Mathematical Methods of Operations Research

, Volume 78, Issue 2, pp 285–299

The prenucleolus and the prekernel for games with communication structures

Original Article

DOI: 10.1007/s00186-013-0444-7

Cite this article as:
Khmelnitskaya, A. & Sudhölter, P. Math Meth Oper Res (2013) 78: 285. doi:10.1007/s00186-013-0444-7

Abstract

It is well-known that the prekernel on the class of TU games is uniquely determined by non-emptiness, Pareto efficiency (EFF), covariance under strategic equivalence (COV), the equal treatment property, the reduced game property (RGP), and its converse. We show that the prekernel on the class of TU games restricted to the connected coalitions with respect to communication structures may be axiomatized by suitably generalized axioms. Moreover, it is shown that the prenucleolus, the unique solution concept on the class of TU games that satisfies singlevaluedness, COV, anonymity, and RGP, may be characterized by suitably generalized versions of these axioms together with a property that is called “independence of irrelevant connections”. This property requires that any element of the solution to a game with communication structure is an element of the solution to the game that allows unrestricted cooperation in all connected components, provided that each newly connected coalition is sufficiently charged, i.e., receives a sufficiently small worth. Both characterization results may be extended to games with conference structures.

Keywords

TU game Solution concept Communication and conference structure Nucleolus Kernel 

JEL Classification

C71 

1 Introduction

In the classical theory of cooperative games one assumes that all players may cooperate, i.e., any coalition may form. However, a more general model for TU games is necessary in order to describe situations in which cooperation is restricted. This model requires to allow restricting the coalition function of a TU game to a set of feasible coalitions. E.g., Faigle (1989) has analyzed the cores of games with restricted cooperation in general. Moreover, in many situations there is a structural restriction on cooperation. E.g., the cooperation may be restricted by some social, economical, hierarchical, or some biological structure. In the present paper we adopt the model of Myerson (1977) who introduces TU games with communication structures. A communication structure on a finite set N, a graph with vertex set N, only allows two players to communicate if they are linked by an edge of the graph. Hence, it is assumed that only members of connected coalitions are able to sign binding agreements via a series of agreements of the linked players in the coalition. As the worth of a non-connected coalition may not be realized, Myerson replaced it by the sum of the worths of the connected components of this coalition [see (6.1) for the precise definition of the corresponding coalition function] in order to define the Shapley value of the arising so-called “Myerson restricted game” as a solution for the game with communication structure. Several other “value-related” solution concepts for this class of games have been introduced and analyzed [see, e.g., Herings et al. (2010)]. In this paper we investigate two famous “core-related” solution concepts, namely the prenucleolus and the prekernel.

In order to generalize the aforementioned core-related solution concepts to a game with communication structure, it is not necessary to extend its coalition function to non-connected coalitions, e.g., by considering the Myerson restricted game. In fact, just the classical definition of the prenucleolus and the prekernel may be directly applied to the coalition function that is restricted to the connected coalitions [cf. Katsev and Yanovskaya (2010) who investigated the prenucleolus for games restricted to systems of coalitions that contain the grand coalition].

It is shown that a suitable modification of Sobolev’s (1975) famous classical characterization of the prenucleolus is still valid for games with communication structures. Only one additional axiom has to be added that we call “independence of irrelevant connections” (IIC). It requires that an element of the solution to a game with communication structure remains an element of the solution to the game that allows internal unrestricted communication in each of the components provided that each of the former non-connected coalitions receives a sufficiently small worth. It is, however, an open problem whether IIC is really needed.

Moreover, suitable versions of the determining axioms in Peleg’s (1986) characterization allow to axiomatize the prekernel for games with communication structures, even without IIC.

Both solution concepts may easily be generalized to TU games with conference structures as introduced by Myerson (1980).

The paper is organized as follows. In Sect. 2 we recall the basic definitions of the general nucleolus, TU games with coalition and communication structures, and related concepts, and propose our definition of the prenucleolus of a TU game with communication structure that is entirely based on the possible cooperation inside the connected coalitions. Indeed, we consider the prenucleolus of a game restricted to the connected coalitions. In Sect. 3 we show that the prenucleolus may be characterized by properties that are similar to those of Kohlberg (1971) for the (pre)nucleolus of classical games. Section 4 is devoted to the characterization of the prenucleolus for games with communication structures that is similar to Sobolev’s (1975) axiomatization in the classical case. Only the aforementioned new property IIC is employed in addition. Section 5 shows by means of examples that each of the axioms except IIC is logically independent of the remaining axioms. Peleg (1986) defines and axiomatizes the prekernel for games with coalition structures. In a completely analogous way we define and axiomatize the prekernel for games with communication structures in Sect. 6. Finally, in Sect. 7 it turns out that both concepts, the prenucleolus and the prekernel, for games with communication structures may easily be extended to games with conference structures.

2 Notation, definitions, and preliminaries

Let \(U,\,|U|\ge 3\), be a set, the universe of players, containing, without loss of generality, \(1,\ldots ,k\) whenever \(|U|\ge k\). A coalition is a finite nonempty subset of \(U\). Let N be a coalition, \(X\subseteq \mathbb{R }^N\), let \(D\) be a finite nonempty set, let \(h:X\rightarrow \mathbb{R }^D\), and \(d:=|D|\). Define \(\theta : X\rightarrow \mathbb{R }^d\) by
$$\begin{aligned} \theta _t(x)=\max _{T\subseteq D,|T|=t}\min _{i\in T}h_i(x)\, \text{ for } \text{ all } x\in X \text{ and } \text{ all } t=1,\ldots ,d, \end{aligned}$$
that is, for any \(x\in X,\,\theta (x)\) is the vector, whose components are the numbers \(h_i(x), i\in D\), arranged in non-increasing order. Let \(\ge _{lex}\) denote the lexicographical order of \(\mathbb{R }^d\). The nucleolus of \(h\) with respect to (w.r.t.) \(X,\,\mathcal{N}(h,X)\), is defined by
$$\begin{aligned} \mathcal{N}(h,X)=\left\{ x\in X\mid \theta (y)\ge _{lex}\theta (x) \text{ for } \text{ all } y\in X\right\} . \end{aligned}$$

Remark 2.1

Justman (1977) proved the following statements.
  1. (1)

    If \(X\) is nonempty and compact and if all \(h_i, i\in D\), are continuous, then \(\mathcal{N}(h,X)\ne \emptyset \).

     
  2. (2)

    If \(X\) is convex and all \(h_i,i\in D\), are convex, then \(\mathcal{N}(h,X)\) is convex and \(h_i(x)=h_i(y)\) for all \(i\in D\) and all \(x,y\in \mathcal{N}(h,X)\).

