On Minimizing Influences Under Multi-Attribute Models

Abstract

In classical transferable-utility models, components typically participate in an all-or-nothing manner and are evaluated under a single criterion. This study generalizes such models by allowing each component to engage through multiple acting measures and by incorporating multiple evaluating attributes simultaneously. We introduce two influence-based assessments, the stable min value and the minimal self-stable value, to evaluate fair assessments of minimal impact across multi-attribute multi-choice environments. These values are rigorously defined via axiomatic characterizations grounded in minimal influence behavior, where coalitions select activity levels that jointly minimize systemic effects. A key theoretical contribution is the identification of a unique, 0-normalized, and efficient multi-attribute potential function corresponding to the minimal self-stable value. The proposed framework enables structured and interpretable evaluation of influence in complex cooperative systems with heterogeneous participation and conflicting objectives.

1. Introduction

In complex organizational systems and policy-making environments, decision-makers often face situations where multiple departments or agents contribute to a shared outcome with varying intensities and under multiple evaluation criteria. For instance, in energy planning, different regional units may invest in alternative energy sources at varying levels, while the effectiveness of such contributions is judged not solely on cost, but also on carbon reduction, reliability, and social equity. Traditional binary participation models fail to capture such heterogeneity in both engagement levels and evaluative dimensions, thus calling for extended frameworks that incorporate multi-dimensional fairness and influence evaluation.

In a classical transferable-utility (TU) model, each component is either fully exclude or included at all under operation with other components. Based on reduced models and relative stability, various axiomatic outcomes have been provided to present that the core, the equal allocation of non-separable costs (EANSC, Ransmeier [1]) and the Shapley value [2] provide fair mechanisms for assessing influences among components on several classes of models, such as Davis and Maschler [3], Hart and Mas-Colell [4], Maschler and Owen [5], Moulin [6], and so on. A multi-choice TU model can be seen as a reasonable extension of a classical TU model in which each component applies distinct acting measures. This extension is motivated by the inherent complexity of real-world interactions where participants often contribute with varying degrees of engagement or resource commitment, moving beyond the binary “all-or-nothing” assumption of classical models. By determining overall assessments for a given component under multi-choice TU models, Hwang and Liao [7], Liao [8], and Nouweland et al. [9] defined different extended assessments and related results for the core, the EANSC, and the Shapley value, respectively, thereby offering a more nuanced and accurate framework for analyzing cooperation and fair allocation in diverse practical conditions.

Prior research has explored the integration of multi-attribute considerations into decision models with multiple acting measures. This line of work reflects the inherent complexity of real-world decision-making, where outcomes are rarely judged by a single criterion. Bednarczuk et al. [10] reformulated the multiple-choice knapsack problem into a bi-objective optimization model, thereby capturing trade-offs across distinct evaluation dimensions. Guarini et al. [11] addressed multi-criteria decision-making with hierarchical actor structures, proposing a systematic method-selection framework under heterogeneous attribute concerns. Similarly, Mustakerov et al. [12] developed a combinatorial optimization approach that supports multi-choice decisions guided by strategic preferences, framed as a multi-attribute linear mixed-integer model. Additional applications appear in Freixas and Marciniak [13], Goli et al. [14], Tirkolaee et al. [15], and Wei et al. [16], demonstrating the growing interest in models that accommodate both choice flexibility and attribute diversity.

However, despite these advances, the literature lacks an axiomatic framework that unifies multi-choice and multi-attribute structures for the purpose of evaluating participant influence under minimal impact objectives. In particular, existing studies do not characterize how fair influence allocations can be constructed when coalitional activity profiles vary across both action intensities and evaluative dimensions. This gap motivates the present study, which proposes two principled allocation assessments, grounded in minimal influence reasoning, for multi-attribute multi-choice environments.

The above-mentioned statements and existing studies raise related motivation:

• Whether different analogs for the EANSC and the Shapley value could be considered under multi-attribute multi-choice models simultaneously. Specifically, we aim to develop mathematical assessments that evaluate and minimize the impacts generated by the acting measures of each component under a multi-attribute context. When each component may engage through more than one acting measure and multiple attributes are jointly considered, the cumulative influence contributed by different acting configurations becomes nontrivial. This motivates a structural need for refined solution schemes that account for minimal influence behavior within the coalition space.

This study is devoted to investigating the establishment of a foundation for assessments under the framework of multi-attribute multi-choice TU models, wherein each acting measure is associated with a vector-valued evaluation. The major contributions are as follows.

• Different from the frameworks of classical TU models and multi-choice TU models, this study extends the framework by simultaneously incorporating multi-attribute consideration and multi-acting measure structure in Section 2.
• In this study, the minimal influence attainable by coalitions that activate non-zero activity measure vectors serves as the basis of the impact analysis. In Section 3 and Section 4, this study introduces new generalizations of the EANSC and the Shapley value under multi-attribute multi-choice TU models, which we name the stable min value and the minimal self-stable value. Accordingly, the assessments introduced in this paper are characterized not only through marginal influence computation but also by axiomatic processes reflecting relative stability with respect to reduced models. To present related rationalities for these two assessments, this study introduces two extended reduced models to characterize the stable min value and the minimal self-stable value, respectively.
• In Section 5, this study aims to construct a representation of an actual operational system within the framework of multi-attribute multi-choice TU models. Based on this constructed model, we apply the two proposed assessments to perform numerical evaluations that demonstrate how minimal impact allocations can be effectively achieved. Furthermore, we illustrate how the axiomatic processes used in the theoretical development correspond to desirable structural properties that are typically required in practical systems. This correspondence highlights the interpretability and applicability of the proposed framework in real-world multi-attribute environments involving multiple acting measures per component.

2. Multi-Attribute Multi-Choice TU Model

Let $\overline{\ddot{\mathsf{\Omega }}}$ be the universe of components. For $i\in \overline{\ddot{\mathsf{\Omega }}}$ and ${\overrightarrow{\mu }}_i\in \mathbb{N}$, we set ${\overline{\mathbf{U}}}_i=\{0,1,\cdots ,{\overrightarrow{\mu }}_i\}$ to be the acting measure set of component i, where 0 denotes non-participation. For $\overline{\mathsf{\Omega }}\subseteq \overline{\ddot{\mathsf{\Omega }}}$, let ${\overline{\mathbf{U}}}^{\overline{\mathsf{\Omega }}}={\prod }_{i\in \overline{\mathsf{\Omega }}}{\overline{\mathbf{U}}}_i$ be the product set of acting measures for every components in $\overline{\mathsf{\Omega }}$, and let ${\overrightarrow{0}}_{\overline{\mathsf{\Omega }}}$ denote the zero vector in ${\overline{\mathbf{U}}}^{\overline{\mathsf{\Omega }}}$. For $m\in \mathbb{N}$, let ${\overrightarrow{0}}_m$ denote the zero vector in ${\mathbb{R}}^m$, and define ${\mathbb{A}}_m=\{1,2,\cdots ,m\}$.

