1 Introduction

In classical cooperative game theory it is assumed that any coalition of players may form, and a fair distribution of total rewards amongst the players takes into account capacities of all coalitions. For example, the most prominent solution of cooperative games, the Shapley value (Shapley 1953), assigns to each player as a payoff the average of the player’s marginal contributions to his predecessors with respect to all possible linear orderings of the players. However, in many practical situations the collection of feasible coalitions is restricted by some social, economical, hierarchical, communicational, or technical structure. The study of transferable utility (TU) games with limited cooperation introduced by means of coalition structures, or in other terms a priori unions, was initiated in the 1970’s first by Aumann and Drèze (1974) and then by Owen (1977). In these papers a coalition structure is given by a partition of the set of players, and both papers introduce as solutions some adaptations of the classical Shapley value to this situation with restricted cooperation. While Aumann and Drèze assume that cooperation is possible only within a priori unions and as a solution they propose the combination of Shapley values in subgames on a priori unions, Owen admits some cooperation between players of different unions. Similar to the Shapley value, the Owen value assigns to each player as a payoff the average of the player’s marginal contributions to his predecessors with respect to linear orderings of the players. But in case of Owen not all possible linear orderings are taken into account, but only those in which the players of the same a priori union appear successively. Another model of a game with limited cooperation presented by means of undirected communication graphs was introduced in Myerson (1977). The main assumption of Myerson is that only connected players are able to cooperate, and the Myerson value is given by the Shapley value in the Myerson restricted game, in which the worth of each disconnected coalition is replaced by the sum of the worths of its connected components. Various studies in both directions were done during the last four decades, but mostly either within one model or another. Vázquez-Brage et al. (1996) is the first study that combines both models by considering a TU game endowed with, independent of each other, both a coalition structure and a communication graph on the set of players. For this class of games they propose a solution by applying the Owen value for games with coalition structure to the Myerson restricted game of the game with communication graph.

Another model of a TU game endowed with both a coalition structure and a communication graph, the so-called game with two-level communication structure, is considered in Khmelnitskaya (2014). In contrast to Vázquez-Brage et al. (1996), in this model a two-level communication structure relates fundamentally to the given coalition structure and consists of a communication graph on the collection of the a priori unions in the coalition structure, as well as a collection of communication graphs within each union. It is assumed that communication is only possible either among the entire a priori unions or among single players within any a priori union. No communication and therefore no cooperation is allowed between proper subcoalitions, in particular single players, of distinct elements of the coalition structure.Footnote 1 This approach allows to model different network situations, in particular, telecommunication problems, distribution of goods among different cities (countries) along highway networks connecting the cities and local road networks within the cities, or sharing an international river with multiple users but without international firms, i.e., when no cooperation is possible among single users located at different levels along the river, and so on. Communication structures under scrutiny are given by combinations of graphs of different types both undirected-arbitrary graphs and cycle-free graphs, and directed-line-graphs with linearly ordered players, rooted forests and sink forests. The proposed solution concepts reflect a two-stage distribution procedure when, first, a priori unions collect their shares through the upper level bargaining based only on the cumulative interests of all members of every involved entire a priori union, and second, the players collect their individual payoffs through the lower level bargaining over the distribution of the unions’ shares within the unions. Following Myerson (1977) it is assumed that cooperation is possible only between connected participants and different combinations of known component efficient values, such as the Myerson value, the position value, the average tree solution, etc., are applied on both communication levels. However, as discussed in Khmelnitskaya (2014), the two-stage distribution procedure based on the application of component efficient values on both levels suffers from severe restrictions in cases when some a priori unions are internally not connected, because each union always has to distribute its total share among the members. Another solution concept for TU games with two-level communication structure introduced by means of undirected graphs, the so-called Owen-type value for games with two-level communication structure, is considered in van den Brink et al. (2016) under a weaker assumption concerning the communication on the level of a priori unions, when on the upper level bargaining between a priori unions, similar as for the Owen value introduced in Owen (1977), one of the a priori unions can be presented by any of its proper subcoalitions. This solution can be seen as an adaptation of the two-step procedure determining the Owen value for games with coalition structure which takes into account the limited cooperation represented by two-level communication structure replacing twice the Shapley value by the Myerson value.Footnote 2

In this paper we assume that a two-level communication structure is given by combinations of graphs of different types, both undirected and directed, and we introduce two families of two-step values for games with two-level communication structure adapting the two-step procedures staying behind two values for games with coalition structure, the Owen value and the two-step Shapley value introduced in Kamijo (2009). Our approach is based on the unified treatment of several component efficient values for games with communication structure and it generates two-stage solution concepts that apply component efficient values for games with communication structure on both distribution levels. In fact the newly introduced family of the Owen-type values is the generalization of the Owen-type value for games with two-level communication structures of van den Brink et al. (2016), when on both communication levels not only the Myerson value, but different component efficient values for games with communication structure can be applied. The incorporation of different solutions for games with communication structure aims not only to enrich the solution concepts for games with two-level communication structure, but it also opens a broad diversity of applications impossible otherwise, because there exists no universal solution concept for games with communication structure that is applicable to the full variety of possible undirected and directed graph structures. Moreover, it allows to choose, depending on types of graph structures under scrutiny, the most preferable, in particular, the most computationally efficient combination of values among others suitable. We provide axiomatic characterizations of the introduced two-step values. These axiomatizations have several common axioms for the both families which allows to compare the two-step values from different families.

The introduced two families of two-step solution concepts may find application in different resource allocation problems with two hierarchical distribution levels. For instance, they may be used for budget allocation within a university, when the budget has to be distributed first among its departments and then among the individuals within each department. An example showing advantages of using the Owen-type value for games with two-level communication structures, in case when on both communication levels only the Myerson value is applied, is discussed in van den Brink et al. (2016).

The rest of the paper is organized as follows. Basic definitions and notation are introduced in Sect. 2. Section 3 provides the uniform approach to several known component efficient values for games with communication structure, which allows also to consider within a unified framework different deletion link properties with respect to the values for games with two-level communication structure. In Sects. 4 and 5 we introduce correspondingly the families of the Kamijo-type and Owen-type values axiomatically and present their explicit formula representations. Section 6 concludes.

2 Preliminaries

2.1 TU games and values

A cooperative game with transferable utility, or TU game, is a pair \(\langle N,v\rangle \), where \(N\subset \mathrm{I\!N}\) is a finite set of n players and \(v:2^N\rightarrow \mathrm{I\!R}\) is a characteristic function with \(v(\emptyset )=0,\) assigning to every coalition \(S\subseteq N\) of s players its worth v(S). The set of TU games with fixed player set N is denoted by \(\mathcal{G}_N\). For simplicity of notation and if no ambiguity appears we write v when we refer to a TU game \(\langle N, v\rangle \). The subgame of a TU game \(v\in \mathcal{G}_N\) with respect to a player set \(T\subseteq N\) is the TU game \(v|_T\in \mathcal{G}_T\) defined by \(v|_T(S)=v(S)\) for every \({S \subseteq T}\). A payoff vector is a vector \(x \in {\mathrm{I\!R}}^N\) with \(x_i\) the payoff to player \(i\in N\). A single-valued solution, called a value, is a mapping \(\xi :\mathcal{G}_N\rightarrow {\mathrm{I\!R}}^N\) that assigns to every finite set \(N\subset \mathrm{I\!N}\) and every TU game \(v\in \mathcal{G}_N\) a payoff vector \(\xi (v)\in {\mathrm{I\!R}}^N\). A value \(\xi \) is efficient if \(\sum _{i\in N}\xi _i(v)=v(N)\) for every \(v\in \mathcal{G}_N\) and \(N\subset \mathrm{I\!N}\). The best-known efficient value is the Shapley value (Shapley 1953) given by

$$\begin{aligned} Sh_i(v)=\sum _{S\subseteq N\backslash \{i\}} \frac{s!(n-s-1)!}{n!}(v(S\cup \{i\})-v(S)), \quad \text{ for } \text{ all }i\in N.\end{aligned}$$

In the sequel we denote the cardinality of a given set A by |A|, along with lower case letters like \(n=|N|\), \(m=|M|\), \(n_k=|N_k|\), \(s=|S|\), \(c=|C|\), \(c'=|C'|\), and so on, and we use the standard notation \(x(S)\!=\!\sum _{i\in S}x_i\) for any \(x\!\in \!{\mathrm{I\!R}}^N\) and \(S\subseteq N\).

2.2 Games with coalition structure

A coalition structure, or in other terms a system of a priori unions, on \(N\subset \mathrm{I\!N}\) is given by a partition \(\mathcal{P}=\{N_1,\ldots ,N_m\}\) of N, i.e., \(N_1\cup \ldots \cup N_m=N\) and \(N_k\cap N_l=\emptyset \) for \(k\ne l\). Let \({\mathfrak {P}}_N\) denote the set of all coalition structures on N, and let \(\mathcal{G}_N^P=\mathcal{G}_N\times {\mathfrak {P}}_N\). A pair \(\langle v,\mathcal{P}\rangle \in \mathcal{G}_N^P\) constitutes a game with coalition structure, or simply P -game, on N. Remark that \(\langle v,\{N\}\rangle \) represents the same situation as v itself. A P -value is a mapping \(\xi :\mathcal{G}_N^P\rightarrow {\mathrm{I\!R}}^N\) that assigns to every \(N\subset \mathrm{I\!N}\) and every P-game \(\langle v,\mathcal{P}\rangle \in \mathcal{G}_N^P\) a payoff vector \(\xi (v,\mathcal{P})\in {\mathrm{I\!R}}^N\). A P-value \(\xi \) is efficient if \(\sum _{i\in N}\xi _i(v,\mathcal{P})=v(N)\) for every \(v\in \mathcal{G}_N^P\) and \(N\subset \mathrm{I\!N}\). In what follows, denote by \(M=\{1,\ldots ,m\}\) the index set of all a priori unions in \(\mathcal P\); for every P-game \(\langle v,\mathcal{P}\rangle \in \mathcal{G}_N^P\) and every \(k\in M\) let \(v_k\) denote the subgame \(v|_{N_k}\); for every \(i\!\in \!N\), let k(i) be defined by the relation \(i\!\in \!N_{k(i)}\); and for every \(x\!\in \!{\mathrm{I\!R}}^N\), let \(x^\mathcal{P}\!=\!\bigl (x(N_k)\bigr )_{k\in M}\!\in \!{\mathrm{I\!R}}^M\) stand for the vector of total payoffs to a priori unions.

