One-bound core games

Abstract

This paper introduces the new class of one-bound core games, where the core can be described by either a lower bound or an upper bound on the payoffs of the players, named lower bound core games and upper bound core games, respectively. We study the relation of the class of one-bound core games with several other classes of games and characterize the new class by the structure of the core and in terms of Davis-Maschler reduced games. We also provide explicit expressions and axiomatic characterizations of the nucleolus for one-bound core games, and show that the nucleolus coincides with the Shapley value and the $\tau$-value when these games are convex.

1 Introduction

In a cooperative game with transferable utility, coalitions of cooperating players are able to attain joint revenues. A characteristic function models this feature by assigning to each possible coalition a real number, called worth, reflecting these joint revenues. Depending on the structure of the characteristic function, different classes of games arise. For a class of games, a main issue in each game is how to allocate the worth of the grand coalition, consisting of all players in the game. Solution concepts assign to each game in a certain class such an allocation. A central benchmark for evaluating solutions is the core, which equals the set of allocations that for each coalition assign in total at least the worth to its members. The nucleolus (cf. Schmeidler 1969) is a particular solution that assigns to each game with a nonempty core the unique core allocation that lexicographically minimizes the maximal excesses over all coalitions.

In this paper, we introduce a new class of cooperative games, called one-bound core games, where the core can be described by either a lower bound or an upper bound on the payoffs of the players. A game is a lower bound core game if the core can be described by a lower bound on the payoffs of the players, and an upper bound core game if the core can be described by an upper bound on the payoffs of the players. A game is both a lower bound core game and an upper bound core game if and only if it has at most two players or a single-valued core. Both lower bound core games and upper bound core games are specific two-bound core games (cf. Gong et al. 2022b), where the core can be described by a lower bound and an upper bound on the payoffs of the players. Moreover, upper bound core games generalize 1-convex games (cf. Driessen 1985), where the worth of the grand coalition is large enough for each nonempty coalition to cover its worth while allocating to all non-members their marginal contributions to the grand coalition. We provide a necessary and sufficient condition for one-bound core games to be convex (cf. Shapley 1971).

We characterize one-bound core games by the structure of the core. A game with nonempty core is a lower bound core game if and only if each player obtains its maximal payoff within the core exactly when the other players obtain their minimal payoffs within the core, or equivalently, in each extreme point of the core one player obtains its maximal payoff within the core and all other players obtain their minimal payoffs within the core. Similarly, a game with nonempty core is an upper bound core game if and only if each player obtains its minimal payoff within the core exactly when the other players obtain their maximal payoffs within the core, or equivalently, in each extreme point of the core one player obtains its minimal payoff within the core and all other players obtain their maximal payoffs within the core. We show that a game with nonempty core is a lower bound core game if and only if all Davis-Maschler reduced games with respect to core allocations have the same lower exact core bound. Similarly, a game with nonempty core is an upper bound core game if and only if all Davis-Maschler reduced games with respect to core allocations have the same upper exact core bound.

We also study the nucleolus for one-bound core games. We show that it is the unique pre-imputation that is a convex combination of the two exact core bounds. We provide axiomatic characterizations based on new properties that require that the difference between the allocation and the minimal payoff or maximal payoff within the core is equal for all players. For convex one-bound core games, the nucleolus is the unique solution satisfying symmetry and translation covariance. This implies that it coincides with the Shapley value (cf. Shapley 1953), another well-known solution for cooperative games, and the $\tau$-value (cf. Tijs 1981).

The remainder of this paper is organized as follows. Section 2 provides preliminary definitions and notation for cooperative games. Section 3 introduces and studies one-bound core games. Section 4 analyzes the nucleolus for one-bound core games. Section 5 concludes.

2 Preliminaries

Let N be a nonempty and finite set of players and let $2^N=\{S\mid S\subseteq N\}$ be the set of all coalitions. For all $x\in {\mathbb {R}}^N$, we denote $x_S=(x_i)_{i\in S}$ for all $S\in 2^N{\setminus }\{\emptyset \}$. For all $x,y\in {\mathbb {R}}^N$, we denote $x\le y$ if $x_i\le y_i$ for all $i\in N$, $x\ge y$ if $x_i\ge y_i$ for all $i\in N$, and $x+y=(x_i+y_i)_{i\in N}$.

A (transferable utility) game is a pair (N, v), where $v:2^N\rightarrow {\mathbb {R}}$ assigns to each coalition $S\in 2^N$ its worth $v(S)\in {\mathbb {R}}$ such that $v(\emptyset )=0$. The class of all games with player set N is denoted by $\Gamma ^N$. For simplicity, we write $v\in \Gamma ^N$ rather than $(N,v)\in \Gamma ^N$.

For each game $v\in \Gamma ^N$, the set of pre-imputations $X(v)\subseteq {\mathbb {R}}^N$ is given by

$$\begin{aligned}X(v)=\left\{ x\in {\mathbb {R}}^N\ \left| \ \sum _{i\in N}x_i=v(N)\right. \right\} ,\end{aligned}$$

and the core $C(v)\subseteq {\mathbb {R}}^N$ is given by

$$\begin{aligned}C(v)=\left\{ x\in X(v)\ \left| \ \forall S\in 2^N:\sum _{i\in S}x_i\ge v(S)\right. \right\} .\end{aligned}$$

A game $v\in \Gamma ^N$ is balanced (cf. Bondareva 1963 and Shapley 1967) if and only if $C(v)\ne \emptyset$. The class of all balanced games with player set N is denoted by $\Gamma _b^N$.

For each game $v\in \Gamma _b^N$, the lower exact core bound $l^*(v)\in {\mathbb {R}}^N$ (cf. Bondareva and Driessen 1994) is given by

$$\begin{aligned}l_i^*(v)=\min _{x\in C(v)}x_i\quad \text {for all }i\in N,\end{aligned}$$

and the upper exact core bound $u^*(v)\in {\mathbb {R}}^N$ (cf. Bondareva and Driessen 1994) is given by

$$\begin{aligned}u_i^*(v)=\max _{x\in C(v)}x_i\quad \text {for all }i\in N.\end{aligned}$$

A game $v\in \Gamma ^N_b$ is a two-bound core game (cf. Gong et al. 2022b) if there exist $l,u\in {\mathbb {R}}^N$ such that $C(v)=\left\{ x\in X(v)\ \left| \ l\le x\le u\right. \right\}$. Gong et al. (2022b) showed that a game $v\in \Gamma ^N_b$ is a two-bound core game if and only if

$$\begin{aligned}C(v)=\left\{ x\in X(v)\ \left| \ l^*(v)\le x\le u^*(v)\right. \right\} .\end{aligned}$$

The class of all two-bound core games with player set N is denoted by $\Gamma _t^N$. Gong et al. (2022b) showed that $\Gamma _t^N=\Gamma _b^N$ if and only if $|N|\le 3$.

A game $v\in \Gamma _b^N$ is a 1-convex game (cf. Driessen 1985) if

$$\begin{aligned}v(S)+\sum _{i\in N{\setminus } S}\left( v(N)-v(N{\setminus }\{i\})\right) \le v(N)\quad \text {for all }S\in 2^N{\setminus }\{\emptyset \}.\end{aligned}$$

The class of all 1-convex games with player set N is denoted by $\Gamma _{1c}^N$.