Let $\overline{\mathsf{\Omega }}\subseteq \overline{\ddot{\mathsf{\Omega }}}$, $\overrightarrow{\alpha }\in {\mathbb{R}}^{\overline{\mathsf{\Omega }}}$, and $p\in \overline{\mathsf{\Omega }}$. We adopt ${\overrightarrow{\alpha }}_{-p}$ to represent the restriction of $\overrightarrow{\alpha }$ to $\overline{\mathsf{\Omega }}\setminus \left\{p\right\}$, and define $\overrightarrow{\beta }=({\overrightarrow{\alpha }}_{-p},h)$ by setting ${\overrightarrow{\alpha }}_{-p}={\overrightarrow{\beta }}_{-p}$ and ${\overrightarrow{\beta }}_p=h$. Moreover, let $q\in \overline{\mathsf{\Omega }}$, and define ${\overrightarrow{\alpha }}_{-pq}$ as the restriction of $\overrightarrow{\alpha }$ to $\overline{\mathsf{\Omega }}\setminus \{p,q\}$, and $({\overrightarrow{\alpha }}_{-pq},h,f)$ to represent $({({\overrightarrow{\alpha }}_{-p},h)}_{-q},f)$. For $\overline{\mathbf{S}},\overline{\mathbf{T}}\subseteq \overline{\mathsf{\Omega }}$, the set difference $\overline{\mathbf{S}}\setminus \overline{\mathbf{T}}$ is defined as $\{i\in \overline{\mathsf{\Omega }}\mid i\in \overline{\mathbf{S}}\mathrm{and}i\notin \overline{\mathbf{T}}\}$.

A multi-choice TU model is $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },\widehat{\kappa })$, where $\overline{\mathsf{\Omega }}$ is a non-empty and finite set of components, $\overrightarrow{\mu }={\left({\overrightarrow{\mu }}_i\right)}_{i\in \overline{\mathsf{\Omega }}}$ represents the numbers of total non-zero measures for each component and $\widehat{\kappa }:{\overline{\mathbf{U}}}^{\overline{\mathsf{\Omega }}}\to \mathbb{R}$ is a map with $\widehat{\kappa }\left({\overrightarrow{0}}_{\overline{\mathsf{\Omega }}}\right)=0$ which renders to each $\overrightarrow{\alpha }={\left({\overrightarrow{\alpha }}_i\right)}_{i\in \overline{\mathsf{\Omega }}}\in {\overline{\mathbf{U}}}^{\overline{\mathsf{\Omega }}}$ the total value induced by the joint acting measures of the components, where each component i engages at level ${\overrightarrow{\alpha }}_i$.

A multi-attribute multi-choice TU model is a triple $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)$, where $m\in \mathbb{N}$, ${\widehat{K}}^m={\left({\widehat{\kappa }}^t\right)}_{t\in {\mathbb{A}}_m}$ and $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{\kappa }}^t)$ is a multi-choice TU model for every $t\in {\mathbb{A}}_m$. Each ${\widehat{\kappa }}^t$ corresponds to a distinct attribute evaluation indexed by t, and jointly these define a vector-valued representation of the overall influence structure. One would write $\widehat{\mathbb{O}}\left(\overrightarrow{\alpha }\right)=\{i\in \overline{\mathsf{\Omega }}|{\overrightarrow{\alpha }}_i\ne 0\}$ and ${\overrightarrow{\alpha }}_{\overline{\mathbf{T}}}$ to be the restriction of $\overrightarrow{\alpha }$ at $\overline{\mathbf{T}}$ for each $\overline{\mathbf{T}}\subseteq \overline{\mathsf{\Omega }}$, and write $(\overline{\mathsf{\Omega }},\overrightarrow{\alpha },\widehat{\kappa })$ for the multi-choice TU submodel and $(\overline{\mathsf{\Omega }},\overrightarrow{\alpha },{\widehat{K}}^m)$ for the multi-attribute multi-choice TU submodel obtained by restricting $\widehat{\kappa }$ and ${\widehat{K}}^m$ to $\{\overrightarrow{\gamma }\in {\overline{\mathbf{U}}}^{\overline{\mathsf{\Omega }}}\mid {\overrightarrow{\gamma }}_i\le {\overrightarrow{\alpha }}_i \mathrm{for} \mathrm{every} i\in \overline{\mathsf{\Omega }}\}$.

Denote the class of all multi-attribute multi-choice TU models by $\overline{\mathbf{MAM}}$. An assessment on $\overline{\mathbf{MAM}}$ is a map $\rho$ rendering to each $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$ an element

$$\rho \left(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m\right)={\left({\rho }^t\left(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m\right)\right)}_{t\in {\mathbb{A}}_m},$$

where ${\rho }^t\left(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m\right)={\left({\rho }_i^t\left(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m\right)\right)}_{i\in \overline{\mathsf{\Omega }}}\in {\mathbb{R}}^{\overline{\mathsf{\Omega }}}$ and ${\rho }_i^t\left(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m\right)$ is the influence assigned to component i under the influence evaluating function ${\widehat{\kappa }}^t$ corresponding to attribute t.

Remark 1.

As we mentioned in the Introduction Section, multi-attribute analysis (also known as multiobjective or multicriteria analysis) concerns the simultaneous evaluating of more than one objective dimension. Under such settings, each component may be associated with multiple performance dimensions that are to be considered collectively, and these dimensions are typically in conflict or require trade-offs.

In parallel, components are also allowed to engage with multiple acting measures, reflecting a generalization beyond binary participation structures. This dual-layered structure, multi-attribute and multi-acting measure, motivates the consideration related to multi-attribute multi-choice TU models adopted in this study. Within this model, each vector $\overrightarrow{\alpha }\in {\overline{\mathbf{U}}}^{\overline{\mathsf{\Omega }}}$ specifies the acting measures of components, while each function ${\widehat{\kappa }}^t$ captures corresponding influences under attribute t.

Therefore, the formulation $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)$ serves as a unified setting for evaluating and minimizing relative influences induced via activity measures adopted by components, where each ${\widehat{\kappa }}^t$ acts as a coordinate-wise characteristic function. In the following sections, we introduce two different assessments that aim to provide well-structured and axiomatized mechanisms for evaluating under this framework. These assessments are designed to reflect minimal influence criteria and preserve fundamental properties under reductions and symmetry, offering theoretical analysis under the context of multi-attribute and multi-choice evaluating.

3. The Stable Min Value

This section introduces a different generalization of the equal allocation of non-separable costs (EANSC) under multi-attribute, multi-choice models.

Definition 1.

