Properties of Solutions for Games on Union-Closed Systems

Abstract

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A solution for TU-games assigns a set of payoff distributions to every TU-game. In the literature, various models of games with restricted cooperation can be found where, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper, we consider games on a union-closed system where the set of feasible coalitions is closed under the union, i.e., for any two feasible coalitions also, their union is feasible. Properties of solutions (the core, the nucleolus, and the prekernel) are discussed for games on a union-closed system.

1. Introduction

A cooperative game with transferable utility, or simply a TU-game, is a finite set of players and for any subset (coalition) of players a worth representing the total payoff that the players in the coalition can obtain by cooperating. A (single-valued) solution is a function that assigns to every game a payoff vector whose components are the individual payoffs of the players.

In its classical interpretation, a TU-game describes a situation in which the players in every coalition can cooperate to form a feasible coalition and earn its worth. In the literature, various restrictions on coalition formation are developed. For example, in the (communication) graph games of [1], see also [2,3], a coalition is feasible if it is connected in a given undirected (communication) graph. This model is generalized in various ways. For example, Refs. [4,5] consider games where the set of feasible coalitions is a union stable system meaning that the union of every pair of nondisjoint feasible coalitions is also feasible. The class of union stable systems contains well-known structures such as the antimatroids being those sets of feasible coalitions (containing the empty set) that are accessible and union-closed, see [6,7]. A set of feasible coalitions is accessibile if every feasible coalition contains at least one player such that without this player, the coalition is still feasible. A set of feasible coalitions is union-closed if the union of every pair of coalitions is feasible. Games in which the collection of feasible coalitions forms an antimatroid are considered in [8,9]. A well-known example of an antimatroid is the collection of feasible coalitions induced by an acyclic permission structure where players need permission from (some of) their superiors in a hierarchical structure when they want to cooperate with others, see, e.g., [10,11,12,13], and see [14] for a survey. Ref. [15] studies the core of such games. (Instead of restricting the feasible coalitions, Ref. [16] introduces games under precedence constraints where a weak order on the player set defines the admissible permutations in which the players can form coalitions. Ref. [17] considers the core of such games). Other models that generalize both the communication graph games as well as the games on an antimatroid are the games on an augmenting system (see [18,19,20], and games on an accessible union stable system (see [21]). Other examples of cooperation structures are games on convex geometries (see [22,23] games on a regular set system (see [24]), games on an intersection closed system (see [25]), and the structures in [26]. Very general structures are studied in [27,28,29]. For surveys, see [30,31].

In the underlying paper, we consider games with restricted cooperation where the collection of feasible coalitions (besides containing the empty set and the grand coalition) is closed under union. In [32] two single-valued solutions for games on a union-closed system that generalize the Shapley value ([33]) are defined and characterized. (The Shapley value for games on a union stable system, also known as Myerson value, is considered in [5]). A generalization of the Shapley value for games on a union stable system, the so-called Harsanyi power solutions, are characterized in [34], generalizing a result for communication graph games in [35]. The first solution in [32] is based on games with a permission structure; the other directly applies the Shapley value to some restricted game. This restricted game is defined by assigning to each coalition the worth of its largest feasible subset in the union-closed system. In the underlying paper, we apply several well-known excess-based solutions, such as the core ([36]), nucleolus ([37]) and the prekernel to this restricted game. We show some properties of these solutions on the class of games on a union-closed system, in particular for monotone games. We also give sufficient conditions to guarantee that the nucleolus is the unique point in the intersection of the prekernel and the core.

Our motivation to study games on a union-closed system is theoretical as well as applied. Theoretically, as mentioned above, these structures generalize well-known structures such as antimatroids, and, thus, permission structures. Moreover, it is the largest class of structures where every coalition has a unique largest feasible subset, and thus, the restricted game as described above is well defined. (The union-closed systems are a special class of union stable systems. An important difference with the partiton systems of [38] is that in a union-closed system, every coalition has exactly one feasible subcoalition, while in the partition systems, as in communication-restricted games, every coalition can be partitioned into feasible subcomponents). Although generalizations of the Shapley value (and other linear values) are studied in this context, as far as we know, this is the first paper on excess-based solutions for union-closed systems. (The core and nucleolus for games on ordered structures are studied in, e.g., [39,40]). Algorithms to compute the nucleolus of certain classes of games with a permission structure can be found in [41,42] generalizing algorithms for peer group games, being a special class of games with a permission structure where the game is additive and the permission structure is a rooted tree (see [43,44]. From an applied point of view, the results of the excess-based solutions in this paper can be used in special cases of union-closed systems, such as a situation where a coalition needs to pass a certain threshold with respect to the number of players to be feasible, see for example, [45,46]. (As an example, we give a generalized version of this model in Section 2).

This paper is organized as follows. Section 2 is a preliminary section on cooperative TU-games. In Section 3, we introduce games on a union-closed system. Section 4 discusses properties of some solutions for monotone games on a union-closed system. Finally, Section 5 gives special attention to the prekernel.

2. Preliminaries

2.1. TU-Games and Solutions

A situation in which a finite set of players can obtain certain payoffs by cooperating can be described by a cooperative game with transferable utility, or simply a TU-game, being a pair $(N,v)$, where $N\subset \mathrm{I}\mathrm{N}$ is a finite set of $n=\left|N\right|$ players and $v:2^N\to R$ is a characteristic function on N such that $v(\varnothing )=0$. For any coalition, $S\subseteq N$, $v\left(S\right)$ is the worth of coalition S with the interpretation that this is what the members of coalition S can obtain by agreeing to cooperate. For ease of notation, we write $v\left(i\right)=v\left(\right\{i\left\}\right)$ for $i\in N$. Since we take the player set N to be fixed, we denote the game $(N,v)$ just by its characteristic function v. We denote the collection of all characteristic functions on N by ${\mathcal{G}}^N$.

A player $i\in N$ is called a veto player if $v\left(S\right)=0$ whenever $i\notin S$. A game v is veto-rich if it contains at least one veto player. A game $v\in {\mathcal{G}}^N$ is monotone if $v\left(S\right)\le v\left(T\right)$ for all $S\subseteq T\subseteq N$. We denote by ${\mathcal{G}}_m^N$ the class of all monotone TU-games on N.

For each nonempty $T\subseteq N$, the unanimity game $u_T$ is given by $u_T\left(S\right)=1$ if $T\subseteq S$, and $u_T\left(S\right)=0$ otherwise. It is well-known that the unanimity games form a basis for ${\mathcal{G}}^N$: for every $v\in {\mathcal{G}}^N$, it holds that $v={\sum }_{T\subseteq N,T\ne \varnothing }{\mathsf{\Delta }}_T\left(v\right)u_T$, where ${\mathsf{\Delta }}_T\left(v\right)={\sum }_{S\subseteq T}{(-1)}^{\left|T\right|-\left|S\right|}v\left(S\right)$ are the Harsanyi dividends, see [47].

A payoff vector is a vector $x\in {\mathrm{I}\mathrm{R}}^n$ assigning a payoff $x_i$ to every $i\in N$. We denote $x\left(S\right)={\sum }_{i\in S}{x}_i$ for $S\subseteq N$. The set of efficient payoff vectors of a game $v\in {\mathcal{G}}^N$ is given by

$$X\left(v\right)=\{x\in {\mathrm{I}\mathrm{R}}^n|x\left(N\right)=v\left(N\right)\}.$$

The imputation set of a game $v\in {\mathcal{G}}^N$ is the set of efficient and individually rational payoff vectors given by

$$I\left(v\right)=\{x\in X\left(v\right)|x_i\ge v\left(i\right)\mathrm{for}\mathrm{every}i\in N\}.$$

A (set-valued) solution is a mapping $F:{\mathcal{G}}^N\to {\mathrm{I}\mathrm{R}}^n$ that assigns a (possibly empty) set $F\left(v\right)\subset {\mathrm{I}\mathrm{R}}^n$ of payoff vectors to every $v\in {\mathcal{G}}^N$. A solution F is said to be single-valued if it assigns a single payoff vector to every $v\in {\mathcal{G}}^N$. Notice that $F=X$ and $F=I$ are set-valued solutions assigning to every v the set of efficient payoff vectors, respectively the imputation set. The most well-known set-valued solution is the core, denoted as C, assigning to every $v\in {\mathcal{G}}^N$ the set

$$C\left(v\right)=\{x\in X(v\left)\right|x\left(S\right)\ge v\left(S\right)\mathrm{for}\mathrm{every}S\subset N\}.$$

Since $C\left(v\right)\subseteq I\left(v\right)$, the core is the set of stable imputations in the sense that no coalition S can improve by separating from the grand coalition N.

For any $B\subseteq N$, let the vector $e^B\in {\mathrm{I}\mathrm{R}}^n$ be defined by $e_j^B=1$ when $j\in B$, and $e_j^B=0$ otherwise. A collection $\mathcal{B}$ of subsets B of N is a balanced collection when the system of equations

$$\sum _{B\in \mathcal{B}}{\lambda }_B{e}^B=e^N$$

has a positive solution. A game $v\in {\mathcal{G}}^N$ is balanced if

$$\sum _{j=1}^m{\lambda }_j^{B}v\left(S_j\right)\le v\left(N\right)$$

for every balanced collection $\mathcal{B}=\{S_1,\dots ,S_m\}\in \mathcal{B}$. A well-known result states that the core of a game is non-empty if and only if the game is balanced, see [48] or [49]. Notice that every veto-rich monotone game has a non-empty core. Specifically, any payoff vector that allocates the worth $v\left(N\right)$ over the veto players is in the core.

