Symmetric and Asymmetric Allocating Concepts Under Multiple-Goals

Abstract

To analyze utility estimating and allocating under real-world interactive modes, various interrelationships formed by members and its operational extents are always investigated by means of multifaceted approaches. To address this, this study first proposes a symmetric allocating concept under multiple-goals. In order to evaluate relative effects result from varying members and its operational extents under different situations, this study introduces several asymmetric generalizations by applying weights related to the members and its operational extents. Several axiomatic outcomes are also adopted to express the mathematical correctness and practicality for these allocating concepts. Moreover, some more explanations for related axiomatic outcomes and applications are also offered throughout this study.

1. Introduction

The concept of marginal contribution is a fundamental element in allocating and decision-making processes. It is often used to describe the impact that an individual, factor, or variable has on a system’s outcome. The underlying principle evaluates how an additional unit of input can alter the overall results or how a reduction in that input affects the system. This concept is applicable across various domains, such as production, decision analysis, and resource allocating. For example, in complex decision-making scenarios involving multiple variables, marginal contribution helps to identify the impact of each variable on the desired outcome, thus supporting the optimization of resource allocating strategies. The importance of marginal contribution lies in its ability to quantify the incremental changes in a system, enabling more precise and rational decision-making.

The link between marginal contribution and symmetry is often explored within game-theoretical analysis, particularly in discussions of allocating theories and fairness principles. Symmetry is a fundamental concept in ensuring that the allocating of resources or rewards is equitable among participants. Symmetry also plays a crucial role in discussions of fairness in cooperative game theory. The combination of symmetry and marginal contribution guarantees that resources are distributed fairly, preventing certain individuals from obtaining disproportionate rewards due to their unique bargaining positions. Under traditional game-theoretical analysis for utility estimating and allocating, the focus usually lies on whether members decide to operate or not. Applying related notion of marginal contribution, the equal allocation of non-separable costs (EANSC, Ransmeier [1]) is considered for utility allocating under traditional modes. Moulin [2] introduced the notions of symmetry and consistency to demonstrate that the EANSC could provide an equitable concept for allocating utility.

Under many real-world interactive modes, however, the greater part of members take varying extents upon different conditions. It is logical for members under the modes to have varying extents of involvement, applying a multi-choice modes where each member has different operational extents of involvement. Hwang and Liao [3], Liao [4], and Nouweland et al. [5] proposed several generalized extensions for the EANSC on multi-choice modes. Inspired by axiomatic analysis of Moulin [2], Hwang and Liao [3] and Liao [4] also applied an extended Moulin’s reduction to characterize several extensions of the EANSC on multi-choice modes. These diverse extents generate various interrelationships among members, necessitating multifaceted approaches for effective estimating and allocating. For instance, administrative and medical staff must apply different medical measures relying on specific medical conditions under healthcare systems. These measures require evaluating how to efficiently adopt limited medical equipment and resources to resolve patients’ conditions and other medical states promptly. Multiple-goals optimization or equilibrium always aims to achieve such issues within any interactive mode in the field of mathematics. Related studies can be found in Bednarczuk et al. [6], Cheng et al. [7], Goli et al. [8], Guarini et al. [9], Mustakerov et al. [10], Tirkolaee et al. [11], Zhang and Wang [12], and so on.

Although symmetry is a desirable principle in many situations, real-world situations often involve asymmetry in contributions. Asymmetry arises when members differ in terms of its resource inputs, extents, skills, or influence related to the system’s outcomes. Furthermore, members and its operational extents exhibit varying grades of consequence relying on the context, influencing estimating and allocating. In above example for healthcare, the significance of administrative staff and doctors differs among administrative evaluations and major surgical operations. Similarly, a dermatologist’s need for surgical intervention varies when treating patients with burns versus skin cancer. Hence, it is rational to allot relevant weights to members or its operational extents during related processes for estimating and allocating under different conditions.

Related outcomes mentioned above generate a key motivation:

• whether the notions of marginal contribution and weights could be adopted to characterize symmetric and asymmetric allocating behavior under the framework of multiple-goals and multi-choice modes.

To verify this motivation, we aim to formulate different necessary mathematical foundations for allocating concepts to analyze allocating issues under multiple-goals and multi-choice modes. Different from the frameworks of traditional and multi-choice modes, this study considers the framework of multiple-goals multi-choice modes, and further proposes different allocating concepts by applying the notion of the average marginal extents-utility under multiple-goals multi-choice modes.

• By extending the symmetric allocating notion of the EANSC to multiple-goals multi-choice modes, the uniform allocating of undifferentiated utility (UAUU) is defined in Section 2. The UAUU involves members receiving its average marginal extents-utility, and then allocating the rest of utility uniformly.
• By incorporating the notion of member-weighted emphasis into the UAUU, the 1-weighted allocating of undifferentiated utility (1-WAUU) is considered in Section 2. In brief, the allocating notion of the 1WAUU involves members first allocating its average marginal extents-utility, followed by allocating the rest of utility based on member-weighted proportions.
• By integrating the notion of extents-weighted emphasis into the UAUU, the 2-weighted allocating of undifferentiated utility (2-WAUU) is introduced in Section 2. In essence, the allocating notion of the 2WAUU entails members first allocating its weighted marginal extents-utility, and then uniformly allocating the rest of utility.
• Combining the allocating notions of the 1WAUU and the 2WAUU gave rise to the bi-weighted allocating of undifferentiated utility (BWAUU) in Section 2. Briefly, the allocating notion of the BWAUU involves members first allocating its weighted marginal extents-utility, and then allocating the rest of utility based on member-weighted proportions.
• However, both the member-weighted and extents-weighted mechanisms appear somewhat artificial and subjective. In Section 4, the interior allocating of undifferentiated utility (IAUU) is generated as an alternative to weighted allocating concepts, applying average marginal extents-utility.

To express the mathematical correctness and practicality for these allocating concepts, this study propose an extended reduction and related properties of consistency, discussed in Section 3.

• In Section 3.1, based on the bilateral consistency for multiple-goals, the UAUU, the 1-WAUU, the 2-WAUU and the BWAUU could be characterized by different forms of criterion for multiple-goals.
• In Section 3.2, based on a specific property of symmetry, another axiomatic result for the UAUU is proposed. It is also shown that the 1-WAUU, the 2-WAUU and the BWAUU are asymmetric.
• While the IAUU violates bilateral consistency for multiple-goals, in Section 3.3, it adheres to the properties of interior criterion for multiple-goals and revised consistency for multiple-goals. It is also shown that the IAUU is symmetric.

Throughout the study, some more discussions and interpretations regarding these properties and axiomatic analysis are interpreted to further elucidate its implications.

2. Preliminaries

Let $\ddot{\mathbf{UM}}$ denote the universe of members, for instance, the set comprised of all members of a country. Any $i\in \ddot{\mathbf{UM}}$ is identified as a member of $\ddot{\mathbf{UM}}$, such as a member in an operational organization. For $i\in \ddot{\mathbf{UM}}$ and ${\tilde{\zeta }}_i\in \mathbb{N}$, we define ${\ddot{\mathbf{T}}}_i=\{0,1,\cdots ,{\tilde{\zeta }}_i\}$ to be the collection of extents for member i, and ${\ddot{\mathbf{T}}}_i^{+}={\ddot{\mathbf{T}}}_i\setminus \left\{0\right\}$, where 0 means no operating.

Let $\ddot{\mathbf{M}}\subseteq \ddot{\mathbf{UM}}$, such as a collection encompassing all employees of an operational organization within a country $\ddot{\mathbf{UM}}$. Let ${\ddot{\mathbf{T}}}^{\ddot{\mathbf{M}}}={\prod }_{i\in \ddot{\mathbf{M}}}{\ddot{\mathbf{T}}}_i$ be the product set of all extents collections for every member in $\ddot{\mathbf{M}}$. For every $H\subseteq \ddot{\mathbf{M}}$, a member alliance H corresponds, in a standard manner, to the multi-choice alliance ${\widehat{z}}^H\in {\ddot{\mathbf{T}}}^{\ddot{\mathbf{M}}}$, which is a vector indicating ${\widehat{z}}_i^H=1$ if $i\in H$, and ${\widehat{z}}_i^H=0$ if $i\in \ddot{\mathbf{M}}\setminus H$. Define $0_{\ddot{\mathbf{M}}}$ to be the zero vector in ${\mathbb{R}}^{\ddot{\mathbf{M}}}$. For $n\in \mathbb{N}$, also define ${\ddot{\mathbf{G}}}_n=\{1,2,\cdots ,n\}$ and $0_n$ as the zero vector in ${\mathbb{R}}^n$.

