The nucleolus and inheritance of properties in communication situations

This paper studies the nucleolus of graph-restricted games as an alternative for the Shapley value to evaluate communication situations. We focus on the inheritance of properties of cooperative games related to the nucleolus: strong compromise admissibility and compromise stability. These two properties allow for a direct, closed formula for the nucleolus. We characterize the families of graphs for which the graph-restricted games inherit these properties from the underlying games. Moreover, for each of these two properties, we characterize the family of graphs for which the nucleolus is invariant

acterized by Myerson (1980) and studied in several other contexts as well: hypergraphs (cf. van den Nouweland et al., 1992), union stable structures (cf. , antimatroids (cf. Algaba et al., 2004), bipartite graphs (cf. van den Brink & Pintér, 2015), two-level communication structures (cf. van den Brink et al., 2016) and communication situations in which the players have different bargaining abilities (cf. Manuel & Martín, 2021). Moreover, several studies are devoted to the inheritance of properties of cooperative games that are related to the Shapley value. In particular, Owen (1986) studied the inheritance of superadditivity, van den Nouweland and Borm (1991) studied convexity and Slikker (2000) studied, among others, average convexity. The inheritance of convexity is also studied in a unified approach by .
Also the nucleolus is studied in the context of communication situations. Potters and Reijnierse (1995) showed that the nucleolus is the unique element of the kernel if the communication graph is a tree. Reijnierse and Potters (1998) and Katsev and Yanovskaya (2013) studied the collection of coalitions that determine the nucleolus and prenucleolus, respectively. Khmelnitskaya and Sudhölter (2013) provided an axiomatic characterization of the prenucleolus for games with communication structures. Instead, we follow the lines initiated by Owen (1986) and focus on the inheritance of two properties of cooperative games that are related to the nucleolus. Moreover, we study the invariance of the nucleolus, that is, the feature that the nucleolus of the graph-restricted game equals the nucleolus of the underlying game of a communication situation.
For the inheritance, we concentrate on the properties strong compromise admissibility and compromise stability. In general, computing the nucleolus of a cooperative game is not straightforward. However, interestingly, for cooperative games satisfying strong compromise admissibility or compromise stability, there exists a direct, closed formula for the nucleolus, based on the Talmud rule for bankruptcy problems (cf. Quant et al., 2005) and (cf. Driessen, 1988). In particular, when these properties are inherited, computation of the nucleolus can still be facilitated. For strongly compromise admissible games it holds that the nucleolus coincides with the compromise value (cf. Tijs, 1981), which can be directly computed using the utopia vector, as shown by Driessen (1988). Furthermore, Driessen (1988) also showed that strongly compromise admissible games are characterized by their core allocations. The class of strongly compromise admissible games contains for example the class of simple games with one veto-player, while the larger class of compromise stable games contains several interesting classes of economic games, like big boss games (cf. Muto et al., 1988), clan games (cf. Potters et al., 1989) and bankruptcy games (cf. O'Neill, 1982) and (cf. Curiel et al., 1987).
With regard to strong compromise admissibility, we show that the graph-restricted game is strongly compromise admissible for every communication situation with an underlying strongly compromise admissible game, if the graph is biconnected. In fact, for every connected graph that is not biconnected, we explicitly construct a communication situation with an underlying strongly compromise admissible game such that the graph-restricted game is not strongly compromise admissible.
To ensure compromise stability for the graph-restricted game of a communication situation with an underlying strongly compromise admissible game, the connected graph needs to be biconnected or a star. Again, for every connected graph that is not biconnected and not a star, we explicitly construct a communication situation with an underlying strongly compromise admissible game such that the graph-restricted game is not compromise stable.
Finally, we show that the graph-restricted game is compromise stable for every communication situation with an underlying compromise stable game, if the graph is complete. For every connected graph that is not complete (and has at least four players), we explicitly con-struct a communication situation with an underlying compromise stable game such that the graph-restricted game is not compromise stable.
For the invariance of the nucleolus, we identify families of graphs for which it is guaranteed that the nucleolus of the graph-restricted game equals the nucleolus of the underlying game for communication situations with an underlying strongly compromise admissible or compromise stable game. We use the inheritance results to employ the direct formula for the nucleolus for both the graph-restricted game and the underlying game. We show that the nucleolus is invariant for all communication situations with an underlying strongly compromise admissible game, if the graph is biconnected. For every connected graph that is not biconnected, we construct a communication situation with an underlying strongly compromise admissible game such that the nucleolus of the graph-restricted game is not equal to the nucleolus of the underlying game. For communication situations with an underlying compromise stable game, we show that the connected graph needs to be complete in order to guarantee the invariance of the nucleolus. That is, we prove the invariance of the nucleolus if the graph is complete and for every connected graph that is not complete, we provide a communication situation with an underlying compromise stable game such that the nucleoli are different.
This paper is structured in the following way. Section 2 provides all relevant preliminaries on cooperative game theory and graph theory. Section 3 studies the inheritance of strong compromise admissibility and compromise stability. Section 4 studies the invariance of the nucleolus. Finally, Section 5 contains the concluding remarks.

