A Solution Concept and Its Axiomatic Results under Non-Transferable-Utility and Multi-Choice Situations

Abstract

In real situations, agents might take different energy levels to participate. On the other hand, agents always face an increasing need to focus on non-transferable-utility situations efficiently in their operational processes. Thus, we introduce the replicated core under non-transferable-utility situations, and analyze non-emptiness of the replicated core by means of a balanced result. In order to express the rationality of the replicated core, we also define different reduced games to axiomatize the replicated core.

1. Introduction

Each agent of a standard game is either completely concerned or departed from all participation with some other members. However, agents may take different energy levels to participate in real situations. In a multi-choice game, each agent is consented to partake with finite different levels (strategies). By focusing on multi-choice games, Nouweland et al. [1] pointed out some applications, such as a vast structure program with a deadline and a fine for each day if this deadline is overtime. The date of finalization hinges on the endeavor of how all of the members participate in the program: the greater they employ themselves, the sooner the program will be finished. This situation could be treated as a multi-choice game. The merit of a coalition where each member participates at a definite energy level is defined as the minus of the fine which requires to be remunerated for giving the date of finalization of the program when each member presents the relative endeavor. Based on the notion of replicated transferable-utility (TU) games, Calvo and Santos [2] pointed out that the extended Shapley value due to Nouweland et al. [1] concurs with the solution concept due to Moulin [3]. Inspired by the notions of the core on continuum market games (Hart [4]) and fuzzy TU games (Aubin [5]) respectively, Hwang and Liao [6] also proposed a different core concept in the context of multi-choice TU games by applying corresponding replicated TU games.

A non-transferable utility (NTU) game is a measurement of payoffs achievable by agents of each coalition via several joint courses of action (decision, strategy). The agents face the problem of choosing a payoff that is feasible for the entire coalition. This is a bargaining condition and its solution may be rationally required to satisfy various criteria, and different collections of axioms will characterize different solutions. The core concept is, possibly, the most instinctive allocation rule. The core can be naturally extended to multi-choice (NTU) games. However, the investigations of the core when there are no side payments may be much harder than in the TU situation. In Section 2, we propose a generalization of the traditional core, the replicated core, in the context of multi-choice NTU games. We present that the replicated core of a multi-choice NTU game coincides with a replicated subset of the traditional core of a replicated NTU game. In Section 3, we adopt a balancedness result to investigate the non-emptiness condition of the replicated core. In order to express the rationality for the replicated core, we define different reduced games to characterize the replicated core in Section 4 and Section 5. Furthermore, some more comparisons and discussions are provided in Section 6.

2. Preliminaries

Assume that U is the universe of agents and $N\subseteq U$ is a collection of agents with $N\ne \varnothing$. Assume that each agent $i\in U$ has $h_i$ energy levels at which he/she/it can take, where $h_i$ is positive integer. Let $H_i=\{0,1,\cdots ,h_i\}$ be the energy level set of agent i, where 0 represents not participating and $H_i^{+}=H_i\backslash \left\{0\right\}$. Let $H^N={\prod }_{i\in N}{H}_i$ be the product set of the energy level sets for agents of N. $\mu \in H^N$ is a vector that presents current energy levels for every agent in N, where ${\mu }_i$ is the energy level that agent i participates at currently for each $i\in N$. Given $K\subseteq N$ and $\mu \in H^N$, we define $B\left(\mu \right)=\{i\in N|{\mu }_i>0\}$ and ${\mu }_K\in {\mathbb{R}}^K$ to be the restriction of $\mu$ to K. Furthermore, let $\left|K\right|$ be the amount of elements in K, and ${\theta }^K\left(N\right)\in {\mathbb{R}}^N$ be the vector satisfying ${\theta }_i^K\left(N\right)=1$ if $i\in K$, and ${\theta }_i^K\left(N\right)=0$ if $i\notin K$. ${\theta }^K\left(N\right)$ is denoted by ${\theta }^K$ if no bewilderment can arise. The zero vector in ${\mathbb{R}}^N$ is denoted by $0_N$.

Let $\mu ,\beta \in {\mathbb{R}}^N$. $\mu \ge \beta$ if ${\mu }_i\ge {\beta }_i$ for each $i\in N$; $\mu >\beta$ if $\mu \ge \beta$ and $\mu \ne \beta$; $\mu \gg \beta$ if ${\mu }_i>{\beta }_i$ for each $i\in N$. We denote ${\mathbb{R}}_{+}^N=\{\mu \in {\mathbb{R}}^N|\mu \ge 0_N\}$. Let $P\subseteq {\mathbb{R}}^N$. P is comprehensive if $\mu \in P$ and $\mu \ge \beta$ imply $\beta \in P$. The comprehensive hull of P is denoted to be $comP$. The interior of P is denoted to be $intP$, and the boundary of P is denoted to be $\partial P$. $\mu +P=\{\mu +\beta |\beta \in P\}$ if $\mu \in {\mathbb{R}}^N$.

Definition 1.