A multi-choice mode is denoted to be $(\ddot{\mathbf{M}},\tilde{\zeta },\theta )$, where $\ddot{\mathbf{M}}\ne \varnothing$ is a finite collection of members, $\tilde{\zeta }={\left({\tilde{\zeta }}_k\right)}_{k\in \ddot{\mathbf{M}}}\in {\ddot{\mathbf{T}}}^{\ddot{\mathbf{M}}}$ is a vector representing the amount of extents for each member, and $\theta :{\ddot{\mathbf{T}}}^{\ddot{\mathbf{M}}}\to \mathbb{R}$ is a mapping with $\theta \left(0_{\ddot{\mathbf{M}}}\right)=0$ that assigns to each extents vector $\tilde{\chi }={\left({\tilde{\chi }}_k\right)}_{k\in \ddot{\mathbf{M}}}\in {\ddot{\mathbf{T}}}^{\ddot{\mathbf{M}}}$ the utility that members can produce if each member k taking extents ${\tilde{\chi }}_k$. A multiple-goals multi-choice mode is denoted to be $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$, where $n\in \mathbb{N}$, ${\Theta }^n={\left({\theta }^t\right)}_{t\in {\ddot{\mathbf{G}}}_n}$ and $(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)$ represents a multi-choice mode for each $t\in {\ddot{\mathbf{G}}}_n$. The family of all multiple-goals multi-choice modes is denoted as $\overline{\mathbf{MGCM}}$.

An allocation is defined to be a function $\rho$ that allots to each $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ a vector

$$\rho \left(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n\right)={\left({\rho }^t\left(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t\right)\right)}_{t\in {\ddot{\mathbf{G}}}_n},$$

where ${\rho }^t\left(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t\right)={\left({\rho }_k^t\left(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t\right)\right)}_{k\in \ddot{\mathbf{M}}}\in {\mathbb{R}}^{\ddot{\mathbf{M}}}$ and ${\rho }_k^t\left(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t\right)$ means the utility of member k if k participates in $\left(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t\right)$. Let $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, $S\subseteq \ddot{\mathbf{M}}$, and $\tilde{\chi }\in {\mathbb{R}}^{\ddot{\mathbf{M}}}$. We define $\mathit{Z}\left(\tilde{\chi }\right)=\{k\in \ddot{\mathbf{M}}|{\tilde{\chi }}_k\ne 0\}$ and ${\tilde{\chi }}_S\in {\mathbb{R}}^S$ as the restriction of $\tilde{\chi }$ to S. Given $k\in \ddot{\mathbf{M}}$, we also define ${\tilde{\chi }}_{-k}$ to be ${\tilde{\chi }}_{\ddot{\mathbf{M}}\setminus \left\{k\right\}}$. Additionally, $\tilde{\delta }=({\tilde{\chi }}_{-k},q)\in {\mathbb{R}}^{\ddot{\mathbf{M}}}$ is defined by ${\tilde{\delta }}_{-k}={\tilde{\chi }}_{-k}$ and ${\tilde{\delta }}_k=q$.

Based on the aim of allocating how to estimate utility during interactive processes, this study introduces derivative the concepts of the EANSC within the framework of multiple-goals multi-choice modes.

Definition 1. 

The  uniform allocating of undifferentiated utility (UAUU),  $\overline{\eta }$, is defined by

$$\overline{{\eta }_i^t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)={\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{1}{|\ddot{\mathbf{M}}|}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-\sum _{k\in \ddot{\mathbf{M}}}{\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]$$

for every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for every $t\in {\ddot{\mathbf{G}}}_n$ and for every $i\in \ddot{\mathbf{M}}$. The value ${\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)={\displaystyle \frac{1}{{\tilde{\zeta }}_i}}{\sum }_{q\in {\ddot{\mathbf{T}}}_i^{+}}\{{\theta }^t({\tilde{\zeta }}_{-i},q)-{\theta }^t({\tilde{\zeta }}_{-i},0)\}$ is the  average marginal extents-utility  among all extents of member i in $(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)$. (This study utilizes bounded multi-choice modes, treated to be the modes $(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)$ such that, there exists $B_{\theta }^t\in \mathbb{R}$ such that ${\theta }^t\left(\tilde{\chi }\right)\le B_{\theta }^t$ for every $\tilde{\chi }\in {\ddot{\mathbf{T}}}^{\ddot{\mathbf{M}}}$. It could be utilized to assure that ${\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)$ is well-defined.) Under the concept of $\overline{\eta }$, all members firstly distribute its average marginal extents-utility, and further distribute the rest of utility uniformly.

As stated in Introduction, the notion of weights has turned into a considerable factor under numerous interactive processes. For instance, weight proportions might be related to medicament allocating, where weights could indicate the damaging risk of various medicament adopted in different illnesses. Weight assigning also could be applied to mitigating measures for pollution, where different mitigating measures might cause varying weighted execution consequent among different polluted environments. Even if the execution events and relative polluted environment of a certain mitigating measure are fixed, the execution consequent relative to different mitigated periods under the environment might vary in weighted proportions. Related applied outcomes also have been discussed in many researches, such as Shapley [13], and so on. Hence, allotting weights to “members” or its “extents” to distinguish relative disparities is worth investigating.

Here we define the positive function $\widehat{\omega }:\ddot{\mathbf{UM}}\to \mathbb{R}$ to be a weight evaluation for members. Also, we define the positive function $\check{\omega }:{\cup }_{k\in \ddot{\mathbf{UM}}}{\ddot{\mathbf{T}}}_k^{+}\to \mathbb{R}$ to be a weight evaluation for extents. Based on these two kinds of weight evaluations, three weighted extensions of the UAUU could be considered as follows.

Definition 2. 

• The  1-weighted allocating of undifferentiated utility (1-WAUU), ${\eta }^{\widehat{\omega }}$, is considered as follows: For every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for every weight evaluation for members $\widehat{\omega }$, for every $t\in {\ddot{\mathbf{G}}}_n$, and for every member $i\in \ddot{\mathbf{M}}$,

  $${\eta }_i^{\widehat{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)={\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{\widehat{\omega }\left(i\right)}{{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}\widehat{\omega }\left(k\right)}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-\sum _{k\in \ddot{\mathbf{M}}}{\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right].$$
  According to the definition of ${\eta }^{\widehat{\omega }}$, all members firstly allocate its average marginal extents-utility, and the rest of utility are allocated proportionally via weights for members.
• The  2-weighted allocating of undifferentiated utility (2-WAUU), ${\eta }^{\check{\omega }}$, is considered as follows: For every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for every weight evaluation for extents $\check{\omega }$, for every $t\in {\ddot{\mathbf{G}}}_n$, and for every member $i\in \ddot{\mathbf{M}}$,

  $${\eta }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)={\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{1}{|\ddot{\mathbf{M}}|}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-\sum _{k\in \ddot{\mathbf{M}}}{\kappa }_k^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right],$$

  where ${\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\displaystyle \frac{1}{{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_i^{+}}}\check{\omega }\left(q\right)}}{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_i^{+}}}\{\check{\omega }\left(q\right)·[{\theta }^t({\tilde{\zeta }}_{-i},q)-{\theta }^t({\tilde{\zeta }}_{-i},0)]\}$ is the weighted marginal extents-utility among all extents of member b. By definition of ${\eta }^{\check{\omega },t}$, all members firstly allocate its weighted marginal extents-utility, and the rest of utility are allocated uniformly.
• The  bi-weighted allocating of undifferentiated utility (BWAUU), ${\eta }^{\widehat{\omega },\check{\omega }}$, is considered by for every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for every weight evaluation for members $\widehat{\omega }$, for every weight evaluation for extents $\check{\omega }$, for every $t\in {\ddot{\mathbf{G}}}_n$ and for every member $i\in \ddot{\mathbf{M}}$,

  $${\eta }_i^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)={\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{\widehat{\omega }\left(i\right)}{{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}\widehat{\omega }\left(k\right)}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-\sum _{k\in \ddot{\mathbf{M}}}{\kappa }_k^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right].$$
  Based on the definition of ${\eta }^{\widehat{\omega },\check{\omega }}$, all members firstly allocate its weighted marginal extents-utility, and the rest of utility are allocated proportionally via weights for members.