     
A (cooperative TU) game is a pair \((N,v)\) such that N is a coalition and \(v: 2^N\rightarrow \mathbb{R },\,v(\emptyset )=0\). Let N be a coalition. A coalition structure for N is a partition of N. A game with coalition structure (Aumann and Drèze 1974) is a triple \((N,v,\mathcal{R})\) such that \((N,v)\) is a game and \(\mathcal{R}\) is a coalition structure for N. We identify a game \((N,v)\) with the game with coalition structure \((N,v,\{N\})\). The subgame on a coalition \(\emptyset \ne S\subseteq N\) is denoted by \((S,v)\). For any game with coalition structure \((N,v,\mathcal{R})\) let
$$\begin{aligned} X^*(N,v,\mathcal{R})&= \left\{ x\in \mathbb{R }^N\mid x(R)\le v(R) \text{ for } \text{ all } R\in \mathcal{R}\right\} ~\text{ and }\\ X(N,v,\mathcal{R})&= \left\{ x\in \mathbb{R }^N\mid x(R)=v(R) \text{ for } \text{ all } R\in \mathcal{R}\right\} \end{aligned}$$
denote the set of feasible and Pareto efficient feasible payoffs (preimputations), respectively. We use \(x(S)=\sum _{i\in S}x_i~(x(\emptyset )=0)\) for every \(S\in 2^N\) and every \(x\in \mathbb{R }^N\) as a convention. Additionally, \(x_S\) denotes the restriction of \(x\) to \(S\), i.e., \(x_S=(x_i)_{i\in S}\), and we write \(x=(x_S,x_{N\setminus S})\). For \(x\in \mathbb{R }^N\) and \(S\subseteq N\) let \(e(S,x,v)=v(S)-x(S)\) denote the excess of \(S\) at \(x\) w.r.t. \((N,v)\). Let \(\mathcal{S}\subseteq 2^N\) such that \(\{i\}\in \mathcal{S}\) for all \(i\in N\). The nucleolus of the game \((N,v)\) w.r.t. \(X\subseteq \mathbb{R }^N\) and \(\mathcal{S}\), denoted by \(\mathcal{N}(N,v,X,\mathcal{S})\), is the set \(\mathcal{N}(h,X)\) where \(h=(e(S,\cdot ,v))_{S\in \mathcal{S}}\).

By Remark 2.1, \(\mathcal{N}(N,v,X,\mathcal{S})\) is a singleton whenever X is nonempty, compact, and convex. The prenucleolus of a game with coalition structure \((N,v,\mathcal{R})\) w.r.t. \(\mathcal{S}\), denoted by \(\mathcal{N}(N,v,\mathcal{R},\mathcal{S})\), is the set \(\mathcal{N}(N,v,X^*(N,v,\mathcal{R}),\mathcal{S})\). Now, let \(z\in X^*(N,v,\mathcal{R}), \mu =\max _{S\in \mathcal{S}}e(S,z,v)\), and \(X=\{x\in X^*(N,v,\mathcal{R})\mid e(S,x,v)\le \mu \text{ for } \text{ all } S\in \mathcal{S}\}\). Then X is nonempty, compact, and convex so that \(\mathcal{N}(N,v,X,\mathcal{S})\) is a singleton. Clearly, \(\mathcal{N}(N,v,X,\mathcal{S})=\mathcal{N}(N,v,X^*(N,v,\mathcal{R}),\mathcal{S})\) so that this set is a singleton whose unique element, denoted by \(\nu (N,v,\mathcal{R},\mathcal{S})\), is the prenucleolus (point) of \((N,v,\mathcal{R})\) w.r.t. \(\mathcal{S}\).

A graph is a pair \((N,g)\), where N is a coalition, called the set of vertices, and g is a set of 2-element subsets of N. An element of g is called link. Let \(\emptyset \ne S\subseteq N\) and \(i,j\in S\). The vertices i and j are connected in S by g if \(i=j\) or there exists a path in S that connects \(i\) and \(j\), that is, if there exist \(\ell \in \mathbb{N }\) and \(k_1,\ldots ,k_\ell \in S\) such that \(i=k_1,\,j=k_\ell \), and \(\{k_t,k_{t+1}\}\in g\) for all \(t\in \mathbb{N }\) with \(1\le t<\ell \). Let \(S{/g}\) denote the set of components of \(S\) w.r.t. \(g\), that is,
$$\begin{aligned} S{/g}=\left\{ \left\{ i\in S\mid i \text{ and } j \text{ are } \text{ connected } \text{ in } S \text{ by } g\right\} \mid j\in S\right\} . \end{aligned}$$
We say that S is connected by g if \(\left| S{/g}\right| =1\). Moreover, let \(\mathcal{S}_{N,g}\) denote the set of all coalitions in N that are connected by g, that is,
$$\begin{aligned} \mathcal{S}_{N,g}=\left\{ S\in 2^N{\setminus }\{\emptyset \}\mid S \text{ is } \text{ connected } \text{ by } g\right\} . \end{aligned}$$
(2.1)
A game with communication structure is a triple \((N,v,g)\) such that \((N,v)\) is a game and \((N,g)\) is a graph. Games with communication structures and the related definitions of the foregoing paragraph are due to Myerson (1977).

In a game with communication structure \((N,v,g)\) a non-connected coalition \(S\) cannot form. Hence one might ex ante exclude \(S\) from the domain of the coalition function \(v\). We adopt, however, Myerson’s model and do not exclude \(S\) from the domain of \(v\). In fact, \(v(S)\) is regarded as “virtual worth” of \(S\).

Now we are ready to define the prenucleolus of a game with communication structure.

Definition 2.2

Let \((N,v,g)\) be a game with communication structure. The prenucleolus of \((N,v,g)\), denoted by \(\mathcal{N}(N,v,g)\), is the set \(\mathcal{N}(N,v,X^*(N,v,N/g),\mathcal{S}_{N,g})\). The unique element of \(\mathcal{N}(N,v,g)\) is denoted by \(\nu (N,v,g)\) and is called prenucleolus (point) of \((N,v,g)\).

Note that for classical games Schmeidler (1969) introduced the nucleolus, the individually rational modification of the prenucleolus.

Let \((N,v,g)\) be a game with communication structure, \(\emptyset \ne S\subseteq N\), and \(x\in \mathbb{R }^N\). In order to define the “reduced game with communication structure” w.r.t. \(S\) and \(x\), we first recall the definition of the “reduced graph” introduced by Albizuri and Zarzuelo (2009). The reduced graph w.r.t \(S\) is the graph \((S,g^S)\) defined by
$$\begin{aligned} g^S=\left\{ \{i,j\}\subseteq S\mid i\ne j \text{ and } i \text{ and } j \text{ are } \text{ connected } \text{ in } \{i,j\} \cup N{\setminus }S \text{ by } g\right\} . \end{aligned}$$
(2.2)
Hence, two players of \(S\) are linked in \(g^S\) if they are either linked already in \(g\) or if there is a path in \(g\) via players outside \(S\) connecting them.

Remark 2.3

Let \((N,v,g)\) be a game with communication structure, \(x=\nu (N,v,g)\), and \(R\in N/g\). Then \(\nu (R,v,g^R)=x_R\).

Now we are ready to defined the reduced game of a game with communication structure.