Two other well-known solutions are the (pre)nucleolus and the (pre)kernel. To define the (pre)nucleolus of a game $v\in {\mathcal{G}}^N$, let the excess $e(S,x)$ of coalition $S\subseteq N$ with respect to payoff vector $x\in {\mathrm{I}\mathrm{R}}^n$ be defined by

$$e(S,x)=v\left(S\right)-x\left(S\right).$$

Further, let $E\left(x\right)$ be the $(2^n-2)$-component vector that is composed of the excesses of all coalitions $S\subset N,S\ne \varnothing$, in a non-increasing order, so $E_1\left(x\right)\ge E_2\left(x\right)\ge \dots \ge E_{2^n-2}\left(x\right)$. Then the prenucleolus $PN\left(v\right)$ of a game $v\in {\mathcal{G}}^N$ is the unique efficient payoff vector which lexicographically minimizes the vector-valued function $E(·)$ over the set of efficient payoff vectors. Formally,

$$PN\left(v\right)=x\mathrm{such} \mathrm{that} x\in X\left(v\right)\mathrm{and}E\left(x\right){⪯}_{L}E\left(y\right)\mathrm{for} \mathrm{all} y\in X\left(v\right),$$

where ${⪯}_L$ denotes the lexicographic order of vectors. (Although defining the prenucleolus as a singleton would be more consistent with other definitions in this section, for convenience we associate the prenucleolus of a game with the unique payoff vector in this singleton. Similar to the nucleolus that follows next). The nucleolus $Nuc\left(v\right)$ of a game $v\in {\mathcal{G}}^N$ is the unique imputation which lexicographically minimizes the vector-valued function $E(·)$ over the imputation set, so

$$Nuc\left(v\right)=x\mathrm{such} \mathrm{that} x\in I\left(v\right)\mathrm{and}E\left(x\right){⪯}_{L}E\left(y\right)\mathrm{for} \mathrm{all} y\in I\left(v\right).$$

Both the prenucleolus and the nucleolus are single-valued solutions.

To define the prekernel and the kernel of a game $v\in {\mathcal{G}}^N$, we first introduce the notion of complaint. For a payoff vector $x\in {\mathrm{I}\mathrm{R}}^n$, the complaint of player $i\in N$ against another player $j\in N$ is given by

$$s_{ij}\left(x\right)=\underset{\{S\subseteq N|i\in S,j\notin S\}}{max}(v\left(S\right)-x\left(S\right)).$$

The prekernel $PK$ assigns to every $v\in {\mathcal{G}}^N$ the set of efficient payoff vectors such that all pairwise complaints are ‘balanced’:

$$PK\left(v\right)=\{x\in X\left(v\right)|s_{ij}\left(x\right)=s_{ji}\left(x\right)\mathrm{for} \mathrm{all} i,j\in N\}$$

and the kernel K assigns to every $v\in {\mathcal{G}}^N$ the set of imputations such that all pairwise complaints are ‘as balanced as possible under the condition of being imputations’:

$$K\left(v\right)=\{x\in I\left(v\right)|[s_{ij}\left(x\right)=s_{ji}\left(x\right)]\mathrm{or}[s_{ij}\left(x\right)>s_{ji}\left(x\right)\mathrm{and}{x}_j=v\left(j\right)]\mathrm{for} \mathrm{all} i,j\in N\}.$$

Finally, we define the least core $LC\left(v\right)$ of a game $v\in {\mathcal{G}}^N$. For an efficient payoff vector $x\in X\left(v\right)$, the excess $e_v\left(x\right)$ of x is defined by

$$e_v\left(x\right)=\underset{\{S\in 2^N|S\ne \varnothing ,N\}}{max}e(S,x)=\underset{\{S\in 2^N|S\ne \varnothing ,N\}}{max}\left(v\left(S\right)-x\left(S\right)\right).$$

Further, the gain $e\left(v\right)$ of v is defined as the largest negative excess, thus

$$e\left(v\right)=\underset{x\in X(N,v)}{max}-e_v\left(x\right).$$

Notice that $e_v\left(x\right)\le 0$ when $x\in C\left(v\right)$ and $e\left(v\right)\ge 0$ if and only if $C\left(v\right)\ne \varnothing$. Then the least core, introduced by [50], see e.g., also [51], is defined as the solution $LC$ that assigns to game v the set of efficient payoff vectors

$$LC\left(v\right)=\{x\in X(v\left)\right|x\left(S\right)\ge v\left(S\right)+e\left(v\right)\mathrm{for} \mathrm{every} S\ne \varnothing ,N\}.$$

Observe that $LC\left(v\right)\subseteq C\left(v\right)$ if $C\left(v\right)\ne \varnothing$, with $LC\left(v\right)=C\left(v\right)$ when $e\left(v\right)=0$. We also have that $Nuc\left(v\right)\in LC\left(v\right)$ and that $LC\left(v\right)\subseteq I\left(v\right)$ when $v\in {\mathcal{G}}_m^N$.

2.2. Games on Union-Closed Systems

A game with restricted cooperation is a tuple $(v,\mathsf{\Omega })$, where v is a TU-game on a player set N and $\mathsf{\Omega }\subseteq 2^N$ is a collection of subsets of N. In such a game, the collection of subsets $\mathsf{\Omega }$ restricts the cooperation possibilities of the players in N. A coalition $S\in 2^N$ is feasible if and only if $S\in \mathsf{\Omega }$. In this paper, we only consider sets of feasible coalitions that are closed under union.

Definition 1 

([32]). A collection $\mathsf{\Omega }\subseteq 2^N$ is a union-closed system of coalitions if

1.

$\varnothing ,N\in \mathsf{\Omega }$,

2.

If $S,T\in \mathsf{\Omega }$, then $S\cup T\in \mathsf{\Omega }$.

We denote the collection of all union-closed systems in $2^N$ by ${\mathcal{C}}^N$.

Example 1. 

1. Both $\mathsf{\Omega }=\{\varnothing ,N\}$ and $\mathsf{\Omega }=2^N$ are union-closed systems. The first one is the smallest union-closed system and the second one is the largest union-closed system of subsets of N, i.e., $\{\varnothing ,N\}\subseteq \mathsf{\Omega }\subseteq 2^N$ for every union-closed system Ω of subsets of N.

2. For some $k\in \{1,\dots |N\left|\right\}$, the collection of coalitions $\mathsf{\Omega }=\{S\subseteq N\mid |S|\ge k\}\cup \{\varnothing \}$ is closed under union. More generally, let the collection $\mathcal{P}=\{P^1,\dots ,P^m\}$ of nonempty subsets of N be a partition of N, and for every $P^k,k\in \{1,\dots ,m\}$, let $q_k\in \{1,\dots ,|P^k\left|\right\}$ be a quotum meaning that a nonempty coaliton $S\subseteq N$ can form if S contains at least $q_k$ players from $P^k$ for every $k=1,\dots ,m$. The collection of feasible coalitions

$$\mathsf{\Omega }=\{S\subseteq N\mid |S\cap P_k|\ge q_k\mathrm{for} \mathrm{all} k\in \{1,\dots ,m\}\}\cup \{\varnothing \}$$

is closed under union. This model generalizes the games with thresholds as considered in [45,46] Notice that, for $k\ge 2$ (or $q_k\ge 2$), these are not antimatroids (since they are not accessible as shown by coalitions whose cardinality equals the threshold) nor augmenting systems (since they do not satisfy the augmentation property as shown by the empty set having no augmenting player).

3. Every antimatroid is a union-closed system by definition. (Note that examples 1 and 2 above are union-closed systems that are not antimatroids). Since the collection of conjunctive, respectively disjunctive, feasible coalitions in an acyclic permission structure (directed graph) are an antimatroid (see [8]), also these are examples of union-closed systems, see also [12,13]. (Let D be an acyclic directed graph on player set N (representing for instance some hierarchical structure). The collection of conjunctive feasible coalitions is the collection of coalitions Ω such that for every $i\in S$ also all predecessors of i in the digraph D belong to S. The collection of disjunctive feasible coalitions is the collection of coalitions Ω such that for every $i\in S$ having a predecesssor in D, at least one of the predecessors is in S).

For notational convenience, we require in Definition 1 that the grand coalition N is feasible. The results of this paper can be modified to hold without this requirement if we distinguish between players that belong to at least one feasible coalition and those that do not belong to any feasible coalition. Note that union-closedness implies that the grand coalition is feasible if every player belongs to at least one feasible coalition. So, instead of assuming that $N\in \mathsf{\Omega }$, we could do with the weaker normality assumption stating that every player belongs to at least one feasible coalition.

