Non-Emptiness, Relative Coincidences and Axiomatic Results for the Precore

Abstract

Taking into account the definition of the precore, we initially analyze its non-emptiness using duality results from linear programming theory. We then introduce the dominance core to investigate the coincident relations between the precore and dominance core. Finally, we propose specific reduction approaches to axiomatize the precore.

1. Introduction

The properties of consistency and its converse play significant roles under axiomatic approaches to traditional games. These two essential properties have been extensively studied across different issues, often employing the concept of reductions. Numerous definitions of reduction have been introduced, each specifying how the agents outside the subgroup should be compensated.

In summary, the core is considered the most intuitive allocation concept in game theory. Axiomatic analysis and nonemptiness are two important research topics concerning the core. There exist two distinct types of fictitious reductions in the axiomatic analysis of the core, namely, the “Davis and Maschler’s [1] reduction” and the “Moulin’s [2] reduction”. Results found in the existing literature include: (1) Axiomatizations of the core under Moulin’s [2] reduction (cf. Tadenuma [3]); (2) axiomatizations of the core under Davis and Maschler’s [1] reduction and its analogues (cf. Peleg [4]; Serrano and Volij [5]; Voorneveld and van den Nouweland [6]). In addition, based on the constructed requirements for the core in the framework of mathematical models, one can have that the core is a sensible solution given to all players under the deliberation of effective distribution of resources that are unsatisfactory but admissible to whole individuals or generated coalitions. Nevertheless, the resource distribution applying the core is mathematically coincident with the intersection of all relevant sets formed via individuals or generated coalitions. Consequently, there is a possibility of generating an empty condition, which, in reality, expresses that a consensus distribution might not be realized. Recently, some results of nonemptiness for core concepts have been discussed, such as Abe [7], Basile and Scalzo [8], Cesco [9], Liu [10], Predtetchinski and Herings [11], and so on.

Based on the incorporation of the core allocating the notion in consideration of agents’ multichoice behavior, a generalized core for multichoice games was proposed by van den Nouweland et al. [12]. Later, Hwang et al. [13] explored relative analogues of the reductions and corresponding consistency measures proposed by Davis and Maschler [1] and Moulin [2] to analyze the extended core proposed by van den Nouweland et al. [12]. However, considering the absence of satisfaction regarding related consistency, Hwang et al. [13] introduced the notion of the precore as a means to minimally expand the core and restore the desired level of consistency. Inspired by the axiomatic notion of Serrano and Volij [5], relevant axiomatic results for the precore have been proposed. Hwang et al. [13] utilized the extended reduction proposed by Davis and Maschler [1] to axiomatize the precore, demonstrating that the precore cannot be axiomatized using the extended reduction proposed by Moulin [2]. Due to the violation of converse consistency by the precore under the extended Moulin’s reduction, Liao [14] introduced a converse consistent extension. This extension represents the minimal set of solutions that satisfy converse consistency and include the precore. Cheng et al. [15] defined an extended reduction based on Serrano and Volij [5] to axiomatize the precore. Nevertheless, the existing research has not explored the conditions related to the nonemptiness of the precore.

Therefore, the main objective of this study was to analyze the nonemptiness of the precore and propose different axiomatic results.

• In Section 3, we begin by establishing the nonemptiness of the precore based on specific requirements through the application of duality results from linear programming theory. We also introduce the concept of domination to define the dominance core;
• Inspired by the related results of Chang [16] and Hwang [17], we explore the interrelations between the precore and the dominance core, examining necessary and sufficient conditions by utilizing the nonemptiness of the precore;
• To assess the rationality of the precore, an extended Voorneveld and van den Nouweland’s [6] reduction is introduced to axiomatize the precore in Section 4. Further, we propose a particular reduction method that combines notions introduced by Serrano and Volij [5] and Voorneveld and van den Nouweland [6] to axiomatize the precore.

2. Preliminaries

Let A be the grand collection of all participating agents. For $t\in A$ and $p_t\in \mathbb{N}$, $P_t=\{0,1,\cdots ,p_t\}$ to be the participating option collection of agent t, where 0 denotes the nonparticipating option. Further, let $P^A={\prod }_{t\in A}{P}_t$ be the product set of the participating option collections of agents under A, and $K^A=\{(t,j_t)\mid t\in A,j_t\in P_t^{+}\}$, where $P_t^{+}=P_t\setminus \left\{0\right\}$. Denote the zero vector by $0_A$ under ${\mathbb{R}}^A$.

Denote $(A,p,U)$ to be a multichoice game, where A is a nonempty and finite collection of agents, $p={\left(p_t\right)}_{t\in A}$ is a vector that expresses the amount of participating options for every agent, and $U:P^A\to \mathbb{R}$ is a measure mapping which expresses to each $\kappa \in P^A$ the merit that the agents could receive if each agent t takes the participating option ${\kappa }_t\in P_t$ with $U\left(0_A\right)=0$. Denote the family of all multichoice games as $\Delta$.

Given $(A,p,U)\in \Delta$, $\kappa \in P^A$ and $T\subseteq A$, we denote ${\kappa }_T\in {\mathbb{R}}^T$ to be the restriction of $\kappa$ to T and $N\left(\kappa \right)=\{t\in A|{\kappa }_t\ne 0\}$. Let $\left|T\right|$ be the amount of components in T, and let ${\delta }^T\left(A\right)\in {\mathbb{R}}^A$ be the vector satisfying ${\delta }_i^T\left(A\right)=1$ if $i\in T$, and ${\delta }_i^T\left(A\right)=0$ if $i\notin T$. ${\delta }^T\left(A\right)$ will be denoted by ${\delta }^T$ if no confusion can occur.

An outcome vector of a game $(A,p,U)$ is a vector $\tau ={\left({\tau }_{t,j_t}\right)}_{(t,j_t)\in K^A}$, where ${\tau }_{t,j_t}$ signifies the outcome of t at option $j_t$. For convenience, one could define ${\tau }_{t,0}=0$ for each $t\in A$. For $\kappa \in P^A$, one would define $\tau \left(\kappa \right)={\sum }_{t\in A}{\sum }_{j_t=1}^{{\kappa }_t}{\tau }_{t,j_t}$. Then, $\tau$ is efficient (EIT) if $\tau \left(p\right)=U\left(p\right)$. $\tau$ is individually rational (IRL) if $\tau \left(j_t{\delta }^{\left\{t\right\}}\right)\ge U\left(j_t{\delta }^{\left\{t\right\}}\right)$ for all $(t,j_t)\in K^A$. $\tau$ is coalitionally rational (CRL) if $\tau \left(\kappa \right)\ge U\left(\kappa \right)$ for all $\kappa \in P^A$.

