Two-step values for games with two-level communication structure

Transcription

1 J Comb Optm (2018) 35: Two-step values for games wth two-level communcaton structure Sylvan Béal 1 Anna Khmelntsaya 2 Phlppe Solal 3 Publshed onlne: 1 ovember 2017 The Author(s) Ths artcle s an open access publcaton Abstract TU games wth two-level communcaton structure, n whch a two-level communcaton structure relates fundamentally to the gven coalton structure and conssts of a communcaton graph on the collecton of the a pror unons n the coalton structure, as well as a collecton of communcaton graphs wthn each unon, are consdered. For such games we ntroduce two famles of two-step values nspred by the two-step procedures stayng behnd the Owen value (Owen, n: Henn, Moeschln (eds) Essays n mathematcal economcs and game theory, Sprnger, Berln, pp 76 88, 1977) and the two-step Shapley value (Kamjo n Int Game Theory Rev 11: , 2009) for games wth coalton structure. Our approach s based on the unfed treatment of several component effcent values for games wth communcaton structure and t generates two-stage soluton concepts that apply component effcent values for games The authors are grateful to two anonymous revewers for valuable comments. The fnancal support of the frst and thrd authors from research program DynaMITE: Dynamc Matchng and Interactons: Theory and Experments, Contract AR-13-BSHS1-0010, and of the second author from RFBR (Russan Foundaton for Basc Research) Grant # s gratefully acnowledged. The research of the second author was done partally durng her stay at the Unversty of Twente whose hosptalty s also hghly apprecated. B Anna Khmelntsaya a.b.hmelntsaya@utwente.nl Sylvan Béal sylvan.beal@unv-fcomte.fr Phlppe Solal phlppe.solal@unv-st-etenne.fr 1 CRESE EA3190, Unv. Bourgogne Franche-Comté, Besançon, France 2 Sant-Petersburg State Unversty, Faculty of Appled Mathematcs, Sant Petersburg, Russa 3 Unversté Jean Monnet, Sant-Étenne, France

2 564 J Comb Optm (2018) 35: wth communcaton structure on both dstrbuton levels. Comparable axomatc characterzatons are provded. Keywords TU game wth two-level communcaton structure Owen value Two-step Shapley value Component effcency Deleton ln property Mathematcs Subject Classfcaton 91A12 91A43 JEL Classfcaton C71 1 Introducton In classcal cooperatve game theory t s assumed that any coalton of players may form, and a far dstrbuton of total rewards amongst the players taes nto account capactes of all coaltons. For example, the most promnent soluton of cooperatve games, the Shapley value (Shapley 1953), assgns to each player as a payoff the average of the player s margnal contrbutons to hs predecessors wth respect to all possble lnear orderngs of the players. However, n many practcal stuatons the collecton of feasble coaltons s restrcted by some socal, economcal, herarchcal, communcatonal, or techncal structure. The study of transferable utlty (TU) games wth lmted cooperaton ntroduced by means of coalton structures, or n other terms a pror unons, was ntated n the 1970 s frst by Aumann and Drèze (1974) and then by Owen (1977). In these papers a coalton structure s gven by a partton of the set of players, and both papers ntroduce as solutons some adaptatons of the classcal Shapley value to ths stuaton wth restrcted cooperaton. Whle Aumann and Drèze assume that cooperaton s possble only wthn a pror unons and as a soluton they propose the combnaton of Shapley values n subgames on a pror unons, Owen admts some cooperaton between players of dfferent unons. Smlar to the Shapley value, the Owen value assgns to each player as a payoff the average of the player s margnal contrbutons to hs predecessors wth respect to lnear orderngs of the players. But n case of Owen not all possble lnear orderngs are taen nto account, but only those n whch the players of the same a pror unon appear successvely. Another model of a game wth lmted cooperaton presented by means of undrected communcaton graphs was ntroduced n Myerson (1977). The man assumpton of Myerson s that only connected players are able to cooperate, and the Myerson value s gven by the Shapley value n the Myerson restrcted game, n whch the worth of each dsconnected coalton s replaced by the sum of the worths of ts connected components. Varous studes n both drectons were done durng the last four decades, but mostly ether wthn one model or another. Vázquez-Brage et al. (1996) s the frst study that combnes both models by consderng a TU game endowed wth, ndependent of each other, both a coalton structure and a communcaton graph on the set of players. For ths class of games they propose a soluton by applyng the Owen value for games wth coalton structure to the Myerson restrcted game of the game wth communcaton graph. Another model of a TU game endowed wth both a coalton structure and a communcaton graph, the so-called game wth two-level communcaton structure, s

3 J Comb Optm (2018) 35: consdered n Khmelntsaya (2014). In contrast to Vázquez-Brage et al. (1996), n ths model a two-level communcaton structure relates fundamentally to the gven coalton structure and conssts of a communcaton graph on the collecton of the a pror unons n the coalton structure, as well as a collecton of communcaton graphs wthn each unon. It s assumed that communcaton s only possble ether among the entre a pror unons or among sngle players wthn any a pror unon. o communcaton and therefore no cooperaton s allowed between proper subcoaltons, n partcular sngle players, of dstnct elements of the coalton structure. 1 Ths approach allows to model dfferent networ stuatons, n partcular, telecommuncaton problems, dstrbuton of goods among dfferent ctes (countres) along hghway networs connectng the ctes and local road networs wthn the ctes, or sharng an nternatonal rver wth multple users but wthout nternatonal frms,.e., when no cooperaton s possble among sngle users located at dfferent levels along the rver, and so on. Communcaton structures under scrutny are gven by combnatons of graphs of dfferent types both undrected-arbtrary graphs and cycle-free graphs, and drected-lne-graphs wth lnearly ordered players, rooted forests and sn forests. The proposed soluton concepts reflect a two-stage dstrbuton procedure when, frst, a pror unons collect ther shares through the upper level barganng based only on the cumulatve nterests of all members of every nvolved entre a pror unon, and second, the players collect ther ndvdual payoffs through the lower level barganng over the dstrbuton of the unons shares wthn the unons. Followng Myerson (1977) t s assumed that cooperaton s possble only between connected partcpants and dfferent combnatons of nown component effcent values, such as the Myerson value, the poston value, the average tree soluton, etc., are appled on both communcaton levels. However, as dscussed n Khmelntsaya (2014), the two-stage dstrbuton procedure based on the applcaton of component effcent values on both levels suffers from severe restrctons n cases when some a pror unons are nternally not connected, because each unon always has to dstrbute ts total share among the members. Another soluton concept for TU games wth two-level communcaton structure ntroduced by means of undrected graphs, the so-called Owen-type value for games wth two-level communcaton structure, s consdered n van den Brn et al. (2016) under a weaer assumpton concernng the communcaton on the level of a pror unons, when on the upper level barganng between a pror unons, smlar as for the Owen value ntroduced n Owen (1977), one of the a pror unons can be presented by any of ts proper subcoaltons. Ths soluton can be seen as an adaptaton of the two-step procedure determnng the Owen value for games wth coalton structure whch taes nto account the lmted cooperaton represented by two-level communcaton structure replacng twce the Shapley value by the Myerson value. 2 1 A smlar model, but wth other and qute specal assumptons concernng the ablty of players to cooperate under gven communcaton constrants s also studed n Kongo (2011). 2 An extenson of the Owen-type value ntroduced n van den Brn et al. (2016), when the underlyng two-step procedure determnng the Owen value for games wth coalton structure s replaced by a smlar two-step procedure, n whch on both steps the applcaton of the Shapley value s replaced by the τ-value, s studed recently n Zhang et al. (2017).