The stable min value, $\widehat{\chi }$, is an assessment on $\overline{\mathbf{MAM}}$ which associates to each $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, each $t\in {\mathbb{A}}_m$ and each component $i\in \overline{\mathsf{\Omega }}$ the value

$${\widehat{\chi }}_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)={\chi }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)+{\displaystyle \frac{1}{|\overline{\mathsf{\Omega }}|}}[{\widehat{\kappa }}_{\#}^t\left(\overline{\mathsf{\Omega }}\right)-{\displaystyle \sum _{k\in \overline{\mathsf{\Omega }}}}{\chi }_k^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)],$$

where ${\chi }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)={\widehat{\kappa }}_{\#}^t\left(\overline{\mathsf{\Omega }}\right)-{\widehat{\kappa }}_{\#}^t\left(\overline{\mathsf{\Omega }}\setminus \left\{i\right\}\right)$. For any coalition $\overline{\mathbf{S}}\subseteq \overline{\mathsf{\Omega }}$ under model $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{\kappa }}^t)$, ${\widehat{\kappa }}_{\#}^t\left(\overline{\mathbf{S}}\right)={min}_{\overrightarrow{\alpha }\in {\overline{\mathbf{U}}}^{\overline{\mathsf{\Omega }}}}\{{\widehat{\kappa }}^t\left(\overrightarrow{\alpha }\right)|\widehat{\mathbb{O}}\left(\overrightarrow{\alpha }\right)=\overline{\mathbf{S}}\}$ is the minimal influence (To ensure that ${\widehat{\kappa }}_{\#}^t$ is well-defined, this study considers bounded models, defined by $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{\kappa }}^t)$ such that, there exists $B_t\in \mathbb{R}$ such that ${\widehat{\kappa }}^t\left(\overrightarrow{\alpha }\right)\le B_t$ for every $\overrightarrow{\alpha }\in {\overline{\mathbf{U}}}^{\overline{\mathsf{\Omega }}}$. ) among all acting measure vectors with $\widehat{\mathbb{O}}\left(\overrightarrow{\alpha }\right)=\overline{\mathbf{S}}$. Namely, ${\widehat{\kappa }}_{\#}^t\left(\overline{\mathbf{S}}\right)$ is the minimal influence attained when all components in $\overline{\mathbf{S}}$ select non-zero acting measures and all other components remain inactive.

Furthermore, ${\chi }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)={\widehat{\kappa }}_{\#}^t\left(\overline{\mathsf{\Omega }}\right)-{\widehat{\kappa }}_{\#}^t\left(\overline{\mathsf{\Omega }}\setminus \left\{i\right\}\right)$ represents the minimal influence of component i related to the grand coalition $\overline{\mathsf{\Omega }}$. Under the assessment $\widehat{\chi }$, each component first evaluates its marginal impact relative to the overall minimal influence, and the remaining residual influence is then distributed equally across all components.

An assessment $\rho$ satisfies multi-attribute entire-optimality (MAEO) if for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$ and for every $t\in {\mathbb{A}}_m$, ${\sum }_{i\in \overline{\mathsf{\Omega }}}{\rho }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)={\widehat{\kappa }}_{\#}^t\left(\overline{\mathsf{\Omega }}\right)$. MAEO requires that the aggregated assessment of all components corresponds exactly to the minimal influence attained by the entire coalition under attribute t.

Lemma 1.

The stable min value satisfies MAEO.

Proof. 

Please see Appendix A.1. □

Inspired by Moulin [6], one would consider an extended reduction to characterize the stable min value.

Definition 2.

Let $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, $\overline{\mathbf{S}}\subseteq \overline{\mathsf{\Omega }}$ and ρ be an assessment. The M-reduced model $(\overline{\mathbf{S}},{\overrightarrow{\mu }}_{\overline{\mathbf{S}}},{\widehat{K}}_{\overline{\mathbf{S}},\rho }^m)$ with respect to $\overline{\mathbf{S}}$ and $\rho$ is defined by ${\widehat{K}}_{\overline{\mathbf{S}},\rho }^m={\left({\widehat{\kappa }}_{\overline{\mathbf{S}},\rho }^t\right)}_{t\in {\mathbb{A}}_m}$, where for every $t\in {\mathbb{A}}_m$ and $\overrightarrow{\alpha }\in {\overline{\mathbf{U}}}^{\overline{\mathbf{S}}}$,

$${\widehat{\kappa }}_{\overline{\mathbf{S}},\rho }^t\left(\overrightarrow{\alpha }\right)=\left\{\begin{array}{cc}0\hfill & if\overrightarrow{\alpha }={\overrightarrow{0}}_{\overline{\mathbf{S}}},\hfill \\ {\widehat{\kappa }}_{\#}^t\left(\widehat{\mathbb{O}}(\overrightarrow{\alpha },{\overrightarrow{\mu }}_{\overline{\mathsf{\Omega }}\setminus \overline{\mathbf{S}}})\right)-{\displaystyle \sum _{i\in \overline{\mathsf{\Omega }}\setminus \overline{\mathbf{S}}}}{\rho }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & otherwise.\hfill \end{array}\right.$$

For any pairwise subset of components, a corresponding reduced model can be constructed by adjusting the influence values based on a reference assessment $\rho$, which determines the payoffs allocated to components outside the subset. An assessment $\rho$ is said to satisfy M-stability (MSTA) if, when applied to such a reduced model, the resulting payoffs within the subset coincide with those in the original model. Formally, $\rho$ satisfies MSTA if ${\rho }_i^t(\overline{\mathbf{S}},{\overrightarrow{\mu }}_{\overline{\mathbf{S}}},{\widehat{K}}_{\overline{\mathbf{S}},\rho }^m)={\rho }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)$ for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, for every $t\in {\mathbb{A}}_m$, for every $\overline{\mathbf{S}}\subseteq \overline{\mathsf{\Omega }}$ with $|\overline{\mathbf{S}}|=2$ and for every $i\in \overline{\mathbf{S}}$.

Lemma 2.

The stable min value $\widehat{\chi }$ satisfies MSTA.

Proof. 

Please see Appendix A.2. □

Next, we characterize the stable min value by means of MSTA. Let $\rho$ be an assessment on $\overline{\mathbf{MAM}}$. $\rho$ satisfies basis for multi-attribute models (BMAM) if $\rho (\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)=\widehat{\chi }(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)$ for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$ with $|\overline{\mathsf{\Omega }}|\le 2$. $\rho$ satisfies symmetry for assessing (SYAS) if ${\rho }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)={\rho }_k^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)$ for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$ with ${\widehat{\kappa }}_{\#}^t(\overline{\mathbf{S}}\cup \left\{i\right\})={\widehat{\kappa }}_{\#}^t(\overline{\mathbf{S}}\cup \left\{k\right\})$ for some $t\in {\mathbb{A}}_m$, for some $i,k\in \overline{\mathsf{\Omega }}$ and for every $\overline{\mathbf{S}}\subseteq \overline{\mathsf{\Omega }}\setminus \{i,k\}$. $\rho$ satisfies synchronized normalization (SYNO) if ${\rho }^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)={\rho }^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },W^m)+\widehat{ϵ}$ for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m),(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{W}}^m)\in \overline{\mathbf{MAM}}$ with ${\widehat{\kappa }}^t\left(\overrightarrow{\alpha }\right)={\widehat{w}}^t\left(\overrightarrow{\alpha }\right)+{\sum }_{i\in \widehat{\mathbb{O}}\left(\overrightarrow{\alpha }\right)}{\widehat{ϵ}}_i$ for some $t\in {\mathbb{A}}_m$, for some $\widehat{ϵ}\in {\mathbb{R}}^{\overline{\mathsf{\Omega }}}$ and for every $\overrightarrow{\alpha }\in {\overline{\mathbf{U}}}^{\overline{\mathsf{\Omega }}}$.