A multi-choice NTU game is $(N,V)$, (Since $h\in \mathbb{N}$ is fixed in the paper, $(N,v)$ could be written rather than $(N,h,v)$.) where $N\ne \varnothing$ is a finite collection of agents and V is a characteristic mapping that assigns to every $\mu \in H^N\backslash \left\{0_N\right\}$ a subset $V\left(\mu \right)\subseteq {\mathbb{R}}^{B\left(\mu \right)}$, such that

$$V\left(\mu \right)isnonempty,comprehensiveandclosed.$$

(1)

$$V\left(\mu \right)\cap (\gamma +{\mathbb{R}}_{+}^{B\left(\mu \right)})isbounded\forall \gamma \in {\mathbb{R}}^{B\left(\mu \right)}.$$

(2)

Situation (1) is standard. Comprehensiveness could be treated as a marked weakening of monotonicity. Situation (2) assures that $V\left(\mu \right)$ is a proper subset of ${\mathbb{R}}^{B\left(\mu \right)}$. It could be treated as a weakening of boundedness. Denote the family of all multi-choice NTU games by $MG$.

A payoff vector of $(N,V)$ is a vector $x={\left(x_i\right)}_{i\in N}\in {\mathbb{R}}^N$ where $x_i$ could be treated as the per unit payoff that agent i obtains for each $i\in N$, hence $h_i{x}_i$ is the entire payoff that agent i obtains in $(N,V)$. x is efficient (EFC) if ${\left(h_i{x}_i\right)}_{i\in N}\in \partial V\left(h\right)$. x is singlely rational (SR) if $j{x}_i\notin intV\left(j{\theta }^{\left\{i\right\}}\right)$ for each $i\in N$ and for each $j\in H_i^{+}$.

If payoff vector x is EFC and SRA, then x is said to be an imputation in $(N,V)$. Denote the collection of imputations in $(N,V)$ to be $IP(N,V)$. Denote the collection of all feasible payoff vectors in $(N,V)$ to be

$$E^{*}(N,V)=\{x\in {\mathbb{R}}^N|{\left(h_i{x}_i\right)}_{i\in N}\in V\left(h\right)\},$$

where as

$$E(N,V)=\{x\in {\mathbb{R}}^N|xisEFC\}$$

is the collection of efficient payoff vectors of $(N,V)$. In the following, we define the replicated core on multi-choice NTU games.

Definition 2.

The replicated core C of a multi-choice NTU game $(N,V)$ is defined by

$$C(N,V)=\{x\in E(N,V)|{\left({\alpha }_i{x}_i\right)}_{i\in B\left(\alpha \right)}\notin intV\left(\alpha \right)foreach\alpha \in H^N\backslash \left\{0_N\right\}\}.$$

A standard NTU game is $(N,V_{NTU})$, where $N\ne \varnothing$ is a finite collection of agents and $V_{NTU}$ is a characteristic mapping that assigns to every $T\in 2^N\backslash \{\varnothing \}$ a subset $V_{NTU}\left(T\right)\subseteq {\mathbb{R}}^T$, such that

• $V_{NTU}\left(T\right)$ is non-empty, comprehensive, and closed.
• $V_{NTU}\left(T\right)\cap (\gamma +{\mathbb{R}}_{+}^T)$ is bounded for each $\gamma \in {\mathbb{R}}^T$.

The traditional core $C_{NTU}$ of a standard NTU game $(N,V_{NTU})$ is

$$C_{NTU}(N,V_{NTU})=\{x\in \partial V\left(N\right)|x_T\notin int{V}_{NTU}\left(T\right)\mathrm{for}\mathrm{each}T\in 2^N\backslash \{\varnothing \}\}.$$

Assume that $(N,V)\in MG$ and $N^h$ is a set of replicated agents with

$$N^h=\bigcup _{i\in N}{N}_i^h,$$

where, for each $i\in N$, $N_i^h=\{i_1,\dots ,i_{h_i}\}$. For any $T\subseteq N^h$, we denote $\alpha \left(T\right)\in H^N$ to be that ${\alpha }_i\left(T\right)=|T\cap N_i^h|$ for each $i\in N$. For $x={\left(x_i\right)}_{i\in B\left(\alpha \right(T\left)\right)}\in {\mathbb{R}}^{B\left(\alpha \right(T\left)\right)}$, we define the corresponding replicated vector $X={\left(X^{T\cap N_i^h}\right)}_{i\in B\left(\alpha \right(T\left)\right)}\in {\mathbb{R}}^T$ to be that for all $i\in B\left(\alpha \right(T\left)\right)$,

$$X^{T\cap N_i^h}\in {\mathbb{R}}^{T\cap N_i^h} \mathrm{with} X_j^{T\cap N_i^h}=x_i\mathrm{for}\mathrm{all}j\in T\cap N_i^h,$$

and vice versa.

Then, we define the replicated NTU game $\left(N^h,V_{NTU}^h\right)$ as follows. For all $T\in 2^{N^h}\backslash \{\varnothing \}$,

$$V_{NTU}^h\left(T\right)=com\{{(X^{T\cap N_i^h})}_{i\in B(\alpha (T))}\in {\mathbb{R}}^T|{\left({\alpha }_i\left(T\right)x_i\right)}_{i\in B\left(\alpha \right(T\left)\right)}\in V\left(\alpha \left(T\right)\right)\}.$$

The following result presents that the replicated core of a multi-choice NTU game coincides with a “replicated subset” of the traditional core of a replicated NTU game.

Proposition 1.