3. Axiomatic Analysis

3.1. Axiomatic Analysis for the UAUU and Its Weighted Extensions

By simultaneously applying related axiomatic notions and techniques due to Hart and Mas-Colell [14] and Moulin [2], this section will utilize several axiomatic outcomes of the UAUU, the 1-WAUU, the 2-WAUU, and the BWAUU to analyze the mathematical correctness and practical applicability for these allocations.

An allocation $\rho$ satisfies the property of effectiveness for multiple-goals (ETCMG) if for every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ and for every $t\in {\ddot{\mathbf{G}}}_n$, ${\sum }_{i\in \ddot{\mathbf{M}}}{\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)={\theta }^t\left(\tilde{\zeta }\right)$. The ETCMG property ensures that all members distribute whole the utility entirely.

Lemma 1. 

The allocations $\overline{\eta }$, ${\eta }^{\widehat{\omega }}$, ${\eta }^{\check{\omega }}$, ${\eta }^{\widehat{\omega },\check{\omega }}$ satisfy ETCMG.

Proof of Lemma 1. 

Let $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, $t\in {\ddot{\mathbf{G}}}_n$, $\widehat{\omega }$ be weight evaluation for members and $\check{\omega }$ be weight evaluation for extents. By Definitions 1 and 2,

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i\in \ddot{\mathbf{M}}}}{\eta }_i^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)& ={\displaystyle \sum _{i\in \ddot{\mathbf{M}}}}{\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \sum _{i\in \ddot{\mathbf{M}}}}\left[{\displaystyle \frac{\widehat{\omega }\left(i\right)}{{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}\widehat{\omega }\left(k\right)}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}{\kappa }_k^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]\right]\hfill \\ & ={\displaystyle \sum _{i\in \ddot{\mathbf{M}}}}{\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{{\displaystyle \sum _{i\in \ddot{\mathbf{M}}}}\widehat{\omega }\left(i\right)}{{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}\widehat{\omega }\left(k\right)}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}{\kappa }_k^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]\hfill \\ & ={\displaystyle \sum _{i\in \ddot{\mathbf{M}}}}{\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\theta }^t\left(\tilde{\zeta }\right)-{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}{\kappa }_k^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\hfill \\ & ={\theta }^t\left(\tilde{\zeta }\right).\hfill \end{array}$$

The proof is completed. If all the weights for members are set to 1 under above proof process, the ETCMG property of the 2-WAUU can be verified. Similarly, if all the weights for extents are set to 1 under above proof process, the ETCMG property of the 1-WAUU can be verified. Moreover, if all the weights for both members and extents are set to 1 under above proof process, the ETCMG property of the UAUU can be verified. □

To axiomatize the EANSC, Moulin [2] defined a specific reduction as follows. If any members within any interactive coalition in the organizing mode do not yield the expected utility, a mechanism could be executed to arise a re-interaction under the sufficient collaboration among all members that accomplish the expected utility. The extended definition of the Moulin’s reduction under the multiple-goals multi-choice mode is considered as follows.

Let $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, $K\subseteq \ddot{\mathbf{M}}$, and $\rho$ be an allocation. The reduced mode $(K,{\tilde{\zeta }}_K,{\Theta }_{K,\rho }^n)$ is defined by ${\Theta }_{K,\rho }^n={\left({\theta }_{K,\rho }^t\right)}_{t\in {\ddot{\mathbf{G}}}_n}$, and for every $\tilde{\chi }\in {\ddot{\mathbf{T}}}^K$,

$${\theta }_{K,\rho }^t\left(\tilde{\chi }\right)=\left\{\begin{array}{cc}0\hfill & if\tilde{\chi }=0_K,\hfill \\ {\theta }^t\left(\tilde{\chi },{\tilde{\zeta }}_{\ddot{\mathbf{M}}\setminus K}\right)-{\displaystyle \sum _{i\in \ddot{\mathbf{M}}\setminus K}}{\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

Moreover, an allocation $\rho$ satisfies the property of bilateral consistency for multiple-goals (BCIYMG) if ${\rho }_i^t(K,{\tilde{\zeta }}_K,{\theta }_{K,\rho }^t)={\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)$ for every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for every $t\in {\ddot{\mathbf{G}}}_n$, for every $K\subseteq \ddot{\mathbf{M}}$ with $\left|K\right|=2$, and for every $i\in K$.

Lemma 2. 

The allocations $\overline{\eta }$, ${\eta }^{\widehat{\omega }}$, ${\eta }^{\check{\omega }}$, ${\eta }^{\widehat{\omega },\check{\omega }}$ satisfy BCIYMG.

Proof of Lemma 2. 

Let $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, $K\subseteq \ddot{\mathbf{M}}$, $t\in {\ddot{\mathbf{G}}}_n$, $\widehat{\omega }$ be weight evaluation for members and $\check{\omega }$ be weight evaluation for extents. Let $|\ddot{\mathbf{M}}|\ge 2$ and $\left|K\right|=2$. By Definitions 1 and 2,

$${\eta }_i^{\widehat{\omega },\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t)={\kappa }_i^{\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t)+{\displaystyle \frac{\widehat{\omega }\left(i\right)}{{\displaystyle \sum _{k\in K}}\widehat{\omega }\left(k\right)}}·\left[{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t\left({\tilde{\zeta }}_K\right)-\sum _{k\in K}{\kappa }_k^{\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t)\right]$$

(1)

for every $i\in K$ and for every $t\in {\ddot{\mathbf{G}}}_n$. By definitions of ${\kappa }^{\check{\omega },t}$ and ${\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t$,

$$\begin{array}{cc}\hfill {\kappa }_i^{\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t)& ={\displaystyle \frac{1}{{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_i^{+}}}\check{\omega }\left(q\right)}}{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_i^{+}}}\{\check{\omega }\left(q\right)·[{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t({\tilde{\zeta }}_{K\setminus \left\{i\right\}},q)-{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t({\tilde{\zeta }}_{K\setminus \left\{i\right\}},0)]\}\hfill \\ \hfill & ={\displaystyle \frac{1}{{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_i^{+}}}\check{\omega }\left(q\right)}}{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_i^{+}}}\{\check{\omega }\left(q\right)·[{\theta }^t({\tilde{\zeta }}_{-i},q)-{\theta }^t({\tilde{\zeta }}_{-i},0)]\}\hfill \\ \hfill & ={\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t).\hfill \end{array}$$

(2)