BMAM generalizes the two-person standardness requirement in the multi-attribute multi-choice setting, following the foundational idea in Hart and Mas-Colell [4]. SYAS ensures that whenever the minimal influence of two components is indistinguishable under coalition extension, their assigned values must coincide. SYNO formalizes a minimal consistency condition under affine shifts in influence functions, and may be interpreted as a degenerate form of additivity. By Definition 1, it is direct to verify that the stable min value satisfies BMAM, SYAS, and SYNO.

Lemma 3.

If an assessment ρ satisfies BMAM and MSTA, then it satisfies MAEO.

Proof. 

Please see Appendix A.3. □

In the following, we adopt the properties of multi-attribute entire-optimality and M-stability to characterize the stable min value.

Theorem 1.

On $\overline{\mathbf{MAM}}$, the stable min value is the only assessment satisfying BMAM and MSTA.

Proof. 

Please see Appendix B.1. □

• Managerial insight.

Theorem 1 demonstrates that the stable min value is the unique influence assessment that satisfies both the basis for multi-attribute models and M-stability. According to the axiomatic structure, the basis for multi-attribute models requires that influence assessments conform to base cases involving small coalitions, which serve as building blocks for generalized outcomes. M-stability ensures that influence values remain consistent when evaluated through reduced systems omitting individual components. From a managerial standpoint, if a decision-making environment demands both granular fairness at the coalition level and robustness under partial participation (such as project teams with rotating involvement), then the stable min value emerges as the sole viable assessment mechanism consistent with these principles.

The following assessments are to verify that each of the axioms in Theorem 1 is logically independent of the remaining axioms.

Example 1.

Define an assessment ρ to be that

$${\rho }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)=\left\{\begin{array}{cc}{\widehat{\chi }}_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & if |\overline{\mathsf{\Omega }}|\le 2,\hfill \\ 0\hfill & otherwise\hfill \end{array}\right.$$

for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, for every $t\in {\mathbb{A}}_m$ and for every $i\in \overline{\mathsf{\Omega }}$. Clearly, ρ satisfies BMAM, but it violates MSTA.

Example 2.

Define an assessment ρ to be that ${\rho }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)=0$ for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, for every $t\in {\mathbb{A}}_m$ and for every $i\in \overline{\mathsf{\Omega }}$. Clearly, ρ satisfies MSTA, but it violates BMAM.

In the following, the properties of multi-attribute entire-optimality, symmetry for assessing, synchronized normalization and M-stability would be applied to characterize the stable min value.

Lemma 4.

If an assessment ρ satisfies MAEO, SYAS, and SYNO, then ρ satisfies BMAM.

Proof. 

Please see Appendix A.4. □

Theorem 2.

On $\overline{\mathbf{MAM}}$, the stable min value is the only assessment satisfying MAEO, SYAS, SYNO, and MSTA.

Proof. 

By Definition 1, $\widehat{\chi }$ satisfies SYAS and SYNO. The remaining proofs follow from Theorem 1 and Lemmas 1–4. □

• Managerial insight.

Theorem 2 establishes that the stable min value is the only assessment that simultaneously satisfies the axioms of multi-attribute entire-optimality (MAEO), symmetry for assessing (SYAS), symmetry across attributes (SYNO), and M-stability. According to the axiomatic structure, MAEO ensures that all minimal influence must be exhaustively allocated across components; SYAS and SYNO, respectively, ensure fairness within and across evaluation dimensions; M-stability guarantees that influence assessments remain consistent when individual components are omitted from the system. In management environments that require these fairness and consistency principles, such as multi-criteria evaluation systems, resource allocation platforms across departments, or modular participation structures, the stable min value is uniquely suitable. When such conditions are fundamental, this assessment becomes not just preferable, but the only one consistent with the specified requirements.

4. The Minimal Self-Stable Value

This section introduces a new generalization of the Shapley value under multi-attribute multi-choice TU models.

Definition 3.

The minimal self-stable value, η, is an assessment on $\overline{\mathbf{MAM}}$ which associates to each $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, each $t\in {\mathbb{A}}_m$ and each component $i\in \overline{\mathsf{\Omega }}$ the value

$${\eta }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)=\sum _{\stackrel{\overline{\mathbf{S}}\subseteq \overline{\mathsf{\Omega }}}{i\in \overline{\mathbf{S}}}}{\displaystyle \frac{(|\overline{\mathbf{S}}|-1)!(|\overline{\mathsf{\Omega }}|-|\overline{\mathbf{S}}|)!}{|\overline{\mathsf{\Omega }}|!}}·\left[{\widehat{\kappa }}_{\#}^t\left(\overline{\mathbf{S}}\right)-{\widehat{\kappa }}_{\#}^t\left(\overline{\mathbf{S}}\setminus \left\{i\right\}\right)\right].$$

Under the assessment η, each component receives the weighted average of marginal contributions computed with respect to the minimal influence values across all coalitions containing that component.

Lemma 5.

The minimal self-stable value satisfies MAEO.

Proof. 

Please see Appendix A.5. □

Next, it would be shown that the minimal self-stable value can be verified as the vector of marginal influences of a multi-attribute potential.

Definition 4.

An assessment ρ admits a multi-attribute potential if there exists a function $\widehat{P}:\overline{\mathbf{MAM}}\to {\mathbb{R}}^m$ with $\widehat{P}\left(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m\right)={\left({\widehat{P}}^t\left(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m\right)\right)}_{t\in {\mathbb{A}}_m}$ such that ${\rho }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)={\widehat{P}}^t\left(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m\right)-{\widehat{P}}^t\left(\overline{\mathsf{\Omega }},({\overrightarrow{\mu }}_{-i},0),{\widehat{K}}^m\right)$ for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, for every $t\in {\mathbb{A}}_m$ and for every $i\in \overline{\mathsf{\Omega }}$.