Preliminaries
A (transferable utility) cooperative game is a pair (N , v) where N is a non-empty, finite set of players and v : 2 N → R is a characteristic function with v(∅) = 0. Here, 2 N is the collection of all subsets (called coalitions) of N and v(S) is the worth of coalition S ∈ 2 N , representing the joint monetary rewards this coalition can obtain on its own. The class of all cooperative games with player set N is denoted by T U N , and a cooperative game (N , v) is also denoted by v ∈ T U N .
For a cooperative game v ∈ T U N , the imputation set is given by the core (cf. Gillies, 1959) is given by and the core cover (cf. Tijs & Lipperts, 1981) is given by Quant et al., 2005); Driessen, 1988), but we adopt the terminology of Quant et al. (2005)].
Note that strong compromise admissibility implies compromise stability, compromise stability implies balancedness, and balancedness implies imputation admissibility. Moreover, for a cooperative game with two players, all notions are equivalent. For a three player game, only balancedness and compromise stability are equivalent, while all notions differ for games with more than three players.
Let v ∈ T U N be an imputation admissible game. The excess of a coalition S ∈ 2 N with respect to an imputation x ∈ I (v) is defined as E xc (S, x, v) = v(S) − i∈S x i , while the excess vector θ(x) ∈ R 2 |N | is defined as the vector consisting of the excesses in nonincreasing order, i.e. θ(x) k ≥ θ(x) k+1 for all k ∈ {1, . . . , 2 |N | − 1}. The nucleolus (cf. Schmeidler, 1969) nuc(v) ∈ R N is the unique imputation for which θ(nuc(v)) θ(x) for all x ∈ I (v), where denotes the lexicographical order. It is known that nuc(v) ∈ C(v) for all balanced games v ∈ T U N .
A collection B ⊆ 2 N \ {∅} is called balanced if there exists a function λ : B → R ++ such that S∈B:i∈S λ(S) = 1 for all i ∈ N . According to the Kohlberg criterion (cf. Kohlberg, 1971), for a balanced game v ∈ T U N and an imputation x ∈ I (v), it holds that x = nuc(v) if and only if the collection s and for all k ∈ {2, . . . , t(x)}: Here, t(x) ∈ N is the unique number such that B k (x, v) = ∅ for all k ∈ {1, . . . , t(x)} and B t(x)+1 (x, v) = ∅.
A bankruptcy problem (cf. O'Neill, 1982) is a triple (N , A, c) where N is a non-empty, finite set of players, A ∈ R + and c ∈ R N + consists of the claims of the players on A such that i∈N c i ≥ A. For a bankruptcy problem (N , A, c), the constrained equal awards rule (CEA) allocates CEA i (N , A, c) = min {α, c i } for all i ∈ N , where α ∈ R is such that i∈N min {α, c i } = A, while the Talmud rule (TAL) (cf. Aumann & Maschler, 1985) allocates For compromise stable games and strongly compromise admissible games, the nucleolus can be described by a direct, closed formula.
(ii) If v is strongly compromise admissible, then, for all i ∈ N , A graph is a pair (N , E), where N is a non-empty, finite set of players, with |N | ≥ 3 and E ⊆ {i, j} i, j ∈ N , i = j is a finite set of edges. For a graph (N , E) and a subset of players S ∈ 2 N \ {∅}, the induced subgraph on S is defined as the graph (S, E S ), where E S = {i, j} ∈ E i, j ∈ S . A path in a graph (N , E) is defined as a sequence of players Note that every complete graph is biconnected and that every biconnected graph is connected. Also a star is connected. For a graph (N , E), a component C ∈ 2 N \ {∅} is defined as a maximal (inclusion-wise) subset of players such that the induced subgraph (C, E C ) is connected. For a graph (N , E) and a subset of players S ∈ 2 N \ {∅}, let S/E denote the set of all components in the induced subgraph (S, E S ).
For a graph (N , E) and a cooperative game v ∈ T U N , the graph-restricted game v E ∈ T U N is (cf. Myerson, 1977) Note that for a connected graph (N , E) and a cooperative game v ∈ T U N , it holds that v E (N ) = v(N ), since N /E = {N }. Next, we formally define a communication situation, using a slightly modified version of the definition as stated by Myerson (1977).
The set of all communication situations is denoted by CS N .
Note that, for any (connected) graph (N , E) and any cooperative game v ∈ T U N that is superadditive, it holds that v E (S) ≤ v(S) for all S ∈ 2 N . In other words, the (additional) inequality in Definition 2.1 is satisfied if the underlying game is superadditive.

