Optimal Solutions for Constrained Bi-matrix Games with Fuzzy Rough Payoffs

Abstract

This study proposes a novel methodology for solving constrained bi-matrix games with payoffs represented by fuzzy rough numbers, addressing uncertainties common in real-world decision-making. By integrating fuzzy rough set theory with α-cut techniques, the method establishes the existence of an fuzzy rough equilibrium value. Five linear programming models are developed to compute the mean equilibrium and its lower-lower, lower-upper, upper-lower, and upper-upper bounds, based on 0-cut and 1-cut representations. For any confidence level α, corresponding equilibrium bounds are determined through α-cut-based optimization. The methodology is validated through a case study on corporate environmental behavior, demonstrating its effectiveness and practical relevance.

Keywords

constrained bi-matrix Fuzzy rough α-cut Matrix game Linear programming

1 Introduction

1.1 Background

Game theory primarily focuses on the strategic and competitive interactions between decision-makers. Understanding how to make effective decisions in a competitive environment is both a significant and common challenge. Over the years, game theory play a vital role in many fields, such as military strategy, finance, economics, strategic negotiations, cartel behavior, management issues, auctions, social challenges, political voting systems, research and development, and competitive races (Hung et al., 1996; Von Neumann & Morgenstern, 1944).

Constrained bi-matrix games are an advanced extension of classical bi-matrix games that incorporate additional restrictions or limitations on players’ strategies, reflecting more realistic and practical decision-making scenarios. Unlike standard bi-matrix games, where each player aims to optimize their payoff based solely on a fixed strategy set, constrained versions introduce conditions such as budget limits, policy rules, or capacity restrictions that must be satisfied during the strategic interaction. These constraints significantly increase the complexity of the solution process, as they often transform the game into a constrained optimization problem. Constrained bi-matrix games are particularly useful in modeling real-world competitive situations in economics, environmental policy, defense, and resource management, where players not only strive for optimal outcomes but must also operate within regulatory or structural boundaries. The integration of constraints into bi-matrix games has motivated the development of specialized solution techniques, including linear and nonlinear programming, fuzzy set theory, and other soft computing approaches to handle imprecise information and uncertainty inherent in many practical applications. Several papers on constrained matrix and bi-matrix games have evolved significantly to incorporate diverse fuzzy and uncertain environments. Early foundational work by Dengfeng and Chuntian (2002) introduced fuzzy multi-objective programming methods for fuzzy-constrained matrix games, later extended to handle triangular (Li & Hong, 2012) and trapezoidal (Li & Hong, 2013) fuzzy payoffs using α-cut based linear programming. Nan and Li (2014) further enhanced this with techniques for interval-valued constraint games. Ammar and Brikaa (2019) introduced a rough interval approach, adding granularity to constraint modeling. The extension to bi-matrix settings emerged with An and Li (2019), who proposed a linear programming method for games with intuitionistic fuzzy payoffs. Brikaa et al. (2019a) addressed constrained matrix games involving fuzzy rough numbers payoffs via a multi-objective framework. Verma (2021) presented a novel solution method for fuzzy-constrained matrix games, while Gaber et al. (2021) addressed constrained bi-matrix games under single-valued trapezoidal neutrosophic environments. More recently, Djebara et al. (2023) explored a new approach for solving constrained matrix games with fuzzy constraints and fuzzy payoffs.

In real-world decision-making scenarios, imprecision and uncertainty are inherent in many parameters. Often, these situations involve a hybrid uncertain environment where roughness and fuzziness coexist. Fuzzy rough numbers have proven effective for modeling such decision-making problems, as roughness and fuzziness are key contributors to uncertainty. Dubois and Prade (1990) explored the fuzzification of rough sets, while Morsi and Yakout (1998) introduced the lower and upper approximations of fuzzy rough sets. Decision-making approaches, such as rough programming and fuzzy programming, have been developed to address uncertainty by treating fuzziness and roughness as distinct factors. Numerous studies have focused on integrating roughness and fuzziness into a unified framework for analyzing fuzzy rough sets. Presently, fuzzy rough set theory has been applied to a wide range of practical problems.

In the context of matrix game theory, numerous fuzzy models have been developed to address the complexity and uncertainty inherent in real-world payoffs. However, these traditional fuzzy approaches primarily capture vagueness arising from imprecise linguistic or numerical data, and often lack the capacity to deal with ambiguity due to indiscernibility or incomplete information. To overcome this limitation, this study adopts the fuzzy rough model, which effectively integrates the descriptive strength of fuzzy set theory with the boundary approximation ability of rough set theory. This hybrid model allows for more nuanced modeling of payoffs by accommodating both the degree of uncertainty in player preferences and the vagueness in constraint boundaries. Such a dual-level treatment of uncertainty is especially critical in constrained bi-matrix games, where strategic decisions are influenced by both subjective judgments and limited information. Thus, the fuzzy rough framework provides a more comprehensive and realistic representation of strategic interactions, justifying its selection for this study.

1.2 Literature Review

In recent years, extensive research has been conducted to enhance decision-making models under uncertainty through various fuzzy and rough set extensions applied to matrix and bi-matrix games. Brikaa et al. (2020) addressed multi-criteria zero-sum matrix games by employing intuitionistic fuzzy goals and a novel indeterminacy resolution technique, contributing to more robust decision frameworks under uncertainty. Singh et al. (2020) proposed the use of 2-tuple linguistic information in matrix games, enabling more refined and human-consistent expressions of vague preferences. Khan and Mehra (2020) introduced a unique equilibrium solution concept for intuitionistic fuzzy bi-matrix games by integrating possibility and necessity expectations, thus enhancing realism in uncertain environments. Seikh et al. (2020) focused on matrix games with hesitant fuzzy payoffs, providing algorithms that accommodate hesitation and ambiguity in strategic evaluations. Meanwhile, Bhaumik et al. (2021a) advanced bi-matrix game analysis in neutrosophic environments using an (α,β,γ)-cut set-based ranking method, enabling multi-criteria and indeterminate factor incorporation. Roy and Maiti (2020) developed reduction methods for type-2 fuzzy variables and applied them to Stackelberg games, improving computational efficiency and solution accuracy under higher-order fuzziness. Additionally, Bhaumik et al. (2020) examined the Prisoner's Dilemma using a hesitant interval-valued intuitionistic fuzzy-linguistic term set combined with TOPSIS, showcasing the applicability of complex fuzzy tools in real-world socio-economic problems such as human trafficking.

Bhaumik et al. (2021b) introduced a multi-objective linguistic-neutrosophic matrix game framework, applying it to tourism management to better handle imprecise and conflicting criteria. Extending this line of work, Bhaumik and Roy (2021) proposed a novel aggregation operator for intuitionistic interval-valued hesitant fuzzy matrix games, specifically designed to support complex management decision-making scenarios. Seikh et al. (2021c) explored matrix games with dense fuzzy payoffs, contributing to a richer modeling of vagueness in payoff structures. Further, Seikh et al. (2021a) tackled the telecom market share problem using rough interval payoffs, combining fuzzy and rough set theory for enhanced realism. In the realm of neutrosophic modeling, Seikh and Dutta (2021a) presented a nonlinear programming model using single-valued neutrosophic numbers to solve matrix games, reflecting on strategic interactions under ambiguous environments. Similarly, they (Seikh & Dutta, 2021b) developed an intuitionistic fuzzy optimization-based method to address interval-valued matrix games. Seikh et al. (2021b) introduced a new defuzzification technique for type-2 fuzzy variables, applying it effectively to the plastic ban problem. Seikh and Karmakar (2021) also proposed a credibility equilibrium strategy for matrix games involving triangular dense fuzzy lock sets, offering deeper insights into equilibrium analysis. Verma and Aggarwal (2021a) and Verma and Aggarwal (2021b) addressed matrix games with linguistic intuitionistic fuzzy and 2-tuple intuitionistic fuzzy linguistic payoffs, respectively, enhancing the linguistic expressiveness in game settings. Additionally, Xue et al. (2021) applied hesitant fuzzy information and the Ambika method to matrix games, demonstrating its utility in counter-terrorism decision-making. Karmakar et al. (2021) introduced a type-2 intuitionistic fuzzy model with a novel distance measure for application in biogas plant implementation. Chauhan and Gupta (2021) explored matrix games with proportional linguistic payoffs, while Jangid and Kumar (2021) dealt with triangular neutrosophic number-based payoffs. Naqvi et al. (2021) applied the Tanaka and Asai approach to I-fuzzy zero-sum games, and Kon (2021) provided a theoretical characterization for equilibrium strategies with LR fuzzy payoffs. Kumar (2021) used piecewise linear programming to address multi-objective games with I-fuzzy goals. Namarta and Gupta (2021) and Mi et al. (2021) explored expert-based evaluation and probabilistic linguistic information in matrix games, expanding the decision-making capabilities under subjective and probabilistic uncertainty.

Seikh and Dutta (2022) addressed uncertainty using single-valued trapezoidal neutrosophic numbers to model matrix games more effectively. Li and Tu (2022a) examined bi-matrix games with intuitionistic fuzzy payoffs, particularly in the context of corporate environmental behavior, demonstrating the relevance of fuzzy game theory in sustainable decision-making. The rough fuzzy environment explored by Jangid and Kumar (2022b) introduced new computational techniques to handle dual-layer uncertainty in two-person zero-sum games, while Brikaa et al. (2022b) proposed the Mehar approach for matrix games with triangular dual hesitant fuzzy payoffs, enhancing modeling flexibility. Additionally, the Ambika method developed by Brikaa et al. (2022a) provided a robust defuzzification mechanism for neutrosophic settings. Qiu and Xiang (2022) introduced 2-tuple linguistic bi-matrix games, extending applicability to qualitative decision contexts. In parallel, Zheng and Brikaa (2022) presented a multi-objective aspiration-level-based model for bi-matrix games with intuitionistic fuzzy goals, offering strategic depth for goal-driven decision-makers. Furthermore, Jangid and Kumar (2022a) introduced hexadecagonal fuzzy numbers and novel ranking methods, pushing the boundaries of fuzzy set applications in game theory. Li and Tu (2022b) complemented this body of work by combining fuzzy envelope models with prospect theory in probabilistic linguistic game environments.

Jana and Roy (2023) introduced linguistic Pythagorean hesitant fuzzy frameworks to tackle multi-criteria decision-making scenarios, offering enhanced expressiveness for vague human judgments. Karmakar and Seikh (2023) explored dense fuzzy environments in bimatrix games, applying their findings to disaster management problems, thus bridging the gap between abstract models and real-world crisis applications. Naqvi et al. (2023b) proposed a novel approach using linguistic interval-valued intuitionistic fuzzy sets to enrich traditional matrix games, while Seikh and Dutta (2023a) and (2023b) formulated single-valued neutrosophic and interval neutrosophic models to address real-life issues such as cybersecurity and market share dynamics. Verma et al. (2023) incorporated self-confidence levels into Pythagorean fuzzy matrix games, capturing player subjectivity in uncertain environments. In the marketing domain, Bisht and Dangwal (2023) developed a fuzzy ranking technique for bi-matrix games with interval payoffs, while Singla et al. (2023) modeled multi-opinion-based intuitionistic fuzzy bi-matrix games to reflect consensus-driven decision-making. Further broadening the application spectrum, Naqvi et al. (2023a) employed interval-valued hesitant fuzzy linguistic sets in modeling electric vehicle adoption strategies. Achemine and Larbani (2023) offered a Nash equilibrium framework under fuzzy randomness, emphasizing stochastic behaviors in bi-matrix contexts. Li et al. (2023) investigated fuzzy weighted Pareto–Nash equilibria in multi-objective bi-matrix games, expanding the theoretical toolkit for trade-off analysis under vagueness. Bigdeli and Mousazadeh (2023) adopted the Analytic Hierarchy Process within neutrosophic environments to address military decision-making problems, highlighting hybrid modeling approaches. Similarly, Chauhan and Gupta (2023) addressed 2-tuple linguistic bimatrix games, while Jangid et al. (2023) developed a fuzzy rough game solution for two-player settings, combining rough sets with fuzzy logic to better capture granulated uncertainty. Naqvi and Sachdev (2023) extended the discussion to linguistic Pythagorean fuzzy sets in uncertain games, offering a more expressive fuzzy logic structure. Khan and Kumar (2023) introduced regret theory into hesitant fuzzy linguistic matrix games, providing psychological realism in player behavior modeling.

Further advancements have been made by Dong and Wan (2024) proposed a Type-2 interval-valued intuitionistic fuzzy matrix game approach, effectively applied to strategic decisions in the energy vehicle industry, showcasing its robustness in handling multilayered uncertainties. Seikh and Dutta (2024) extended this direction by formulating a nonlinear solution method for matrix games with picture fuzzy payoffs, with practical application to mitigating cyberterrorism attacks, thus bridging game theory with cybersecurity challenges. Karmakar and Seikh (2024) introduced a nonlinear programming model for interval-valued intuitionistic hesitant fuzzy matrix games, contributing a symmetry-based perspective to noncooperative decision-making frameworks. Ahuja and Kumar (2024) offered the Mehar approach to solve hesitant fuzzy linear programming problems, enhancing the precision of solutions under hesitation and vagueness. Kirti et al. (2024) explored modified strategies for solving matrix games with single-valued trapezoidal neutrosophic payoffs, delivering a comprehensive technique suitable for more nuanced and imprecise payoff environments. In the realm of stochastic analysis, İzgi et al. (2024) proposed a hybrid Shapley and iterative method based on matrix norms, enabling effective handling of stochastic matrix games where uncertainty arises from probabilistic behavior. Djebara et al. (2024) dealt with fuzzy random constraints in matrix games, providing a practical optimization approach that combines fuzziness and randomness to model real-world constraints. Bisht et al. (2024) introduced a method for solving interval-valued matrix games, supported by a MATLAB implementation, offering accessibility to both theorists and practitioners. Wan and Qiu (2024) presented a timely application by integrating sentiment analysis with trapezoidal Type-2 fuzzy linguistic intuitionistic matrix games, demonstrating its relevance in pandemic response management where emotions and uncertainties coexist. Kumar and Garg (2025) proposed a novel two-level fuzzy set theoretic framework to solve multi-objective matrix games, enhancing decision-making under layered uncertainty and offering broad applicability across domains. Similarly, Devi and Sowmiya (2025) introduced the use of octagonal neutrosophic fuzzy numbers to address game problems, enriching the modeling of indeterminacy in strategic interactions. Still, despite these advancements, most works remain centered on single matrix or zero-sum frameworks, lacking the comprehensive treatment of constrained bi-matrix games under hybrid uncertainties like fuzzy rough sets, which limits their applicability in highly complex, real-world multi-agent systems underscoring the novelty and necessity of the constrained fuzzy rough bi-matrix model proposed in this study.