Assume that $(N,V)\in MG$ and $\left(N^h,V_{NTU}^h\right)$ is the corresponding replicated NTU game. If $x={\left(x_i\right)}_{i\in N}\in C(N,V)$, then $X={\left(X^{N_i^h}\right)}_{i\in N}\in C_{NTU}\left(N^h,V_{NTU}^h\right)$, where, for all $i\in N$, $X^{N_i^h}\in {\mathbb{R}}^{N_i^h}$ with $X_j^{N_i^h}=x_i$ for all $j\in N_i^h$.

Proof of Proposition 1.

The proof is trivial. Hence, we omit it. □

3. Balancedness and Non-Emptiness

Scarf [7] showed that each balanced standard NTU game has a nonempty traditional core. Here, we offer an analogue result on multi-choice NTU games.

Definition 3.

Assume that A is a collection of $2^N$. A is called balanced if there exists $\Gamma \left(S\right)>0$ for each $S\in A$ such that

$$\sum _{\stackrel{S\in A}{i\in S}}\Gamma \left(S\right)=1foreachi\in N.$$

(3)

Definition 4.

A standard NTU game $(N,V_{NTU})$ is called balanced if it holds that

$$\bigcap _{S\in A}{V}_{NTU}^S\subseteq V_{NTU}\left(N\right)foreachbalancedcollectionsA,$$

(4)

where $V_{NTU}^S=V_{NTU}\left(S\right)\times {\mathbb{R}}^{N\backslash S}foreachS$.

Remark 1.

Scarf [7] showed that the traditional core is nonempty for each balanced standard NTU game.

Subsequently, we provide an analogue balancedness on multi-choice NTU games.

Definition 5.

Assume that Θ is a collection of $H^N$. Θ is called m-balanced if there exists $\lambda \left(\alpha \right)>0$ for each $\alpha \in \Theta$ such that, for all $i\in N$,

$$\sum _{\alpha \in \Theta }{\alpha }_i·\lambda \left(\alpha \right)=h_i.$$

(5)

Definition 6.

A multi-choice NTU game $(N,V)$ is m-balanced if for all balanced collections Θ, and it holds that

$$\bigcap _{\alpha \in \Theta }{V}^{\alpha }\subseteq V\left(h\right),$$

(6)

where $V^{\alpha }=V\left(\alpha \right)\times {\mathbb{R}}^{N\backslash B\left(\alpha \right)}forall\alpha$.

The following theorem presents that the balancedness of a multi-choice NTU game concurs with the balancedness of the corresponding replicated NTU game.

Theorem 1.

Assume that $(N,V)\in MG$ and $\left(N^h,V_{NTU}^h\right)$ is the corresponding replicated NTU game. Then, $(N,V)$ is m-balanced if and only if $\left(N^h,V_{NTU}^h\right)$ is balanced.

Proof of Theorem 1.

Assume that $(N,V)\in MG$ is a m-balanced. Let $A=\{S^1,S^2,\cdots ,S^k\}$ be a balanced collection of $2^{N^h}$. That is, there exists $\Gamma \left(S^t\right)>0$ for all $t=1,2,\cdots ,k$ such that for all $j\in N^h$,

$$\sum _{\stackrel{S^t\in A}{j\in S^t}}\Gamma \left(S^t\right)=1.$$

For $t=1,2,\cdots ,k$, define the vector ${\alpha }^t\in {\mathbb{R}}^N$ by

$${\alpha }_i^t=|S^t\cap N_i^h|$$

for all $i\in N$. Let $\Theta =\{{\alpha }^1,{\alpha }^2,\cdots ,{\alpha }^k\}$ be a collection of $H^N$. Taking $\lambda \left({\alpha }^t\right)=\Gamma \left(S^t\right)$ for all $t=1,2,\cdots ,k$, we see that for all $i\in N$,

$$\sum _{{\alpha }^t\in \Theta }{\alpha }_i^t·\lambda \left({\alpha }^t\right)=h_i.$$

Hence, $\Theta$ is a balanced collection of $H^N$. By the replication special nature of $\left(N^h,V_{NTU}^h\right)$ and the balancedness of $(N,V)$, it is easy to see that $\left(N^h,V_{NTU}^h\right)$ is balanced.

Conversely, let $\left(N^h,V_{NTU}^h\right)$ be balanced. Let $\Theta =\{{\alpha }^1,{\alpha }^2,\cdots ,{\alpha }^k\}$ be a m-balanced collection of $H^N$. For $t=1,2,\cdots ,k$, let $L^t$ be the least common multiple of $C_a^{h_i}$, where $C_a^{h_i}=\frac{h_i!}{a!(h_i-a)!}$, for all $1\le a\le {\alpha }_i^t$ and for all $i\in N$.

For $t=1,2,\cdots ,k$, define $S_p^t\in 2^N$ for all $1\le p\le L^t$ by for all $i\in N$,

$$|S_p^t\cap N_i^h|={\alpha }_i^{t}and$$

(7)

$$\sum _{p=1}^{L^t}|S_p^t\cap \left\{i_b\right\}|=\sum _{p=1}^{L^t}|S_p^t\cap \left\{i_c\right\}|,$$

(8)

where $i_b,i_c\in N_i^h$. Note that Condition (8) can be done because that $L^t$ is the least common multiple.