EIT denotes the condition where all agents collectively allocate the entire resource. IRL denotes the condition where the cumulative outcomes ${\sum }_{x=1}^{j_t}{\tau }_{t,x}$ for agent t are at least as good as the outcome that agent t can achieve when operating alone with option $j_t$. CRL denotes the condition where the sum of the outcomes $\tau \left(\kappa \right)$ for all agents is at least as good as the collective outcome that all agents can achieve when each agent t operates with participating option ${\kappa }_t$. The collection of all efficient vectors of $(A,p,U)$ is denoted by $\Theta (A,p,U)$. The collection of all practical outcome vectors of $(A,p,U)$ is denoted by

$${\Theta }^{*}(A,p,U)=\{\tau \in {\mathbb{R}}^{K^A}|\tau \left(p\right)\le U\left(p\right)\}.$$

$\tau$ is said to be an imputation of $(A,p,U)$ if $\tau$ is EIT and IRL under $(A,p,U)$, and the collection of all imputations under $(A,p,U)$ is denoted as $\mathbb{I}(A,p,U)=\{\tau \in \Theta (A,p,U\left)\right|\tau \text{ is IRL}\}$.

A solution on $\Delta$ is a function $\sigma$, which associates with each game $(A,p,U)$ a subset $\sigma (A,p,U)\subseteq {\Theta }^{*}(A,p,U)$. Hwang et al. [13] introduced an extended core as follows:

Definition 1. 

(Hwang et al. [13]) Let $(A,p,U)\in \Delta$. The precore $C_{Pre}(A,p,U)$ consists of all efficient vectors of $(A,p,U)$, which satisfy CRL, i.e.,

$$C_{Pre}(A,p,U)=\{\tau \in \Theta (A,p,U)|\tau is \text{CRL}\}.$$

3. Nonemptiness, Dominance Core, and Coincident Relations

By applying the related results of Bondareva [18] and Shapley [19], this section presents a necessary and sufficient condition for the nonemptiness of the precore.

Definition 2. 

A multichoice game $(A,p,U)$ is balanced if, for every mapping $\eta :P^A\to {\mathbb{R}}_{+}$ satisfying

$$\sum _{\stackrel{\kappa \in P^A}{{\kappa }_t\ge j_t}}\eta \left(\kappa \right)=1for all(t,j_t)\in K^A,$$

(1)

it holds that ${\displaystyle \sum _{\kappa \in P^A}}\eta \left(\kappa \right)·U\left(\kappa \right)\le U\left(p\right).$

Theorem 1. 

The precore for the multichoice game $(A,p,U)$ is not empty if and only if $(A,p,U)$ is balanced.

Proof. 

Categorically, the precore for the multichoice game $(A,p,U)$ is not empty if and only if there exists $\tau ={\left({\tau }_{t,j_t}\right)}_{(t,j_t)\in K^A}$ such that

$$\tau \left(p\right)=U\left(p\right),$$

(2)

and

$$\tau \left(\kappa \right)\ge U\left(\kappa \right)\text{for all}\kappa \in P^A.$$

(3)

Let $\mathbb{W}=\left\{\tau \right|\tau \left(\kappa \right)\ge U\left(\kappa \right)\text{ for all }\kappa \in P^A\}$. Then, there exists $\tau ={\left({\tau }_{t,j_t}\right)}_{(t,j_t)\in K^A}$ matches for Equations (2) and (3) if and only if there exists ${\tau }^{*}\in \mathbb{W}$ such that

$$U\left(p\right)={\tau }^{*}\left(p\right)=min\left\{\tau \left(p\right)\right|\tau \in \mathbb{W}\}.$$

(4)

Let $\Omega =\left\{\eta \right|\eta :P^A\to {\mathbb{R}}_{+} \text{and} \eta \text{matches }\left(1\right)\}$. By applying the duality results of linear programming theory, Equation (4) coincides with

$$U\left(p\right)=max\{\sum _{\kappa \in P^A}\eta \left(\kappa \right)·U\left(\kappa \right)|\eta \in \Omega \}.$$

That is, $(A,p,U)$ is balanced. □

Let ${\Delta }_C$ be the collection of all balanced multichoice games. Let $(A,p,U)$ be a multichoice game, $\tau ,\zeta$ be imputations and $\kappa$ be a participating option vector. $\zeta$ dominates $\tau$ via $\kappa$, denoted as $\zeta$ $do{m}_{\kappa }$$\tau$ if $\zeta \left(\kappa \right)\le U\left(\kappa \right)$ and ${\zeta }_{t,j_t}>{\tau }_{t,j_t}$ for all $t\in N\left(\kappa \right)$ and for all $j_t\in \{1,\cdots ,{\kappa }_t\}$. $\zeta$ dominates $\tau$ if there exists $\kappa \in P^A$ such that $\zeta$ $do{m}_{\kappa }$$\tau$.

Definition 3. 

The  dominance core  $C_{Dom}(A,p,U)$ consists of all imputation τ of $(A,p,U)$ for which there does not exist imputation ζ of $(A,p,U)$ such that ζ dominates τ.

Lemma 1. 

The precore is a subset of the dominance core for all multichoice games.

Proof. 

Assume that there exists $(A,p,U)\in \Delta$ and $\tau \in C_{Pre}(A,p,U)$ such that $\tau \notin C_{Dom}(A,p,U)$. Therefore, there exists $\zeta \in \mathbb{I}(A,p,U)$ and $\kappa \in P^A$ with $\kappa \ne 0_A$ such that $\zeta$ $do{m}_{\kappa }$$\tau$. Then,

$$U\left(\kappa \right)\ge \zeta \left(\kappa \right)>\tau \left(\kappa \right)\ge U\left(\kappa \right),$$

which demonstrates a clear contradiction. Therefore, $\tau \in C_{Dom}(A,p,U)$. □

Remark 1. 

The following example can present that the dominance core is not empty, but the precore is empty for a multichoice game. Let $(A,p,U)\in \Delta$ with $A=\{i,t,k\}$, $p=(1,1,1)$, $U(1,1,1)=1$, $U(1,1,0)=2$, $U\left(\kappa \right)=0$ for all $\kappa \in P^A\setminus \{(1,1,1),(1,1,0)\}$. It is easy to determine that $C_{Dom}=\{\tau \in \mathbb{I}(A,p,U)|{\tau }_{k,1}={\tau }_{k,0}=0\}$ and $C_{Pre}(A,p,U)=\varnothing$.