1.3 Motivation

The increasing complexity of real-world decision-making scenarios, particularly in fields like environmental regulation, cybersecurity, and resource allocation, necessitates game-theoretic models that can effectively handle multiple layers of uncertainty and interdependent strategies. Traditional matrix games and their constrained extensions have evolved significantly, incorporating fuzzy numbers, intuitionistic fuzzy sets, neutrosophic logic, and rough set theory to model imprecision and vagueness in payoffs or constraints. However, most of these models are restricted to single-player constrained games or zero-sum frameworks, lacking the flexibility to represent non-cooperative interactions between two strategic players with mutual but distinct objectives as is the case in bi-matrix games. Furthermore, while some research has addressed constrained bi-matrix games using intuitionistic fuzzy payoffs (An & Li, 2019) or neutrosophic numbers (Gaber et al., 2021), these approaches often fail to account for dual types of uncertainty namely fuzziness (degree-based vagueness) and roughness (boundary-based indiscernibility). The fuzzy rough set framework offers a powerful hybrid mechanism to handle such complex uncertainty, yet it remains underutilized in game theory, particularly for constrained bi-matrix games, where both players operate under strategic constraints and vague, partially observable information. Additionally, most existing solution methods are limited to linear or crisp programming techniques, which may not be robust enough for handling non-linear, multi-objective, or hybrid fuzzy rough constraints encountered in practice. This reveals a critical research gap: the lack of comprehensive methodologies that integrate fuzzy rough approximations into constrained bi-matrix game models, enabling a more realistic and holistic treatment of multi-agent strategic decision-making under high uncertainty. Developing such models can significantly enhance the applicability of game theory to emerging complex systems, where decision-making environments are inherently uncertain, imprecise, and constrained.

1.4 The Contribution and Structure of this Article

This article introduces a novel type of constraint bi-matrix games grounded in the concept of α-cut sets. Specifically, it focuses on fuzzy rough constraint bi-matrix games where the payoffs for each player are represented by fuzzy rough numbers. These games are referred to as constraint bi-matrix games with fuzzy rough numbers payoffs. To the best of our knowledge, no prior research has investigated a methodology for solving constraint bi-matrix games within a fuzzy rough environment. The primary contributions and innovations of this study are summarized as follows:

This paper introduces a novel class of constraint bi-matrix games models formulated under a fuzzy rough environment, expanding the scope of game theory in uncertain settings.

The α-cut approach is systematically developed to address the constraint bi-matrix games problem under fuzzy rough conditions.

The equilibrium values for each player are derived in fuzzy rough form, which enhances decision-making under uncertainty.

To achieve optimal results, the problem is transformed into five distinct crisp constraint bi-matrix games models derived using data from the 1-cut and 0-cut sets of fuzzy rough payoffs, facilitating computational efficiency.

The effectiveness of the proposed framework is demonstrated through its application to a corporate environmental behavior problem.

Furthermore, the proposed method is adaptable for solving non-cooperative matrix games involving specialized forms of fuzzy rough numbers, including Gaussian fuzzy rough numbers, neutrosophic fuzzy rough numbers, intuitionistic fuzzy rough numbers, and hesitant fuzzy rough numbers. This broad applicability makes it a valuable tool for addressing a wide range of real-world strategic decision-making problems.

The structure of this paper is as follows: Section 2 provides a brief overview of fundamental concepts, including triangular fuzzy numbers, rough intervals, and fuzzy rough numbers. Section 3 defines classical constraint bi-matrix games. Section 4 formulates constraint bi-matrix games with fuzzy rough payoffs and develops α-cut solution. Section 5 presents a numerical example for corporate environmental behavior, and discusses the results to validate the proposed methodologies. Finally, Section 6 concludes the paper with a concise summary.

2 Preliminaries

In this section, certain fundamental definitions of triangular fuzzy numbers (Zadeh, 1965), rough interval numbers (Pawlak, 1982) and fuzzy rough numbers (Ammar & Muamer, 2016) are presented.

2.1 Triangular Fuzzy Number

Definition 1:
A fuzzy number $\tilde{P} = (p^{L}, p^{M}, p^{U})$ is said to be a triangular fuzzy number if its membership function is given as follows:
$μ_{\tilde{P}} (z) = {\begin{array}{ll} \frac{z - p^{L}}{p^{M} - p^{L}} & i f p^{L} \leq z < p^{M} \\ 1 & i f z = p^{M} \\ \frac{p^{U} - z}{p^{U} - p^{M}} & i f p^{M} < z \leq p^{U} \\ 0 & e l s e, \end{array}$
where $p^{M}$ is the mean of $\tilde{P} p^{L}$ and $p^{U}$ are the lower and upper limits of $\tilde{P}$ , respectively.
Definition 2:
The $α$ -cut set of the triangular fuzzy number $\tilde{P} = (p^{L}, p^{M}, p^{U})$ is given as $\tilde{P} (α) = z | μ_{\tilde{p}} (z) \geq α$ , where $α \in [0, 1]$ . Then, for any $α \in [0, 1]$ , we can get a $α$ -cut set of the triangular fuzzy number $\tilde{P} = (p^{L}, p^{M}, p^{U})$ , which is an interval, represented by $\tilde{P} (α) = [α p^{M} + (1 - α) p^{L}, α p^{M} + (1 - α) p^{U}]$ .
2.2 Rough Interval

Definition 3:
Suppose $D^{R} \subset R$ be a qualitative value, $D^{U} = [D^{- U}, D^{+ U}]$ and $D^{L} = [D^{- L}, D^{+ L}]$ are upper and lower approximations intervals of $D^{R}$ , respectively. $D^{R} = [D^{U} : D^{L}] = [[D^{- U}, D^{+ U}] : [D^{- L}, D^{+ L}]]$ is named as rough interval if the following conditions are satisfied:
i
If $z \in D^{L}$ then $z \in D^{R}$ (i.e. $D^{R}$ surely take z).
ii
If $z \in D^{U}$ then $D^{R}$ possibly takes z.
iii
If $z \notin D^{U}$ then $z \notin D^{R}$ (i.e. $D^{R}$ surely does not take z).
Definition 4:
Let $P^{R} = [P^{U} : P^{L}] = [[P^{- U}, P^{+ U}] : [P^{- L}, P^{+ L}]]$ and $Q^{R} = [Q^{U} : Q^{L}] = [[Q^{- U}, Q^{+ U}] : [Q^{- L}, Q^{+ L}]]$ be two non-negative rough intervals number, then
i
$P^{R} + Q^{R} = [(P^{U} + Q^{U}) : (P^{L} + Q^{L})] = [[P^{- U} + Q^{- U}, P^{+ U} + Q^{+ U}] : [P^{- L} + Q^{- L}, P^{+ L} + Q^{+ L}]]$ ,
ii
$P^{R} - Q^{R} = [[P^{- U} - Q^{+ U}, P^{+ U} - Q^{- U}] : [P^{- L} - Q^{+ L}, P^{+ L} - Q^{- L}]]$ ,
iii
$P^{R} \times Q^{R} = [[P^{- U} \times Q^{- U}, P^{+ U} \times Q^{+ U}] : [P^{- L} \times Q^{- L}, P^{+ L} \times Q^{+ L}]]$ ,
iv
$P^{R} / Q^{R} = [[P^{- U} / Q^{+ U}, P^{+ U} / Q^{- U}] : [P^{- L} / Q^{+ L}, P^{+ L} / Q^{- L}]]$ ,
2.3 Fuzzy Rough Number

Definition 5:
Let Y denote a compact set of real numbers. A fuzzy rough variable ${\tilde{Y}}^{R}$ is given as ${\tilde{Y}}^{R} = [{\tilde{Y}}^{L} : {\tilde{Y}}^{U}]$ where ${\tilde{Y}}^{L}$ and ${\tilde{Y}}^{U}$ are fuzzy numbers called lower and upper approximation fuzzy numbers of ${\tilde{Y}}^{R}$ with ${\tilde{Y}}^{L} ⊑ {\tilde{Y}}^{U}$ .
Definition 6:
A fuzzy rough number ${\tilde{P}}^{R}$ is a triangular fuzzy rough number given by ${\tilde{P}}^{R} = [(p^{L L}, p^{M}, p^{U L}) : (p^{L U}, p^{M}, p^{U U})]$ where $p^{L L}, p^{M}, p^{U L}, p^{L U} and p^{U U} \in R$ such that $p^{L U} \leq p^{L L} \leq p^{M} \leq p^{U L} \leq p^{U U}$ and the membership function is given as follows:
$μ_{{\tilde{P}}^{R}} (z) = {\begin{array}{ll} μ_{{\tilde{P}}^{L}} (z) = {\begin{array}{ll} \frac{z - p^{L L}}{p^{M} - p^{L L}} & if p^{L L} \leq z < p^{M} \\ 1 & if z = p^{M} \\ \frac{p^{U L} - z}{p^{U L} - p^{M}} & if p^{M} < z \leq p^{U L} \\ 0 & else, \end{array} \\ μ_{{\tilde{P}}^{U}} (z) = {\begin{array}{ll} \frac{z - p^{L U}}{p^{M} - p^{L U}} & if p^{L U} \leq z < p^{M} \\ 1 & if z = p^{M} \\ \frac{p^{U U} - z}{p^{U U} - p^{M}} & if p^{M} < z \leq p^{U U} \\ 0 & else, \end{array} \end{array}$
where ${\tilde{P}}^{L} = (p^{L L}, p^{M}, p^{U L}), {\tilde{P}}^{U} = (p^{L U}, p^{M}, p^{U U})$ and ${\tilde{P}}^{L} ⊑ {\tilde{P}}^{U}$ . Where $μ_{{\tilde{P}}^{U}} (z)$ and $μ_{{\tilde{P}}^{L}} (z)$ are membership functions of upper and lower approximation triangular fuzzy numbers.
Definition 7:
The $α$ -cut set of any triangular fuzzy rough number ${\tilde{P}}^{R} = [(p^{L L}, p^{M}, p^{U L}) : (p^{L U}, p^{M}, p^{U U})]$ is described as rough interval:
${\tilde{P}}^{R} (α) = [{\tilde{P}}^{L} (α) : {\tilde{P}}^{U} (α)] = [[p^{L L} (α), p^{U L} (α)] : [p^{L U} (α), p^{U U} (α)]]$
where ${\tilde{P}}^{L} (α) and {\tilde{P}}^{U} (α)$ are intervals with ${\tilde{P}}^{L} (α) ⊑ {\tilde{P}}^{U} (α)$ .
Definition 8:
Suppose ${\tilde{P}}^{R} = [{\tilde{P}}^{L} : {\tilde{P}}^{U}] = [(p^{L L}, p^{M}, p^{U L}) : (p^{L U}, p^{M}, p^{U U})]$ and ${\tilde{Q}}^{R} = [{\tilde{Q}}^{L} : {\tilde{Q}}^{U}] = [(q^{L L}, q^{M}, q^{U L}) : (q^{L U}, q^{M}, q^{U U})]$ be two non-negative triangular fuzzy rough numbers, then i
${\tilde{P}}^{R} + {\tilde{Q}}^{R} = [({\tilde{P}}^{L} + {\tilde{Q}}^{L}) : ({\tilde{P}}^{U} + {\tilde{Q}}^{U})] = [(p^{L L} + q^{L L}, p^{M} + q^{M}, p^{U L} + q^{U L}) : (p^{L U} + q^{L U}, p^{M} + q^{M}, p^{U U} + q^{U U})]$ .
ii
${\tilde{P}}^{R} - {\tilde{Q}}^{R} = [({\tilde{P}}^{L} - {\tilde{Q}}^{L}) : ({\tilde{P}}^{U} - {\tilde{Q}}^{U})] = [(p^{L L} - q^{U L}, p^{M} - q^{M}, p^{U L} - q^{L L}) : (p^{L U} - q^{U U}, p^{M} - q^{M}, p^{U U} - q^{L U})]$ .
iii
${\tilde{P}}^{R} \times {\tilde{Q}}^{R} = [({\tilde{P}}^{L} \times {\tilde{Q}}^{L}) : ({\tilde{P}}^{U} \times {\tilde{Q}}^{U})] = [(p^{L L} \times q^{L L}, p^{M} \times q^{M}, p^{U L} \times q^{U L}) : (p^{L U} \times q^{L U}, p^{M} \times q^{M}, p^{U U} \times q^{U U})]$ .
iv
${\tilde{P}}^{R} \div {\tilde{Q}}^{R} = [({\tilde{P}}^{L} \div {\tilde{Q}}^{L}) : ({\tilde{P}}^{U} \div {\tilde{Q}}^{U})] = [(p^{L L} \div q^{U L}, p^{M} \div q^{M}, p^{U L} \div q^{L L}) : (p^{L U} \div q^{U U}, p^{M} \div q^{M}, p^{U U} \div q^{L U})]$ .

