = {j}, E j = } and v(j) = w,j 2, for every j N, j, whle C = {F j 1 = {}, E j = } and v() = w,. As a sum of n proper GAGs such that M(N) = N, Proposton 6 mmedately mples the non-emptness of the core of game v. Moreover, notce that accordng to Proposton 4 each basc GAG v C can be decomposed as the sum v C = j N vc j, for each, j N, and then the repeated applcaton of Proposton 7 on each v C j can be used to effcently derve allocatons n the core of the sum game v. Followng smlar ntutons, we argue that the smple structure of basc GAGs could be useful to generalze some of the complexty results about the problem of fndng core allocatons provded n [13], for nstance, consderng classes of more sophstcated games that can be generated as a postve lnear combnaton of basc GAGs. 6 Semvalues and GAGs It s well known that the problem of dentfyng nfluental users on a socal networkng web ste plays a key role to fnd strateges amed at ncreasng the ste s overall vew [29]. The man ssue s to target advertsement to the ste members of the onlne socal network whose actvtes levels have a sgnfcant mpact on the actvty of the other ste members. The overall nfluence of a user can be seen as the combnaton of two ngredents: 1) the ndvdual ablty to get the attenton of other ste members, and 2) the personal characterstc of the socal profle, that can be represented n terms of groups or communtes to whch users belong. A basc GAS N, v, C can represent an onlne socal network as descrbed above (see also Example 14). More specfcally, each player N of the basc GAS s assocated to a value v() representng her/hs ndvdual actvty n a socal networkng web ste (for nstance, measured n terms of the productve 2 Snce the graph s undrected, we assume by conventon that, f an edge between and j s present, w,j 0 and w j, = 0 for < j.

16 16 Gula Cesar et al. tme spent n uploadng content fles), and the partcpaton of the ndvduals to the global actvty of the socal network s based on a coaltonal structure C of frends and enemes that s determned by players socal profles. Thus, t s nterestng to analyze, for ths type of games, the behavor of ndces amed at measurng the nfluence of the players n the game: n partcular we consder the Shapley value [30], the Banzhaf value [4] and other semvalues [15]. Snce we are nterested n evaluatng addtve power ndces for the players n basc GAGs, thanks to Proposton 4 t becomes possble to evaluate the ndces on the games v C defned va relaton (6). Frst of all, we present some results concernng the Shapley and Banzhaf values on nterestng subclasses of basc GAGs, where each player has a unque set of frends. Furthermore, we extend our analyss to a generc basc GAG, wth multple sets of frends. In what follows, n order to smplfy the notaton, we fx N and denote by f the cardnalty of F and by e the cardnalty of E (n order to smplfy the notaton, f E = we assume by conventon that e = 0 and 1 e = 0). When we consder a basc GAG wth a sngle set of frends for each player, the game v C reduces to: { v() f S F, S E v C (S) = = 0 otherwse and the followng proposton holds. Proposton 9 Let us consder a basc GAS on N, v, {C = {F, E }} N. Then the Shapley and Banzhaf values for the game v C are gven, respectvely, by: 0 f j N \ (F E ) σ j (v C v() ) = f+e f j F f v() j E and β j (v C ) = e(f+e) f 0 f j N \ (F E ) v() 2 f+e 1 f j F v() 2f 1 2 f+e 1 f j E. Proof Clearly, players not n F E are null players. Now, let us call a player decsve for a coalton S f v C (S {}) v C (S) 0. Consder a player j F. He s decsve for S (wth worth v()) f and only f no player n F and no player k n E s n S. Now, let j E. He s decsve for S (wth worth v()) f and only f at least one player n F s n S and no player k n E s n S. These facts provde the formulas for β, and σ s derved by consderng all the possble orderngs of the players.