A game $v\in \Gamma ^N$ is convex (cf. Shapley 1971) if and only if

$$\begin{aligned}v(S\cup \{i\})-v(S)\le v(T\cup \{i\})-v(T)\quad \text {for all }i\in N\text { and all }S\subseteq T\subseteq N{\setminus }\{i\}.\end{aligned}$$

The class of all convex games with player set N is denoted by $\Gamma _c^N$. Shapley (1971) showed that $\Gamma _c^N\subseteq \Gamma _b^N$, and $\Gamma _c^N=\Gamma _b^N$ if and only if $|N|\le 2$. Moreover, for each $v\in \Gamma _c^N$, $l_i^*(v)=v(\{i\})$ and $u_i^*(v)=v(N)-v(N{\setminus }\{i\})$ for all $i\in N$.

A solution $\varphi$ on a domain of games assigns to each game v in this domain an allocation $\varphi (v)\in X(v)$. The nucleolus $\eta$ (cf. Schmeidler 1969) is the solution that assigns to each game $v\in \Gamma _b^N$ the allocation $x\in X(v)$ that lexicographically minimizes the excesses $v(S)-\sum _{i\in S}x_i$ for all $S\in 2^N{\setminus }\{\emptyset \}$ arranged in non-increasing order. Clearly, $\eta (v)\in C(v)$ for all $v\in \Gamma _b^N$. Gong et al. (2022b) showed that the nucleolus of a two-bound core game $v\in \Gamma _t^N$ is for each $i\in N$ given by

$$\begin{aligned} \eta _i(v)= {\left\{ \begin{array}{ll} l_i^*(v)+\min \left\{ \frac{1}{2}(u_i^*(v)-l_i^*(v)),\lambda \right\} &{} \text {if }\frac{1}{2}\sum _{i\in N}(u_i^*(v)+l_i^*(v))\ge v(N); \\ l_i^*(v)+\max \left\{ \frac{1}{2}(u_i^*(v)-l_i^*(v)),u_i^*(v)-l_i^*(v)-\lambda \right\} &{} \text {if }\frac{1}{2}\sum _{i\in N}(u_i^*(v)+l_i^*(v))\le v(N), \end{array}\right. } \end{aligned}$$

where $\lambda \in {\mathbb {R}}$ is such that $\sum _{i\in N}\eta _i(v)=v(N)$. This implies that for each $v\in \Gamma _t^N$, $\eta _i(v)-l_i^*(v)=\eta _j(v)-l_j^*(v)$ for all $i,j\in N$ with $u_i^*(v)-l_i^*(v)=u_j^*(v)-l_j^*(v)$. The Shapley value $\phi$ (cf. Shapley 1953) is the solution that assigns to each game $v\in \Gamma ^N$ the allocation given by

$$\begin{aligned}\phi _i(v)=\sum _{S\in 2^N:i\notin S}\frac{|S|!(|N|-|S|-1)!}{|N|!}\left( v(S\cup \{i\})-v(S)\right) \quad \text {for all }i\in N.\end{aligned}$$

Shapley (1971) showed that $\phi (v)\in C(v)$ for all $v\in \Gamma _c^N$. The $\tau$-value $\tau$ (cf. Tijs 1981) is the solution that assigns to each game $v\in \Gamma _b^N$ the allocation given by

$$\begin{aligned}\tau _i(v)=\lambda a_i(v)+(1-\lambda )b_i(v)\quad \text {for all }i\in N,\end{aligned}$$

where

$$\begin{aligned}a_i(v)=\max _{S\in 2^N:i\in S}\left\{ v(S)-\sum _{j\in S{\setminus }\{i\}}\left( v(N)-v(N{\setminus }\{j\})\right) \right\} ,\end{aligned}$$

$$\begin{aligned}b_i(v)=v(N)-v(N{\setminus }\{i\}),\end{aligned}$$

and $\lambda \in [0,1]$ is such that $\sum _{i\in N}\tau _i(v)=v(N)$. Tijs (1981) showed that $a(v)=l^*(v)$ and $b(v)=u^*(v)$ for all $v\in \Gamma _c^N$. Driessen and Tijs (1983) showed that $\eta (v)=\tau (v)$ for all $v\in \Gamma _{1c}^N$. Driessen and Tijs (1985) showed that $\eta (v)=\phi (v)=\tau (v)$ for all $v\in \Gamma _{1c}^N\cap \Gamma _c^N$.

A solution $\varphi$ on a domain of games satisfies symmetry if for each game v in this domain, it holds that $\varphi _i(v)=\varphi _j(v)$ for all $i,j\in N$ with $v(S\cup \{i\})=v(S\cup \{j\})$ for all $S\subseteq N{{\setminus }}\{i,j\}$. A solution $\varphi$ on a domain of games satisfies translation covariance if for each game v in this domain and each $\alpha \in {\mathbb {R}}^N$ such that $v+\alpha$ is in this domain, it holds that $\varphi (v+\alpha )=\varphi (v)+\alpha$, where $v+\alpha$ is defined by $(v+\alpha )(S)=v(S)+\sum _{i\in S}\alpha _i$ for all $S\in 2^N$. It is known that the nucleolus, the Shapley value, and the $\tau$-value each satisfy symmetry and translation covariance on each subdomain of balanced games.

3 One-bound core games

In this section, we introduce the new class of one-bound core games, where the core can be described by either a lower bound or an upper bound on the payoffs of the players. A game is a lower bound core game if the core can be described by a lower bound on the payoffs of the players, and an upper bound core game if the core can be described by an upper bound on the payoffs of the players.

Definition 1

A game $v\in \Gamma _b^N$ is a lower bound core game if there exists $l\in {\mathbb {R}}^N$ such that

$$\begin{aligned}C(v)=\left\{ x\in X(v)\ \left| \ l\le x\right. \right\} .\end{aligned}$$

A game $v\in \Gamma _b^N$ is an upper bound core game if there exists $u\in {\mathbb {R}}^N$ such that

$$\begin{aligned}C(v)=\left\{ x\in X(v)\ \left| \ x\le u\right. \right\} .\end{aligned}$$

A game is a one-bound core game if it is a lower bound core game or an upper bound core game.

The class of all lower bound core games with player set N is denoted by $\Gamma _l^N$. The class of all upper bound core games with player set N is denoted by $\Gamma _u^N$. It turns out that each lower bound core game with at least two players can only be described by the lower exact core bound, and each upper bound core game with at least two players can only be described by the upper exact core bound.¹

Lemma 1

Let $v\in \Gamma _b^N$ and let $l,u\in {\mathbb {R}}^N$. Then the following statements hold:

• (i) If $C(v)=\{x\in X(v)\mid l\le x\}$, then $|N|=1$ or $l=l^*(v)$.

• (ii) If $C(v)=\{x\in X(v)\mid x\le u\}$, then $|N|=1$ or $u=u^*(v)$.

Proof

(i) Assume that $C(v)=\{x\in X(v)\mid l\le x\}$. Then $l\le l^*(v)$. Assume that $|N|\ge 2$ and suppose for the sake of contradiction that there exists $i\in N$ such that $l_i<l_i^*(v)$. Define $x\in {\mathbb {R}}^N$ by $x_i=l_i$ and $x_j=l_j+\frac{1}{|N|-1}(v(N)-\sum _{k\in N}l_k)$ for all $j\in N{{\setminus }}\{i\}$. Then $x\in C(v)$, which contradicts the definition of $l_i^*(v)$.

(ii) The proof is analogous to the proof of (i). $\square$

The following result follows directly from Lemma 1.

Corollary 1

• (i) A game $v\in \Gamma _b^N$ is a lower bound core game if and only if $C(v)=\{x\in X(v)\mid l^*(v)\le x\}$.

• (ii) A game $v\in \Gamma _b^N$ is an upper bound core game if and only if $C(v)=\{x\in X(v)\mid x\le u^*(v)\}$.

Moreover, the lower exact core bound of each lower bound core game with infinitely many core elements is given by the individual worths of the players, and the upper exact core bound of each upper bound core game with infinitely many core elements is given by the marginal contributions to the grand coalition.