Definition 9:
Suppose ${\tilde{P}}^{R} = [{\tilde{P}}^{L} : {\tilde{P}}^{U}]$ and ${\tilde{Q}}^{R} = [{\tilde{Q}}^{L} : {\tilde{Q}}^{U}]$ be two non-negative triangular fuzzy rough numbers, then
i
${\tilde{P}}^{R} \geq {\tilde{Q}}^{R},$ iff ${\tilde{P}}^{L} \geq {\tilde{Q}}^{L}$ and ${\tilde{P}}^{U} \geq {\tilde{Q}}^{U},$
ii
${\tilde{P}}^{R} \leq {\tilde{Q}}^{R},$ iff ${\tilde{P}}^{L} \leq {\tilde{Q}}^{L}$ and ${\tilde{P}}^{U} \leq {\tilde{Q}}^{U}$ ,
iii
${\tilde{P}}^{R} = {\tilde{Q}}^{R},$ iff ${\tilde{P}}^{L} = {\tilde{Q}}^{L}$ and ${\tilde{P}}^{U} = {\tilde{Q}}^{U} .$
3 Mathematical Model of the Constrained bi-matrix Games

In this section, we shall describe the classical constrained bi-matrix game problem discussed by An and Li (2019). The pure strategies sets for both players I and II are represented by $T_{1} = {ξ_{1}, ξ_{2}, \dots ., ξ_{κ}}$ and $T_{2} = {η_{1}, η_{2}, \dots ., η_{ℓ}}$ , respectively. The payoff matrices of both players are denoted as $C = (c_{i j})_{κ \times ℓ}$ and $D = (d_{i j})_{κ \times ℓ}$ , respectively. The mixed strategies are represented as the vectors $r = (r_{1}, r_{2}, \dots ., r_{κ})^{T}$ and $s = (s_{1}, s_{2}, \dots ., s_{ℓ})^{T}$ , where $r_{i} (i = 1, 2, \dots, κ)$ and $s_{j} (j = 1, 2, \dots, ℓ)$ are probabilities in which two players II and I select their pure strategies $ξ_{i} \in T_{1}$ and $η_{j} \in T_{2}$ , respectively. Let $R = {r : G^{T} r \geq p, r \geq 0}$ represent the strategy constraint set of player I, where $p = (p_{1}, p_{2}, \dots ., p_{e})^{T}$ , $G = (g_{i n})_{κ \times e}$ and e is a positive integer. Let $S = {s : H s \geq q, s \geq 0}$ express the strategy constraint set of player II, where $= (q_{1}, q_{2}, \dots ., q_{b})^{T}$ , $H = (h_{m j})_{b \times ℓ}$ and b is a positive integer. Note that $G^{T} r \geq p$ contains $\sum_{i = 1}^{κ} r_{i} = 1$ since $\sum_{i = 1}^{κ} r_{i} = 1$ is equivalent to $\sum_{i = 1}^{κ} r_{i} \geq 1$ and $- \sum_{i = 1}^{κ} r_{i} \geq - 1$ . Similarly, $H s \geq q$ contains $\sum_{j = 1}^{ℓ} s_{j} = 1$ .

Without loss of generality, suppose that the both players I and II, respectively, select mixed strategies $r \in R$ and $s \in S$ in order to maximize her/ his own payoffs, then their expected payoffs can be obtained as follows:

\begin{aligned} E_{1} (r, s, C) = r^{T} C s = \sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} c_{i j} s_{j} \end{aligned}

(1)and

\begin{aligned} E_{2} (r, s, D) = r^{T} D s = \sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} d_{i j} s_{j} \end{aligned}

(2)

Definition 10:

If $(r^{*}, s^{*}) \in R \times S$ satisfies the conditions as follows:

r^{* T} C s^{*} = \begin{array}{ll} m i n \\ s \in S \end{array} r^{* T} C s = \begin{array}{ll} m a x \\ r \in R \end{array} \begin{array}{ll} m i n \\ s \in S \end{array} r^{T} C s

and

r^{* T} D s^{*} = \begin{array}{ll} m i n \\ r \in R \end{array} r^{T} D s^{*} = \begin{array}{ll} m a x \\ s \in S \end{array} \begin{array}{ll} m i n \\ r \in R \end{array} r^{T} D s

for any mixed strategies

r \in R

and

s \in S

. Then,

r^{*}

and

s^{*}

are called optimal strategies,

V * = r^{* T} C s^{*}

and

W * = r^{* T} D s^{*}

are called equilibrium values of the both players, respectively.

Theorem 1:

If $(r^{*}, y^{*})$ and $(s^{*}, z^{*})$ are the optimal solutions of the following linear programming problems as follows:

\begin{aligned} \begin{aligned} max {q^{T} y} \\ s . t . & {\begin{array}{l} H^{T} y \leq C^{T} r \\ G^{T} r \geq p \\ r \geq 0 \\ y \geq 0 \end{array} \end{aligned} \end{aligned}

(3)and

\begin{aligned} \begin{aligned} max {p^{T} z} \\ s . t . & {\begin{cases} G z \leq D s \\ H s \geq q \\ s \geq 0 \\ z \geq 0 \end{cases} \end{aligned} \end{aligned}

(4)respectively. Then,

r^{*}

and

s^{*}

are optimal strategies,

V * = q^{T} y^{*} = r^{* T} C s^{*}

and

W * = p^{T} z^{*} = r^{* T} D s^{*}

are equilibrium values of the both players, respectively.

4 The Fuzzy Rough Constrained bi-Matrix Games and Solution Approaches

In this section, three subsections are allowed. First one describes the mathematical model of fuzzy rough constrained bi-matrix game, whereas the second provide the proposed approach for solving constrained bi-matrix game with fuzzy rough payoffs and the third introduce the solution procedure to solve such type of matrix games using ranking function method.

4.1 Constrained bi-Matrix Games with Fuzzy Rough Payoffs

Let's suppose that the constrained bi-matrix game with fuzzy rough payoffs, where the pure strategies sets $T_{1}$ and $T_{1}$ and strategies constrained sets $R$ and $S$ for both players I and II are described as previous Section 3. Without loss of generality, suppose that the payoff matrix of both players are ${\tilde{C}}^{R} = ({\tilde{c}}^{R}_{i j})_{κ \times ℓ}$ and ${\tilde{D}}^{R} = ({\tilde{d}}^{R}_{i j})_{κ \times ℓ}$ , where ${\tilde{c}}^{R}_{i j} = [({c_{i j}}^{L L}, {c_{i j}}^{M}, {c_{i j}}^{U L}) : ({c_{i j}}^{L U}, {c_{i j}}^{M}, {c_{i j}}^{U U})]$ and ${\tilde{d}}^{R}_{i j} = [({d_{i j}}^{L L}, {d_{i j}}^{M}, {d_{i j}}^{U L}) : ({d_{i j}}^{L U}, {d_{i j}}^{M}, {d_{i j}}^{U U})] (i = 1, 2, . ., κ; j = 1, 2, . ., ℓ)$ are fuzzy rough numbers. In the following, we call the constrained bi-matrix game with payoffs of fuzzy rough numbers simply as a fuzzy rough constrained bi-matrix game FRCBG.

The fuzzy rough expected payoffs of both players I and II can be obtained as follows:

E_{1} (r, s, {\tilde{C}}^{R}) = r^{T} {\tilde{C}}^{R} s = \sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} {\tilde{c}}^{R}_{i j} s_{j}

and

E_{2} (r, s, {\tilde{D}}^{R}) = r^{T} {\tilde{D}}^{R} s = \sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} {\tilde{d}}^{R}_{i j} s_{j}

which are a fuzzy rough numbers.

Computing the equilibrium strategies $r^{*}, s^{*}$ , and the equilibrium values $V *, W *$ of the fuzzy rough constrained bi-matrix games are equivalent to calculating the following two fuzzy rough programming models.

\begin{aligned} \begin{aligned} max {q^{T} y} \\ s . t . & {\begin{array}{l} H^{T} y \leq {({\tilde{C}}^{R})}^{T} r \\ G^{T} r \geq p \\ r \geq 0 \\ y \geq 0 \end{array} \end{aligned} \end{aligned}

(5)and

\begin{aligned} \begin{aligned} max {p^{T} z} \\ s . t . & {\begin{array}{l} G z \leq {\tilde{D}}^{R} s \\ H s \geq q \\ s \geq 0 \\ z \geq 0 \end{array} \end{aligned} \end{aligned}

(6)

4.2 Proposed Algorithm for Solving Fuzzy Rough Constrained bi-Matrix Game

In this section, we will discuss the linear programming algorithm and models for solving constrained bi-matrix games with fuzzy rough payoffs based on the $α$ -cut concept of fuzzy rough number. Stated as earlier, for any $α \in [0, 1]$ , $α$ -cut sets of fuzzy rough payoffs ${\tilde{C}}^{R} = ({\tilde{c}}^{R}_{i j})_{κ \times ℓ}$ and ${\tilde{D}}^{R} = ({\tilde{d}}^{R}_{i j})_{κ \times ℓ}$ are rough intervals, i.e., ${\tilde{C}}^{R} (α) = ({\tilde{c}}^{R}_{i j} (α))_{κ \times ℓ} = ([{\tilde{c}}^{L}_{i j} (α) : {\tilde{c}}^{U}_{i j} (α)])_{κ \times ℓ}$ and ${\tilde{D}}^{R} (α) = ({\tilde{d}}^{R}_{i j} (α))_{κ \times ℓ} = ([{\tilde{d}}^{L}_{i j} (α) : {\tilde{d}}^{U}_{i j} (α)])_{κ \times ℓ}$ , where ${\tilde{c}}^{R}_{i j} (α) = [[α {c_{i j}}^{M} + (1 - α) {c_{i j}}^{L L}, α {c_{i j}}^{M} + (1 - α) {c_{i j}}^{U L}] : [α {c_{i j}}^{M} + (1 - α) {c_{i j}}^{L U}, α {c_{i j}}^{M} + (1 - α) {c_{i j}}^{U U}]]$ and ${\tilde{d}}^{R}_{i j} (α) = [[α {d_{i j}}^{M} + (1 - α) {d_{i j}}^{L L}, α {d_{i j}}^{M} + (1 - α) {d_{i j}}^{U L}] : [α {d_{i j}}^{M} + (1 - α) {d_{i j}}^{L U}, α {d_{i j}}^{M} + (1 - α) {d_{i j}}^{U U}]]$ .

It is clear from equations (1) and (2) that the player I's equilibrium value V is a function of $c_{i j} (α)$ in the rough intervals payoff ${\tilde{C}}^{R}_{i j} (α)$ , represented by $V = V (c_{i j} (α))$ . Also the corresponding player I's optimal strategy $r * \in R$ is a function of $c_{i j} (α)$ , represented by $r * = r * (c_{i j} (α)) .$ Likewise, the player II's equilibrium value W and optimal strategy $s * \in S$ are functions of $d_{i j} (α)$ in the rough intervals payoff ${\tilde{D}}^{R}_{i j} (α)$ , represented by $W = W (d_{i j} (α))$ and $s * = s * (d_{i j} (α))$ , respectively.

Theorem 2:
The equilibrium values $V = V (c_{i j} (α))$ and $W = W (d_{i j} (α))$ of the two players are non-decreasing and monotonic functions of $c_{i j} (α)$ and $d_{i j} (α) (i = 1, 2, . ., κ; j = 1, 2, . ., ℓ)$ in the rough interval payoffs ${\tilde{c}}^{R}_{i j} (α)$ and ${\tilde{d}}^{R}_{i j} (α)$ , respectively.
Proof:
For any values $(c_{i j}, d_{i j})$ and $(c_{i j}^{'}, d_{i j}^{'})$ in the rough interval payoffs ${\tilde{c}}^{R}_{i j} (α)$ and ${\tilde{d}}^{R}_{i j} (α)$ . If $(c_{i j}, d_{i j}) \leq (c_{i j}^{'}, d_{i j}^{'})$ for all $i = 1, 2, . ., κ and\; j = 1, 2, . ., ℓ$ , i.e., $c_{i j} \leq c_{i j}^{'}$ and $d_{i j} \leq d_{i j}^{'}$ , then
$\begin{aligned} \sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} c_{i j} (α) s_{j} \leq \sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} c_{i j}^{'} (α) s_{j} \end{aligned}$
(7)and
$\begin{aligned} \sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} d_{i j} (α) s_{j} \leq \sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} d_{i j}^{'} (α) s_{j} \end{aligned}$
(8)since $r_{i} (i = 1, 2, \dots, κ)$ and $s_{j} (j = 1, 2, \dots, ℓ)$ , where $r \in R$ and $s \in S$ . Hence,
$\begin{aligned} \begin{array}{ll} m i n \\ s \in S \end{array} {\sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} c_{i j} (α) s_{j}} \leq \begin{array}{ll} m i n \\ s \in S \end{array} {\sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} {c^{'}}_{i j} (α) s_{j}} \end{aligned}$
(9)and
$\begin{aligned} \begin{array}{ll} min \\ r \in R \end{array} {\sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} d_{ij} (α) s_{j}} \leq \begin{array}{ll} min \\ r \in R \end{array} {\sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} {d^{'}}_{ij} (α) s_{j}} \end{aligned}$
(10)which implies that
$\begin{aligned} \begin{array}{ll} max \\ r \in R \end{array} \begin{array}{ll} min \\ s \in S \end{array} {\sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} c_{ij} (α) s_{j}} \leq \begin{array}{ll} max \\ r \in R \end{array} \begin{array}{ll} min \\ s \in S \end{array} {\sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} {c^{'}}_{ij} (α) s_{j}} \end{aligned}$
(11)and
$\begin{aligned} \begin{array}{ll} max \\ s \in S \end{array} \begin{array}{ll} min \\ r \in R \end{array} {\sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} d_{ij} (α) s_{j}} \leq \begin{array}{ll} max \\ s \in S \end{array} \begin{array}{ll} min \\ r \in R \end{array} {\sum_{i = 1}^{κ} \sum_{j = 1}^{ℓ} r_{i} {d^{'}}_{ij} (α) s_{j}} \end{aligned}$
(12)i.e.,
$\begin{aligned} V (c_{i j} (α)) \leq V ({c^{'}}_{i j} (α)) \end{aligned}$
(13)and
$\begin{aligned} W (d_{i j} (α)) \leq W ({d^{'}}_{i j} (α)) \end{aligned}$
(14)
4.2.1 Solution Process for Player I

As stated earlier, the equilibrium values of the two players in the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ should be rough intervals (Morsi & Yakout, 1998; Kumar, 2021), expressed by ${\tilde{V}}^{R} (α) = [[V^{L L} (α), V^{U L} (α)] : [V^{L U} (α), V^{U U} (α)]]$ and ${\tilde{W}}^{R} (α) = [[W^{L L} (α), W^{U L} (α)] : [W^{L U} (α), W^{U U} (α)]]$ , respectively. Then, the lower lower bound $V^{L L} (α)$ of player I's equilibrium value ${\tilde{V}}^{R} (α)$ and the equilibrium strategy $r^{* L L} (α) \in R$ in the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ are $V^{L L} (α) = V^{L L} ({c_{i j}}^{L L} (α))$ and $r^{* L L} = r^{* L L} ({c_{i j}}^{L L} (α))$ , respectively. From equation (3), $(V^{L L} (α), r^{* L L})$ can be computed from solving the linear programming problem as follows:

\begin{aligned} \begin{aligned} max {q^{T} y^{L L} (α)} \\ s . t . & {\begin{array}{ll} H^{T} y^{L L} (α) \leq {(C^{L L} (α))}^{T} r^{L L} (α) \\ G^{T} r^{L L} (α) \geq p \\ r^{L L} (α) \geq 0 \\ y^{L L} (α) \geq 0 \end{array} \end{aligned} \end{aligned}

(15)

Using the Lingo software to solve equation (15), we get the optimal solution $(y^{* L L} (α), r^{* L L} (α))$ . Then, we get the lower lower bound $V^{L L} (α) = q^{T} y^{* L L} (α)$ of player I's equilibrium value ${\tilde{V}}^{R} (α)$ and the corresponding equilibrium strategy $r^{* L L} (α)$ for the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ .

Analogously, the upper lower bound $V^{U L} (α)$ of player I's equilibrium value ${\tilde{V}}^{R} (α)$ and the equilibrium strategy $r^{* U L} (α) \in R$ in the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ are $V^{U L} (α) = V^{U L} ({c_{i j}}^{U L} (α))$ and $r^{* U L} = r^{* U L} ({c_{i j}}^{U L} (α))$ , respectively. From equation (3), $(V^{U L} (α), r^{* U L})$ can be computed from solving the linear programming problem as follows:

\begin{aligned} \begin{aligned} max {q^{T} y^{U L} (α)} \\ s . t . & {\begin{array}{l} H^{T} y^{U L} (α) \leq {(C^{U L} (α))}^{T} r^{U L} (α) \\ G^{T} r^{U L} (α) \geq p \\ r^{U L} (α) \geq 0 \\ y^{U L} (α) \geq 0 \end{array} \end{aligned} \end{aligned}

(16)

Using the lingo software to solve equation (16), we get the optimal solution $(y^{* U L} (α), r^{* U L} (α))$ . Then, we get the upper lower bound $V^{U L} (α) = q^{T} y^{* U L} (α)$ of player I's equilibrium value ${\tilde{V}}^{R} (α)$ and the corresponding equilibrium strategy $r^{* U L} (α)$ for the $rough\; interval$ constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ .

Similarly, the lower upper bound $V^{L U} (α)$ of player I's equilibrium value ${\tilde{V}}^{R} (α)$ and the equilibrium strategy $r^{* L U} (α) \in R$ in the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ are $V^{L U} (α) = V^{L U} ({c_{i j}}^{L U} (α))$ and $r^{* L U} = r^{* L U} ({c_{i j}}^{L U} (α))$ , respectively. From equation (3), $(V^{L U} (α), r^{* L U})$ can be computed from solving the linear programming problem as follows:

\begin{aligned} \begin{aligned} max {q^{T} y^{L U} (α)} \\ s . t . & {\begin{array}{l} H^{T} y^{L U} (α) \leq {(C^{L U} (α))}^{T} r^{L U} (α) \\ G^{T} r^{L U} (α) \geq p \\ r^{L U} (α) \geq 0 \\ y^{L U} (α) \geq 0 \end{array} \end{aligned} \end{aligned}

(17)

Using the lingo software to solve equation (17), we get the optimal solution $(y^{* L U} (α), r^{* L U} (α))$ . Then, we get the lower upper bound $V^{L U} (α) = q^{T} y^{* L U} (α)$ of player I's equilibrium value ${\tilde{V}}^{R} (α)$ and the corresponding equilibrium strategy $r^{* L U} (α)$ for the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ .

Also, the upper upper bound $V^{U U} (α)$ of player I's equilibrium value ${\tilde{V}}^{R} (α)$ and the equilibrium strategy $r^{* U U} (α) \in R$ in the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ are $V^{U U} (α) = V^{U U} ({c_{i j}}^{U U} (α))$ and $r^{* U U} = r^{* U U} ({c_{i j}}^{U U} (α))$ , respectively. From equation (3), $(V^{U U} (α), r^{* U U})$ can be computed from solving the linear programming problem as follows:

\begin{aligned} \begin{aligned} max {q^{T} y^{U U} (α)} \\ s . t . & {\begin{array}{l} H^{T} y^{U U} (α) \leq {(C^{U U} (α))}^{T} r^{U U} (α) \\ G^{T} r^{U U} (α) \geq p \\ r^{U U} (α) \geq 0 \\ y^{U U} (α) \geq 0 \end{array} \end{aligned} \end{aligned}

(18)

Using the lingo software to solve equation (18), we get the optimal solution $(y^{* U U} (α), r^{* U U} (α))$ . Then, we get the upper upper bound $V^{U U} (α) = q^{T} y^{* U U} (α)$ of player I's equilibrium value ${\tilde{V}}^{R} (α)$ and the corresponding equilibrium strategy $r^{* U U} (α)$ for the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ . Thus, we obtain the lower lower bound $V^{L L} (α)$ , the upper lower bound $V^{U L} (α)$ , the lower upper bound $V^{L U} (α)$ and the upper upper bound $V^{U U} (α)$ of player I's rough interval type value. Therefore, the player I's equilibrium value of the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ is a rough interval ${\tilde{V}}^{R} (α) = [[V^{L L} (α), V^{U L} (α)] : [V^{L U} (α), V^{U U} (α)]]$ .

4.2.2 Solution Process for Player II

In a similar way, the lower lower bound $W^{L L} (α)$ of player II's equilibrium value ${\tilde{W}}^{R} (α)$ and the equilibrium strategy $s^{* L L} (α) \in S$ in the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ are $W^{L L} (α) = W^{L L} ({d_{i j}}^{L L} (α))$ and $s^{* L L} = s^{* L L} ({d_{i j}}^{L L} (α))$ , respectively. From equation (4), $(w^{L L} (α), s^{* L L})$ can be computed from solving the linear programming problem as follows:

\begin{aligned} \begin{aligned} max {p^{T} z^{L L} (α)} \\ s . t . & {\begin{array}{l} G z \leq D^{L L} (α) s^{L L} (α) \\ H s^{L L} (α) \geq q \\ s^{L L} (α) \geq 0 \\ z^{L L} (α) \geq 0 \end{array} \end{aligned} \end{aligned}

(19)

Using the lingo software to solve equation (19), we get the optimal solution $(z^{* L L} (α), s^{* L L} (α))$ . Then, we get the lower lower bound $W^{L L} (α) = p^{T} z^{* L L} (α)$ of player II's equilibrium value ${\tilde{W}}^{R} (α)$ and the equilibrium strategy $s^{* L L} (α)$ for the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ .

Likewise, the upper lower bound $W^{U L} (α)$ of player II's equilibrium value ${\tilde{W}}^{R} (α)$ and the equilibrium strategy $s^{* U L} (α) \in S$ in the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ are $W^{U L} (α) = W^{U L} ({d_{i j}}^{U L} (α))$ and $s^{* U L} = s^{* U L} ({d_{i j}}^{U L} (α)$ , respectively. From equation (4), $(w^{U L} (α), s^{* U L})$ can be computed from solving the linear programming problem as follows:

\begin{aligned} \begin{aligned} max {p^{T} z^{U L} (α)} \\ s . t . & {\begin{array}{l} G z \leq D^{U L} (α) s^{U L} (α) \\ H s^{U L} (α) \geq q \\ s^{U L} (α) \geq 0 \\ z^{U L} (α) \geq 0 \end{array} \end{aligned} \end{aligned}

(20)

Using the lingo software to solve equation (20), we get the optimal solution $(z^{* U L} (α), s^{* U L} (α))$ . Then, we get the upper lower bound $W^{U L} (α) = p^{T} z^{* U L} (α)$ of player II's equilibrium value ${\tilde{W}}^{R} (α)$ and the equilibrium strategy $s^{* U L} (α)$ for the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ .

Similarly, the lower upper bound $W^{L U} (α)$ of player II's equilibrium value ${\tilde{W}}^{R} (α)$ and the equilibrium strategy $s^{* L U} (α) \in S$ in the rough interval constraint bi-matrix game are $W^{L U} (α) = W^{L U} ({d_{i j}}^{L U} (α))$ and $s^{* L U} = s^{* L U} ({d_{i j}}^{L U} (α))$ , respectively. From equation (4), $(w^{L U} (α), s^{* L U})$ can be computed from solving the linear programming problem as follows:

\begin{aligned} \begin{aligned} max {p^{T} z^{L U} (α)} \\ s . t . & {\begin{array}{ll} G z \leq D^{L U} (α) s^{L U} (α) \\ H s^{L U} (α) \geq q \\ s^{L U} (α) \geq 0 \\ z^{L U} (α) \geq 0 \end{array} \end{aligned} \end{aligned}

(21)

Using the lingo software to solve equation (21), we get the optimal solution $(z^{* L U} (α), s^{* L U} (α))$ . Then, we get the lower upper bound $W^{L U} (α) = p^{T} z^{* L U} (α)$ of player II's equilibrium value ${\tilde{W}}^{R} (α)$ and the equilibrium strategy $s^{* L U} (α)$ for the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ .

Also, the upper upper bound $W^{U U} (α)$ of player II's equilibrium value ${\tilde{W}}^{R} (α)$ and the equilibrium strategy $s^{* U U} (α) \in S$ in the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ are $W^{U U} (α) = W^{U U} ({d_{i j}}^{U U} (α))$ and $s^{* U U} = s^{* U U} ({d_{i j}}^{U U} (α)$ , respectively. From equation (4), $(w^{U U} (α), s^{* U U})$ can be computed from solving the linear programming problem as follows:

\begin{aligned} \begin{aligned} max {p^{T} z^{U U} (α)} \\ s . t . & {\begin{array}{l} G z \leq D^{U U} (α) s^{U U} (α) \\ H s^{U U} (α) \geq q \\ s^{U U} (α) \geq 0 \\ z^{U U} (α) \geq 0 \end{array} \end{aligned} \end{aligned}

(22)

Using the lingo software to solve equation (22), we get the optimal solution $(z^{* U U} (α), s^{* U U} (α))$ . Then, we get the upper upper bound $W^{U U} (α) = p^{T} z^{* U U} (α)$ of player II's equilibrium value ${\tilde{W}}^{R} (α)$ and the equilibrium strategy $s^{* U U} (α)$ for the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ .Thus, we obtain the lower lower bound $W^{L L} (α)$ , the upper lower bound $W^{U L} (α)$ , the lower upper bound $W^{L U} (α)$ and the upper upper bound $W^{U U} (α)$ of player II's rough interval type value. Therefore, the player II's equilibrium value of the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ is a rough interval ${\tilde{W}}^{R} (α) = [[W^{L L} (α), W^{U L} (α)] : [W^{L U} (α), W^{U U} (α)]]$ .

Theorem 3:

For any $α \in [0, 1]$ , the rough interval constraint bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ has rough intervals equilibrium value for the two players expressed as ${\tilde{V}}^{R} (α) = [[V^{L L} (α), V^{U L} (α)] : [V^{L U} (α), V^{U U} (α)]]$ and ${\tilde{W}}^{R} (α) = [[W^{L L} (α), W^{U L} (α)] : [W^{L U} (α), W^{U U} (α)]]$ , whose lower lower, upper lower, lower upper and upper upper bounds can be obtained by solving equations (15)–(18) and equations (19)–(22), respectively.

In particular, for $α = 1$ , from equations (15), (16), (17) and (18) or equations (19), (20), (21) and (22), the linear programming models are formulated as follows:

\begin{aligned} \begin{aligned} max {q^{T} y^{L L} (1)} \\ s . t . & {\begin{array}{l} H^{T} y^{L L} (1) \leq {(C^{L L} (1))}^{T} r^{L L} (1) \\ G^{T} r^{L L} (1) \geq p \\ r^{L L} (1) \geq 0 \\ y^{L L} (1) \geq 0 \end{array} \end{aligned} \end{aligned}

(23)

\begin{aligned} \begin{aligned} max {q^{T} y^{U L} (1)} \\ s . t . & {\begin{array}{l} H^{T} y^{U L} (1) \leq {(C^{U L} (1))}^{T} r^{U L} (1) \\ G^{T} r^{U L} (1) \geq p \\ r^{U L} (1) \geq 0 \\ y^{U L} (1) \geq 0 \end{array} \end{aligned} \end{aligned}

(24)

\begin{aligned} \begin{aligned} max {q^{T} y^{L U} (1)} \\ s . t . & {\begin{array}{l} H^{T} y^{L U} (1) \leq {(C^{L U} (1))}^{T} r^{L U} (1) \\ G^{T} r^{L U} (1) \geq p \\ r^{L U} (1) \geq 0 \\ y^{L U} (1) \geq 0 \end{array} \end{aligned} \end{aligned}

(25)

and

\begin{aligned} \begin{aligned} max {q^{T} y^{U U} (1)} \\ s . t . & {\begin{array}{l} H^{T} y^{U U} (1) \leq {(C^{U U} (1))}^{T} r^{U U} (1) \\ G^{T} r^{U U} (1) \geq p \\ r^{U U} (1) \geq 0 \\ y^{U U} (1) \geq 0 \end{array} \end{aligned} \end{aligned}

(26)or

\begin{aligned} \begin{aligned} max {p^{T} z^{L L} (1)} \\ s . t . & {\begin{array}{l} G z \leq D^{L L} (1) s^{L L} (1) \\ H s^{L L} (1) \geq q \\ s^{L L} (1) \geq 0 \\ z^{L L} (1) \geq 0 \end{array} \end{aligned} \end{aligned}

(27)

\begin{aligned} \begin{aligned} max {p^{T} z^{U L} (1)} \\ s . t . & {\begin{array}{l} G z \leq D^{U L} (1) s^{U L} (1) \\ H s^{U L} (1) \geq q \\ s^{U L} (1) \geq 0 \\ z^{U L} (1) \geq 0 \end{array} \end{aligned} \end{aligned}

(28)

\begin{aligned} \begin{aligned} max {p^{T} z^{L U} (1)} \\ s . t . & {\begin{array}{l} G z \leq D^{L U} (1) s^{L U} (1) \\ H s^{L U} (1) \geq q \\ s^{L U} (1) \geq 0 \\ z^{L U} (1) \geq 0 \end{array} \end{aligned} \end{aligned}

(29)and

\begin{aligned} \begin{aligned} max {p^{T} z^{U U} (1)} \\ s . t . & {\begin{array}{ll} G z \leq D^{U U} (1) s^{U U} (1) \\ H s^{U U} (1) \geq q \\ s^{U U} (1) \geq 0 \\ z^{U U} (1) \geq 0 \end{array} \end{aligned} \end{aligned}

(30)respectively.

Using the Simplex technique, we can obtain the mean of player I's fuzzy rough equilibrium value ${\tilde{V}}^{R}$ by solving one of equations (23)–(26) and the mean of player II's fuzzy rough equilibrium value ${\tilde{W}}^{R}$ also can be obtained by solving one of equations (24)–(30).