4 566 J Comb Optm (2018) 35: In ths paper we assume that a two-level communcaton structure s gven by combnatons of graphs of dfferent types, both undrected and drected, and we ntroduce two famles of two-step values for games wth two-level communcaton structure adaptng the two-step procedures stayng behnd two values for games wth coalton structure, the Owen value and the two-step Shapley value ntroduced n Kamjo (2009). Our approach s based on the unfed treatment of several component effcent values for games wth communcaton structure and t generates two-stage soluton concepts that apply component effcent values for games wth communcaton structure on both dstrbuton levels. In fact the newly ntroduced famly of the Owen-type values s the generalzaton of the Owen-type value for games wth two-level communcaton structures of van den Brn et al. (2016), when on both communcaton levels not only the Myerson value, but dfferent component effcent values for games wth communcaton structure can be appled. The ncorporaton of dfferent solutons for games wth communcaton structure ams not only to enrch the soluton concepts for games wth two-level communcaton structure, but t also opens a broad dversty of applcatons mpossble otherwse, because there exsts no unversal soluton concept for games wth communcaton structure that s applcable to the full varety of possble undrected and drected graph structures. Moreover, t allows to choose, dependng on types of graph structures under scrutny, the most preferable, n partcular, the most computatonally effcent combnaton of values among others sutable. We provde axomatc characterzatons of the ntroduced two-step values. These axomatzatons have several common axoms for the both famles whch allows to compare the twostep values from dfferent famles. The ntroduced two famles of two-step soluton concepts may fnd applcaton n dfferent resource allocaton problems wth two herarchcal dstrbuton levels. For nstance, they may be used for budget allocaton wthn a unversty, when the budget has to be dstrbuted frst among ts departments and then among the ndvduals wthn each department. An example showng advantages of usng the Owen-type value for games wth two-level communcaton structures, n case when on both communcaton levels only the Myerson value s appled, s dscussed n van den Brn et al. (2016). The rest of the paper s organzed as follows. Basc defntons and notaton are ntroduced n Sect. 2. Secton 3 provdes the unform approach to several nown component effcent values for games wth communcaton structure, whch allows also to consder wthn a unfed framewor dfferent deleton ln propertes wth respect to the values for games wth two-level communcaton structure. In Sects. 4 and 5 we ntroduce correspondngly the famles of the Kamjo-type and Owen-type values axomatcally and present ther explct formula representatons. Secton 6 concludes. 2 Prelmnares 2.1 TU games and values A cooperatve game wth transferable utlty, or TU game, s a par,v, where I s a fnte set of n players and v : 2 IR s a characterstc functon wth v( ) = 0, assgnng to every coalton S of s players ts worth v(s). The set of

5 J Comb Optm (2018) 35: TU games wth fxed player set s denoted by G. For smplcty of notaton and f no ambguty appears we wrte v when we refer to a TU game,v. Thesubgame of a TU game v G wth respect to a player set T s the TU game v T G T defned by v T (S) = v(s) for every S T.Apayoff vector s a vector x IR wth x the payoff to player. A sngle-valued soluton, called a value, s a mappng ξ : G IR that assgns to every fnte set I and every TU game v G a payoff vector ξ(v) IR.Avalueξ s effcent f ξ (v) = v() for every v G and I. The best-nown effcent value s the Shapley value (Shapley 1953) gven by Sh (v) = S \{} s!(n s 1)! (v(s {}) v(s)), for all. n! In the sequel we denote the cardnalty of a gven set A by A, along wth lower case letters le n =, m = M, n =, s = S, c = C, c = C, and so on, and we use the standard notaton x(s)= S x for any x IR and S. 2.2 Games wth coalton structure A coalton structure, or n other terms a system of a pror unons,on I s gven by a partton P ={ 1,..., m } of,.e., 1... m = and l = for = l.letp denote the set of all coalton structures on, and let G P = G P. A par v, P G P consttutes a game wth coalton structure, orsmplyp-game, on. Remar that v, {} represents the same stuaton as v tself. A P-value s a mappng ξ : G P IR that assgns to every I and every P-game v, P G P a payoff vector ξ(v,p) IR.AP-value ξ s effcent f ξ (v, P) = v() for every v G P and I. In what follows, denote by M ={1,...,m} the ndex set of all a pror unons n P; for every P-game v, P G P and every M let v denote the subgame v ; for every,let() be defned by the relaton () ; and for every x IR,letx P = ( x( ) ) M IR M stand for the vector of total payoffs to a pror unons. One of the best-nown values for games wth coalton structure s the Owen value (Owen 1977) that can be seen as a two-step procedure n whch the Shapley value apples twce. amely, the Owen value assgns to player hs Shapley value n the game v (),.e., Ow (v, P) = Sh ( v () ), for all, whle for every a pror unon, M, thegame v G on the player set s gven by v (S) = Sh ( ˆv S ), S,

6 568 J Comb Optm (2018) 35: where for every S the game ˆv S G M on the player set M of a pror unons s defned by ˆv S (Q) = { v( h Q h ), / Q, v( h Q\{} h S), Q, for all Q M. The Owen value s effcent,.e., Ow (v, P) = v(), and satsfes the quotent game property,.e., for every a pror unon the total payoff to the players wthn that unon s determned by applyng the Shapley value to the so-called quotent game beng the game v P G M n whch the unons act as ndvdual players, v P (Q) = v( Q ), for all Q M. otce that for every M the game ˆv s equal to the quotent game v P. Another value for games wth coalton structure that also can be seen as a two-step procedure n whch the Shapley value apples twce s the so-called two-step Shapley value ψ ntroduced n Kamjo (2009). The two-step Shapley value frst allocates to player hs Shapley value n the subgame on the a pror unon () he belongs to and then dstrbutes what remans of the Shapley value of ts unon n the quotent game equally among the unon s members,.e., for any P-game v, P G P, Ka (v, P) = Sh (v () ) + Sh ()(v P ) v( () ) n (). (1) The Kamjo s two-step Shapley value s effcent and meets the quotent game property. 2.3 Games wth communcaton structure A communcaton structure on s specfed by a graph Ɣ, undrected or drected, on. Agraph on conssts of as the set of nodes and for an undrected graph a collecton of unordered pars Ɣ {{, j}, j, = j} as the set of lns between two nodes n, and for a drected graph, oradgraph, a collecton of ordered pars Ɣ {(, j), j, = j} as the set of drected lns from one node to another node n. When t s necessary to specfy the set of nodes nagraphɣ, we wrte Ɣ nstead of Ɣ.LetG denote the set of all communcaton structures, undrected or drected, on, and let G Γ = G G. A par v, Ɣ G Γ consttutes a game wth graph (communcaton) structure, orsmplyagraph game, oraγ-game, on. A Γ -value s a mappng ξ : G Γ IR that assgns to every I and every Γ -game v, Ɣ G Γ a payoff vector ξ(v,ɣ) IR. In a graph Ɣ a sequence of dfferent nodes ( 1,..., r ), r 2, s a path n Ɣ from node 1 to node r f for h = 1,...,r 1 t holds that { h, h+1 } Ɣ when Ɣ s undrected and {( h, h+1 ), ( h+1, h )} Ɣ = when Ɣ s drected. In a dgraph Ɣ a path ( 1,..., r ) s a drected path from node 1 to node r f ( h, h+1 ) Ɣ for all h = 1,...,r 1. In a dgraph Ɣ, j = s a successor of and s a predecessor of j f there exsts a drected path from to j. Gven a dgraph Ɣ on and, the