Definition 2.4

Let \((N,v,g)\) be a game with communication structure, \(\emptyset \ne S\subseteq N\), and \(x\in \mathbb{R }^N\). The reduced game with communication structure of \((N,v,g)\) w.r.t. \(S\) and \(x\) is the game with communication structure \(\left( S,v^{S,x}_g,g^S\right) \) whose coalition function, for \(\emptyset \ne T\subseteq S\), is defined by
$$\begin{aligned} v^{S,x}_g(T)= \left\{ \begin{array}{ll} v(R)-x(R{\setminus }T),&{} \text{ if } T=S\cap R \text{ for } \text{ some } R\in N{/g},\\ \max \nolimits _{Q\subseteq N{\setminus }S,T\cup Q\in \mathcal{S}_{N,g}}v(T\cup Q)-x(Q),&{} \text{ if } T\in \mathcal{S}_{S,g^S},T\notin S{/g^S}\!,\\ \max \nolimits _{Q\subseteq N{\setminus }S}v(T\cup Q)-x(Q),&{} \text{ otherwise }. \end{array}\right. \end{aligned}$$
(2.3)
The definition of the reduced game with communication structure is similar to the definition of the Davis and Maschler (1965) reduced game with coalition structure as given by Peleg and Sudhölter (2007, Definition 3.8.8): If \(\mathcal{R}\) is a coalition structure for N, then \(\mathcal{R}^S=\{R\cap S\mid R\cap S\ne \emptyset ,R\in \mathcal{R}\}\) and the coalition function of the reduced game \(\left( N,v^{S,x}_\mathcal{R},\mathcal{R}^S\right) \) is defined, for \(\emptyset \ne T\subseteq S\), by
$$\begin{aligned} v^{S,x}_\mathcal{R}(T)=\left\{ \begin{array}{ll} v(R)-x(R{\setminus }T),&{} \text{ if } T=S\cap R \text{ for } \text{ some } R\in \mathcal{R},\\ \max \nolimits _{Q\subseteq N{\setminus }S}v(T\cup Q)-x(Q),&{} \text{ otherwise }.\end{array}\right. \end{aligned}$$
(2.4)
In order to interpret (2.3), we compare it with (2.4) for \(\mathcal{R}=N/g\). In a game with coalition structure each of the coalitions in \(\mathcal{R}\) may distribute its worth among its members. However, cooperation in any other coalition is still possible. Now, in the reduced game the players of \(S\) play their reduced game assuming that the players in \(N{\setminus }S\) are ready to cooperate. In a game with communication structure only the members of a connected coalition may cooperate—as mentioned earlier, the worth of a non-connected coalition is regarded as virtual worth. Hence, in the reduced game the worth of any disconnected coalition is still virtual (there is no need to change the definition), whereas the members of any coalition T that may be connected with the help of players in \(Q\subseteq N{\setminus }S\) (i.e., \(T\cup Q\) is connected by g) may in fact cooperate with the members of Q, thereby receiving \(v(T\cup Q)\). This difference in the interpretations is reflected by the difference of the definitions of the reduced games.

3 Kohlberg’s characterization

This section is devoted to present a suitable modification of Kohlberg’s (1971) characterization of the prenucleolus of games with coalition structures by balanced collections of coalitions.

First, we recall his characterization: Let \((N,v,\mathcal{R})\) be a game with coalition structure. Denote by \(\nu (N,v,\mathcal{R})\) the prenucleolus of this game, i.e., the unique element of \(\mathcal{N}(N,v,X^*(N,v,\mathcal{R}),2^N)\). For every \(x\in \mathbb{R }^N\) and any \(\alpha \in \mathbb{R }\) denote
$$\begin{aligned} \mathcal{D}(\alpha ,x,v)=\left\{ S\in 2^N\mid e(S,x,v)\ge \alpha \right\} . \end{aligned}$$
Moreover, \(\mathcal{B}\subseteq 2^N\) is called balanced (over N) if there are \(\delta ^S>0,S\in \mathcal{B}\), such that \(\sum _{S\in \mathcal{B}}\delta ^S\chi ^S=\chi ^N\), where for any \(T\subseteq N,\,\chi ^T\) is the indicator vector of \(T\), i.e., \(\chi ^T\in \mathbb{R }^N\) is defined by \(\chi ^T_i=1\) if \(i\in T\) and \(\chi ^T_i=0\) if \(i\in N{\setminus }T\).

The following remark is an immediate generalization of Kohlberg’s characterization of the nucleolus. For an explicit proof see Peleg and Sudhölter (2007, Theorems 5.2.6 and 6.4.1).

Remark 3.1

Let \((N,v,\mathcal{R})\) be a game with coalition structure and \(x\in X(N,v,\mathcal{R})\). Then the following statements are equivalent:
  1. (1)

    \(x=\nu (N,v,\mathcal{R})\).

     
  2. (2)

    For all \(\alpha \in \mathbb{R }\) the following property is satisfied: If \(y\in \mathbb{R }^N\) satisfies \(y(R)=0\) for all \(R\in \mathcal{R}\) and \(y(S)\ge 0\) for all \(S\in \mathcal{D}(\alpha ,x,v)\), then \(y(S)=0\) for all \(S\in \mathcal{D}(\alpha ,x,v)\).

     
  3. (3)

    For all \(\alpha \in \mathbb{R }\), the set \(\mathcal{D}(\alpha ,x,v)\cup \mathcal{R}\) is balanced.

     
Now, the foregoing characterization of the prenucleolus of a game with coalition structure is modified. Let \((N,v,g)\) be a game with communication structure. For every \(x\in \mathbb{R }^N\) and any \(\alpha \in \mathbb{R }\) denote
$$\begin{aligned} \mathcal{D}(\alpha ,x,v,g)=\left\{ S\in \mathcal{S}_{N,g}\mid e(S,x,v)\ge \alpha \right\} . \end{aligned}$$

Proposition 3.2

Let \((N,v,g)\) be a game with communication structure and \(x\in X(N,v,N/g)\). Then the following statements are equivalent:
  1. (1)

    \(x=\nu (N,v,g)\).

     
  2. (2)

    For all \(\alpha \in \mathbb{R }\) the following property is satisfied: If \(y\in \mathbb{R }^N\) satisfies \(y(R)=0\) for all \(R\in N/g\), and \(y(S)\ge 0\) for all \(S\in \mathcal{D}(\alpha ,x,v,g)\), then \(y(S)=0\) for all \(S\in \mathcal{D}(\alpha ,x,v,g)\).

     
  3. (3)

    For all \(\alpha \in \mathbb{R }\), the set \(\mathcal{D}(\alpha ,x,v,g)\cup N/g\) is balanced.

     

The proof of Proposition 3.2 is similar to the proof of the equivalence of the statements (1)–(3) in Remark 3.1 and, hence, it is omitted.

In view of Remark 2.3, the statements (2) and (3) of Proposition 3.2 may be reformulated as follows. For any \(R\in N/g\),
  1. (2’)

    For all \(\alpha \in \mathbb{R }\) the following property is satisfied: If \(y\in \mathbb{R }^R\) satisfies \(y(R)=0\) and \(y(S)\ge 0\) for all \(S\in \mathcal{D}(\alpha ,x_R,v,g^R)\), then \(y(S)=0\) for all \(S\in \mathcal{D}(\alpha ,x_R,v,g^R)\).

     
  2. (3’)

    For all \(\alpha \in \mathbb{R }\) the set \(\mathcal{D}(\alpha ,x_R,v,g^R)\cup \{R\}\) is balanced over \(R\).

