The Replicated Core under Multi-Choice Non-Transferable- Utility Situations: Converse Reduction Axiomatic Enlargements

Abstract

Since the replicated core counters the (inferior) converse reduction axiom under multi-choice non-transferable-utility (NTU) situations, two converse reduction axiomatic enlargements of the replicated core are generated. These two enlargements are the smallest (inferior) converse reduction axiomatic solutions that contain the replicated core. Finally, relative axiomatic results are also provided.

1. Introduction

Reduction axiom (Reduced game property) and converse reduction axiom (converse reduced game property) are crucial properties of feasible solutions in the characteristic formulation of cooperative situations. These two properties have been analyzed in numerous kinds of issues by applying reductions, such as apportionment problems, cost allocation problems, bankruptcy and taxation problems, resource problems, matching problems, bargaining problems, fair assignment problems of indivisible goods, and so on. Several definitions of a reduction have been generated relying upon how the participants outside of the subdivision of a group are handed over. For instance, Sobolev [1] and Peleg [2,3] characterized the core, the prekernel, and the prenucleolus, respectively, by considering the reduced situation and relative reduction axiom due to Davis and Maschler [4]. Hart and Mas-Colell [5] proposed a notion of a reduced situation to characterize the Shapley value. In addition, Moulin [6] considered a different notion of a reduced situation under the issue of quasi-linear cost allocation problems. “Averaging notion” and “summing notion” instead of “maximizing behavior” under the reduced situation were terms used by Davis and Maschler [4] and Hwang and Liao [7]; Liao [8] characterized several analogues of the Banzhaf–Coleman index and the Banzhaf–Coleman index by adopting different extended reductions. Reduction axiom permits one to infer, from the attractiveness of a consequence for some issue, the attractiveness of its restraint to each subdivision of a group for the correlative reduction of the subdivision confronts; as its name demonstrates, the converse reduction axiom empowers a converse functioning: to infer the attractiveness of a consequence for several issues from the attractiveness of its restraint to every subdivision for the correlative reduction this subdivision confronts. The converse reduction axiom might be regarded as an attribute of “decentralizability”. Taken several situations, it should be picked for the situation involving the entire collection if an alternative is picked for every one of its correlative reductions. The core is, probably, the most intuitional solution concept. There exist two distinct forms of reductions among the axiomatic results for the core in the existing research: the “max-reduction” (Davis and Maschler [4]) and the “complement-reduction” (Moulin [6]). For instance, Peleg [2,3] as well as Serrano and Volij [9] characterized the core by means of the notion of the max-reductions. In addition, Tadenuma [10] characterized the core by applying the complement reductions.

A non-transferable-utility (NTU) situation could be regarded as a measuring of outcomes achievable by participants of every alliance through some joint procedures of decision (strategy). The participants confront the issue of picking an outcome that is feasible for the whole environment. On the other hand, a multi-choice NTU situation is a generalized analogue of a traditional NTU situation. Under a traditional NTU situation, each participant is either thoroughly involved or not involved radically under operation with some other participants, while under a multi-choice situation, every participant is allowed to apply a finite number of distinct participation levels. Solution concepts under multi-choice situations have been used in many fields, such as economics, environmental sciences, management sciences, and even political sciences. Several extended core concepts under multi-choice NTU situations and relative results have been introduced, such as Hwang and Li [11], Hwang and Liao [12], Liu et al. [13], Tian et al. [14], and so on. Inspired by van den Nouweland et al. [15], Hwang and Li [11] introduced an extended core on multi-choice NTU situations. Liu et al. [13] generated an extended existence result of the payoff-dependent balanced core due to Bonnisseau and Iehlé [16] under multi-choice NTU situations. Based on a specific K-K-M-S result, Tian et al. [14] presented the non-emptiness for a socially stable core concept under multi-choice NTU structured situations. Here, we focus on the extended core concept due to Hwang and Liao [12]. Hwang and Liao [12] proposed the replicated core by extending the replicated notion of Calvo and Santos [17] and Hwang and Liao [18] to multi-choice NTU situations. Furthermore, Hwang and Liao [12] defined a multi-choice analogue of Peleg’s [2] reduction to characterize the replicated core. Hwang and Liao [12] also provided an extended complement reduction on multi-choice NTU situations. However, Hwang and Liao [12] presented that the replicated core is “not” the unique solution matching individual rationality, non-emptiness, and the complement-reduction axiom under the domain of multi-choice NTU balanced situations. The reason is that the replicated core violates the (inferior) converse complement-reduction axiom.