For $α = 0$ , from equations (15), (16), (17) and (18) or equations (19), (20), (21) and (22), the linear programming models are formulated as follows:

\begin{aligned} \begin{aligned} max {q^{T} y^{U L} (0)} \\ s . t & {\begin{array}{l} H^{T} y^{U L} (0) \leq {(C^{U L} (0))}^{T} r^{U L} (0) \\ G^{T} r^{U L} (0) \geq p \\ r^{U L} (0) \geq 0 \\ y^{U L} (0) \geq 0 \end{array} \end{aligned} \end{aligned}

(31)

\begin{aligned} \begin{aligned} max {q^{T} y^{L U} (0)} \\ s . t . & {\begin{array}{l} H^{T} y^{L U} (0) \leq {(C^{L U} (0))}^{T} r^{L U} (0) \\ G^{T} r^{L U} (0) \geq p \\ r^{L U} (0) \geq 0 \\ y^{L U} (0) \geq 0 \end{array} \end{aligned} \end{aligned}

(32)

\begin{aligned} \begin{aligned} max {q^{T} y^{L U} (0)} \\ s . t . & {\begin{array}{l} H^{T} y^{L U} (0) \leq {(C^{L U} (0))}^{T} r^{L U} (0) \\ G^{T} r^{L U} (0) \geq p \\ r^{L U} (0) \geq 0 \\ y^{L U} (0) \geq 0 \end{array} \end{aligned} \end{aligned}

(33)

and

\begin{aligned} \begin{aligned} max {q^{T} y^{U U} (0)} \\ s . t . & {\begin{array}{l} H^{T} y^{U U} (0) \leq {(C^{U U} (0))}^{T} r^{U U} (0) \\ G^{T} r^{U U} (0) \geq p \\ r^{U U} (0) \geq 0 \\ y^{U U} (0) \geq 0 \end{array} \end{aligned} \end{aligned}

(34)or

\begin{aligned} \begin{aligned} max {p^{T} z^{L L} (0)} \\ s . t . & {\begin{array}{l} G z \leq D^{L L} (0) s^{L L} (0) \\ H s^{L L} (0) \geq q \\ s^{L L} (0) \geq 0 \\ z^{L L} (0) \geq 0 \end{array} \end{aligned} \end{aligned}

(35)

\begin{aligned} \begin{aligned} max {p^{T} z^{U L} (0)} \\ s . t . & {\begin{array}{l} G z \leq D^{U L} (0) s^{U L} (0) \\ H s^{U L} (0) \geq q \\ s^{U L} (0) \geq 0 \\ z^{U L} (0) \geq 0 \end{array} \end{aligned} \end{aligned}

(36)

\begin{aligned} \begin{aligned} max {p^{T} z^{L U} (0)} \\ s . t . & {\begin{array}{l} G z \leq D^{L U} (0) s^{L U} (0) \\ H s^{L U} (0) \geq q \\ s^{L U} (0) \geq 0 \\ z^{L U} (0) \geq 0 \end{array} \end{aligned} \end{aligned}

(37)

and

\begin{aligned} \begin{aligned} max {p^{T} z^{U U} (0)} \\ s . t . & {\begin{array}{l} G z \leq D^{U U} (0) s^{U U} (0) \\ H s^{U U} (0) \geq q \\ s^{U U} (0) \geq 0 \\ z^{U U} (0) \geq 0 \end{array} \end{aligned} \end{aligned}

(38)respectively.

Using the Simplex technique, we can obtain the lower and the upper limits of player I's fuzzy rough equilibrium value ${\tilde{V}}^{R}$ by solving equations (31)–(34) and the lower and the upper limits of player II's fuzzy rough equilibrium value ${\tilde{W}}^{R}$ also can be obtained by solving equations (35)–(38).

Theorem 4:

The fuzzy rough interval constraint bi-matrix game $({\tilde{C}}^{R}, {\tilde{D}}^{R})$ always has fuzzy rough equilibrium values.

Proof:

The proof follows a similar structure to that of Theorem 5 in Li and Hong (2012) and is therefore omitted for brevity.

Algorithmic Framework

Based on the preceding discussion, the procedure for solving a constrained bi-matrix game with fuzzy rough payoffs can be systematically outlined as follows:

Step 1. Define the decision-making agents, referred to as Player I and Player II.

Step 2. Specify the sets of pure strategies available to Player I and Player II, denoted by $T_{1} = ξ_{1}, ξ_{2}, \dots ., ξ_{κ}$ and $T_{2} = η_{1}, η_{2}, \dots ., η_{ℓ}$ , respectively.

Step 3. Determine the sets of feasible (constrained) strategies for both players, represented by R for Player I and S for Player II.

Step 4. Formulate and solve the linear programming models as defined in Equations (15)–(18) to compute the equilibrium value ${\tilde{V}}^{R} (α)$ and the corresponding equilibrium strategies for Player I in the fuzzy rough interval-constrained bi-matrix game $({\tilde{C}}^{R} (α), {\tilde{D}}^{R} (α))$ , where $α \in [0, 1]$ .

Step 5. Develop and solve the linear programming models given in Equations (19)–(22) to derive the equilibrium value ${\tilde{W}}^{R} (α)$ and associated equilibrium strategies for Player II in the same game structure.

Step 6. Construct and solve the appropriate linear programming model using one of Equations (23)–(26) to determine $V^{M}$ , the refined game value for Player I.

Step 7. Similarly, formulate and solve a linear programming model based on one of Equations (27)–(30) to compute $W^{M}$ , the refined game value for Player II.

Step 8. Establish and solve four linear programming models corresponding to Equations (31)–(34) to obtain the lower and upper bounds of the equilibrium values for Player I, namely $V^{L L}$ , $V^{U L}$ , $V^{L U}$ and $V^{U U}$ , respectively.

Step 9. Similarly, construct and solve four linear programming models based on Equations (35)–(38) to derive the corresponding bounds for Player II, denoted as $W^{L L}$ , $W^{U L}$ , $W^{L U}$ and $W^{U U}$ , respectively.

Step 10. Finally, integrate the results to determine the fuzzy rough game value and the optimal constrained strategies for both players within the fuzzy rough bi-matrix game framework.

5 Numerical Application

In practical scenarios, decision makers (DMs), such as governmental bodies and corporations, often rely on multiple, and occasionally conflicting, sources of information. Much of this information is derived from secondary data, which may lack precision and introduce ambiguity into the decision-making process. Moreover, the data involved in strategic environmental assessments is frequently characterized by vagueness and uncertainty. In such contexts, fuzzy rough sets offer a robust and appropriate tool for modeling imprecise, incomplete, and vague information. Applying fuzzy rough set theory within a constrained bi-matrix game framework allows for more realistic modeling of the strategic interactions between the government and corporations. This approach is particularly useful in analyzing complex decisions related to environmental regulation and compliance, where both players operate under uncertainty and limited information. The proposed model demonstrates how fuzzy rough environments can be effectively utilized to support decision-making in corporate environmental behavior.

5.1 Description of Strategy Choice Problem in Corporate Environmental Behavior

With the accelerating pace of industrial development, environmental challenges have intensified significantly. In pursuit of economic gains, some corporations compromise environmental protection efforts, leading to deteriorating ecological conditions. Within this context, environmental regulation emerges as a critical area where strategic interaction takes place between governmental authorities and corporations.

These two agents act as decision-makers in a non-cooperative game. The government seeks to maximize overall social and ecological welfare, while the corporation aims to maximize its own economic interests. Their conflicting objectives, constrained by regulatory and financial limitations, create a suitable environment for modeling via fuzzy rough constrained bi-matrix games, where uncertainty and vagueness in payoffs are naturally handled through fuzzy rough theory.

Environmental governance is influenced by various strategic choices:

Government strategies:

$a_{1}$ : Strict environmental supervision .

$a_{2}$ : Neutral stance (status quo).

$a_{3}$ : Lenient or passive regulation .

Corporate strategies:

$b_{1}$ : Proactive adoption of environmentally responsible practices.

$b_{1}$ : Maintain existing environmental policies .

$b_{1}$ : Neglect or downgrade environmental standards.

When the government enforces stringent policies, ecological welfare improves, although this often comes with economic trade-offs. Conversely, relaxed regulation may lead to a decrease in environmental performance, government penalties, and reduced public trust. Similarly, if a corporation avoids implementing sustainable practices, it risks reputational damage, regulatory penalties, and reduced competitiveness in green markets, hindering high-quality economic development. To account for the uncertainty and boundary vagueness in evaluating social and economic outcomes, we formulate the problem as a constrained bi-matrix game with fuzzy rough payoffs. This approach enables a more nuanced representation of players’ strategies under incomplete information and external limitations.

Let Player $P_{1}$ (government) and Player $P_{2}$ (corporation) select their mixed strategies from constrained sets:

Government's constrained strategy set:

R = {r | 80 r_{1} + 60 r_{2} \leq 75, r_{1} + r_{2} = 1, r_{1} \geq 0, r_{2} \geq 0},

Corporation's constrained strategy set:

S = {s | 50 s_{1} + 75 s_{2} \leq 55, s_{1} + s_{2} = 1, s_{1} \geq 0, s_{2} \geq 0},

These constraints may reflect resource availability, legal boundaries, or policy limitations.

Let the payoff matrices under fuzzy rough conditions be:

Government's fuzzy rough payoff matrix $C^{R}$ :

{\tilde{C}}^{R} = (\begin{array}{ll} [(60, 70, 90) : (40, 70, 110)] & [(35, 50, 70) : (20, 50, 80)] \\ [(30, 50, 60) : (10, 50, 90)] & [(50, 65, 80) : (40, 65, 100)] \end{array}),

Corporation's fuzzy rough payoff matrix $D^{R}$ :

{\tilde{D}}^{R} = (\begin{array}{ll} [(50, 65, 80) : (30, 65, 105)] & [(40, 50, 70) : (25, 50, 120)] \\ [(35, 45, 60) : (30, 45, 75)] & [(60, 70, 80) : (45, 70, 95)] \end{array}),

The constraints on the strategy vectors can be compactly formulated using matrix-vector notation:

Government:

G = (\begin{array}{lll} - 80 & 1 & - 1 \\ - 60 & 1 & - 1 \end{array}), p = {(\begin{array}{lll} - 75 & 1 & - 1 \end{array})}^{T},

Corporation:

H^{T} = (\begin{array}{lll} - 50 & 1 & - 1 \\ - 75 & 1 & - 1 \end{array}), q = {(\begin{array}{lll} - 55 & 1 & - 1 \end{array})}^{T} .

5.2 The Solution Procedure by the Proposed Approach

According to equation (23), the linear programming problem is formulated as follows:

\begin{aligned} \begin{aligned} m a x {- 65 {y_{1}}^{L L} (1) + {y_{2}}^{L L} (1) - {y_{3}}^{L L} (1)} \\ s . t . & {\begin{array}{l} - 50 {y_{1}}^{L L} (1) + {y_{2}}^{L L} (1) - {y_{3}}^{L L} (1) - 70 {r_{1}}^{L L} (1) - 50 {r_{2}}^{L L} (1) \leq 0 \\ - 75 {y_{1}}^{L L} (1) + {y_{2}}^{L L} (1) - {y_{3}}^{L L} (1) - 50 {r_{1}}^{L L} (1) - 65 {r_{2}}^{L L} (1) \leq 0 \\ 80 {r_{1}}^{L L} (1) + 60 {r_{2}}^{L L} (1) \leq 75 \\ {r_{1}}^{L L} (1) + {r_{2}}^{L L} (1) = 1 \\ {y_{1}}^{L L} (1), {y_{2}}^{L L} (1), {y_{3}}^{L L} (1), {r_{1}}^{L L} (1), {r_{2}}^{L L} (1) \geq 0. \end{array} \end{aligned} \end{aligned}

(42)

Using the Simplex technique, we can compute the optimal solution $(y^{* L L} (1), r^{* L L} (1))$ of equation (42), where $y^{* L L} (1) = (0, 58.57, 0)$ and $r^{* L L} (1) = (0.43, 0.57)$ . Then, the mean of the equilibrium value and the equilibrium strategy for player I are $V^{M} = V^{L L} (1) = q^{T} y^{* L L} (1) = 58.57$ and $r^{* M} = r^{* L L} (1) = (0.43, 0.57)$ , respectively.

According to equation (31), the linear programming problem is formulated as follows:

\begin{aligned} \begin{aligned} m a x {- 65 {y_{1}}^{L L} (0) + {y_{2}}^{L L} (0) - {y_{3}}^{L L} (0)} \\ s . t . & {\begin{array}{l} - 50 {y_{1}}^{L L} (0) + {y_{2}}^{L L} (0) - {y_{3}}^{L L} (0) - 60 {r_{1}}^{L L} (0) - 30 {r_{2}}^{L L} (0) \leq 0 \\ - 75 {y_{1}}^{L L} (0) + {y_{2}}^{L L} (0) - {y_{3}}^{L L} (0) - 35 {r_{1}}^{L L} (0) - 50 {r_{2}}^{L L} (0) \leq 0 \\ 80 {r_{1}}^{L L} (0) + 60 {r_{2}}^{L L} (0) \leq 7.5 \\ {r_{1}}^{L L} (0) + {r_{2}}^{L L} (0) = 0 \\ {y_{1}}^{L L} (0), {y_{2}}^{L L} (0), {y_{3}}^{L L} (0), {r_{1}}^{L L} (0), {r_{2}}^{L L} (0) \geq 0. \end{array} \end{aligned} \end{aligned}

(43)

Using the Simplex technique, we can compute the optimal solution $(y^{* L L} (0), r^{* L L} (0))$ of equation (43), where $y^{* L L} (0) = (0.55, 80, 0)$ and $r^{* L L} (0) = (0.75, 0.25)$ . Then, the lower lower limit of the equilibrium value and the equilibrium strategy for player I are $V^{L L} = V^{L L} (0) = q^{T} y^{* L L} (0) = 44.25$ and $r^{* L L} = r^{* L L} (0) = (0.75, 0.25)$ , respectively.