7 J Comb Optm (2018) 35: sets of predecessors and successors of n Ɣ we denote correspondngly by P Ɣ () and S Ɣ (); moreover, P Ɣ () = P Ɣ () {} and S Ɣ () = S Ɣ () {}. Gven a graph Ɣ on, two nodes and j n are connected f ether there exsts a path from node to node j,or and j concde. Graph Ɣ on s connected f any two nodes n are connected. For a graph Ɣ on and a coalton S, thesubgraph of Ɣ on S s the graph Ɣ S ={{, j} Ɣ, j S} on S when Ɣ s undrected and the dgraph Ɣ S ={(, j) Ɣ, j S} on S when Ɣ s drected. Gven a graph Ɣ on, a coalton S s connected f the subgraph Ɣ S s connected. For a graph Ɣ on and coalton S, C Ɣ (S) s the set of all connected subcoaltons of S, S/Ɣ s the set of maxmal connected subcoaltons of S, called the components of S, and (S/Ɣ) s the component of S contanng player S. otce that S/Ɣ s a partton of S. For any v, Ɣ G Γ, a payoff vector x IR s component effcent f x(c) = v(c), for every C /Ɣ. Followng Myerson (1977), we assume that for Γ -games cooperaton s possble only among connected players and concentrate on component effcent Γ -values. A Γ -value ξ s component effcent (CE) f for any v, Ɣ G Ɣ, for all C /Ɣ, C ξ (v, Ɣ) = v(c). BelowforaΓ-game v, Ɣ G Γ we also consder the ntroduced n Myerson (1977) restrcted game v Ɣ G defned as v Ɣ (S) = C S/Ɣ v(c), for all S. (2) Herenafter along wth communcaton structures gven by arbtrary undrected graphs we consder also those gven by cycle-free undrected graphs and by drected graphs lnear graphs wth lnearly ordered players, rooted and sn forests. In an undrected graph Ɣ a path ( 1,..., r ), r 3, s a cycle n Ɣ f { r, 1 } Ɣ. An undrected graph s cycle-free f t contans no cycles. A drected graph Ɣ s a rooted tree f there s one node n, called a root, havng no predecessors n Ɣ and there s a unque drected path n Ɣ from ths node to any other node n. A drected graph Ɣ s a sn tree f the drected graph composed by the same set of lns as Ɣ but wth the opposte orentaton s a rooted tree; n ths case the root of a tree changes ts meanng to the absorbng sn. A drected graph s a rooted/sn forest f t s composed by a number of dsjont rooted/sn trees. A lnear graph s a drected graph that contans lns only between subsequent nodes. Wthout loss of generalty we may assume that n a lnear graph nodes are ordered accordng to the natural order from 1 to n,.e., lnear graph Ɣ {(, + 1) = 1,...,n 1}. For ease of notaton gven graph Ɣ and ln {, j} Ɣ f Ɣ s undrected, or (, j) Ɣ f Ɣ s drected, the subgraph Ɣ\{{, j}}, correspondngly Ɣ\{(, j)}, s denoted by Ɣ j. 2.4 Games wth two-level communcaton structure We now consder stuatons n whch the players are parttoned nto a coalton structure P and are lned to each other by communcaton graphs. Frst, there s a communcaton graph Ɣ M on the set of a pror unons determned by the partton

8 570 J Comb Optm (2018) 35: P. Second, for each a pror unon, M, there s a communcaton graph Ɣ between the players n. In what follows for smplcty of notaton and when t causes no ambguty we denote graphs Ɣ wthn a pror unons, M, by Ɣ. Gven a player set I and a coalton structure P P,atwo-level graph (communcaton) structure on s a tuple Ɣ P = Ɣ M, {Ɣ } M. For every I and P P by G P we denote the set of all two-level graph structures on wth fxed P.LetG P = P P G P be the set of all two-level graph structures on, and let G PΓ = G G P. A par v, Ɣ P G PΓ consttutes a game wth two-level graph (communcaton) structure, orsmplyatwo-level graph game or a PΓ -game, on. A PΓ -value s a mappng ξ : G PΓ IR that assgns to every I and every PΓ -game v, Ɣ P G PΓ a payoff vector ξ(v,ɣ P ) IR. Gven a PΓ -game v, Ɣ P G PΓ, one can consder the quotent Γ -game v P,Ɣ M GM Γ and the Γ -games wthn a pror unons v,ɣ G Γ wth v = v, M, that model the barganng between a pror unons for ther total shares and the barganng wthn each a pror unon for the dstrbuton of ts total worth among the members tang also nto account lmted cooperaton at both communcaton levels ntroduced by the communcaton graphs Ɣ M and Ɣ, M. Moreover, gven a Γ -value φ, for any v, Ɣ P G PΓ wth a graph structure Ɣ M on the level of a pror unons sutable for applcaton of φ to the correspondng quotent Γ -game v P,Ɣ M, along wth a subgame v wthn a pror unon, M, one can also consder a φ -game v φ defned as v φ (S) = { φ (v P,Ɣ M ), S =, v(s), S =, for all S, where φ (v P,Ɣ M ) s the payoff to gven by φ n v P,Ɣ M. Then a Γ -game v φ,ɣ G Γ models the barganng wthn unon for the dstrbuton of ts total share among the members tang nto account restrctons on cooperaton n gven by Ɣ, when the share s obtaned by the applcaton of Γ -value φ at the upper level barganng between a pror unons. 3 Deleton ln propertes for two-level graph games As t s dscussed n Khmelntsaya (2014), a number of nown component effcent Γ -values for games wth undrected or drected communcaton structure such as for undrected graph games the Myerson value μ (cf., Myerson 1977) and the poston value π (cf., Meessen 1988; Borm et al. 1992; Sler 2005) for arbtrary undrected graph games, the average tree soluton AT (cf., Herngs et al. 2008) and the compensaton soluton CS (cf., Béal et al. 2012) for undrected cycle-free graph games, or for drected graph games the upper equvalent soluton UE, the lower equvalent soluton LE and the equal loss soluton EL for lnear graph games (cf., van den Brn et al. 2007),thetreevaluet for rooted forest and the sn value s for sn forest dgraph games (cf., Khmelntsaya 2010), can be approached wthn the unfed framewor. Indeed, each one of these Γ -values s defned for Γ -games wth sutable graph struc-