One of the best-known values for games with coalition structure is the Owen value (Owen 1977) that can be seen as a two-step procedure in which the Shapley value applies twice. Namely, the Owen value assigns to player \(i\in N\) his Shapley value in the game \(\bar{v}_{k(i)}\), i.e.,

$$\begin{aligned} Ow_i(v,\mathcal{P})=Sh_i(\bar{v}_{k(i)}),\quad \text{ for } \text{ all }i\in N, \end{aligned}$$

while for every a priori union \(N_k\), \(k\in M\), the game \(\bar{v}_k\in \mathcal{G}_{N_k}\) on the player set \(N_k\) is given by

$$\begin{aligned} \bar{v}_k(S)= Sh_k(\hat{v}_S),\quad S\subseteq N_k, \end{aligned}$$

where for every \(S\subseteq N_k\) the game \(\hat{v}_S\in \mathcal{G}_M\) on the player set M of a priori unions is defined by

$$\begin{aligned}\hat{v}_S(Q)=\left\{ \begin{array}{ll} {v(\cup _{h\in Q}N_h)},&{}\quad k\notin Q,\\ {v(\cup _{h\in Q\backslash \{k\}}N_h\cup S)},&{}\quad k\in Q,\\ \end{array} \right. \quad \text{ for } \text{ all } Q\subseteq M. \end{aligned}$$

The Owen value is efficient, i.e., \(\sum _{i\in N}Ow_i(v,\mathcal{P})=v(N)\), and satisfies the quotient game property, i.e., for every a priori union the total payoff to the players within that union is determined by applying the Shapley value to the so-called quotient game being the game \(v_\mathcal{P} \in \mathcal{G}_M\) in which the unions act as individual players,

$$\begin{aligned} v_{\mathcal{P}}(Q)=v(\cup _{k\in Q}N_k),\quad \text{ for } \text{ all } Q\subseteq M. \end{aligned}$$

Notice that for every \(k \in M\) the game \(\hat{v}_{N_k}\) is equal to the quotient game \(v_{\mathcal{P}}\).

Another value for games with coalition structure that also can be seen as a two-step procedure in which the Shapley value applies twice is the so-called two-step Shapley value \(\psi \) introduced in Kamijo (2009). The two-step Shapley value first allocates to player \(i\in N\) his Shapley value in the subgame on the a priori union \(N_{k(i)}\) he belongs to and then distributes what remains of the Shapley value of its union in the quotient game equally among the union’s members, i.e., for any P-game \(\langle v,\mathcal{P}\rangle \in \mathcal{G}_N^P\),

$$\begin{aligned} Ka_i(v,\mathcal{P})=Sh_i(v_{k(i)})+ \frac{Sh_{k(i)}(v_\mathcal{P})-v(N_{k(i)})}{n_{k(i)}}. \end{aligned}$$
(1)

The Kamijo’s two-step Shapley value is efficient and meets the quotient game property.

2.3 Games with communication structure

A communication structure on N is specified by a graph \(\Gamma \), undirected or directed, on N. A graph on N consists of N as the set of nodes and for an undirected graph a collection of unordered pairs \(\Gamma \subseteq \{\{i,j\}\,|\,i,j\in N,\ i\ne j\}\) as the set of links between two nodes in N, and for a directed graph, or a digraph, a collection of ordered pairs \(\Gamma \subseteq \{(i,j)\,|\,i,j\in N,\ i\ne j\}\) as the set of directed links from one node to another node in N. When it is necessary to specify the set of nodes N in a graph \(\Gamma \), we write \(\Gamma _N\) instead of \(\Gamma \). Let \({\mathfrak {G}}_N\) denote the set of all communication structures, undirected or directed, on N, and let \(\mathcal{G}_N^{\varGamma }=\mathcal{G}_N\times {\mathfrak {G}}_N\). A pair \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\) constitutes a game with graph (communication) structure, or simply a graph game, or a \(\varGamma \)-game, on N. A \(\varGamma \)-value is a mapping \(\xi :\mathcal{G}_N^{\varGamma }\rightarrow {\mathrm{I\!R}}^N\) that assigns to every \(N\subset \mathrm{I\!N}\) and every \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\) a payoff vector \(\xi (v,\Gamma )\in {\mathrm{I\!R}}^N\).

In a graph \(\Gamma \) a sequence of different nodes \((i_1,\ldots ,i_r)\), \(r\ge 2\), is a path in \(\Gamma \) from node \(i_1\) to node \(i_r\) if for \(h\!=\!1,\ldots ,r\!-\!1\) it holds that \(\{i_h,i_{h+1}\}\!\in \!\Gamma \) when \(\Gamma \) is undirected and \(\{(i_h,i_{h+1}),(i_{h+1},i_h)\}\cap \Gamma \ne \emptyset \) when \(\Gamma \) is directed. In a digraph \(\Gamma \) a path \((i_1,\ldots ,i_r)\) is a directed path from node \(i_1\) to node \(i_r\) if \((i_h,i_{h+1})\in \Gamma \) for all \(h\!=\!1,\ldots ,r\!-\!1\). In a digraph \(\Gamma \), \(j\ne i\) is a successor of i and i is a predecessor of j if there exists a directed path from i to j. Given a digraph \(\Gamma \) on N and \(i\in N\), the sets of predecessors and successors of i in \(\Gamma \) we denote correspondingly by \(P^\Gamma (i)\) and \(S^\Gamma (i)\); moreover, \(\bar{P}^\Gamma (i)=P^\Gamma (i)\cup \{i\}\) and \(\bar{S}^\Gamma (i)=S^\Gamma (i)\cup \{i\}\).

Given a graph \(\Gamma \) on N, two nodes i and j in N are connected  if either there exists a path from node i to node j, or i and j coincide. Graph \(\Gamma \) on N is connected  if any two nodes in N are connected. For a graph \(\Gamma \) on N and a coalition \(S\subseteq N\), the subgraph of \(\Gamma \) on S is the graph \(\Gamma |_S=\{\{i,j\}\!\in \!\Gamma \mid i,j\!\in \!S\}\) on S when \(\Gamma \) is undirected and the digraph \(\Gamma |_S=\{(i,j)\!\in \!\Gamma \mid i,j\!\in \!S\}\) on S when \(\Gamma \) is directed. Given a graph \(\Gamma \) on N, a coalition \(S\subseteq N\) is connected  if the subgraph \(\Gamma |_S\) is connected. For a graph \(\Gamma \) on N and coalition \(S\subseteq N\), \(C^\Gamma (S)\) is the set of all connected subcoalitions of S, \(S/\Gamma \) is the set of maximal connected subcoalitions of S, called the components of S, and \((S/\Gamma )_i\) is the component of S containing player \(i\in S\). Notice that \(S/\Gamma \) is a partition of S. For any \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\), a payoff vector \(x\in {\mathrm{I\!R}}^N\) is component efficient if \(x(C)=v(C)\), for every \(C\in N/\Gamma \).

Following Myerson (1977), we assume that for \(\varGamma \)-games cooperation is possible only among connected players and concentrate on component efficient \(\varGamma \)-values. A \(\varGamma \)-value \(\xi \) is component efficient (CE) if for any \(\langle v,\Gamma \rangle \in \mathcal{G}_N^\Gamma \), for all \(C\!\in \!N/\Gamma \), \(\sum _{i\in C}\xi _i (v,\Gamma )=v(C)\). Below for a \(\varGamma \)-game \(\langle v,\Gamma \rangle \!\in \!\mathcal{G}_N^{\varGamma }\) we also consider the introduced in Myerson (1977) restricted game \(v^\Gamma \!\in \!\mathcal{G}_N\) defined as

$$\begin{aligned} v^\Gamma (S)=\sum _{C\in S/\Gamma }v(C),\quad \text{ for } \text{ all }S\subseteq N. \end{aligned}$$
(2)

Hereinafter along with communication structures given by arbitrary undirected graphs we consider also those given by cycle-free undirected graphs and by directed graphs—linear graphs with linearly ordered players, rooted and sink forests. In an undirected graph \(\Gamma \) a path \((i_1,\ldots ,i_r)\), \(r\ge 3\), is a cycle  in \(\Gamma \) if \(\{i_r,i_1\}\!\in \!\Gamma \). An undirected graph is cycle-free if it contains no cycles. A directed graph \(\Gamma \) is a rooted tree if there is one node in N, called a root, having no predecessors in \(\Gamma \) and there is a unique directed path in \(\Gamma \) from this node to any other node in N. A directed graph \(\Gamma \) is a sink tree if the directed graph composed by the same set of links as \(\Gamma \) but with the opposite orientation is a rooted tree; in this case the root of a tree changes its meaning to the absorbing sink. A directed graph is a rooted/sink forest if it is composed by a number of disjoint rooted/sink trees. A linear graph is a directed graph that contains links only between subsequent nodes. Without loss of generality we may assume that in a linear graph nodes are ordered according to the natural order from 1 to n, i.e., linear graph \(\Gamma \subseteq \{(i,i+1)\,|\ i=1,\ldots ,n-1\}\).

For ease of notation given graph \(\Gamma \) and link \(\{i,j\}\in \Gamma \) if \(\Gamma \) is undirected, or \((i,j)\in \Gamma \) if \(\Gamma \) is directed, the subgraph \(\Gamma \backslash \{\{i,j\}\}\), correspondingly \(\Gamma \backslash \{(i,j)\}\), is denoted by \(\Gamma |_{-ij}\).