17 Generalzed Addtve Games 17 We comment the results wth the help of the followng example. Example 14 As a toy example, consder an onlne socal network wth four users N = {1, 2, 3, 4} where each user spends the same amount of tme T n uploadng new content fles and, accordng to her/hs socal profle, each user N belongs to a sngle communty F N (e.g., the set of users wth whom ntends to share her/hs content fles) whch s n conflct wth the complementary one E = N \ F (here, enemes n E are nterpreted as those members that have no permsson to access the content fles of player ). Suppose, for nstance, that F 1 = {1, 2, 3}, F 2 = {2, 3}, F 3 = {3} and F 4 = {1, 2, 3, 4}. Followng the dscusson about socal networkng web stes ntroduced n Secton 1, we can represent such a stuaton as a basc GAS N, v, {C = {F, E = N \F }} N. How to dentfy the most nfluental users? Accordng to Proposton 9, the nfluence vector provded by the Shapley value s: σ(v C ) = ( T 6, 2T 5T 3, T, 6 ). So, user 3 results the most nfluental one, followed by 2, then 1 and fnally 4, who s the only user to get a negatve ndex. Suppose now that user 2 wants to mprove her/hs nfluence as measured by the Shapley value. It s worth notng that f user 2 removes 3 from her/hs set of frends (and all the other sets of frends and enemes reman the same), then player 2 gets exactly the same Shapley value of user 3. Precsely, f now F 2 = {2} and E 2 = {1, 3, 4}, then σ 2 (v C ) = σ 3 (v C ) = 2T 3. Notce that the fact that an nfluental player has been removed from hs/her lst of frends does not mpact drectly the nfluence of player 2, but t determnes an mportant reducton of the nfluence of player 3. When E =, the formulas n Proposton 9 are further smplfed and the followng corollary holds. Corollary 2 Let us consder a basc GAS on N, v, {C = {F, E = }} N. Then the Shapley value σ and the Banzhaf value β for the game v C are gven, respectvely, by: { 0 f j N \ F σ j (v C ) = v() f f j F and { 0 f j N \ F β j (v C ) = v() f j F 2 f 1. Proposton 9 and Corollary 2 represent a useful tool for computng the Shapley and Banzhaf values of a subclass of basc GAGs. Ther advantage reles on the fact that, once a game s descrbed n terms of basc GAGs, the formulas can be derved n a straghtforward way from the ndvdual values of the players and the cardnaltes of the sets of frends and enemes. As an example, consder the game ntroduced n Example 7. The Shapley value of such a game has been proposed as a measure of centralty n networks n [32], where an approxmate algorthm for ts computaton s provded. Moreover, n [1], an exact formula for the Shapley value of such game s provded, but ts

18 18 Gula Cesar et al. proof reles on elaborate combnatoral and probablstc arguments. On the other hand, the descrpton of that game as a basc GAG, whch can be easly derved from the defnton of the game tself, leads to the same formula n a drect and ntutve way, snce the Shapley value (and the Banzhaf value) can be drectly derved from Corollary 2 and relaton (7). When generalzng the prevous results to the case of a basc GAG wth multple sets of frends, t s natural to extend the analyss to other solutons, beyond the Shapley and Banzhaf values. In what follows, we focus on the class of semvalues. Let F = m k=1 F k and let Γ = {1,..., f 1}... {1,..., f m }, where f and f k are the cardnaltes of F and F k, for each k = 1,..., m. The followng Theorem generalzes the results n Proposton 9. Theorem 2 Consder a GAS stuaton N, v, {F 1,..., F m, E } N wth F j F k = for all N and j, k = 1,, m, j k. For all j N \ (F E ), we have that π p j (vc ) = 0. Take j F b, wth b {1,..., m}. Then πp j (vc ) s equal to the followng expresson: v() n e f ( f 1) ( (k 1,...,kb 1,0,k b+1,...,k m) Γ l=0... f m) ( n e f ) l ph+l where e = E and h = m j=1 kj. Now, take j E. Then π p j ) = v() n t f ( f 1) ( (vc (k 1,...,km ) Γ l=0... f m k 1 k 1 k m k m ) ( n e f l ) ph+l. (14) (15) Proof Players n N \(F E ) are dummy players, so they receve nothng. Now, consder the case j F b and take a coalton S N \{j} that does not contan j. The margnal contrbuton of j to coalton S s v C (S {j}) v C (S) = v() f S contans at least one frend from each set of frends F t wth t b (.e, S F t for t b), and S does not contan nether any element of F b nor any element of E ; otherwse, v C (S {j}) v C (S) = 0. Gven a vector (k 1,..., kb 1, 0, k b+1,..., k Γ (.e., k b = 0) and l {0,..., n e f}, the product ( f ) ( 1 k... f m) ( 1 k n e f ) m l represents the number of sets S contanng k t elements of F t, for each t {1,..., m } wth t b, l elements of N \(F E ) and such that v C (S {j}) v C (S) = v(). Of course, the probablty of such a set S to form s p h+l, and relaton (14) follows. Now, consder the case j T and take a coalton S N \ {j} that does not contan j. The margnal contrbuton of j to coalton S v C (S {j}) v C (S) = v() f S contans at least one frend from each set of frends F t for each t (.e, S F t for each t = 1,..., m ), and S does not contan any element of E ; otherwse, S v C (S {j}) v C (S) = 0. Gven a vector (k 1,..., km ) Γ and l {0,..., n e f}, the product ( f ) ( 1 k... f m) ( 1 k n e f ) m l represents the number of sets S contanng k t elements of F t, for each t = 1,..., m, l elements of N \(F E ) and such that v C (S {j}) v C (S) = v(). Of course, the probablty of such a set S to form s p h+l, and relaton (15) follows.