Based on Equations (1) and (2) and definitions of ${\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t$ and ${\eta }^{\widehat{\omega },\check{\omega }}$,

$$\begin{array}{cc}& {\eta }_i^{\widehat{\omega },\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t)\hfill \\ \hfill =& {\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{\widehat{\omega }\left(i\right)}{{\displaystyle \sum _{k\in K}}\widehat{\omega }\left(k\right)}}\left[{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t\left({\tilde{\zeta }}_K\right)-{\displaystyle \sum _{k\in K}}{\kappa }_k^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]\hfill \\ \hfill =& {\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{\widehat{\omega }\left(i\right)}{{\displaystyle \sum _{k\in K}}\widehat{\omega }\left(k\right)}}\left[{\theta }^t\left(\tilde{\zeta }\right)-{\displaystyle \sum _{k\in \ddot{\mathbf{M}}\setminus K}}{\eta }_k^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)-{\displaystyle \sum _{k\in K}}{\kappa }_k^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]\hfill \\ \hfill =& {\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{\widehat{\omega }\left(i\right)}{{\displaystyle \sum _{k\in K}}\widehat{\omega }\left(k\right)}}\left[{\displaystyle \sum _{k\in K}}{\eta }_k^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)-{\displaystyle \sum _{k\in K}}{\kappa }_k^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]\hfill \\ & \left(\mathbf{ETCMG}\mathbf{of}{\eta }^{\widehat{\omega },\check{\omega }}\right)\hfill \\ \hfill =& {\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{\widehat{\omega }\left(i\right)}{{\displaystyle \sum _{k\in K}}\widehat{\omega }\left(k\right)}}\left[{\displaystyle \frac{{\displaystyle \sum _{k\in K}}\widehat{\omega }\left(k\right)}{{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}\widehat{\omega }\left(p\right)}}\left[{\theta }^t\left(\tilde{\zeta }\right)-{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}{\kappa }_{\ddot{\mathbf{M}}}^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]\right]\hfill \\ \hfill =& {\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{\widehat{\omega }\left(i\right)}{{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}\widehat{\omega }\left(p\right)}}\left[{\theta }^t\left(\tilde{\zeta }\right)-{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}{\kappa }_{\ddot{\mathbf{M}}}^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]\hfill \\ \hfill =& {\eta }_i^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\hfill \end{array}$$

for every $i\in K$ and for every $t\in {\ddot{\mathbf{G}}}_n$. If all the weights for members are set to 1 under above proof process, the BCIYMG property of the 2-WAUU can be verified. Similarly, if all the weights for extents are set to 1 under above proof process, the BCIYMG property of the 1-WAUU can be verified. Moreover, if all the weights for both members and extents are set to 1 under above proof process, the BCIYMG property of the UAUU can be verified. □

An allocation $\rho$ satisfies the property of criterion for multiple-goals (CMG) if $\rho (\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)=\overline{\eta }(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$ for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ with $|\ddot{\mathbf{M}}|\le 2$. An allocation $\rho$ satisfies the property of 1-weighted criterion for multiple-goals (1WCMG) if $\rho (\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\eta }^{\widehat{\omega }}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$ for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ with $|\ddot{\mathbf{M}}|\le 2$ and for all weight evaluation $\widehat{\omega }$ for members. An allocation $\rho$ satisfies the property of 2-weighted criterion for multiple-goals (2WCMG) if $\rho (\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\eta }^{\check{\omega }}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$ for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ with $|\ddot{\mathbf{M}}|\le 2$ and for all weight evaluation $\check{\omega }$ for extents. An allocation $\rho$ satisfies the property of bi-weighted criterion for multiple-goals (BWCMG) if $\rho (\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\eta }^{\widehat{\omega },\check{\omega }}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$ for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ with $|\ddot{\mathbf{M}}|\le 2$, for all weight evaluation for members $\widehat{\omega }$ and for all weight evaluation for extents $\check{\omega }$. The properties of the CMG, the 1WCMG, the 2WCMG, and the BWCMG are extended analogues in modes involving only two members interacting, defined by Hart and Mas-Colell [14] under axiomatizing the Shapley value [15].

Applying the axiomatic notions techniques due to Hart and Mas-Colell [14] and Moulin [2], the property of the BCIYMG is adopted to axiomatize these allocations as follows.

Theorem 1. 

1. 

On $\overline{\mathbf{MGCM}}$, the UAUU is the unique allocation satisfying CMG and BCIYMG.

2. 

On $\overline{\mathbf{MGCM}}$, the 1-WAUU is the unique allocation satisfying 1WCMG and BCIYMG.

3. 

On $\overline{\mathbf{MGCM}}$, the 2-WAUU is the unique allocation satisfying 2WCMG and BCIYMG.

4. 

On $\overline{\mathbf{MGCM}}$, the BWAUU is the unique allocation satisfying BWCMG and BCIYMG.

Proof of Theorem 1. 

By Lemma 2, the allocations $\overline{\eta }$, ${\eta }^{\widehat{\omega }}$, ${\eta }^{\check{\omega }}$, ${\eta }^{\widehat{\omega },\check{\omega }}$ satisfy BCIYMG. Clearly, the allocations $\overline{\eta }$, ${\eta }^{\widehat{\omega }}$, ${\eta }^{\check{\omega }}$, ${\eta }^{\widehat{\omega },\check{\omega }}$ satisfy CMG, 1WCMG, 2WCMG and BWCMG respectively.

To present the uniqueness of result 4, suppose that $\rho$ satisfies BWCMG and BCIYMG. By BWCMG and BCIYMG of $\rho$, it is easy to clarify that $\rho$ also satisfies ETCMG, thus we omit it. Let $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, $\widehat{\omega }$ be weight evaluation for members and $\check{\omega }$ be weight evaluation for extents. By BWCMG of $\rho$, $\rho (\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\eta }^{\widehat{\omega },\check{\omega }}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$ if $|\ddot{\mathbf{M}}|\le 2$. The situation $|\ddot{\mathbf{M}}|>2$: Let $i\in \ddot{\mathbf{M}}$, $t\in {\ddot{\mathbf{G}}}_n$ and $K=\{i,p\}$ with $p\in \ddot{\mathbf{M}}\setminus \left\{i\right\}$.

$$\begin{array}{cc}\hfill {\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)-{\eta }_i^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)& ={\rho }_i^t(K,{\tilde{\zeta }}_K,{\theta }_{K,\rho }^t)-{\eta }_i^{\widehat{\omega },\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t)\hfill \\ \hfill & \left(\mathbf{BCIYMG}\mathbf{of}{\eta }^{\widehat{\omega },\check{\omega },\mathbf{t}}\mathbf{and}\rho \right)\hfill \\ \hfill & ={\eta }_i^{\widehat{\omega },\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,\rho }^t)-{\eta }_i^{\widehat{\omega },\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega },}}^t).\hfill \\ \hfill & \left(\mathbf{BWCMG}\mathbf{of}\rho \right)\hfill \end{array}$$

(3)

Similar to Equation (2)

$${\kappa }_i^{\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,\rho }^t)={\kappa }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)={\kappa }_i^{\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,{\kappa }^{\eta ,\check{\omega },}}^t).$$

(4)

By Equations (3) and (4),

$$\begin{array}{cc}& {\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)-{\eta }_i^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\hfill \\ \hfill =& {\eta }_i^{\widehat{\omega },\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,\rho }^t)-{\eta }_i^{\widehat{\omega },\check{\omega },t}(K,{\tilde{\zeta }}_K,{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega },}}^t)\hfill \\ \hfill =& {\displaystyle \frac{\widehat{\omega }\left(i\right)}{\widehat{\omega }\left(i\right)+\widehat{\omega }\left(p\right)}}\left[{\theta }_{K,\rho }^t\left({\tilde{\zeta }}_K\right)-{\theta }_{K,{\eta }^{\widehat{\omega },\check{\omega }}}^t\left({\tilde{\zeta }}_K\right)\right]\hfill \\ \hfill =& {\displaystyle \frac{\widehat{\omega }\left(i\right)}{\widehat{\omega }\left(i\right)+\widehat{\omega }\left(p\right)}}\left[{\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\rho }_{\ddot{\mathbf{M}}}^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)-{\eta }_i^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)-{\eta }_{\ddot{\mathbf{M}}}^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right].\hfill \end{array}$$

Thus,

$$\widehat{\omega }\left(p\right)·\left[{\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)-{\eta }_i^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]=\widehat{\omega }\left(i\right)·\left[{\rho }_{\ddot{\mathbf{M}}}^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)-{\eta }_{\ddot{\mathbf{M}}}^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right].$$

By ETCMG of ${\eta }^{\widehat{\omega },\check{\omega },t}$ and $\rho$,

$$\begin{array}{cc}\hfill \left[{\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)-{\eta }_i^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]·{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}\widehat{\omega }\left(p\right)& =\widehat{\omega }\left(i\right)·{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}\left[{\rho }_{\ddot{\mathbf{M}}}^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)-{\eta }_{\ddot{\mathbf{M}}}^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right]\hfill \\ & =\widehat{\omega }\left(i\right)·\left[{\theta }^t\left(\tilde{\zeta }\right)-{\theta }^t\left(\tilde{\zeta }\right)\right]\hfill \\ & =0.\hfill \end{array}$$

Hence, ${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)={\eta }_i^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)$ for every $i\in \ddot{\mathbf{M}}$ and for every $t\in {\ddot{\mathbf{G}}}_n$. If all the weights for members are set to 1 under above proof process, the proof of outcome 3 could be verified. Similarly, if all the weights for extents are set to 1 under above proof process, the proof of outcome 2 could be verified. Moreover, if all the weights for both members and extents are set to 1 under above proof process, the proof of outcome 1 could be verified. □

In the following some instances are exhibited to show that each of the properties applied in Theorem 1 is independent of the rest of properties.