Assessments that admit a multi-attribute potential provide a scalar representation over models such that each component’s evaluation corresponds to the marginal influence exerted under the multi-attribute structure. A function $\widehat{P}:\overline{\mathbf{MAM}}\to {\mathbb{R}}^m$ is multi-attribute 0-normalized if $\widehat{P}(\overline{\mathsf{\Omega }},{\overrightarrow{0}}_{\overline{\mathsf{\Omega }}},{\widehat{K}}^m)={\left({\overrightarrow{0}}_{\overline{\mathsf{\Omega }}}^t\right)}_{t\in {\mathbb{A}}_m}$ for each $\overline{\mathsf{\Omega }}\subseteq \overline{\ddot{\mathsf{\Omega }}}$. $\widehat{P}$ is multi-attribute efficient if for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$ and for every $t\in {\mathbb{A}}_m$,

$$\sum _{i\in \overline{\mathsf{\Omega }}}\left[{\widehat{P}}^t\left(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m\right)-{\widehat{P}}^t\left(\overline{\mathsf{\Omega }},({\overrightarrow{\mu }}_{-i},0),{\widehat{K}}^m\right)\right]={\widehat{\kappa }}_{\#}^t\left(\overline{\mathsf{\Omega }}\right).$$

(1)

The existence of a multi-attribute potential and that of a multi-attribute 0-normalized potential are equivalent, since the transformation

$${\widehat{P}}^0(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)=\widehat{P}(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)-\widehat{P}(\overline{\mathsf{\Omega }},{\overrightarrow{0}}_{\overline{\mathsf{\Omega }}},{\widehat{K}}^m)$$

defines a multi-attribute 0-normalized potential whenever $\widehat{P}$ is a multi-attribute potential. Furthermore, for any assessment $\rho$, there exists at most one multi-attribute 0-normalized potential admitting $\rho$.

Hart and Mas-Colell [4] were the first to condider the potential approach under classical models. The following outcome provides an extension of Theorem A in Hart and Mas-Colell [4] to the multi-attribute multi-choice TU setting, within which the assessment reflects marginal influences associated with minimal impact across attributes and action measures.

Theorem 3.

There exists a uniquely multi-attribute 0-normalized and multi-attribute efficient potential $\widehat{P}$ such that η admits the potential $\widehat{P}$.

Proof. 

Please see Appendix B.2. □

• Managerial insight.

Theorem 3 shows that there exists a unique multi-attribute potential function that is both 0-normalized and multi-attribute efficient, and that the marginal differences in this function correspond exactly to the minimal self-stable value for each participant. In management systems, such a potential function can be understood as a generalized energy or benefit transformation function—analogous to a potential energy model in physics or an accumulated value function in performance evaluation. The 0-normalized condition ensures that the system begins from a defined baseline (e.g., no participation implies zero potential), while multi-attribute efficiency ensures that the marginal gains from each participant, across all attributes, are fully captured and correctly distributed. This result implies that once a manager or analyst identifies an appropriate system-wide potential function meeting these two structural criteria, the minimal self-stable value can be directly obtained through marginal comparisons across participants. Thus, it offers a parsimonious and interpretable mechanism for deriving fair influence values from a latent, system-level function, suitable for data-driven settings or structured resource allocation platforms.

Inspired by Hart and Mas-Colell [4], we introduce an extended reduction to characterize the minimal self-stable value. The H-reduced model $(\overline{\mathbf{S}},{\overrightarrow{\mu }}_{\overline{\mathbf{S}}},{\widehat{K}}_{\overline{\mathbf{S}},\overrightarrow{\mu },\rho }^m)$ is defined by ${\widehat{K}}_{\overline{\mathbf{S}},\overrightarrow{\mu },\rho }^m={\left({\widehat{\kappa }}_{\overline{\mathbf{S}},\overrightarrow{\mu },\rho }^t\right)}_{t\in {\mathbb{A}}_m}$,

$${\widehat{\kappa }}_{\overline{\mathbf{S}},\overrightarrow{\mu },\rho }^t\left(\overrightarrow{\alpha }\right)={\widehat{\kappa }}_{\#}^t\left(\widehat{\mathbb{O}}(\overrightarrow{\alpha },{\overrightarrow{\mu }}_{\overline{\mathsf{\Omega }}\setminus \overline{\mathbf{S}}})\right)-\sum _{k\in \overline{\mathsf{\Omega }}\setminus \overline{\mathbf{S}}}{\rho }_k^t\left(\overline{\mathsf{\Omega }},(\overrightarrow{\alpha },{\overrightarrow{\mu }}_{\overline{\mathsf{\Omega }}\setminus \overline{\mathbf{S}}}),{\widehat{K}}^m\right).$$

for every $t\in {\mathbb{A}}_m$ and for every $\overrightarrow{\alpha }\in {\overline{\mathbf{U}}}^{\overline{\mathbf{S}}}$. This reduction isolates the influence of a specific subset $\overline{\mathbf{S}}$ of components by deducting the assessed values of all others in $\overline{\mathsf{\Omega }}\setminus \overline{\mathbf{S}}$, based on the marginal impact relative to the full model. The construction enables the evaluation of stability in the assessment $\rho$ under variation in coalition scope, while maintaining consistency in total minimal influence.

An assessment $\rho$ satisfies H-stability (HSTA) if for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, for every $t\in {\mathbb{A}}_m$, for every $\overline{\mathbf{S}}\subseteq \overline{\mathsf{\Omega }}$ and for every $i\in \overline{\mathbf{S}}$,

$${\rho }_i^t(\overline{\mathbf{S}},{\overrightarrow{\mu }}_{\overline{\mathbf{S}}},{\widehat{K}}_{\overline{\mathbf{S}},\overrightarrow{\mu },\rho }^m)={\rho }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m).$$

(2)

That is, $\rho$ is said to be H-stable if each component’s assigned value remains invariant when evaluated through the H-reduced model, thereby ensuring consistency of influence attribution across multi-attribute minimal impact assessments under varying coalition restrictions.

The following lemma establishes the equivalence between two methods of excluding a component’s acting measure from the evaluation process under H-reduction. Specifically, it shows that removing the acting measure of a component before or after constructing the H-reduced model yields the same result. This property ensures order-independence when assessing the minimal-impact influence of each component under multi-attribute considerations.

Lemma 6.

Let ρ be an assessment, $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, $\overline{\mathbf{S}}\subseteq \overline{\mathsf{\Omega }}$, and $\overrightarrow{\alpha }\in {\overline{\mathbf{U}}}^{\overline{\mathbf{S}}}$. Then, $\left(\overline{\mathbf{S}},\overrightarrow{\alpha },{\widehat{K}}_{\overline{\mathbf{S}},\overrightarrow{\mu },\rho }^m\right)=\left(\overline{\mathbf{S}},\overrightarrow{\alpha },{\widehat{K}}_{\overline{\mathbf{S}},(\overrightarrow{\alpha },{\overrightarrow{\mu }}_{\overline{\mathsf{\Omega }}\setminus \overline{\mathbf{S}}}),\rho }^m\right)$.

Proof. 

It is clear to derive this proof via definitions of a submodel and a H-reduction. □

Lemma 7.

ρ is H-stable if and only if Equation (2) is satisfied for every $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$ and for every $\overline{\mathbf{S}}\subseteq \overline{\mathsf{\Omega }}$ with $|\overline{\mathsf{\Omega }}\setminus \overline{\mathbf{S}}|=1$.

Proof. 

Please see Appendix A.6. □

Lemma 8.