19 Generalzed Addtve Games 19 Formulas for the semvalues on v C when each N has only one set of frends can be derved drectly from the prevous theorem. Indeed, the followng corollares hold. Corollary 3 Let us consder a basc GAS on N, v, {C = {F, E }} N. Then a semvalue π p for the game v C s gven by: 0 f j N \ (F E ) π p j (vc ) = v() n f e ( n f e ) k=0 k pk f j F v() f ( f ) n f e ( n f e ) k=1 k h=0 h pk+h f j E. Corollary 4 Let us consder a basc GAS on N, v, {C = {F, E = }} N. A semvalue π p for the game v C s gven by: { π p 0 f j N \ F j (vc ) = v() n f ) k=0 pk f j F. ( n f k Note that, n the basc GASs N, v, {C = {F, E }} N consdered n Corollary 3, a semvalue π p for the game v C assgns to a player j F a postve share of v, proportonally to the probablty (accordng to p) that j enters a coalton not contanng any player of the set F E ; on the contrary, each player l E receves a negatve share of v, proportonally to the probablty that l enters a coalton contanng at least one player of F. In partcular, the unque semvalue such that F π p ) = (vc E π p ), (vc for every F, E N, F E =, F and E, s the Shapley value. We fnally observe that the equvalence of the formula n Corollary 3 wth the formulas of Proposton 9 for the Shapley value can be verfed through the followng combnatoral denttes: n ( n ) k=0 k (k)!(n + t k)! = (n+t+1)! t+1, n ( n k=0 k) (k + 1)!(n + t k 1)! = (n+t+1)! t(t+1), whch hold for all n, t N and can be proved by means of products of formal seres. 7 Conclusons In the present paper, the class of generalzed addtve games s ntroduced, where the worth of a coalton of players s evaluated by means of a map M that selects the valuable players n the coalton. Several examples from the lterature of classcal coaltonal games that can be descrbed wthn our approach are presented, n partcular for the class of basc GAGs, where the worth of each coalton s calculated addtvely over the ndvdual contrbutons and keepng nto account socal relatonshps among groups of players. Our approach enables to classfy exstng games based on the propertes of M. The nterest of the classfcaton s not only taxonomcal, snce t also allows to study the propertes of solutons for several classes of games known from the lterature.