     
Here the function \(v\) in \(\mathcal{D}(\alpha ,x_R,v,g^R)\) is the coalition function of the subgame \((R,v)\) of \((N,v)\).
Let \(\Delta ^{\mathrm{cmm}}_U\) and \(\Delta ^{\mathrm{clt}}_U\) be the sets of games with communication structures and of games with coalition structures, respectively, and let N be a coalition. To any coalition structure \(\mathcal{R}\) for N we may assign its associated graph \(g=G(\mathcal{R})\) defined by \(g=\{\{i,j\}\mid i,j\in R, i\ne j, \text{ for } \text{ some } R\in \mathcal{R}\}\). Thus, to any game with coalition structure, \((N,v,\mathcal{R})\), we may assign its associated game with communication structure, \((N,v,G(\mathcal{R}))\). We conclude that
$$\begin{aligned} \Delta ^{\mathrm{clt}}_U\hookrightarrow \Delta ^{\mathrm{cmm}}_U \text{ defined } \text{ by } (N,v,\mathcal{R})\mapsto \left( N,v,G(\mathcal{R})\right) \end{aligned}$$
is an embedding and we write “\(\Delta ^{\mathrm{clt}}_U\subseteq \Delta ^{\mathrm{cmm}}_U\)”.

The following example shows that there is \((N,v,\mathcal{R})\in \Delta ^{\mathrm{clt}}_U\) such that \(\nu (N,v,\mathcal{R})\) does not coincide with \(\nu (N,v,G(\mathcal{R}))\).

Example 3.3

Let \(N=\{1,2,3\},\mathcal{R}=\{\{1,2\},\{3\}\}\), and \((N,v)\) be defined by \(v(\{1,3\})=2\) and \(v(S)=0\) for all \(S\in 2^N{\setminus }\{\{1,3\}\}\). We may easily conclude from Proposition 3.2(3) that \(\nu (N,v,G(\mathcal{R}))=(0,0,0)\). Indeed, as \(\{1,3\}\) is not connected, all connected coalitions have the excess of 0. Let \(x=\nu (N,v,\mathcal{R})\). As \(x\) is a preimputation, \(x=(t,-t,0)\) for some \(t\in \mathbb{R }\). By (3) of Remark 3.1, \(t=1\) so that
$$\begin{aligned} \nu (N,v,G(\mathcal{R}))=(0,0,0)\ne (1,-1,0)=\nu (N,v,\mathcal{R}). \end{aligned}$$

Remark 3.4

Nevertheless, there is the following relation between the prenucleoli of games with communication structures and games with coalition structures. Let \((N,v,\mathcal{R})\in \Delta ^{\mathrm{clt}}_U\). Define \((N,w)\) by
$$\begin{aligned} w(S)=\sum _{R\in \mathcal{R}}v(S\cap R) \quad \text{ for } \text{ all } S\subseteq N. \end{aligned}$$
(3.1)
Then
$$\begin{aligned} \nu \left( N,w,\mathcal{R}\right) =\nu \left( N,v,G(\mathcal{R})\right) . \end{aligned}$$
(3.2)
In order to show (3.2), we apply the well-known reduced game property for the prenucleolus of games with coalition structures:
$$\begin{aligned} \text{ If } (N,v,\mathcal{R})\!\in \!\Delta ^{\mathrm{clt}}_U,x\!=\!\nu (N,v,\mathcal{R}), \text{ and } \emptyset \!\ne \! S\!\subseteq \! N, \text{ then } x_S\!=\!\nu (S,v^{S,x}_\mathcal{R},\mathcal{R}^S).\qquad \,\,\, \end{aligned}$$
(3.3)
Indeed, let \(R\in \mathcal{R}\) and \(x=\nu (N,w,\mathcal{R})\). By (3.3), \(x_R=\nu (R,w^{R,x}_\mathcal{R})\). Now, \(w^{R,x}_\mathcal{R}(R)=v(R)\) and, for any \(\emptyset \ne S\subsetneqq R,\,w^{R,x}_\mathcal{R}(S)=v(S)+c\), where \(c=\max _{Q\subseteq N{\setminus }R}w(Q)-x(Q)\). Hence, for any \(\alpha \in \mathbb{R },\,\mathcal{D}(\alpha +c,x_R,w^{R,x}_\mathcal{R})\cup \{\emptyset ,R\}=\mathcal{D}(\alpha ,x_R,v)\cup \{\emptyset ,R\}\) so that the proof is finished by Remarks 3.1(3) and 2.3.

4 Characterization of the prenucleolus for games with communication structures

Let \(\Delta \subseteq \Delta ^{\mathrm{cmm}}_U\). A solution on \(\Delta \) is a function \(\sigma \) which associates with any \((N,v,g)\in \Delta \) a subset \(\sigma (N,v,g)\) of \(X^*(N,v,N/g)\). A solution \(\sigma \) on \(\Delta \) satisfies
  1. (1)

    efficiency (EFF) if \(\sigma (N,v,g)\in X(N,v,N/g)\) for every game \((N,v,g)\in \Delta \);

     
  2. (2)

    covariance under strategic equivalence (COV) if the following condition is satisfied for all \((N,v,g)\) and all \((N,w,g)\in \Delta \): If \(\alpha >0,\beta \in \mathbb{R }^N\), and \(w=\alpha v+\beta \), then \(\sigma (N,w,g)=\alpha \sigma (N,v,g)+\beta ;\)

     
  3. (3)

    anonymity (AN) if the following condition is satisfied: If \((N,v,g)\in \Delta ,\pi :N\rightarrow U\) is an injection, and if \((\pi (N),\pi v,\pi g)\in \Delta \), then \(\sigma (\pi (N),\pi v,\pi g) =\pi (\sigma (N,v,g))\), where \((\pi v)(\pi (S))=v(S)\) for all \(S\subseteq N,\,\pi g=\{\{\pi (i),\pi (j)\}\mid \{i,j\}\in g\}\), and \(\pi (x)=y\in \mathbb{R }^{\pi (N)}\) is defined by \(y_{\pi (i)}=x_{i}\) for all \(x\in \mathbb{R }^N\) and all \(i\in N\);

     
  4. (4)

    singlevaluedness (SIVA) if \(|\sigma (N,v,g)|=1\) for all \((N,v,g)\in \Delta \);

     
  5. (5)

    the reduced game property (RGP\(^{\mathrm{cmm}}\)) if the following condition is satisfied for all \((N,v,g)\in \Delta \): If \(x\in \sigma (N,v,g)\) and \(\emptyset \ne S\subseteq N\), then \((S,v^{S,x}_g,g^S)\in \Delta \) and \(x_S\in \sigma (S,v^{S,x}_g,g^S)\).

     
We recall that a solution \(\sigma \) on a set \(\Delta \subseteq \Delta ^{\mathrm{clt}}_U\) satisfies the reduced game property in the sense of Davis and Maschler (\(\text{ RGP }^{\mathrm{clt}}\)) if it satisfies the property that differs from \(\text{ RGP }^{\mathrm{cmm}}\) only inasmuch as \(g\) is replaced by \(\mathcal{R}\) in the displayed part of (5) wherever it occurs [cf. (3.3)].