From Lemma 1 and the above example, it can be deduced that the precore is a subset of the dominance core for all multichoice games, and the precore is not equal to the dominance core for some multichoice games. Furthermore, the precore and the dominance core are formed by distinct definitions and possess their own characteristics. Therefore, if the precore coincides with the dominance core under certain reasonable conditions, it signifies that the definitions and characteristics attributed to the precore and the dominance core can mutually describe each other. In other words, in those reasonable conditions, both the precore and the dominance core can have multiple interpretations. This serves as the basis for the forthcoming investigation and analysis.

A multichoice game $(A,p,U)$ is zero-normalized if $U\left(j_t{\delta }^{\left\{t\right\}}\right)=0$ for all $(t,j_t)\in K^A$. The normalization of a multichoice game $(A,p,U)$, denoted by $(A,p,U_{*})$, is defined by

$$U_{*}\left(\kappa \right)=U\left(\kappa \right)-\sum _{t\in A}U\left({\kappa }_t{\delta }^{\left\{t\right\}}\right)\text{for all}\kappa \in P^A.$$

Remark 2. 

1. 

By Lemma 1 and Definitions 1, 3, $C_{Pre}(A,p,U)\subseteq C_{Dom}(A,p,U)\subseteq \mathbb{I}(A,p,U)$. Thus, $C_{Pre}(A,p,U)=C_{Dom}(A,p,U)=\varnothing$ if $\mathbb{I}(A,p,U)=\varnothing$. In the remaining portion of this section, one can assume that $\mathbb{I}(A,p,U)\ne \varnothing$;

2. 

It is easy to have that $\tau \left(j_t{\delta }^{\left\{t\right\}}\right)\ge 0$ for all zero-normalized games $(A,p,U)$, for all $\tau \in \mathbb{I}(A,p,U)$, and for all $(t,j_t)\in K^A$. That is, $(A,p,U)=(A,p,U_{*})$ for all zero-normalized games $(A,p,U)$.

Lemma 2. 

Let $(A,p,U)\in \Delta$, $(A,p,U_{*})$ be the normalization of $(A,p,U)$ and τ be an outcome vector under $(A,p,U)$. Define $\zeta :K^A\to \mathbb{R}$ to be that ${\zeta }_{t,j_t}={\tau }_{t,j_t}-U\left(j_t{\delta }^{\left\{t\right\}}\right)+U\left((j_t-1){\delta }^{\left\{t\right\}}\right)$ for all $(t,j_t)\in K^A$.

1. 

$\tau \in \mathbb{I}(A,p,U)$ if and only if $\zeta \in \mathbb{I}(A,p,U_{*})$;

2. 

$\tau \in C_{Pre}(A,p,U)$ if and only if $\zeta \in C_{Pre}(A,p,U_{*})$;

3. 

$\tau \in C_{Dom}(A,p,U)$ if and only if $\zeta \in C_{Dom}(A,p,U_{*})$.

Proof. 

Based on the related definitions of the imputation set, the precore, the dominance core, and the normalization of a game, the proofs could be easily derived. Thus, we omitted it. □

Lemma 3. 

Let $(A,p,U)$ be a multichoice game such that the dominance core is not empty. Then, the precore coincides with the dominance core under multichoice game $(A,p,U)$ if and only if $(A,p,U_{*})$ matches $U_{*}\left(p\right)\ge U_{*}\left(\kappa \right)$ for all $\kappa \in P^A$.

Proof. 

Based on Lemma 2, it suffices to finish this lemma under zero-normalized games. Assume that the multichoice game $(A,p,U)$ is zero-normalized.

Let $(A,p,U)$ be a multichoice game such that the dominance core is not empty. Suppose that the precore coincides with the dominance core under a multichoice game $(A,p,U)$ and let $\tau \in C_{Pre}(A,p,U)$. Based on Remark 2, ${\tau }_{t,j_t}\ge 0$ for all $(t,j_t)\in K^A$. Since $\tau \in C_{Pre}(A,p,U)$,

$$U\left(p\right)=\tau \left(p\right)=\sum _{t\in A}\sum _{j_t=1}^{{\kappa }_t}{\tau }_{t,j_t}+\sum _{t\in A}\sum _{j_t={\kappa }_t+1}^{p_t}{\tau }_{t,j_t}\ge \tau \left(\kappa \right)\ge U\left(\kappa \right)\forall \kappa \in P^A.$$

Now assume that $U\left(p\right)\ge U\left(\kappa \right)$ for all $\kappa \in P^A$. Based on Lemma 1, it suffices to have that $\tau \notin C_{Dom}(A,p,U)$ for all $\tau \in \mathbb{I}(A,p,U)\setminus C_{Pre}(A,p,U)$. Let $\tau \in \mathbb{I}(A,p,U)\setminus C_{Pre}(A,p,U)$. Since $\tau \in \mathbb{I}(A,p,U)\setminus C_{Pre}(A,p,U)$, there exists $\kappa \in P^A$ such that $U\left(\kappa \right)>\tau \left(\kappa \right)$. Define $\zeta :K^A\to \mathbb{R}$ as follows:

$${\zeta }_{i,j_i}=\left\{\begin{array}{cc}{\tau }_{i,j_i}+\frac{U\left(\kappa \right)-\tau \left(\kappa \right)}{{\displaystyle \sum _{t\in A}}{\kappa }_t}\hfill & \text{if} i\in A,j_i\in \{1,\cdots ,{\kappa }_i\}\hfill \\ \frac{U\left(p\right)-U\left(\kappa \right)}{{\displaystyle \sum _{t\in A}}(p_t-{\kappa }_t)}\hfill & \text{if} i\in A,j_i\in \{{\kappa }_i+1,\cdots ,p_i\}.\hfill \end{array}\right.$$

By definition of $\zeta$, $\zeta \left(p\right)=U\left(p\right)$. Since ${\tau }_{i,j_i}\ge 0$ for all $(i,j_i)\in K^A$, $U\left(\kappa \right)>\tau \left(\kappa \right)$ and $U\left(p\right)\ge U\left(\kappa \right)$, we have that for all $(i,j_i)\in K^A$, ${\zeta }_{i,j_i}\ge 0$. Thus, for all $(i,j_i)\in K^A$, $\zeta \left(j_i{\delta }^{\left\{i\right\}}\right)\ge 0=U\left(j_i{\delta }^{\left\{i\right\}}\right)$. Hence, $\zeta$ is IRL under $(A,p,U)$. Since $\zeta$ is EIT and IRL under $(A,p,U)$, $\zeta \in \mathbb{I}(A,p,U)$.