As mentioned above, one motivation could be considered worthy:

• We would like to evaluate how severe the contravention of the (inferior) converse reduction axiom is if a solution violates (inferior) the complement-converse reduction axiom.

To resolve this motivation, we would build on the results of Hwang and Liao [12]. One approach to assess them is to appraise the range to which the solution would have to be revised in order to match the axiom. Relative results are as follows.

1.

In Section 4, we minimally expand the replicated core so as to reinstate the (inferior) converse reduction axiom, that is, the smallest (inferior) converse reduction axiomatic solution that contains the replicated core.

2.

To present the applied expedience and the mathematical correctness simultaneously, these two enlargements would be characterized by means of relative properties of the reduction axiom and (inferior) converse reduction axiom.

2. The Replicated Core

Let $UP$ be the universe of participants. Assume that every participant t has $h_t\in \mathbb{P}$ participation levels at which it can operate. Let $h^{UP}\in {\mathbb{P}}^{UP}$ be the vector that presents the amount of participation levels for every participant. Let $P\subseteq UP$ be a collection of participants. Denote $h_P^{UP}\in {\mathbb{R}}^P$ to be the restriction of $h^{UP}$ to P. For $t\in UP$, we set $H_t=\{0,1,\cdots ,h_t\}$ as the participation level space of participant t, where 0 means not participating and $H_t^{+}=H_t\backslash \left\{0\right\}$. For $P\subseteq UP$, $P\ne \varnothing$, let $H^P={\prod }_{t\in P}{H}_t$ be the product collection of the participation level spaces for participants P. Given $\alpha \in {\mathbb{R}}^P$, we define that $E\left(\alpha \right)=\{t\in P|{\alpha }_t>0\}$ and ${\alpha }_T\in {\mathbb{R}}^T$ to be the restriction of $\alpha$ to coalition T.

Let $\alpha ,\beta \in {\mathbb{R}}^P$. $\alpha \ge \beta$ if ${\alpha }_i\ge {\beta }_i$ for every $i\in P$; $\alpha >\beta$ if $\alpha \ge \beta$ and $\alpha \ne \beta$; $\alpha \gg \beta$ if ${\alpha }_i>{\beta }_i$ for every $i\in P$. Denote that ${\mathbb{R}}_{+}^P=\{\alpha \in {\mathbb{R}}^P|\alpha \ge 0_P\}$. Let $K\subseteq {\mathbb{R}}^P$. K is comprehensive if $\alpha \in K$ and $\alpha \ge \beta$ imply $\beta \in K$. Denote the comprehensive hull of K to be $\mathbf{com}K$, the boundary of K to be $\partial K$, and the interior of K to be $\mathbf{int}K$. If $\alpha \in {\mathbb{R}}^P$, then $\alpha +K=\{\alpha +k|k\in K\}$.

Definition 1.

Amulti-choice NTU situation is denoted by$(P,M)$. As$h^{UP}\in {\mathbb{P}}^{UP}$is fixed in the note, we could write$(P,M)$rather than$(P,h_P^{UP},M)$. Thus,$h_P^{UP}$will be denoted by h if no confusion can occur. Here,$P\ne \varnothing$is a finite collection of participants and M is a measuring function that allocates to every$\alpha ={\left({\alpha }_i\right)}_{i\in P}\in H^P\backslash \left\{0_P\right\}$a subset$M\left(\alpha \right)$of${\mathbb{R}}^{E\left(\alpha \right)}$matching

1. 