A set $S\subseteq N$ of players can attain its value $v\left(S\right)$ if $S\in \mathsf{\Omega }$. When $S\notin \mathsf{\Omega }$ then coalition S can not be formed, and the set S can not realize its worth $v\left(S\right)$. For $\mathsf{\Omega }\subseteq 2^N$, let ${\sigma }_{\mathsf{\Omega }}\left(S\right)={\cup }_{\{E\in \mathsf{\Omega }\mid E\subseteq S\}}E$ be the largest feasible subset of S in the system $\mathsf{\Omega }$. By union-closedness, this largest feasible subset is unique. The restricted game $r_{v,\mathsf{\Omega }}\in {\mathcal{G}}^N$ of game on union-closed system $(v,\mathsf{\Omega })$ assigns to each coalition $S\subseteq N$ the worth of its largest feasible subset, and thus, is defined by

$$r_{v,\mathsf{\Omega }}\left(S\right)=v\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right).$$

Notice that when $\mathsf{\Omega }=\{\varnothing ,N\}$, then ${\sigma }_{\mathsf{\Omega }}\left(N\right)=N$ and ${\sigma }_{\mathsf{\Omega }}\left(S\right)=\varnothing$ for all $S\ne N$, and thus $r_{v,\mathsf{\Omega }}\left(N\right)=v\left(N\right)$ and $r_{v,\mathsf{\Omega }}\left(S\right)=v(\varnothing )=0$ for every $S\ne N$. In these cases, the restricted game $r_{v,\mathsf{\Omega }}$ is a multiple of the unanimity game of N, being a game in which every player is a veto-player. When $\mathsf{\Omega }=2^N$, then ${\sigma }_{\mathsf{\Omega }}\left(S\right)=S$ and $r_{v,\mathsf{\Omega }}\left(S\right)=v\left(S\right)$ for every $S\subseteq N$. In this case the restricted game $r_{v,\mathsf{\Omega }}$ coincides with v.

3. Properties of Games on a Union-Closed System

Before discussing solutions for games on a union-closed system, we give some definitions and properties for union-closed systems.

Definition 2 

([32]). For two players $i,j\in N$, $i\ne j$, player i is a superior of player j in $\mathsf{\Omega }\in {\mathcal{C}}^N$, if $i\in S$ for every $S\in \mathsf{\Omega }$ such that $j\in S$. In that case, player j is a subordinate of i.

Corollary 1. 

If i is a superior of j in Ω and k is a superior of i in Ω then k is a superior of j in Ω.

Further, for $i\in N$ and $\mathsf{\Omega }\in {\mathcal{C}}^N$, define

$$S_i^{\mathsf{\Omega }}=\{j\in N\mid j=i \mathrm{or} i \text{is a superior of} j\},$$

i.e., $S_i^{\mathsf{\Omega }}\subseteq N$ denotes the set containing player i and all subordinates of i in $\mathsf{\Omega }$. The next proposition says that when $\mathsf{\Omega }$ is a union-closed system, for every $i\in N$ the complement of i and all its subordinates are in $\mathsf{\Omega }$.

Proposition 1. 

When $\mathsf{\Omega }\in {\mathcal{C}}^N$, then $N\backslash S_i^{\mathsf{\Omega }}\in \mathsf{\Omega }$ for every $i\in N$.

Proof. 

Let U be the union of all feasible sets not containing i. Since $\mathsf{\Omega }\in {\mathcal{C}}^N$, it follows that $U\in \mathsf{\Omega }$. Further, by definition of U, we have that $i\notin U$. Consider a player $j\notin U$ with $j\ne i$. It holds that any feasible set without i does not contain j. So, i is superior to j, and thus, $j\in S_i^{\mathsf{\Omega }}$. Hence, $N\backslash U\subseteq S_i^{\mathsf{\Omega }}$. On the other hand, consider some player $j\in S_i^{\mathsf{\Omega }}$. If $j=i$ then $j\notin U$ by definition of U. If $j\ne i$, then any feasible set containing j also contains i. Hence, $j\notin U$, which shows that $S_i^{\mathsf{\Omega }}\subseteq N\backslash U$. Hence, $N\backslash S_i^{\mathsf{\Omega }}=U\in \mathsf{\Omega }$. □

Next, we state some inheritance properties of the restricted game which generalize known results for games with a permission structure and games on antimatroids.

Proposition 2. 

Let$\mathsf{\Omega }\in {\mathcal{C}}^N$and let$v\in {\mathcal{G}}_m^N$be a monotone game. Then

1.

the restricted game $r_{v,\mathsf{\Omega }}$ is monotone;

2.

if v is superadditive then $r_{v,\mathsf{\Omega }}$ is superadditive;

3.

if v is balanced then $r_{v,\mathsf{\Omega }}$ is balanced;

4.

if v is convex and Ω is closed under intersection (i.e., $S,T\in \mathsf{\Omega }$ implies that $S\cap T\in \mathsf{\Omega }$), then $r_{v,\mathsf{\Omega }}$ is convex.

Proof. 

Let $\mathsf{\Omega }\in {\mathcal{C}}^N$ and let $v\in {\mathcal{G}}^N$ be a monotone game.

1

By definition of ${\sigma }_{\mathsf{\Omega }}$ it is obvious that $S\subseteq T$ implies that ${\sigma }_{\mathsf{\Omega }}\left(S\right)\subseteq {\sigma }_{\mathsf{\Omega }}\left(T\right)$, and thus, by monotonicity of v, $S\subseteq T$ implies that $r_{v,\mathsf{\Omega }}\left(S\right)=v\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right)\le v\left({\sigma }_{\mathsf{\Omega }}\left(T\right)\right)=r_{v,\mathsf{\Omega }}\left(T\right)$. This shows monotonicity of $r_{v,\mathsf{\Omega }}$.

2

By union-closedness, ${\sigma }_{\mathsf{\Omega }}\left(S\right)\cup {\sigma }_{\mathsf{\Omega }}\left(T\right)\in \mathsf{\Omega }$ for all $S,T\in \mathsf{\Omega }$. Since ${\sigma }_{\mathsf{\Omega }}\left(S\right)\subseteq {\sigma }_{\mathsf{\Omega }}(S\cup T)$ and ${\sigma }_{\mathsf{\Omega }}\left(T\right)\subseteq {\sigma }_{\mathsf{\Omega }}(S\cup T)$, we then have ${\sigma }_{\mathsf{\Omega }}\left(S\right)\cup {\sigma }_{\mathsf{\Omega }}\left(T\right)\subseteq {\sigma }_{\mathsf{\Omega }}(S\cup T)$. If $S\cap T=\varnothing$ then ${\sigma }_{\mathsf{\Omega }}\left(S\right)\cap {\sigma }_{\mathsf{\Omega }}\left(T\right)=\varnothing$, and thus, $r_{v,\mathsf{\Omega }}\left(S\right)+r_{v,\mathsf{\Omega }}\left(T\right)=v\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right)+v\left({\sigma }_{\mathsf{\Omega }}\left(T\right)\right)\le v({\sigma }_{\mathsf{\Omega }}\left(S\right)\cup {\sigma }_{\mathsf{\Omega }}\left(T\right))\le v\left({\sigma }_{\mathsf{\Omega }}(S\cup T)\right)=r_{v,\mathsf{\Omega }}(S\cup T)$, where the first inequality follows from superadditivity, and the second inequality follows from monotonicity of v. This shows the superadditivity of $r_{v,\mathsf{\Omega }}$.

3

This follows straightforward since for monotone v, it holds that $r_{v,\mathsf{\Omega }}\left(S\right)\le v\left(S\right)$ for all $S\subseteq N$, and thus, $C\left(v\right)\subseteq C\left(r_{v,\mathsf{\Omega }}\right)$.

4

In 2, we already showed that ${\sigma }_{\mathsf{\Omega }}\left(S\right)\cup {\sigma }_{\mathsf{\Omega }}\left(T\right)\subseteq {\sigma }_{\mathsf{\Omega }}(S\cup T)$ by union-closedness. Similar, ${\sigma }_{\mathsf{\Omega }}\left(S\right)\cap {\sigma }_{\mathsf{\Omega }}\left(T\right)\in \mathsf{\Omega }$ for all $S,T\in \mathsf{\Omega }$ by intersection closedness. Since ${\sigma }_{\mathsf{\Omega }}\left(S\right)\cap {\sigma }_{\mathsf{\Omega }}\left(T\right)\subseteq S\cap T$, we have ${\sigma }_{\mathsf{\Omega }}\left(S\right)\cap {\sigma }_{\mathsf{\Omega }}\left(T\right)\subseteq {\sigma }_{\mathsf{\Omega }}(S\cap T)$. This implies $r_{v,\mathsf{\Omega }}\left(S\right)+r_{v,\mathsf{\Omega }}\left(T\right)=v\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right)+v\left({\sigma }_{\mathsf{\Omega }}\left(T\right)\right)\le v({\sigma }_{\mathsf{\Omega }}\left(S\right)\cup {\sigma }_{\mathsf{\Omega }}\left(T\right))+v({\sigma }_{\mathsf{\Omega }}\left(S\right)\cap {\sigma }_{\mathsf{\Omega }}\left(T\right))\le v\left({\sigma }_{\mathsf{\Omega }}(S\cup T)\right)+v\left({\sigma }_{\mathsf{\Omega }}(S\cap T)\right)=r_{v,\mathsf{\Omega }}(S\cup T)+r_{v,\mathsf{\Omega }}(S\cap T)$, where the first inequality follows from the convexity of v, and the second inequality follows from the above inclusions and monotonicity of v. This shows the convexity of $r_{v,\mathsf{\Omega }}$.