Let $A=\left\{S_p^t\right|1\le t\le kand1\le p\le L^t\}$ be a collection of N. Taking $\Gamma \left(S_p^t\right)=\frac{\lambda \left({\alpha }^t\right)}{L^t}$ for all $1\le t\le k$ and for all $1\le p\le L^t$, we see that for all $j\in N^h$,

$$\sum _{\stackrel{S_p^t\in A}{j\in S_p^t}}\Gamma \left(S_p^t\right)=1.$$

Hence, A is a balanced collection of N. Similarly, by the replication special nature of $\left(N^h,V_{NTU}^h\right)$ and the balancedness of $\left(N^h,V_{NTU}^h\right)$, $(N,V)$ is m-balanced. □

A multi-choice NTU extension of the Scarf’s [7] result is as follows.

Theorem 2.

The replicated core is nonempty for each m-balanced multi-choice NTU game.

Proof of Theorem 2.

Assume that $(N,V)\in MG$ and $\left(N^h,V_{NTU}^h\right)$ is the corresponding replicated NTU game. By Proposition 1, $C(N,V)\ne \varnothing$ if and only if $C_{NTU}\left(N^h,V_{NTU}^h\right)\ne \varnothing$. Hence, this theorem immediately follows Remark 1 and Theorem 1. □

4. Axiomatic Result

A solution on $MG$ is a map $\sigma$ which assigns a subset $\sigma (N,V)\subseteq E^{*}(N,V)$ to each $(N,V)$. One would make use of some more axioms as follows. $\sigma$ satisfies efficiency (EFCY) if $\sigma (N,V)\subseteq E(N,V)$ for each $(N,V)\in MG$. $\sigma$ satisfies single rationality (SRY) if x is SR for each $(N,V)\in MG$ and for each $x\in \sigma (N,V)$. $\sigma$ satisfies standard for one-person game (SOPG) if $\sigma (N,V)=IP(N,V)$ for each $(N,V)\in MG$ with $\left|N\right|=1$.

Next, we provide a multi-choice NTU extension of the reduction proposed by Peleg [8]. Assume that $(N,V)\in MG$, $T\in 2^N\backslash \{\varnothing \}$ and x is a payoff vector. The reduced game in regard to T and x is $(T,V_{T,x})$ defined as follows. For all $\mu \in H^T\backslash \left\{0_T\right\}$,

$$V_{T,x}\left(\mu \right)=\left\{\begin{array}{cc}\{y\in {\mathbb{R}}^T|\left(y,{\left(h_i{x}_i\right)}_{i\in N\backslash T}\right)\in V\left(h\right)\}\hfill & ,\mathrm{if}\mu =h_T,\hfill \\ \bigcup _{\beta \in H^{N\backslash T}}\{y\in {\mathbb{R}}^{B\left(\mu \right)}|\left(y,{\left({\beta }_i{x}_i\right)}_{i\in B\left(\beta \right)}\right)\in V(\mu ,\beta )\}\hfill & ,\mathrm{otherwise}.\hfill \end{array}\right.$$

Consistency asserts that if x is appointed by $\sigma$ for $(N,V)\in MG$, then the projection of x to S should be appointed by $\sigma$ in the reduction $V_{T,x}$ for each $T\subseteq N$. Thus, the projection of x to S should be consistent with the expectations among the agents of T as mirrored by its reduction. Converse consistency asserts that if the projection of a single rational vector x to each $T\subseteq N$ is consistent with the expectations among the agents of T as mirrored by its reduction, then x itself should be supported for total game.

• Consistency (CONY): For each $(N,V)\in MG$, for each $T\in 2^N\backslash \{\varnothing \}$, and for each $x\in \sigma (N,V)$, $(T,V_{T,x})\in MG$ and $x_T\in \sigma (T,V_{T,x})$.
• Converse consistency (CCONY): For each $(N,V)\in MG$ with $\left|N\right|\ge 2$ and for each $x\in IP(N,V)$, if for each $T\subset N$ such that $0<\left|T\right|<\left|N\right|$, $(T,V_{T,x})\in MG$ and $x_T\in \sigma (T,V_{T,x})$, then $x\in \sigma (N,V)$.

Here, we confine our consideration to non-level multi-choice NTU games, defined as those games $(N,V)$ such that

$$x=y\mathrm{if}x,y\in \partial V\left(h\right)\mathrm{and}x\ge y.$$

(9)

Situation (9) states that $\partial V\left(h\right)$ has no level segments, i.e., segments parallel to a coordinate hyperplane. We adopt it to ensure that, if $x\in {\mathbb{R}}^N$ is efficient, then $x_T$ is efficient in $(T,V_{T,x})$.

Next, we would like to adopt SRY, SOPG, CONY, and CCONY to characterize the replicated core. The proofs that the replicated core satisfies CONY and CCONY are in the following steps:

Lemma 1.

Assume that $(N,V)\in MG$, $x\in E^{*}(N,V)$ and $S\in 2^N\backslash \{\varnothing \}$. Then, the reduction $(S,V_{S,x})\in MG$.

Proof of Lemma 1.

It is trivial to verify the condition (9) and Definition 1. □

Lemma 2.

Assume that $(N,V)\in MG$, $x\in E^{*}(N,V)$ and $T\in 2^N\backslash \{\varnothing \}$. Then, $x_T\in E(T,V_{T,x})$ if and only if $x\in E(N,V)$.

Proof of Lemma 2. 