$M\left(\alpha \right)$is closed, non-empty, and comprehensive.

2. 

$M\left(\alpha \right)\cap (\lambda +{\mathbb{R}}_{+}^{E\left(\alpha \right)})$is bounded for all$\lambda \in {\mathbb{R}}^{E\left(\alpha \right)}$.

Denote the class of total multi-choice NTU situations to be $\overline{NG}$. Taken $T\subseteq P$, let $\left|T\right|$ be the amount of elements in T, and let $e^T\left(P\right)={\left(e_i^T\left(P\right)\right)}_{i\in P}\in {\mathbb{R}}^P$ with

$$e_i^T\left(P\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if} i\in T,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Here, $e^T\left(P\right)$ would be denoted to be $e^T$ if no confusion can arise.

Let $(P,M)\in \overline{NG}$. A payoff vector of $(P,M)$ is a vector ${\left({\lambda }_i\right)}_{i\in P}\in {\mathbb{R}}^P$, where ${\lambda }_i$ shows the per-unit payoff that participant i obtains for every $i\in P$; hence, $h_i{\lambda }_i$ is the global payoff that participant i obtains at $(P,M)$. Then, a payoff vector $\lambda$ is

• Efficient if ${\left(h_t{\lambda }_t\right)}_{t\in P}\in \partial M\left(h\right)$.
• Individually expedient if for all $t\in P$ and for all $k_t\in H_t^{+}$, $k_t{\lambda }_i\notin \mathbf{int}M\left(k_t{e}^{\left\{t\right\}}\right)$.

In addition, $\lambda$ is an imputation of $(P,M)$ if it is efficient and expedient. Denote the collection of feasible payoff vectors of $(P,M)$ to be

$$X^{*}(P,M)=\{\lambda \in {\mathbb{R}}^P|{\left(h_i{\lambda }_i\right)}_{i\in P}\in M\left(h\right)\},$$

whereas

$$X(P,M)=\{\lambda \in {\mathbb{R}}^P|\lambda \mathrm{is}\mathrm{EFF}\}$$

is the collection of preimputations of $(P,M)$ and the collection of imputations of $(P,M)$ is

$$I(P,M)=\{\lambda \in {\mathbb{R}}^P|\lambda \mathrm{is}\mathrm{an}\mathrm{imputation}\mathrm{of} (P,M)\}.$$

A solution on $\overline{NG}$ is a mapping $\tau$ which associates with every $(P,M)\in \overline{NG}$ a subset $\tau (P,M)$ of $X^{*}(P,M)$. Hwang and Liao [12] proposed a solution concept as follows.

Definition 2.

(Hwang and Liao [12]). Let $(P,M)\in \overline{NG}$. The replicated core $RC(P,M)$ of $(P,M)\in \overline{NG}$ is defined by

$$RC(P,M)=\{\lambda \in X(P,M)|{\left({\alpha }_i{\lambda }_i\right)}_{i\in E\left(\alpha \right)}\notin \mathbf{int}M\left(\alpha \right)for all \alpha \in M^P\backslash \left\{0_P\right\}\}.$$

3. Complement Reduction

Let $\tau$ be a solution on $\overline{NG}$. $\tau$ matches non-emptiness (NES) if for all $(P,M)\in \overline{NG}$, $\tau (P,M)\ne \varnothing$. $\tau$ matches efficiency (EIY) if for all $(P,M)\in \overline{NG}$, $\tau (P,M)\subseteq X(P,M)$. $\tau$ matches individual expediency (IEY) if for all $(P,M)\in \overline{NG}$, $\tau (P,M)\subseteq I(P,M)$. $\tau$ matches one-person expediency (OPEY) if for all $(P,M)\in \overline{NG}$ with $\left|P\right|=1$, $\tau (P,M)=I(P,M)$.