The minimal self-stable value η satisfies HSTA.

Proof. 

Please see Appendix A.7. □

In the following, H-stability would be applied to characterize the minimal self-stable value. Similarly to Lemmas 3 and 4 and Theorems 1 and 2, the following results hold.

Theorem 4.

1. 

An assessment ρ satisfies MAEO if it satisfies BMAM and HSTA.

2. 

On $\overline{\mathbf{MAM}}$, the minimal self-stable value is the only assessment satisfying BMAM and HSTA.

3. 

On $\overline{\mathbf{MAM}}$, the minimal self-stable value is the only assessment satisfying MAEO, SYAS, SYNO, and HSTA.

• Managerial insight.

Theorem 4 demonstrates that the minimal self-stable value is the only assessment that satisfies H-stability. According to the axiomatic structure, H-stability ensures that influence assessments remain consistent when evaluated through recursively reduced systems, which reflects adaptability in contexts with nested coalitions or hierarchical decision layers. This contrasts with M-stability, which requires stability only under single-level component omission. In managerial systems where recursive stability is essential, such as layered decision-making environments, stakeholder networks with shifting subgroup composition, or federated collaborations, the minimal self-stable value is the uniquely consistent mechanism. If such systems require adherence to H-stability, then this assessment becomes the sole option consistent with that structural principle.

Remark 2.

It is clear from Theorems 1, 2 and 4 that both the stable min value and the minimal self-stable value simultaneously uphold the principles of fairness and impartiality, specifically MAEO, SYAS, SYNO, and BMAM, while these two concepts exhibit distinct forms of stability, the difference stems from the notion of “dissatisfaction reevaluation.” Nevertheless, the underlying concept of derived stability is coincident for both.

5. Application on Influence Evaluating Systems

Under the consideration of multi-attribute multi-choice models, it is essential to assess how each component, through multiple acting measures, contributes to the overall minimal influence generated by the system. This section applies the theoretical foundations and axiomatic results presented earlier to illustrate how the proposed minimal impact assessments function in practice.

Let $\overline{\mathit{N}}$ represent a collection of components (e.g., departments) participating in a system where each can adopt various levels of acting measures. The influence generated by their joint engagement, encoded by a vector $\overrightarrow{\alpha }={\left({\overrightarrow{\alpha }}_i\right)}_{i\in \overline{\mathit{N}}}$, is measured by a function ${\widehat{K}}^m\left(\overrightarrow{\alpha }\right)$, which assigns a scalar outcome to each acting measure configuration. The acting measure set for each component i is denoted by ${\overline{\mathit{H}}}_i$. The resulting model is formalized as a multi-attribute multi-choice model, with ${\widehat{K}}^m={\left({\widehat{\kappa }}^t\right)}_{t\in {\mathbb{A}}_m}$ serving as its characteristic influence functions. We illustrate the operational meaning of Definitions 1 and 3 through the following computational example.

Example 3.

Consider an operational system $\overline{\mathit{N}}$ with three components Manufacturing (M), Marketing (K), and Research and Development (R). Related acting measure vector is $\overrightarrow{\mu }=({\overrightarrow{\mu }}_M,{\overrightarrow{\mu }}_K,{\overrightarrow{\mu }}_R)=(2,2,1)$. The influence functions with 2-attributes ${\widehat{K}}^m=({\widehat{\kappa }}^1,{\widehat{\kappa }}^2)$ is specified as

$$\begin{array}{cccccccccccc}{\widehat{\kappa }}^1(2,2,1)\hfill & =& \hfill 126,& {\widehat{\kappa }}^1(2,2,0)\hfill & =& \hfill 108,& {\widehat{\kappa }}^1(2,1,1)\hfill & =& \hfill 150,& {\widehat{\kappa }}^1(2,1,0)\hfill & =& \hfill 84,\\ {\widehat{\kappa }}^1(2,0,1)\hfill & =& \hfill 96,& {\widehat{\kappa }}^1(2,0,0)\hfill & =& \hfill 90,& {\widehat{\kappa }}^1(1,2,1)\hfill & =& \hfill 102,& {\widehat{\kappa }}^1(1,2,0)\hfill & =& \hfill 60,\\ {\widehat{\kappa }}^1(1,1,1)\hfill & =& \hfill 84,& {\widehat{\kappa }}^1(1,1,0)\hfill & =& \hfill 60,& {\widehat{\kappa }}^1(1,0,1)\hfill & =& \hfill 114,& {\widehat{\kappa }}^1(1,0,0)\hfill & =& \hfill 72,\\ {\widehat{\kappa }}^1(0,2,1)\hfill & =& \hfill 108,& {\widehat{\kappa }}^1(0,2,0)\hfill & =& \hfill 54,& {\widehat{\kappa }}^1(0,1,1)\hfill & =& \hfill 66,& {\widehat{\kappa }}^1(0,1,0)\hfill & =& \hfill 108,\\ {\widehat{\kappa }}^1(0,0,1)\hfill & =& \hfill 102,& {\widehat{\kappa }}^1(0,0,0)\hfill & =& \hfill 0,\end{array}$$

and

$$\begin{array}{cccccccccccc}{\widehat{\kappa }}^2(2,2,1)\hfill & =& \hfill 102,& {\widehat{\kappa }}^2(2,2,0)\hfill & =& \hfill 108,& {\widehat{\kappa }}^2(2,1,1)\hfill & =& \hfill 84,& {\widehat{\kappa }}^2(2,1,0)\hfill & =& \hfill 96,\\ {\widehat{\kappa }}^2(2,0,1)\hfill & =& \hfill 126,& {\widehat{\kappa }}^2(2,0,0)\hfill & =& \hfill 108,& {\widehat{\kappa }}^2(1,2,1)\hfill & =& \hfill 90,& {\widehat{\kappa }}^2(1,2,0)\hfill & =& \hfill 60,\\ {\widehat{\kappa }}^2(1,1,1)\hfill & =& \hfill 54,& {\widehat{\kappa }}^2(1,1,0)\hfill & =& \hfill 36,& {\widehat{\kappa }}^2(1,0,1)\hfill & =& \hfill 114,& {\widehat{\kappa }}^2(1,0,0)\hfill & =& \hfill 102,\\ {\widehat{\kappa }}^2(0,2,1)\hfill & =& \hfill 150,& {\widehat{\kappa }}^2(0,2,0)\hfill & =& \hfill 84,& {\widehat{\kappa }}^2(0,1,1)\hfill & =& \hfill 60,& {\widehat{\kappa }}^2(0,1,0)\hfill & =& \hfill 72,\\ {\widehat{\kappa }}^2(0,0,1)\hfill & =& \hfill 108,& {\widehat{\kappa }}^2(0,0,0)\hfill & =& \hfill 0.\end{array}$$