20 20 Gula Cesar et al. We showed that, n many cases, basc GAGs allow for an easy computaton of several classcal solutons from cooperatve game theory and, at the same tme, provde qute smple representatons of practcal stuatons (for nstance, arsng from onlne socal networks). One of the goal of our future research s to apply these models on real socal network data. As shown by Example 14, the nformaton requred to compute classcal power ndces on basc GAGs representng onlne socal networks (lke the users actvty tme or the users socal profles and socal affntes) s not very demandng and can be obtaned by avalable records and models from the lterature [28]. Moreover, as t has been stressed n the same example, t would be nterestng to explore the strategc ssues related to the attempt of players to ncrease ther nfluence (as measured by the Shapley value or by other power ndces) on a socal network. An nterestng drecton for future research s ndeed that of coalton formaton, snce for generc basc GAGs assocated to GASs wth nonnegatve v, where the sets of enemes are not empty, the grand coalton s not lkely to form. In general, we beleve that the ssue about whch coaltons are more lkely to form n a basc GAG s not trval and deserves to be further explored. To conclude, observe that further extensons could be ntroduced, generalzng the dea of coaltonal map, n order to embrace a wder range of games represented n a compact way nto our framework. The defnton of GAG s based on the coaltonal map M, assgnng a coalton M(S) N to each coalton S N of players. But one can nstead consder a more general map M : 2 N 2 2N, whch assgns to each coalton S N, a subset M(S) 2 N. Ths can embrace the dea of graph-restrcted game ntroduced by Myerson (1977), where the worth of a coalton s evaluated on the connected components nduced by an underlyng graph. Analogously, the defnton of a basc GAG s based on a collecton of sets C = {C } N, one for each player, where C = {F 1,..., F m, E } s a collecton of subsets of N that satsfy some partcular propertes. If we provde each player wth multple collectons {C 1,, Ck }, wth C k = {F k1,..., F km, E k }, we are then able to represent those games that are assocated to margnal contrbuton nets (MC-nets), ntroduced by Ieong and Shoham (2005). The basc dea behnd margnal contrbuton nets s to represent n a compact way the characterstc functon of a game, as a set of rules of the form: pattern value, where a pattern s a Boolean formula over a set of n varables (one for each player) and a value s a real number (see also [10] for a formal defnton and a deeper analyss). Every coaltonal game can be represented through MC-nets by defnng one rule for each coalton S N, where the pattern contans all the varables correspondng to the players n S and the negaton of all the other varables, and the correspondng value s equal to the value of S n the game.

21 Generalzed Addtve Games 21 A game dervng from a MC-nets representaton can be descrbed as a generalzaton of a basc GAG, where each player has multple collectons of set of frends and enemes, one for each rule, and s assgned a vector of values v() = {v 1 (),, v m ()}. See [9] for more detals. In ths way, we are ndeed able to descrbe every TU-game, snce the representaton of MC-nets s complete. The computatonal complexty of such representaton s n general hgh. However, when a game can be descrbed by a small collecton of rules, and therefore the assocated extended GAS s descrbed n a relatvely compact way, the complexty of ts representaton and of the computaton of solutons s consequently reduced. Acknowledgements We thank two anonymous referees and the Assocate Edtor for ther valuable suggestons and comments on a former verson of ths paper. The research of S. Morett benefted from the support of the projects CoCoRICo- CoDec ANR-14-CE and NETLEARN ANR-13-INFR-004 of the French Natonal Research Agency (ANR). References 1. Aadthya, K. V., Ravndran, B., Mchalak, T. P., Jennngs, N. R. (2010). Effcent computaton of the Shapley value for centralty n networks. In Internatonal Workshop on Internet and Network Economcs, Sprnger Berln Hedelberg. 2. Amer, R., Gménez, J. M. (2004). A connectvty game for graphs. Mathematcal Methods of Operatons Research, 60(3), Aumann, R. J., Maschler, M. (1985). Game theoretc analyss of a bankruptcy problem from the Talmud. Journal of Economc Theory, 36(2), Banzhaf III, J. F. (1964). Weghted votng doesn t work: A mathematcal analyss. Rutgers L. Rev., 19, Bondareva, O. (1962). The theory of the core n an n-person game. Vestnk Lenngrad. Unv, 13, Bonzon, E., Maudet N., Morett S. Coaltonal games for abstract argumentaton. In Proceedngs of the 5th Internatonal Conference on Computatonal Models of Argument (COMMA 14), Borm, P., Hamers, H., Hendrckx, R. (2001). Operatons research games: A survey. Top, 9(2), Brânze, R., Fragnell, V., Tjs, S. (2002) Tree-connected peer group stuatons and peer group games Mathematcal Methods of Operatons Research 55(1), Cesar, G., Lucchett, R., Morett, S. (2015). Generalzed Addtve Games. Caher du LAMSADE Chalkadaks, G., Elknd, E.,Wooldrdge, M. (2011). Computatonal aspects of cooperatve game theory. Synthess Lectures on Artfcal Intellgence and Machne Learnng, 5(6), Curel, I., Potters, J., Prasad, R., Tjs, S., Veltman, B. (1993). Cooperaton n one machne schedulng. Zetschrft für Operatons Research, 38(2), Curel, I., Hamers, H., Tjs, S., Potters, J. (1997). Restrcted component addtve games. Mathematcal methods of operatons research, 45(2), Deng, X., Papadmtrou, C. H. (1994). On the complexty of cooperatve soluton concepts. Mathematcs of Operatons Research, 19(2),

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