Hwang and Liao [12] introduced and extended complement-reduction as follows. Given $(P,M)\in \overline{NG}$, $T\subseteq P$ with $T\ne \varnothing$ and a payoff vector $\lambda$, the M-reduction is the situation $(T,M_{T,\lambda })$ defined by for all $\alpha \in H^T$,

$$M_{T,\lambda }\left(\alpha \right)=\{\omega \in {\mathbb{R}}^{E\left(\alpha \right)}|\left(\omega ,{\left(h_i{\lambda }_i\right)}_{i\in P\backslash T}\right)\in M(\alpha ,h_{P\backslash T})\}.$$

The reduction axiom, originally defined by Harsanyi [19] under the name of bilateral equilibrium, claims that the projection of $\lambda$ to P should be stipulated by $\tau$ for the reduced condition with regard to P and $\lambda$ if $\lambda$ is stipulated by $\tau$ for a situation $(P,M)$. Hence, the projection of $\lambda$ to P should be reduction axiomatic with the expectations of the components of P as reflected by its reduced condition.

• Reduction axiom (RDA): If $(P,M)\in \overline{NG}$, $T\subseteq P$, $T\ne \varnothing$, and $\lambda \in \tau (P,M)$, then $(T,M_{T,\lambda })\in \overline{NG}$ and ${\lambda }_T\in \tau (T,M_{T,\lambda })$.

The converse reduction axiom claims that $\lambda$ itself should be appointed for the entire situation if the projection of an efficient payoff vector $\lambda$ to each proper P is reduction axiomatic with the expectations of the components of P as reflected by its reduced condition.

• Converse reduction axiom (CRDA): If $(P,M)\in \overline{NG}$ with $\left|P\right|\ge 2$, $\lambda \in X(P,M)$, and for all $T\subset P$, $0<\left|T\right|<\left|P\right|$, $(T,M_{T,\lambda })\in \overline{NG}$ and ${\lambda }_T\in \tau (T,M_{T,\lambda })$, then $\lambda \in \tau (P,M)$.

The following axiom is a weakening of converse reduction axiom, since it claims that $\lambda$ be individually expedient as well.

• Inferior converse reduction axiom (ICRDA): If $(P,M)\in \overline{NG}$ with $\left|P\right|\ge 2$, $\lambda \in I(P,M)$, and for all $T\subset P$, $0<\left|T\right|<\left|P\right|$, $(T,M_{T,\lambda })\in \overline{NG}$ and ${\lambda }_T\in \tau (T,M_{T,\lambda })$, then $\lambda \in \tau (P,M)$.

Hwang and Liao [12] showed that the replicated core matches the reduction axiom and violates the (inferior) converse reduction axiom.

4. Minimal Conversely Reduction Axiomatic Enlargement

In this section, the minimal conversely reduction axiomatic enlargement of the replicated core would be considered. Let $\Lambda$ be the family of whole solutions on $\overline{NG}$, ${\Lambda }^C$ be the family of whole solutions on $\overline{NG}$ matching CRDA. Let $\tau \in \Lambda$. Define that

$${\Lambda }_{\tau }^m=\{\kappa \in {\Lambda }^C|\tau (P,M)\subseteq \kappa (P,M)\mathrm{for}\mathrm{all}(P,M)\in \overline{NG}\}.$$

The minimal conversely reduction axiomatic enlargement of $\tau$ on $\overline{NG}$, $mcra{e}^{\tau }$, is defined by

$$mcra{e}^{\tau }=\bigcap _{\kappa \in {\Lambda }_{\tau }^m}\kappa .$$

Evidently, the minimal conversely reduction axiomatic enlargement is well-defined and uniquely defined, since the feasible collection matches CRDA, and CRDA is maintained under the intersection of conversely reduction axiomatic solutions.