Lemma 2

• (i) Let $v\in \Gamma _l^N$. Then $|C(v)|=1$ or $l_i^*(v)=v(\{i\})$ for all $i\in N$.

• (ii) Let $v\in \Gamma _u^N$. Then $|C(v)|=1$ or $u_i^*(v)=v(N)-v(N{\setminus }\{i\})$ for all $i\in N$.

Proof

(i) Assume that $|C(v)|\ne 1$. Then $v(N)>\sum _{i\in N}l_i^*(v)$. Suppose for the sake of contradiction that there exists $i\in N$ such that $l_i^*(v)>v(\{i\})$. Let $\varepsilon >0$ be small. Define $x\in {\mathbb {R}}^N$ by $x_i=l_i^*(v)-\varepsilon$ and $x_j=l_j^*(v)+\frac{1}{|N|-1}(v(N)-\sum _{k\in N}l_k^*(v)+\varepsilon )$ for all $j\in N{{\setminus }}\{i\}$. Then $x\in C(v)$, which contradicts the definition of $l_i^*(v)$.

(ii) The proof is analogous to the proof of (i). $\square$

All lower bound core games with at most two players are upper bound core games, and all upper bound core games with at most two players are lower bound core games, but this does not hold for more players.

Theorem 1

• (i) If $|N|\le 2$, then $\Gamma _l^N=\Gamma _u^N$.

• (ii) If $|N|\ge 3$, then $\Gamma _l^N\not \subseteq \Gamma _u^N$ and $\Gamma _l^N\not \supseteq \Gamma _u^N$.

Proof

(i) Assume that $|N|\le 2$. Let $v\in \Gamma _b^N$. Then $v\in \Gamma _c^N$, so $l_i^*(v)=v(\{i\})$ and $u_i^*(v)=v(N)-v(N{\setminus }\{i\})$ for all $i\in N$. This implies that $C(v)=\{x\in X(v)\mid l^*(v)\le x\}=\{x\in X(v)\mid x\le u^*(v)\}$, so $v\in \Gamma _l^N\cap \Gamma _u^N$. Hence, $\Gamma _l^N=\Gamma _u^N=\Gamma _b^N$.

(ii) Let $v\in \Gamma _b^N$ with $|N|\ge 3$ be defined by

$$\begin{aligned} v(S)= {\left\{ \begin{array}{ll} 1 &{} \text {if }S=N\text {;} \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Then $l_i^*(v)=0$ and $u_i^*(v)=1$ for all $i\in N$. This implies that $C(v)=\{x\in X(v)\mid l^*(v)\le x\}$ and $C(v)\ne \{x\in X(v)\mid x\le u^*(v)\}$. Hence, $v\in \Gamma _l^N{\setminus }\Gamma _u^N$.

Let $v\in \Gamma _b^N$ with $|N|\ge 3$ be defined by

$$\begin{aligned} v(S)= {\left\{ \begin{array}{ll} |N|-1 &{} \text {if }S=N\text {;} \\ |N|-2 &{} \text {if }|S|=|N|-1\text {;} \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Then $l_i^*(v)=0$ and $u_i^*(v)=1$ for all $i\in N$. This implies that $C(v)\ne \{x\in X(v)\mid l^*(v)\le x\}$ and $C(v)=\{x\in X(v)\mid x\le u^*(v)\}$. Hence, $v\in \Gamma _u^N{\setminus }\Gamma _l^N$. $\square$

The following result follows from Lemmas 1 and 2 and Theorem 1.

Theorem 2

A game $v\in \Gamma _b^N$ is both a lower bound core game and an upper bound core game if and only if $|N|\le 2$ or $|C(v)|=1$.

Proof

For the if-part, assume that $|N|\le 2$ or $|C(v)|=1$. If $|N|\le 2$, then Theorem 1 implies that $v\in \Gamma _l^N\cap \Gamma _u^N$. If $|C(v)|=1$, then Lemma 2 implies that $v\in \Gamma _l^N\cap \Gamma _u^N$.

For the only-if part, assume that $v\in \Gamma _l^N\cap \Gamma _u^N$. Let $x\in C(v)$. Then $v(\{i\})\le x_i\le v(N)-v(N{{\setminus }}\{i\})$ for all $i\in N$. Assume that $|N|>2$. Then Lemmas 1 and 2 imply that

$$\begin{aligned}C(v)= & {} \left\{ x\in X(v)\ \left| \ \forall i\in N:v(\{i\})\le x_i\right. \right\} \\= & {} \left\{ x\in X(v)\ \left| \ \forall i\in N:x_i\le v(N)-v(N{\setminus }\{i\})\right. \right\} .\end{aligned}$$

This implies that $v(\{i\})=x_i=v(N)-v(N{\setminus }\{i\})$ for all $i\in N$. Hence, $|C(v)|=1$. $\square$

As the following example shows, one-bound core games are not necessarily convex, and convex games are not necessarily one-bound core games.

Example 1

Let $v\in \Gamma _l^N\cap \Gamma _u^N$ with $|N|\ge 3$ be defined by

$$\begin{aligned} v(S)= {\left\{ \begin{array}{ll} |S| &{} \text {if }|S|\in \{1,|N|\}\text {;} \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Then $v(\{i\})-v(\emptyset )=1>-1=v(\{i,j\})-v(\{j\})$ for all distinct $i,j\in N$, so $v\notin \Gamma _c^N$.

Now, let $v\in \Gamma _c^N$ with $|N|\ge 3$ be defined by

$$\begin{aligned} v(S)= {\left\{ \begin{array}{ll} 3 &{} \text {if }S=N\text {;} \\ 1 &{} \text {if }|S|=|N|-1\text {;} \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Then $l_i^*(v)=0$ and $u_i^*(v)=2$ for all $i\in N$, which implies that $C(v)\ne \{x\in X(v)\mid l^*(v)\le x\}$ and $C(v)\ne \{x\in X(v)\mid x\le u^*(v)\}$, so $v\notin \Gamma _l^N\cup \Gamma _u^N$. $\triangle$

As the following theorem shows, all 1-convex games are upper bound core games, and all one-bound core games are two-bound core games. Strict inclusion depends on the number of players in the game.

Theorem 3

• (i) If $|N|=1$, then $\Gamma _{1c}^N=\Gamma _l^N=\Gamma _u^N=\Gamma _t^N=\Gamma _b^N=\Gamma ^N$.

• (ii) If $|N|=2$, then $\Gamma _{1c}^N=\Gamma _l^N=\Gamma _u^N=\Gamma _t^N=\Gamma _b^N\subsetneq \Gamma ^N$.

• (iii) If $|N|=3$, then $\Gamma _l^N\subsetneq \Gamma _t^N=\Gamma _b^N\subsetneq \Gamma ^N$ and $\Gamma _{1c}^N\subsetneq \Gamma _u^N\subsetneq \Gamma _t^N=\Gamma _b^N\subsetneq \Gamma ^N$.

• (iv) If $|N|\ge 4$, then $\Gamma _l^N\subsetneq \Gamma _t^N\subsetneq \Gamma _b^N\subsetneq \Gamma ^N$ and $\Gamma _{1c}^N\subsetneq \Gamma _u^N\subsetneq \Gamma _t^N\subsetneq \Gamma _b^N\subsetneq \Gamma ^N$.

Proof

First, we show that $\Gamma _l^N\subseteq \Gamma _t^N\subseteq \Gamma _b^N\subseteq \Gamma ^N$ and $\Gamma _{1c}^N\subseteq \Gamma _u^N\subseteq \Gamma _t^N\subseteq \Gamma _b^N\subseteq \Gamma ^N$. Clearly, $\Gamma _t^N\subseteq \Gamma _b^N\subseteq \Gamma ^N$.

