A Resolution Under Interval Uncertainty
Abstract
:1. Introduction
- Can related results for PEANSC and PWANSC be described under the interval TU setting?
- How do interval TU solutions compare to existing stochastic and fuzzy cooperative models in handling uncertainty?
- What computational frameworks can be developed to implement interval-based resolutions efficiently in large-scale applications?
- An extended resolution: We introduce novel generalizations of the pseudo equal allocation of non-separable costs (PEANSC) and the pseudo weighted allocation of non-separable costs (PWANSC) under interval TU settings, ensuring allocation stability under uncertain environments.
- Interval-based reduction and consistency: We establish a systematic framework to adapt classical reduction and consistency principles to interval TU games, thereby enhancing the theoretical foundations of interval resolutions.
- A weighted generalization: A novel weighted resolution is proposed to refine interval allocation by incorporating variable importance factors among players, allowing for more flexible and adaptive allocation strategies.
- A normalization mechanism: We develop a normalization method to standardize individual contributions under interval uncertainty, ensuring robustness and fairness in allocation processes.
- Axiomatic characterizations: By extending and modifying classical axioms, we provide rigorous axiomatic results that justify the proposed interval-based resolutions, establishing consistency, efficiency, and fairness criteria for interval TU games.
- Comparative analysis and distinct contributions: Unlike previous works that primarily focus on deterministic or fuzzy TU models, this study explicitly integrates interval-based methodologies, offering a more comprehensive framework for allocation mechanisms under uncertainty.
- Application in real-world situations: Through case studies and computational validation, we demonstrate how the proposed resolutions can be implemented in resource distribution, coalition formation, and decision-making under uncertainty.
2. Preliminaries
3. Axiomatic Results
- satisfies efficiency (EFF) if for all . Axiom EFF states that the total utility of the grand coalition is entirely allocated among all players.
- satisfies standard for games (SFG) if for all with , and
- satisfies symmetry (SYM) if for all with for some and for all . Axiom SYM describes that if the marginal contributions of any two players are equal, their final allocations should also be the same.
- satisfies zero-independence (ZI) if for all with for some and for all . Mathematically, Axiom ZI represents a weakened analogue of additivity.
- Based on Definition 1, it is easy to check that IPEANSC satisfies EFF, SFG, SYM, and ZI.
- Given , , and a resolution , the reduced game with respect to S and is defined by for all ,
- The property of consistency could be described in the following condition. For any coalition of players within a interval TU game, a “reduction” could be defined by considering the value left after the other players have obtained its outcomes distributed via the resolution . The resolution is said to be consistent if, when applied to each reduction, it always begets outcomes that coincide with those in the beginning status. Formally, a resolution satisfies consistency (CON) if for all with , for all with , and for all .
4. A Weighted Generalization
5. A Normalization
- It is easy to show that the interval normalized index satisfies EFF and SYM, but it violates ZI.
- One could also consider two normalized resolutions as follows.
- -
- Resolution is defined by
- -
- Resolution is defined by
However, resolution , due to considerations regarding the definition of intervals as well as multiplication and division operations, necessitates the imposition of multiple conditions and constraints to ensure its existence. On the other hand, although resolution , based on the definition of the norm of interval, does not impose existential restrictions, its correlation with relative position of the interval related to each player’s individual contribution is relatively weak. In future research, it may be possible to refine both resolutions and by incorporating additional interval arithmetic properties to enhance their theoretical consistency and applicability.
Application Example
- Problem context: Consider a scenario where a technology company collaborates with multiple partners to develop an innovative product. Each partner’s contribution is influenced by resource limitations and uncertainties, such as market demands and technological breakthroughs. These uncertainties can be modeled using interval utilities, representing the minimum and maximum potential contributions of each partner.
- Interval TU game-theoretical modeling: Based on the definitions, this situation can be modeled as an interval TU game , where:
- -
- is the set of participants.