2.4 Games with two-level communication structure

We now consider situations in which the players are partitioned into a coalition structure \(\mathcal{P}\) and are linked to each other by communication graphs. First, there is a communication graph \(\Gamma _M\) on the set of a priori unions determined by the partition \(\mathcal P\). Second, for each a priori union \(N_k\), \(k \in M\), there is a communication graph \(\Gamma _{N_k}\) between the players in \(N_k\). In what follows for simplicity of notation and when it causes no ambiguity we denote graphs \(\Gamma _{N_k}\) within a priori unions \(N_k\), \(k\in M\), by \(\Gamma _k\). Given a player set \(N\subset \mathrm{I\!N}\) and a coalition structure \(\mathcal{P}\in {\mathfrak {P}}_N\), a two-level graph (communication) structure on N is a tuple \(\Gamma _\mathcal{P}\!=\!\langle \Gamma _M,\{\Gamma _k\}_{k\in M}\rangle \). For every \(N\subset \mathrm{I\!N}\) and \(\mathcal{P}\in {\mathfrak {P}}_N\) by \({\mathfrak {G}}_N^\mathcal{P}\) we denote the set of all two-level graph structures on N with fixed \(\mathcal P\). Let \({\mathfrak {G}}_N^P=\bigcup _{\mathcal{P}\in {\mathfrak {P}}_N}{\mathfrak {G}}_N^\mathcal{P}\) be the set of all two-level graph structures on N, and let \(\mathcal{G}_N^{P\varGamma }=\mathcal{G}_N\times {\mathfrak {G}}_N^P\). A pair \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{P\varGamma }\) constitutes a game with two-level graph (communication) structure, or simply a two-level graph game or a \(P\varGamma \)-game, on N. A \(P\varGamma \)-value is a mapping \(\xi :\mathcal{G}_N^{P\varGamma }\rightarrow {\mathrm{I\!R}}^N\) that assigns to every \(N\subset \mathrm{I\!N}\) and every \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{P\varGamma }\) a payoff vector \(\xi (v,\Gamma _\mathcal{P})\in {\mathrm{I\!R}}^N\).

Given a \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{P\varGamma }\), one can consider the quotient \(\varGamma \)-game \(\langle v_\mathcal{P},\Gamma _M\rangle \in \mathcal{G}_M^{\varGamma }\) and the \(\varGamma \)-games within a priori unions \(\langle v_k,\Gamma _k\rangle \in \mathcal{G}_{N_k}^{\varGamma }\) with \(v_k=v|_{N_k}\), \(k\in M\), that model the bargaining between a priori unions for their total shares and the bargaining within each a priori union for the distribution of its total worth among the members taking also into account limited cooperation at both communication levels introduced by the communication graphs \(\Gamma _M\) and \(\Gamma _k\), \(k\in M\). Moreover, given a \(\varGamma \)-value \(\phi \), for any \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{P\varGamma }\) with a graph structure \(\Gamma _M\) on the level of a priori unions suitable for application of \(\phi \) to the corresponding quotient \(\varGamma \)-game \(\langle v_\mathcal{P},\Gamma _M\rangle \), along with a subgame \(v_k\) within a priori union \(N_k\), \(k\in M\), one can also consider a \(\phi _k\)-game \(v_k^{\phi }\) defined as

$$\begin{aligned}v_k^{\phi }(S)=\left\{ \begin{array}{ll} \phi _k(v_\mathcal{P},\Gamma _M),&{}\quad S=N_k,\\ v(S),&{}\quad S\ne N_k,\\ \end{array} \right. \quad \text{ for } \text{ all } S\subseteq N_k, \end{aligned}$$

where \(\phi _k(v_\mathcal{P},\Gamma _M\!)\) is the payoff to \(N_k\) given by \(\phi \) in \(\langle v_\mathcal{P},\Gamma _M\!\rangle \). Then a \(\varGamma \)-game \(\langle v_k^{\phi },\Gamma _k\rangle \in \mathcal{G}_{N_k}^{\varGamma }\) models the bargaining within union \(N_k\) for the distribution of its total share among the members taking into account restrictions on cooperation in \(N_k\) given by \(\Gamma _k\), when the share is obtained by the application of \(\varGamma \)-value \(\phi \) at the upper level bargaining between a priori unions.

3 Deletion link properties for two-level graph games

As it is discussed in Khmelnitskaya (2014), a number of known component efficient \(\varGamma \)-values for games with undirected or directed communication structure such as for undirected graph games the Myerson value \(\mu \) (cf., Myerson 1977) and the position value \(\pi \) (cf., Meessen 1988; Borm et al. 1992; Slikker 2005) for arbitrary undirected graph games, the average tree solution AT (cf., Herings et al. 2008) and the compensation solution CS (cf., Béal et al. 2012) for undirected cycle-free graph games, or for directed graph games the upper equivalent solution UE, the lower equivalent solution LE and the equal loss solution EL for linear graph games (cf., van den Brink et al. 2007), the tree value t for rooted forest and the sink value s for sink forest digraph games (cf., Khmelnitskaya 2010), can be approached within the unified framework. Indeed, each one of these \(\varGamma \)-values is defined for \(\varGamma \)-games with suitable graph structure and is characterized by two axioms, CE and one or another deletion link (DL) property (axiom), reflecting the relevant reaction of a \(\varGamma \)-value on the deletion of a link in the communication graph. The corresponding DL properties are fairness (F), balanced link contributions (BLC), component fairness (CF), relative fairness (RF), upper equivalence (UE), lower equivalence (LE), equal loss property (EL), successor equivalence (SE), and predecessor equivalence (PE), and the characterization results are as follows:

$$\begin{aligned} \mathrm{CE} + \hbox {F}\quad \text{ for } \text{ all } \text{ undirected }\;\varGamma \hbox {-games}\Longleftrightarrow & {} \mu (v,\Gamma ),\\ \mathrm{CE} + \hbox {BLC}\quad \text{ for } \text{ all } \text{ undirected }\;\varGamma \hbox {-games}\Longleftrightarrow & {} \pi (v,\Gamma ),\\ \mathrm{CE} + \hbox {CF}\quad \text{ for } \text{ undirected } \text{ cycle-free }\;\varGamma \hbox {-games}\Longleftrightarrow & {} AT(v,\Gamma ),\\ \mathrm{CE} + \hbox {RF}\quad \text{ for } \text{ undirected } \text{ cycle-free }\;\varGamma \hbox {-games}\Longleftrightarrow & {} CS(v,\Gamma ),\\ \mathrm{CE} + \hbox {UE}\quad \text{ for } \text{ linear } \text{ graph }\;\varGamma \hbox {-games}\Longleftrightarrow & {} UE(v,\Gamma ),\\ \mathrm{CE} + \hbox {LE}\quad \text{ for } \text{ linear } \text{ graph }\;\varGamma \hbox {-games}\Longleftrightarrow & {} LE(v,\Gamma ),\\ \mathrm{CE} + \hbox {EL}\quad \text{ for } \text{ linear } \text{ graph }\;\varGamma \hbox {-games}\Longleftrightarrow & {} EL(v,\Gamma ),\\ \mathrm{CE} + \hbox {SE}\quad \text{ for } \text{ rooted } \text{ forest }\;\varGamma \hbox {-games}\Longleftrightarrow & {} t(v,\Gamma ),\\ \mathrm{CE} + \hbox {PE}\quad \text{ for } \text{ sink } \text{ forest }\;\varGamma \hbox {-games}\Longleftrightarrow & {} s(v,\Gamma ). \end{aligned}$$

This observation allows to identify each of the listed above \(\varGamma \)-values with the corresponding DL axiom. Given a DL axiom, let \(\mathcal{G}_N^{DL}\subseteq \mathcal{G}_N^{\varGamma }\) denote a set of all \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\) with \(\Gamma \) suitable for DL application. Then

$$\begin{aligned} \mathrm{CE\ +\ DL\quad \text{ on } \mathcal{G}_N^{DL}}\ \Longleftrightarrow \ DL(v,\Gamma ). \end{aligned}$$

Whence it simply follows that \(F(v,\Gamma )=\mu (v,\Gamma )\) and \(BLC(v,\Gamma )=\pi (v,\Gamma )\) for all undirected \(\varGamma \)-games, \(CF(v,\Gamma )=AT(v,\Gamma )\) and \(RF(v,\Gamma )=CS(v,\Gamma )\) for all undirected cycle-free \(\varGamma \)-games, \(UE(v,\Gamma )\), \(LE(v,\Gamma )\), and \(EL(v,\Gamma )\) are UE, LE, and EL solutions correspondingly for all linear graph \(\varGamma \)-games, \(SE(v,\Gamma )=t(v,\Gamma )\) for all rooted forest \(\varGamma \)-games, and \(PE(v,\Gamma )=s(v,\Gamma )\) for all sink forest \(\varGamma \)-games. Remark that all just discussed values are additive.

Next notice that every discussed DL axiom can be equivalently defined by an equality

$$\begin{aligned} \Psi ^{DL}(\xi (v,\Gamma ),\Gamma ')=0, \end{aligned}$$
(3)

where \(\Psi ^{DL}\) is an operator which for a \(\varGamma \)-value \(\xi \) defined on \(\mathcal{G}_N^{DL}\) and applied to a \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{DL}\) assigns a real number representing the numerical evaluation of players’ payoff reaction on the deletion of links in \(\Gamma \) from a chosen set of links \(\Gamma '\subseteq \Gamma \) according to the considered DL axiom. For the above mentioned DL-axioms, we have \(\Gamma ' = \Gamma \), and the corresponding operators of the DL-axioms are:

Fairness (F) For any player set \(N\subset \mathrm{I\!N}\), for every \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\),

$$\begin{aligned} \Psi ^F(\xi (v,\Gamma ),\Gamma ')\!=\! \!\sum _{i,j\in N|\{i,j\}\in \Gamma ' }\! \Bigl |\Bigl (\xi _i(v,\Gamma )-\xi _i(v,\Gamma |_{-ij})\Bigr )\!-\! \Bigl (\xi _j(v,\Gamma )-\xi _j(v,\Gamma |_{-ij})\Bigr )\Bigr |, \end{aligned}$$
(4)

Balanced link contributions (BLC) For any player set \(N\subset \mathrm{I\!N}\), for every \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\),

$$\begin{aligned} \Psi ^{BLC}(\xi (v,\Gamma ),\Gamma ')= & {} \!\sum _{i,j\in N} \left| \!\sum _{h\in N|\{i,h\}\!\in \!\Gamma '} \Bigl (\xi _j(v,\Gamma )-\xi _j(v,\Gamma |_{-ih})\Bigr )\right. \nonumber \\&\left. - \sum _{h\in N|\{j,h\}\in \Gamma '}\! \Bigl (\xi _i(v,\Gamma )-\xi _i(v,\Gamma |_{-jh})\Bigr )\right| , \end{aligned}$$
(5)