According to equation (32), the linear programming problem is formulated as follows:

\begin{aligned} \begin{aligned} m a x {- 65 {y_{1}}^{U L} (0) + {y_{2}}^{U L} (0) - {y_{3}}^{U L} (0)} \\ s . t . & {\begin{array}{l} - 50 {y_{1}}^{U L} (0) + {y_{2}}^{U L} (0) - {y_{3}}^{U L} (0) - 90 {r_{1}}^{U L} (0) - 60 {r_{2}}^{U L} (0) \leq 0 \\ - 75 {y_{1}}^{U L} (0) + {y_{2}}^{U L} (0) - {y_{3}}^{U L} (0) - 70 {r_{1}}^{U L} (0) - 80 {r_{2}}^{U L} (0) \leq 0 \\ 80 {r_{1}}^{U L} (0) + 60 {r_{2}}^{U L} (0) \leq 75 \\ {r_{1}}^{U L} (0) + {r_{2}}^{U L} (0) = 0 \\ {y_{1}}^{U L} (0), {y_{2}}^{U L} (0), {y_{3}}^{U L} (0), {r_{1}}^{U L} (0), {r_{2}}^{U L} (0) \geq 0. \end{array} \end{aligned} \end{aligned}

(44)

Using the Simplex technique, we can compute the optimal solution $(y^{* U L} (0), r^{* U L} (0))$ of equation (44), where $y^{* U L} (0) = (0.4, 102.5, 0)$ and $r^{* U L} (0) = (0.75, 0.25)$ . Then, the upper lower limit of the equilibrium value and the equilibrium strategy for player I are $V^{U L} = V^{U L} (0) = q^{T} y^{* U L} (0) = 76.5$ and $r^{* U L} = r^{* U L} (0) = (0.75, 0.25)$ , respectively.

According to equation (33), the linear programming problem is formulated as follows:

\begin{aligned} \begin{aligned} max {- 65 {y_{1}}^{L U} (0) + {y_{2}}^{L U} (0) - {y_{3}}^{L U} (0)} \\ s . t . & {\begin{array}{l} - 50 {y_{1}}^{L U} (0) + {y_{2}}^{L U} (0) - {y_{3}}^{L U} (0) - 40 {r_{1}}^{L U} (0) - 10 {r_{2}}^{L U} (0) \leq 0 \\ - 75 {y_{1}}^{L U} (0) + {y_{2}}^{L U} (0) - {y_{3}}^{L U} (0) - 20 {r_{1}}^{L U} (0) - 40 {r_{2}}^{L U} (0) \leq 0 \\ 80 {r_{1}}^{L U} (0) + 60 {r_{2}}^{L U} (0) \leq 75 \\ {r_{1}}^{L U} (0) + {r_{2}}^{L U} (0) = 1 \\ {y_{1}}^{L U} (0), {y_{2}}^{L U} (0), {y_{3}}^{L U} (0), {r_{1}}^{L U} (0), {r_{2}}^{L U} (0) \geq 0. \end{array} \end{aligned} \end{aligned}

(45)

Using the Simplex technique, we can compute the optimal solution $(y^{* L U} (0), r^{* L U} (0))$ of equation (45), where $y^{* L U} (0) = (0, 28, 0)$ and $r^{* L U} (0) = (0.6, 0.4)$ . Then, the lower upper limit of the equilibrium value and the equilibrium strategy for player I are $V^{L U} = V^{L U} (0) = q^{T} y^{* L U} (0) = 28$ and $r^{* L U} = r^{* L U} (0) = (0.6, 0.4)$ , respectively.

According to equation (34), the linear programming problem is formulated as follows:

\begin{aligned} \begin{aligned} m a x {- 65 {y_{1}}^{U U} (0) + {y_{2}}^{U U} (0) - {y_{3}}^{U U} (0)} \\ s . t . & {\begin{array}{l} - 50 {y_{1}}^{U U} (0) + {y_{2}}^{U U} (0) - {y_{3}}^{U U} (0) - 110 {r_{1}}^{U U} (0) - 90 {r_{2}}^{U U} (0) \leq 0 \\ - 75 {y_{1}}^{U U} (0) + {y_{2}}^{U U} (0) - {y_{3}}^{U U} (0) - 80 {r_{1}}^{U U} (0) - 100 {r_{2}}^{U U} (0) \leq 0 \\ 80 {r_{1}}^{U U} (0) + 60 {r_{2}}^{U U} (0) \leq 75 \\ {r_{1}}^{U U} (0) + {r_{2}}^{U U} (0) = 1 \\ {y_{1}}^{U U} (0), {y_{2}}^{U U} (0), {y_{3}}^{U U} (0), {r_{1}}^{U U} (0), {r_{2}}^{U U} (0) \geq 0. \end{array} \end{aligned} \end{aligned}

(46)

Using the Simplex technique, we can compute the optimal solution $(y^{* U U} (0), r^{* U U} (0))$ of equation (46), where $y^{* U U} (0) = (0, 95, 0)$ and $r^{* U U} (0) = (0.25, 0.75)$ . Then, the upper upper limit of the equilibrium value and the equilibrium strategy for player I are $V^{U U} = V^{U U} (0) = q^{T} y^{* U U} (0) = 95$ and $r^{* U U} = r^{* U U} (0) = (0.25, 0.75)$ , respectively. Therefore, the fuzzy rough equilibrium value for player I is $[(44.25, 58.57, 76.5) : (28, 58.57, 95)]$ .

According to equation (27), the linear programming problem is formulated as follows:

\begin{aligned} \begin{aligned} m a x {- 75 {z_{1}}^{L L} (1) + {z_{2}}^{L L} (1) - {z_{3}}^{L L} (1)} \\ s . t . & {\begin{array}{l} - 80 {z_{1}}^{L L} (1) + {z_{2}}^{L L} (1) - {z_{3}}^{L L} (1) - 65 {s_{1}}^{L L} (1) - 50 {s_{2}}^{L L} (1) \leq 0 \\ - 60 {z_{1}}^{L L} (1) + {z_{2}}^{L L} (1) - {z_{3}}^{L L} (1) - 45 {s_{1}}^{L L} (1) - 70 {s_{2}}^{L L} (1) \leq 0 \\ 50 {s_{1}}^{L L} (1) + 75 {s_{2}}^{L L} (1) \leq 65 \\ {s_{1}}^{L L} (1) + {s_{2}}^{L L} (1) = 1 \\ {z_{1}}^{L L} (1), {z_{2}}^{L L} (1), {z_{3}}^{L L} (1), {s_{1}}^{L L} (1), {s_{2}}^{L L} (1) \geq 0. \end{array} \end{aligned} \end{aligned}

(47)

Using the Simplex technique, we can compute the optimal solution $(z^{* L L} (1), s^{* L L} (1))$ of equation (47), where $z^{* L L} (1) = (0, 57.5, 0)$ and $s^{* L L} (1) = (0.5, 0.5)$ . Then, the mean of the equilibrium value and the equilibrium strategy for player II are $W^{M} = W^{L L} (1) = p^{T} z^{* L L} (1) = 57.5$ and $s^{* M} = s^{* L L} (1) = (0.5, 0.5)$ , respectively.

According to equation (35), the linear programming problem is formulated as follows:

\begin{aligned} \begin{aligned} m a x {- 75 {z_{1}}^{L L} (0) + {z_{2}}^{L L} (0) - {z_{3}}^{L L} (0)} \\ s . t . & {\begin{array}{l} - 80 {z_{1}}^{L L} (0) + {z_{2}}^{L L} (0) - {z_{3}}^{L L} (0) - 50 {s_{1}}^{L L} (0) - 40 {s_{2}}^{L L} (0) \leq 0 \\ - 60 {z_{1}}^{L L} (0) + {z_{2}}^{L L} (0) - {z_{3}}^{L L} (0) - 35 {s_{1}}^{L L} (0) - 60 {s_{2}}^{L L} (0) \leq 0 \\ 50 {s_{1}}^{L L} (0) + 75 {s_{2}}^{L L} (0) \leq 65 \\ {s_{1}}^{L L} (0) + {s_{2}}^{L L} (0) = 1 \\ {z_{1}}^{L L} (0), {z_{2}}^{L L} (0), {z_{3}}^{L L} (0), {s_{1}}^{L L} (0), {s_{2}}^{L L} (0) \geq 0. \end{array} \end{aligned} \end{aligned}

(48)

Using the Simplex technique, we can compute the optimal solution $(z^{* L L} (0), s^{* L L} (0))$ of equation (48), where $z^{* L L} (0) = (0, 45.71, 0)$ and $s^{* L L} (0) = (0.57, 0.43)$ . Then, the lower lower limit of the equilibrium value and the equilibrium strategy for player II are $W^{L L} = W^{L L} (0) = p^{T} z^{* L L} (0) = 45.71$ and $s^{* L L} = s^{* L L} (0) = (0.57, 0.43)$ , respectively.

According to equation (36), the linear programming problem is formulated as follows:

\begin{aligned} \begin{aligned} max {- 75 {z_{1}}^{U L} (0) + {z_{2}}^{U L} (0) - {z_{3}}^{U L} (0)} \\ s . t . & {\begin{array}{l} - 80 {z_{1}}^{U L} (0) + {z_{2}}^{U L} (0) - {z_{3}}^{U L} (0) - 80 {s_{1}}^{U L} (0) - 70 {s_{2}}^{U L} (0) \leq 0 \\ - 60 {z_{1}}^{U L} (0) + {z_{2}}^{U L} (0) - {z_{3}}^{U L} (0) - 60 {s_{1}}^{U L} (0) - 80 {s_{2}}^{U L} (0) \leq 0 \\ 50 {s_{1}}^{U L} (0) + 75 {s_{2}}^{U L} (0) \leq 65 \\ {s_{1}}^{U L} (0) + {s_{2}}^{U L} (0) = 1 \\ {z_{1}}^{U L} (0), {z_{2}}^{U L} (0), {z_{3}}^{U L} (0), {s_{1}}^{U L} (0), {s_{2}}^{U L} (0) \geq 0. \end{array} \end{aligned} \end{aligned}

(49)

Using the Simplex technique, we can compute the optimal solution $(z^{* U L} (0), s^{* U L} (0))$ of equation (49), where $z^{* U L} (0) = (0, 72, 0)$ and $s^{* U L} (0) = (0.4, 0.6)$ . Then, the upper lower limit of the equilibrium value and the equilibrium strategy for player II are $W^{U L} = W^{U L} (0) = p^{T} z^{* U L} (0) = 72$ and $s^{* U L} = s^{* U L} (0) = (0.4, 0.6)$ , respectively.

According to equation (37), the linear programming problem is formulated as follows:

\begin{aligned} \begin{aligned} m a x {- 75 {z_{1}}^{L U} (0) + {z_{2}}^{L U} (0) - {z_{3}}^{L U} (0)} \\ s . t . & {\begin{array}{l} - 80 {z_{1}}^{L U} (0) + {z_{2}}^{L U} (0) - {z_{3}}^{L U} (0) - 30 {s_{1}}^{L U} (0) - 25 {s_{2}}^{L U} (0) \leq 0 \\ - 60 {z_{1}}^{L U} (0) + {z_{2}}^{L U} (0) - {z_{3}}^{L U} (0) - 30 {s_{1}}^{L U} (0) - 45 {s_{2}}^{L U} (0) \leq 0 \\ 50 {s_{1}}^{L U} (0) + 75 {s_{2}}^{L U} (0) \leq 65 \\ {s_{1}}^{L U} (0) + {s_{2}}^{L U} (0) = 1 \\ {z_{1}}^{L U} (0), {z_{2}}^{L U} (0), {z_{3}}^{L U} (0), {s_{1}}^{L U} (0), {s_{2}}^{L U} (0) \geq 0. \end{array} \end{aligned} \end{aligned}

(50)

Using the Simplex technique, we can compute the optimal solution $(z^{* L U} (0), s^{* L U} (0))$ of equation (50), where $z^{* L U} (0) = (0, 30, 0)$ and $s^{* L U} (0) = (1, 0)$ . Then, the lower upper limit of the equilibrium value and the equilibrium strategy for player II are $W^{L U} = W^{L U} (0) = p^{T} z^{* L U} (0) = 30$ and $s^{* L U} = s^{* L U} (0) = (1, 0)$ , respectively.

According to equation (38), the linear programming problem is formulated as follows:

\begin{aligned} \begin{aligned} m a x {- 75 {z_{1}}^{U U} (0) + {z_{2}}^{U U} (0) - {z_{3}}^{U U} (0)} \\ s . t . & {\begin{array}{ll} - 80 {z_{1}}^{U U} (0) + {z_{2}}^{U U} (0) - {z_{3}}^{U U} (0) - 105 {s_{1}}^{U U} (0) - 120 {s_{2}}^{U U} (0) \leq 0 \\ - 60 {z_{1}}^{U U} (0) + {z_{2}}^{U U} (0) - {z_{3}}^{U U} (0) - 75 {s_{1}}^{U U} (0) - 95 {s_{2}}^{U U} (0) \leq 0 \\ 50 {s_{1}}^{U U} (0) + 75 {s_{2}}^{U U} (0) \leq 65 \\ {s_{1}}^{U U} (0) + {s_{2}}^{U U} (0) = 1 \\ {z_{1}}^{U U} (0), {z_{2}}^{U U} (0), {z_{3}}^{U U} (0), {s_{1}}^{U U} (0), {s_{2}}^{U U} (0) \geq 0. \end{array} \end{aligned} \end{aligned}

(51)

Using the Simplex technique, we can compute the optimal solution $(z^{* U U} (0), s^{* U U} (0))$ of equation (51), where $z^{* U U} (0) = (0, 87, 0)$ and $s^{* U U} (0) = (0.4, 0.6)$ . Then, the upper upper limit of the equilibrium value and the equilibrium strategy for player II are $W^{U U} = W^{U U} (0) = p^{T} z^{* U U} (0) = 87$ and $s^{* U U} = s^{* U U} (0) = (0.4, 0.6)$ , respectively. Therefore, the fuzzy rough equilibrium value for player II is $[(45.71, 57.5, 72) : (30, 57.5, 87)]$ .

5.3 Comparative Analysis and Discussion

The numerical example presented in this study effectively demonstrates the practical application of constrained bi-matrix games with fuzzy rough payoffs in modeling strategic decision-making in corporate environmental behavior. The dual-player model, consisting of a governmental regulator and a profit-oriented corporation, captures a typical conflict of interest: social welfare maximization versus profit maximization. By formulating and solving a series of linear programming problems (equations 42–51), the study systematically derives the equilibrium values and optimal strategies for both players using five α-cut–based fuzzy rough constructs: lower-lower, lower-upper, upper-lower, upper-upper, and the mean. These outcomes offer a complete fuzzy rough characterization of the equilibrium space, allowing decision-makers to assess risk and robustness in strategy selection.

The numerical results obtained from solving the fuzzy rough constrained bi-matrix game demonstrate the efficacy of incorporating both fuzziness and roughness to handle uncertainty in real-world decision-making, particularly in the context of corporate environmental behavior. The fuzzy rough equilibrium value for Player I (the government) is $[(44.25, 58.57, 76.5) : (28, 58.57, 95)]$ , and for Player II (the corporation) it is $[(45.71, 57.5, 72) : (30, 57.5, 87)]$ . These equilibrium values are expressed as fuzzy rough numbers, where each consists of a lower fuzzy number (first triplet) and an upper fuzzy number (second triplet), capturing the range of possible payoffs under incomplete and imprecise information.