Let $v\in \Gamma _l^N$. Then $C(v)\subseteq \{x\in X(v)\mid l^*(v)\le x\le u^*(v)\}\subseteq \{x\in X(v)\mid l^*(v)\le x\}=C(v)$, so $C(v)=\{x\in X(v)\mid l^*(v)\le x\le u^*(v)\}$ and $v\in \Gamma _t^N$. Hence, $\Gamma _l^N\subseteq \Gamma _t^N$.

Let $v\in \Gamma _{1c}^N$. If $x\in X(v)$ and $x\le u^*(v)$, then for each $S\in 2^N{\setminus }\{\emptyset \}$,

$$\begin{aligned}\sum _{i\in S}x_i &= \sum _{i\in N}x_i-\sum _{i\in N{\setminus } S}x_i\ge v(N)-\sum _{i\in N{\setminus } S}u_i^*(v)\\ &\ge v(N)-\sum _{i\in N{\setminus } S}\left( v(N)-v(N{\setminus }\{i\})\right) \ge v(S),\end{aligned}$$

so $x\in C(v)$. This implies that $C(v)=\{x\in X(v)\mid x\le u^*(v)\}$, so $v\in \Gamma _u^N$. Hence, $\Gamma _{1c}^N\subseteq \Gamma _u^N$.

Let $v\in \Gamma _u^N$. Then $C(v)\subseteq \{x\in X(v)\mid l^*(v)\le x\le u^*(v)\}\subseteq \{x\in X(v)\mid x\le u^*(v)\}=C(v)$, so $C(v)=\{x\in X(v)\mid l^*(v)\le x\le u^*(v)\}$ and $v\in \Gamma _t^N$. Hence, $\Gamma _u^N\subseteq \Gamma _t^N$.

$(i)\ \& \ (ii)$ Assume that $|N|\in \{1,2\}$. Clearly, $\Gamma _b^N=\Gamma ^N$ if and only if $|N|=1$. Let $v\in \Gamma _b^N$. Then for each $S\in 2^N{\setminus }\{\emptyset \}$,

$$\begin{aligned}v(S)+\sum _{i\in N{\setminus } S}(v(N)-v(N{\setminus }\{i\}))=v(N).\end{aligned}$$

This implies that $v\in \Gamma _{1c}^N$. Hence, $\Gamma _{1c}^N=\Gamma _l^N=\Gamma _u^N=\Gamma _t^N=\Gamma _b^N$.

$(iii)\ \& \ (iv)$ Assume that $|N|\ge 3$. By Gong et al. (2022b), $\Gamma _t^N=\Gamma _b^N$ if and only if $|N|\le 3$. Let $v\in \Gamma _l^N\cap \Gamma _u^N$ be the one-bound core game from Example 1. Then $v(N)-v(N{\setminus }\{i\})=|N|$ for all $i\in N$. This implies that for each $S\in 2^N$ with $|S|=1$,

$$\begin{aligned}v(S)+\sum _{i\in N{\setminus } S}\left( v(N)-v(N{\setminus }\{i\})\right) =1+\left( |N|-1\right) |N|>|N|=v(N),\end{aligned}$$

so $v\notin \Gamma _{1c}^N$. Hence, $\Gamma _{1c}^N\subsetneq \Gamma _u^N$.

Let $v\in \Gamma _t^N$ be the convex game from Example 1. Then $v\notin \Gamma _l^N\cup \Gamma _u^N$. Hence, $\Gamma _l^N\subsetneq \Gamma _t^N$ and $\Gamma _u^N\subsetneq \Gamma _t^N$. $\square$

We provide a necessary and sufficient condition for one-bound core games to be convex.

Theorem 4

• (i) A lower bound core game $v\in \Gamma _l^N$ is convex if and only if

  $$\begin{aligned}\sum _{i\in S}l_i^*(v)=v(S)\quad \text {for all }S\in 2^N{\setminus }\{N\}.\end{aligned}$$

• (ii) An upper bound core game $v\in \Gamma _u^N$ is convex if and only if

  $$\begin{aligned}\sum _{i\in N{\setminus } S}u_i^*(v)=v(N)-v(S)\quad \text {for all }S\in 2^N{\setminus }\{\emptyset \}.\end{aligned}$$

Proof

(i) Let $v\in \Gamma _l^N$. Assume that $\sum _{i\in S}l_i^*(v)=v(S)$ for all $S\in 2^N{{\setminus }}\{N\}$. Let $i\in N$ and let $S\subseteq N{\setminus }\{i\}$. If $S=N{\setminus }\{i\}$, then

$$\begin{aligned} v(S\cup \{i\})-v(S) &= v(N)-v(N{\setminus }\{i\})=v(N)-\sum _{j\in N{\setminus }\{i\}}l_j^*(v)\\ &\ge \sum _{j\in N}l_j^*(v)-\sum _{j\in N{\setminus }\{i\}}l_j^*(v)=l_i^*(v).\end{aligned}$$

If $S\ne N{{\setminus }}\{i\}$, then $v(S\cup \{i\})-v(S)=\sum _{j\in S\cup \{i\}}l_j^*(v)-\sum _{j\in S}l_j^*(v)=l_i^*(v)$. This implies that $v(S\cup \{i\})-v(S)\le v(T\cup \{i\})-v(T)$ for all $S\subseteq T\subseteq N{\setminus }\{i\}$. Hence, $v\in \Gamma _c^N$.

Let $v\in \Gamma _l^N$. Then $(v(N)-\sum _{j\in N{{\setminus }}\{i\}}l_j^*(v),l_{N{{\setminus }}\{i\}}^*(v))\in C(v)$ for all $i\in N$, which implies that $\sum _{i\in S}l_i^*(v)\ge v(S)$ for all $S\in 2^N{\setminus }\{N\}$. Assume that $v\in \Gamma _c^N$. Let $S\in 2^N{\setminus }\{\emptyset ,N\}$. Denote $S=\{i_1,\ldots ,i_{|S|}\}$. Then

$$\begin{aligned}\sum _{i\in S}l_i^*(v)= & {} \sum _{i\in S}v(\{i\})=\sum _{k=1}^{|S|}\left( v(\{i_k\})-v(\emptyset )\right) \\\le & {} \sum _{k=1}^{|S|}\left( v(\{i_1,\ldots ,i_k\})-v(\{i_1,\ldots ,i_{k-1}\})\right) =v(S),\end{aligned}$$

where the first equality and the inequality follow from convexity. Hence, $\sum _{i\in S}l_i^*(v)=v(S)$ for all $S\in 2^N{\setminus }\{N\}$.

(ii) Let $v\in \Gamma _u^N$. Assume that $\sum _{i\in N{\setminus } S}u_i^*(v)=v(N)-v(S)$ for all $S\in 2^N{\setminus }\{\emptyset \}$. Let $i\in N$ and let $S\subseteq N{\setminus }\{i\}$. If $S=\emptyset$, then

$$\begin{aligned}v(S\cup \{i\})-v(S)= & {} v(\{i\})=v(N)-\sum _{j\in N{\setminus }\{i\}}u_j^*(v)\\\le & {} \sum _{j\in N}u_j^*(v)-\sum _{j\in N{\setminus }\{i\}}u_j^*(v)=u_i^*(v).\end{aligned}$$

If $S\ne \emptyset$, then

$$\begin{aligned}v(S\cup \{i\})-v(S)=\left( v(N)-\sum _{j\in N{\setminus }(S\cup \{i\})}u_j^*(v)\right) -\left( v(N)-\sum _{j\in N{\setminus } S}u_j^*(v)\right) =u_i^*(v).\end{aligned}$$

This implies that $v(S\cup \{i\})-v(S)\le v(T\cup \{i\})-v(T)$ for all $S\subseteq T\subseteq N{\setminus }\{i\}$. Hence, $v\in \Gamma _c^N$.