□

4. Properties of the Core, Least Core and Nucleolus for the Class of Monotone Games on Union-Closed Systems

A solution for games on a union-closed system is a mapping F that assigns a set of payoff vectors $F(v,\mathsf{\Omega })\subset {\mathrm{I}\mathrm{R}}^n$ to every $v\in {\mathcal{G}}^N$ and $\mathsf{\Omega }\in {\mathcal{C}}^N$. In this paper, we only consider solutions for games on a union-closed system that assigns to each tuple $(v,\mathsf{\Omega })\in {\mathcal{G}}^N\times {\mathcal{C}}^N$ the set of payoff vectors $F\left(r_{v,\mathsf{\Omega }}\right)$ of a solution $F:{\mathcal{G}}^N\to {\mathrm{I}\mathrm{R}}^n$ applied to the restricted game $r_{v,\mathsf{\Omega }}$. For ease of notation, we denote $F(v,\mathsf{\Omega })=F\left(r_{v,\mathsf{\Omega }}\right)$.

In this section, we specifically apply the solutions for TU games given in Section 2 and consider their properties for games on a union-closed system. In particular, we consider the relation between the payoffs of some player j and its superior i for monotone games. Notice that, when v is monotone, it holds that for every $\mathsf{\Omega }\in {\mathcal{C}}^N$ also the restricted game $r_{v,\mathsf{\Omega }}$ is monotone (by Proposition 2). Further, it should be noticed that $r_{v,\mathsf{\Omega }}\left(\left\{j\right\}\right)=v(\varnothing )=0$ when j has a superior, because $\left\{j\right\}$ is not feasible when j has a superior.

First, we consider the core and the least core of the restricted game. For a tuple $(v,\mathsf{\Omega })$, let $C(v,\mathsf{\Omega })$ be given by

$$C(v,\mathsf{\Omega })=C\left(r_{v,\mathsf{\Omega }}\right)=\{x\in X\left(r_{v,\mathsf{\Omega }}\right)|x\left(S\right)\ge v\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right),S\subset N\},$$

i.e., $C(v,\mathsf{\Omega })$ is the core of the restricted game.

Alternatively, for a tuple $(v,\mathsf{\Omega })$, let $C^{*}(v,\mathsf{\Omega })$ be given by

$$C^{*}(v,\mathsf{\Omega })=\{x\in X\left(v\right)|x\left(S\right)\ge v\left(S\right) \text{for any} S\in \mathsf{\Omega } \mathrm{and} x_j\ge 0 \text{for any} j\in N\}.$$

i.e., $C^{*}(v,\mathsf{\Omega })$ is the set of nonnegative efficient payoff vectors satisfying the core inequalities corresponding to the feasible coalitions in $\mathsf{\Omega }$. It turns out that the above two core definitions coincide on the class of monotone games on a union-closed system.

Proposition 3. 

For every $v\in {\mathcal{G}}_m^N$ and $\mathsf{\Omega }\in {\mathcal{C}}^N$, we have $C(v,\mathsf{\Omega })=C^{*}(v,\mathsf{\Omega })$.

Proof. 

Since $N\in \mathsf{\Omega }$, we have that ${\sigma }_{\mathsf{\Omega }}\left(N\right)=N$, and thus, $v\left({\sigma }_{\mathsf{\Omega }}\left(N\right)\right)=v\left(N\right)$ and $X\left(r_{v,\mathsf{\Omega }}\right)=X\left(v\right)$. Let $x\in C(v,\mathsf{\Omega })$. When the singleton player set $\left\{j\right\}\in \mathsf{\Omega }$, then ${\sigma }_{\mathsf{\Omega }}\left(\left\{j\right\}\right)=\left\{j\right\}$, and thus, $v\left({\sigma }_{\mathsf{\Omega }}\left(\left\{j\right\}\right)\right)=v\left(\left\{j\right\}\right)\ge 0$ since $v\in {\mathcal{G}}_m^N$. Otherwise, ${\sigma }_{\mathsf{\Omega }}\left(\left\{j\right\}\right)=\varnothing$, and thus, $v\left({\sigma }_{\mathsf{\Omega }}\left(\left\{j\right\}\right)\right)=v(\varnothing )=0$. Hence, for every $x\in C(v,\mathsf{\Omega })$ we have that $x_j\ge v\left({\sigma }_{\mathsf{\Omega }}\left(\left\{j\right\}\right)\right)\ge 0$ for every $j\in N$. Further, since ${\sigma }_{\mathsf{\Omega }}\left(S\right)=S$ if $S\in \mathsf{\Omega }$, the inequalities $x\left(S\right)\ge v\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right)$ for every $S\subset N$, imply that $x\left(S\right)\ge v\left(S\right) \text{for every} S\in \mathsf{\Omega }$. Thus, $x\in C^{*}(v,\mathsf{\Omega })$.

Next, let $x\in C^{*}(v,\mathsf{\Omega })$. Obviously, $x\left(S\right)\ge v\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right)$ for every $S\in \mathsf{\Omega }$. Since ${\sigma }_{\mathsf{\Omega }}\left(S\right)\subseteq S$ and $x_j\ge 0$ for all $j\in N$, we have that $x\left(S\right)\ge x\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right)\ge v\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right)$ for every $S\subset N$ and $v\in {\mathcal{G}}_m^N$. Thus, $x\in C(v,\mathsf{\Omega })$. □

Recall that the least core of a monotone game is contained in the imputation set of the game. Since the restricted game of a monotone game is also monotone (see Property 1 of Proposition 2), it follows that $x\in I\left(r_{v,\mathsf{\Omega }}\right)$ for every $v\in {\mathcal{G}}_m^N$ and $x\in LC\left(r_{v,\mathsf{\Omega }}\right)$. Since, for every $j\in N$ and $x\in LC\left(r_{v,\mathsf{\Omega }}\right)$, either $\left\{j\right\}$ is feasible in $\mathsf{\Omega }$ and thus, $x_j\ge v\left(\left\{j\right\}\right)\ge 0$, or j is not feasible and $x_j\ge r_{v,\mathsf{\Omega }}\left(\left\{j\right\}\right)=v(\varnothing )=0$, we have the following proposition.

Proposition 4. 

Let $v\in {\mathcal{G}}_m^N$ be monotone and $\mathsf{\Omega }\in {\mathcal{C}}^N$. Then $x_j\ge 0$ for every $x\in LC\left(r_{v,\mathsf{\Omega }}\right)$ and $j\in N$.

For a monotone game v, it is straightforward that for two union-closed systems ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_2$ such that ${\mathsf{\Omega }}_1\subseteq {\mathsf{\Omega }}_2$, we have $r_{v,{\mathsf{\Omega }}_1}\left(S\right)\le r_{v,{\mathsf{\Omega }}_2}\left(S\right)$ for every $S\in 2^N$. Therefore, the next proposition follows immediately without proof.

Proposition 5. 

Let v be monotone and ${\mathsf{\Omega }}_1$, ${\mathsf{\Omega }}_2$ be two union-closed systems such that ${\mathsf{\Omega }}_1\subseteq {\mathsf{\Omega }}_2$. Then $C(v,{\mathsf{\Omega }}_2)\subseteq C(v,{\mathsf{\Omega }}_1)$.

Since $r_{v,\mathsf{\Omega }}=v$, and thus, $C(v,\mathsf{\Omega })=C\left(v\right)$ when $\mathsf{\Omega }=2^N$, Proposition 5 yields that $C(v,\mathsf{\Omega })\ne \varnothing$ for any $\mathsf{\Omega }\in {\mathcal{C}}^N$ and balanced $v\in {\mathcal{G}}_m^N$.

In the following, let i and j be two fixed players such that i is a superior of j in $\mathsf{\Omega }$ (and thus, $r_{v,\mathsf{\Omega }}\left(\left\{j\right\}\right)=0$). For a vector x with $x_j>0$ and some number $0\le a\le x_j$, we denote for fixed i and j the vector $x^a$ by

$$\left\{\begin{array}{cc}{x}_i^a=x_i+a,\hfill & \\ x_j^a=x_j-a,\hfill & \\ x_k^a=x_k\hfill & \mathrm{when} k\ne i,j.\hfill \end{array}\right.$$

(1)

(There is some abuse of notation, actually $x^a$ also depends on i and j).

Clearly, since $x_j^a=x_j-a\ge 0=r_{v,\mathsf{\Omega }}\left(\left\{j\right\}\right)$, we have that $x^a\in I(N,r_{v,\mathsf{\Omega }})$ when $x\in I(N,r_{v,\mathsf{\Omega }})$. Moreover, for $S\subset N$

$$\left\{\begin{array}{cc}{x}^a\left(S\right)=x\left(S\right)+a>x\left(S\right)\hfill & \mathrm{if}i\in S,j\notin S,\hfill \\ x^a\left(S\right)=x\left(S\right)-a<x\left(S\right)\hfill & \mathrm{if}j\in S,i\notin S,\hfill \\ x^a\left(S\right)=x\left(S\right)\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

So, for every $S\in \mathsf{\Omega }$ it is true that $x^a\left(S\right)\ge x\left(S\right)$ because i is a superior of j and thus, there does not exist $S\in \mathsf{\Omega }$ with $j\in S$ and $i\notin S$. We now have the following proposition.

Proposition 6. 