Example 1. 

Consider the allocation ρ as follows. For every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for every weight evaluation for members $\widehat{\omega }$, for every weight evaluation for extents $\check{\omega }$, for every $t\in {\ddot{\mathbf{G}}}_n$ and for every member $i\in \ddot{\mathbf{M}}$,

$${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)=\left\{\begin{array}{cc}{\eta }_i^{\widehat{\omega },\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\hfill & if|\ddot{\mathbf{M}}|\le 2,\hfill \\ 0\hfill & otherwise.\hfill \end{array}\right.$$

Clearly, ρ satisfies BWCMG, but it does not satisfy BCIYMG.

Example 2. 

Consider the allocation ρ as follows. For every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for every weight evaluation for members $\widehat{\omega }$, for every weight evaluation for extents $\check{\omega }$, for every $t\in {\ddot{\mathbf{G}}}_n$ and for every member $i\in \ddot{\mathbf{M}}$,

$${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)=\left\{\begin{array}{cc}{\eta }_i^{\check{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\hfill & if|\ddot{\mathbf{M}}|\le 2,\hfill \\ 0\hfill & otherwise.\hfill \end{array}\right.$$

Clearly, ρ satisfies 2WCMG, but it does not satisfy BCIYMG.

Example 3. 

Consider the allocation ρ as follows. For every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for every weight evaluation for members $\widehat{\omega }$, for every weight evaluation for extents $\check{\omega }$, for every $t\in {\ddot{\mathbf{G}}}_n$ and for every member $i\in \ddot{\mathbf{M}}$,

$${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)=\left\{\begin{array}{cc}{\eta }_i^{\widehat{\omega },t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\hfill & if|\ddot{\mathbf{M}}|\le 2,\hfill \\ 0\hfill & otherwise.\hfill \end{array}\right.$$

Clearly, ρ satisfies 1WCMG, but it does not satisfy BCIYMG.

Example 4. 

Consider the allocation ρ as follows. For every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for every weight evaluation for members $\widehat{\omega }$, for every weight evaluation for extents $\check{\omega }$, for every $t\in {\ddot{\mathbf{G}}}_n$ and for every member $i\in \ddot{\mathbf{M}}$,

$${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)=\left\{\begin{array}{cc}\overline{{\eta }_i^t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\hfill & if|\ddot{\mathbf{M}}|\le 2,\hfill \\ 0\hfill & otherwise.\hfill \end{array}\right.$$

Clearly, ρ satisfies CMG, but it does not satisfy BCIYMG. ${\eta }^{\widehat{\omega }}$

Example 5. 

Consider the allocation ρ as follows. For every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for every weight evaluation for members $\widehat{\omega }$, for every weight evaluation for extents $\check{\omega }$, for every $t\in {\ddot{\mathbf{G}}}_n$ and for every member $i\in \ddot{\mathbf{M}}$, ${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)=0$. Clearly, ρ satisfies BCIYMG, but it does not satisfy CMG, 1WCMG, 2WCMG and BWCMG.

3.2. Symmetry for Multiple-Goals

To analyze related symmetry for these allocations in this section, some more properties are needed.

• An allocation $\rho$ satisfies symmetric for multiple-goals (SYMMG) if ${\rho }_i(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\rho }_k(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$ for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ with ${\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$ for some $i,k\in \ddot{\mathbf{M}}$ and for all $t\in {\ddot{\mathbf{G}}}_n$. Property SYMMG asserts that the utilities should be coincident if the average marginal extents-utilities are coincident.
• An allocation $\rho$ satisfies covariance for multiple-goals (COVMG) if $\rho (\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)=\rho (\ddot{\mathbf{M}},\tilde{\zeta },Q^n)+{\left(h^t\right)}_{t\in {\ddot{\mathbf{G}}}_n}$ for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n),(\ddot{\mathbf{M}},\tilde{\zeta },D^n)\in \overline{\mathbf{MGCM}}$ with ${\theta }^t\left(\tilde{\chi }\right)=d^t\left(\tilde{\chi }\right)+{\sum }_{i\in \mathit{Z}\left(\tilde{\chi }\right)}{h}_i^t$ for some $h^t\in {\mathbb{R}}^{\ddot{\mathbf{M}}}$, for all $t\in {\ddot{\mathbf{G}}}_n$ and for all $\tilde{\chi }\in {\ddot{\mathbf{T}}}^{\ddot{\mathbf{M}}}$. Property COVMG can be treated to be a weak representation of additivity.

Next, we will show that the UAUU satisfies SYMMG and COVMG.

Lemma 3. 

On $\overline{\mathbf{MGCM}}$, the UAUU $\overline{\eta }$ satisfies symmetric for multiple-goals.

Proof of Lemma 3. 

Let $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ with ${\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$ for some $i,k\in \ddot{\mathbf{M}}$ and for all $t\in {\ddot{\mathbf{G}}}_n$. By Definition 1,

$$\begin{array}{cc}\hfill \overline{{\eta }_i^t}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)& ={\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)+{\displaystyle \frac{1}{|\ddot{\mathbf{M}}|}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}{\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\right]\hfill \\ & ={\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)+{\displaystyle \frac{1}{|\ddot{\mathbf{M}}|}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}{\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\right]\hfill \\ & =\overline{{\eta }_k^t}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n).\hfill \end{array}$$

Hence, the UAUU $\overline{\eta }$ satisfies SYMMG. □

Lemma 4. 

On $\overline{\mathbf{MGCM}}$, the UAUU $\overline{\eta }$ satisfies covariance for multiple-goals.

Proof of Lemma 4. 

Let $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n),(\ddot{\mathbf{M}},\tilde{\zeta },D^n)\in \overline{\mathbf{MGCM}}$ with ${\theta }^t\left(\tilde{\chi }\right)=d^t\left(\tilde{\chi }\right)+{\sum }_{k\in \mathit{Z}\left(\tilde{\chi }\right)}{h}_k^t$ for all $t\in {\ddot{\mathbf{G}}}_n$ and for all $\tilde{\chi }\in {\ddot{\mathbf{T}}}^{\ddot{\mathbf{M}}}$, where $h^t\in {\mathbb{R}}^{\ddot{\mathbf{M}}}$. By Definition 1, for all $p\in \ddot{\mathbf{M}}$ and for all $t\in {\ddot{\mathbf{G}}}_n$,

$$\begin{array}{cc}\hfill {\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)& ={\displaystyle \frac{1}{{\tilde{\zeta }}_p}}{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_p^{+}}}\{{\theta }^t({\tilde{\zeta }}_{-p},q)-{\theta }^t({\tilde{\zeta }}_{-p},0)\}\hfill \\ \hfill & ={\displaystyle \frac{1}{{\tilde{\zeta }}_p}}{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_p^{+}}}\{d^t({\tilde{\zeta }}_{-p},q)+{\displaystyle \sum _{k\in \mathit{Z}({\tilde{\zeta }}_{-p},q)}}{h}_k^t-d^t({\tilde{\zeta }}_{-p},0)-{\displaystyle \sum _{k\in \mathit{Z}({\tilde{\zeta }}_{-p},0)}}{h}_k^t\}\hfill \\ \hfill & ={\displaystyle \frac{1}{{\tilde{\zeta }}_p}}{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_p^{+}}}\{d^t({\tilde{\zeta }}_{-p},q)-d^t({\tilde{\zeta }}_{-p},0)+h_p^t\}\hfill \\ \hfill & ={\displaystyle \frac{1}{{\tilde{\zeta }}_p}}{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_p^{+}}}\{d^t({\tilde{\zeta }}_{-p},q)-d^t({\tilde{\zeta }}_{-p},0)\}\hfill \\ \hfill & ={\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)+h_i^t.\hfill \end{array}$$