Based on definition of $\zeta$, ${\zeta }_{i,j_i}>{\tau }_{i,j_i}$ for $i\in A$ and $j_i\in \{1,\cdots ,{\kappa }_i\}$. Hence, $\zeta \left({\kappa }_i{\delta }^{\left\{i\right\}}\right)\ge \tau \left({\kappa }_i{\delta }^{\left\{i\right\}}\right)$ for all $i\in A$. In addition,

$$\zeta \left(\kappa \right)=\tau \left(\kappa \right)+\sum _{i\in A}\sum _{j_i=1}^{{\kappa }_i}\frac{U\left(\kappa \right)-\tau \left(\kappa \right)}{{\displaystyle \sum _{k\in A}}{\kappa }_k}=U\left(\kappa \right).$$

Since $\zeta \in \mathbb{I}(A,p,U)$, $\zeta \left({\kappa }_i{\delta }^{\left\{i\right\}}\right)\ge \tau \left({\kappa }_i{\delta }^{\left\{i\right\}}\right)$ for all $i\in A$ and $\zeta \left(\kappa \right)=U\left(\kappa \right)$, $\zeta$ $do{m}_{\kappa }$$\tau$. Hence, $\tau \notin C_{Dom}(A,p,U)$. □

Theorem 2. 

The precore coincides with the dominance core for all balanced multichoice games.

Proof. 

Based on Lemma 2, it suffices to finish this theorem under zero-normalized games. Assume that the multichoice game $(A,p,U)$ is zero-normalized. Based on the proof of Lemma 3, $C_{Pre}(A,p,U)\ne \varnothing$ implies that $U\left(p\right)\ge U\left(\kappa \right)$ for all $\kappa \in P^A$. Since $(A,p,U)$ is balanced, the precore for the multichoice game $(A,p,U)$ is not empty. Based on Lemma 1, the dominance core for the multichoice game $(A,p,U)$ is not empty. By applying Lemma 3, the precore coincides with the dominance core for the multichoice game $(A,p,U)$. □

Inspired by the related results of Chang [16] and Hwang [17], we adopt a specific game to analyze some coincidences among the precore and the dominance core. Let $(A,p,U)$ be a multichoice games. Define a new game $(A,p,U^{\prime })$ by $U^{\prime }\left(\kappa \right)=min\{U\left(p\right),U\left(\kappa \right)\}$ for all $\kappa \in P^A$.

Remark 3. 

By definition of $\mathbb{I}$ and Remark 2, $U\left(p\right)\ge 0$, $U\left(p\right)=U^{\prime }\left(p\right)$, $U\left(j_t{\delta }^{\left\{t\right\}}\right)=U^{\prime }\left(j_t{\delta }^{\left\{t\right\}}\right)=0$ for all $(t,j_t)\in K^A$, $U^{\prime }\left(p\right)\ge U^{\prime }\left(\kappa \right)$ for all $\kappa \in P^A$ and $\mathbb{I}(A,p,U)=\mathbb{I}(A,p,U^{\prime })$ if $(A,p,U)$ is zero-normalized.

Lemma 4. 

Let $(A,p,U)$ be a multichoice game, $\tau ,\zeta$ be imputations, and κ be a participating option vector. Then, ζ $do{m}_{\kappa }$τ in $(A,p,U)$ if and only if ζ $do{m}_{\kappa }$τ in $(A,p,U^{\prime })$.

Proof. 

Based on Lemma 2, it suffices to finish this lemma under zero-normalized games. Assume that the multichoice game $(A,p,U)$ is zero-normalized. If $\zeta$ $do{m}_{\kappa }$$\tau$ in $(A,p,U^{\prime })$, then $\zeta \left(\kappa \right)\le U^{\prime }\left(\kappa \right)$ and ${\zeta }_{t,j_t}>{\tau }_{t,j_t}$ for all $t\in N\left(\kappa \right)$ and for all $j_t\in \{1,\cdots ,{\kappa }_t\}$. Therefore, $\zeta \left(\kappa \right)\le U\left(\kappa \right)$ and $\zeta$ $do{m}_{\kappa }$$\tau$ in $(A,p,U)$. On the other hand, if $\zeta$ $do{m}_{\kappa }$$\tau$ in $(A,p,U)$, then $\zeta \left(\kappa \right)\le U\left(\kappa \right)$ and ${\zeta }_{t,j_t}>{\tau }_{t,j_t}$ for all $t\in N\left(\kappa \right)$ and for all $j_t\in \{1,\cdots ,{\kappa }_t\}$. Since $\zeta \in \mathbb{I}$,

$$U\left(p\right)=\zeta \left(p\right)=\sum _{t\in A}\sum _{j_t=1}^{{\kappa }_t}{\zeta }_{t,j_t}+\sum _{t\in A}\sum _{j_t={\kappa }_t+1}^{p_t}{\zeta }_{t,j_t}\ge \zeta \left(\kappa \right).$$

These imply that $\zeta \left(\kappa \right)\le U^{\prime }\left(\kappa \right)$ and $\zeta$ $do{m}_{\kappa }$$\tau$ in $(A,p,U^{\prime })$. □

Lemma 5. 

For all multichoice games $(A,p,U)$, $C_{Dom}(A,p,U)=C_{Dom}(A,p,U^{\prime })$.

Proof. 

It follows from Lemma 4. □

Lemma 6. 

For all multichoice games $(A,p,U)$, $C_{Pre}(A,p,U^{\prime })=C_{Dom}(A,p,U^{\prime })$.

Proof. 

It follows from Lemma 3 and Remark 3. □

Lemma 7. 

For all multichoice games $(A,p,U)$, $C_{Dom}(A,p,U)=C_{Pre}(A,p,U^{\prime })$.

Proof. 

It follows from Lemmas 5 and 6. □

Lemma 8. 

For all multichoice games $(A,p,U)$, $C_{Dom}(A,p,U)\ne \varnothing$ if and only if $(A,p,U^{\prime })$ is balanced.

Proof. 

It follows from Lemma 7 and Theorem 1. □

Lemma 9. 