Let $(v,\mathsf{\Omega })$ be a monotone game on a union-closed system and, for a vector x and two players i and j such that i is a superior of j, let $x^a$ be as defined in Equation (1). Then

(i) 

if $x\in C(v,\mathsf{\Omega })$, then $x^a\in C(v,\mathsf{\Omega })$ for all $a\in (0,x_j]$.

(ii) 

if $x\in LC(v,\mathsf{\Omega })$ and $x_i<x_j$, then $x^a\in LC(v,\mathsf{\Omega })$ for all $a\in (0,\frac{1}{2}(x_j-x_i)]$.

Proof. 

To prove (i), recall from Proposition 3 that

$$C(v,\mathsf{\Omega })=\{x\in X\left(v\right)\mid x\left(S\right)\ge v\left(S\right) \text{for any} S\in \mathsf{\Omega }and{x}_j\ge 0 \text{for any} j\in N\}.$$

Clearly, $x^a\left(S\right)\ge x\left(S\right)\ge v\left(S\right)$ for every $S\in \mathsf{\Omega }$. Further, we have for every $k\ne j$ that $x_k^a\ge x_k\ge 0$ and that $x_j^a=x_j-a\ge 0$. Since i is a superior of j, and thus, $\left\{j\right\}\notin \mathsf{\Omega }$, it follows that $x^a\in C(v,\mathsf{\Omega })$.

To prove (ii), notice that $x_i^a\le x_j^a$ for all $a\in (0,\frac{1}{2}(x_j-x_i)]$. Suppose that $x^a$ is not in $LC(v,\mathsf{\Omega })$. Then there exists a coalition $S\subset N$ such that

$$x^a\left(S\right)-r_{v,\mathsf{\Omega }}\left(S\right)<e\left(r_{v,\mathsf{\Omega }}\right).$$

(2)

Since $x\in LC(v,\mathsf{\Omega })$, we have that

$$x\left(S\right)-r_{v,\mathsf{\Omega }}\left(S\right)\ge e\left(r_{v,\mathsf{\Omega }}\right).$$

Hence $x\left(S\right)>x^a\left(S\right)$, implying that S contains j but not i. Let $T=S\backslash \left\{j\right\}$ and $S^{\prime }=T\cup \left\{i\right\}$. Then

$$x^a\left(S\right)-r_{v,\mathsf{\Omega }}\left(S\right)=x^a\left(S\right)-v\left({\sigma }_{\mathsf{\Omega }}\left(T\right)\right)$$

because $i\notin S$ and thus, $j\notin {\sigma }_{\mathsf{\Omega }}\left(S\right)$. Hence

$$x^a\left(S\right)-r_{v,\mathsf{\Omega }}\left(S\right)=x^a\left(S\right)-v\left({\sigma }_{\mathsf{\Omega }}\left(T\right)\right)\ge x^a\left(T\right)+x_j^a-v\left({\sigma }_{\mathsf{\Omega }}(T\cup \left\{i\right\})\right)\ge$$

$$x^a\left(T\right)+x_i^a-v\left({\sigma }_{\mathsf{\Omega }}(T\cup \left\{i\right\})\right)=x^a\left(S^{\prime }\right)-r_{v,\mathsf{\Omega }}\left(S^{\prime }\right),$$

where the second inequality follows because $x_i^a\le x_j^a$. With Equation (2) it follows that

$$x^a\left(S^{\prime }\right)-r_{v,\mathsf{\Omega }}\left(S^{\prime }\right)\le x^a\left(S\right)-r_{v,\mathsf{\Omega }}\left(S\right)<e\left(r_{v,\mathsf{\Omega }}\right).$$

Since $x^a\left(S^{\prime }\right)>x\left(S^{\prime }\right)$ whenever $i\in S^{\prime }$ and $j\notin S^{\prime }$, this implies that $x\left(S^{\prime }\right)-r_{v,\mathsf{\Omega }}\left(S^{\prime }\right)<e\left(r_{v,\mathsf{\Omega }}\right)$. This contradicts with the fact that $x\left(S^{\prime }\right)-r_{v,\mathsf{\Omega }}\left(S^{\prime }\right)\ge e\left(r_{v,\mathsf{\Omega }}\right)$ for $x\in LC(v,\mathsf{\Omega })$. □

From Part (i) of Proposition 6, we obtain the following corollary, saying that when the core of the resticted game is non-empty, there exist core stable payoff vectors that give zero payoffs to every player j that has a superior in $\mathsf{\Omega }$.

Corollary 2. 

If $C\left(r_{v,\mathsf{\Omega }}\right)\ne \varnothing$, then there exists $x\in C\left(r_{v,\mathsf{\Omega }}\right)$ such that $x_j=0$ for every j that has a superior.

The final proposition in this section states that for monotone games on a union-closed system, a player gets at most the same payoff as its superior when applying the nucleolus to the restricted game. It should be noticed that when v is monotone, $Nu{c}_j(v,\mathsf{\Omega })\ge 0$ for all j, because $Nuc(v,\mathsf{\Omega })$ is in the least core of $r_{v,\mathsf{\Omega }}$ and thus, also in $I(N,r_{v,\mathsf{\Omega }})$.

Proposition 7. 

Let $(v,\mathsf{\Omega })$ be a monotone game on a union-closed system. Then for every two players i and j such that i is a superior of j, it holds that $Nu{c}_i(v,\mathsf{\Omega })\ge Nu{c}_j(v,\mathsf{\Omega })$.

Proof. 

Let $w\in {\mathcal{G}}^N$ be a game such that for every $S\subseteq N\backslash \{i,j\}$ it holds that $w(S\cup \{i\left\}\right)\ge w(S\cup \{j\left\}\right)$. Then we know from [52], Theorem 5.3.5) that $x_i\ge x_j$ for every x in the prekernel of w. Since the nucleolus of a game is in the prekernel of a game, it is sufficient to show that for every $S\subseteq N\backslash \{i,j\}$ it holds that $r_{v,\mathsf{\Omega }}(S\cup \left\{i\right\})\ge r_{v,\mathsf{\Omega }}(S\cup \left\{j\right\})$ when i a superior of j. Indeed, in that case, we have that

$$r_{v,\mathsf{\Omega }}(S\cup \left\{i\right\})=v\left({\sigma }_{\mathsf{\Omega }}(S\cup \left\{i\right\})\right)\ge v\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right)=v\left({\sigma }_{\mathsf{\Omega }}(S\cup \left\{j\right\})\right)=r_{v,\mathsf{\Omega }}(S\cup \left\{j\right\}),$$

where the second equality follows from the fact that $i\notin S$ and there does not exist a feasible set containing j but not i. □

5. Properties of the Prekernel of Monotone Games on Union-Closed Systems

In this section, we focus on the prekernel for games on a union-closed system. Ref. [53] proved that the kernel of a game $v\in {\mathcal{G}}^N$ consists of only one point (and coincides with the nucleolus) when the game is veto-rich and $I\left(v\right)$ is non-empty. When in the tuple $(v,\mathsf{\Omega })$ there exists a player $i\in N$ such that $i\in S$ for every $S\in \mathsf{\Omega }$, then i is a veto-player in the restricted game $r_{v,\mathsf{\Omega }}$. When $v\in {\mathcal{G}}_m^N$, we have that $I\left(r_{v,\mathsf{\Omega }}\right)\ne \varnothing$ and thus, it follows from [53] that the kernel of $r_{v,\mathsf{\Omega }}$ has the nucleolus of $r_{v,\mathsf{\Omega }}$ as its unique element. It is also well-known that for every game $(N,v)$ with $\left|N\right|\le 3$, the intersection of the prekernel and the core consists of at most one point. In this section, we generalize these results and give a sufficient condition to guarantee that the prekernel and the core of a monotone game on a union-closed system have at most one point in common. Of course, when such a point exists, then it is the nucleolus of the restricted game. We first introduce some new notions.

Definition 3. 

For two players $i,j\in N$, $i\ne j$, player i is a strong superior of player j in $\mathsf{\Omega }\in {\mathcal{C}}^N$ if i is a superior of j and j is not a superior of i.

Definition 4. 

A player $i\in N$ is a free player in $\mathsf{\Omega }\in {\mathcal{C}}^N$ if i has no superiors; player $i\in N$ is a weakly free player in $\mathsf{\Omega }\in {\mathcal{C}}^N$ if i has no strong superiors.

Notice that a free player is also a weakly free player and that a weakly free player i is superior to j when j is superior to i. For $\mathsf{\Omega }\in {\mathcal{C}}^N$, we denote the set of weakly free players by

$$W_{\mathsf{\Omega }}=\{i\in N\mid i \text{is a weakly free player in} \mathsf{\Omega }\}.$$

The next proposition gives three properties of the set $W_{\mathsf{\Omega }}$.

Proposition 8. 

1. For every player $j\notin W_{\mathsf{\Omega }}$, there is a player $i\in W_{\mathsf{\Omega }}$ such that i is a strong superior of j.

2. When j is a superior of a player $i\in W_{\mathsf{\Omega }}$, then i is a superior of j.

3. When j is a superior of a player $i\in W_{\mathsf{\Omega }}$, then $j\in W_{\mathsf{\Omega }}$.

Proof. 