(5)

By Definition 1 and Equation (5), for all $i\in \ddot{\mathbf{M}}$ and for all $t\in {\ddot{\mathbf{G}}}_n$,

$$\begin{array}{cc}\hfill \overline{{\eta }_i^t}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)& ={\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)+{\displaystyle \frac{1}{|\ddot{\mathbf{M}}|}}·[{\theta }^t(\tilde{\zeta })-{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}{\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)]\hfill \\ & ={\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)+h_i^t+{\displaystyle \frac{1}{|\ddot{\mathbf{M}}|}}·[d^t(\tilde{\zeta })+{\displaystyle \sum _{k\in \mathit{Z}(\tilde{\zeta })}}{h}_k^t-{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}[{\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)+h_p^t]]\hfill \\ & \left(\mathbf{by} \mathbf{Equation} (\mathbf{5})\right)\hfill \\ & ={\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)+h_i^t+{\displaystyle \frac{1}{|\ddot{\mathbf{M}}|}}·[d^t(\tilde{\zeta })+{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}{h}_k^t-{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}{\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)-{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}{h}_p^t]\hfill \\ & \left(\mathbf{since}\mathit{Z}(\tilde{\zeta })=\ddot{\mathbf{M}}\right)\hfill \\ & ={\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)+{\displaystyle \frac{1}{|\ddot{\mathbf{M}}|}}·[d^t(\tilde{\zeta })-{\displaystyle \sum _{p\in \ddot{\mathbf{M}}}}{\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)]+h_i^t\hfill \\ & =\overline{{\eta }_i^t}(\ddot{\mathbf{M}},\tilde{\zeta },D^n)+h_i^t.\hfill \end{array}$$

Hence, the UAUU $\overline{\eta }$ satisfies COVMG. □

Remark 1. 

Similar to related proof processes of Lemmas 3 and 4, it is easy to check that the 1-WAUU, the 2-WAUU and the BWAUU violates SYMMG. That is, the 1-WAUU, the 2-WAUU and the BWAUU are not symmetric allocation. Furthermore, similar to related proof processes of Lemmas 3 and 4, it is easy to check that the 1-WAUU satisfies COVMG, but the 2-WAUU and the BWAUU violates COVMG.

Next, we characterize the UAUU by means of related properties of ETCMG, SYMMG, COVMG and BCIYMG.

Lemma 5. 

On $\overline{\mathbf{MGCM}}$, an allocation $\rho$ satisfies CMG if it satisfies ETCMG, SYMMG and COVMG.

Proof of Lemma 5. 

Assume that an allocation $\rho$ satisfies ETCMG, SYMMG and COVMG on $\overline{\mathbf{MGCM}}$. Let $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$. The proof is completed by ETCMG of $\rho$ if $|\ddot{\mathbf{M}}|=1$. Let $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ with $\ddot{\mathbf{M}}=\{p,k\}$ for some $p\ne k$. Define an mode $(\ddot{\mathbf{M}},\tilde{\zeta },D^n)$ to be that $d^t\left(\tilde{\chi }\right)={\theta }^t\left(\tilde{\chi }\right)-{\sum }_{i\in \mathit{Z}\left(\tilde{\chi }\right)}{\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$ for all $\tilde{\chi }\in {\ddot{\mathbf{T}}}^{\ddot{\mathbf{M}}}$ and for all $t\in {\ddot{\mathbf{G}}}_n$. By definition of $D^n$,

$$\begin{array}{cc}\hfill {\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)& ={\displaystyle \frac{1}{{\tilde{\zeta }}_p}}{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_p^{+}}}\{d^t({\tilde{\zeta }}_{-p},q)-d^t({\tilde{\zeta }}_{-p},0)\}\hfill \\ & ={\displaystyle \frac{1}{{\tilde{\zeta }}_p}}{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_p^{+}}}\{{\theta }^t(q,{\tilde{\zeta }}_k)-{\theta }^t(0,{\tilde{\zeta }}_k)-{\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\}\hfill \\ & \left(\mathbf{by}\mathbf{definition}\mathbf{of}d\right)\hfill \\ & ={\displaystyle \frac{1}{{\tilde{\zeta }}_p}}{\displaystyle \sum _{q\in {\ddot{\mathbf{T}}}_p^{+}}}\{{\theta }^t(j,{\widetilde{zeta}}_k)-{\theta }^t(0,{\widetilde{zeta}}_k)\}-{\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\hfill \\ & ={\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)-{\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\hfill \\ & =0.\hfill \end{array}$$

Similarly, ${\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)=0$. Therefore, ${\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)={\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)$. By SYMMG of $\rho$,

$${\rho }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)={\rho }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n).$$

By ETCMG of $\rho$,

$$d^t\left(\tilde{\zeta }\right)={\rho }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)+{\rho }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)=2·{\rho }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n).$$

Therefore,

$$\begin{array}{cc}\hfill {\rho }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },D^n)& ={\displaystyle \frac{d^t\left(\tilde{\zeta }\right)}{2}}\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-{\eta }_p(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)-{\eta }_k(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\right].\hfill \end{array}$$

(6)

By Equation (6) and COVMG of $\rho$,

$$\begin{array}{cc}\hfill {\rho }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)& ={\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)+{\displaystyle \frac{1}{2}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-{\eta }_p^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)-{\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\right]\hfill \\ & =\overline{{\eta }_p^t}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n).\hfill \end{array}$$

Similarly, ${\rho }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)=\overline{{\eta }_k^t}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$. Hence, $\rho$ satisfies CMG. □

Theorem 2. 

On $\overline{\mathbf{MGCM}}$, the UAUU $\overline{\eta }$ is the only allocation satisfying ETCMG, SYMMG, COVMG and BCIYMG.

Proof. 

By Lemmas 1–4, the UAUU $\overline{\eta }$ satisfies ETCMG, SYMMG, COVMG and BCIYMG. The remaining proofs follow from Theorem 1 and Lemma 5. □

The subsequent examples illustrate the logical independence of each axiom utilized in Theorem 2 from the remaining axioms.

Example 6. 

Define an allocation ρ to be that

$${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$$

for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for all $t\in {\ddot{\mathbf{G}}}_n$ and for all $i\in \ddot{\mathbf{M}}$. Clearly, ρ satisfies SYMMG, COVMG and BCIYMG, but it violates ETCMG.

Example 7. 

Define an allocation ρ to be that

$${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\displaystyle \frac{{\theta }^t\left(\tilde{\zeta }\right)}{|\ddot{\mathbf{M}}|}}$$

for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for all $t\in {\ddot{\mathbf{G}}}_n$ and for all $i\in \ddot{\mathbf{M}}$. Clearly, ρ satisfies ETCMG, SYMMG and BCIYMG, but it violates COVMG.

Example 8. 

Define an allocation ρ by for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for all $t\in {\ddot{\mathbf{G}}}_n$ and for all $i\in \ddot{\mathbf{M}}$,

$${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)=\left[{\theta }^t\left(\tilde{\zeta }\right)-{\theta }^t({\tilde{\zeta }}_{-i},0)\right]+{\displaystyle \frac{1}{|\ddot{\mathbf{M}}|}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-\sum _{k\in \ddot{\mathbf{M}}}\left[{\theta }^t\left(\tilde{\zeta }\right)-{\theta }^t({\tilde{\zeta }}_{-k},0)\right]\right].$$

Clearly, ρ satisfies ETCMG, COVMG and BCIYMG, but it violates SYMMG.

Example 9. 