9 J Comb Optm (2018) 35: ture and s characterzed by two axoms, CE and one or another deleton ln (DL) property (axom), reflectng the relevant reacton of a Γ -value on the deleton of a ln n the communcaton graph. The correspondng DL propertes are farness (F), balanced ln contrbutons (BLC), component farness (CF), relatve farness (RF), upper equvalence (UE), lower equvalence (LE), equal loss property (EL), successor equvalence (SE), and predecessor equvalence (PE), and the characterzaton results are as follows: CE + F for all undrected Γ -games μ(v, Ɣ), CE + BLC for all undrected Γ -games π(v,ɣ), CE + CF for undrected cycle-free Γ -games AT(v, Ɣ), CE + RF for undrected cycle-free Γ -games CS(v, Ɣ), CE + UE for lnear graph Γ -games UE(v, Ɣ), CE + LE for lnear graph Γ -games LE(v, Ɣ), CE + EL for lnear graph Γ -games EL(v, Ɣ), CE + SE for rooted forest Γ -games t(v, Ɣ), CE + PE for sn forest Γ -games s(v, Ɣ). Ths observaton allows to dentfy each of the lsted above Γ -values wth the correspondng DL axom. Gven a DL axom, let G DL GΓ denote a set of all v, Ɣ GΓ wth Ɣ sutable for DL applcaton. Then CE + DL on G DL DL(v, Ɣ). Whence t smply follows that F(v, Ɣ) = μ(v, Ɣ) and BLC(v, Ɣ) = π(v,ɣ) for all undrected Γ -games, CF(v, Ɣ) = AT(v, Ɣ) and RF(v, Ɣ) = CS(v, Ɣ) for all undrected cycle-free Γ -games, UE(v, Ɣ), LE(v, Ɣ), and EL(v, Ɣ) are UE, LE, and EL solutons correspondngly for all lnear graph Γ -games, SE(v, Ɣ) = t(v, Ɣ) for all rooted forest Γ -games, and PE(v, Ɣ) = s(v, Ɣ) for all sn forest Γ -games. Remar that all just dscussed values are addtve. ext notce that every dscussed DL axom can be equvalently defned by an equalty DL (ξ(v, Ɣ), Ɣ ) = 0, (3) where DL s an operator whch for a Γ -value ξ defned on G DL and appled to a Γ -game v, Ɣ G DL assgns a real number representng the numercal evaluaton of players payoff reacton on the deleton of lns n Ɣ from a chosen set of lns Ɣ Ɣ accordng to the consdered DL axom. For the above mentoned DL-axoms, we have Ɣ = Ɣ, and the correspondng operators of the DL-axoms are: Farness (F) For any player set I, for every Γ -game v, Ɣ G Γ, F (ξ(v, Ɣ), Ɣ )=, j {, j} Ɣ ( ) (, ξ (v, Ɣ) ξ (v, Ɣ j ) ξ j (v, Ɣ) ξ j (v, Ɣ j )) (4)

10 572 J Comb Optm (2018) 35: Balanced ln contrbutons (BLC) For any player set I, for every Γ -game v, Ɣ G Γ, BLC (ξ(v, Ɣ), Ɣ ) =, j h {,h} Ɣ h { j,h} Ɣ ( ) ξ j (v, Ɣ) ξ j (v, Ɣ h ) ( ξ (v, Ɣ) ξ (v, Ɣ jh )), (5) Component farness (CF) For any player set I, for every cycle-free Γ -game v, Ɣ G Γ, CF (ξ(v, Ɣ), Ɣ ) =, j {, j} Ɣ 1 (/Ɣ j ) j 1 (/Ɣ j ) h (/Ɣ j ) j h (/Ɣ j ) ( ) ξ h (v, Ɣ) ξ h (v, Ɣ j ) ( ξh (v, Ɣ) ξ h (v, Ɣ j ) ), (6) Relatve farness (RF) For any player set I, for every cycle-free Γ -game v, Ɣ G Γ, ( RF (ξ(v, Ɣ), Ɣ 1 ) ) = ξ (v, Ɣ) ξ h (v, Ɣ j ) (/Ɣ, j {, j} Ɣ j ) h (/Ɣ j ) ( 1 ), ξ j (v, Ɣ) ξ h (v, Ɣ j ) (7) (/Ɣ j ) j h (/Ɣ j ) j Upper equvalence (UE) For any player set I, for every lnear graph Γ -game v, Ɣ G Γ, UE (ξ(v, Ɣ), Ɣ ) = n 1 =1 j=1 ξ j (v, Ɣ) ξ j (v, Ɣ,+1 ). (8) Lower equvalence (LE) For any player set I, for every lnear graph Γ -game v, Ɣ G Γ, LE (ξ(v, Ɣ), Ɣ ) = n 1 n =1 j=+1 ξ j (v, Ɣ) ξ j (v, Ɣ,+1 ). (9)

11 J Comb Optm (2018) 35: Equal loss property (EL) For any player set I, for every lnear graph Γ -game v, Ɣ G Γ, EL (ξ(v, Ɣ), Ɣ ) n 1 ( ) = ξ j (v, Ɣ) ξ j (v, Ɣ,+1 ) =1 j=1 n j=+1 ( )) ξ j (v, Ɣ) ξ j (v, Ɣ,+1. Successor equvalence (SE) For any player set I, for every rooted forest Γ - game v, Ɣ G Γ, SE (ξ(v, Ɣ), Ɣ ) =, j {, j} Ɣ h S Ɣ ( j) (10) ξh (v, Ɣ) ξ h (v, Ɣ j ). (11) Predecessor equvalence (PE) For any player set I, for every sn forest Γ -game v, Ɣ G Γ, PE (ξ(v, Ɣ), Ɣ ) =, j {, j} Ɣ h P Ɣ ( j) ξ h (v, Ɣ) ξ h (v, Ɣ j ). (12) The defnton of deleton ln axoms for Γ -values n terms of equalty (3) allows to consder the correspondng deleton ln axoms wthn a unfed framewor also for PΓ -values wth respect to both communcaton levels, at the upper level wth respect to the quotent Γ -game determnng the total payoffs to a pror unons and at the lower level wth respect to Γ -games wthn a pror unons determnng the dstrbuton of total payoffs wthn each a pror unon. Let ξ be a PΓ -value. By defnton ξ s a mappng ξ : G PΓ IR that assgns a payoff vector to any PΓ -game on the player set I. For every I a mappng ξ ={ξ } generates on the doman of PΓ -games on a mappng ξ P : G PΓ IR M, ξ P ={ξ P} M, wth ξ P = ξ, M, that assgns to every PΓ -game on a vector of total payoffs to all a pror unons, and m mappngs ξ : G PΓ IR, ξ ={ξ }, M, assgnng payoffs to players wthn a pror unons. For a gven (m + 1)-tuple of deleton ln axoms DL P, {DL } M for Γ -values, let G DLP,{DL } M G PΓ be the set of PΓ -games composed of PΓ -games v, Ɣ P wth graph structures Ɣ P = Ɣ M, {Ɣ } M such that v P,Ɣ M GM DLP and v DLP,Ɣ G DL, M. Gvena(m + 1)-tuple of deleton ln axoms DL P, {DL } M for Γ -values and a PΓ -game v, Ɣ P G DLP,{DL } M, we may consder the operators (4) (12) appled to the Γ -value ξ P assgnng the total payoffs to a pror unons to evaluate ts reacton on deleton of lns n graph Ɣ M, or appled to the Γ -values ξ, M, assgnng payoffs to sngle players wthn a pror unons to evaluate ther reacton on deleton of lns n the correspondng graphs Ɣ. Based on the last observaton, we defne axoms of Quotent DL property and Unon DL property for PΓ -values defned on G DLP,{DL } M as follows:

12 574 J Comb Optm (2018) 35: Quotent DL property (QDL) For any player set I and PΓ -game v, Ɣ P G DLP,{DL } M, Ɣ P = Ɣ M, {Ɣ h } h M, DLP (ξ P (v, Ɣ P ), Ɣ M ) = 0, (13) where DLP s gven by one of the operators (4) (12) and n the correspondng formulas Ɣ P l = Ɣ M l, {Ɣ h } h M for every ln {, l} Ɣ M f Ɣ M s undrected, or (, l) Ɣ M f Ɣ M s drected. Unon DL property (UDL) For any player set I, PΓ -game v, Ɣ P G DLP,{DL } M, Ɣ P = Ɣ M, {Ɣ h } h M, and M, DL (ξ (v, Ɣ P ), Ɣ ) = 0, (14) where DL s gven by one of the operators (4) (12) and n the correspondng formulas Ɣ P j = Ɣ P j = Ɣ M, { Ɣ h } h M wth Ɣ h = Ɣ h for h = and Ɣ = Ɣ j for every ln {, j} Ɣ f Ɣ s undrected, or (, j) Ɣ f Ɣ s drected. For example, f we consder the Myerson farness (F) as a DL axom, QDL and UDL can be denoted QF and UF correspondngly. In fact n ths case QF concdes wth quotent farness (QF) and m-tuple of axoms (UF 1,...,UF m ) concdes wth unon farness (UF) employed n van den Brn et al. (2016). QDL and UDL axoms for PΓ -values provde the unform approach to varous deleton ln propertes on both barganng levels. Ths allows to ntroduce wthn the unfed framewor two famles of PΓ -values based on the adaptaton of the two-step dstrbuton procedures of Owen and Kamjo respectvely for games wth coalton structure to the case when the cooperaton between and wthn a pror unons s restrcted by communcaton graphs and when dfferent combnatons of nown component effcent soluton concepts on both communcaton levels could be appled. Moreover, ths allows to nclude nto consderaton not only combnatons of undrected communcaton graphs but also combnatons ncludng some types of dgraphs. 4 Kamjo-type values for two-level graph games In ths secton we consder the famly of PΓ -values based on the adaptaton of the Kamjo s two-step dstrbuton procedure for P-games. We ntroduce these values axomatcally by means of sx axoms among whch QDL and UDL as defned n the prevous secton. The other four axoms are defned below. To begn wth, Quotent component effcency requres that each component on the upper level between a pror unons dstrbutes fully ts total worth among the players of the a pror unons formng ths component.

13 J Comb Optm (2018) 35: Quotent component effcency (QCE) For any, any v, Ɣ P G PΓ, Ɣ P = Ɣ M, {Ɣ } M, ξ (v, Ɣ P ) = v P (K ), for all K M/Ɣ M. (15) K The second axom s a straghtforward adaptaton of the standard covarance under strategc equvalence to the case of PΓ -games. Two games v, w G are strategcally equvalent f there are a IR ++ and b IR n such that w(s) = av(s) + b(s), for all S. Covarance under strategc equvalence (COV) For any player set I and PΓ -game v, Ɣ P G PΓ, Ɣ P = Ɣ M, {Ɣ } M, for any a IR ++ and b IR n t holds that ξ(av + b,ɣ P ) = aξ(v,ɣ P ) + b, where the game (av + b) G s defned by (av + b)(s) = av(s) + b(s) for all S. The next axom requres equal payoffs to all members of every a pror unon, M, for whch all subcoaltons S wth nonzero worth v (S) = 0are dsconnected. In ths case every connected coalton possesses zero worth,.e., communcaton between the members of every connected coalton s useless, and so, the asymmetres among the players created by game v and by ther locatons n graph Ɣ on vansh. Therefore, t maes sense to treat all players of symmetrcally. Remar that the condton that v(s) = 0 mples S to be dsconnected s equvalent to v Ɣ 0,.e., the axom requres that all players n obtan the same payoffs f the Myerson restrcted game v Ɣ s a null game. The latter observaton determnes the name of the axom. Unon null restrcted game property (URGP) For any player set I and PΓ -game v, Ɣ P G PΓ, Ɣ P = Ɣ M, {Ɣ } M,fforsome M, for all S, v (S) = 0 mples S / C Ɣ ( ), then t holds that for all, j, = j, ξ (v, Ɣ P ) = ξ j (v, Ɣ P ). The last axom determnes the dstrbuton of the total shares obtaned by nternally dsconnected a pror unons at the upper level barganng between a pror unons among the components of these unons. Imagne that each a pror unon, M, s a publc nsttuton (e.g. unversty, hosptal, or frm) of whch every component C /Ɣ s an ndependent unt (e.g. the facultes wthn a unversty, medcal departments wthn a hosptal, or producton plants wthn a frm). Frst publc nsttutons, M compete among themselves for ther annual budgets from the government. Once obtaned the budget, nsttuton has to decde how much to gve

14 576 J Comb Optm (2018) 35: to each of ts ndependent unts. At ths stage the ndependent unts of an nsttuton compete aganst each other for the best possble shares from the nsttuton s budget. Smlarly as n the competton among the publc nsttutons, the total payoff to a unt depends on the total productvty of each of the unts, but not on the productvty of the smaller collaboratng teams wthn the unts. Our last axom requres the total payoff to a component of any a pror unon to be ndependent of the so-called nternal coaltons, each of whch s a proper subcoalton of some component of one of the gven a pror unons, or more precsely, a coalton = S s nternal f there s M such that S C for some C /Ɣ. From now on, gven a player set I, a partton P ={ 1,..., m } of, and a set of communcaton graphs {Ɣ } M on a pror unons, M, the set of all nternal coaltons we denote by Int(, P, {Ɣ } M ). It s worth to remar that, of course, the worths of nternal coaltons play a crucal role n the redstrbuton of the total payoff obtaned by a component among ts members, but the axom does not concern ths. Unon component payoff ndependence of nternal coaltons (UCPIIC) For any player set I and two PΓ -games v, Ɣ P, w, Ɣ P G PΓ wth the same Ɣ P = Ɣ M, {Ɣ } M and such that w(s) = v(s) for all S, S / Int(, P, {Ɣ } M ), t holds that for every M, for all C /Ɣ, ξ (v, Ɣ P ) = ξ (w, Ɣ P ). C C Our frst theorem provdes an axomatc characterzaton of the a famly of PΓ - values. Theorem 1 For any (m + 1)-tuple of deleton ln axoms DL P, {DL } M such that DL P s not RF and the set of DL, M, axoms s restrcted to F and CF, there s a unque PΓ -value defned on G DLP,{DL } M that meets axoms QCE, QDL, UDL, COV, URGP, and UCPIIC, and for every PΓ -game v, Ɣ P G DLP,{DL } M t s gven by ξ (v, Ɣ P ) = DL () (v (),Ɣ () ) + DLP () (v P,Ɣ M ) v Ɣ() Before provng Theorem 1, we formulate some remars. () ( ()) n (), for all. (16) Remar 1 It s not dffcult to trace a relaton between the PΓ -value ξ gven by (16) and the two-step Shapley value (1). Indeed, for a PΓ -game v, Ɣ P G FP,{F } M, ξ(v,ɣ P ) = Ka(v Ɣ P, P), where v Ɣ P G s determned as v Ɣ M v Ɣ P (Q), Q M : S = q Q q, P (S) = v Ɣ (S), S, M, 0, otherwse, for all S,

15 J Comb Optm (2018) 35: e., when S s the unon of number of a pror unons ts worth s defned accordng to the Myerson restrcted quotent game, when S s a subset of some a pror unon ts worth s defned accordng to the Myerson restrcted game wthn the unon, otherwse the worth of S s zero. Because of the mentoned smlarty, from now on we refer to the PΓ -value (16)as to the Kamjo-type DL P, {DL } M -value, denoted further by Ka DLP,{DL } M. Remar 2 Axom RF cannot be used at any level n Theorem 1. The reason s that t gves rse to the compensaton soluton at the level where RF s appled and the compensaton soluton does not satsfy COV on ts doman. Furthermore, the PΓ - value Ka DLP,{DL } M volates the UDL property f n the (m + 1)-tuple of deleton ln axoms DL P, {DL } M among the axoms DL, M there are axoms BLC, UE, LE, EL, SE, and PE. The reason s that the second summand n the numerator of the second term n the rght-hand sde of (16) s senstve to the deleton of dfferent lns n graph Ɣ, and therefore, for these cases the equalty DL (ξ (v, Ɣ P ), Ɣ ) = DL (DL (v,ɣ ), Ɣ ) = 0 n general does not hold. Proof (Theorem 1)I.[Exstence]. We show that under the hypothess of the theorem the PΓ -value ξ = Ka DLP,{DL } M defned on G DLP,{DL } M by (16) meets the axoms QCE, QDL, UDL, COV, URGP and UCPIIC. Consder an arbtrary PΓ - game v, Ɣ P G DLP,{DL } M. QCE By the defnton (16)ofξ and component effcency of each DL -value for all M t holds that ξ (v, Ɣ P ) = DL P (v P,Ɣ M ). (17) Thus, for any K M/Ɣ M we have ξ (v, Ɣ P ) = DL P (v P,Ɣ M ) = v P (K ), K K where the second equalty follows from component effcency of DL P -value. QDL From (17) we obtan that ξ P (v, Ɣ P ) = DL P (v P,Ɣ M ). Whence t follows that ( ) ( ) DLP ξ P (v, Ɣ P ), Ɣ M = DLP DL P (v P,Ɣ M ), Ɣ M = 0, where the last equalty holds true snce the DL P -value for Γ -games meets DL P. UDL We need to show that f the set of axoms DL, M, s restrcted to F and CF, then for all M, DL (ξ (v, Ɣ P ), Ɣ ) = 0. Let for some M, DL = F. Then snce by defnton for any ln {, j} Ɣ, Ɣ P j = Ɣ P j = Ɣ M, { Ɣ h } h M wth Ɣ h = Ɣ h for h =, and Ɣ = Ɣ j,we