For all multichoice games $(A,p,U)$ with $C_{Pre}(A,p,U)\ne \varnothing$, $C_{Pre}(A,p,U)=C_{Pre}(A,p,U^{\prime })$.

Proof. 

By applying Lemmas 1 and 7, it is known that $C_{Pre}(A,p,U)\subseteq C_{Pre}(A,p,U^{\prime })$. It remains to show that $C_{Pre}(A,p,U^{\prime })\subseteq C_{Pre}(A,p,U)$. If $\tau \in C_{Pre}(A,p,U^{\prime })$, then $\tau \in \mathbb{I}(A,p,U^{\prime })=\mathbb{I}(A,p,U)$ and $\tau \left(\kappa \right)\ge U^{\prime }\left(\kappa \right)$ for all $\kappa \in P^A$. By applying Remark 2 and $C_{Pre}(A,p,U)\ne \varnothing$, there exists $\zeta \in C_{Pre}(A,p,U)$ such that for all $\kappa \in P^A$, $\zeta \left(\kappa \right)\ge U\left(\kappa \right)$ and

$$U\left(\kappa \right)\le \zeta \left(\kappa \right)=\sum _{i\in A}\sum _{j=i}^{{\kappa }_t}{\zeta }_{i,j}\le =\sum _{i\in A}\sum _{j=i}^{p_t}{\zeta }_{i,j}.$$

Thus, $U\left(p\right)\ge U\left(\kappa \right)$ for all $\kappa \in P^A$. Therefore, $\tau \left(\kappa \right)\ge U^{\prime }\left(\kappa \right)=U\left(\kappa \right)$ for all $\kappa \in P^A$ and $\tau \in C_{Pre}(A,p,U)$. The proof is completed. □

Lemma 10. 

For all multichoice games $(A,p,U)$, $C_{Pre}(A,p,U)=C_{Pre}(A,p,U^{\prime })$ if and only if $(A,p,U)$ is balanced or $(A,p,U^{\prime })$ is not balanced.

Proof. 

For any multichoice game $(A,p,U)$. If $C_{Pre}(A,p,U)=C_{Pre}(A,p,U^{\prime })$, then either both $C_{Pre}(A,p,U)$ and $C_{Pre}(A,p,U^{\prime })$ are empty or both are nonempty. If both $C_{Pre}(A,p,U)$ and $C_{Pre}(A,p,U^{\prime })$ are empty, then $(A,p,U^{\prime })$ is not balanced. If both $C_{Pre}(A,p,U)$ and $C_{Pre}(A,p,U^{\prime })$ are nonempty, then $(A,p,U)$ is balanced. On the other hand, if $(A,p,U^{\prime })$ is not balanced, then $C_{Pre}(A,p,U)\subseteq C_{Pre}(A,p,U^{\prime })=\varnothing$. This implies that $C_{Pre}(A,p,U)=C_{Pre}(A,p,U^{\prime })$. If $(A,p,U)$ is balanced, then $C_{Pre}(A,p,U)\ne \varnothing$. By applying Lemma 9, $C_{Pre}(A,p,U)=C_{Pre}(A,p,U^{\prime })$. □

Theorem 3. 

For all multichoice games $(A,p,U)$, $C_{Pre}(A,p,U)=C_{Dom}(A,p,U)$ if and only if $(A,p,U)$ is balanced or $(A,p,U^{\prime })$ is not balanced.

Proof. 

Since it is known that $C_{Dom}(A,p,U)=C_{Pre}(A,p,U^{\prime })$ for any multichoice game $(A,p,U)$ by Lemma 7, it suffices to show $C_{Pre}(A,p,U)=C_{Pre}(A,p,U^{\prime })$ if and only if $(A,p,U)$ is balanced or $(A,p,U^{\prime })$ is not balanced. By applying Lemma 10, we obtain that $C_{Pre}(A,p,U)=C_{Dom}(A,p,U)$ if and only if $(A,p,U)$ is balanced or $(A,p,U^{\prime })$ is not balanced. □

Remark 4. 

A set $K\subseteq {\mathbb{R}}^n$ is said to be convex if $\delta \tau +(1-\delta )\zeta \in K$ for all $\tau ,\zeta \in K$ and for all $0\le \delta \le 1$. By definition of the precore, it is easy to verify that $C_{Pre}(A,p,U)$ is convex for all $(A,p,U)\in \Delta$. By applying Lemma 7, it is easy to have that $C_{Dom}(A,p,U)$ is convex for all $(A,p,U)\in \Delta$.

4. Reductions and Axiomatic Results

Hwang et al. [13] and Cheng et al. [15] extended the reduction defined by Davis and Maschler [1], Moulin [2], and Serrano and Volij [5] to multichoice situations as follows: given a multichoice game $(A,p,U)$, a nonempty subset $H\in 2^A$, and an outcome vector $\tau$, the DM-reduction $(H,p_H,U_{DM,H}^{\tau })$ is defined by for all $\kappa \in P^H$,

$$U_{DM,H}^{\tau }\left(\kappa \right)=\left\{\begin{array}{cc}0\hfill & ,\kappa =0_H,\hfill \\ U\left(p\right)-\tau (p_{A\setminus H},0_H)\hfill & ,\kappa =p_H,\hfill \\ \underset{\mu \in P^{A\setminus H}}{max}\{U(\kappa ,\mu )-\tau (\mu ,0_H)\}\hfill & ,\mathrm{otherwise}.\hfill \end{array}\right.$$

The M-reduction $(H,p_H,U_{M,H}^{\tau })$ is defined by for all $\kappa \in P^H$,

$$U_{M,H}^{\tau }\left(\kappa \right)=\left\{\begin{array}{cc}0\hfill & ,\kappa =0_H,\hfill \\ U(\kappa ,p_{A\setminus H})-\tau (p_{A\setminus H},0_H)\hfill & ,\mathrm{otherwise}.\hfill \end{array}\right.$$

The SV-reduction $(H,p_H,U_{SV,H}^{\tau })$ is defined by for all $\kappa \in P^H$,

$$U_{SV,H}^{\tau }\left(\kappa \right)=\left\{\begin{array}{cc}0\hfill & ,\kappa =0_H,\hfill \\ \underset{\mu \in P^{A\setminus H}}{max}\{U(\kappa ,\mu )-\tau (\mu ,0_H)\}\hfill & ,\mathrm{otherwise}.\hfill \end{array}\right.$$

When a group of agents in a game questions a solution, a reduction is a mechanism that allows the questioning agents to re-engage. In the three aforementioned reductions, if there are no questioning agents willing to participate, the reduction naturally does not generate any benefit. In the DM-reduction, if all questioning agents choose to participate, all satisfied agents cooperate to the fullest extent. However, if only some questioning agents participate, the reduction selects the group of satisfied agents that can generate the maximum benefit to assist. The SV-reduction is a variant of the DM-reduction, where regardless of whether all questioning agents choose to participate, it selects the group of satisfied agents that can generate the maximum benefit to assist. In the M-reduction, regardless of whether all questioning agents choose to participate, all satisfied agents cooperate to the fullest extent. In these three reductions, the satisfied agents receive a reward for their assistance in returning the solution, and the remaining benefits are the gains from the questioning agents’ re-engagement.