- -
- is a characteristic function assigning an interval of utilities to each subset . For example:The joint utility of subsets is defined as:
- Application of interval allocating resolutions: Based on Definitions 1 and 2, the interval utilities can be allocated using two main approaches:
- -
- The interval pseudo equal allocation of non-separable costs (IPEANSC)The allocation of player is defined as:For the Company:Simplifying:Similarly, the allocations for Partner A and Partner B can be computed.
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- The interval pseudo weighted allocation of non-separable costs (IPWANSC)If weights are introduced to represent the importance of participants, e.g.,For the Company:
- -
- The interval normalized indexThe allocation of player is given by:For the Company:
- These allocating processes provide a robust framework for fair and weighted distribution of utilities in collaborative scenarios with interval uncertainties. IPEANSC emphasizes fairness by equal division, while IPWANSC incorporates weights to reflect participants’ relative importance. Such methodologies are applicable in investment planning, resource allocation, and risk management under uncertainty.
6. Conclusions
- 1.
- Unlike classical TU games, where utilities and payoffs are precisely defined, interval TU games offer a flexible framework to model the inherent uncertainties in practical settings. This study has extended the classical transferable utility (TU) game framework to incorporate interval uncertainty, addressing practical scenarios where precise utilities and payoffs are not always available. By building upon the concepts of the pseudo equal allocation of non-separable costs (PEANSC) and the pseudo weighted allocation of non-separable costs (PWANSC), this study has developed a structured methodology for managing uncertainty in cooperative allocation settings. The key contributions and findings of this study are summarized as follows:
- Development of interval-based reduction and related consistency: This study introduced a novel reduction and related consistency property specifically adapted for interval TU games, ensuring the stability and coherence of allocation decisions under uncertain conditions.
- Extensions of PEANSC and PWANSC: The study systematically extends these classical allocation concepts to the interval setting, providing a more robust and flexible allocating mechanism applicable to diverse real-world challenges, including economic planning and resource distribution.
- Axiomatic justifications: The proposed interval-based resolutions have been rigorously justified through a series of axiomatic results, reinforcing their mathematical soundness and alignment with classical game-theoretical principles.
- Introduction of a weighted generalization: By incorporating variable importance factors among players, this study developed a weighted generalization that allows for a more tailored and context-sensitive distribution of resources, aligning with strategic and economic considerations. This weighted approach aligns with real-world scenarios where contributions vary based on strategic importance and availability of resources.
- Normalization for fairness: Additionally, this study introduces a normalization process to ensure comparability and fairness in interval allocations. By standardizing the distribution of resources across players with varying contribution magnitudes, the normalization mechanism enhances the robustness of the proposed allocation methods. This adjustment allows for a more balanced approach in settings where extreme variations in coalition contributions might otherwise skew the allocation outcomes, further strengthening the practical applicability of the proposed framework.
- Theoretical and practical integration: The study bridges the gap between theoretical advancements and practical applicability, offering a comprehensive framework for implementing interval-based resolutions in cooperative game-theoretical situations.
- Comparative analysis with classical and fuzzy approaches: Our results highlight the advantages of interval-based resolution over traditional deterministic and fuzzy TU models, demonstrating its superiority in handling real-world uncertainty.
- Application and computational validation: Through a detailed application example, this study has illustrated the practical implementation of the proposed methods, showcasing their effectiveness in managing coalition-based decision-making under interval uncertainty.
By addressing key limitations in classical TU models and incorporating interval-based refinements, this study provides a novel, theoretically grounded, and practically applicable approach to cooperative allocation under uncertainty. The proposed resolutions contribute to the growing body of research on interval game theory, offering a solid foundation for further advancements and real-world implementations. - 2.
- Introduced by Branzei et al. [1,2], interval TU games model coalition utilities and payoffs as closed intervals, accommodating real-world imprecision while maintaining mathematical rigor. Foundational techniques in interval arithmetic, as explored by Moore [3], provide the necessary tools to handle interval uncertainty. Various extensions of interval TU games have been applied in resource allocation, political negotiations, and economic planning [4,5,6]. Building upon the foundational works of Branzei et al. [1,2], Alparslan Gök et al. [5,8], and Liao [19,20], this study extends key allocating concepts such as the Shapley value and the core into interval settings, preserving fairness and stability. Additionally, the interval-based consistency property explored in this work complements the reduction methodologies introduced by Harsanyi [14] and refined in subsequent studies by Hart and Mas-Colell [18] and Moulin [17], providing a unified approach to analyzing allocation processes. The introduction of interval-based allocation enhances decision-making processes in environments where coalition values fluctuate due to uncertain external conditions. One should proceed with a comparison and analysis of the relevant existing studies.