1. Consider some player $i_0\in N$. If $i_0$ is not in $W_{\mathsf{\Omega }}$, then $i_0$ has a strong superior, say $i_1$. Then, either $i_1\in W_{\mathsf{\Omega }}$ and thus, $i_0$ has a strong superior in $W_{\mathsf{\Omega }}$, or not. In the latter case, $i_1$ has a strong superior, say $i_2$. When $i_2$ is not in $W_{\mathsf{\Omega }}$, it also has a strong superior. Continuing this we get a sequence of players $i_0,i_1,i_2,\dots ,i_m$ such that for $h=1,\dots ,m-1$, player $i_{h+1}$ is a strong superior of $i_h$ and thus, $i_h\notin W_{\mathsf{\Omega }}$ and either $i_m\in W_{\mathsf{\Omega }}$ or $m\ge 2$ and $i_m=i_k$ for some $k=0,\dots ,m-2$. In the latter case, by Corollary 1 every pair $i_j,i_{\ell }$ with $j,\ell \in \{k,k+1,\dots ,m-1\}$ are superiors of each other, contradicting that $i_{h+1}$ is strong superior of $i_h$, $h=k,\dots ,m-1$. Hence, every next player in the sequence is different from all preceding players. Since the number of players is finite, this case can not happen and thus, within a finite number of steps, some player $i_m\in W_{\mathsf{\Omega }}$ is generated. By Corollary 1 $i_m$ is a superior of $i_0$. When $i_0$ is a superior of $i_m$, then again by Corollary 1 we have that $i_0$ is a superior of $i_1$, contradicting that $i_1$ is a strong superior of $i_0$. Hence $i_m\in W_{\mathsf{\Omega }}$ and is a strong superior of $i_0$.

2. By definition, i is a superior of j, since otherwise j is a strong superior of i, which contradicts that $i\in W_{\mathsf{\Omega }}$.

3. Suppose $j\notin W_{\mathsf{\Omega }}$. Then by the first property, j has a strong superior k in $W_{\mathsf{\Omega }}$. By Corollary 1 player k is also a superior of i, and thus, by Property 2 of Proposition 8, we have that player i is also a superior of k. However, this implies that also j is a superior of k, contradicting that k is a strong superior of j. □

The first property of Proposition 8 yields the following corollary.

Corollary 3. 

For every $\mathsf{\Omega }\in {\mathcal{C}}^N$, $W_{\mathsf{\Omega }}\ne \varnothing$.

Next, for $i\in W_{\mathsf{\Omega }}$, define $T_{\mathsf{\Omega }}\left(i\right)=\{j\in N|j=i \mathrm{or} j \text{is a superior of} i\}$, and let ${\mathcal{T}}_{\mathsf{\Omega }}$ be the collection of sets defined by

$${\mathcal{T}}_{\mathsf{\Omega }}=\left\{T_{\mathsf{\Omega }}\left(i\right)\right|i\in W_{\mathsf{\Omega }}\}.$$

Notice that, for every $j\in T_{\mathsf{\Omega }}\left(i\right)\backslash \left\{i\right\}$, i is a superior of j, because $i\in W_{\mathsf{\Omega }}$. This implies, $T_{\mathsf{\Omega }}\left(i\right)\subseteq S_i^{\mathsf{\Omega }}=\{j\in N|j=i \mathrm{or} i \text{is a superior of} j\}$. The next proposition describes the set $W_{\mathsf{\Omega }}$.

Proposition 9. 

The collection ${\mathcal{T}}_{\mathsf{\Omega }}$ is a partition of the set $W_{\mathsf{\Omega }}$.

Proof. 

First, by Property 3 of Proposition 8 we have that $j\in W_{\mathsf{\Omega }}$ when $j\in T_{\mathsf{\Omega }}\left(i\right)$ for some $i\in W_{\mathsf{\Omega }}$, and thus, $T_{\mathsf{\Omega }}\left(i\right)\subseteq W_{\mathsf{\Omega }}$. Next, let $R\subseteq W_{\mathsf{\Omega }}\times W_{\mathsf{\Omega }}$ be the binary relation on $W_{\mathsf{\Omega }}$ defined by $(j,i)\in R$ if and only if $j\in T_{\mathsf{\Omega }}\left(i\right)$. It is sufficient to show that this relation is an equivalence relation on $W_{\mathsf{\Omega }}$, i.e., the relation is reflexive, symmetric, and transitive. First, R is reflexive, since by definition $(i,i)\in R$ for all $i\in W_{\mathsf{\Omega }}$. Second, for $j\ne i$, when $(j,i)\in R$, then j is a superior of i. By Property 2 of Proposition 8, then also i is a superior of j, and thus, $(i,j)\in R$, showing that R is symmetric. Third, when $(k,j)\in R$ and $(j,i)\in R$, then k is a superior of j and j of i and thus, by Corollary 1, also k is a superior of i. Hence, $(k,i)\in R$, and thus, R is transitive. Since R is an equivalence relation, it follows that the sets $T_{\mathsf{\Omega }}\left(i\right)$, $i\in W_{\mathsf{\Omega }}$, are equivalence classes of $W_{\mathsf{\Omega }}$, and thus, the collection ${\mathcal{T}}_{\mathsf{\Omega }}$ partitions $W_{\mathsf{\Omega }}$. □

Proposition 9 implies that $j\in T_{\mathsf{\Omega }}\left(i\right)$ if and only if $i\in T_{\mathsf{\Omega }}\left(j\right)$. When, for two different agents $i,j\in W_{\mathsf{\Omega }}$, i is not a superior of j, then $T_{\mathsf{\Omega }}\left(i\right)$ and $T_{\mathsf{\Omega }}\left(j\right)$ are two different equivalence classes.

Proposition 10. 

Let Ω be a union-closed system. When $j\in W_{\mathsf{\Omega }}$ is a superior of $i\in N$, then every $k\in T_{\mathsf{\Omega }}\left(j\right)$ is a superior of i.

Proof. 

For $i\in W_{\mathsf{\Omega }}$, the proposition follows from Proposition 9, because $T_{\mathsf{\Omega }}\left(i\right)=T_{\mathsf{\Omega }}\left(j\right)$ when j is a superior of i. Let $i\notin W_{\mathsf{\Omega }}$ and $j\in W_{\mathsf{\Omega }}$ be a superior of i. Then every $k\ne j$ in $T_{\mathsf{\Omega }}\left(j\right)$ is a superior of j, and thus, a superior of i by Corollary 1. □

Proposition 11. 

Let $(v,\mathsf{\Omega })$ be a game on a union-closed system. When ${\mathcal{T}}_{\mathsf{\Omega }}$ consists of only one set, then every player in $W_{\mathsf{\Omega }}$ is a veto-player in the restricted game $r_{v,\mathsf{\Omega }}$.

Proof. 

First, when ${\mathcal{T}}_{\mathsf{\Omega }}$ consists of only one set, say T, then, by Proposition 9, $T=W_{\mathsf{\Omega }}$. So, $T_{\mathsf{\Omega }}\left(i\right)=W_{\mathsf{\Omega }}$ for every $i\in W_{\mathsf{\Omega }}$ and thus, by definition of $T_{\mathsf{\Omega }}\left(i\right)$ and Proposition 10, every player $k\in W_{\mathsf{\Omega }}$ is a superior to every other i in $W_{\mathsf{\Omega }}$. Moreover, by Property 1 of Proposition 8 every player not in $W_{\mathsf{\Omega }}$ has a player i in $W_{\mathsf{\Omega }}$ as it is superior. Thus, again by Proposition 10, every player in $W_{\mathsf{\Omega }}$ is superior to every player not in $W_{\mathsf{\Omega }}$. So, every player in $W_{\mathsf{\Omega }}$ is superior to every other player in N, so that every $S\in \mathsf{\Omega }$ contains all players in $W_{\mathsf{\Omega }}$. □

Notice that $T_{\mathsf{\Omega }}\left(i\right)=\left\{i\right\}$ when i is a free player. So, every free player i gives a single element equivalence class $T_{\mathsf{\Omega }}\left(i\right)=\left\{i\right\}$ in the partition ${\mathcal{T}}_{\mathsf{\Omega }}$ of $W_{\mathsf{\Omega }}$. When there is a free player i and ${\mathcal{T}}_{\mathsf{\Omega }}$ consists of only one set, then $W_{\mathsf{\Omega }}=\left\{i\right\}$. In the sequel, we call the number of sets in ${\mathcal{T}}_{\mathsf{\Omega }}$ the weakly free player cardinality of $\mathsf{\Omega }$. Since by Corollary 3 the set of weakly free players is non-empty, this cardinality is at least one. It follows from Proposition 11 that $r_{v,\mathsf{\Omega }}$ is a veto-rich game when this cardinality is equal to one. Then the next corollary follows from [53].

Corollary 4. 

If $(v,\mathsf{\Omega })$ is a game on a union-closed system, then the kernel $K\left(r_{v,\mathsf{\Omega }}\right)$ contains the nucleolus $Nuc\left(r_{v,\mathsf{\Omega }}\right)$ as its unique element when the weakly free player cardinality is one.