Let $v\in \Gamma _u^N$. Then $(v(N)-\sum _{j\in N{\setminus }\{i\}}u_j^*(v),u_{N{\setminus }\{i\}}^*(v))\in C(v)$ for all $i\in N$, which implies that $\sum _{i\in N{\setminus } S}u_i^*(v)\le v(N)-v(S)$ for all $S\in 2^N{\setminus }\{\emptyset \}$. Assume that $v\in \Gamma _c^N$. Let $S\in 2^N{\setminus }\{\emptyset ,N\}$. Denote $N{\setminus } S=\{i_1,\ldots ,i_{|N{\setminus } S|}\}$. Then

$$\begin{aligned} \sum _{i\in N{\setminus } S}u_i^*(v)&= \sum _{i\in N{\setminus } S}\left( v(N)-v(N{\setminus }\{i\})\right) \\ {}&= \sum _{k=1}^{|N{\setminus } S|}\left( v(N)-v(N{\setminus }\{i_k\})\right) \\ {}&= \sum _{k=1}^{|N{\setminus } S|}\left( v((N{\setminus }\{i_k\})\cup \{i_k\})-v(N{\setminus }\{i_k\})\right) \\ {}&\ge \sum _{k=1}^{|N{\setminus } S|}\left( v((N{\setminus }\{i_1,\ldots ,i_k\})\cup \{i_k\})-v(N{\setminus }\{i_1,\ldots ,i_k\})\right) \\ {}&= v(N)-v(N{\setminus }(N{\setminus } S)) \\ {}&= v(N)-v(S), \end{aligned}$$

where the first equality and the inequality follow from convexity. Hence, $\sum _{i\in N{\setminus } S}u_i^*(v)=v(N)-v(S)$ for all $S\in 2^N{\setminus }\{\emptyset \}$. $\square$

One-bound core games are characterized by the structure of the core. A balanced game is a lower bound core game if and only if each player obtains its maximal payoff within the core exactly when the other players obtain their minimal payoffs within the core, or equivalently, in each extreme point of the core one player obtains its maximal payoff within the core and all other players obtain their minimal payoffs within the core. Similarly, a balanced game is an upper bound core game if and only if each player obtains its minimal payoff within the core exactly when the other players obtain their maximal payoffs within the core, or equivalently, in each extreme point of the core one player obtains its minimal payoff within the core and all other players obtain their maximal payoffs within the core. These observations are captured by the following theorem.

Theorem 5

• (i) A game $v\in \Gamma _b^N$ is a lower bound core game if and only if $u_i^*(v)+\sum _{j\in N{\setminus }\{i\}}l_j^*(v)=v(N)$ for all $i\in N$, or equivalently, $C(v)=conv\{(u_i^*(v),l_{N{\setminus }\{i\}}^*(v))\mid i\in N\}$.²

• (ii) A game $v\in \Gamma _b^N$ is an upper bound core game if and only if $l_i^*(v)+\sum _{j\in N{\setminus }\{i\}}u_j^*(v)=v(N)$ for all $i\in N$, or equivalently, $C(v)=conv\{(l_i^*(v),u_{N{\setminus }\{i\}}^*(v))\mid i\in N\}$.

Proof

(i) The only-if part follows directly from Lemma 2. For the if-part, let $v\in \Gamma _b^N$ and assume that $u_i^*(v)+\sum _{j\in N{{\setminus }}\{i\}}l_j^*(v)=v(N)$ for all $i\in N$. For each $i\in N$ and each $x\in C(v)$ such that $x_i=u_i^*(v)$, we have $x_j=l_j^*(v)$ for all $j\in N{\setminus }\{i\}$, which implies that $(u_i^*(v),l_{N{\setminus }\{i\}}^*(v))\in C(v)$. Convexity of the core implies that $conv\{(u_i^*(v),l_{N{{\setminus }}\{i\}}^*(v))\mid i\in N\}\subseteq C(v)$. Let $x\in C(v)$. Define $\lambda \in [0,1]^N$ by

$$\begin{aligned}\lambda _i=\frac{x_i-l_i^*(v)}{v(N)-\sum _{j\in N}l_j^*(v)}\quad \text {for all }i\in N.\end{aligned}$$

Then $\sum _{i\in N}\lambda _i=1$ and $x=\sum _{i\in N}\lambda _i(u_i^*(v),l_{N{{\setminus }}\{i\}}^*(v))$, so $x\in conv\{(u_i^*(v),l_{N{\setminus }\{i\}}^*(v))\mid i\in N\}$. Hence, $C(v)=conv\{(u_i^*(v),l_{N{\setminus }\{i\}}^*(v))\mid i\in N\}$. Then $\sum _{i\in S}l_i^*(v)\ge v(S)$ for all $S\in 2^N{\setminus }\{N\}$. Now, let $x\in X(v)$ be such that $l^*(v)\le x$. For each $S\in 2^N{\setminus }\{N\}$,

$$\begin{aligned}\sum _{i\in S}x_i\ge \sum _{i\in S}l_i^*(v)\ge v(S),\end{aligned}$$

which implies that $x\in C(v)$, so $C(v)=\{x\in X(v)\mid l^*(v)\le x\}$. Hence, $v\in \Gamma _l^N$.

(ii) The only-if part follows directly from Lemma 2. For the if-part, let $v\in \Gamma _b^N$ and assume that $l_i^*(v)+\sum _{j\in N{\setminus }\{i\}}u_j^*(v)=v(N)$ for all $i\in N$. For each $i\in N$ and each $x\in C(v)$ such that $x_i=l_i^*(v)$, we have $x_j=u_j^*(v)$ for all $j\in N{\setminus }\{i\}$, which implies that $(l_i^*(v),u_{N{\setminus }\{i\}}^*(v))\in C(v)$. Convexity of the core implies that $conv\{(l_i^*(v),u_{N{{\setminus }}\{i\}}^*(v))\mid i\in N\}\subseteq C(v)$. Let $x\in C(v)$. Define $\lambda \in [0,1]^N$ by

$$\begin{aligned}\lambda _i=\frac{u_i^*(v)-x_i}{\sum _{j\in N}u_j^*(v)-v(N)}\quad \text {for all }i\in N.\end{aligned}$$

Then $\sum _{i\in N}\lambda _i=1$ and $x=\sum _{i\in N}\lambda _i(l_i^*(v),u_{N{\setminus }\{i\}}^*(v))$, so $x\in conv\{(l_i^*(v),u_{N{\setminus }\{i\}}^*(v))\mid i\in N\}$. Hence, $C(v)=conv\{(l_i^*(v),u_{N{\setminus }\{i\}}^*(v))\mid i\in N\}$. Then $v(N)-\sum _{i\in N{\setminus } S}u_i^*(v)\ge v(S)$ for all $S\in 2^N{\setminus }\{\emptyset \}$. Now, let $x\in X(v)$ be such that $x\le u^*(v)$. For each $S\in 2^N{\setminus }\{\emptyset \}$,

$$\begin{aligned}\sum _{i\in S}x_i=\sum _{i\in N}x_i-\sum _{i\in N{\setminus } S}x_i=v(N)-\sum _{i\in N{\setminus } S}x_i\ge v(N)-\sum _{i\in N{\setminus } S}u_i^*(v)\ge v(S),\end{aligned}$$

which implies that $x\in C(v)$, so $C(v)=\{x\in X(v)\mid x\le u^*(v)\}$. Hence, $v\in \Gamma _u^N$. $\square$