Thus, it is clear that

$$\begin{array}{ccccccccc}{\widehat{\kappa }}_{\#}^1\left(\{M,K,R\}\right)\hfill & =& \hfill 84,& {\widehat{\kappa }}_{\#}^1\left(\{M,K\}\right)\hfill & =& \hfill 60,& {\widehat{\kappa }}_{\#}^1\left(\{M,R\}\right)\hfill & =& \hfill 96,\\ {\widehat{\kappa }}_{\#}^1\left(\{K,R\}\right)\hfill & =& \hfill 66,& {\widehat{\kappa }}_{\#}^1\left(\left\{M\right\}\right)\hfill & =& \hfill 72,& {\widehat{\kappa }}_{\#}^1\left(\left\{K\right\}\right)\hfill & =& \hfill 54,\\ {\widehat{\kappa }}_{\#}^1\left(\left\{R\right\}\right)\hfill & =& \hfill 102,& {\widehat{\kappa }}_{\#}^1(\varnothing )\hfill & =& \hfill 0,\end{array}$$

and

$$\begin{array}{ccccccccc}{\widehat{\kappa }}_{\#}^2\left(\{M,K,R\}\right)\hfill & =& \hfill 54,& {\widehat{\kappa }}_{\#}^2\left(\{M,K\}\right)\hfill & =& \hfill 36,& {\widehat{\kappa }}_{\#}^2\left(\{M,R\}\right)\hfill & =& \hfill 114,\\ {\widehat{\kappa }}_{\#}^2\left(\{K,R\}\right)\hfill & =& \hfill 60,& {\widehat{\kappa }}_{\#}^2\left(\left\{M\right\}\right)\hfill & =& \hfill 102,& {\widehat{\kappa }}_{\#}^2\left(\left\{K\right\}\right)\hfill & =& \hfill 72,\\ {\widehat{\kappa }}_{\#}^2\left(\left\{R\right\}\right)\hfill & =& \hfill 108,& {\widehat{\kappa }}_{\#}^2(\varnothing )\hfill & =& \hfill 0.\end{array}$$

The assessment $\widehat{\chi }$ from Definition 1 (the stable min value) on $\overline{\mathbf{MAM}}$ is defined as for each $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, for each $t\in {\mathbb{A}}_m$ and for each component $i\in \overline{\mathsf{\Omega }}$

$${\widehat{\chi }}_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)={\chi }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)+{\displaystyle \frac{1}{|\overline{\mathsf{\Omega }}|}}[{\widehat{\kappa }}_{\#}^t\left(\overline{\mathsf{\Omega }}\right)-\sum _{k\in \overline{\mathsf{\Omega }}}{\chi }_k^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)],$$

where ${\chi }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)={\widehat{\kappa }}_{\#}^t\left(\overline{\mathsf{\Omega }}\right)-{\widehat{\kappa }}_{\#}^t\left(\overline{\mathsf{\Omega }}\setminus \left\{i\right\}\right)$. By computation,

$$\begin{array}{ccccccccc}{\chi }_M^1(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 18,& {\chi }_K^1(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill -12,& {\chi }_R^1(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 24,\\ {\widehat{\chi }}_M^1(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 36,& {\widehat{\chi }}_K^1(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 6,& {\widehat{\chi }}_R^1(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 42,\\ {\chi }_M^2(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill -6,& {\chi }_K^2(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill -60,& {\chi }_R^2(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 18,\\ {\widehat{\chi }}_M^2(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 28,& {\widehat{\chi }}_K^2(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill -26,& {\widehat{\chi }}_R^2(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 52.\end{array}$$

Further, the assessment η from Definition 1 (the minimal self-stable value) on $\overline{\mathbf{MAM}}$ is defined as for each $(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\in \overline{\mathbf{MAM}}$, for each $t\in {\mathbb{A}}_m$ and for each component $i\in \overline{\mathsf{\Omega }}$

$${\eta }_i^t(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)=\sum _{\stackrel{\overline{\mathbf{S}}\subseteq \overline{\mathsf{\Omega }}}{i\in \overline{\mathbf{S}}}}{\displaystyle \frac{(|\overline{\mathbf{S}}|-1)!(|\overline{\mathsf{\Omega }}|-|\overline{\mathbf{S}}|)!}{|\overline{\mathsf{\Omega }}|!}}·\left[{\widehat{\kappa }}_{\#}^t\left(\overline{\mathbf{S}}\right)-{\widehat{\kappa }}_{\#}^t\left(\overline{\mathbf{S}}\setminus \left\{i\right\}\right)\right].$$

By computation,

$$\begin{array}{ccccccccc}{\eta }_M^1(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 30,& {\eta }_K^1(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 6,& {\eta }_R^1(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 48,\\ {\eta }_M^2(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 27,& {\eta }_K^2(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill -15,& {\eta }_R^2(\overline{\mathsf{\Omega }},\overrightarrow{\mu },{\widehat{K}}^m)\hfill & =& \hfill 42.\end{array}$$

This confirms the application of the stable min value $\widehat{\mathsf{\Psi }}$ in evaluating minimal influence values across components under multi-attribute multiple acting measure structures.

5.1. Visual Illustration

To enhance the inter-pretability and accessibility of the proposed evaluation principles, this section provides a visual and tabular synthesis of key concepts and results. We begin with a geometric interpretation of the minimal influence concept and continue with an axiomatic comparison between the two proposed allocation values.

Figure 1 shows the surface of ${\kappa }_t\left(\overrightarrow{\alpha }\right)$ for all action vectors $\overrightarrow{\alpha }$ with fixed coalition activation, highlighting the minimal influence point ${\kappa }_t^{♯}\left(S\right)$. This geometric insight reveals how the value is obtained through optimization over feasible activity vectors, reflecting non-binary influence profiles across active components.

Table 1. Evaluating values under the stable min value ($\widehat{\chi }$) and the minimal self-stable value ($\eta$).

Component	$\widehat{\mathit{\chi }}$		$\mathit{\eta }$
	Attribute 1	Attribute 2	Attribute 1	Attribute 2
M	36	28	30	27
K	6	−26	6	−15
R	42	52	48	42

and Figure 2 and Figure 3 present the distributional differences between $\widehat{\chi }$ and $\eta$ across two attributes. These comparative views further emphasize the distinct impact assessments that arise from their respective theoretical foundations.

5.2. Comparative Insights on the Stable Min Value and the Minimal Self-Stable Value

Although both the stable min value $\widehat{\chi }$ and the minimal self-stable value $\eta$ satisfy fundamental axioms such as MAEO, BMAM, SYAS, and SYNO, their distinctive structural characteristics yield differentiated practical interpretations. Theorems 1 and 4 already illustrate their axiomatic gaps, particularly regarding $\widehat{\chi }$’s M-stability and $\eta$’s H-stability. To deepen this contrast, we emphasize the following.