Consistency demonstrates that the projection of an outcome vector $\tau$ to a coalition $H\subseteq A$ should be provided via a solution $\sigma$ for the reduction for all H if $\tau$ is provided via $\sigma$ for a game $(A,p,U)$. Hence, the projection of $\tau$ to H should be consistent with the expectancy of the agents of H as declared via its reduction.

• A solution $\sigma$ matches DM-consistency (DMCSY) if $(H,p_H,U_{DM,H}^{\tau })\in \Delta$ and ${\tau |}_H\in \sigma (H,p_H,U_{DM,H}^{\tau })$ for all $(A,p,U)\in \Delta$, for all $H\in 2^A\setminus \{\varnothing \}$, and for all $\tau \in \sigma (A,p,U)$.

Converse consistency demonstrates that an efficient outcome vector $\tau$ itself should be appointed for the entire game if the projection of $\tau$ to each coalition $H\subseteq A$ is consistent with the expectancy of the agents of H as declared via its reduction.

• A solution $\sigma$ matches converse DM-consistency (CVDMCSY) if $(H,p_H,U_{DM,H}^{\tau })\in \Delta$ and ${\tau |}_H\in \sigma (H,p_H,U_{DM,H}^{\tau })$ for all $H\subset A$ such that $0<\left|H\right|<\left|A\right|$. Thus, it holds that $\tau \in \sigma (A,p,U)$ for all $(A,p,U)\in \Delta$ with $\left|A\right|\ge 2$ and for all $\tau \in \Theta (A,p,U)$.

By replacing “DM-reduction” with“M-reduction” and “SV-reduction”, one can define the M-consistency (MCSY), the converse M-consistency (CVMCSY), the SV-consistency (SVCSY), and the converse M-consistency (CVSVCSY), respectively.

To present the relative axiomatic results, one should make use of the following properties: Let $\sigma$ be a solution on $\Delta$. $\sigma$ matches efficiency (EICY) if $\sigma (A,p,U)\subseteq \Theta (A,p,U)$ for all $(A,p,U)\in \Delta$. $\sigma$ matches individual rationality (IRLY) if $\tau \in \sigma (A,p,U)$ implies that $\tau$ is individually rational for all $(A,p,U)\in \Delta$. $\sigma$ matches one-personal rationality (OPRLY) if $\sigma (A,p,U)=\mathbb{I}(A,p,U)$ for all $(A,p,U)\in \Delta$ with $\left|A\right|=1$.

Remark 5. 

Inspired by the axiomatic notion of Serrano and Volij [5], the related axiomatic results of the precore are as follows:

• Hwang et al. [13] axiomatized the precore by means of OPRLY, DMCSY, and CVDMCSY. They also showed that the precore violates CVMCSY;
• Cheng et al. [15] axiomatized the precore by means of OPRLY, SVCSY, and CVSVCSY.

In the following, we provide a different reduction by extending the reduced notions of Voorneveld and van den Nouweland [6].

Definition 4. 

Given a multichoice game $(A,p,U)$, a nonempty subset $H\in 2^A$ and an outcome vector τ, the VN-reduction$(H,p_H,U_{S,\tau }^{VN})$ is defined by for all $\kappa \in P^H$,

$$U_{H,\tau }^{VN}\left(\kappa \right)=\left\{\begin{array}{cc}0\hfill & ,\kappa =0_H,\hfill \\ U\left(p\right)-\tau (p_{A\setminus H},0_H)\hfill & ,\kappa =p_H,\hfill \\ \underset{\mu \in P^{A\setminus H},\mu \ne 0_{A\setminus H}}{max}\{U(\kappa ,\mu )-\tau (\mu ,0_H)\}\hfill & ,otherwise.\hfill \end{array}\right.$$

It is evident that VN-reduction is derived from DM-reduction. The main distinction between the two lies in the fact that if someone in H is willing to initiate VN-reduction, then individuals in $U\setminus H$ cannot refrain from taking action in coordination. By replacing “DM-reduction” with “VN-reduction”, one can define the VN-consistency (VNCSY) and the converse VN-consistency (CVVNSY), respectively. Subsequently, one can characterize the precore by applying VNCSY and CVVNCSY.

Theorem 4. 

On Δ, the precore is the only solution matching OPRLY, CVNSY, and CVVNCSY.

Proof. 

The proof technique and process for this theorem are nearly identical to those of Hwang et al. [13], hence it was omitted. □

By combining the reduced notions of Serrano and Volij [5] and Voorneveld and van den Nouweland [6], we provide a different reduction to axiomatize the precore.

Definition 5. 

Given a multichoice game $(A,p,U)$, a nonempty subset $H\in 2^A$, and an outcome vector τ, the combined reduction$(H,p_H,U_{H,\tau }^C)$ is defined by for all $\kappa \in P^H$,

$$U_{H,\tau }^C\left(\kappa \right)=\left\{\begin{array}{cc}0\hfill & ,\kappa =0_H,\hfill \\ \underset{\mu \in P^{A\setminus H},\mu \ne 0_{A\setminus H}}{max}\{U(\kappa ,\mu )-\tau (\mu ,0_H)\}\hfill & ,otherwise.\hfill \end{array}\right.$$

The major difference between the combined reduction and the above reductions is the fact that the coalition H is also permitted to envisage the latent interplay with the arbitrary subsets of $A\setminus H$. That is, in order to achieve the maximal utility, the whole coalitions composed with the agents of $A\setminus H$ should be considered to operate with H. By replacing “DM-reduction” with “combined reduction”, one can define the C-consistency (CCSY) and the converse C-consistency (CVCCSY), respectively. Subsequently, one can characterize the precore by applying CCSY and CVCCSY.

Lemma 11. 

On Δ, the precore $C_{Pre}$ matches CCSY.

Proof. 