The reduced game (cf. Davis and Maschler 1965) of $v\in \Gamma _b^N$ on $T\in 2^N{\setminus }\{\emptyset \}$ with respect to $x\in {\mathbb {R}}^N$, denoted by $v^x_T\in \Gamma ^T$, is defined by

$$\begin{aligned} v_T^x(S)= {\left\{ \begin{array}{ll} v(N)-\sum _{i\in N{\setminus } T}x_i &{}\text {if }S=T; \\ \max _{Q\subseteq N{\setminus } T}\left\{ v(S\cup Q)-\sum _{i\in Q}x_i\right\} &{}\text {if }S\in 2^T{\setminus }\{\emptyset ,T\}; \\ 0 &{} \text {if }S=\emptyset . \end{array}\right. } \end{aligned}$$

In other words, the worth of a coalition in a reduced game is defined as the maximal remainder in cooperation with any subgroup of players in the original game that are not present in the reduced game. We show that a balanced game is a lower bound core game if and only if all reduced games with respect to core allocations have the same lower exact core bound. Similarly, a balanced game is an upper bound core game if and only if all reduced games with respect to core allocations have the same upper exact core bound. We use the following lemma, which follows from Peleg (1986) and Hwang and Sudhölter (2001).

Lemma 3

Let $v\in \Gamma _b^N$, let $T\in 2^N{\setminus }\{\emptyset \}$, and let $x\in C(v)$. Then $C(v_T^x)=\{y\in X(v_T^x)\mid (y,x_{N{\setminus } T})\in C(v)\}$.

Theorem 6

• (i) A game $v\in \Gamma _b^N$ is a lower bound core game if and only if $l^*(v_T^x)=l_T^*(v)$ for all $T\in 2^N$ with $|T|\ge 2$ and all $x\in C(v)$.

• (ii) A game $v\in \Gamma _b^N$ is an upper bound core game if and only if $u^*(v_T^x)=u_T^*(v)$ for all $T\in 2^N$ with $|T|\ge 2$ and all $x\in C(v)$.

Proof

(i) Let $v\in \Gamma _b^N$. For the only-if part, assume that $v\in \Gamma _l^N$. Let $T\in 2^N$ with $|T|\ge 2$ and let $x\in C(v)$. By Lemma 3, $l^*(v_T^x)\ge l_T^*(v)$. For each $i\in T$, define $y^i\in X(v_T^x)$ by $y_i^i=\sum _{j\in T}x_j-\sum _{j\in T{\setminus }\{i\}}l_j^*(v)$ and $y_j^i=l_j^*(v)$ for all $j\in T{\setminus }\{i\}$. For each $i\in T$,

$$\begin{aligned}y_i^i=\sum _{j\in T}x_j-\sum _{j\in T{\setminus }\{i\}}l_j^*(v)=x_i+\sum _{j\in T{\setminus }\{i\}}x_j-\sum _{j\in T{\setminus }\{i\}}l_j^*(v)\ge x_i\ge l_i^*(v),\end{aligned}$$

which implies that $l^*(v)\le (y^i,x_{N{{\setminus }} T})$, so $(y^i,x_{N{\setminus } T})\in C(v)$. By Lemma 3, $y^i\in C(v_T^x)$ for all $i\in T$, so $l^*(v_T^x)\le l_T^*(v)$. Hence, $l^*(v_T^x)=l_T^*(v)$.

For the if-part, assume that $l^*(v_T^x)=l_T^*(v)$ for all $T\in 2^N$ with $|T|\ge 2$ and all $x\in C(v)$. If $|N|\le 2$, then $v\in \Gamma _l^N$ by Theorem 3. Suppose that $|N|\ge 3$. Denote $N=\{1,\ldots ,|N|\}$. Let $x^1\in C(v)$ be such that $x_1^1=l_1^*(v)$. Then $l^*(v_{N{\setminus }\{1\}}^{x^1})=l_{N{\setminus }\{1\}}^*(v)$. Let $x^2\in C(v_{N{\setminus }\{1\}}^{x^1})$ be such that $x_2^2=l_2^*(v_{N{\setminus }\{1\}}^{x^1})=l_2^*(v)$. By Lemma 3, $(x^2,x_1^1)\in C(v)$. Moreover, $l^*(v_{N{\setminus }\{1,2\}}^{(x^2,x_1^1)})=l_{N{\setminus }\{1,2\}}^*(v)$ if $|N|>3$. If $|N|>3$, let $x^3\in C(v_{N{{\setminus }}\{1,2\}}^{(x^2,x_1^1)})$ be such that $x_3^3=l_3^*(v_{N{\setminus }\{1,2\}}^{(x^2,x_1^1)})=l_3^*(v)$. By Lemma 3, $(x^3,x_2^2,x_1^1)\in C(v)$. Continuing this reasoning, $(v(N)-\sum _{i=1}^{|N|-1}l_i^*(v),l_{\{1,\ldots ,|N|-1\}}^*(v))\in C(v)$. This holds for all permutations, so $(u_i^*(v),l_{N{{\setminus }}\{i\}}^*(v))\in C(v)$ for all $i\in N$. Convexity of the core implies that $conv\{(u_i^*(v),l_{N{{\setminus }}\{i\}}^*(v))\mid i\in N\}\subseteq C(v)$. Now, let $x\in C(v)$. Define $\lambda \in [0,1]^N$ by

$$\begin{aligned}\lambda _i=\frac{x_i-l_i^*(v)}{v(N)-\sum _{j\in N}l_j^*(v)}\quad \text {for all }i\in N.\end{aligned}$$

Then $\sum _{i\in N}\lambda _i=1$ and $x=\sum _{i\in N}\lambda _i(u_i^*(v),l_{N{{\setminus }}\{i\}}^*(v))$, so $x\in conv\{(u_i^*(v),l_{N{\setminus }\{i\}}^*(v))\mid i\in N\}$. This implies that $C(v)=conv\{(u_i^*(v),l_{N{\setminus }\{i\}}^*(v))\mid i\in N\}$. Hence, by Theorem 5, $v\in \Gamma _l^N$.

(ii) Let $v\in \Gamma _b^N$. For the only-if part, assume that $v\in \Gamma _u^N$. Let $T\in 2^N$ with $|T|\ge 2$ and let $x\in C(v)$. Then $u^*(v_T^x)\le u_T^*(v)$. For each $i\in T$, define $y^i\in X(v_T^x)$ by $y_i^i=\sum _{j\in T}x_j-\sum _{j\in T{{\setminus }}\{i\}}u_j^*(v)$ and $y_j^i=u_j^*(v)$ for all $j\in T{{\setminus }}\{i\}$. For each $i\in T$,

$$\begin{aligned}y_i^i=\sum _{j\in T}x_j-\sum _{j\in T{\setminus }\{i\}}u_j^*(v)=x_i+\sum _{j\in T{\setminus }\{i\}}x_j-\sum _{j\in T{\setminus }\{i\}}u_j^*(v)\le x_i\le u_i^*(v),\end{aligned}$$

which implies that $(y^i,x_{N{{\setminus }} T})\le u^*(v)$, so $(y^i,x_{N{{\setminus }} T})\in C(v)$. By Lemma 3, $y^i\in C(v_T^x)$ for all $i\in T$, so $u^*(v_T^x)\ge u_T^*(v)$. Hence, $u^*(v_T^x)=u_T^*(v)$.