To generate the main results, some notations are needed. Let $P\subseteq UP$, $P\ne \varnothing$. The collection of operational vectors with the full participation level of one participant, ${\Lambda }^P$, is defined by

$${\Lambda }^P=\{\alpha \in H^P|\alpha =(h_i,0_{P\backslash \left\{i\right\}})\mathrm{for}\mathrm{all} i\in P\}.$$

The collection of operational vectors with the non-full participation level of every participants, ${\Theta }^P$, is defined by

$${\Theta }^P=\{\alpha \in H^P|{\alpha }_i\ne h_i\mathrm{for}\mathrm{all} i\in P\}.$$

Let $H_{*}^P=H^P\backslash ({\Lambda }^P\cup {\Theta }^P)$, the minimal replicated core $R{C}_m$ is defined as for every $(P,M)\in \overline{NG}$,

$$\begin{array}{cc}\hfill & R{C}_m(P,M)\hfill \\ \hfill =& \left\{\begin{array}{cc}RC(P,M),\hfill & \mathrm{if} \left|P\right|=1,\hfill \\ \{\lambda \in I(P,M)|{\left({\alpha }_i{\lambda }_i\right)}_{i\in E(\alpha ,N)}\notin \mathbf{int}M\left(\alpha \right)\mathrm{for}\mathrm{all}\alpha \in H_{*}^P\},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

Clearly, $RC(P,M)\subseteq R{C}_m(P,M)$ for every $(P,M)\in \overline{NG}$. Subsequently, we would show that the minimal replicated core is the minimal conversely reduction axiomatic enlargement of the replicated core on $\overline{NG}$.

Lemma 1.

Let $(P,M)\in \overline{NG}$, $\lambda \in M\left(h\right)$ and $T\subseteq P$, $T\ne \varnothing$. Then, the reduction $(T,M_{T,\lambda })\in \overline{NG}$.

Proof. 

The verification of this lemma is straightforward. Thus, we omit it. □

Lemma 2.

Let $(P,M)\in \overline{NG}$, $\lambda \in M\left(m\right)$ and $T\subseteq P$, $T\ne \varnothing$. Then, $\lambda \in X(P,M)$ if and only if ${\lambda }_T\in X(T,M_{T,\lambda })$.

Proof. 

Clearly, ${\lambda }_T\in M_{T,\lambda }\left(h_T\right)$. If ${\lambda }_T\notin \lambda (T,M_{T,\lambda })$; then, there exists ${\omega }_T\in M_{T,\lambda }\left(h_T\right)$ such that ${\omega }_T>{\lambda }_T$. Hence, $({\omega }_T,{\lambda }_{P\backslash T})\in M\left(h\right)$ and $({\omega }_T,{\lambda }_{P\backslash T})>\lambda$. Therefore, $\lambda \notin X(P,M)$. Similarly, there exists $\omega \in M\left(h\right)$ such that $\omega \gg \lambda$ if $\lambda \notin X(P,M)$. Hence, $({\omega }_T,{\lambda }_{P\backslash T})\in M\left(h\right)$. Thus, ${\omega }_T\in M_{T,\lambda }\left(h_T\right)$ and ${\omega }_T\gg {\lambda }_T$. Therefore, ${\lambda }_T\notin X(S,M_{T,\lambda })$. □

Lemma 3.

On $\overline{NG}$, the minimal replicated core $R{C}_m$ matches RDA and CRDA.

Proof. 

First, we would prove that the minimal replicated core matches RDA. Let $(P,M)\in \overline{NG}$, $T\subseteq P$ with $0<\left|T\right|<\left|P\right|$ and $\lambda \in R{C}_m(P,M)$. Since $\lambda \in X(P,M)$, ${\lambda }_T\in X(T,M_{T,\lambda })$ by Lemma 2. It remains to show that for all $\alpha \in H_{*}^T\backslash \left\{0_T\right\}$, ${\left({\alpha }_i{\lambda }_i\right)}_{i\in E\left(\alpha \right)}\notin int{M}_{T,\lambda }\left(\alpha \right)$. Let $\alpha \in H_{*}^T\backslash \left\{0_T\right\}$. Since $\alpha \in H_{*}^T$ and $0<\left|T\right|<\left|P\right|$, $(\alpha ,m_{P\backslash T})\in H_{*}^P$. Since $\lambda \in R{C}_m(P,M)$ and $(\alpha ,h_{P\backslash T})\in H_{*}^P$,

$$\left({\left({\alpha }_i{\lambda }_i\right)}_{i\in E\left(\alpha \right)},{\left(h_i{\lambda }_i\right)}_{i\in P\backslash T}\right)\notin intM\left((\alpha ,h_{P\backslash T})\right).$$