- Interval-Based Extensions: While Alparslan Gök et al. [8] explored interval-based Shapley values and core concepts, this study advances interval TU frameworks by incorporating weighted allocations and normalization mechanisms.
- Axiomatic and Reduction-Based Justifications: Prior studies, such as Guerrero and Olvera-Lopez [10], focused on axiomatic characterizations of interval Myerson values. This study builds upon axiomatic principles and introduces interval-based consistency and reduction methodologies.
- 3.
- This study builds upon the framework established by Liao [19] while extending the results of Hsieh [15].
- Specifically, Liao’s [19] work provided foundational principles for interval game-theoretical resolutions, but it did not incorporate an explicit weighting mechanism or normalization process. This study extends Liao’s [19] approach by integrating these features, thereby ensuring a more balanced allocation process that better reflects real-world economic and strategic conditions.
- Furthermore, compared to Hsieh [15], which focused on classical TU game extensions, this study bridges the gap between interval-based frameworks and non-separable cost allocation.
- 4.
- By applying the related discussions and comparisons mentioned above, the distinct contributions of this study are as follows.
- Novel interval-based allocating principles: Introducing extended PEANSC and PWANSC formulations tailored for interval TU environments.
- Weighted generalization and normalization: Ensuring comparability and fairness by refining allocation processes under uncertainty.
- Theoretical and practical integration: Bridging theoretical advancements with real-world applications through structured resolution mechanisms.
- Enhanced fairness and stability: Demonstrating superior allocation stability compared to classical TU and fuzzy models.
By systematically integrating interval-based reductions, axiomatic refinements, and normalization techniques, this study presents a structured and theoretically sound advancement in cooperative game theory. Its applications extend to economic modeling, strategic decision-making, and coalition stability, reinforcing its significance in modern uncertainty-driven environments. - 5.
- The findings of this study open several avenues for future research:
- Exploring additional subclasses of interval TU games, such as those involving fuzzy or stochastic utilities, to further enhance their applicability. This would provide a broader theoretical foundation for uncertainty modeling in cooperative games.
- Investigating the computational efficiency of interval-based resolutions for large-scale cooperative problems to improve feasibility in practical applications. Developing algorithmic approaches to efficiently compute interval solutions will be crucial for handling high-dimensional cooperative games.
- Conducting a deeper analysis of the role of weights in resource allocation and developing tailored strategies for specific industry situations. The introduction of customized weighting mechanisms may lead to more equitable and efficient allocation methods for economic, political, and industrial applications.
- Integrating interval game-theoretical analysis with other uncertainty-handling methodologies, such as fuzzy sets or stochastic processes, to broaden the theoretical framework’s applicability. Combining interval approaches with existing uncertainty modeling techniques could lead to a more comprehensive decision-support framework for real-world applications.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Hwang, Y.-A.; Liao, Y.-H. A Resolution Under Interval Uncertainty. Mathematics 2025, 13, 762. https://doi.org/10.3390/math13050762
Hwang Y-A, Liao Y-H. A Resolution Under Interval Uncertainty. Mathematics. 2025; 13(5):762. https://doi.org/10.3390/math13050762
Chicago/Turabian StyleHwang, Yan-An, and Yu-Hsien Liao. 2025. "A Resolution Under Interval Uncertainty" Mathematics 13, no. 5: 762. https://doi.org/10.3390/math13050762
APA StyleHwang, Y.-A., & Liao, Y.-H. (2025). A Resolution Under Interval Uncertainty. Mathematics, 13(5), 762. https://doi.org/10.3390/math13050762