Component fairness (CF) For any player set \(N\subset \mathrm{I\!N}\), for every cycle-free \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\),

$$\begin{aligned} \Psi ^{CF}(\xi (v,\Gamma ),\Gamma ')= & {} \!\sum _{i,j\in N|\{i,j\}\in \Gamma '} \Bigl |\frac{1}{|(N/\Gamma |_{-ij})_i|} \sum _{h\in (N/\Gamma |_{-ij})_i} \Bigl (\xi _h(v,\Gamma )-\xi _h(v,\Gamma |_{-ij})\Bigr )\nonumber \\&- \frac{1}{|(N/\Gamma |_{-ij})_j|} \sum _{h\in (N/\Gamma |_{-ij})_j} \left( \xi _h(v,\Gamma )-\xi _h(v,\Gamma |_{-ij})\right) \Bigr |, \end{aligned}$$
(6)

Relative fairness (RF) For any player set \(N\subset \mathrm{I\!N}\), for every cycle-free \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\),

$$\begin{aligned} \Psi ^{RF}(\xi (v,\Gamma ),\Gamma ')= & {} \!\sum _{i,j\in N|\{i,j\}\in \Gamma '} \Bigl |\Bigl (\xi _i(v,\Gamma )-\frac{1}{|(N/\Gamma |_{-ij})_i|} \sum _{h\in (N/\Gamma |_{-ij})_i} \xi _h(v,\Gamma |_{-ij})\Bigr )\nonumber \\&-\Bigl (\xi _j(v,\Gamma )-\frac{1}{|(N/\Gamma |_{-ij})_j|} \sum _{h\in (N/\Gamma |_{-ij})_j} \xi _h(v,\Gamma |_{-ij})\Bigr )\Bigr |, \end{aligned}$$
(7)

Upper equivalence (UE) For any player set \(N\subset \mathrm{I\!N}\), for every linear graph \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\),

$$\begin{aligned} \Psi ^{UE}(\xi (v,\Gamma ),\Gamma ')= \sum _{i=1}^{n-1}\sum _{j=1}^i\, \bigl |\xi _j(v,\Gamma )-\xi _j(v,\Gamma |_{-i,i+1})\bigr |. \end{aligned}$$
(8)

Lower equivalence (LE) For any player set \(N\subset \mathrm{I\!N}\), for every linear graph \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\),

$$\begin{aligned} \Psi ^{LE}(\xi (v,\Gamma ),\Gamma ')= \sum _{i=1}^{n-1}\sum _{j=i+1}^n \bigl |\xi _j(v,\Gamma )-\xi _j(v,\Gamma |_{-i,i+1})\bigr |. \end{aligned}$$
(9)

Equal loss property (EL) For any player set \(N\subset \mathrm{I\!N}\), for every linear graph \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\),

$$\begin{aligned}&\Psi ^{EL}(\xi (v,\Gamma ),\Gamma ')\nonumber \\&\quad = \sum _{i=1}^{n-1}\left| \sum _{j=1}^i\Bigl (\xi _j(v,\Gamma )- \xi _j(v,\Gamma |_{-i,i+1})\Bigl )- \sum _{j=i+1}^n\Bigl (\xi _j(v,\Gamma )- \xi _j(v,\Gamma |_{-i,i+1})\Bigl )\right| .\nonumber \\ \end{aligned}$$
(10)

Successor equivalence (SE) For any player set \(N\subset \mathrm{I\!N}\), for every rooted forest \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\),

$$\begin{aligned} \Psi ^{SE}(\xi (v,\Gamma ),\Gamma ')= \!\sum _{i,j\in N|\{i,j\}\in \Gamma '} \sum _{h\in \bar{S}^\Gamma (j)} \bigl |\xi _h(v,\Gamma )-\xi _h(v,\Gamma |_{-ij})\bigr |. \end{aligned}$$
(11)

Predecessor equivalence (PE) For any player set \(N\subset \mathrm{I\!N}\), for every sink forest \(\varGamma \)-game \(\langle v,\Gamma \rangle \in \mathcal{G}_N^{\varGamma }\),

$$\begin{aligned} \Psi ^{PE}(\xi (v,\Gamma ),\Gamma ')= \!\sum _{i,j\in N|\{i,j\}\in \Gamma '} \sum _{h\in \bar{P}^\Gamma (j)} \bigl |\xi _h(v,\Gamma )-\xi _h(v,\Gamma |_{-ij})\bigr |. \end{aligned}$$
(12)

The definition of deletion link axioms for \(\varGamma \)-values in terms of equality (3) allows to consider the corresponding deletion link axioms within a unified framework also for \(P\varGamma \)-values with respect to both communication levels, at the upper level with respect to the quotient \(\varGamma \)-game determining the total payoffs to a priori unions and at the lower level with respect to \(\varGamma \)-games within a priori unions determining the distribution of total payoffs within each a priori union. Let \(\xi \) be a \(P\varGamma \)-value. By definition \(\xi \) is a mapping \(\xi :\mathcal{G}_N^{P\varGamma }\rightarrow {\mathrm{I\!R}}^N\) that assigns a payoff vector to any \(P\varGamma \)-game on the player set \(N\subset \mathrm{I\!N}\). For every \(N\subset \mathrm{I\!N}\) a mapping \(\xi \!=\!\{\xi _i\}_{i\in N}\) generates on the domain of \(P\varGamma \)-games on N a mapping \(\xi ^\mathcal{P}:\mathcal{G}_N^{P\varGamma }\rightarrow {\mathrm{I\!R}}^M\), \(\xi ^\mathcal{P}\!=\!\{\xi ^\mathcal{P}_k\}_{k\in M}\), with \(\xi ^\mathcal{P}_k\!=\!\sum _{i\in N_k}\xi _i\), \(k\!\in \!M\), that assigns to every \(P\varGamma \)-game on N a vector of total payoffs to all a priori unions, and m mappings \(\xi ^k:\mathcal{G}_N^{P\varGamma }\rightarrow {\mathrm{I\!R}}^{N_k}\), \(\xi ^k\!=\!\{\xi _i\}_{i\in N_k}\), \(k\!\in \!M\), assigning payoffs to players within a priori unions. For a given \((m+1)\)-tuple of deletion link axioms \(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle \) for \(\varGamma \)-values, let \(\mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\in M}}\!\subseteq \mathcal{G}_N^{P\varGamma }\) be the set of \(P\varGamma \)-games composed of \(P\varGamma \)-games \(\langle v,\Gamma _\mathcal{P}\rangle \) with graph structures \(\Gamma _\mathcal{P}=\langle \Gamma _M,\{\Gamma _k\}_{k\in M}\rangle \) such that \(\langle v_\mathcal{P},\Gamma _M\rangle \in \mathcal{G}_M^{{DL}^\mathcal{P}}\) and \(\langle v^{DL^\mathcal{P}}_k,\Gamma _k\rangle \in \mathcal{G}_{N_k}^{{DL}^k}\), \(k\in M\). Given a \((m+1)\)-tuple of deletion link axioms \(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle \) for \(\varGamma \)-values and a \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\in M}}\), we may consider the operators (4)–(12) applied to the \(\varGamma \)-value \(\xi ^\mathcal{P}\) assigning the total payoffs to a priori unions to evaluate its reaction on deletion of links in graph \(\Gamma _M\), or applied to the \(\varGamma \)-values \(\xi ^k\), \(k\in M\), assigning payoffs to single players within a priori unions to evaluate their reaction on deletion of links in the corresponding graphs \(\Gamma _k\). Based on the last observation, we define axioms of Quotient DL property and Union DL property for \(P\varGamma \)-values defined on \(\mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\in M}}\) as follows:

Quotient DL property (QDL) For any player set \(N\subset \mathrm{I\!N}\) and \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\in M}}\), \(\Gamma _\mathcal{P}=\langle \Gamma _M,\{\Gamma _h\}_{h\in M}\rangle \),

$$\begin{aligned} \Psi ^{DL^\mathcal{P}}(\xi ^\mathcal{P}(v,\Gamma _\mathcal{P}),\Gamma _M)=0, \end{aligned}$$
(13)

where \(\Psi ^{DL^\mathcal{P}}\) is given by one of the operators (4)–(12) and in the corresponding formulas \(\Gamma _\mathcal{P}|_{-kl}\!=\!\langle \Gamma _M|_{-kl},\{\Gamma _h\}_{h\in M}\rangle \) for every link \(\{k,l\}\in \Gamma _M\) if \(\Gamma _M\) is undirected, or \((k,l)\in \Gamma _M\) if \(\Gamma _M\) is directed.

Union DL property (UDL) For any player set \(N\subset \mathrm{I\!N}\), \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\in M}}\), \(\Gamma _\mathcal{P}=\langle \Gamma _M,\{\Gamma _h\}_{h\in M}\rangle \), and \(k\in M\),

$$\begin{aligned} \Psi ^{DL^k}(\xi ^k(v,\Gamma _\mathcal{P}),\Gamma _k)=0, \end{aligned}$$
(14)

where \(\Psi ^{DL^k}\) is given by one of the operators (4)–(12) and in the corresponding formulas \(\Gamma _\mathcal{P}|_{-ij}=\Gamma _\mathcal{P}|^k_{-ij}=\langle \Gamma _M,\{\widehat{\Gamma }_h\}_{h\in M}\rangle \) with \(\widehat{\Gamma }_h=\Gamma _h\) for \(h\ne k\) and \(\widehat{\Gamma }_k=\Gamma _k|_{-ij}\) for every link \(\{i,j\}\in \Gamma _k\) if \(\Gamma _k\) is undirected, or \((i,j)\in \Gamma _k\) if \(\Gamma _k\) is directed.

For example, if we consider the Myerson fairness (F) as a DL axiom, QDL and \(\mathrm{UDL}^k\) can be denoted QF and \(\mathrm{UF}^k\) correspondingly. In fact in this case QF coincides with quotient fairness (QF) and m-tuple of axioms \((\mathrm{UF}^1,\ldots ,\mathrm{UF}^m)\) coincides with union fairness (UF) employed in van den Brink et al. (2016).