(1)

Assume that ${\left({\alpha }_i{\lambda }_i\right)}_{i\in E\left(\alpha \right)}\in int{M}_{T,\lambda }\left(\alpha \right)$. By definition of $M_{T,\lambda }$,

$$\left({\left({\alpha }_i{\lambda }_i\right)}_{i\in E\left(\alpha \right)},{\left(h_i{\lambda }_i\right)}_{i\in P\backslash T}\right)\in intM\left((\alpha ,h_{P\backslash T})\right).$$

(2)

From Equations (1) and (2), the desired contradiction has been obtained. Hence, for all $\alpha \in H_{*}^T\backslash \left\{0_T\right\}$, ${\left({\alpha }_i{\lambda }_i\right)}_{i\in E\left(\alpha \right)}\notin int{M}_{T,\lambda }\left(\alpha \right)$.

Next, we would show that the minimal replicated core matches CRDA. Let $(P,M)\in \overline{NG}$ with $\left|P\right|\ge 2$ and $\lambda \in X(P,M)$. Suppose that for every $T\subset P$ with $0<\left|T\right|<\left|P\right|$, $(T,M_{T,\lambda })\in \overline{NG}$ and ${\lambda }_T\in R{C}_m(T,M_{T,\lambda })$. Let $\alpha \in H_{*}^P\backslash \left\{0_P\right\}$ and ${\alpha }_K=h_K$ for some $K\subset P$ with $1\le \left|K\right|<\left|P\right|$. Since $1\le \left|K\right|<\left|P\right|$ and ${\lambda }_{P\backslash K}\in R{C}_m(P\backslash K,M_{P\backslash K,\lambda })$, ${\left({\alpha }_i{\lambda }_i\right)}_{i\in E\left(\alpha \right)}\notin int{M}_{P\backslash K,\lambda }\left({\alpha }_{P\backslash K}\right)$. By definition of $M_{P\backslash K,\lambda }$, ${\left({\alpha }_i{\lambda }_i\right)}_{i\in E\left(\alpha \right)}\notin intM\left(\alpha \right)$. Assume that $\alpha =h$, since ${\left(h_i{\lambda }_i\right)}_{i\in P}\in \partial M\left(h\right)$, ${\left(h_i{\lambda }_i\right)}_{i\in P}\notin intM\left(h\right)$. Hence, for all $\alpha \in H_{*}^P$, ${\left({\alpha }_i{\lambda }_i\right)}_{i\in E\left(\alpha \right)}\notin intM\left(\alpha \right)$. Therefore, $\lambda \in R{C}_m(P,M)$. □

Theorem 1.

The minimal replicated core $R{C}_m$ is the minimal conversely reduction axiomatic enlargement of the replicated core on $\overline{NG}$.

Proof. 

Let $(P,M)\in \overline{NG}$. Since $RC(P,M)\subseteq R{C}_m(P,M)$ and $R{C}_m$ matches CRDA, $mcra{e}^{RC}(P,M)\subseteq R{C}_m(P,M)$ by definition of $mcra{e}^{RC}$.

It remains to prove that for every $(P,M)\in \overline{NG}$, $R{C}_m(P,M)\subseteq mcra{e}^{RC}(P,M)$. Let $(P,M)\in \overline{NG}$. The proof proceeds by induction on $\left|P\right|$. Suppose that $\left|P\right|=1$. Since $RC(P,M)\subseteq mcra{e}^{RC}(P,M)\subseteq R{C}_m(P,M)$ and $R{C}_m(P,M)=RC(P,M)=I(P,M)$, $mcra{e}^{RC}(P,M)=R{C}_m(P,M)$. Assume that $R{C}_m(P,M)\subseteq mcra{e}^{RC}(P,M)$ if $\left|P\right|\le l-1$, where $l\ge 2$.