Clearly, $x_T\in E^{*}(T,V_{T,x})$. If $x_T\notin E(T,V_{T,x})$, then there exists $y_T\in E^{*}(T,V_{T,x})$ such that $y_T>x_T$. Thus, $(y_T,x_{N\backslash T})>x$ and $(y_T,x_{N\backslash T})\in E^{*}(N,V)$. Thus, $x\notin E(N,V)$. Similarly, there exists $y\in E^{*}(N,V)$ such that $y\gg x$ if $x\notin E(N,V)$. Thus, $(y_T,x_{N\backslash T})\in E^{*}(N,V)$. Therefore, $y_T\gg x_T$ and $y_T\in E^{*}(T,V_{T,x})$. Thus, $x_T\notin E(T,V_{T,x})$. □

Lemma 3.

Assume that $(N,V)\in MG$, $x\in C(N,V)$ and $T\in 2^N\backslash \{\varnothing \}$. Then, for each $\mu \in H^T\backslash \left\{0_T\right\}$,

$${\left({\mu }_i{x}_i\right)}_{i\in B\left(\mu \right)}\notin int\left[\bigcup _{\gamma \in H^{N\backslash T}}\{y\in {\mathbb{R}}^{B\left(\mu \right)}|\left(y,{\left({\gamma }_i{x}_i\right)}_{i\in B\left(\gamma \right)}\right)\in V(\mu ,\gamma )\}\right].$$

Proof of Lemma 3. 

Let $(N,V)\in MG$, $x\in C(N,V)$ and $T\in 2^N\backslash \{\varnothing \}$. Let $\mu \in H^T\backslash \left\{0_T\right\}$. Assume that

$${\left({\mu }_i{x}_i\right)}_{i\in B\left(\mu \right)}\in int\left[\bigcup _{\gamma \in H^{N\backslash T}}\{y\in {\mathbb{R}}^{B\left(\mu \right)}|\left(y,{\left({\gamma }_i{x}_i\right)}_{i\in B\left(\gamma \right)}\right)\in V(\mu ,\gamma )\}\right].$$

Then, there exists $\gamma \in H^{N\backslash T}$ such that

$$\left({\left({\mu }_i{x}_i\right)}_{i\in B\left(\mu \right)},{\left({\gamma }_i{x}_i\right)}_{i\in B\left(\gamma \right)}\right)\in intV(\mu ,\gamma ).$$

Let $\overline{\mu }=(\mu ,\gamma )\in H^N\backslash \left\{0_N\right\}$. Hence, ${\left({\overline{\mu }}_i{x}_i\right)}_{i\in B\left(\overline{\mu }\right)}\in intV\left(\overline{\mu }\right)$. This contradicts to the fact that $x\in C(N,V)$. □

Lemma 4.

The replicated core satisfies CONY.

Proof of Lemma 4. 

Let $(N,V)\in MG$. Let $x\in C(N,V)$ and $T\in 2^N\backslash \{\varnothing \}$. By Lemma 1, $(T,V_{T,x})\in MG$. Since $x\in C(N,V)$, $x\in E(N,V)$. Thus, $x_T\in E(T,V_{T,x})$ by Lemma 2. It remains to show that ${\left({\alpha }_i{x}_i\right)}_{i\in B\left(\alpha \right)}\notin int{V}_{T,x}\left(\alpha \right)$ for all $\alpha \in H^T\backslash \{0_T,h_T\}$. Let $\alpha \in H^T\backslash \{0_T,h_T\}$. Since

$$V_{T,x}\left(\alpha \right)=\bigcup _{\beta \in H^{N\backslash T}}\{y\in {\mathbb{R}}^{B\left(\alpha \right)}|\left(y,{\left({\beta }_i{x}_i\right)}_{i\in B\left(\beta \right)}\right)\in V(\alpha ,\beta )\},$$

${\left({\alpha }_i{x}_i\right)}_{i\in B\left(\alpha \right)}\notin int{V}_{T,x}\left(\alpha \right)$ by Lemma 3. Thus, $x_T\in C(T,V_{T,x})$ for each $T\in 2^N\backslash \{\varnothing \}$ if $x\in C(N,V)$. □

Lemma 5.

The replicated core satisfies CCONY.

Proof of Lemma 5. 

Assume that $(N,V)\in MG$ with $\left|N\right|\ge 2$ and $x\in IP(N,V)$. Suppose that $(T,V_{T,x})\in MG$ and $x_T\in C(T,V_{T,x})$ for each $T\subset N$ with $0<\left|T\right|<\left|N\right|$. We will prove that $x\in C(N,V)$. Since $x\in IP(N,V)$, it remains to prove that ${\left({\mu }_i{x}_i\right)}_{i\in B\left(\mu \right)}\notin intV\left(\mu \right)$ for all $\mu \in H^N\backslash [\{0_N,h\}\cup \left\{k_i{\theta }^{\left\{i\right\}}\right|i\in Nand{k}_i\in H_i^{+}\}]$. Two conditions could be distinguished:

Condition 1: $\left|N\right|=2$:

Let $\mu \in H^N\backslash [\{0_N,h\}\cup \left\{k_i{\theta }^{\left\{i\right\}}\right|i\in Nand{k}_i\in H_i^{+}\}]$. Suppose $N=\{i,j\}$.