16 578 J Comb Optm (2018) 35: obtan that F (ξ (v, Ɣ P ), Ɣ ) (4) ( ) ( = ξ (v, Ɣ P) ξ (v, Ɣ P j ) ξ j (v, Ɣ P) ξ j (v, Ɣ P j )) (16) =, j {, j} Ɣ, j {, j} Ɣ ( F (v,ɣ ) + DLP (v P,Ɣ M ) v Ɣ ( ) F (v,ɣ j ) + DLP (v P,Ɣ M ) v Ɣ j ( )) n (Fj (v,ɣ ) + DLP (v P,Ɣ M ) v Ɣ ( ) Fj n (v,ɣ j ) + DLP (v P,Ɣ M ) v Ɣ j ( )) n ( ) ( = F (v,ɣ ) F (v,ɣ j ) F j (v,ɣ ) F j (v,ɣ j )), j {, j} Ɣ (4) = F (F(v,Ɣ ), Ɣ ) = 0, where the last equalty holds true snce the F-value for Γ -games (the Myerson value) meets farness F. Usng the smlar arguments as above we prove that for all M for whch DL = CF, CF (ξ (v, Ɣ P ), Ɣ ) = 0. COV Pc any a IR ++ and b IR n. Then, for all, n ξ (av+b,ɣ P ) (16),(2) = DL () ((av + b) (),Ɣ () ) + DLP () ((av + b) P,Ɣ M ) C () /Ɣ () (av + b)(c) n () = adl () (v (),Ɣ () ) + b + adlp () (v P,Ɣ M )+b( () ) C () /Ɣ () (av(c)+b(c)) n () = aξ (v, Ɣ P ) + b, where the second equalty s true because each of the consdered DL -values, M, and DL P -values meets COV on ts doman, and (av+b) P (Q)=av P (Q)+b( Q ) for all Q M; and the thrd equalty s due to the equalty C () /Ɣ () b(c) = b( () ), snce () /Ɣ () forms a partton of (). URGP Assume that for the chosen PΓ -game v, Ɣ P there exsts M such that for all S, v (S) = 0 mples S / C Ɣ ( ). Every of the consdered DL -values,

17 J Comb Optm (2018) 35: the Myerson value and the average tree soluton, are determned only by worths of connected coaltons, and therefore, DL (v,ɣ ) = 0 for all. Moreover, as t was already mentoned earler, the above assumpton s equvalent to v Ɣ 0, whch mples that v(c) = 0 for every C /Ɣ. Hence, from (16) t follows that for all, ξ (v, Ɣ P ) = DLP (v P,Ɣ M ) n, where the rght sde s ndependent of, from whch t follows that PΓ -value ξ meets URGP. UCPIIC. Tae any M and C /Ɣ. The component effcency of each of the consdered DL -values mples DL (v,ɣ )(C) = v(c). Then from (16) t follows that ξ(v,ɣ P ) = v(c) + c ( ) DL P n (v P,Ɣ M ) v Ɣ ( ), C where the rght sde s ndependent of worths of nternal coaltons,.e. PΓ -value ξ meets UCPIIC. II. [Unqueness]. Assume that a (m + 1)-tuple of deleton ln axoms DL P, {DL } M such that the set of axoms DL, M, s restrcted to F and CF, s gven. We show that there exsts at most one PΓ -value on G DLP,{DL } M that satsfes axoms QCE, QDL, UDL, COV, URGP, and UCPIIC. Let φ be such PΓ -value on G DLP,{DL } M. Tae an arbtrary PΓ -game v, Ɣ P G DLP,{DL } M.Fxsome M. We start by determnng the unon payoffs φ P(v, Ɣ P), M, by nducton on the number of lns n Ɣ M smlarly as t s done n the proof of unqueness of the Myerson value for Γ -games, cf., Myerson (1977). Intalzaton: If Ɣ M =0then for all M the set of neghborng unons {h M {h, } Ɣ} =, and therefore by QCE and defnton of the quotent game v P, φ P(v, Ɣ P) = v P ({}) = v( ). Inducton hypothess: Assume that the values φ P(v, Ɣ P ), M, have been determned for all two-level graph structures Ɣ P = Ɣ, {Ɣ h } h M wth Ɣ such that Ɣ < Ɣ M. Inducton step:letq M/Ɣ M be a component n graph Ɣ M on M.IfQ M s a sngleton, let Q ={}, then from QCE t follows that φ P(v, Ɣ P) = v( ).If q 2, then there exsts a spannng tree Ɣ Ɣ M Q on Q wth the number of lns Ɣ =q 1. By QDL t holds that DLP (φ P (v, Ɣ P ), Ɣ M ) = 0. (18) The above equalty n fact provdes for each ln {, l} Ɣ M some equalty relatng values of φ P h (v, Ɣ P), h M, wth values of dstnct φ P h (v, Ɣ P l ), h M. Snce

18 580 J Comb Optm (2018) 35: Ɣ M l = Ɣ M 1, from the nducton hypothess t follows that for all lns {, l} Ɣ M ((, l) Ɣ M ) the payoffs φh P(v, Ɣ P l ), h M, are already determned. Thus wth respect to q 1 lns {, l} Ɣ, (18) yelds q 1 lnearly ndependent lnear equatons n the q unnown payoffs φ P(v, Ɣ P), Q. Moreover, by QCE t holds that φ P (v, Ɣ P) = v P (Q). Q All these q equatons are lnearly ndependent. Whence t follows that for every Q M/Ɣ, all payoffs φ P(v, Ɣ P), Q, are unquely determned. otce that n the proof of the nducton step, every possble spannng tree Ɣ yelds the same soluton for the values φ P(v, Ɣ P), Q, because otherwse a soluton does not exst, whch contradcts the already proved exstence part of the proof of the theorem. ext, we show that the ndvdual payoffs φ (v, Ɣ P ),, are unquely determned. Ths part of the proof s also by nducton, now on the number of lns n Ɣ. Intalzaton: Assume that Ɣ =0. Let v 0 be the 0-normalzaton of the TU game v,.e., v 0 (S) = v(s) S v({}) for all S, and let (v 0) = v 0. As already shown above, the unon payoffs φ P(w, Ɣ P), M, are unquely determned for any PΓ -game w, Ɣ P G DLP,{DL } M. In partcular, the unon payoffs φ P(v 0,Ɣ P ), M, are unquely determned. By defnton (v 0 ) ({}) = 0 for every, and therefore, (v 0 ) Ɣ 0 snce Ɣ =0. Then, from URGP t follows that Whence by COV we obtan φ (v 0,Ɣ P ) = φp (v 0,Ɣ P ) n, for all. φ (v, Ɣ P ) = v({}) + φp (v0,ɣ P ) n, for all,.e., for every, φ (v, Ɣ P ) s unquely determned. Inducton hypothess: Let Ɣ P denote the two-level graph structure Ɣ M, {Ɣ h } h M wth Ɣ h = Ɣ h f h = and Ɣ = Ɣ for some graph Ɣ on. Assume that the values φ (v, Ɣ P ) have been determned for every Ɣ wth Ɣ < Ɣ. Inducton step: For every S Int(, P, {Ɣ } M ) let C S /Ɣ be the unque component such that S C S. Consder a game w G defned as { v(s), S / Int(, P, {Ɣ } M ), w(s) = s v(c S ) c S, S Int(, P, {Ɣ } M ), for all S. (19) For the 0-normalzaton w 0 of w, (w 0 ) = (w ) 0. The subgame w s an addtve game and, therefore, (w ) 0 0. Then, due to URGP, smlar as n the Intalzaton step, t follows that