To generalize this, we use the famous theorem of [54] giving a sufficient and necessary condition for a payoff vector to be in the prenucleolus of a game. For game $v\in {\mathcal{G}}^N$, a payoff vector $x\in {\mathrm{I}\mathrm{R}}^n$ and real number $\alpha$, let $\mathcal{B}(\alpha ,x)$ be the collection of coalitions given by $\mathcal{B}(\alpha ,x)=\{S\in N|e(S,x)\ge \alpha \}$.

Theorem 1 

([54]). For game $v\in {\mathcal{G}}^N$, a payoff vector x is in $PN\left(v\right)$ if and only if for any real number α the collection of coalitions $\mathcal{B}(\alpha ,x)$ is either balanced or empty.

In [55] an analog of this theorem for the prekernel is proved in terms of 2-balancedness. We first give the notion of k-balancedness for $2\le k\le n$.

Definition 5 

([55]). A collection $\mathcal{S}$ of coalitions $S\in 2^N$ is k-balanced if for every coalition $K\subseteq N$ with $\left|K\right|=k$, the collection ${\mathcal{S}}_K=\{S^{\prime }\subset K|S^{\prime }=S\cap K,S\in \mathcal{S}\}$ is balanced on K.

Theorem 2 

([55]). For $v\in {\mathcal{G}}^N$, a payoff vector x is in $PK\left(v\right)$ if and only if for any real number α the collection of coalitions $\mathcal{B}(\alpha ,x)$ is either 2-balanced or empty.

Recall from the standard definition of balancedness that, when a collection ${\mathcal{S}}_K$ is balanced on K, then there exist strictly positive weights ${\lambda }_T$, $T\in {\mathcal{S}}_K$, such that for every $i\in K$ the total weight of the sets $T\in {\mathcal{S}}_K$ that contain i is equal to one. From this, the following corollary follows immediately.

Corollary 5. 

Let $K=\{i,j\}\subseteq N$ be a two-player coalition and $\mathcal{S}$ be a collection of coalitions such that ${\mathcal{S}}_K$ is balanced on K. When $\mathcal{S}$ contains a set T such that $i\in T$ and $j\notin T$, then $\mathcal{S}$ contains a coalition $T^{\prime }$ such that $j\in T^{\prime }$ and $i\notin T^{\prime }$.

Moreover, notice that a k-balanced collection $\mathcal{S}$ is balanced when $k=n$. Moreover, when $\left|N\right|=3$, any 2-balanced collection is also balanced. The next lemma generalizes this fact and will be used to prove the main result of this section.

Lemma 1. 

For a union-closed system Ω with weakly free player cardinality of at most three, let $\mathcal{B}\subset 2^N$ be a 2-balanced collection that only contains feasible sets in Ω and singletons. Then $\mathcal{B}$ is balanced.

Proof. 

Let $c\in \{1,2,3\}$ be the weakly free player cardinality of $\mathsf{\Omega }$. Without loss of generality, let the players be numbered in such way that $W_{\mathsf{\Omega }}\supset \{1,\dots ,c\}$ and that $T_{\mathsf{\Omega }}\left(k\right)$, $k=1,\dots ,c$, are the equivalence classes of ${\mathcal{T}}_{\mathsf{\Omega }}$. By Property 2 of Proposition 8, every player $j\ne k$ in $T_{\mathsf{\Omega }}\left(k\right)$ has player k as its superior. Moreover, by Property 1 of Proposition 8 and by Proposition 10, every player $j\in N\backslash W_{\mathsf{\Omega }}$ has at least one of the players k, $k\in W_{\mathsf{\Omega }}$, as one of its superiors. For $k\in W_{\mathsf{\Omega }}$, suppose that there exists j in the set

$$S_k^{\mathsf{\Omega }}\backslash \left\{k\right\}=\{i\in N|k \text{is a superior of} i\}$$

such that there is some T in $\mathcal{B}$ containing k, but not j. Take $K=\{k,j\}$. By the 2-balancedness of $\mathcal{B}$, the collection $\{S\cap K|S\in \mathcal{B}\}$ is balanced on K. So, by Corollary 5 there exists a set $T^{\prime }\in \mathcal{B}$ such that $j\in T^{\prime }$ and $k\notin T^{\prime }$. Since $\mathcal{B}$ only contains feasible sets and singletons, and k is a superior of j, it follows that $T^{\prime }=\left\{j\right\}$. Let

$$S_k={\cap }_{\{S\in \mathcal{B}|k\in S\}}S,k\in W_{\mathsf{\Omega }}.$$

From above it follows that $\left\{j\right\}\in \mathcal{B}$ for every $j\notin {\cup }_{k\in W_{\mathsf{\Omega }}}{S}_k$. Now, let

$${\mathcal{B}}^{\prime }=\{U\in \mathcal{B}|U\cap W_{\mathsf{\Omega }}\ne \varnothing \}$$

and consider the collection of subsets of $W_{\mathsf{\Omega }}$ given by

$${\mathcal{B}}^{″}=\{W_{\mathsf{\Omega }}\cap U|U\in {\mathcal{B}}^{\prime }\}.$$

This is a balanced collection on $W_{\mathsf{\Omega }}$. This is trivial when $c=1$, and it follows by the 2-balancedness of $\mathcal{B}$ when $c=2$. When $c=3$ this follows from the fact that every 2-balanced collection on a three-player set is balanced. So, for the sets $U\in {\mathcal{B}}^{\prime }$ there are weights, say ${\lambda }_U^{\mathcal{B}}$, such that

$$\sum _{\{U\in {\mathcal{B}}^{\prime }|k\in U\}}{\lambda }_U^{\mathcal{B}}=1,k\in W_{\mathsf{\Omega }}.$$

Since every feasible set has a nonempty intersection with $W_{\mathsf{\Omega }}$, this yields weight ${\lambda }_U^{\mathcal{B}}>0$ for every feasible set $U\in \mathcal{B}$. Moreover,

$$\sum _{\{U\in {\mathcal{B}}^{\prime }|j\in U\}}{\lambda }_U^{\mathcal{B}}=1,\text{for every}j\in {\cup }_{k=1,\dots ,c}{S}_k,$$

since if $j\in S_k$ for some $k=1,\dots ,c$, then the collection of sets from ${\mathcal{B}}^{\prime }$ containing j coincides with the collection of sets from $\mathcal{B}$’ containing k. Finally, consider some $j\in N\backslash \left({\cup }_{k=1,\dots ,c}{S}_k\right)$. Recall that such a player j has at least one of the players from the set $W_{\mathsf{\Omega }}$ as one of its superiors, say player k. So, when j is contained in some set $U\in {\mathcal{B}}^{\prime }$, then also $k\in U$. Moreover, there exists at least one $U\in {\mathcal{B}}^{\prime }$ containing k and not j, otherwise $j\in S_k$. Therefore,

$$\sum _{\{U\in {\mathcal{B}}^{\prime }|j\in U\}}{\lambda }_U^{\mathcal{B}}<1 \text{for every} j\in N\backslash \left({\cup }_{k=1,\dots ,c}{S}_k\right),$$

i.e., the total weight of the feasible sets containing such a player j is less than one. However, for every such j we also have the singleton $\left\{j\right\}\in \mathcal{B}$. This yields weight ${\lambda }_{\left\{j\right\}}^{\mathcal{B}}=1-{\displaystyle \sum _{\{U\in {\mathcal{B}}^{\prime }|j\in U\}}}{\lambda }_U^{\mathcal{B}}$ for every singleton set $\left\{j\right\}\in \mathcal{B}$, $j\in N\backslash \left({\cup }_{k=1,\dots ,c}{S}_k\right)$. Since for every $j\in {\cup }_{k=1,\dots ,c}{S}_k$, every set in $\mathcal{B}$ containing j also contains one of the players from $\{1,\dots ,c\}$, there are no other singletons in $\mathcal{B}$. So, we have determined weights for all sets in $\mathcal{B}$ satisfying that

$$\sum _{\{S\in \mathcal{B}|j\in S\}}{\lambda }_S^{\mathcal{B}}=1, \text{for every} j\in N,$$

and thus, $\mathcal{B}$ is balanced. □

We are now ready to formulate the main result of this section.

Theorem 3. 

Let $(v,\mathsf{\Omega })$ be a monotone game on a union-closed system. Then the intersection of $PK(v,\mathsf{\Omega })$ and $C(v,\mathsf{\Omega })$ consists of at most one point if the weakly free player cardinality of Ω is at most three.

Proof. 

Clearly, the statement of the theorem is true when $C(v,\mathsf{\Omega })=\varnothing$. So, we only consider the case that $C(v,\mathsf{\Omega })\ne \varnothing$. Then $PN(v,\mathsf{\Omega })=Nuc(v,\mathsf{\Omega })$ and lies in the core. Suppose there is a payoff vector $y\in PK(v,\mathsf{\Omega })\cap C(v,\mathsf{\Omega })$ with $y\ne x=Nuc(v,\mathsf{\Omega })$. Since $y\ne PN(v,\mathsf{\Omega })$, according to Kohlberg’s theorem there is some $\alpha$ for which $\mathcal{B}(\alpha ,y)$ is not balanced. Since $x=PN(v,\mathsf{\Omega })$, also according to Kohlberg’s theorem, we have that $\mathcal{B}(\alpha ,x)$ is balanced and thus, $\mathcal{B}(\alpha ,x)\ne \mathcal{B}(\alpha ,y)$. Since for $\alpha$ big enough we have that $\mathcal{B}(\alpha ,x)=\mathcal{B}(\alpha ,y)=\varnothing$, there exists some value $\alpha$ with the properties that

(i) $\mathcal{B}(\alpha ,x)\ne \mathcal{B}(\alpha ,y)$, and

(ii) for every $\beta >\alpha$, it holds that either $\mathcal{B}(\beta ,x)=\mathcal{B}(\beta ,y)$ or both $\mathcal{B}(\alpha ,x)=\mathcal{B}(\beta ,x)$

and $\mathcal{B}(\alpha ,y)=\mathcal{B}(\beta ,y)$.