Given a multichoice game $(A,p,U)$, a nonempty subset $H\in 2^A$. and an outcome vector $\tau$, first one would show that ${\tau |}_H$ is CRL under $(H,p_H,U_{H,\tau }^C)$ if $\tau$ is CRL under $(A,p,U)$. Since $\tau \in C_{Pre}(A,p,U)$, $\tau$ is CRL under $(A,p,U)$. For all $\kappa \in P^S$,

$$\begin{array}{cc}\hfill U_{H,\tau }^C\left(\kappa \right)-\tau \left(\kappa \right)& =\underset{\mu \in P^{A\setminus H},\mu \ne 0_{A\setminus H}}{max}\{U(\kappa ,\mu )-\tau (\mu ,0_H)\}-\tau (\kappa ,0_{A\setminus H})\hfill \\ & =\underset{\mu \in P^{A\setminus H},\mu \ne 0_{A\setminus H}}{max}\{U(\kappa ,\mu )-\tau (\mu ,0_H)-\tau (\kappa ,0_{A\setminus H})\}\hfill \\ & =\underset{\mu \in P^{A\setminus H},\mu \ne 0_{A\setminus H}}{max}\{U(\kappa ,\mu )-\tau (\kappa ,\mu )\}.\hfill \end{array}$$

Since $\tau$ is CRL under $(A,p,U)$, $U(\kappa ,\mu )-\tau (\kappa ,\mu )\le 0$. That is, ${\tau |}_H\left(\kappa \right)\ge U_{H,\tau }^C\left(\kappa \right)$ for all $\kappa \in P^A$. Hence, ${\tau |}_H$ is CRL under $(H,p_H,U_{H,\tau }^C)$.

Next, one would show that ${\tau |}_H$ is EIT under $(H,p_H,U_{H,\tau }^C)$. By definition of $U_{H,\tau }^C$ and $\tau \in \Theta (A,p,U)$,

$$\begin{array}{ccc}& U_{H,\tau }^C\left(p_H\right)-\tau (p_H,0_{A\setminus \left\{H\right\}})\hfill & \\ \hfill =& \underset{\mu \in P^{A\setminus H},\mu \ne 0_{A\setminus H}}{max}\{U(p_H,\mu )-\tau (\mu ,0_H)\}-\tau (p_H,0_{A\setminus H})\hfill & \\ \hfill \ge & U\left(p\right)-\tau (p_{A\setminus H},0_H)-\tau (p_H,0_{A\setminus H})\hfill & (\mathbf{Take}\mu =m_{A\setminus H})\hfill \\ \hfill =& U\left(p\right)-\tau \left(p\right)\hfill & \\ \hfill =& 0.\hfill & (\mathbf{Since}\tau \in \Theta (A,p,U\left)\right)\hfill \end{array}$$

Therefore, $U_{H,\tau }^C\left(p_H\right)\ge \tau (p_H,0_{A\setminus \left\{H\right\}})$. Since ${\tau |}_H$, $U_{H,\tau }^C\left(p_H\right)\le \tau (p_H,0_{A\setminus \left\{H\right\}})$ is CRL. That is, $U_{H,\tau }^C\left(p_H\right)=\tau (p_H,0_{A\setminus \left\{H\right\}})$. Hence, ${\tau |}_H$ is EIT under $(H,p_H,U_{H,\tau }^C)$. □

Lemma 12. 

On Δ, the precore $C_{Pre}$ matches CVCCSY.

Proof. 

Let $(A,p,U)\in \Delta$ with $\left|A\right|\ge 2$ and let $\tau \in \Theta (A,p,U)$. Suppose for all $H\subset A$ such that $0<\left|H\right|<\left|A\right|$, $(H,p_H,U_{H,\tau }^C)\in {\Delta }^{\prime }$ and ${\tau }_H\in C_{Pre}(H,p_H,U_{H,\tau }^C)$. We will show that $\tau \in C_{Pre}(A,p,U)$, i.e., for all $\kappa \in P^A$, $\tau \left(\kappa \right)\ge U\left(\kappa \right)$. Let $i\in A$ and $\kappa \in P^A$, $\kappa \ne 0_A$. Consider the reduction $(\left\{i\right\},p_i,U_{\left\{i\right\},\tau }^C)$. Then,

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{k_i=1}^{{\kappa }_i}}{\tau }_{i,k_i}& \ge U_{\left\{i\right\},\tau }^C\left({\kappa }_i\right)\hfill & \left(\mathbf{By}{\tau }_i\in C_{\mathrm{Pre}}\left(\left\{i\right\},U_{\left\{i\right\},\tau }^C\right)\right)\hfill \\ & =\underset{\mu \in P^{A\setminus \left\{i\right\}},\mu \ne 0_{A\setminus \left\{i\right\}}}{max}\{U({\kappa }_i,\mu )-\tau (\mu ,0)\}\hfill & \left(\mathbf{By} \mathbf{definition} \mathbf{of} U_{\left\{i\right\},\tau }^C\left({\kappa }_i\right)\right)\hfill \\ & \ge U\left(\kappa \right)-\tau ({\kappa }_{A\setminus \left\{i\right\}},0).\hfill & \left(\mathbf{Take}\mu ={\kappa }_{A\setminus \left\{i\right\}}\right)\hfill \end{array}$$

Hence, $\tau \left(\kappa \right)\ge U\left(\kappa \right)$. □

Lemma 13. 

A solution σ matches EICY if σ matches OPRLY and CCSY.

Proof. 

Let $(A,p,U)\in \Delta$, $\tau \in \sigma (A,p,U)$ and $i\in A$. Assume that $\left|A\right|=1$. The proof is trivial since OPRLY implies EICY under the condition $\left|A\right|=1$. Assume that $\left|A\right|>1$. Consider the reduction $(\left\{i\right\},p_i,U_{\left\{i\right\},\tau }^C)$. By CCSY of $\sigma$, ${\tau |}_{\left\{i\right\}}\in \sigma (\left\{i\right\},p_i,U_{\left\{i\right\},\tau }^C)$. By OPRLY of $\sigma$,

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{k_i=1}^{p_i}}{\tau }_{i,k_i}& \ge U_{\left\{i\right\},\tau }^C\left(p_i\right)\hfill & \\ & =\underset{\mu \in P^{A\setminus \left\{i\right\}},\mu \ne 0_{A\setminus \left\{i\right\}}}{max}\{U(p_i,\mu )-\tau (\mu ,0)\}\hfill & \left(\mathbf{By} \mathbf{definition} \mathbf{of} U_{\left\{i\right\},\tau }^C\left(p_i\right)\right)\hfill \\ & \ge U\left(p\right)-\tau (p_{A\setminus \left\{i\right\}},0).\hfill & \left(\mathbf{Take}\mu =p_{A\setminus \left\{i\right\}}\right)\hfill \end{array}$$

Therefore, $\tau \left(p\right)\ge U\left(p\right)$. Since $\sigma$ is a solution, $\tau \in \sigma (A,p,U)\subseteq {\Theta }^{*}(A,p,U)$. Hence, $\tau \left(p\right)=U\left(p\right)$. □

Theorem 5. 