Remark 4.1

The prenucleolus on \(\Delta ^{\mathrm{clt}}_U\) is the unique solution that satisfies COV, AN, SIVA, and \(\text{ RGP }^{\mathrm{clt}}\), provided that \(|U|=\infty \). Sobolev (1975) proved this famous result for the restricted set of games “without coalition structures”, that is, for \(\Gamma _U=\{(N,v,\mathcal{R})\in \Delta ^{\mathrm{clt}}_U\mid \mathcal{R}=\{N\}\}\), but his proof may easily be extended to \(\Delta ^{\mathrm{clt}}_U\) [see Peleg and Sudhölter (2007)].

Lemma 4.2

The prenucleolus on \(\Delta ^{\mathrm{cmm}}_U\) satisfies COV, AN, SIVA, and \(\mathrm{RGP}^{\mathrm{cmm}}\).

Proof

COV, AN, and SIVA are immediate. In order to show \(\text{ RGP }^{\mathrm{cmm}}\), let \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U,\,\emptyset \ne S\subseteq N,\,x=\nu (N,v,g),\,w=v^{S,x}_g,\,R^{\prime }\in S/g^S,\,\alpha \in \mathbb{R }\), and \(y_{R^{\prime }}\in \mathbb{R }^{R^{\prime }}\) such that \(y(R^{\prime })=0\) and \(y(T)\ge 0\) for all \(T\in \mathcal{D}(\alpha ,x_{R^{\prime }},w,g^{R^{\prime }})\). Let \(R\in N/g\) such that \(R^{\prime }=R\cap S\). Then
$$\begin{aligned} \left\{ T\cap R^{\prime }\mid T\in \mathcal{D}(\alpha ,x_R,v,g^R), \emptyset \!\ne \! T\cap R\ne R^{\prime }\right\} \!=\!\mathcal{D}\left( \alpha ,x_{R^{\prime }},w,g^{R^{\prime }}\right) \big \backslash \left\{ \emptyset ,R^{\prime }\right\} .\nonumber \\ \end{aligned}$$
(4.1)
Let \(y_R=(y_{R^{\prime }},0_{R{\setminus }R^{\prime }})\). Then \(y_R\in \mathbb{R }^R,y(R)=0\), and, by (4.1), \(y(T)\ge 0\) for all \(T\in \mathcal{D}(\alpha ,x_R,v,g^R)\). By (2) of Proposition 3.2, \(y(T)=0\) for all \(T\in \mathcal{D}(\alpha ,x_R,v,g^R)\). Therefore \(y(T)=0\) for all \(T\in \mathcal{D}(\alpha ,x_{R^{\prime }},w,g^{R^{\prime }})\) and Proposition 3.2 completes the proof. \(\square \)

Lemma 4.3

If \(\sigma \) is a solution on \(\Delta ^{\mathrm{cmm}}_U\) that satisfies COV, SIVA, and \(\mathrm{RGP}^{\mathrm{cmm}}\), then \(\sigma \) satisfies EFF.

The proof of Lemma 4.3 is similar as in the classical case [where \(\Delta ^{\mathrm{cmm}}_U\) is replaced by \(\Delta ^{\mathrm{clt}}_U\) and \(\text{ RGP }^{\mathrm{cmm}}\) is replaced by \(\text{ RGP }^{\mathrm{clt}}\), see, e.g., Peleg and Sudhölter (2007, Lemma 6.2.11)] and, hence, it is skipped.

In order to modify Sobolev’s ingenious characterization proof of the prenucleolus also to games with communication structures, we now introduce one new additional property. This property first requires that the solution to a game is determined by the restriction of the coalition function to the connected coalitions. Secondly, this property requires that the solution to a game whose communication structure is connected coincides with the solution to the game with unrestricted cooperation in which the virtual worth of each formerly disconnected coalition is replaced by some small “real” worth. The formal definition of this property is as follows. For \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\) and \(\beta \in \mathbb{R }\) let \(\Delta ^\beta _{N,v,g}\) denote the set of all \((N,w,G(N/g))\in \Delta ^{\mathrm{cmm}}_U\) that satisfy, for all \(\emptyset \ne S\subseteq N\),
$$\begin{aligned} w(S)=v(S)\quad \text{ for } \text{ all } S\in \mathcal{S}_{N,g}\quad \text{ and }\quad w(S)\le \beta , \text{ otherwise }. \end{aligned}$$

Definition 4.4

A solution \(\sigma \) on \(\Delta ^{\mathrm{cmm}}_U\) satisfies independence of irrelevant connections (IIC) if for any \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\) and any \(x\in \sigma (N,v,g)\) there exists \(\beta \in \mathbb{R }\) such that \(x\in \sigma (N,w,G(N/g))\) for all \((N,w,G(N/g))\in \Delta ^\beta _{N,v,g}\).

Thus, IIC requires from \(\sigma \) that, if two games with communication structures \((N,v,g)\) and \((N,v^{\prime },g)\) coincide on all connected coalitions, then \(\sigma (N,v,g)=\sigma (N,v^{\prime },g)\). Moreover, if \((N,g)\) is connected, then each element of the solution to the game belongs to the solution to the game with unrestricted cooperation provided that the worth of each formerly disconnected coalition is small enough. An interpretation of this property is as follows. In our model, a disconnected coalition S is not allowed or able to form. But instead of prohibiting S to form we may prevent S from forming by charging S sufficiently if it forms nevertheless. I.e., if we charge S a sufficiently large “fee” for forming, then it may be expected that the members of S decide not to form the coalition so that the new connectedness of S becomes irrelevant. Formally this charging of a fee is modeled by replacing the former virtual worth of S by a sufficiently small “real” worth of S. Hence, S does not form in any case so that both cases (where S is not allowed to form or where S has a sufficiently small worth so that it does not form and its connectedness becomes, hence, irrelevant) may be regarded as similar. A solution that satisfies IIC takes care of this similarity: If a proposal belongs to the solution to the game with communication structure, then it also belongs to the solution to any of the corresponding games with unrestricted cooperation provided that the formerly disconnected coalitions are sufficiently charged for the right to form; i.e., if the connectedness of S is irrelevant.

Lemma 4.5

The prenucleolus on \(\Delta ^{\mathrm{cmm}}_U\) satisfies IIC.

Proof

Let \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U,\,x=\nu (N,v,g)\),
$$\begin{aligned} \beta <\min _{S\in \mathcal{S}_{N,g}}e(S,x,v)+ \min _{S\subseteq N}x(S), \end{aligned}$$
(4.2)
and \((N,w)\) satisfy \(w(S)=v(S)\) for all coalitions that are connected by g and \(w(S)\le \beta \) for all non-connected coalitions. Let
$$\begin{aligned} \gamma =\min _{S\in \mathcal{S}_{N,g}}e(S,x,v). \end{aligned}$$
(4.3)
Then \(e(S,x,w)\ge \gamma \) for all \(S\in \mathcal{S}_{N,g}\cup \{\emptyset \}\) and \(e(T,x,w)<\gamma \) for all other \(T\in 2^N\). Hence, for any \(\alpha \ge \gamma ,\,\mathcal{D}(\alpha ,x,w)=\mathcal{D}(\alpha ,x,v,g)\) so that, by Proposition 3.2(3), \(\mathcal{D}(\alpha ,x,w)\cup N/g\) is balanced. Moreover, all singletons are connected by definition so that, for any \(T\subseteq N,\,\chi _T\) is in the linear span of \(\{\chi _S\mid S\in \mathcal{D}(\gamma ,x,w)\}\). It is straightforward to show [see, e.g., Peleg and Sudhölter (2007, Lemma 6.1.2)] that \(\{T\}\cup \mathcal{D}(\gamma ,x,w)\) is balanced. We conclude that \(\mathcal{D}(\alpha ,x,w)\cup N/g\) is balanced for any \(\alpha \in \mathbb{R }\) so that the proof is finished by (3) of Remark 3.1. \(\square \)

Now we are able to prove the main result of this section.