The condition $\left|P\right|=l$: Let $\lambda \in R{C}_m(P,M)$. By RDA of $R{C}_m$, for all $T\subset P$ with $0<\left|T\right|<\left|P\right|$, ${\lambda }_T\in R{C}_m(T,M_{T,\lambda })$. By the induction hypotheses, ${\lambda }_T\in R{C}_m(T,M_{T,\lambda })\subseteq mcra{e}^{RC}(T,M_{T,\lambda })$. By CRDA of $mcra{e}^{RC}$, $\lambda \in mcra{e}^{RC}(P,M)$. Hence, $R{C}_m(P,M)\subseteq mcra{e}^{RC}(P,M)$. □

Remark 1.

The multi-choice NTU situation $(P,M)\in \overline{NG}$ is balanced if $RC(P,M)\ne \varnothing$. Let ${\overline{NG}}^B$ be the collection of whole balanced multi-choice NTU situations. The minimal replicated core $R{C}_m$ is still the minimal conversely reduction axiomatic enlargement of the replicated core on ${\overline{NG}}^B$ if one restricts the domain to ${\overline{NG}}^B$. It is also possible to introduce the notion of the minimal inferior conversely reduction axiomatic enlargement of the replicated core. It is the smallest solution containing the replicated core which matches the inferior converse reduction axiom. Define solution $R{C}_{mi}$ on $\overline{NG}$ by for every $(P,M)\in \overline{NG}$,

$$R{C}_{mi}(P,M)=\{\lambda \in I(P,M)|{\left({\alpha }_i{\lambda }_i\right)}_{i\in E\left(\alpha \right)}\notin \mathbf{int}M\left(\alpha \right)forall\alpha \in \left(H^P\backslash {\Theta }^P\right)\}.$$

Similarly, we could show that the solution $R{C}_{mi}$ coincides with the minimal inferior conversely reduction axiomatic enlargement of the replicated core on $\overline{NG}$ and ${\overline{NG}}^B$.

Intuitively, due to the axiomatic techniques of the core concept on traditional NTU situations, it is easy to derive an axiomatic result relative to the minimal (inferior) conversely reduction axiomatic enlargement of the replicated core as follows.

Lemma 4.

A solution τ on $\overline{NG}$ matches EIY if it matches OPEY and RDA.

Proof. 

The proof can easily be deduced from Lemma 5.4 in Peleg [2]. □

Theorem 2. 

1. 

A solution τ on $\overline{NG}$ matches OPEY, RDA, and CRDA if and only if for every $(P,M)\in \overline{NG}$, $\tau (P,M)=R{C}_m(P,M)$.

2. 

A solution τ on $\overline{NG}$ matches OPEY, IEY, RDA, and ICRDA if and only if for every $(P,M)\in \overline{NG}$, $\tau (P,M)=R{C}_{mi}(P,M)$.

Proof. 

By Lemma 3, the solution $R{C}_m$ matches RDA and CRDA. By Remark 1, the solution $R{C}_{iw}$ matches RDA and ICRDA. Clearly, the solutions $R{C}_m$ and $R{C}_{mw}$ match OPEY and IEY.

To present the uniqueness of 1, assume that a solution τ matches OPEY, RDA, and CRDA. By Lemma 4, τ matches EIY. Let $(P,M)\in \overline{NG}$. The proof proceeds by induction on $\left|P\right|$. By OPEY of τ, $\tau (P,M)=I(P,M)=R{C}_m(P,M)$ if $\left|P\right|=1$. Assume that $\tau (P,M)=R{C}_m(P,M)$ if $\left|P\right|<k$, $k\ge 2$.