• Consider $\mu =({\mu }_i,{\mu }_j)$ with ${\mu }_i\ne h_i$. Then ${\mu }_i{x}_i\notin int{V}_{\left\{i\right\},x}\left({\mu }_i\right)$ by $x_i\in C(\left\{i\right\},V_{\left\{i\right\},x})$. Since

  $$V_{\left\{i\right\},x}\left({\mu }_i\right)=\bigcup _{\gamma \in H^{\left\{i\right\}}}\{y\in {\mathbb{R}}^{\left\{i\right\}}|\left(y,{\left(\gamma x_k\right)}_{k\in B\left(\gamma \right)}\right)\in V({\mu }_i,\gamma )\},$$
  ${\mu }_i{x}_i\notin int\bigcup _{\gamma \in H^{\left\{i\right\}}}\{y\in {\mathbb{R}}^{\left\{i\right\}}|\left(y,{\left(\gamma x_k\right)}_{k\in B\left(\gamma \right)}\right)\in V({\mu }_i,\gamma )\}$. Thus, ${\left({\mu }_k{x}_k\right)}_{k\in B\left(\mu \right)}\notin intV\left(\mu \right)$.
• Consider $\mu =({\mu }_i,{\mu }_j)$ with ${\mu }_j\ne h_j$. The proof is similar to the above processes by applying the reduction $(\left\{j\right\},V_{\left\{j\right\},x})$.

Condition 2: $\left|N\right|>2$:

Let $\mu \in H^N\backslash [\{0_N,h\}\cup \left\{k_i{\theta }^{\left\{i\right\}}\right|i\in Nand{k}_i\in H_i^{+}\}]$.

• If there exists $i\in N$ such that $0<{\mu }_i<h_i$, consider the reduction $(\left\{i\right\},V_{\left\{i\right\},x})$. Then, ${\mu }_i{x}_i\notin int{V}_{\left\{i\right\},x}\left({\mu }_i\right)$ by $x_i\in C(\left\{i\right\},V_{\left\{i\right\},x})$. Since

  $$V_{\left\{i\right\},x}\left({\mu }_i\right)=\bigcup _{\gamma \in H^{N\backslash \left\{i\right\}}}\{y\in {\mathbb{R}}^{\left\{i\right\}}|\left(y,{\left({\gamma }_j{x}_j\right)}_{j\in B\left(\gamma \right)}\right)\in V({\mu }_i,\gamma )\},$$
  ${\mu }_i{x}_i\notin int\bigcup _{\gamma \in H^{N\backslash \left\{i\right\}}}\{y\in {\mathbb{R}}^{\left\{i\right\}}|\left(y,{\left({\gamma }_j{x}_j\right)}_{j\in B\left(\gamma \right)}\right)\in V({\mu }_i,\gamma )\}$. Hence, ${\left({\mu }_k{x}_k\right)}_{k\in B\left(\mu \right)}\notin intV\left(\mu \right)$.
• Let ${\mu }_i=0$ or ${\mu }_i=h_i$ for all $i\in N$. Assume that ${\mu }_i=h_i$ and ${\mu }_j=0$ where $i,j\in N$. Consider the reduction $(\{i,j\},V_{\{i,j\},x})$. By $x_{\{i,j\}}\in C(\{i,j\},V_{\{i,j\},x})$, ${\mu }_i{x}_i\notin int{V}_{\{i,j\},x}({\mu }_i,0)$. Since

  $$V_{\{i,j\},x}({\mu }_i,0)=\bigcup _{\gamma \in H^{N\backslash \{i,j\}}}\{y\in {\mathbb{R}}^{\left\{i\right\}}|\left(y,{\left({\gamma }_k{x}_k\right)}_{k\in B\left(\gamma \right)}\right)\in V({\mu }_i,0,\gamma )\},$$
  ${\mu }_i{x}_i\notin int\bigcup _{\gamma \in H^{N\backslash \{i,j\}}}\{y\in {\mathbb{R}}^{\left\{i\right\}}|\left(y,{\left({\gamma }_k{x}_k\right)}_{k\in B\left(\gamma \right)}\right)\in V({\mu }_i,0,\gamma )\}$. Thus, ${\left({\mu }_k{x}_k\right)}_{k\in B\left(\mu \right)}\notin intV\left(\mu \right)$. □

Lemma 6.

If a solution σ satisfies CONY and SOPG, then it also satisfies EFCY.

Proof of Lemma 6. 

Assume that there exist $(N,V)\in MG$ and $x\in \sigma (N,v)$ such that $x\notin E(N,V)$. Let $t\in N$. Consider the reduction $(\left\{t\right\},V_{\left\{t\right\},x})$. By CONY of $\sigma$, $x_t\in \sigma (\left\{t\right\},V_{\left\{t\right\},x})$. By SOPG of $\sigma$, $x_t\in E(\left\{t\right\},V_{\left\{t\right\},x})$. By Lemma 2, $x\in E(N,V)$. Therefore, the desired contradiction could be presented. □

Theorem 3.

A solution σ on $MG$ satisfies CONY, CCONY, SOPG, and SRY if and only if $C(N,V)=\sigma (N,V)$ for each $(N,V)\in MG$.

Proof of Theorem 3. 

By Lemmas 4 and 5, the replicated core satisfies CONY and CCONY. Absolutely, the replicated core satisfies SOPG and SRY.