Theorem 4.6

Let \(|U|\) be infinite. Then there is a unique solution \(\sigma \) on \(\Delta ^{\mathrm{cmm}}_U\) that satisfies COV, AN, SIVA, \(\mathrm{RGP}^{\mathrm{cmm}}\), and IIC, and it is the prenucleolus.

Proof

By Lemmas 4.2 and 4.5 the prenucleolus satisfies the desired properties. Thus, it remains to prove the uniqueness part. Let \(\sigma \) be a solution on \(\Delta ^{\mathrm{cmm}}_U\) that satisfies COV, AN, SIVA, \(\text{ RGP }^{\mathrm{cmm}}\), and IIC, let \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\) and \(x=\nu (N,v,g)\). It remains to prove that \(\sigma (N,v,g)=\{x\}\). By COV, we may assume that \(x=0\in \mathbb{R }^N\). By IIC, we may assume that g is component-complete, i.e., \(g=G(N/g)\). By \(\text{ RGP }^{\mathrm{cmm}}\) we may assume that \(|N/g|=1\). Let \(\mathcal{R}=N/g\) and \((N,w)\) be defined by (3.1). As \(N/g=\{N\},\,w=v\). By Remark 2.3, \(\nu (N,v,g)=\nu (N,v,\mathcal{R})=\nu (N,w)\). Now, according to Sobolev (1975) there is a game \((M,u)\) with the following properties [see Peleg and Sudhölter (2007, Section 6.3) for an English version of Sobolev’s proof]:
  1. (1)

    \((M,u)\) is transitive, i.e., the symmetry group of \((M,u)\) is transitive.

     
  2. (2)

    \(N\subseteq M,u(M)=0\).

     
  3. (3)

    With \(y=0\in \mathbb{R }^M,\,u^{N,y}=u^{N,y}_{\{M\}}=v\).

     
By SIVA, \(\sigma (M,u,G(\{M\}))=\{z\}\) for some \(z\in \mathbb{R }^M\). By AN and the transitivity of the symmetry group of \((M,u,G(\{M\})),\,z_i=z_j\) for all \(i,j\in M\). By Lemma 4.3, \(z(M)=u(M)=0\), hence \(z=y\). As \(G(\{M\})^N=G(\{N\})\) is complete, \(\text{ RGP }^{\mathrm{cmm}}\) and SIVA yield \(\{y_N\}=\sigma (N,v,g)\). \(\square \)

5 On the logical independence of the axioms

This section serves to show that each of the first four axioms in Theorem 4.6 is logically independent of the remaining axioms and that this theorem is no longer valid if the infinity assumption on the cardinality of \(U\) is deleted. For each \(k=1,\ldots ,5\), we construct a solution \(\sigma ^k\) that exclusively violates the \(k\)th axiom.

The “equal split solution” \(\sigma ^1\) that assigns to each \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\) the unique element \(x\in \mathbb{R }^N\) defined by \(x_i=\frac{v(R)}{|R|}\) for all \(i\in R\in N/g\) satisfies AN, SIVA, \(\text{ RGP }^{\mathrm{cmm}}\), and IIC, but does not coincide with the prenucleolus, hence violates COV.

We now construct an example of a solution that satisfies SIVA, COV, \(\text{ RGP }^{\mathrm{cmm}}\), and IIC, but violates AN. For this purpose we use the notation \(t_+=\max \{0,t\}\) for any \(t\in \mathbb{R }\) and define, for any \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\) (see Sect. 2 for the definition of the general nucleolus),
$$\begin{aligned} \mathcal{C}_+(N,v,g)=\mathcal{N}((e(S,\cdot ,v)_+)_{S\in \mathcal{S}_{N,g}},X^*(N,v,N/g)). \end{aligned}$$
(5.1)
That is, with \(x\!=\!\nu (N,v,g)\) we have \(y\in \mathcal{C}_+(N,v,g)\) iff \(y\!\in \!\mathbb{R }^N\) and \(e(S,y,v)_+\!=\!e(S,x,v)_+\) for all coalitions \(S\) that are connected by \(g\). Applied to a game \((N,v)\), this solution, i.e., \(\mathcal{C}_+(N,v)\!=\!\mathcal{C}_+(N,v,G(\{N\}))\), is called the positive core (Orshan and Sudhölter 2010) of \((N,v)\). As singletons are connected, \(\mathcal{C}_+(N,v,g)\) is a nonempty compact convex polyhedral set. Select any total order \(\succeq \) of \(U\) and define
$$\begin{aligned} \sigma ^2(N,v,g)=\left\{ x\in \mathcal{C}_+(N,v,g)\mid x\succeq _{lex} y \text{ for } \text{ all } y\in \mathcal{C}_+(N,v,g)\right\} , \end{aligned}$$
where \(\succeq _{lex}\) is the lexicographic order on \(\mathbb{R }^N\) induced by \(\succeq \), i.e., if \(x,y\in \mathbb{R }^N\), then \(x\succeq _{lex}y\) is defined by the requirement that, if \(y_i>x_i\) for some \(i\in N\), then there exists \(j\in N\) with \(x_j>y_i\) and \(j\succeq i\). Clearly \(\sigma ^2\) satisfies COV and SIVA, and it is straightforward to verify IIC. The proof of \(\text{ RGP }^{\mathrm{cmm}}\) is a straightforward generalization of the proof in the classical case [see, e.g., Peleg and Sudhölter (2007, Lemma 6.3.15)], and hence, it is skipped.

The positive core \(\sigma ^3=\mathcal{C}_+\) satisfies all axioms except SIVA.

In order to give an example of a solution \(\sigma ^4\) that satisfies COV, AN, SIVA, and IIC, but violates \(\text{ RGP }^{\mathrm{cmm}}\), we generalize a well-known solution concept for TU games (Driessen and Funaki (1991) called it the center of the imputation set) to TU games with communication structures. For \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\) let \(\sigma ^4(N,v,g)=\{x\}\) be defined by the requirement that \(x\in \mathbb{R }^N\) is given by
$$\begin{aligned} x_i=v(\{i\})+\frac{v(R)-\sum _{j\in R}v(\{j\})}{|R|} \text{ for } \text{ all } i\in R \text{ and } \text{ all } R\in N/g. \end{aligned}$$
Clearly, \(\sigma ^4\) satisfies COV, AN and SIVA. As \(w(\{i\})=v(\{i\})\) and \(w(R)=v(R)\) for all \(i\in R\in N/g\) for all \((N,w,N,G(N/g))\in \Delta ^\beta _{N,v,g}\), it also satisfies and IIC. Three-person examples show that \(\sigma ^4(N,v,g)\) may not coincide with \(\nu (N,v,g)\) so that \(\sigma ^4\) violates \(\text{ RGP }^{\mathrm{cmm}}\).