Define an allocation ρ by for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for all $t\in {\ddot{\mathbf{G}}}_n$ and for all $i\in \ddot{\mathbf{M}}$,

$${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)+{\displaystyle \frac{f^t\left(i\right)}{{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}{f}^t\left(k\right)}}·\left[{\theta }^t\left(\tilde{\zeta }\right)-{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}{\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\right],$$

where for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, $f^t:P\to {\mathbb{R}}^{+}$ is defined by $f^t\left(i\right)=f^t\left(k\right)$ if ${\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$. Define an allocation ψ by for all $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for all $t\in {\ddot{\mathbf{G}}}_n$ and for all $i\in \ddot{\mathbf{M}}$,

$${\psi }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)=\left\{\begin{array}{cc}\overline{{\eta }_i^t}(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\hfill & if|\ddot{\mathbf{M}}|\le 2,\hfill \\ {\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\hfill & otherwise.\hfill \end{array}\right.$$

Clearly, ψ satisfies ETCMG, SYMMG and COVMG, but it violates BCIYMG.

3.3. Different Generalization and Revised Consistency

Throughout Section 2, Section 3.1 and Section 3.2, this study presents corresponding weight evaluations for members and its relevant extents to allot corresponding interaction weights. However, the validity and representativeness for these weight evaluations might be questioned, as the relative weight assigning for members or its extents might appear somewhat artificial. Hence, it seems more rational and natural to replace the weight evaluations with relative average marginal extents-utility under different modes. This section proposes applying the average marginal extents-utility generated via participation to replace these weight evaluations, which appears more rational and natural.

Definition 3. 

The  interior allocating of undifferentiated utility (IAUU), ${\eta }^I$, is defined as follows. For every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in {\overline{\mathbf{MGCM}}}^{*}$, for every $t\in {\ddot{\mathbf{G}}}_n$, and for every member $i\in \ddot{\mathbf{M}}$,

$${\eta }_i^{I,t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)={\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)+{\displaystyle \frac{{\eta }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)}{{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}{\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)}}\left[{\theta }^t\left(\tilde{\zeta }\right)-\sum _{k\in \ddot{\mathbf{M}}}{\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\right],$$

where ${\overline{\mathbf{MGCM}}}^{*}=\{(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}|{\displaystyle \sum _{k\in \ddot{\mathbf{M}}}}{\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\ne 0\mathit{for}\mathit{every}t\in {\ddot{\mathbf{G}}}_n\}$. Based on definition of ${\eta }^I$, all members firstly allocate its average marginal extents-utility, and the rest of utility then allocated proportionally based on these average marginal extents-utility.

Next, one would like to characterize the IAUU by means of consistency. An allocation $\rho$ satisfies the interior criterion for multiple-goals (ICMG) if $\rho (\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)={\eta }^I(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)$ for every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ with $|\ddot{\mathbf{M}}|\le 2$.

It is easy to check that ${\sum }_{k\in K}{\eta }_k^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)=0$ for some $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for some $K\subseteq \ddot{\mathbf{M}}$, and for some $t\in {\ddot{\mathbf{G}}}_n$, i.e., ${\eta }^{I,t}(K,{\tilde{\zeta }}_K,{\theta }_{K,\eta }^t)$ doesn’t exist for some $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for some $K\subseteq \ddot{\mathbf{M}}$, and for some $t\in {\ddot{\mathbf{G}}}_n$. Therefore, we focus on the revised consistency for multiple-goals as follows. an allocation $\rho$ satisfies the revised consistency for multiple-goals (RCIYMG) if $(K,{\tilde{\zeta }}_K,{\Theta }_{K,\rho }^n)$ and $\rho (K,{\tilde{\zeta }}_K,{\Theta }_{K,\rho }^n)$ exist for some $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$, for some $K\subseteq \ddot{\mathbf{M}}$, and for some $t\in {\ddot{\mathbf{G}}}_n$, and it holds that ${\rho }_i(K,{\tilde{\zeta }}_K,{\theta }_{K,\rho }^t)={\rho }_i(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)$ for every $i\in K$.

Similar to Theorem 1, several axiomatic outcomes for ${\eta }^I$ would also be provided as follows.

Theorem 3. 

1. 

The allocation ${\eta }^I$ satisfies ETCMG on ${\overline{\mathbf{MGCM}}}^{*}$.

2. 

The allocation ${\eta }^I$ satisfies RCIYMG on ${\overline{\mathbf{MGCM}}}^{*}$.

3. 

The allocation ${\eta }^I$ satisfies SYMMG on ${\overline{\mathbf{MGCM}}}^{*}$.

4. 

On ${\overline{\mathbf{MGCM}}}^{*}$, the IAUU is the only allocation satisfying ICMG and RCIYMG.

Proof of Theorem 2. 

The proofs are similar to Lemmas 1–3, and Theorem 1. □

Remark 2. 

It is shown that the IAUU satisfies SYMMG, i.e., the IAUU is symmetric allocation. Similar to related proof processes of Lemma 4, however, the IAUU violates COVMG. Thus, related axiomatic notion technique of Theorem 2 could not be applied to characterize the IAUU.

In the following some examples are exhibited to display that every of the properties applied in Theorem 3 is independent of the rest of properties.

Example 10. 

Consider the allocation ρ as follows. For every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in {\overline{\mathbf{MGCM}}}^{*}$, for every $t\in {\ddot{\mathbf{G}}}_n$ and for every member $i\in \ddot{\mathbf{M}}$,

$${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)=\left\{\begin{array}{cc}{\eta }_i^{I,t}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)\hfill & if|\ddot{\mathbf{M}}|\le 2,\hfill \\ 0\hfill & otherwise.\hfill \end{array}\right.$$

Clearly, ρ satisfies ICMG, but it does not satisfy RCIYMG.

Example 11. 

Consider the allocation ρ as follows. For every $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in {\overline{\mathbf{MGCM}}}^{*}$, for every $t\in {\ddot{\mathbf{G}}}_n$ and for every member $i\in \ddot{\mathbf{M}}$, ${\rho }_i^t(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^t)=0$. Clearly, ρ satisfies RCIYMG, but it does not satisfy ICMG.

In the following, an instance is offered to present (*) how the new allocations would allocate utility differently than the previous allocations and (**) differently from each other. Let $(\ddot{\mathbf{M}},\tilde{\zeta },{\Theta }^n)\in \overline{\mathbf{MGCM}}$ with $\ddot{\mathbf{M}}=\{j,i,k\}$, $m=2$, $\tilde{\zeta }=(2,1,1)$, ${\ddot{\mathbf{T}}}_j=\{0,1_j,2_j\}$, ${\ddot{\mathbf{T}}}_i=\{0,1_i\}$, ${\ddot{\mathbf{T}}}_k=\{0,1_k\}$, $\widehat{\omega }\left(j\right)=4$, $\widehat{\omega }\left(i\right)=2$, $\widehat{\omega }\left(k\right)=3$, $\check{\omega }\left(1_j\right)=2$, $\check{\omega }\left(2_j\right)=1$, $\check{\omega }\left(1_i\right)=4$, $\check{\omega }\left(1_k\right)=3$.