QDL and UDL axioms for \(P\varGamma \)-values provide the uniform approach to various deletion link properties on both bargaining levels. This allows to introduce within the unified framework two families of \(P\varGamma \)-values based on the adaptation of the two-step distribution procedures of Owen and Kamijo respectively for games with coalition structure to the case when the cooperation between and within a priori unions is restricted by communication graphs and when different combinations of known component efficient solution concepts on both communication levels could be applied. Moreover, this allows to include into consideration not only combinations of undirected communication graphs but also combinations including some types of digraphs.

4 Kamijo-type values for two-level graph games

In this section we consider the family of \(P\varGamma \)-values based on the adaptation of the Kamijo’s two-step distribution procedure for P-games. We introduce these values axiomatically by means of six axioms among which QDL and UDL as defined in the previous section. The other four axioms are defined below. To begin with, Quotient component efficiency requires that each component on the upper level between a priori unions distributes fully its total worth among the players of the a priori unions forming this component.

Quotient component efficiency (QCE) For any \(N\subset \mathbb {N}\), any \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{P\varGamma }\), \(\Gamma _\mathcal{P}=\langle \Gamma _M,\{\Gamma _k\}_{k\in M}\rangle \),

$$\begin{aligned} \sum _{k\in K}\sum _{i\in N_k}\xi _i(v,\Gamma _\mathcal{P})=v_\mathcal{P}(K),\quad \text{ for } \text{ all } K\in M/\Gamma _M. \end{aligned}$$
(15)

The second axiom is a straightforward adaptation of the standard covariance under strategic equivalence to the case of \(P\varGamma \)-games. Two games \(v,w\in \mathcal{{G}}_N\) are strategically equivalent  if there are \(a\in \mathrm{I\!R}_{++}\) and \(b\in \mathrm{I\!R}^n\) such that

$$\begin{aligned} w(S)=av(S)+b(S),\quad \text{ for } \text{ all }S\subseteq N. \end{aligned}$$

Covariance under strategic equivalence (COV) For any player set \(N\subset \mathrm{I\!N}\) and \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{P\varGamma }\), \(\Gamma _\mathcal{P}=\langle \Gamma _M,\{\Gamma _k\}_{k\in M}\rangle \), for any \(a\in \mathrm{I\!R}_{++}\) and \(b\in \mathrm{I\!R}^n\) it holds that

$$\begin{aligned} \xi (av+b,\Gamma _\mathcal{P})=a\xi (v,\Gamma _\mathcal{P})+b, \end{aligned}$$

where the game \((av+b)\in \mathcal{G}_N\) is defined by \((av+b)(S)=av(S)+b(S)\) for all \(S\subseteq N\).

The next axiom requires equal payoffs to all members of every a priori union \(N_k\), \(k\in M\), for which all subcoalitions \(S\subseteq N_k\) with nonzero worth \(v_k(S)\ne 0\) are disconnected. In this case every connected coalition possesses zero worth, i.e., communication between the members of every connected coalition is useless, and so, the asymmetries among the players created by game \(v_k\) and by their locations in graph \(\Gamma _k\) on \(N_k\) vanish. Therefore, it makes sense to treat all players of \(N_k\) symmetrically. Remark that the condition that \(v(S)\ne 0\) implies S to be disconnected is equivalent to \(v_k^{\Gamma _k}\equiv \mathbf{0}\), i.e., the axiom requires that all players in \(N_k\) obtain the same payoffs if the Myerson restricted game \(v_k^{\Gamma _k}\) is a null game. The latter observation determines the name of the axiom.

Union null restricted game property (UNRGP) For any player set \(N\subset \mathrm{I\!N}\) and \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{P\varGamma }\), \(\Gamma _\mathcal{P}=\langle \Gamma _M,\{\Gamma _k\}_{k\in M}\rangle \), if for some \(k\in M\), for all \(S\subseteq N_k\), \(v_k(S)\ne 0\) implies \(S\notin C^{\Gamma _k}(N_k)\), then it holds that for all \( i,j\in N_k\), \( i\ne j\),

$$\begin{aligned} \xi _i(v,\Gamma _\mathcal{P})=\xi _j(v,\Gamma _\mathcal{P}). \end{aligned}$$

The last axiom determines the distribution of the total shares obtained by internally disconnected a priori unions at the upper level bargaining between a priori unions among the components of these unions. Imagine that each a priori union \(N_k\), \(k\in M\), is a public institution (e.g. university, hospital, or firm) of which every component \(C\in N_k/\Gamma _k\) is an independent unit (e.g. the faculties within a university, medical departments within a hospital, or production plants within a firm). First public institutions \(N_k\), \(k\in M\) compete among themselves for their annual budgets from the government. Once obtained the budget, institution \(N_k\) has to decide how much to give to each of its independent units. At this stage the independent units of an institution compete against each other for the best possible shares from the institution’s budget. Similarly as in the competition among the public institutions, the total payoff to a unit depends on the total productivity of each of the units, but not on the productivity of the smaller collaborating teams within the units. Our last axiom requires the total payoff to a component of any a priori union to be independent of the so-called internal coalitions, each of which is a proper subcoalition of some component of one of the given a priori unions, or more precisely, a coalition \(\emptyset \ne S\subseteq N\) is internal if there is \(k\in M\) such that \(S\subset C\) for some \(C\in N_k/\Gamma _k\). From now on, given a player set \(N\subset \mathrm{I\!N}\), a partition \(\mathcal{P}=\{N_1,\ldots ,N_m\}\) of N, and a set of communication graphs \(\{\Gamma _k\}_{k\in M}\) on a priori unions \(N_k\), \(k\in M\), the set of all internal coalitions we denote by \(\mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M})\). It is worth to remark that, of course, the worths of internal coalitions play a crucial role in the redistribution of the total payoff obtained by a component among its members, but the axiom does not concern this.

Union component payoff independence of internal coalitions (UCPIIC) For any player set \(N\subset \mathrm{I\!N}\) and two \(P\varGamma \)-games \(\langle v,\Gamma _\mathcal{P}\rangle ,\langle w,\Gamma _\mathcal{{P}}\rangle \in \mathcal{G}_N^{P\varGamma }\) with the same \(\Gamma _\mathcal{P}=\langle \Gamma _M,\{\Gamma _k\}_{k\in M}\rangle \) and such that \(w(S)=v(S)\) for all \(S\subseteq N\), \(S\notin \mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M})\), it holds that for every \(k\in M\), for all \(C\in N_k/\Gamma _k\),

$$\begin{aligned} \sum _{i\in C}\xi _i(v,\Gamma _\mathcal{{P}})=\sum _{i\in C}\xi _i(w,\Gamma _\mathcal{{P}}). \end{aligned}$$

Our first theorem provides an axiomatic characterization of the a family of \(P\varGamma \)-values.

Theorem 1

For any \((m+1)\)-tuple of deletion link axioms \(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle \) such that \(\mathrm{DL}^\mathcal{P}\) is not RF and the set of  \(\mathrm{DL}^k\), \(k\in M\), axioms is restricted to F and CF, there is a unique \(P\varGamma \)-value defined on \(\mathcal{G}_N^{{DL}^\mathcal{P},\{{DL}^k\}_{k\in M}}\) that meets axioms QCE, QDL, UDL, COV, UNRGP, and UCPIIC, and for every \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{{DL}^\mathcal{P},\{{DL}^k\}_{k\in M}}\) it is given by

$$\begin{aligned} \xi _i(v,\Gamma _\mathcal{P})=DL_i^{k(i)}(v_{k(i)},\Gamma _{k(i)})+ \frac{DL_{k(i)}^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _{k(i)}}_{k(i)}(N_{k(i)})}{n_{k(i)}}, \quad \text{ for } \text{ all }i\in N. \end{aligned}$$
(16)

Before proving Theorem 1, we formulate some remarks.

Remark 1

It is not difficult to trace a relation between the \(P\varGamma \)-value \(\xi \) given by (16) and the two-step Shapley value (1). Indeed, for a \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{F^\mathcal{P},\{F^k\}_{k\in M}}\), \(\xi (v,\Gamma _\mathcal{P})=Ka(v^{\Gamma _\mathcal{P}},\mathcal P)\), where \(v^{\Gamma _\mathcal{P}}\in \mathcal{G}_N\) is determined as

$$\begin{aligned} v^{\Gamma _\mathcal{P}}(S)=\left\{ \begin{array}{ll} v_\mathcal{P}^{\Gamma _M}(Q),&{}\quad Q\subseteq M:S=\cup _{q\in Q} N_q,\\ \,v_k^{\Gamma _k}(S),&{}\quad S\subseteq N_k,\ k\in M,\\ 0,&{}\quad \text{ otherwise }, \end{array} \right. \quad \text{ for } \text{ all }S\subseteq N, \end{aligned}$$

i.e., when S is the union of number of a priori unions its worth is defined according to the Myerson restricted quotient game, when S is a subset of some a priori union its worth is defined according to the Myerson restricted game within the union, otherwise the worth of S is zero.

Because of the mentioned similarity, from now on we refer to the \(P\varGamma \)-value (16) as to the Kamijo-type \(\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle \)-value, denoted further by \(Ka^{\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle }\).

Remark 2

Axiom RF cannot be used at any level in Theorem 1. The reason is that it gives rise to the compensation solution at the level where RF is applied and the compensation solution does not satisfy COV on its domain. Furthermore, the \(P\varGamma \)-value \(Ka^{\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle }\) violates the UDL property if in the \((m+1)\)-tuple of deletion link axioms \(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle \) among the axioms \(\mathrm{DL}^k\), \(k\in M\) there are axioms BLC, UE, LE, EL, SE, and PE. The reason is that the second summand in the numerator of the second term in the right-hand side of (16) is sensitive to the deletion of different links in graph \(\Gamma _k\), and therefore, for these cases the equality \(\Psi ^{DL^k}(\xi ^k(v,\Gamma _\mathcal{P}),\Gamma _k)=\Psi ^{DL^k}({DL^k}(v_k,\Gamma _k),\Gamma _k)=0\) in general does not hold.