To analyze uniqueness, suppose that a solution $\sigma$ satisfies SOPG, SRY, CONY, and CCONY. By Lemma 6, $\sigma$ satisfies EFCY. Let $(N,V)\in MG$. The following proof proceeds by induction on $\left|N\right|$. By SOPG of $\sigma$, $\sigma (N,V)=IP(N,V)=C(N,V)$ if $\left|N\right|=1$. Suppose that $\sigma (N,V)=C(N,V)$ if $\left|N\right|<k$, $k\ge 2$. The condition $\left|N\right|=k$:

We first prove that $\sigma (N,V)\subseteq C(N,V)$. Assume that $x\in \sigma (N,V)$. Since $\sigma$ satisfies SRY and EFCY, $x\in IP(N,V)$. Clearly, $x_T\in \sigma (T,V_{T,x})$ for each $T\subset N$ with $0<\left|T\right|<\left|N\right|$ by CONY of $\sigma$. Based on the induction hypothesis, $x_T\in \sigma (T,V_{T,x})=C(T,V_{T,x})$ for each $T\subset N$ with $0<\left|T\right|<\left|N\right|$. Thus, $x\in C(N,V)$ by CCONY of the replicated core. The remaining proof could be completed analogously by switching the roles of C and $\sigma$. Thus, $C(N,V)=\sigma (N,V)$. □

The following examples present that each of the properties adopted in Theorem 3 should be logically independent of the others. In order to present the independence of the adopted properties, $\left|U\right|\ge 2$ is necessary.

Example 1.

Assume that $\sigma (N,V)=\varnothing$ for each $(N,V)\in MG$. Absolutely, σ satisfies SRY, CONY, and CCONY, but σ violates SOPG.

Example 2.

Here, we define a solution σ on $MG$ to be

$$\sigma (N,V)=\left\{\begin{array}{cc}IP(N,V)\hfill & ,if\left|N\right|=1,\hfill \\ E(N,V)\hfill & ,otherwise.\hfill \end{array}\right.$$

Absolutely, σ satisfies SOPG, CONY, and CCONY, but σ violates SRY.

Example 3.

Assume that $\sigma (N,V)=IP(N,V)$ for each $(N,V)\in MG$. Absolutely, σ satisfies SOPG, SRY, and CCONY, but σ violates CONY.

Example 4.

Here, we define a solution σ on $MG$ to be

$$\sigma (N,V)=\left\{\begin{array}{cc}IP(N,V)\hfill & ,if\left|N\right|=1,\hfill \\ \varnothing \hfill & ,otherwise.\hfill \end{array}\right.$$

Absolutely, σ satisfies SOPG, SRY, and CONY, but σ violates CCONY.

5. Another Reduction

In this section, we provide a multi-choice NTU extension of the reduction due to Moulin [9].

Given a payoff vector of some game assigned by a solution, and given a subgroup of agents, Moulin [9] introduced the reduction as that in which every coalition in the subgroup could reach payoffs to its agents only if they are harmonious with the original payoffs to “all” the agents outside of the subgroup. A natural multi-choice extension of Moulin’s reduction is as follows. Assume that $(N,V)\in MG$, $T\in 2^N\backslash \{\varnothing \}$ and x is a payoff vector. The M-reduced game in regard to T and x is $(T,V_{T,x}^M)$ defined by for all $\mu \in H^T\backslash \left\{0_T\right\}$,

$$V_{T,x}^M\left(\mu \right)=\{y\in {\mathbb{R}}^{B\left(\mu \right)}|\left(y,{\left(h_i{x}_i\right)}_{i\in T^c}\right)\in V(\mu ,h_{T^c})\}.$$

By replacing “reduction” with “ M-reduction”, we define the properties of M-consistency (MCONY) and converse M-consistency (CMCONY).

Lemma 7.

The replicated core satisfies MCONY.

Proof of Lemma 7. 

Similar to Lemmas 1–4, the proof could be completed. □

In the following, we adopt an example to present that the replicated core violates CMCONY.

Example 5.

Define $(N,V)\in MG$ to be $N=\{i,k\}$, $h=(3,2)$, $V(3,2)=\left\{\right(p,q\left)\right|p+q\le 18\}$, $V(3,1)=\left\{\right(p,q\left)\right|p+q\le 14\}$, $V(2,2)=\left\{\right(p,q\left)\right|p+q\le 12\}$, $V(2,1)=\left\{\right(p,q\left)\right|p+q\le 10\}$, $V(1,2)=\left\{\right(p,q\left)\right|p+q\le 6\}$, $V(1,1)=\left\{\right(p,q\left)\right|p+q\le 7\}$ and $V\left(\alpha \right)=\left\{h\right|h\le 0\}$ for all $\alpha \in H^N$ with $\left|B\right(\alpha \left)\right|=1$. Clearly, $C(N,V)\ne \varnothing$. Assume that $x\in I(N,V)$ with $x_i=6$, $x_k=0$. Clearly, $V_{\left\{i\right\},x}^M\left(t\right)=\left\{p\right|(p,0)\in V(t,2)\}$ for all $t=1,2,3$ and $V_{\left\{k\right\},x}^M\left(t\right)=\left\{q\right|(18,q)\in V(3,t)\}$ for each $t=1,2$. Clearly, $x_i\in C(\left\{i\right\},V_{\left\{i\right\},x}^M)$ and $x_k\in C(\left\{k\right\},V_{\left\{k\right\},x}^M)$. However, $(x_i,x_k)=(6,0)\in intV(1,1)$. Thus, $x\notin C(N,V)$. The replicated core violates converse M-consistency.