19 J Comb Optm (2018) 35: Whence by COV we obtan φ (w 0,Ɣ P ) = φp (w 0,Ɣ P ) n, for all. φ (w, Ɣ P ) = w({}) + φp (w 0,Ɣ P ) n, for all. Consder a component C /Ɣ. From the above equalty t follows that C φ (w, Ɣ P ) = C w({}) + c n φ P (w 0,Ɣ P ) (19) = v(c) + c n φ P (w 0,Ɣ P ). By the defnton (19) ofw, w(s) = v(s) for all S, S / Int(, P, {Ɣ } M ). Whence by UCPIIC t follows that C φ (v, Ɣ P ) = v(c) + c n DL P (w 0,P,Ɣ M ), for all C /Ɣ. (20) ext, f c = 1, then C s a sngleton and the payoff φ (v, Ɣ P ) of the only player C s unquely determned by (20). If c 2, then there exsts a spannng tree Ɣ Ɣ C on C wth the number of lns Ɣ =c 1. By UDL t holds that DL ( φ (v, Ɣ P ), Ɣ ) = 0. (21) The above equalty n fact provdes for any ln {, j} Ɣ some equalty relatng values of φ h (v, Ɣ P ), h, wth values of dstnct φ h (v, Ɣ P j ). Snce Ɣ j = Ɣ 1, by the nducton hypothess t follows that for all lns {, j} Ɣ the payoffs φ h (v, Ɣ P j ), h, are already determned. Thus wth respect to c 1 lns {, j} Ɣ, (21) yelds c 1 lnearly ndependent lnear equatons n the c unnown payoffs φ (v, Ɣ P ), C. These c 1 equatons together wth (20) form asystemofc lnearly ndependent equatons n the c unnown payoffs φ (v, Ɣ P ), C. Hence, for every C /Ɣ, all payoffs φ (v, Ɣ P ), C, are unquely determned. Logcal ndependence of the axoms Gven a (m + 1)-tuple of deleton ln axoms DL P, {DL } M, such that the set of DL, M, axoms s restrcted to F, CF, and RF, the logcal ndependence of the axoms n Theorem 1 s demonstrated by the followng examples of PΓ -values: The PΓ -value ξ (1) assgnng n every v, Ɣ P G DLP,{DL } M a payoff to every player ξ (1) (v, Ɣ P ) = DLP () (v P,Ɣ M ) n ()

20 582 J Comb Optm (2018) 35: satsfes all axoms except COV. The PΓ -value ξ (2) assgnng n every v, Ɣ P G DLP,{DL } M to every player a payoff ξ (2) (v, Ɣ P ) = v({}) + DLP () (v P,Ɣ M ) n () j () v({ j}) satsfes every axom except UCPIIC. The PΓ -value ξ (3) assgnng n every v, Ɣ P G DLP,{DL } M a payoff to every player ξ (3) (v, Ɣ P ) = DL () (v (),Ɣ () ) + Sh ()(v (M/ƔM ) () ) v Ɣ ()( ) n () satsfes every axom except QDL. The PΓ -value ξ (4) assgnng n every v, Ɣ P G DLP,{DL } M a payoff to every player ξ (4) (v, Ɣ P ) = EL () (v (),Ɣ () ) + DLP () (v P,Ɣ M ) v Ɣ ()( ) n () satsfes every axom except UDL. The PΓ -value ξ (5) assgnng n every v, Ɣ P G DLP,{DL } M a payoff to every player ξ (5) (v, Ɣ P ) = DL () (v (),Ɣ () ) + v( ()) v Ɣ ()( () ) n () satsfes every axom except QCE. The PΓ -value ξ (6) assgnng n every v, Ɣ P G DLP,{DL } M to every player a payoff ξ (6) (v, Ɣ P ) = Ka DLP,{DL } M (v, Ɣ P ) + a 1 n () j () a j, where (a j ) j IR n s any vector of real numbers such that not all coordnates are equal, satsfes every axom except URGP. 5 Owen-type values for two-level graph games In ths secton we consder another famly of PΓ -values based on the adaptaton of the Owen s two-step dstrbuton procedure for P-games. We ntroduce these values axomatcally by means of four axoms. The frst three axoms are QCE, QDL, and

21 J Comb Optm (2018) 35: UDL already employed n the prevous secton for characterzaton of the Kamjotype DL P, {DL } M -values. The fourth axom of far dstrbuton of the surplus wthn unons was frst ntroduced n van den Brn et al. (2016) where t was used for the axomatzaton of the Owen-type value for PΓ -games whch s based on the applcaton of the Myerson value on both communcaton levels. Ths axom requres balanced average payoff varaton for all components wthn a pror unon n case the other components leave the game. Far dstrbuton of the surplus wthn unons (FDSU) For any I, any v, Ɣ P G PΓ, Ɣ P = Ɣ M, {Ɣ } M, M,anyC, C /Ɣ, 1 c C (ξ (v, Ɣ P ) ξ ( v C,Ɣ P C )) = 1 c C ( )) (ξ (v, Ɣ P ) ξ vc,ɣ P, (22) C where for M and component C /Ɣ, v C denotes the subgame v (\ ) C of v wth respect to the coalton ( \ ) C, P C denotes the partton on ( \ ) C consstng of unon C and all unons h n P, h =, and Ɣ P C = Ɣ M, { Ɣ h } h M wth Ɣ = Ɣ C and Ɣ h = Ɣ h for all h M \{}, denotes the two-level communcaton structure that s obtaned from Ɣ M, {Ɣ h } h M by replacng the communcaton graph Ɣ by ts restrcton on C. The next theorem extends the Owen-type value for PΓ -games studed n van den Brn et al. (2016) by allowng the applcaton of dfferent combnatons of nown component effcent soluton concepts for Γ -games on both communcaton levels. Theorem 2 For any (m + 1)-tuple of deleton ln axoms DL P, {DL } M such that the set of DL, M, axoms s restrcted to F, CF, and RF, there s a unque PΓ -value defned on G DLP,{DL } M that meets axoms QCE, FDSU, QDL, and UDL. For every PΓ -game v, Ɣ P G DLP,{DL } M t s gven by ξ (v, Ɣ P ) = DL () (ṽ (),Ɣ () )+ where for all M, ṽ G s defned as DL P () (v P,Ɣ M ) ṽ () (C) C () /Ɣ () ṽ (S) = DL P ( ˆv S,Ɣ M ), for all S, and for every S, ˆv S G M s gven by n (),, (23) ˆv S (Q) = { v( h Q h ), / Q, v( h Q\{} h S), Q, for all Q M. The proof strategy s smlar to that appled n van den Brn et al. (2016), a careful reader may fnd the proof n Appendx. We sp the proof of logcal ndependence of axoms snce t can be easly obtaned by modfcaton of the examples used for the