Proof

(Theorem 1) I. [Existence]. We show that under the hypothesis of the theorem the \(P\varGamma \)-value \(\xi \!=\!Ka^{\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\!\in \!M}\rangle }\) defined on \(\mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\in M}}\) by (16) meets the axioms QCE, QDL, UDL, COV, UNRGP and UCPIIC. Consider an arbitrary \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\!\in \!M}}\).

QCE By the definition (16) of \(\xi \) and component efficiency of each \(DL^k\)-value for all \(k\in M\) it holds that

$$\begin{aligned} \sum _{i\in N_k}\xi _i(v,\Gamma _\mathcal{P})=DL^\mathcal{P}_k(v_\mathcal{P},\Gamma _M). \end{aligned}$$
(17)

Thus, for any \(K\in M/\Gamma _M\) we have

$$\begin{aligned} \sum _{k\in K}\sum _{i\in N_k}\xi _i(v,\Gamma _\mathcal{P})=\sum _{k\in K}DL^\mathcal{P}_k(v_\mathcal{P},\Gamma _M) = v_\mathcal{P}(K), \end{aligned}$$

where the second equality follows from component efficiency of \(DL^\mathcal{P}\)-value.

QDL From (17) we obtain that \(\xi ^\mathcal{P}(v,\Gamma _\mathcal{P})=DL^\mathcal{P}(v_\mathcal{P},\Gamma _M)\). Whence it follows that

$$\begin{aligned} \Psi ^{DL^\mathcal{P}}\left( \xi ^\mathcal{P}(v,\Gamma _\mathcal{P}),\Gamma _M\right) =\Psi ^{DL^\mathcal{P}}\left( DL^\mathcal{P}(v_\mathcal{P},\Gamma _M),\Gamma _M\right) =0, \end{aligned}$$

where the last equality holds true since the \(DL^\mathcal{P}\)-value for \(\varGamma \)-games meets \(DL^\mathcal{P}\).

UDL We need to show that if the set of axioms \(\mathrm{DL}^k\), \(k\in M\), is restricted to F and CF, then for all \(k\in M\), \(\Psi ^{DL^k}(\xi ^k(v,\Gamma _\mathcal{P}),\Gamma _k)=0\).

Let for some \(k\in M\), \(\mathrm{DL}^k=\mathrm{F}\). Then since by definition for any link \(\{i,j\}\in \Gamma _k\), \(\Gamma _\mathcal{P}|_{-ij}=\Gamma _\mathcal{P}|^k_{-ij}=\langle \Gamma _M,\{\widehat{\Gamma }_h\}_{h\in M}\rangle \) with \(\widehat{\Gamma }_h=\Gamma _h\) for \(h\ne k\), and \(\widehat{\Gamma }_k=\Gamma _k|_{-ij}\), we obtain that

$$\begin{aligned}&\Psi ^F(\xi ^k(v,\Gamma _\mathcal{P}),\Gamma _k)\\&\quad {\mathop {=}\limits ^{(4)}}\sum _{i,j\in N_k|\{i,j\}\in \Gamma _k }\! \Bigl |\Bigl (\xi ^k_i(v,\Gamma _\mathcal{P})- \xi ^k_i(v,\Gamma _\mathcal{P}|_{-ij})\Bigr )\!-\! \Bigl (\xi ^k_j(v,\Gamma _\mathcal{P})- \xi ^k_j(v,\Gamma _\mathcal{P}|_{-ij})\Bigr )\Bigr |\\&\quad {\mathop {=}\limits ^{(16)}} \sum _{i,j\in N_k|\{i,j\}\in \Gamma _k }\! \Bigl |\Bigl (F_i^k(v_k,\Gamma _k)+ \frac{DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _k}_k(N_k)}{n_k}\\&\quad \quad \quad -\,F_i^k(v_k,\Gamma _k|_{-ij})+\frac{DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _k|_{-ij}}_k(N_k)}{n_k}\Bigr )\\&\quad \quad \quad -\,\Bigl (F_j^k(v_k,\Gamma _k)+ \frac{DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _k}_k(N_k)}{n_k}-F_j^k(v_k,\Gamma _k|_{-ij})\\&\quad \quad \quad +\, \frac{DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _k|_{-ij}}_k(N_k)}{n_k}\Bigr )\Bigr |\\&\quad =\sum _{i,j\in N_k|\{i,j\}\in \Gamma _k }\! \Bigl |\Bigl (F_i(v_k,\Gamma _k)-F_i(v_k,\Gamma _k|_{-ij})\Bigr )- \Bigl (F_j(v_k,\Gamma _k)-F_j(v_k,\Gamma _k|_{-ij})\Bigr )\Bigr |\\&\quad {\mathop {=}\limits ^{(4)}}\Psi ^F(F(v_k,\Gamma _k),\Gamma _k)=0, \end{aligned}$$

where the last equality holds true since the F-value for \(\varGamma \)-games (the Myerson value) meets fairness F. Using the similar arguments as above we prove that for all \(k\in M\) for which \(\mathrm{DL}^k=\mathrm{CF}\), \(\Psi ^{CF}(\xi ^k(v,\Gamma _\mathcal{P}),\Gamma _k)=0\).

COV Pick any \(a\in \mathrm{I\!R}_{++}\) and \(b\in \mathrm{I\!R}^n\). Then, for all \(i\in N\),

$$\begin{aligned}&\xi _i(av\!+\!b,\Gamma _\mathcal{P}){{\mathop {=}\limits ^{(16),(2)}}} DL_i^{k(i)}((av+b)_{k(i)},\Gamma _{k(i)})\\&\qquad \qquad \qquad + \frac{DL_{k(i)}^\mathcal{P}((av+b)_\mathcal{P},\Gamma _M)-\! \sum \nolimits _{C\!\in \!N_{k(i)}/\Gamma _{k(i)}}\!(av+b)(C)}{n_{k(i)}}\\&\qquad \qquad \quad = aDL_i^{k(i)}(v_{k(i)},\Gamma _{k(i)})+b_i\\&\qquad \qquad \qquad \quad +\frac{aDL_{k(i)}^\mathcal{P}(v_\mathcal{P},\Gamma _M)\!+\!b(N_{k(i)})\!-\! \sum \nolimits _{C\!\in \!N_{k(i)}/\Gamma _{k(i)}}\!(av(C)\!+\!b(C))}{n_{k(i)}}\\&\qquad \qquad \quad = a\xi _i(v,\Gamma _\mathcal{P})+b_i, \end{aligned}$$

where the second equality is true because each of the considered \(DL^k\)-values, \(k\in M\), and \(DL^\mathcal{{P}}\)-values meets COV on its domain, and \((av\!+\!b)_\mathcal{P}(Q)\!=\!av_\mathcal{P}(Q)\!+\!b(\cup _{k\in Q}N_k)\) for all \(Q\subseteq M\); and the third equality is due to the equality \(\sum _{C\in N_{k(i)}/\Gamma _{k(i)}}b(C)=b(N_{k(i)})\), since \(N_{k(i)}/\Gamma _{k(i)}\) forms a partition of \(N_{k(i)}\).

UNRGP Assume that for the chosen \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \) there exists \(k\in M\) such that for all \(S\subseteq N_k\), \(v_k(S)\ne 0\) implies \(S\notin C^{\Gamma _k}(N_k)\). Every of the considered \(DL^k\)-values, the Myerson value and the average tree solution, are determined only by worths of connected coalitions, and therefore, \(DL_i^k(v_k,\Gamma _k)=0\) for all \(i\in N_k\). Moreover, as it was already mentioned earlier, the above assumption is equivalent to \(v_k^{\Gamma _k}\equiv \mathbf{0}\), which implies that \(v(C)=0\) for every \(C\in N_k/\Gamma _k\). Hence, from (16) it follows that for all \(i\in N_k\),

$$\begin{aligned} \xi _i(v,\Gamma _\mathcal{P})=\frac{DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)}{n_k}, \end{aligned}$$

where the right side is independent of i, from which it follows that \(P\varGamma \)-value \(\xi \) meets UNRGP.

UCPIIC. Take any \(k\in M\) and \(C\in N_k/\Gamma _k\). The component efficiency of each of the considered \(DL^k\)-values implies \(DL^k(v_k,\Gamma _k)(C)=v(C)\). Then from (16) it follows that

$$\begin{aligned} \sum \limits _{i\in C}\xi (v,\Gamma _\mathcal{P})=v(C)+ \frac{c}{n_k}\Bigl (DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _k}(N_k)\Bigr ), \end{aligned}$$

where the right side is independent of worths of internal coalitions, i.e. \(P\varGamma \)-value \(\xi \) meets UCPIIC.

II. [Uniqueness]. Assume that a \((m+1)\)-tuple of deletion link axioms \(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle \) such that the set of axioms  \(\mathrm{DL}^k\), \(k\in M\), is restricted to F and CF, is given. We show that there exists at most one \(P\varGamma \)-value on \(\mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\!\in \!M}}\) that satisfies axioms QCE, QDL, UDL, COV, UNRGP, and UCPIIC. Let \(\phi \) be such \(P\varGamma \)-value on \(\mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\!\in \!M}}\). Take an arbitrary \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\!\in \!M}}\). Fix some \(k\in M\). We start by determining the union payoffs \(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})\), \(k\in M\), by induction on the number of links in \(\Gamma _M\) similarly as it is done in the proof of uniqueness of the Myerson value for \(\varGamma \)-games, cf., Myerson (1977).

Initialization: If \(|\Gamma _M|=0\) then for all \(k \in M\) the set of neighboring unions \(\{h\in M\mid \{h,k\}\in \Gamma \}=\emptyset \), and therefore by QCE and definition of the quotient game \(v_\mathcal{P}\), \(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})=v_\mathcal{P}(\{k\})=v(N_k)\).

Induction hypothesis: Assume that the values \(\phi ^\mathcal{P}_k(v,\Gamma '_\mathcal{P})\), \(k\in M\), have been determined for all two-level graph structures \(\Gamma '_\mathcal{P}=\langle \Gamma ',\{\Gamma _h\}_{h\in M}\rangle \) with \(\Gamma '\) such that \(|\Gamma '|<|\Gamma _M|\).