• Definition mode: $\widehat{\chi }$ is implicitly defined via an M-reduction and residual allocation scheme based on global influence quantities, while $\eta$ is explicitly represented through marginal coalition contributions under the Shapley-type formula.
• Stability interpretation: M-stability (underpinning $\widehat{\chi }$) emphasizes internal consistency across pairwise subsystems; H-stability (underpinning $\eta$) focuses on recursive cohesion under component withdrawal, making $\eta$ robust to layered subsystem reevaluations.
• Decision-theoretic implication: $\widehat{\chi }$ reflects centralized consistency when evaluating participation deviations among few actors; by contrast, $\eta$ captures influence dynamics that adjust recursively across full coalition hierarchies.

These theoretical distinctions are numerically manifested in the evaluation results from Example 3 (Table 1). For instance, ${\eta }_R^1=48$ exceeds ${\widehat{\chi }}_R^1=42$, reflecting that $\eta$ accounts for greater marginal impact across nested coalitions containing R, while $\widehat{\chi }$ dilutes the impact by residual averaging. Conversely, ${\widehat{\chi }}_K^2=-26$ is more extreme than ${\eta }_K^2=-15$, illustrating how $\widehat{\chi }$’s symmetric remainder rule may accentuate under-performance in marginal contributions compared to $\eta$’s global weighting scheme.

Therefore, while both assessments uphold fairness under multi-attribute influences, they are strategically suited for different evaluating logic. Specifically, $\widehat{\chi }$ is appropriate for synchronized systems with homogeneous action evaluating across subsystems, whereas $\eta$ may better reflect asynchronous, adaptive influence propagation in complex or nested participation frameworks.

5.3. Interpretation of Numerical Results and Practical Influence Evaluation

To link the numerical evaluations with real-world influence interpretations, we analyze the outcome from both $\widehat{\chi }$ and $\eta$ across the two attributes.

• M department (Manufacturing) obtains the highest allocation under $\eta$ in both attributes (${\eta }_M^1=30$, ${\eta }_M^2=27$), suggesting that its multi-attribute contributions are stable and consistently recognized under recursive evaluation logic. This indicates that M plays a structurally significant and stable role across diverse coalition structures.
• R department (R&D) receives the largest share under $\widehat{\chi }$ (${\widehat{\chi }}_R^1=42$, ${\widehat{\chi }}_R^2=52$), indicating that R provides key marginal gains in several coalitions, particularly in configurations where its presence creates dominant shifts. This behavior highlights its context-sensitive but decisive influence.
• K department (Marketing) exhibits weak or even negative influence assessments, with ${\eta }_K^2=-15$ and ${\widehat{\chi }}_K^2=-26$. These values imply that K’s participation may occasionally disrupt synergy, suggesting limited cooperative value or possible counterproductive effects within certain coalitions, particularly in Attribute 2.

From a fairness and design perspective, $\widehat{\chi }$ emphasizes residual marginality and group-level redistribution, making it preferable when robustness to coalition instability or marginal consistency is prioritized. In contrast, $\eta$ aggregates influence across all possible sub-coalitions, thus better reflecting systemic participation and suitability for systems with recursive delegation or nested collaborative structures. These differences affirm that $\widehat{\chi }$ and $\eta$ encode distinct logics for evaluating influence under multi-attribute environments, offering adaptable frameworks depending on fairness criteria or governance architecture.

5.4. Axiomatic Comparison

The stable min value and the minimal self-stable value provide evaluative mechanisms that account for heterogeneity in acting measures while ensuring that influence allocations remain efficient and fair. Their foundational axioms correspond to structural properties in multi-attribute minimal influence systems.

• Multi-attribute entire-optimality: The total minimal influence exerted by components should be exhaustively allocated.
• The basis for multi-attribute models: Influence assessments must conform to base cases involving small coalitions, which serve as building blocks for generalized outcomes.
• M-stability and H-stability: Influence values remain consistent when evaluated through reduced systems omitting individual components.
• Symmetry for assessing: Members with equal marginal impact under minimal influence should receive equal assessments.
• Synchronized normalization: Uniform adjustments across the system must induce parallel changes in individual valuations.

As established in Theorems 1, 2, and 4, and supported by this example, the stable min value and the minimal self-stable value satisfy these axioms and provide a robust evaluating framework within influence-driven multi-attribute multi-choice systems.

Table 2. Axiomatic comparison between $\widehat{\chi }$ and $\eta$.

Axiom	The Stable Min Value ($\widehat{\mathit{\chi }}$)	The Minimal Self-Stable Value ($\mathit{\eta }$)
Multi-attribute entire-optimality (MAEO)	V	V
Symmetry for assessing (SYAS)	V	V
Synchronized normalization (SYNO)	V	V
Basis for multi-attribute models (BMAM)	V	V
M-stability (MSTA)	V	X
H-stability (HSTA)	X	V

highlights the theoretical differences between the two proposed assessments, while both assessments satisfy multi-attribute entire-optimality, symmetry for assessing and synchronized normalization, the stable min value satisfies the basis for multi-attribute models, whereas the stable min value and the minimal self-stable value are uniquely characterized by, respectively, M-stability and H-stability that allows recursive influence evaluations. These distinctions reflect differing fairness rationales within the same modeling context.

6. Concluding Remarks

• The principal contributions of this study are summarized as follows.
  • This study applies the framework of multi-attribute multi-choice TU models to simultaneously account for heterogeneous acting measures per component and multiple evaluative dimensions within influence-driven systems.
  • Based on this framework, this study proposes and formally characterizes the stable min value and the minimal self-stable value. These concepts are supported by the development of extended reduction procedures and axiomatic results concerning the minimal influence consideration.
  • It is clear from Theorems 1, 2, and 4 that both the stable min value and the minimal self-stable value simultaneously uphold the principles of fairness and impartiality, specifically MAEO, SYAS, SYNO, and BMAM, while these two concepts exhibit distinct forms of stability, the difference stems from the notion of “dissatisfaction reevaluation.” Nevertheless, the underlying concept of derived stability is coincident for both.
• The results obtained herein extend and differentiate from existing literature in several ways.
  • Previous work by Nouweland et al. [9] focused on extensions of the Shapley value in multi-choice settings. Hwang and Liao [7] and Liao [8] developed reduction-based approaches for two extensions of the core and the EANSC under multi-choice frameworks. However, these studies do not incorporate multi-attribute structures in the evaluation of influence.
  • The approach in this paper diverges from those studies by integrating both multi-attribute structure and multi-choice participation, and by formulating all results based on minimal influence evaluation.
• In contrast with classical TU models, where each component’s role is limited to binary participation, our model allows each component to engage with multiple acting measures across several attributes. The results in this paper generalize several classical solution concepts by embedding them into this more expressive model structure. In particular:
  • The construction of reduced models and related axiomatic notions are adapted from the foundational work of Moulin [6] and Hart and Mas-Colell [4].
• The multi-attribute minimal influence framework developed in this paper lays the groundwork for extending classical solution concepts. It is natural to consider how other classical assessments, such as the Banzhaf value, the weighted Shapley value, or bargaining-based concepts, could be generalized using the same minimal influence structure under multi-attribute and multi-choice consideration. This presents an avenue for future investigation.