On Δ, the precore is the only solution matching OPRLY, CCSY, and CVCCSY.

Proof. 

By Lemmas 11 and 12, the precore matches CCSY and CVCCSY. Clearly, the precore matches OPRLY.

To analyze uniqueness, assume that a solution $\sigma$ matches OPRLY, CCSY, and CVCCSY. By Lemma 13, $\sigma$ matches EICY. Let $(A,p,U)\in \Delta$. Relative proof proceeds via induction on $\left|A\right|$. By the OPRLY of $\sigma$, $\sigma (A,p,U)=\mathbb{I}(A,p,U)=C_{Pre}(A,p,U)$ if $\left|A\right|=1$. Assume that $\sigma (A,p,U)=C_{Pre}(A,p,U)$ if $\left|A\right|<k$, $k\ge 2$.

The condition $\left|A\right|=k$ :

First, one would show that $\sigma (A,p,U)\subseteq C_{Pre}(A,p,U)$. Let $\tau \in \sigma (A,p,U)$. Since $\sigma$ matches EICY, $\tau \in \Theta (A,p,U)$. By the CCSY of $\sigma$, ${\tau |}_H\in \sigma (H,U_{H,\tau }^C)$ for all $H\subset A$ with $0<\left|H\right|<\left|A\right|$. By the induction hypothesis, ${\tau |}_H\in \sigma (H,p_H,U_{H,\tau }^C)=C_{Pre}(H,p_H,U_{H,\tau }^C)$ for all $H\subset A$ with $0<\left|H\right|<\left|A\right|$. By the CVCCSY of the precore, $\tau \in C_{Pre}(A,p,U)$. The reverse inclusion could be generated similarly via exchanging the parts of $\sigma$ and $C_{Pre}$. Thence, $\sigma (A,p,U)=C_{Pre}(A,p,U)$. □

The following instances present that each of the properties utilized in Theorem 5 is logically independent of the remainder.

Example 1. 

Let $\sigma (A,p,U)=\varnothing$ for all $(A,p,U)\in \Delta$. Categorically, σ matches CCSY and CVCCSY, but it does not match OPRLY.

Example 2. 

For all $(A,p,U)\in \Delta$, one would define a solution σ on Δ to be

$$\sigma (A,p,U)=\left\{\begin{array}{cc}\mathbb{I}(A,p,U)\hfill & ,if\left|A\right|=1\hfill \\ \Theta (A,p,U)\hfill & ,otherwise.\hfill \end{array}\right.$$

Categorically, σ matches OPRLY and CVCCSY, but it does not match CCSY.

Example 3. 

For all $(A,p,U)\in \Delta$, one would define a solution σ on Δ to be

$$\sigma (A,p,U)=\left\{\begin{array}{cc}\mathbb{I}(A,p,U)\hfill & ,if\left|A\right|=1\hfill \\ \varnothing \hfill & ,otherwise.\hfill \end{array}\right.$$

Absolutely, σ matches OPRLY and CCSY, but it does not match CVCCSY.

Remark 6. 

Given $(A,p,U)\in \Delta$, $H\in 2^A\setminus \{\varnothing \}$ and an outcome vector τ:

1. 

By applying Theorems 1 and 5, it is easily to verify that $(H,p_H,U_{S,\tau }^C)$ is balanced if $(A,p,U)$ is balanced and $\tau \in C_{Pre}(A,p,U)$. Thence, Theorem 5 holds on all balanced multichoice games as well. Based on similar concepts, the related axiomatic results of Theorem 4 and Remark 5 also hold on all balanced multichoice games;

2. 

The following example can present that $(H,p_H,U_{S,\tau }^C)$ might be not balanced. Let $(A,p,U)\in \Delta$ with $A=\{i,t,k\}$, $p=(1,1,1)$, $U(1,1,1)=2$, $U(1,1,0)=U(0,1,1)=U(1,0,1)=1$, $U(1,0,0)=U(0,1,0)=U(0,0,1)=\frac{2}{3}$ and $U(0,0,0)=0$. Clearly, $(\frac{2}{3},\frac{2}{3},\frac{2}{3})\in C_{Pre}(A,p,U)$, i.e., $(A,p,U)$ is balanced. By taking $S=\{i,k\}$ and $\tau =(1,0,1)$, however, it is easy to determine that $C_{Pre}(H,p_H,U_{S,\tau }^C)$ is empty, i.e., $(H,p_H,U_{S,\tau }^C)$ is not balanced.

5. Conclusions

1.

By extending the findings of Cheng et al. [15], Hwang et al. [13], and Liao [14], this paper presents novel results regarding the precore. A comparison is made between the results of Cheng et al. [15], Hwang et al. [13], Liao [14], and the outcomes of this study.

• In previous studies, Cheng et al. [15], Hwang et al. [13], and Liao [14] extended the reductions proposed by Davis and Maschler [1], Moulin [2], and Serrano and Volij [5], respectively, to axiomatize the precore. However, in this paper, we present a novel approach by introducing a relative axiomatic result through the extension of reduced notions from Voorneveld and van den Nouweland [6]. By combining the reduced notions from Serrano and Volij [5] and Voorneveld and van den Nouweland [6], we propose a different axiomatic result;
• The nonemptiness and convexity of the precore are determined using the duality results of linear programming theory. Additionally, the dominance core is introduced to explore coincidences between the precore and the dominance core, considering nonemptiness and certain necessary and sufficient conditions. These findings are not present in Cheng et al. [15], Hwang et al. [13], and Liao [14].

2.

Convexity plays a crucial role in analyzing optimization problems across various fields, and core concepts are frequently employed in economic theories as solution concepts for allocation scenarios. According to Remark 4, the precore and the dominance core are demonstrated to be convex under certain conditions. Future research could explore the application of the precore and the dominance core in analyzing optimization problems within allocation considerations.