Inheritance of properties
This section studies the inheritance of two properties: strong compromise admissibility and compromise stability. For each of these properties, we identify the family of graphs for which the inheritance of this property from the underlying game to the graph-restricted game is guaranteed. First, we want to remark that both balancedness and imputation admissibility are always inherited. That is, for every communication situation with an underlying balanced (imputation admissible) game it holds that the graph-restricted is balanced (imputation admissible) as well. This was first observed by van den Nouweland and Borm (1991).
For strong compromise admissibility, the graph needs to be biconnected in order to guarantee the inheritance. In Theorem 3.1, we show that for every communication situation with an underlying strongly compromise admissible game it holds that the graph-restricted game is strongly compromise admissible as well, if the graph is biconnected. Moreover, for every connected graph that is not biconnected, we construct a communication situation with an underlying strongly compromise admissible game such that the graph-restricted game is not strongly compromise admissible. Thus, we can conclude that the family of biconnected graphs guarantees the inheritance of strong compromise admissibility.
In the proof of Theorem 3.1, we use the following lemma.
for all i ∈ N . Using this, we have that Theorem 3.1 The following two statements hold: Here, the second inequality is due to the fact that v is strongly compromise admissible. Consequently, v E is strongly compromise admissible.
Consequently, v E 1 is not strongly compromise admissible. This finishes the construction of the communication situation (N , v 1 , E) ∈ CS N where v 1 is strongly compromise admissible, while v E 1 is not strongly compromise admissible. Theorem 3.1 characterizes the family of graphs that guarantees the inheritance of strong compromise admissibility from the underlying game to the graph-restricted game. The next theorem characterizes the family of graphs that guarantees compromise stability for the graphrestricted game of any communication situation with an underlying strongly compromise admissible game. This family includes of course all biconnected graphs (cf. Theorem 3.1) and, in addition, it is seen that it contains all stars.
Theorem 3.2 The following two statements hold:

be a communication situation with (N , E) is biconnected or a star
and v is strongly compromise admissible. Then, v E is compromise stable; (ii) Let (N , E) be a connected graph that is not biconnected and not a star. Then, there exists a communication situation (N , v, E) It can be readily checked that M(v 2 ) = m(v 2 ) = (2, 2, 2, 2, 0, . . . , 0).
Hence, v 2 is strongly compromise admissible.
We show that v E 2 is not compromise stable, by showing that For all S ∈ 2 N with |S| ≤ n − 2 and {1, 2} ⊆ S, we have that v E 2 (S) ≤ v 2 (S) = 3 and 2 ∈ S, and, consequently, Similarly, one can show that m 2 (v E 2 ) ≤ 1 and thus 2 is not compromise stable. This finishes the construction of the communication situation (N , v 2 , E) ∈ CS N where v 2 is strongly compromise admissible, while v E 2 is not compromise stable.
The next theorem provides a characterization of the family of graphs for which the inheritance of compromise stability is guaranteed. Note that for a three player game, balancedness and compromise stability are equivalent. Therefore, for every communication situation with an underlying compromise stable game with three players it holds that the graph-restricted is compromise stable as well, since balancedness is always inherited. For more than three players, the graph needs to be complete in order to guarantee compromise stability for the graph-restricted game, for every communication situation with an underlying compromise stable game.