For completeness reasons, it should be remarked that Theorem 4.6 is no longer valid if the infinity assumption of \(U\) is deleted. Indeed, if \(4\le |U|<\infty \), then the example \(\sigma \) defined by Peleg and Sudhölter (2007, Remark 6.3.3) may easily be extended to as solution on \(\Delta ^{\mathrm{cmm}}_U\) that satisfies all five axioms but does not coincide with the prenucleolus on games with communication structures.

Finally, it should be noted that it is not known if IIC is logically independent of the remaining axioms in Theorem 4.6.

6 The prekernel

The prekernel introduced by Maschler et al. (1972) [see also Davis and Maschler (1965)] may also be generalized to games with communication structures. For \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U,\,k,\ell \in N, k\ne \ell \), and \(x\in \mathbb{R }^N\) let \(s_{k\ell }(x,v,g)\) denote the maximum surplus of \(k\) over \(\ell \) at \(x\), i.e.,
$$\begin{aligned} s_{k\ell }(x,v,g)=\max \left\{ e(S,x,v)\mid S\in \mathcal{S}_{N,g},k\in S\not \ni \ell \right\} . \end{aligned}$$
Then the prekernel of \((N,v,g)\) is the set
$$\begin{aligned} \mathcal{PK}(N,v,g)&= \{x\in X(N,v,N/g)\mid s_{k\ell }(x,v,g)=s_{\ell k}(N,v,g)\\&\text{ for } \text{ all } k,\ell \in R\in N/g, k\ne \ell \}. \end{aligned}$$

Remark 6.1

By literally copying the proof for games with coalition structures [see Peleg and Sudhölter (2007, Theorem 5.1.17)] it may be shown that \(\nu (N,v,g)\in \mathcal{PK}(N,v,g)\) for any game with communication structure \((N,v,g)\). Moreover, similarly to the classical context it may be shown that \(\mathcal{PK}\) satisfies COV, AN, and \(\text{ RGP }^{\mathrm{cmm}}\).

Proposition 6.2

The prekernel on the set of games with communication structures satisfies IIC.

Proof

Let \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\), and \(x\in \mathcal{PK}(N,v,g)\). Let \(\beta \) satisfy (4.2), let \(\gamma \) be defined by (4.3), and let \((N,w,G(N/g))\in \Delta ^\beta _{N,v,g}\) so that \(e(S,x,w)\ge \gamma \) for all coalitions \(S\) that are connected by \(g\) and \(e(T,x,w)<\gamma \) for all other coalitions. Note that \(N/g=N/G(N/g)\). Hence, for any \(k,\ell \in R\in N/g\) with \(k\ne \ell ,\,s_{k\ell }(x,w,G(N/g))\) can only be attained by a connected coalition so that \(s_{k\ell }(x,w,G(N/g))=s_{k\ell }(x,v,g)\). Thus, \(x\in \mathcal{PK}(N,w,G(N/g))\). \(\square \)

Let \((N,v,g)\in \Delta _U^{\mathrm{cnm}}\) and \(k,\ell \in N\). We say that \(k\) and \(\ell \) are substitutes w.r.t. \((N,v,g)\) if, for all \(S\subseteq N{\setminus }\{k,\ell \}\), (a) \(v(S\cup \{k\})=v(S\cup \{\ell \})\) and (b) \(S\cup \{k\}\in \mathcal{S}_{N,g}\) if and only if \(S\cup \{\ell \}\in \mathcal{S}_{N,g}\). Note that (b) is equivalent to the requirement that, for any \(j\in N{\setminus }\{k,\ell \},\,\{j,k\}\in g\) if and only if \(\{j,\ell \}\in g\).

We may now adjust Peleg’s (1986) axiomatization of the prekernel on games with coalition structures to games with communications structures. To this end let \(\sigma \) be a solution on \(\Delta ^{\mathrm{cmm}}_U\). Then \(\sigma \) satisfies
  1. (1)

    non-emptiness (NE) if \(\sigma (N,v,g)\ne \emptyset \) for all \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\);

     
  2. (2)

    the restricted equal treatment property (RETP) if, for any \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\) and any substitutes \(k,\ell \in N\) w.r.t. \((N,v,g),\,x_k=x_\ell \) for all \(x\in \sigma (N,v,g)\);

     
  3. (3)

    the converse reduced game property (\(\text{ CRGP }^{\mathrm{cmm}}\)) if the following condition holds for any \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\) with \(|N|\ge 2\): If \(x\in X(N,v,N/g)\) and \(x_S\in \sigma (S,v^{S,x}_g,g^S)\) for all \(S\subseteq R\) with \(|S|=2\) for all \(R\in N/g\), then \(x\in \sigma (N,v,g)\).

     
The proof of the following theorem is similar to Peleg’s proof so that it is skipped.

Theorem 6.3

There is a unique solution on \(\Delta ^{\mathrm{cmm}}_U\) that satisfies NE, EFF, COV, RETP, \(\mathrm{RGP}^{\mathrm{cmm}}\), and \(\mathrm{CRGP}^{\mathrm{cmm}}\), and it is the prekernel.

As in the classical case, each of the employed axioms in Theorem 6.3 is logically independent of the remaining axioms, provided \(|U|\ge 4\). Indeed, the empty solution exclusively violates NE, the “non-efficient prekernel” (defined by
$$\begin{aligned} \begin{aligned} (N,v,g)&\mapsto \{x\in X^*(N,v,N/g)\mid s_{k\ell }(x,v,g)=s_{\ell k}(N,v,g)\\&\qquad \text{ for } \text{ all } k,\ell \in R\in N/g, k\ne \ell \}) \end{aligned} \end{aligned}$$
exclusively violates EFF, the equal split solution \(\sigma ^1\) exclusively violates COV, and the “preimputation solution” (defined by \((N,v,g)\mapsto X(N,v,N/g)\)) exclusively violates RETP.
Note that, for any \((N,v,g)\in \Delta ^{\mathrm{cmm}}_U\) with \(|R|\le 2\) for all \(R\in N/g,\,\mathcal{PK}(N,v,g)=\{\nu (N,v,g)\}\). We say that a solution \(\sigma \) is a standard solution \(\sigma (N,v,g)=\{\nu (N,v,g)\}\) for any game with communication structure that satisfies \(|R|\le 2\) for all \(R\in N/g\). Note that the Myerson value, defined by \(\mathcal{M}(N,v,g)=\{\phi (N,v/g)\}\) where \(\phi \) denotes the Shapley value in the sense of Shapley (1953) and
$$\begin{aligned} (v/g)(S)=\sum _{T\in S/g}v(T), \end{aligned}$$
(6.1)
is a standard solution as well as the solution defined by \((N,v,g)\mapsto \{\nu (N,v/g,N/g)\}\). Already for games with \(|R|=3\) for some connected component, \(\nu (N,v,g)\) may differ from \(\nu (N,v/g,N/g)\) as the following example shows. Let \(N=\{1,2,3\},\,(N,v)\) be defined by \(v(S)=6\) for all \(\emptyset \ne S\subseteq N\), and let \(g=\{\{1,2\},\{2,3\}\}\). Then \(N/g=\{N\}\) and \(S=\{1,3\}\) is the unique non-connected coalition so that \((v/g)(S)=12\) and \((v/g)(T)=v(T)\) for all \(T\in 2^N{\setminus }\{S\}\). It is straightforward to verify that
$$\begin{aligned} \nu (N,v,g)=(2,2,2)\ne (3,0,3)=\nu (N,v/g)=\nu (N,v/g,N/g). \end{aligned}$$
Clearly, both aforementioned standard solutions (i.e., \(\mathcal{M}\) and \((N,v,g)\mapsto \{\nu (N,v/g,N/g)\}\)) satisfy EFF, RETP, and COV. Hence, the union of the prekernel with any other standard solution that satisfies EFF, RETP, and COV, exclusively violates \(\text{ RGP }^{\mathrm{cmm}}\), e.g., the solution defined by \((N,g,v)\mapsto \mathcal{PK}(N,v,g)\cup \{\nu (N,v/g,N/g)\}\).