For a coalition S and payoff vector x, let $e(S,x)=r_{v,\mathsf{\Omega }}\left(S\right)-x\left(S\right)=v\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right)-x\left(S\right)$ be the excess of coalition S at x in the restricted game $r_{v,\mathsf{\Omega }}$, and let ${\alpha }^{*}$ be a value satisfying the two properties (i) and (ii). Now, suppose that there exists $S\in \mathcal{B}({\alpha }^{*},x)$ such that $e(S,x)<e(S,y)$. Then, for $\beta =e(S,y)>e(S,x)\ge {\alpha }^{*}$, we have that $S\in \mathcal{B}(\beta ,y)$ and $S\notin \mathcal{B}(\beta ,x)$. So, $\mathcal{B}(\beta ,x)\ne \mathcal{B}(\beta ,y)$ and $\mathcal{B}({\alpha }^{*},x)\ne \mathcal{B}(\beta ,x)$, which contradicts that property (ii) holds for ${\alpha }^{*}$. Hence

$$e(S,x)\ge e(S,y) \text{for every} S\in \mathcal{B}({\alpha }^{*},x).$$

(3)

Further, for $S\in \mathcal{B}({\alpha }^{*},x)$, let ${\lambda }_S$ be the weight of S in the balanced system of collection $\mathcal{B}({\alpha }^{*},x)$. Since both x and y are efficient, it follows that

$$\sum _{\left\{S\right|S\in \mathcal{B}({\alpha }^{*},x)\}}{\lambda }_{S}e(S,x)=\sum _{\left\{S\right|S\in \mathcal{B}({\alpha }^{*},x)\}}{\lambda }_S(r_{v,\mathsf{\Omega }}\left(S\right)-x\left(S\right)).$$

Since (i) x is efficient, and (ii) the weights are balanced, we have that ${\sum }_{\{S\in \mathcal{B}({\alpha }^{*},x)|i\in S\}}{\lambda }_S=1$ for every $i\in N$, it follows that

$$\sum _{S\in \mathcal{B}({\alpha }^{*},x)}{\lambda }_{S}x\left(S\right)=\sum _{S\in \mathcal{B}({\alpha }^{*},x)}{\lambda }_S\sum _{i\in S}{x}_i=\sum _{i\in N}{x}_i\sum _{\{S\in \mathcal{B}({\alpha }^{*},x)|i\in S\}}{\lambda }_S=\sum _{i\in N}{x}_i=r_{v,\mathsf{\Omega }}\left(N\right),$$

and thus

$$\sum _{\left\{S\right|S\in \mathcal{B}({\alpha }^{*},x)\}}{\lambda }_{S}e(S,x)=\sum _{\left\{S\right|S\in \mathcal{B}({\alpha }^{*},x)\}}{\lambda }_S(r_{v,\mathsf{\Omega }}\left(S\right)-x\left(S\right)).$$

$$=\sum _{\left\{S\right|S\in \mathcal{B}({\alpha }^{*},x)\}}{\lambda }_S{r}_{v,\mathsf{\Omega }}\left(S\right)-r_{v,\mathsf{\Omega }}\left(N\right).$$

Analogously

$$\sum _{\left\{S\right|S\in \mathcal{B}({\alpha }^{*},x)\}}{\lambda }_{S}e(S,y)=\sum _{\left\{S\right|S\in \mathcal{B}({\alpha }^{*},x)\}}{\lambda }_S{r}_{v,\mathsf{\Omega }}\left(S\right)-r_{v,\mathsf{\Omega }}\left(N\right).$$

So,

$$\sum _{\left\{S\right|S\in \mathcal{B}({\alpha }^{*},x)\}}{\lambda }_{S}e(S,x)=\sum _{\left\{S\right|S\in \mathcal{B}({\alpha }^{*},x)\}}{\lambda }_{S}e(S,y).$$

With inequalities (3) this implies $e(S,y)=e(S,x)$ for every $S\in \mathcal{B}({\alpha }^{*},x)$, and thus $\mathcal{B}({\alpha }^{*},x)\subseteq \mathcal{B}({\alpha }^{*},y)$.

Now, suppose that also the collection $B({\alpha }^{*},y)$ is balanced. Then, by the same reasoning as above, we obtain that $e(S,x)=e(S,y)$ for every $S\in \mathcal{B}({\alpha }^{*},y)$ and thus, also $\mathcal{B}({\alpha }^{*},y)\subseteq \mathcal{B}({\alpha }^{*},x)$. Since this contradicts with $\mathcal{B}({\alpha }^{*},x)\ne \mathcal{B}({\alpha }^{*},y)$, $B({\alpha }^{*},y)$ is not balanced.

On the other hand, by Theorem 2 and $y\in PK(v,\mathsf{\Omega })$, we have that $\mathcal{B}({\alpha }^{*},y)$ is 2-balanced. So, $\mathcal{B}({\alpha }^{*},y)$ is 2-balanced, but not balanced. Then, according to Lemma 1, $\mathcal{B}({\alpha }^{*},y)$ contains a non-feasible coalition S with $\left|S\right|>1$. By definition of ${\sigma }_{\mathsf{\Omega }}\left(S\right)$ and $\mathsf{\Omega }$ being union-closed, we have that $r_{v,\mathsf{\Omega }}\left(T\right)=0$ for every $T\subseteq S\backslash {\sigma }_{\mathsf{\Omega }}\left(S\right)$. Then, for every $i\in S\backslash {\sigma }_{\mathsf{\Omega }}\left(S\right)$ it follows that

$$e(S,y)=r_{v,\mathsf{\Omega }}\left(S\right)-y\left(S\right)=r_{v,\mathsf{\Omega }}\left({\sigma }_{\mathsf{\Omega }}\left(S\right)\right)-\sum _{j\in {\sigma }_{\mathsf{\Omega }}\left(S\right)}{y}_j-\sum _{h\in S\backslash {\sigma }_{\mathsf{\Omega }}\left(S\right)}{y}_h\le$$

$$e({\sigma }_{\mathsf{\Omega }}\left(S\right),y)-y_i=e({\sigma }_{\mathsf{\Omega }}\left(S\right),y)+e(\left\{i\right\},y),$$

because $y\in C\left(r_{v,\mathsf{\Omega }}\right)$ and thus, $y_h\ge r_{v,\mathsf{\Omega }}\left(\left\{h\right\}\right)=0$ for all $h\in S\backslash {\sigma }_{\mathsf{\Omega }}\left(S\right)$. Since both $e({\sigma }_{\mathsf{\Omega }}\left(S\right),y)\le 0$ and $e\left(\right\{i\},y)\le 0$ (again because $y\in C\left(r_{v,\mathsf{\Omega }}\right)$), it follows that

$$e(S,y)\le e({\sigma }_{\mathsf{\Omega }}\left(S\right),y) \mathrm{and} e(S,y)\le e\left(\left\{i\right\}\right),y).$$

Hence both $\sigma \left(S\right)\in \mathcal{B}({\alpha }^{*},y)$ and $\left\{i\right\}\in \mathcal{B}({\alpha }^{*},y)$ for every $i\in S\backslash {\sigma }_{\mathsf{\Omega }}\left(S\right)$. However, then also the collection $\mathcal{B}({\alpha }^{*},y)\backslash \left\{S\right\}$ is 2-balanced and not balanced. Let $NF=\{T\in \mathcal{B}({\alpha }^{*},y)|Tisnon-feasibleand\left|T\right|>1\}$. Repeating the reasoning above for every $T\in NF$ it follows that ${\mathcal{B}}^{\prime }=\mathcal{B}({\alpha }^{*},y)\backslash NF$ is 2-balanced and not balanced. However, since ${\mathcal{B}}^{\prime }$ only consists of feasible sets and singletons, this contradicts Lemma 1. So, there is no $y\in PK(v,\mathsf{\Omega })\cap C(v,\mathsf{\Omega })$ with $y\ne x=Nuc(v,\mathsf{\Omega })$. □

The intersection of the core and the prekernel is an attractive solution since the payoff vectors in this intersection satisfy the core group stability requirements as well as the prekernel pairwise balancedness properties. The above theorem clarifies the role of weakly free players.

6. Concluding Remarks

In this paper, we generalized the model of games on an antimatroid to games on a union-closed system. This is a subclass of games on a union stable system and, specifically, is the largest class of network structures such that every coalition has a unique largest feasible subset. As far as we know, this is the first paper that studies excess-based solutions for this class of games. Whether the solutions considered in this paper are more suitable than linear solutions such as the Shapley value and related solutions, depends on the application one has in mind. Although studying specific applications of union-closed systems is beyond the scope of this paper, the results of this paper are useful when one wants to analyze or compute these solutions in applications where excess-based solutions are more appropriate.