Invariance of the nucleolus
In this section, we study the invariance of the nucleolus. That is, we focus on necessary and sufficient conditions on a communication situation such that the nucleolus of the graphrestricted game equals the nucleolus of the game underlying the communication situation. In particular, we reconsider communication situations with an underlying strongly compromise admissible game and an underlying compromise stable game, respectively. We start out with the former. Recall from Theorem 3.1 that the family of biconnected graphs guarantees the inheritance of strong compromise admissibility from the underlying game to the graph-restricted game. For communication situations with an underlying strongly compromise admissible game, we show that this family of biconnected graphs also guarantees the invariance of the nucleolus. Moreover, for every connected graph that is not biconnected, we explicitly construct a communication situation with an underlying strongly compromise admissible game for which the nucleolus of the graph-restricted game is not equal to the nucleolus of the underlying game. For this, we benefit from the construction in the proof of Theorem 3.1.

Theorem 4.1 The following two statements hold: (i) Let (N , v, E) ∈ CS N be a communication situation with (N , E) is biconnected and v is strongly compromise admissible. Then, nuc(v E ) = nuc(v); (ii) Let (N , E) be a connected graph that is not biconnected. Then, there exists a communication situation (N , v, E) ∈ CS N where v is strongly compromise admissible such that nuc(v E ) = nuc(v).
Proof (i) First, note that v E is strongly compromise admissible, according to part (i) of Theorem 3.1. Hence, by using Proposition 2.1, for all i ∈ N ,  (N , v 1 Recall that v 1 is strongly compromise admissible and since M(v 1 ) = (1, 0, 0, . . . , 0), we have that, using Proposition 2.1, nuc(v 1 ) = (1, 0, 0, . . . , 0).
and consequently, Hence, nuc(v E 1 ) = nuc(v 1 ). This concludes the construction of the communication situa- Next, we reconsider the class of communication situations with an underlying compromise stable game. We need a stronger condition than biconnectedness to guarantee the invariance of the nucleolus for this class, since this class of communication situations is larger than the class of communication situations with an underlying strongly compromise admissible game. It turns out that to guarantee invariance, we need the strongest condition possible, a complete graph. For every connected graph that is not complete, one can construct a communication situation with an underlying compromise stable game for which the nucleolus is not invariant. This construction, however, is quite intricate.

Theorem 4.2
The following two statements hold:

be a communication situation with (N , E) is complete and v is compromise stable. Then, nuc(v E ) = nuc(v); (ii) Let (N , E) be a connected graph that is not complete. Then, there exists a communication situation
Proof (i) It is left for the reader (ii) We distinguish between two cases: either |N | = 3 or |N | ≥ 4. First, suppose that |N | = 3 and set N = {1, 2, 3}. Assume w.l.o.g. that Consequently, nuc(v E 1 ) = ( 1 2 , 1 2 , 0) = (1, 0, 0) = nuc(v 1 ). Secondly, suppose that |N | ≥ 4. Set N = {1, 2, 3, 4, . . . , n} and assume w.l.o.g. that {1, 2} / ∈ E and {1, 3} ∈ E. Reconsider the communication situation (N , v 3 Then, M(v 3 ) = (1, 2, 4, 3, 4, . . . , 4) and m(v 3 ) = (1, 2, 2, 0, 0, . . . , 0), and using Proposition 2.1, To show that nuc(v E 3 ) = nuc(v 3 ), we use the Kohlberg criterion and show that is not balanced. For this, we need to identify the coalitions (non-empty and not the grand coalition) with the highest excess. To structure this identification process, for S ∈ 2 N we distinguish between seven cases, in which players 1, 2 and 3 play an important role: Case i), vi) and vii) deal with all coalitions with exactly 1 or n − 1 players, respectively. Case i) also includes three 2-player coalitions. For the other coalitions, we distinguish whether {1, 2} ⊆ S (and 3 / ∈ S) or {1, 3} ⊆ S or neither of the two inclusions. In particular, case ii) deals with the 2-player coalitions that contains both players 1 and 2, but not 3 and case iii) deals with similar coalitions that contain at least 4 players. In case iv), {1, 3} ⊆ S and finally, case v) deals with all coalitions such that both {1, 2} S and {1, 3} S. Case (i) For this first case, we know that and hence, one readily checks that 3, if the induced subgraph on S is connected; 0, otherwise, and hence, and hence, since nuc i (v 3 ) > 0 for all i ∈ N . Subsequently, these coalitions can not be coalitions with the highest excess. Case (iv) For all S ∈ 2 N with 2 < |S| < n − 1 and {1, 3} ⊆ S, it holds that v E 3 (S) = 3 and hence, Case (vi) For all S ∈ 2 N with |S| = n − 1 and S = N \ { j} for j ∈ N \ {1, 2, 3, 4}, it holds that v E 3 (S) = 3, since {1, 3} ⊆ S and hence, , the worth of the coalition S in the graph-restricted game depends on whether the induced subgraph is connected or not: Consequently, In order to determine which of the above excesses is the highest, note that, if n ≥ 4, and, if n > 4, We may conclude that, if the induced subgraph on N \ {1} or the induced subgraph on N \ {2} is connected, the highest excess equals 0 and Clearly, for these cases, is not a balanced collection and nuc(v E 3 ) = nuc(v 3 ). Note that, if n = 4, it holds that the induced subgraph on N \ {1} or the induced subgraph on N \ {2} is connected, due to the connectedness of the graph and the fact that {1, 2} / ∈ E.