For the if-part, assume that $u^*(v_T^x)=u_T^*(v)$ for all $T\in 2^N$ with $|T|\ge 2$ and all $x\in C(v)$. If $|N|\le 2$, then $v\in \Gamma _u^N$ by Theorem 3. Suppose that $|N|\ge 3$. Denote $N=\{1,\ldots ,|N|\}$. Let $x^1\in C(v)$ be such that $x_1^1=u_1^*(v)$. Then $u^*(v_{N{\setminus }\{1\}}^{x^1})=u_{N{\setminus }\{1\}}^*(v)$. Let $x^2\in C(v_{N{\setminus }\{1\}}^{x^1})$ be such that $x_2^2=u_2^*(v_{N{\setminus }\{1\}}^{x^1})=u_2^*(v)$. By Lemma 3, $(x^2,x_1^1)\in C(v)$. Moreover, $u^*(v_{N{\setminus }\{1,2\}}^{(x^2,x_1^1)})=u_{N{\setminus }\{1,2\}}^*(v)$ if $|N|>3$. If $|N|>3$, let $x^3\in C(v_{N{{\setminus }}\{1,2\}}^{(x^2,x_1^1)})$ be such that $x_3^3=u_3^*(v_{N{\setminus }\{1,2\}}^{(x^2,x_1^1)})=u_3^*(v)$. By Lemma 3, $(x^3,x_2^2,x_1^1)\in C(v)$. Continuing this reasoning, $(v(N)-\sum _{i=1}^{|N|-1}u_i^*(v),u_{\{1,\ldots ,|N|-1\}}^*(v))\in C(v)$. This holds for all permutations, so $(l_i^*(v),u_{N{{\setminus }}\{i\}}^*(v))\in C(v)$ for all $i\in N$. Convexity of the core implies that $conv\{(l_i^*(v),u_{N{{\setminus }}\{i\}}^*(v))\mid i\in N\}\subseteq C(v)$. Now, let $x\in C(v)$. Define $\lambda \in [0,1]^N$ by

$$\begin{aligned}\lambda _i=\frac{u_i^*(v)-x_i}{\sum _{j\in N}u_j^*(v)-v(N)}\quad \text {for all }i\in N.\end{aligned}$$

Then $\sum _{i\in N}\lambda _i=1$ and $x=\sum _{i\in N}\lambda _i(l_i^*(v),u_{N{{\setminus }}\{i\}}^*(v))$, so $x\in conv\{(l_i^*(v),u_{N{{\setminus }}\{i\}}^*(v))\mid i\in N\}$. This implies that $C(v)=conv\{(l_i^*(v),u_{N{\setminus }\{i\}}^*(v))\mid i\in N\}$. Hence, by Theorem 5, $v\in \Gamma _u^N$. $\square$

By Lemma 3 and Theorem 6, reduced games of one-bound core games with respect to core allocations are again one-bound core games with the same exact core bound. Typically, the exact core bounds of reduced balanced games are different from the exact core bounds of the original game. Theorem 6 actually states that if all reduced games have the same exact core bound, then the original game has to be a one-bound core game.

4 Nucleolus

In this section, we analyze the nucleolus for one-bound core games. The nucleolus for one-bound core games is the unique pre-imputation that is a convex combination of the lower exact core bound and the upper exact core bound.

Theorem 7

• (i) Let $v\in \Gamma _l^N$ be a lower bound core game. Then

  $$\begin{aligned}\eta (v)=\frac{1}{|N|}u^*(v)+\left( 1-\frac{1}{|N|}\right) l^*(v).\end{aligned}$$

• (ii) Let $v\in \Gamma _u^N$ be an upper bound core game. Then

  $$\begin{aligned}\eta (v)=\frac{1}{|N|}l^*(v)+\left( 1-\frac{1}{|N|}\right) u^*(v).\end{aligned}$$

Proof

(i) By Theorem 3, $v\in \Gamma _t^N$. By Theorem 5, $u_i^*(v)-l_i^*(v)=v(N)-\sum _{j\in N}l_j^*(v)$ for all $i\in N$. This implies that $\eta _i(v)-l_i^*(v)=\eta _j(v)-l_j^*(v)$ for all $i,j\in N$, so for each $i\in N$, we have

$$\begin{aligned} \eta _i(v)&= l_i^*(v)+\frac{1}{|N|}\left( v(N)-\sum _{j\in N}l_j^*(v)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}\left( u_i^*(v)-l_i^*(v)\right) \\ {}&= \frac{1}{|N|}u_i^*(v)+\left( 1-\frac{1}{|N|}\right) l_i^*(v). \end{aligned}$$

(ii) By Theorem 3, $v\in \Gamma _t^N$. By Theorem 5, $u_i^*(v)-l_i^*(v)=\sum _{j\in N}u_j^*(v)-v(N)$ for all $i\in N$. This implies that $\eta _i(v)-l_i^*(v)=\eta _j(v)-l_j^*(v)$ for all $i,j\in N$, so for each $i\in N$, we have

$$\begin{aligned} \eta _i(v)&= l_i^*(v)+\frac{1}{|N|}\left( v(N)-\sum _{j\in N}l_j^*(v)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}\left( l_i^*(v)+\sum _{j\in N{\setminus }\{i\}}u_j^*(v)-\sum _{j\in N}l_j^*(v)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}\sum _{j\in N{\setminus }\{i\}}\left( u_j^*(v)-l_j^*(v)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}\left( |N|-1\right) \left( u_i^*(v)-l_i^*(v)\right) \\ {}&= \frac{1}{|N|}l_i^*(v)+\left( 1-\frac{1}{|N|}\right) u_i^*(v). \end{aligned}$$

$\square$

Corollary 2

Let $v\in \Gamma _l^N\cup \Gamma _u^N$ be a one-bound core game. Then

$$\begin{aligned}\eta (v)=\lambda l^*(v)+\left( 1-\lambda \right) u^*(v),\end{aligned}$$

where $\lambda \in [0,1]$ is such that $\sum _{i\in N}\eta _i(v)=v(N)$.

The nucleolus for one-bound core games is characterized by the properties that require that the difference between the allocation and the minimal payoff or maximal payoff within the core is equal for all players. We refer to these properties as balanced lower gaps and balanced upper gaps, respectively.

Definition 2

A solution $\varphi$ on a subdomain of balanced games satisfies balanced lower gaps if for each game v in this domain, it holds that $\varphi _i(v)-l_i^*(v)=\varphi _j(v)-l_j^*(v)$ for all $i,j\in N$.

A solution $\varphi$ on a subdomain of balanced games satisfies balanced upper gaps if for each game v in this domain, it holds that $u_i^*(v)-\varphi _i(v)=u_j^*(v)-\varphi _j(v)$ for all $i,j\in N$.

Theorem 8

• (i) The nucleolus is the unique solution for one-bound core games satisfying balanced lower gaps.

• (ii) The nucleolus is the unique solution for one-bound core games satisfying balanced upper gaps.³

Proof

(i) Let $v\in \Gamma _l^N\cup \Gamma _u^N$. Let $i,j\in N$. If $v\in \Gamma _l^N$, then Theorem 5 implies that

$$\begin{aligned}u_i^*(v)-l_i^*(v)=v(N)-\sum _{k\in N}l_k^*(v)=u_j^*(v)-l_j^*(v).\end{aligned}$$

If $v\in \Gamma _u^N$, then Theorem 5 implies that

$$\begin{aligned}u_i^*(v)-l_i^*(v)=\sum _{k\in N}u_k^*(v)-v(N)=u_j^*(v)-l_j^*(v).\end{aligned}$$

By Theorem 7,

$$\begin{aligned}\eta _i(v)-l_i^*(v)=\frac{1}{|N|}\left( u_i^*(v)-l_i^*(v)\right) =\frac{1}{|N|}\left( u_j^*(v)-l_j^*(v)\right) =\eta _j(v)-l_j^*(v).\end{aligned}$$

Hence, the nucleolus satisfies balanced lower gaps.