Induction step: Let \(Q\in M/\Gamma _M\) be a component in graph \(\Gamma _M\) on M. If \(Q\subseteq M\) is a singleton, let \(Q=\{k\}\), then from QCE it follows that \(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})=v(N_k)\). If \(q\ge 2\), then there exists a spanning tree \(\widetilde{\Gamma }\subseteq \Gamma _M|_Q\) on Q with the number of links \(|\widetilde{\Gamma }|=q-1\). By QDL it holds that

$$\begin{aligned} \Psi ^{DL^\mathcal{P}}(\phi ^\mathcal{P}(v,\Gamma _\mathcal{P}),\Gamma _M)=0. \end{aligned}$$
(18)

The above equality in fact provides for each link \(\{k,l\}\in \Gamma _M\) some equality relating values of \(\phi ^\mathcal{P}_h(v,\Gamma _\mathcal{P})\), \(h\in M\), with values of distinct \(\phi ^\mathcal{P}_h(v,\Gamma _\mathcal{P}|_{-kl})\), \(h\in M\). Since \(|\Gamma _M|_{-kl}|=|\Gamma _M|-1\), from the induction hypothesis it follows that for all links \(\{k,l\}\in \Gamma _M\) (\((k,l)\in \Gamma _M\)) the payoffs \(\phi ^\mathcal{P}_h(v,\Gamma _\mathcal{P}|_{-kl})\), \(h\in M\), are already determined. Thus with respect to \(q-1\) links \(\{k,l\}\in \widetilde{\Gamma }\), (18) yields \(q-1\) linearly independent linear equations in the q unknown payoffs \(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})\), \(k\in Q\). Moreover, by QCE it holds that

$$\begin{aligned} \sum _{k\in Q}\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P}) = v_\mathcal{P}(Q). \end{aligned}$$

All these q equations are linearly independent. Whence it follows that for every \(Q \in M / \Gamma \), all payoffs \(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})\), \(k \in Q\), are uniquely determined. Notice that in the proof of the induction step, every possible spanning tree \(\widetilde{\Gamma }\) yields the same solution for the values \(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})\), \(k \in Q\), because otherwise a solution does not exist, which contradicts the already proved “existence” part of the proof of the theorem.

Next, we show that the individual payoffs \(\phi _i(v,\Gamma _\mathcal{P})\), \(i\in N_k\), are uniquely determined. This part of the proof is also by induction, now on the number of links in \(\Gamma _k\).

Initialization: Assume that \(|\Gamma _k|=0\). Let \(v_0\) be the 0-normalization of the TU game v, i.e., \(v_0(S)=v(S)-\sum _{i\in S}v(\{i\})\) for all \(S\subseteq N\), and let \((v_0)_k=v_0|_{N_k}\). As already shown above, the union payoffs \(\phi ^\mathcal{P}_k(w,\Gamma _\mathcal{P})\), \(k\in M\), are uniquely determined for any \(P\varGamma \)-game \(\langle w,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\!\in \!M}}\). In particular, the “union payoffs” \(\phi ^\mathcal{P}_k(v_0,\Gamma _\mathcal{P})\), \(k\in M\), are uniquely determined. By definition \((v_0)_k(\{i\})=0\) for every \(i\in N_k\), and therefore, \((v_0)_k^{\Gamma _k}\equiv \mathbf{0}\) since \(|\Gamma _k|=0\). Then, from UNRGP it follows that

$$\begin{aligned} \phi _i(v_0,\Gamma _\mathcal{P})=\frac{\phi ^\mathcal{P}_k(v_0,\Gamma _\mathcal{P})}{n_k},\quad \text{ for } \text{ all }i\in N_k. \end{aligned}$$

Whence by COV we obtain

$$\begin{aligned} \phi _i(v,\Gamma _\mathcal{P})=v(\{i\})+\frac{\phi ^\mathcal{P}_k(v^0,\Gamma _\mathcal{P})}{n_k},\quad \text{ for } \text{ all }i\in N_k, \end{aligned}$$

i.e., for every \(i\in N_k\), \(\phi _i(v,\Gamma _\mathcal{P})\) is uniquely determined.

Induction hypothesis: Let \(\Gamma '_\mathcal{P}\) denote the two-level graph structure \(\langle \Gamma _M,\{\Gamma '_h\}_{h\in M}\rangle \) with \(\Gamma '_h=\Gamma _h\) if \(h\ne k\) and \(\Gamma '_k = \Gamma '\) for some graph \(\Gamma '\) on \(N_k\). Assume that the values \(\phi _i(v,\Gamma '_\mathcal{P})\) have been determined for every \(\Gamma '\) with \(|\Gamma '|<|\Gamma _k|\).

Induction step: For every \(S\in \mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M})\) let \(C_S\in N_k/\Gamma _k\) be the unique component such that \(S\subset C_S\). Consider a game \(w\in \mathcal{G}_N\) defined as

$$\begin{aligned} w(S)=\left\{ \begin{array}{ll} v(S),&{}\quad S\notin \mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M}),\\ {\frac{s\,v(C_S)}{c_S}},&{}\quad S\in \mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M}),\\ \end{array} \right. \quad \text{ for } \text{ all } S\subseteq N. \end{aligned}$$
(19)

For the 0-normalization \(w_0\) of w, \((w_0)_k=(w_k)_0\). The subgame \(w_k\) is an additive game and, therefore, \((w_k)_0\equiv \mathbf{0}\). Then, due to UNRGP, similar as in the Initialization step, it follows that

$$\begin{aligned} \phi _i(w_0,\Gamma _\mathcal{P})=\frac{\phi ^\mathcal{P}_k(w_0,\Gamma _\mathcal{P})}{n_k},\quad \text{ for } \text{ all }i\in N_k. \end{aligned}$$

Whence by COV we obtain

$$\begin{aligned} \phi _i(w,\Gamma _\mathcal{P})=w(\{i\})+\frac{\phi ^\mathcal{P}_k(w_0,\Gamma _\mathcal{P})}{n_k},\quad \text{ for } \text{ all }i\in N_k. \end{aligned}$$

Consider a component \(C\in N_k/\Gamma _k\). From the above equality it follows that

$$\begin{aligned} \sum _{i\in C}\phi _i(w,\Gamma _\mathcal{P})=\sum _{i\in C}w(\{i\})+ \frac{c}{n_k}\phi ^\mathcal{P}_k(w_0,\Gamma _\mathcal{P}) {\mathop {=}\limits ^{(19)}} v(C)+\frac{c}{n_k}\phi ^\mathcal{P}_k(w_0,\Gamma _\mathcal{P}). \end{aligned}$$

By the definition (19) of w, \(w(S)=v(S)\) for all \(S\subseteq N\), \(S\notin \mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M})\). Whence by UCPIIC it follows that

$$\begin{aligned} \sum _{i\in C}\phi _i(v,\Gamma _\mathcal{P})=v(C)+\frac{c}{n_k}\mathrm{DL}_k^\mathcal{P}(w_{0,\mathcal{P}},\Gamma _M),\quad \text {for all}C\in N_k/\Gamma _k. \end{aligned}$$
(20)

Next, if \(c=1\), then C is a singleton and the payoff \(\phi _i (v,\Gamma _\mathcal{P})\) of the only player \(i\in C\) is uniquely determined by (20). If \(c\ge 2\), then there exists a spanning tree \(\widetilde{\Gamma }\subseteq \Gamma _k|_C\) on C with the number of links \(|\widetilde{\Gamma }|=c-1\). By UDL it holds that

$$\begin{aligned} \Psi ^{DL^k}\left( \phi ^k(v,\Gamma _\mathcal{P}),\Gamma _k\right) =0. \end{aligned}$$
(21)

The above equality in fact provides for any link \(\{i,j\}\in \Gamma _k\) some equality relating values of \(\phi _h(v,\Gamma _\mathcal{P})\), \(h\in N_k\), with values of distinct \(\phi _h(v,\Gamma _\mathcal{P}|^k_{-ij})\). Since \(|\Gamma _k|_{-ij}|=|\Gamma _k|-1\), by the induction hypothesis it follows that for all links \(\{i,j\}\in \Gamma _k\) the payoffs \(\phi _h(v,\Gamma _\mathcal{P}|^k_{-ij})\), \(h\in N_k\), are already determined. Thus with respect to \(c-1\) links \(\{i,j\}\in \widetilde{\Gamma }\), (21) yields \(c-1\) linearly independent linear equations in the c unknown payoffs \(\phi _i(v,\Gamma _\mathcal{P})\), \(i\in C\). These \(c-1\) equations together with (20) form a system of c linearly independent equations in the c unknown payoffs \(\phi _i(v,\Gamma _\mathcal{P})\), \(i\in C\). Hence, for every \(C \in N_k/ \Gamma _k\), all payoffs \(\phi _i(v,\Gamma _\mathcal{P})\), \(i \in C\), are uniquely determined. \(\square \)

Logical independence of the axioms

Given a \((m+1)\)-tuple of deletion link axioms \(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle \), such that the set of \(\mathrm{DL}^k\), \(k\in M\), axioms is restricted to F, CF, and RF, the logical independence of the axioms in Theorem 1 is demonstrated by the following examples of \(P\varGamma \)-values:

  • The \(P\varGamma \)-value \(\xi ^{(1)}\) assigning in every \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{\mathrm{DL}^\mathcal{P},{\{\mathrm{DL}^k\}}_{k\in M}}\) to every player \(i\in N\) a payoff

    $$\begin{aligned} \xi ^{(1)}_i(v,\Gamma _\mathcal{P})=\frac{DL_{k(i)}^\mathcal{P}(v_\mathcal{P},\Gamma _M)}{n_{k(i)}} \end{aligned}$$

    satisfies all axioms except COV.

  • The \(P\varGamma \)-value \(\xi ^{(2)}\) assigning in every \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{\mathrm{DL}^\mathcal{P},{\{\mathrm{DL}^k\}}_{k\in M}}\) to every player \(i\in N\) a payoff

    $$\begin{aligned} \xi ^{(2)}_i(v,\Gamma _\mathcal{P})=v(\{i\})+\frac{DL_{k(i)}^\mathcal{P}(v_\mathcal{P},\Gamma _M)-\sum _{j\in N_{k(i)}}v(\{j\})}{n_{k(i)}} \end{aligned}$$

    satisfies every axiom except UCPIIC.