To finalize this section, we show that there exists an unique solution other than the replicated core satisfying SOPG, SRY, MCONY, and CMCONY on $MG$.

Lemma 8.

On $MG$, there exists an unique solution other than the replicated core satisfying SOPG, SRY, MCONY and CMCONY.

Proof of Lemma 8. 

Clearly, it is shown that the replicated core satisfies SOPG, SRY, and MCONY. Here, we define $C_{mw}$ by for each $(N,V)\in MG$,

$$C_{mw}(N,V)=\{x\in IP(N,V)|{\left({\mu }_i{x}_i\right)}_{i\in B\left(\mu \right)}\notin intV\left(\mu \right)forall\mu \in H_{*}^N\},$$

where $H_{*}^N=\{\mu \in H^N|{\mu }_i=h_{i}forsomei\in N\}$. Clearly, $C(N,V)\subseteq C_{mw}(N,V)$ for each $(N,V)\in MG$. It is easy to verify that the solution $C_{mw}$ satisfies SOPG and SRY. To verify that $C_{mw}$ satisfies MCONY and CMCONY, it could similarly be completed by the proof of Lemmas 1–5. Similar to Lemma 6 and Theorem 3, it is easy to show that $C_{mw}$ is the only solution satisfying SOPG, SRY, MCONY, and CMCONY on $MG$. □

6. Discussion

Some multi-choice NTU extensions of the core and related results have been introduced, such as Hwang and Li [10], Liu et al. [11], Tian et al. [12], and so on. One should compare our works with the works of Hwang and Li [10], Liu et al. [11], and Tian et al. [12].

• The core due to Hwang and Li [10] was defined by determining a payoff for a given agent adopting at a given energy level. Based on the notion of replicated games, we propose the replicated core by considering an entire payoff for a given agent.
• Liu et al. [11] derived an extended existence result of the payoff-dependent balanced core proposed by Bonnisseau and Iehlé [13] to multi-choice NTU games which implies a multi-choice generalization of the existence result due to Scarf [7]. By adopting an extended K-K-M-S theorem, Tian et al. [12] showed that a socially stable core is non-empty in a multi-choice NTU structured game if the transfer rate rule contains several values of power mappings, and this game is payoff-dependent balanced with reference to the rule. Different from the works of Liu et al. [11] and Tian et al. [12], we define the replicated core on multi-choice NTU games by means of “replicated NTU games”. The non-emptiness of the replicated core is analyzed by extending the existence result due to Scarf [7] and the notion of replication simultaneously. Inspired by Peleg [8], we characterize the replicated core by means of consistency and its converse.

Multi-choice games permit a finite number of energy levels, while fuzzy games admit a continuous level. In a fuzzy game, every agent i can adopt level $t_i\in [0,1]$ to participate. Fuzzy games and multi-choice games could be treated as continuous generalization and discrete generalization of standard games, respectively. In the framework of fuzzy NTU games, an extended core was introduced by Hwang [14]. The extended core due to Hwang [14] and the replicated core are based on the notions of the core on continuum market games (Hart [4]) and fuzzy TU games (Aubin [5]). The major difference is that our works are based on the “multi-choice” NTU games and “replicated NTU games”, but Hwang’s [14] works are based on the “fuzzy” NTU games. Although the notions of axiomatic results are similar, the techniques of the proofs are still different. Furthermore, the notion of replication can not be applied in Hwang [14].

By applying the replicated core, we would like to analyze cost distribution problems in which the indivisible commodities might be only available under finite natural numbers (energy levels). The replicated core is introduced to offer per unit payoffs (prices) among commodities in multi-choice NTU situations. Assume that $N=\{1,2,\cdots ,n\}$ is a collection of commodities (or projects) that could be equipped conjunctly by various departments. Let $K\left(T\right)$ be the cost of offering the components in $T\subseteq N$ conjunctly. The mapping K is called a cost-distributing problem (alternatively, K could be considered as a production mapping that appears to be the outcome for any coalition). Modeled by this conception, a cost distribution problem could be considered as a cooperative game, with K being its characteristic mapping. Another condition is investigated when the outcome could be altered continuously. In this condition, it is supposed that commodities are entirely divisible commodities and then quantities of commodities could be estimated with real numbers. This is an adequate approach for situations such as various rural commodities, petroleum commodities (olive oil, gas), and so on. However, there exist many others types of commodities for which this is not possible, such as buildings, cars, and so on. This class of indivisible commodities are only available under finite natural numbers. This is the type of condition that the replicated core would like to analyze: cost distribution problems in which commodities could be provided at finite different levels.

7. Conclusions

In view of the techniques of the analysis processes for the core concepts, consistency and converse consistency play essential roles. In this paper, the replicated core and its non-emptiness are investigated. In order to analyze the rationality, an axiomatic result of the replicated core is also proposed by adopting consistency and its converse. However, we also show that the replicated core violates converse M-consistency. In future investigation, we would like to propose axiomatic results by forsaking consistency or its converse.

On the other hand, Iñarra et al. [15] provided an update on the bargaining set, the kernel and the nucleolus, respectively, by investigating consistency, non-emptiness, and uniqueness of the nucleolus. Inspired by Iñarra et al. [15], it is reasonable that some more different solutions and related axiomatic results could be extended to multi-choice NTU games, such as the bargaining set, the kernel, the nucleolus, and so on. This is left to the readers.