The modal values 58.57 for Player I and 57.5 for Player II represent the most credible expected payoffs in this uncertain environment, while the lower-lower (28 and 30) and upper-upper bounds (95 and 87) delineate the most pessimistic and optimistic outcomes, respectively. These wide bounds highlight the variability and ambiguity in strategic decision-making under complex regulatory, environmental, and economic factors. For instance, the government's equilibrium value suggests that strict or positive supervision can lead to high social welfare under favorable conditions (up to 95), but in unfavorable situations (such as lack of compliance or poor enforcement), it may yield only 28 units of utility. Similarly, the corporation can achieve up to 87 units in best-case scenarios when aligning with environmental norms, but this value could drop to 30 if regulations tighten or market pressure intensifies.

The associated equilibrium strategies, computed through the fuzzy rough linear programming models under the given constraints, reflect the optimal mixed strategies that both players should adopt to maximize their respective payoffs. The constraints $80 r_{1} + 60 r_{2} \leq 75$ for the government and $50 s_{1} + 75 s_{2} \leq 55$ for the corporation) simulate real-world budgetary or policy limitations, forcing players to allocate their strategies wisely. These results showcase how incorporating fuzzy rough sets enables a more comprehensive evaluation of strategic interactions, accounting not only for imprecise preferences and evaluations (fuzziness) but also for vagueness due to boundary knowledge or incomplete classification (roughness).

In summary, the fuzzy rough equilibrium values provide a nuanced and informative depiction of each player's potential outcomes, ensuring that decision-makers can better assess risks and opportunities within a dual-layer uncertainty framework. This approach significantly enhances traditional game models by offering greater realism and flexibility in complex environmental policy planning and corporate strategic management. The obtained results and the derived strategies show consistency with rational decision-making behavior while providing more flexible and realistic outcomes than models limited to either fuzzy or rough paradigms alone. The alignment with environmental strategy games makes this methodology well-suited for multi-criteria decision analysis and regulatory policy planning.

The proposed fuzzy rough constrained bi-matrix game model distinguishes itself through its capability to handle complex environmental decision-making scenarios under dual-layer uncertainty, offering a more practical extension of fuzzy rough matrix games compared to existing approaches. Compared to the model in An and Li (2019), which uses intuitionistic fuzzy values and yields a single crisp solution (e.g., optimal strategies (0.75, 0.25) for Player I and (0.45, 0.6055 for Player I with expected game value around 4.033 for Player I and 4.193 for Player II), the current study incorporates fuzzy rough intervals, allowing for a richer representation of uncertainty and generating a range of outcomes (e.g., fuzzy rough equilibrium value for Player I is $[(44.25, 58.57, 76.5) : (28, 58.57, 95)]$ , and for Player II it is $[(45.71, 57.5, 72) : (30, 57.5, 87)]$ ) that reflect real-world variability. In Ammar and Brikaa (2017), the focus was on continuous differential games, primarily modeling dynamic interactions over time but without considering explicit strategic constraints, making it less suitable for resource-limited decision environments like environmental governance. Brikaa et al. (2019a) introduced a fuzzy multi-objective programming model with fuzzy rough numbers, which incorporated goal-based preferences but did not address equilibrium strategy derivation under linear inequality constraints, limiting its applicability in policy-based decision settings. Jangid et al. (2023) proposed a novel solution for fuzzy rough matrix games but focused exclusively on zero-sum games without considering real-world non-zero-sum interactions or strategy restrictions, making it less applicable to cooperative-competitive contexts like government–corporation environmental negotiations. Similarly, Jangid and Kumar (2022b) used rough fuzzy environments to analyze two-person zero-sum matrix games, employing ranking techniques but not extending their methodology to constrained or fuzzy rough bounded strategies. In contrast, the present model addresses these limitations by incorporating both fuzzy rough payoffs and constrained strategy sets using linear programming techniques, enabling the calculation of fuzzy rough equilibrium values and strategies under realistic limitations. This dual consideration of fuzziness and roughness, combined with bounded strategic feasibility, offers a more comprehensive and applicable framework for real-world decision problems, particularly in sustainable environmental management.

The key differences and novel aspects of our study and the earlier studies by An and Li (2019) and Li and Hong (2012) are summarized below:

Use of Fuzzy Rough Numbers:

While An and Li (2019) applied intuitionistic fuzzy numbers and Li and Hong (2012) utilized triangular fuzzy numbers for representing uncertainty in constrained matrix bi-matrix and constrained matrix games, our study introduces fuzzy rough numbers as a new way of modeling ambiguity and vagueness in the payoffs. Fuzzy rough sets integrate the concepts of both fuzziness and roughness, enabling a more flexible and granular representation of uncertain information, especially when exact membership values are difficult to determine due to incomplete or overlapping data.

2.
Game Structure – Bi-matrix with Constraints:

Similar to An and Li (2019), our model addresses constrained bi-matrix games, but extends the framework by adopting a new class of payoff structure (i.e., fuzzy rough numbers), which has not been considered in either of the referenced studies. This incorporation allows for richer modeling in environments where both vagueness (fuzziness) and boundary uncertainty (roughness) coexist. 3.
Modeling and Computational Approach:

We develop a hybrid modeling framework that constructs fuzzy models based on fuzzy rough sets and subsequently transforms them into crisp linear programming problems using α-cut and rough boundary approximations. While both An and Li (2019) and Li and Hong (2012) relied on α-cut or linear programming techniques tailored to specific fuzzy environments, our approach is novel in the way it adapts these methods to handle the dual uncertainty layers inherent in fuzzy rough structures. 4.
Generalization and Application:

Our method generalizes previous models by enabling the solution of a broader class of constrained bi-matrix games under more complex uncertainty settings. In addition, we validate the proposed approach through a real-world case study on corporate environmental behavior, which demonstrates the flexibility and effectiveness of fuzzy rough sets in capturing nuanced decision-making scenarios. This type of application was not considered in the aforementioned studies.
Advantages, disadvantages, and limitations

Advantages
Handles Dual Uncertainty: Combines the strengths of fuzzy sets (handling vagueness) and rough sets (handling ambiguity due to lack of knowledge), allowing for a more robust modeling of real-world imprecise payoffs.

Incorporates Constraints: Allows the incorporation of realistic constraints (e.g., budget, capacity, or policy limits) into the players’ strategy sets, increasing applicability to real decision-making scenarios.

Captures Subjective Judgments: Useful in situations where expert opinion or linguistic assessments dominate (e.g., sustainability evaluations, policy-making, environmental strategies).

Flexible Strategy Representation: α-cut and fuzzy rough interval representations offer flexibility to decision-makers with different confidence or risk levels.

Equilibrium Analysis Under Uncertainty: Provides structured equilibrium strategies even when data is incomplete or uncertain, which is crucial for sectors like environmental regulation, economics, or negotiation models.

Disadvantages
Computational Complexity: Solving multiple linear programming models (especially across α-cuts) for each strategy and payoff interval can be time-consuming and computationally expensive.

Requires Advanced Mathematical Knowledge: Understanding and implementing fuzzy rough sets, constraint models, and equilibrium concepts may be too complex for non-technical users.

Dependence on α-Level Choice: The outcomes can vary depending on the choice of α-level, which may introduce subjective bias or inconsistency if not justified.

Limitations
Data Quality Sensitivity: The accuracy of fuzzy rough payoffs relies heavily on the quality and structure of the input data. Poor or biased data leads to unreliable outputs.

Limited Real-Time Application: Due to computational requirements and the need for detailed modeling, it might not be feasible for time-sensitive or real-time decision-making processes.

Interpretability Challenges: Fuzzy rough outputs (e.g., lower-upper bounds) may be harder to interpret for decision-makers unfamiliar with fuzzy logic or set theory, reducing the model's accessibility.

Not Always Uniquely Solvable: Depending on the structure of the constraints and the payoffs, the game may have multiple or non-dominant equilibria, complicating solution selection.

6 Conclusion

In addressing real-world constraint bi-matrix games problems, uncertainty and hesitation frequently arise due to various uncontrollable factors. To mitigate these challenges, the fuzzy rough approach is employed as a viable solution methodology. In this paper, we present linear programming models and methodologies for solving fuzzy rough constraint bi-matrix games, grounded in the fuzzy rough sets theorem and α-cut sets. The main scientific contributions of this study can be outlined as follows:

A novel formulation of constrained bi-matrix games is introduced, wherein the payoff values are characterized using fuzzy rough numbers to capture deeper uncertainty and vagueness.

Fuzzy modeling frameworks are constructed based on the proposed fuzzy rough representation to facilitate the mathematical handling of imprecision.

An efficient computational algorithm grounded in the α-cut methodology (Li & Hong, 2012) is proposed to derive optimal strategic solutions for the formulated game models.

This work extends the existing constrained bi-matrix game frameworks with fuzzy payoffs, as previously examined by Bigdeli et al. (2018) and An and Li (2019), by incorporating fuzzy rough structures.

The fuzzy rough models are further transformed into equivalent crisp linear programming models to enable tractable computation.

These crisp reformulations are solved using LINGO 14.0 (Lindo Systems, Chicago, IL, USA), ensuring practical implementation.

The developed method guarantees the existence of an equilibrium value expressed in terms of fuzzy rough numbers, which can be precisely determined by solving the linear programming formulations provided in equations (15)–(18) and (19)–(22).

A numerical case study related to corporate environmental decision-making is carried out to validate the applicability and robustness of the proposed framework.

The experimental findings suggest that fuzzy rough sets provide a more nuanced and comprehensive representation of uncertainty compared to traditional fuzzy sets, particularly when addressing ambiguity in game-theoretic environments.

In the future, the proposed methodology is anticipated to be applicable to various related domains, including ecological management, environmental studies, military science, medical science, forest management, telecom market competition, plastic ban policies, tourism environment management, and biogas plant implementation. Moreover, an intriguing research direction involves extending the proposed approach to address other categories of game-theoretic problems, such as cooperative games, matrix games, evolutionary games, multi-objective games, bi-matrix games, non-zero-sum games, constrained matrix games, and other complex decision-making scenarios.

Footnotes

ORCID iD

Mohamed Gaber Brikaa

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Achemine

Larbani

(2023). Nash equilibrium in Fuzzy random bi-matrix games. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 31(6), 1005–1031. https://doi.org/10.1142/S0218488523500459

Ahuja

Kumar

(2024). Mehar approach to solve hesitant fuzzy linear programming Problems. The Journal of Analysis, 32(1), 335–371. https://doi.org/10.1007/s41478-023-00629-9

Ammar

E. S.

Brikaa

M. G.

(2017). A study on fuzzy rough continuous differential games. Journal of Fuzzy Set Valued Analysis, 2017(3), 102–119. https://doi.org/10.5899/2017/jfsva-00361

Ammar

E. S.

Brikaa

M. G.

(2019). On solution of constraint matrix games under rough interval approach. Granular Computing, 4(3), 601–614. https://doi.org/10.1007/s41066-018-0123-4

Ammar

Muamer

(2016). On solving fuzzy rough linear fractional programming problem. Int. Res. J. Eng. Technol, 30(4), 2099–2120.

J. J.

D. F.

(2019). A linear programming approach to solve constrained bi-matrix games with intuitionistic fuzzy payoffs. International Journal of Fuzzy Systems, 21(3), 908–915. https://doi.org/10.1007/s40815-018-0573-5

Bhaumik

Roy

S. K.

(2021). Intuitionistic interval-valued hesitant fuzzy matrix games with a new aggregation operator for solving management problem. Granular Computing, 6(2), 359–375. https://doi.org/10.1007/s41066-019-00191-5

Bhaumik

Roy

S. K.

(2021a). (α,β,γ) -Cut set based ranking approach to solving bi-matrix games in neutrosophic environment. Soft Computing, 25(4), 2729–2739. https://doi.org/10.1007/s00500-020-05332-6

Bhaumik

Roy

S. K.

Weber

G. W.

(2020). Hesitant interval-valued intuitionistic fuzzy-linguistic term set approach in Prisoners’ dilemma game theory using TOPSIS: A Case Study on human-trafficking. Central European Journal of Operations Research, 28(2), 797–816. https://doi.org/10.1007/s10100-019-00638-9

10.

Bhaumik

Roy

S. K.

Weber

G. W.

(2021b). Multi-objective linguistic-neutrosophic matrix game and its applications to tourism management. Journal of Dynamics & Games, 8(2), 101–118. https://doi.org/10.3934/jdg.2020031

11.

Bigdeli

Hassanpour

Tayyebi

(2018). Constrained bimatrix games with fuzzy goals and its application in nuclear negotiations. Iranian Journal of Numerical Analysis and Optimization, 8(1), 81–110. https://doi.org/10.22067/ijnao.v8i1.55385

12.

Bigdeli

Mousazadeh

(2023). Analytical hierarchy process in modeling and solving matrix games in neutrosophic environment and its application in military problems. Military Science and Tactics, 19(64), 5–33. https://doi.org/10.22034/qjmst.2023.544038.1627

13.

Bisht

Beg

Rawat

(2024). A new method to solve matrix game with interval payoffs and its MATLAB code. International Game Theory Review, 26(3), 2450001. https://doi.org/10.1142/S0219198924500014

14.

Bisht

Dangwal

(2023). Fuzzy ranking approach to bi-matrix games with interval payoffs in marketing-management problem. International Game Theory Review, 25(1), 2250016. https://doi.org/10.1142/S0219198922500165

15.

Brikaa

M. G.

Zheng

Dagestani

A. A.

Ammar

E. S.

AlNemer

Zakarya

(2022a). Ambika approach for solving matrix games with payoffs of single-valued trapezoidal neutrosophic numbers. Journal of Intelligent & Fuzzy Systems, 42(6), 5139–5153. https://doi.org/10.3233/JIFS-211604

16.

Brikaa

M. G.

Zheng

Ammar

E.-S.

(2019a). Fuzzy multi-objective programming approach for constrained matrix games with payoffs of fuzzy rough numbers. Symmetry, 11(5), 702. https://doi.org/10.3390/sym11050702

17.

Brikaa

M. G.

Zheng

Ammar

E.-S.

(2020). Resolving indeterminacy approach to solve multi-criteria zero-sum matrix games with intuitionistic fuzzy goals. Mathematics, 8(3), 305. https://doi.org/10.3390/math8030305

18.

Brikaa

M. G.

Zheng

Ammar

E.-S.