  • The \(P\varGamma \)-value \(\xi ^{(3)}\) assigning in every \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{\mathrm{DL}^\mathcal{P},{\{\mathrm{DL}^k\}}_{k\in M}}\) to every player \(i\in N\) a payoff

    $$\begin{aligned} \xi ^{(3)}_i(v,\Gamma _\mathcal{P})=DL_i^{k(i)}(v_{k(i)},\Gamma _{k(i)})+\frac{Sh_{k(i)}(v_{(M/\Gamma _M)_{k(i)}})-v^{\Gamma _{k(i)}}(N_k)}{n_{k(i)}} \end{aligned}$$

    satisfies every axiom except QDL.

  • The \(P\varGamma \)-value \(\xi ^{(4)}\) assigning in every \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{\mathrm{DL}^\mathcal{P},{\{\mathrm{DL}^k\}}_{k\in M}}\) to every player \(i\in N\) a payoff

    $$\begin{aligned} \xi ^{(4)}_i(v,\Gamma _\mathcal{P})=EL_i^{k(i)}(v_{k(i)},\Gamma _{k(i)})+\frac{DL_{k(i)}^\mathcal{P}(v_\mathcal{P},\Gamma _M)-v^{\Gamma _{k(i)}}(N_k)}{n_{k(i)}} \end{aligned}$$

    satisfies every axiom except UDL.

  • The \(P\varGamma \)-value \(\xi ^{(5)}\) assigning in every \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{\mathrm{DL}^\mathcal{P},{\{\mathrm{DL}^k\}}_{k\in M}}\) to every player \(i\in N\) a payoff

    $$\begin{aligned} \xi ^{(5)}_i(v,\Gamma _\mathcal{P})= DL_i^{k(i)}(v_{k(i)},\Gamma _{k(i)})+ \frac{v(N_{k(i)})-v^{\Gamma _{k(i)}}(N_{k(i)})}{n_{k(i)}} \end{aligned}$$

    satisfies every axiom except QCE.

  • The \(P\varGamma \)-value \(\xi ^{(6)}\) assigning in every \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{\mathrm{DL}^\mathcal{P},{\{\mathrm{DL}^k\}}_{k\in M}}\) to every player \(i\in N\) a payoff

    $$\begin{aligned} \xi ^{(6)}_i(v,\Gamma _\mathcal{P})=Ka^{\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle }_i(v,\Gamma _\mathcal{P})+a_i-\frac{1}{n_{k(i)}}\sum _{j\in N_{k(i)}}a_j, \end{aligned}$$

    where \((a_j)_{j\in N}\in \mathrm{I\!R}^n\) is any vector of real numbers such that not all coordinates are equal, satisfies every axiom except UNRGP.

5 Owen-type values for two-level graph games

In this section we consider another family of \(P\varGamma \)-values based on the adaptation of the Owen’s two-step distribution procedure for P-games. We introduce these values axiomatically by means of four axioms. The first three axioms are QCE, QDL, and UDL already employed in the previous section for characterization of the Kamijo-type \(\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle \)-values. The fourth axiom of fair distribution of the surplus within unions was first introduced in van den Brink et al. (2016) where it was used for the axiomatization of the Owen-type value for \(P\varGamma \)-games which is based on the application of the Myerson value on both communication levels. This axiom requires balanced average payoff variation for all components within a priori union in case the other components leave the game.

Fair distribution of the surplus within unions (FDSU) For any \(N\subset \mathrm{I\!N}\), any \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{P\varGamma }\), \(\Gamma _\mathcal{P}=\langle \Gamma _M,\{\Gamma _k\}_{k\in M}\rangle \), \(k\in M\), any \(C,C'\in N_k/\Gamma _k\),

$$\begin{aligned} \frac{1}{c}\sum _{i\in C} \left( \xi _i(v,\Gamma _\mathcal{P})- \xi _i\left( v^k_C,\Gamma _{\mathcal{P}^k_C}\right) \right) = \frac{1}{c'}\sum _{i\in C'}\left( \xi _i (v,\Gamma _\mathcal{P})- \xi _i\left( v^k_{C'},\Gamma _{\mathcal{P}^k_{C'}}\right) \right) , \end{aligned}$$
(22)

where for \(k\in M\) and component \(C\in N_k/\Gamma _k\), \(v^k_C\) denotes the subgame \(v|_{(N\setminus N_k)\cup C}\) of v with respect to the coalition \((N\setminus N_k)\cup C\), \(\mathcal{P}^k_C\) denotes the partition on \((N\setminus N_k)\cup C\) consisting of union C and all unions \(N_h\) in \(\mathcal{P}\), \(h\ne k\), and \(\Gamma _{\mathcal{P}^k_C}=\langle \Gamma _M,\{\widetilde{\Gamma }_h\}_{h\in M}\rangle \) with \(\widetilde{\Gamma }_k=\Gamma _k|_C\) and \(\widetilde{\Gamma }_h=\Gamma _h\) for all \(h\in M\setminus \{k\}\), denotes the two-level communication structure that is obtained from \(\langle \Gamma _M,\{\Gamma _h\}_{h \in M}\rangle \) by replacing the communication graph \(\Gamma _k\) by its restriction on \(C\subseteq N_k\).

The next theorem extends the Owen-type value for \(P\varGamma \)-games studied in van den Brink et al. (2016) by allowing the application of different combinations of known component efficient solution concepts for \(\varGamma \)-games on both communication levels.

Theorem 2

For any \((m+1)\)-tuple of deletion link axioms \(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle \) such that the set of  \(\mathrm{DL}^k\), \(k\in M\), axioms is restricted to F, CF, and RF, there is a unique \(P\varGamma \)-value defined on \(\mathcal{G}_N^{{DL}^\mathcal{P},\{{DL}^k\}_{k\in M}}\) that meets axioms QCE, FDSU, QDL, and UDL. For every \(P\varGamma \)-game \(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{{DL}^\mathcal{P},\{{DL}^k\}_{k\in M}}\) it is given by

$$\begin{aligned} \xi _i(v,\Gamma _\mathcal{P})= DL^{k(i)}_i(\tilde{v}_{k(i)},\Gamma _{k(i)})+ \frac{DL^\mathcal{P}_{k(i)}(v_\mathcal{P},\Gamma _M)-\! \sum \limits _{C\in N_{k(i)}/\Gamma _{k(i)}}\tilde{v}_{k(i)}(C)}{n_{k(i)}},\quad i\in N, \end{aligned}$$
(23)

where for all \(k\in M\), \(\tilde{v}_k\in \mathcal{G}_{N_k}\) is defined as

$$\begin{aligned} \tilde{v}_k(S)=DL^\mathcal{P}_k(\hat{v}_S,\Gamma _M), \quad \text{ for } \text{ all }S\subseteq N_k, \end{aligned}$$

and for every \(S\subseteq N_k\), \(\hat{v}_S\in \mathcal{G}_M\) is given by

$$\begin{aligned}\hat{v}_S(Q)=\left\{ \begin{array}{ll} v(\cup _{h\in Q}\ N_h),&{}\quad k\notin Q,\\ v(\cup _{h\in Q\backslash \{k\}}\ N_h\cup S),&{}\quad k\in Q,\\ \end{array} \right. \quad \text{ for } \text{ all } Q\subseteq M. \end{aligned}$$

The proof strategy is similar to that applied in van den Brink et al. (2016), a careful reader may find the proof in “Appendix”. We skip the proof of logical independence of axioms since it can be easily obtained by modification of the examples used for the proof in van den Brink et al. (2016) in case of \(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle =\langle \mathrm{F}^\mathcal{P},\{\mathrm{F}^k\}_{k\in M}\rangle \), i.e., when the Myerson fairness F is applied on both levels.

Note that in case when \(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle =\langle \mathrm{F}^\mathcal{P},\{\mathrm{F}^k\}_{k\in M}\rangle \), the statement of Theorem 2 coincides with the statement of both Proposition 4.1 and Theorem 4.2 proved in van den Brink et al. (2016) together and the \(P\varGamma \)-value \(\xi \) given by (23) coincides as well with the Owen-type \(P\varGamma \)-value introduced there. Because of the latter coincidence, from now on we refer to the \(P\varGamma \)-value (23) as to the Owen-type \(\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle \)-value denoted further by \(Ow^{\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle }\).

Observe also that the Owen-type \(\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle \)-value admits the similar two-step construction procedure as the Owen value for games with coalition structure. The difference is that instead of two applications of the Shapley value used in the case of the classical Owen value, in the Owen-type \(\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle \)-value different known component efficient \(\varGamma \)-values can be applied on both communication levels.

Remark 3

Note that different from Theorem 1, in Theorem 2 RF can be used at any level due to the fact that COV employed in Theorem 1 is replaced by FDSU in Theorem 2, and the latter axiom is compatible with the compensation solution characterized by RF. By the reasons similar to those mentioned in Remark 2, the \(P\varGamma \)-value \(Ow^{\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle }\) violates the UDL property when there are axioms BLC, UE, LE, EL, SE, and PE among the axioms \(\mathrm{DL}^k\), \(k\in M\) in the \((m+1)\)-tuple \(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle \).

6 Conclusion

We conclude by a final comparison of the two families of values introduced by Theorems 1 and 2. Observe that all considered \(P\varGamma \)-values meet QCE, QDL, UDL, and COV (the last one only if RF is not involved). However, while the Owen-type \(\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle \)-values satisfy FDSU, but violate UNRGP and UCPIIC, the Kamijo-type \(\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle \)-values vice versa satisfy UNRGP and UCPIIC, but violate FDSU. The summary of the properties is given in the following table, where the axioms needed for our axiomatizations are marked by \(*\).

Value

QCE

QDL

UDL

COV

FDSU

UNRGP

UCPIIC

Owen-type \(\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle \)-value

\(+^*\)

\(+^*\)

\(+^*\)

\(+\)

\(+^*\)

−

−

Kamijo-type \(\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\in M}\rangle \)-value

\(+^*\)

\(+^*\)

\(+^*\)

\(+^*\)

−

\(+^*\)

\(+^*\)