Further, let ${\theta }^1(2,1,1)=7$, ${\theta }^1(1,1,1)=9$, ${\theta }^1(2,1,0)=4$, ${\theta }^1(2,0,1)=2$, ${\theta }^1(2,0,0)=5$, ${\theta }^1(1,1,0)=-2$, ${\theta }^1(1,0,1)=-3$, ${\theta }^1(0,1,1)=6$, ${\theta }^1(1,0,0)=2$, ${\theta }^1(0,1,0)=3$, ${\theta }^1(0,0,1)=-2$, ${\theta }^2(2,1,1)=9$, ${\theta }^2(1,1,1)=3$, ${\theta }^2(2,1,0)=2$, ${\theta }^2(2,0,1)=3$, ${\theta }^2(2,0,0)=1$, ${\theta }^2(1,1,0)=7$, ${\theta }^2(1,0,1)=7$, ${\theta }^2(0,1,1)=1$, ${\theta }^2(1,0,0)=5$, ${\theta }^2(0,1,0)=5$, ${\theta }^2(0,0,1)=-2$ and ${\theta }^1(0,0,0)=0={\theta }^2(0,0,0)$. By Definitions 1–3,

$$\begin{array}{cccccc}\hfill \overline{{\eta }_j^1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill 1,& \hfill \overline{{\eta }_i^1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill 4,& \hfill \overline{{\eta }_k^1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill 2,\\ \hfill {\eta }_j^{\widehat{\omega },1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{2}{3}},& \hfill {\eta }_i^{\widehat{\omega },1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{13}{3}},& \hfill {\eta }_k^{\widehat{\omega },1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill 2,\\ \hfill {\eta }_j^{\check{\omega },1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{11}{9}},& \hfill {\eta }_i^{\check{\omega },1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{35}{9}},& \hfill {\eta }_k^{\check{\omega },1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{17}{9}},\\ \hfill {\eta }_j^{\widehat{\omega },\check{\omega },1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{23}{27}},& \hfill {\eta }_i^{\widehat{\omega },\check{\omega },1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{115}{27}},& \hfill {\eta }_k^{\widehat{\omega },\check{\omega },1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{51}{27}},\\ \hfill {\eta }_j^{I,1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{7}{5}},& \hfill {\eta }_i^{I,1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{7}{2}},& \hfill {\eta }_k^{I,1}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^1)=& \hfill {\displaystyle \frac{21}{10}},\\ \hfill \overline{{\eta }_j^2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill 2,& \hfill \overline{{\eta }_i^2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill 3,& \hfill \overline{{\eta }_k^2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill 4,\\ \hfill {\eta }_j^{\widehat{\omega },2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill 1,& \hfill {\eta }_i^{\widehat{\omega },2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill 4,& \hfill {\eta }_k^{\widehat{\omega },2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill 4,\\ \hfill {\eta }_j^{\check{\omega },2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill {\displaystyle \frac{4}{3}},& \hfill {\eta }_i^{\check{\omega },2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill {\displaystyle \frac{10}{3}},& \hfill {\eta }_k^{\check{\omega },2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill {\displaystyle \frac{13}{3}},\\ \hfill {\eta }_j^{\widehat{\omega },\check{\omega },2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill {\displaystyle \frac{4}{9}},& \hfill {\eta }_i^{\widehat{\omega },\check{\omega },2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill {\displaystyle \frac{38}{9}},& \hfill {\eta }_k^{\widehat{\omega },\check{\omega },2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill {\displaystyle \frac{39}{9}},\\ \hfill {\eta }_j^{I,2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill {\displaystyle \frac{5}{2}},& \hfill {\eta }_i^{I,2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill 3,& \hfill {\eta }_k^{I,2}(\ddot{\mathbf{M}},\tilde{\zeta },{\theta }^2)=& \hfill {\displaystyle \frac{7}{2}}.\end{array}$$

4. Conclusions

1.

Differing from pre-existing outcomes, this study considers different types of allocating notions by applying weights and multiple-goals multi-choice consideration. Several extended allocations along with related axiomatic analysis are also formed. In contrast to more artificial weight assigning, this study reasonably adopts the average marginal extents-utility to replace weight evaluations, introducing a different generalization and related axiomatic analysis within the framework of multiple-goals multi-choice modes. Further, the symmetric analysis for these allocations are also presented.

• The utility allocating concepts of the UAUU, the 1-WAUU, the 2-WAUU, the BWAUU, and the UAUU, along with related axiomatic analysis, have not been appeared in game-theoretical literature related to traditional modes and multiple-goals multi-choice modes.

  −

  Under allocating processes of the UAUU and the 2-WAUU, different kinds of the marginal extents-utility for members are firstly allocated, and the rest of utility are uniformly allocated among all members.

  −

  ’ Based on a specific symmetry related to marginal contributions, it is shown that the UAUU and the IAUU are symmetric allocations and the 1-WAUU, 2-WAUU and BWAUU are asymmetric allocations.

  −

  Under the allocating processes of the 1-WAUU and the BWAUU, different kinds of the marginal extents-utility of members are first allocated, and the rest of utility are allocated among all members based on different weighted proportions.

  −

  Under the allocating concepts of the 2-WAUU and the BWAUU, the marginal extents-utility of all members are allocated considering the weight assigning of extents, while the UAUU and the 1-WAUU do not consider the weight assigning of extents.

  −

  The importance of members and its extents under multiple-goals multi-choice modes is paramount. Hence, weights assigning should focus on both members and its extents. Under the allocating process of the BWAUU, the average marginal weighted utility of members are firstly allocated, and then the rest of utility are allocated among all members based on relative weighted proportions.

  −

  However, weights assigning via weight evaluations might lack rationality and representativeness. Hence, under the allocating process of the IAUU, the average marginal extents-utility of members are firstly allocated, and then the rest of utility are allocated naturally among all members based on relative proportions due to its average marginal extents-utility.

2.

Although symmetry is a desirable principle in many situations, real-world situations often involve asymmetry in contributions. Asymmetry arises when participants differ in terms of their resource inputs, skills, or influence on the system’s outcomes. In such cases, the symmetry axiom may no longer apply, and other theories and approaches are required to manage these disparities. Related studies can be found in Kumar and Singh [16], Li and Chen [17], and so on. Asymmetry challenges traditional notions of fairness, as equal distribution may not always be justifiable when contributions differ significantly. This issue is explored in equity theory and risk exposure, which argue that fairness is not necessarily about equal distribution but about aligning rewards with individuals’ contributions, needs, or risks.

• Equity theory suggests that individuals assess fairness by comparing their inputs to their rewards. In asymmetric scenarios, a person who invests more resources or adds more value expects a corresponding return. If their compensation does not reflect their input, dissatisfaction may arise, potentially undermining cooperation.
• In some cases, asymmetry results from differences in risk exposure. For instance, in business partnerships, some participants may assume greater financial or market risks than others, warranting a larger share of the profits. **Risk-sharing theory** emphasizes that, in situations where participants face unequal risks, the distribution of gains should be proportional to the risks each party has taken on.

The concept of asymmetry is applicable in numerous areas. Examples include

• Corporate profit sharing: Within companies, employees’ contributions may vary, making asymmetric distribution of bonuses or profits necessary. The allocation of rewards should reflect each person’s actual input and role.
• International trade negotiations: In global trade talks, countries differ in terms of economic size, bargaining power, and available resources. Trade agreements must account for these differences, with results adjusted to reflect the relative strength and influence of each country.
• Resource allocating in public policy: Public resource distribution must consider the varying needs and efficiency levels of different regions. For example, rural areas may require more resources to achieve the same level of development as urban areas, necessitating an asymmetric allocation of resources.

3.

The allocations proposed throughout this study present several advantages.

• Allocations under traditional modes always focus on whether members are involved or not. Under the framework of multiple-goals multi-choice modes applied throughout this study, however, all members can take different extents relying upon different conditions.
• In some studies related to multi-choice modes, although allocations presented that members can take different extents, they allocate the interaction utility derived from specific members at specific extents. In contrast, the allocating concepts of this study focus on the overall interaction utility derived from whole the extents of each member.
• To comply with real-world interactive conditions, the BWAUU focuses on simultaneously the weight assigning for members and its related extents during estimating and allocating processes. Based on potential concerns about the rationality or representativeness for weight assigning, the UAUU considers relative average marginal extents-utility instead of weight assigning naturally.

4.

However, the allocations proposed in this study generate some limitations. As mentioned above, each member can take different extents under different conditions. Although it is possible to allocate the overall interaction utility derived from the extents of each member, it is unable to allocate the interaction utility derived from specific members at specific extents. Future analytic directions should focus on proposing different allocating concepts that consider both specific extents utility and overall utility.

5.

Related outcomes introduced throughout this study have also led to further motivation.

• Is it possible to replace the EANSC with other traditional allocating concepts under multiple-goals multi-choice modes to derive different symmetric or asymmetric allocating concepts?

The motivation mentioned above can offer avenues for further investigations.