Concluding remarks
In this paper, we studied both the inheritance of strong compromise admissibility and compromise stability and the invariance of the nucleolus. With regard to the inheritance, the results are summarized in Table 1. Loosely speaking, in Theorem 3.1, we showed that strong compromise admissibility is always inherited from the underlying game to the graph-restricted game if and only if the graph is biconnected. Moreover, to go from strong compromise admissibility for the underlying game to compromise stability for the graph-restricted game, the graph needs to be either biconnected or a star, as shown in Theorem 3.2. Finally, in Theorem 3.3, we showed that compromise stability is always inherited if and only if there are only three players or the graph is complete. To finalize Table 1, it remains to check whether we can obtain a strongly compromise admissible graph-restricted game if the underlying game is compromise stable. Example 5.1 shows that this is not possible. More precisely, for every connected graph there exists a communication situation with a compromise stable underlying game such that the graph-restricted game is not strongly compromise admissible. Note that M(v 4 ) = (1, 1, 0, . . . , 0) and m(v 4 ) = (0, 0, 0, . . . , 0), and hence, CC(v 4 ) = ∅. With regard to the invariance of the nucleolus, Table 2 summarizes our results. For both properties, it identifies the weakest condition on the graph for which invariance of the nucleolus is guaranteed for all communication situations with an underlying game satisfying this property.
Interestingly, the condition on the graph with regard to compromise stability can be relaxed if we restrict attention to communication situations with an underlying simple game. A cooperative game v ∈ T U N is called simple if v(S) ∈ {0, 1} for all S ∈ 2 N , v(N ) = 1 and v(S) ≤ v(T ) for all S, T ∈ 2 N with S ⊆ T . Moreover, for a simple game v ∈ T U N , the set of veto-players is given by veto(v) = S ∈ 2 N v(S) = 1 .
For simple games, having veto-players is equivalent to balancedness, which in turn is equivalent to compromise stability. Moreover, if a simple game has veto-players, then the nucleolus divides the worth of the grand coalition equally among these veto-players. Note that if the game underlying a communication situation is simple, then the graph-restricted game is simple too.
If, in addition to compromise stability, we also require that the underlying game is simple, it turns out that the nucleolus is invariant for all such communication situations if the graph is biconnected. Furthermore, for every connected graph that is not biconnected, we construct a communication situation with an underlying game that is both compromise stable and simple for which the nucleolus of the graph-restricted game is not equal to the nucleolus of the underlying game. Again, we benefit from the construction in the proof of Theorem 3.1 (and Theorem 4.1).

Proposition 5.1 The following two statements hold: (i) Let (N , v, E) ∈ CS N be a communication situation with (N , E) is biconnected and v is both compromise stable and simple. Then, nuc(v E ) = nuc(v); (ii) Let (N , E) be a connected graph that is not biconnected. Then, there exists a communi-
cation situation (N , v, E) ∈ CS N where v is both compromise stable and simple such that nuc(v E ) = nuc(v).
Proof (i) It suffices to show that veto(v) = veto(v E ).
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