Let $\varphi$ be a solution on $\Gamma _l^N\cup \Gamma _u^N$ satisfying balanced lower gaps. Let $i\in N$. If $v\in \Gamma _l^N$, then Theorem 5 implies that $v(N)-u_i^*(v)=\sum _{j\in N{{\setminus }}\{i\}}l_j^*(v)$, so

$$\begin{aligned} \varphi _i(v)&= u_i^*(v)+\left( v(N)-u_i^*(v)\right) +\left( \varphi _i(v)-v(N)\right) \\ {}&= u_i^*(v)+\sum _{j\in N{\setminus }\{i\}}l_j^*(v)-\sum _{j\in N{\setminus }\{i\}}\varphi _j(v) \\ {}&= u_i^*(v)-\sum _{j\in N{\setminus }\{i\}}\left( \varphi _j(v)-l_j^*(v)\right) \\ {}&= u_i^*(v)-\sum _{j\in N{\setminus }\{i\}}\left( \varphi _i(v)-l_i^*(v)\right) \\ {}&= u_i^*(v)-\left( |N|-1\right) \left( \varphi _i(v)-l_i^*(v)\right) \\ {}&= u_i^*(v)+\left( |N|-1\right) l_i^*(v)-\left( |N|-1\right) \varphi _i(v), \end{aligned}$$

where the fourth equality follows from balanced lower gaps, and rewriting yields

$$\begin{aligned}\varphi _i(v)=\frac{1}{|N|}u_i^*(v)+\left( 1-\frac{1}{|N|}\right) l_i^*(v).\end{aligned}$$

If $v\in \Gamma _u^N$, then Theorem 5 implies that $l_j^*(v)=v(N)-\sum _{k\in N{\setminus }\{j\}}u_k^*(v)$ for all $j\in N$, so

$$\begin{aligned} \varphi _i(v)&= l_i^*(v)+\frac{1}{|N|}|N|\left( \varphi _i(v)-l_i^*(v)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}\sum _{j\in N}\left( \varphi _j(v)-l_j^*(v)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}\left( \sum _{j\in N}\varphi _j(v)-\sum _{j\in N}l_j^*(v)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}\left( v(N)-\sum _{j\in N}\left( v(N)-\sum _{k\in N{\setminus }\{j\}}u_k^*(v)\right) \right) \\ {}&= l_i^*(v)+\frac{1}{|N|}\left( v(N)-|N|v(N)+\sum _{j\in N}\sum _{k\in N{\setminus }\{j\}}u_k^*(v)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}\left( (1-|N|)v(N)+(|N|-1)\sum _{j\in N}u_j^*(v)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}(|N|-1)\left( \sum _{j\in N}u_j^*(v)-v(N)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}(|N|-1)\left( u_i^*(v)-l_i^*(v)\right) \\ {}&= \frac{1}{|N|}l_i^*(v)+\left( 1-\frac{1}{|N|}\right) u_i^*(v), \end{aligned}$$

where the second equality follows from balanced lower gaps. Hence, by Theorem 7, $\varphi _i(v)=\eta _i(v)$.

(ii) The proof is analogous to the proof of (i). $\square$

As the following example shows, in contrast to the nucleolus, the Shapley value and the $\tau$-value do not assign to each one-bound core game a core allocation.

Example 2

Let $N=\{1,2,3,4\}$ and let $v\in \Gamma _l^N\cap \Gamma _u^N$ be defined by

$$\begin{aligned} v(S)= {\left\{ \begin{array}{ll} 36 &{} \text {if }S=N\text {;} \\ 24 &{} \text {if }S\in \{\{1,2\},\{1,3\},\{2,3\}\}\text {;} \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Then $l^*(v)=u^*(v)=(12,12,12,0)$, so $\eta (v)=(12,12,12,0)$ and $\eta (v)\in C(v)$. However, $\phi (v)=(11,11,11,3)$ and $\tau (v)=(9,9,9,9)$, so $\phi (v)\notin C(v)$ and $\tau (v)\notin C(v)$. $\triangle$

However, the Shapley value and the $\tau$-value assign to each convex one-bound core game a specific core allocation. In fact, the nucleolus, the Shapley value, and the $\tau$-value coincide for convex one-bound core games. This follows directly from the following theorem.

Theorem 9

The nucleolus is the unique solution for convex one-bound core games satisfying symmetry and translation covariance.

Proof

It is known that the nucleolus satisfies symmetry and translation covariance on each subdomain of balanced games. Let $\varphi$ be a solution for convex one-bound core games satisfying symmetry and translation covariance. Let $v\in \Gamma _l^N\cap \Gamma _c^N$. By Theorem 4, $v(S)=\sum _{i\in S}l_i^*(v)$ for all $S\in 2^N{\setminus }\{N\}$. This implies that $(v-l^*(v))(S)=0$ for all $S\in 2^N{\setminus }\{N\}$. Let $i\in N$. By symmetry,

$$\begin{aligned}\varphi _i(v-l^*(v))=\frac{1}{|N|}\left( v(N)-\sum _{j\in N}l_j^*(v)\right) .\end{aligned}$$

By translation covariance and Theorem 5,

$$\begin{aligned} \varphi _i(v)&= l_i^*(v)+\varphi _i(v-l^*(v)) \\ {}&= l_i^*(v)+\frac{1}{|N|}\left( v(N)-\sum _{j\in N}l_j^*(v)\right) \\ {}&= l_i^*(v)+\frac{1}{|N|}\left( u_i^*(v)-l_i^*(v)\right) \\ {}&= \frac{1}{|N|}u_i^*(v)+\left( 1-\frac{1}{|N|}\right) l_i^*(v). \end{aligned}$$

Hence, by Theorem 7, $\varphi _i(v)=\eta _i(v)$. The case $v\in \Gamma _u^N\cap \Gamma _c^N$ follows analogously. $\square$

Theorem 9 implies that the nucleolus for convex one-bound core games coincides with each other solution satisfying symmetry and translation covariance. This observation is in line with Yokote et al. (2017). In particular, the nucleolus for convex one-bound core games coincides with the Shapley value and the $\tau$-value.

Corollary 3

The nucleolus, the Shapley value, and the $\tau$-value coincide for convex one-bound core games.

Example 3

Let $N=\{1,2,3\}$ and let $v\in \Gamma _l^N\cap \Gamma _c^N$ be defined by

$$\begin{aligned} v(S)= {\left\{ \begin{array}{ll} 3 &{} \text {if }S=N\text {;} \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Then $\eta (v)=\phi (v)=\tau (v)=(1,1,1)$. $\triangle$

The game from Example 3 is not a 1-convex game, so Corollary 3 does not follow from Driessen and Tijs (1985). Kar et al. (2009) and Trudeau and Vidal-Puga (2020) showed that the nucleolus and the Shapley value coincide for so-called PS-games and clique games, respectively. It can be shown that the game from Example 3 is not a PS-game or a clique game. This implies that it is neither a 2-additive game (cf. Deng and Papadimitriou 1994), so Corollary 3 does neither follow from their results, nor from Van den Nouweland et al. (1996), Chun and Hokari (2007), or Chun et al. (2016).

5 Concluding remarks

In this paper, the new class of one-bound core games is introduced. By Theorem 3, all one-bound core games are two-bound core games. By Lemma 3 and Theorem 6, all reduced games of one-bound core games with respect to core allocations are one-bound core games. This implies that the axiomatic characterizations of the core, the nucleolus, and the egalitarian core (cf. Arin and Iñarra 2001) provided by Gong et al. (2022a) on the class of two-bound core games can be reformulated on the class of one-bound core games. However, the exact core bounds of reduced two-bound core games are not necessarily the same. By Theorem 6, if all reduced games of a balanced game have the same exact core bound, then it is necessarily a one-bound core game.