Finally, the prenucleolus exclusively violates \(\text{ CRGP }^{\mathrm{cmm}}\).

7 Extension to games with conference structures

According to Myerson (1980) a conference structure is a pair \((N,Q)\) where N is a coalition and \(Q\subseteq 2^N\) satisfies \(|R|\ge 2\) for all \(R\in Q\). A game with conference structure is a triple \((N,v,Q)\) such that \((N,v)\) is a game and \((N,Q)\) is a conference structure. Let \(\Delta ^{\mathrm{cnf}}_U\) denote the set of games with conference structures. According to Myerson (1980, p. 178) “complete cooperation within the coalition \(S\)” is reflected by \(\{\{i,j\}\mid i,j\in S,i\ne j\}=g\) for a graphical communication structure and by \(\{T\subseteq S\mid |T|\ge 2\}=Q\) for a conference structure. So we identify a graph \(g\) with the conference structure of all coalitions that contain at least two elements and are connected by \(g\). This identification is in contrast to that of Albizuri and Zarzuelo (2009) who identify any graph \((N,g)\) with the conference structure \((N,Q)\) defined by \(Q=g\).

Thus, if \((N,g)\) is a communication structure, then the conference structure corresponding to \((N,g),\,(N,Q(g))\), is defined by
$$\begin{aligned} Q(g)=\left\{ S\subseteq N\mid |S|\ge 2 \text{ and } S \text{ is } \text{ connected } \text{ by } g\right\} . \end{aligned}$$
(7.1)
Hence, “\(\Delta ^{\mathrm{cmm}}_U\subseteq \Delta ^{\mathrm{cnf}}_U\)” throughout means the embedding
$$\begin{aligned} \Delta ^{\mathrm{cmm}}_U\hookrightarrow \Delta ^{\mathrm{cnf}}_U \text{ defined } \text{ by } (N,v,g)\mapsto \left( N,v,Q(g)\right) . \end{aligned}$$
Let \((N,v,Q)\) be a game with conference structure. We adopt the notion of connectedness from the aforementioned articles: Let \(S\subseteq N\). The elements \(i,j\in S\) are connected in \(S\) by \(Q\) if there exits \(S_1,\ldots ,S_\ell \in Q\) such that \(i\in S_1,j\in S_\ell , S_j\subseteq S\) for all \(j=1,\ldots ,\ell \), and \(S_t\cap S_{t+1}\ne \emptyset \) for all \(t=1,\ldots ,\ell -1\). Moreover, \(S/Q\) denotes the components of \(S\) w.r.t. \(Q\), and a coalition \(S\) that has only one component is called connected. Furthermore, \(\mathcal{S}_{N,Q}\) denotes the set of all coalitions in N that are connected by \(Q\).

A solution on \(\Delta \subseteq \Delta ^{\mathrm{cnf}}_U\) is a function \(\sigma \) which associates with any \((N,v,Q)\in \Delta \) a subset \(\sigma (N,v,Q)\) of \(X^*(N,v,N/Q)\). EFF, COV, AN, SIVA are defined similarly as in Sect. 4; just “graph \((N,g)\)” has to be replaced by “conference structure \((N,Q)\)”.

In order to define \(\text{ RGP }^{\mathrm{cnf}}\), we adopt the definition of the reduced conference structure of Albizuri and Zarzuelo (2009). Let \((N,Q)\) be a conference structure and \(\emptyset \ne S\subseteq N\). Then the reduced conference structure \((S,Q^S)\) is defined by
$$\begin{aligned} Q^S&= \left\{ S\cap \bigcup _{t=1}^\ell S_t~\Big |~\ell \in \mathbb{N },S_t, S_\ell \in Q, S_t\cap S_{t+1}\ne \emptyset \right. \nonumber \\&\quad \left. \left. \text{ for } \text{ all } t=1,\ldots ,\ell -1,\left| S\cap \bigcup _{t=1}^\ell S_t\right| \ge 2\right\} \right. . \end{aligned}$$
(7.2)
Note that \((N,Q^N)\) may not coincide with \((N,Q)\). Indeed, \(Q^N\) contains \(Q\) and all coalitions in N that are connected by \(Q\) and contain at least two elements. However, for any graph \((N,g),\,Q(g)=Q(g)^N\).

The coalition function of the reduced game with conference structure w.r.t. \(S\) and \(x\in \mathbb{R }^N,\,(S,v^{S,x}_Q,Q^S)\), may now be defined analogously to (2.3) by replacing \(g\) with \(Q\) whenever it occurs. Now, again by replacing \(g\) with \(Q\) whenever this is needed, \(\text{ RGP }^{\mathrm{cnf}}\) is defined analogously to \(\text{ RGP }^{\mathrm{cmm}}\) and IIC is generalized in a similar way. The prenucleolus of \((N,v,Q),\,\nu (N,v,Q)\), is defined by literally copying Definition 2.2 with the exception that “\(g\)” has to be replaced by “\(Q\)” wherever it occurs.

Now it is straightforward to generalize Theorem 4.6 to games with conference structures.

We should like to remark that there are examples of games with conference structures, \((N,v,Q)\), such that \(Q^N\ne Q\), whereas and \(v^{N,x}_Q=v,\,v^{S,x}_Q=v^{S,x}_{Q^N}\), and \(g^N=g\) for any \(x\in \mathbb{R }^N,\,(N,v,Q)\in \Delta ^{\mathrm{cnf}}_U,\,\emptyset \ne S\subseteq N\), and any graph \((N,g)\).

It should finally be remarked that the prekernel for games with communication structures together with Proposition 6.2 and Theorem 6.3 may be generalized to games with conference structures.

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of Applied MathematicsSaint-Petersburg State UniversitySaint-PetersburgRussia
  2. 2.Department of Business and Economics, COHEREUniversity of Southern DenmarkOdense MDenmark

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