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

First, we would prove that $\tau (P,M)\subseteq R{C}_m(P,M)$. Let $\lambda \in \tau (P,M)$. By RDA of τ, for all $T\subset P$ with $0<\left|T\right|<\left|P\right|$, ${\lambda }_T\in \tau (T,M_{T,\lambda })$. By the induction hypothesis, ${\lambda }_T\in \tau (T,M_{T,\lambda })=R{C}_m(T,M_{T,\lambda })$. Since τ matches EIY, $\lambda \in X(P,M)$. By CRDA of the minimal replicated core, $\lambda \in R{C}_m(P,M)$. The opposite inclusion could be completed analogously by exchanging the parts of τ and $R{C}_m$. Thus, $\tau (P,M)=R{C}_m(P,M)$. The proof of the uniqueness of 2 is similar. □

Remark 2.

The union of two conversely reduction axiomatic solutions might not be conversely reduction axiomatic; thus, it is not appropriate to propose the maximal conversely reduction axiomatic sub-solution. The maximal conversely reduction axiomatic sub-solution is proposed as follows. Given $\tau \in \Gamma$. Define that

$${\Lambda }_{\tau }^M=\{\kappa \in {\Lambda }^C|\tau (P,M)\supseteq \kappa (P,M)forall(P,M)\in \overline{NG}\}.$$

Themaximal conversely reduction axiomatic sub-solutionof τ on $\overline{NG}$, $Mcra{e}^{\tau }$, is defined by

$$Mcra{e}^{\tau }=\bigcup _{\kappa \in {\Lambda }_{\tau }^M}\kappa .$$

The following examples present that every of the axioms taken in Theorem 2 is logically independent of the others. Clearly, $\left|UP\right|\ge 2$ is needed.

Example 1.

Let $\tau (P,M)=\varnothing$ for every $(P,M)\in \overline{NG}$. Then, τ matches IEY, RDA, and (I)CRDA, but it violates OPEY.

Example 2.

Define a solution τ on $\overline{NG}$ by

$$\tau (P,M)=\left\{\begin{array}{cc}I(P,M),\hfill & if \left|P\right|=1\hfill \\ X(P,M),\hfill & otherwise.\hfill \end{array}\right.$$

Then, τ matches OPEY, RDA, and ICRDA, but it violates IEY.

Example 3.

Let $\tau (P,M)=I(P,M)$ for every $(P,M)\in \overline{NG}$. Then, τ matches IEY and ICRDA, but it violates RDA.

Example 4.

Liao (2008) showed that the replicated core matches IEY, OPEY, and RDA, but it violates (I)CRDA.

5. Conclusions

1.

In this note, the main results of Hwang and Liao [12] are extended to multi-choice NTU situations. Some comparisons among the results of this note and relative results of Hwang and Liao [12] are as follows.

• Under multi-choice NTU situations, Hwang and Liao [12] proposed the replicated core by extending the core concept of Hwang and Liao [18]. The main results of Hwang and Liao [12] are as follows.

  –

  Hwang and Liao [12] defined a multi-choice analogue of Peleg’s [2] reduction to characterize the replicated core.

  –

  Hwang and Liao [12] also defined a multi-choice analogue of Moulin’s [6] reduction to present that this extended Moulin’s [6] reduction “could not” be applied to characterize the replicated core.

• Different from the results of Hwang and Liao [12], the extended Moulin’s [6] reduction due to Hwang and Liao [12] is applied to propose two converse reduction axiomatic enlargements of the replicated core. The main results of this note are as follows.

  –

  The (inferior) converse reduction axiomatic enlargement is the smallest (inferior) converse reduction axiomatic solution that contains the replicated core.

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  The extended Moulin’s [6] reduction due to Hwang and Liao [12] could be applied to characterize these two converse reduction axiomatic enlargements.

• The results of this note do not appear in Hwang and Liao [12] and existing results on multi-choice NTU games. Moreover, the axiomatic techniques of Hwang and Liao [12] and this note are exactly corresponding to the relative techniques of Serrano and Volij [9].

2.

The reduction axiom and (inferior) converse reduction axiom of a solution are indispensable under axiomatic techniques of existing research of core concepts. However, some extended core concepts violate the reduction axiom or (inferior) converse reduction axiom under multi-choice NTU situations. In future research, we could attempt to characterize these core concepts by dropping the reduction axiom or (inferior) converse reduction axiom.