22 584 J Comb Optm (2018) 35: proof n van den Brn et al. (2016) n case of DL P, {DL } M = F P, {F } M,.e., when the Myerson farness F s appled on both levels. ote that n case when DL P, {DL } M = F P, {F } M, the statement of Theorem 2 concdes wth the statement of both Proposton 4.1 and Theorem 4.2 proved n van den Brn et al. (2016) together and the PΓ -value ξ gven by (23) concdes as well wth the Owen-type PΓ -value ntroduced there. Because of the latter concdence, from now on we refer to the PΓ -value (23) as to the Owen-type DL P, {DL } M -value denoted further by Ow DLP,{DL } M. Observe also that the Owen-type DL P, {DL } M -value admts the smlar twostep constructon procedure as the Owen value for games wth coalton structure. The dfference s that nstead of two applcatons of the Shapley value used n the case of the classcal Owen value, n the Owen-type DL P, {DL } M -value dfferent nown component effcent Γ -values can be appled on both communcaton levels. Remar 3 ote that dfferent from Theorem 1, n Theorem 2 RF can be used at any level due to the fact that COV employed n Theorem 1 s replaced by FDSU n Theorem 2, and the latter axom s compatble wth the compensaton soluton characterzed by RF. By the reasons smlar to those mentoned n Remar 2, the PΓ -value Ow DLP,{DL } M volates the UDL property when there are axoms BLC, UE, LE, EL, SE, and PE among the axoms DL, M n the (m+1)-tuple DL P, {DL } M. 6 Concluson We conclude by a fnal comparson of the two famles of values ntroduced by Theorems 1 and 2. Observe that all consdered PΓ -values meet QCE, QDL, UDL, and COV (the last one only f RF s not nvolved). However, whle the Owentype DL P, {DL } M -values satsfy FDSU, but volate URGP and UCPIIC, the Kamjo-type DL P, {DL } M -values vce versa satsfy URGP and UCPIIC, but volate FDSU. The summary of the propertes s gven n the followng table, where the axoms needed for our axomatzatons are mared by. Value QCE QDL UDL COV FDSU URGP UCPIIC Owen-type DL P, {DL } M -value Kamjo-type DL P, {DL } M -value Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4.0 Internatonal Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s) and the source, provde a ln to the Creatve Commons lcense, and ndcate f changes were made.

23 J Comb Optm (2018) 35: Appendx Proof (Theorem 2)I.[Exstence]. We show that under the hypothess of the theorem the PΓ -value ξ = Ow DLP,{DL } M defned on G DLP,{DL } M by (23) meets the axoms QCE, QDL, FDSU, and UDL. Consder an arbtrary PΓ -game v, Ɣ P G DLP,{DL } M. QCE, QDL, UDL The proof of the fact that the Owen-type DL P, {DL } M -value meets these axoms s smlar to the proof of same statement for the Kamjo-type DL P, {DL } M -value n Theorem 1, and so we sp t. FDSU From (23) we obtan that for every C /Ɣ t holds that ξ (v, Ɣ P ) (23) = DL (ṽ,ɣ ) + c DL P n (v P,Ɣ M ) ṽ (H) C C H /Ɣ =ṽ (C) + c DL P n (v P,Ɣ M ) ṽ (H), H /Ɣ where the second equalty s due to component effcency of DL -value for Γ -games. Further, C ξ (vc,ɣ PC )(23) = DL ( (v C ),Ɣ C ) C + c DL P n ((v C ) P,Ɣ M) (v C C ) (H) H C/Ɣ C = (vc ) (C) + c (DL P n ((vc ) P C,Ɣ M) (v ) C ) (C) =ṽ (S), where the second equalty s due to component effcency of DL -value for Γ - games and C beng the only component n Ɣ C, and the thrd equalty follows from equalty (vc ) (C) = ṽ (C) whch holds true because ˆv C = (vc ) C, and equaltes DL P ((v C ) PC,Ɣ M) = DL P ( ˆv C,Ɣ M ) def = ṽ (C) frst of whch holds true because (v C ) P C = (v C ) C =ˆv C. Thus, 1 c ) (ξ (v, Ɣ P ) ξ (vc,ɣ P C ) = 1 DL P n (v P,Ɣ M ) ṽ (H), C H /Ɣ where the rght sde of the latter equalty s ndependent of C.

24 586 J Comb Optm (2018) 35: II. [Unqueness]. Assume that a (m + 1)-tuple of deleton ln axoms DL P, {DL } M such that the set of axoms DL, M, s restrcted to F, CF, and RF, s gven. We show that there exsts at most one PΓ -value on G DLP,{DL } M that satsfes axoms QCE, FDSU, QDL, and UDL. Let φ be such PΓ -value on G DLP,{DL } M. Gven an arbtrary PΓ -game v, Ɣ P G DLP,{DL } M we show that the ndvdual payoffs φ (v, Ɣ P ),, are unquely determned. Smlar to the proof of Theorem 1 we may show that the unon payoffs φ P(v, Ɣ P), M, are unquely determned. Applyng the same strategy, for every M and any subset C we determne the unon payoffs φ P(v C,Ɣ PC ) n the game (v C,Ɣ P ), n whch v C C denotes the subgame v (\ ) C of v wth respect to the coalton ( \ ) C, and Ɣ P denotes the two-level communcaton structure C Ɣ M, {Ɣ h } h M, where Ɣ M s the communcaton graph on the partton (P\{ }) {C} and the communcaton graph Ɣ s replaced by ts restrcton on C. otce that now for M, the unon payoff φ P(v C,Ɣ PC ) s the total payoff to the players n C n the game (vc,ɣ PC ). ext we determne the ndvdual payoffs n each a pror unon, M. For ths frst we show that for each component C /Ɣ the total payoff to the players n C s unquely determned. The payoff φ P (v, Ɣ P) to the a pror unon has been already determned, so φ (v, Ɣ P ) = φ P (v, Ɣ P). (24) If s the unque component n /Ɣ, then FDSU does not state any requrement. When /Ɣ conssts of multple components, then for every component C /Ɣ, from FDSU t follows that ( ) C φ (v, Ɣ P ) φ P vc,ɣ φ PC P(v, Ɣ P) ( ) φ P v K,Ɣ PK K /Ɣ =. c n (25) otce that every payoff φ P n ths equaton has already been determned, and therefore, C φ (v, Ɣ P ) s unquely determned. The rest of the proof we proceed by nducton smlar to the proof of Theorem 1. Tae some M. LetƔ P denote the two-level graph structure Ɣ M, {Ɣ h } h M wth Ɣ h = Ɣ h f h = and Ɣ = Ɣ for some graph Ɣ on. Intalzaton: If Ɣ =0then {} /Ɣ for all. FDSU mples that φ (v, Ɣ P ) φ P φ ( ) P (v, Ɣ P ) φ P v{},ɣ j P{} = ( ) v{ j},ɣ P{ j}, for all. n (26) We have already determned φ P (v, Ɣ P) and φ P (v { j},ɣ P { j} ), for all j. So, Eq. (26) determnes φ (v, Ɣ P ) for all.