(2022b). Mehar approach for solving matrix games with triangular dual hesitant fuzzy payoffs. Granul. Comput, 7(3), 731–750.

19.

Chauhan

Gupta

(2021). Matrix games with proportional linguistic payoffs. Soft Computing, 25(24), 15067–15081. https://doi.org/10.1007/s00500-021-06363-3

20.

Chauhan

Gupta

(2023). A methodology for solving bimatrix games under 2-tuple linguistic environment. International Journal of Systems Science: Operations & Logistics, 10(1), 2199368. https://doi.org/10.1080/23302674.2023.2199368

21.

Dengfeng

Chuntian

(2002). Fuzzy multiobjective programming methods for fuzzy constrained matrix games with fuzzy numbers. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10(4), 385–400. https://doi.org/10.1142/S0218488502001545

22.

Devi

R. N.

Sowmiya

(2025). On solving game problem using octagonal neutrosophic fuzzy number. International Journal of Operational Research, 52(1), 116–127. https://doi.org/10.1504/IJOR.2025.143961

23.

Djebara

Achemine

Larbani

(2024). Solving constrained matrix games with fuzzy random linear constraints. The Yugoslav Journal of Operations Research, 35(1), 209–227.

24.

Djebara

Achemine

Zerdani

(2023). A new approach for solving constrained matrix games with fuzzy constraints and fuzzy payoffs. Journal of Mathematical Modeling, 11(3), 425–439.

25.

Dong

Wan

(2024). Type-2 interval-valued intuitionstic fuzzy matrix game and application to energy vehicle industry development. Expert Systems with Applications, 249(1), 123398. https://doi.org/10.1016/j.eswa.2024.123398

26.

Dubois

Prade

(1990). Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems, 17(2), 191–209. https://doi.org/10.1080/03081079008935107

27.

Gaber

Alharbi

M. G.

Dagestani

A. A.

Ammar

E. S.

(2021). Optimal solutions for constrained bimatrix games with payoffs represented by single-valued trapezoidal neutrosophic numbers. J. Math, 1(2021), 5594623. https://doi.org/10.1155/2021/5594623

28.

Hung

I. C.

Hsia

K. H.

Chen

L. W.

(1996). Fuzzy differential game of guarding a movable territory. Inf. Sci, 91(1-2), 113–131. https://doi.org/10.1016/0020-0255(95)00299-5

29.

İzgi

Özkaya

Üre

N. K.

Perc

(2024). Matrix norm based hybrid shapley and iterative methods for the solution of stochastic matrix games. Applied Mathematics and Computation, 473(2024), 128638.

30.

Jana

Roy

S. K.

(2023). Linguistic pythagorean hesitant fuzzy matrix game and its application in multi-criteria decision making. Applied Intelligence, 53(1), 1–22. https://doi.org/10.1007/s10489-022-03442-2

31.

Jangid

Kumar

Sharama

Mishra

V. N.

(2023). A novel approach to solve fuzzy rough matrix game with two players. Applications and Applied Mathematics, 18(2).

32.

Jangid

Kumar

(2021). Matrix games with single-valued triangular neutrosophic numbers as pay-offs. Neutrosophic Sets and Systems, 45(2021), 196–217.

33.

Jangid

Kumar

(2022a). Hexadecagonal fuzzy numbers: novel ranking and defuzzification techniques for fuzzy matrix game problems. Fuzzy Information and Engineering, 14(1), 84–122. https://doi.org/10.1080/16168658.2021.2019969

34.

Jangid

Kumar

(2022b). A novel technique for solving two-person zero-sum matrix games in a rough fuzzy environment. Yugoslav Journal of Operations Research, 32(2), 251–278. https://doi.org/10.2298/YJOR210617003J

35.

Karmakar

Seikh

M. R.

(2023). Bimatrix games under dense fuzzy environment and its application to natural disaster management. Artificial Intelligence Review, 56(3), 2241–2278. https://doi.org/10.1007/s10462-022-10220-6

36.

Karmakar

Seikh

M. R.

(2024). A nonlinear programming approach to solving interval-valued intuitionistic hesitant noncooperative fuzzy matrix games. Symmetry, 16(5), 573. https://doi.org/10.3390/sym16050573

37.

Karmakar

Seikh

M. R.

Castillo

(2021). Type-2 intuitionistic fuzzy matrix games based on a new distance measure: application to biogas-plant implementation problem. Applied Soft Computing, 106(2021), 107357. https://doi.org/10.1016/j.asoc.2021.107357

38.

Khan

Kumar

(2023). Solving hesitant fuzzy linguistic matrix game with regret theory. Granular Computing, 8(6), 1325–1340. https://doi.org/10.1007/s41066-023-00375-0

39.

Khan

Mehra

(2020). A novel equilibrium solution concept for intuitionistic fuzzy bi-matrix games considering proportion mix of possibility and necessity expectations. Granular Computing, 5(4), 461–483. https://doi.org/10.1007/s41066-019-00170-w

40.

Kirti

Verma

Kumar

(2024). Modified approaches to solve matrix games with payoffs of single-valued trapezoidal neutrosophic numbers. Soft Computing, 28(1), 1–50. https://doi.org/10.1007/s00500-023-09133-5

41.

Kon

(2021). On characterization of equilibrium strategy for matrix games with LR fuzzy payoffs. Journal of the Operations Research Society of Japan, 64(3), 158–174. https://doi.org/10.15807/jorsj.64.158

42.

Kumar

(2021). Piecewise linear programming approach to solve multi-objective matrix games with I-fuzzy goals. Journal of Control and Decision, 8(1), 1–13. https://doi.org/10.1080/23307706.2019.1619491

43.

Kumar

Garg

(2025). A novel two-level fuzzy set theoretic approach to solve multi-objective matrix games and its applications. Ann. Oper. Res, 1–41. https://doi.org/10.1007/s10479-025-06524-9

44.

D. F.

Hong

F. X.

(2012). Solving constrained matrix games with payoffs of triangular fuzzy numbers. Computers & Mathematics with Applications, 64(4), 432–446. https://doi.org/10.1016/j.camwa.2011.12.009

45.

D. F.

Hong

(2013). Alfa-cut based linear programming methodology for constrained matrix games with payoffs of trapezoidal fuzzy numbers. Fuzzy Optimization and Decision Making, 12(2), 191–213. https://doi.org/10.1007/s10700-012-9148-3

46.

(2022a). Bi-matrix games with general intuitionistic fuzzy payoffs and application in corporate environmental behavior. Symmetry, 14(4), 671. https://doi.org/10.3390/sym14040671

47.

(2022b). Probabilistic linguistic matrix game based on fuzzy envelope and prospect theory with its application. Mathematics, 10(7), 1070. https://doi.org/10.3390/math10071070

48.

Feng

Zou

(2023). Fuzzy weighted Pareto–Nash equilibria of multi-objective bi-matrix games with fuzzy payoffs and their applications. Mathematics, 11(20), 4266. https://doi.org/10.3390/math11204266

49.

Liao

Zeng

X. J.

(2021). The two-person and zero-sum matrix game with probabilistic linguistic information. Information Sciences, 570(1), 487–499. https://doi.org/10.1016/j.ins.2021.05.019

50.

Morsi

N. N.

Yakout

M. M.

(1998). Axiomatics for fuzzy rough sets. Fuzzy Sets and Systems, 100(1), 327–342. https://doi.org/10.1016/S0165-0114(97)00104-8

51.

Namarta

P. K.

Gupta

U. C.

(2021). A new method to solve intuitionistic fuzzy matrix game based on the evaluation of different experts. J. Intell. Fuzzy Syst, 41(2), 2665–2674. https://doi.org/10.3233/JIFS-202232

52.

Nan

J.-X.

D.-F.

(2014). Linear programming technique for solving interval-valued constraint matrix games. Journal of Industrial & Management Optimization, 10(4), 1059–1070. https://doi.org/10.3934/jimo.2014.10.1059

53.

Naqvi

Aggarwal

Sachdev

Khan

(2021). Solving I-fuzzy two person zero-sum matrix games: Tanaka and Asai approach. Granular Computing, 6(2), 399–409. https://doi.org/10.1007/s41066-019-00200-7

54.

Naqvi

D. R.

Sachdev

(2023). On uncertain matrix games involving linguistic pythagorean fuzzy sets. In Combinatorial Optimization Under Uncertainty (pp. 95–128). CRC Press.

55.

Naqvi

D. R.

Sachdev

Ahmad

(2023a). Matrix games involving interval-valued hesitant fuzzy linguistic sets and its application to electric vehicles. Journal of Intelligent & Fuzzy Systems, 44(3), 5085–5105. https://doi.org/10.3233/JIFS-222466

56.

Naqvi

D. R.

Verma

Aggarwal

Sachdev

(2023b). Solutions of matrix games involving linguistic interval-valued intuitionistic fuzzy sets. Soft Computing, 27(2), 783–808. https://doi.org/10.1007/s00500-022-07609-4

57.

Pawlak

(1982). Rough sets. International Journal of Computer & Information Sciences, 11(5), 341–356. https://doi.org/10.1007/BF01001956

58.

Qiu

Xiang

(2022). Bi-matrix games with 2-Tuple linguistic information. IAENG International Journal of Applied Mathematics, 52(3).

59.

Roy

S. K.

Maiti

S. K.

(2020). Reduction methods of Type-2 fuzzy variables and their applications to stackelberg game. Applied Intelligence, 50(5), 1398–1415. https://doi.org/10.1007/s10489-019-01578-2

60.

Seikh

M. R.

Dutta

(2021a). A nonlinear programming model to solve matrix games with pay-offs of single-valued neutrosophic numbers. Neutrosophic Sets Syst, 47(2021), 366–383. https://digitalrepository.unm.edu/nss_journal/vol47/iss1/24

61.

Seikh

M. R.

Dutta

(2021b). Solution of interval-valued matrix games using intuitionistic fuzzy optimisation technique: an effective approach. International Journal of Mathematics in Operational Research, 20(3), 297–322. https://doi.org/10.1504/IJMOR.2021.119951

62.

Seikh

M. R.

Dutta

(2022). Solution of matrix games with payoffs of single-valued trapezoidal neutrosophic numbers. Soft Computing, 26(2022), 921–936. https://doi.org/10.1007/s00500-021-06559-7

63.

Seikh

M. R.

Dutta

(2023a). Interval neutrosophic matrix game-based approach to counter cybersecurity issue. Granular Computing, 8(2), 271–292. https://doi.org/10.1007/s41066-022-00327-0

64.

Seikh

M. R.

Dutta

(2023b). Solution of matrix games with pay-offs of single-valued neutrosophic numbers and its application to market share problem. Math. Comput. Sci, 1(2023), 213–230. https://doi.org/10.1002/9781119879831.ch10

65.

Seikh

M. R.

Dutta

(2024). A non-linear mathematical approach for solving matrix games with picture fuzzy payoffs with application to cyberterrorism attacks. Decision Analytics Journal, 10(1), 100394. https://doi.org/10.1016/j.dajour.2023.100394

66.

Seikh

M. R.

Dutta

D. F.

(2021a). Solution of matrix games with rough interval pay-offs and its application in the telecom market share problem. International Journal of Intelligent Systems, 36(10), 6066–6100. https://doi.org/10.1002/int.22542

67.

Seikh

M. R.

Karmakar

(2021). Credibility equilibrium strategy for matrix games with payoffs of triangular dense fuzzy lock sets. Sādhanā, 46(3), 1–17. https://doi.org/10.1007/s12046-021-01666-5

68.

Seikh

M. R.

Karmakar

Castillo

(2021b). A novel defuzzification approach of Type-2 fuzzy variable to solving matrix games: an application to plastic ban problem. International Journal of Applied Mathematics, 18(5), 155–172. https://doi.org/10.22111/ijfs.2021.6262

69.

Seikh

M. R.

Karmakar

Nayak

P. K.

(2021c). Matrix games with dense fuzzy payoffs. International Journal of Intelligent Systems, 36(4), 1770–1799. https://doi.org/10.1002/int.22360

70.

Seikh

M. R.

Karmakar

Xia

(2020). Solving matrix games with hesitant fuzzy pay-offs. Iranian Journal of Fuzzy Systems, 17(4), 25–40. https://doi.org/10.22111/ijfs.2020.5404

71.

Singh

Gupta

Mehra

(2020). Matrix games with 2-Tuple linguistic information. Annals of Operations Research, 287(2), 895–910. https://doi.org/10.1007/s10479-018-2810-6

72.

Singla

Kaur

Gupta

U. C.

(2023). A new approach to solve intuitionistic fuzzy bi-matrix games involving multiple opinions. Iranian Journal of Fuzzy Systems, 20(1), 185–197. https://doi.org/10.22111/ijfs.2023.7354

73.

Verma

(2021). A novel method for solving constrained matrix games with fuzzy payoffs. J. Intell. Fuzzy Syst, 40(1), 191–204. https://doi.org/10.3233/JIFS-191192

74.

Verma

Aggarwal

(2021a). Matrix games with linguistic intuitionistic fuzzy payoffs: basic results and solution methods. Artificial Intelligence Review, 54(7), 5127–5162. https://doi.org/10.1007/s10462-021-10014-2

75.

Verma

Aggarwal

(2021b). On matrix games with 2-Tuple intuitionistic fuzzy linguistic payoffs. Iranian Journal of Fuzzy Systems, 18(4), 149–167. https://doi.org/10.1007/s00500-023-08785-7

76.

Verma

Singla

Yager

R. R.

(2023). Matrix games under a pythagorean fuzzy environment with self-confidence levels: formulation and solution approach. Soft Comput, (2023), 1–23. https://doi.org/10.1007/s00500-023-08785-7

77.

Von Neumann

Morgenstern

(1944). The theory of games in economic bahavior. J. Wiley.

78.

Wan

Qiu

(2024). Pandemic management by using sentiment analysis and trapezoidal Type-2 fuzzy linguistic intuitionistic fuzzy matrix games. J. Intell. Fuzzy Syst, 46(4), 8677–8695. https://doi.org/10.3233/JIFS-237319

79.

Xue

Zeng

X. J.

(2021). Solving matrix games based on Ambika method with hesitant fuzzy information and its application in the counter-terrorism issue. Applied Intelligence, 51(3), 1227–1243. https://doi.org/10.1007/s10489-020-01759-4

80.

Zadeh

L. A.

(1965). Fuzzy sets. Information and Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X

81.

Zheng

Brikaa

M. G.

(2022). Solving multi-objective bi-matrix games with intuitionistic fuzzy goals through an aspiration level approach. International Journal of Computing Science and Mathematics, 16(4), 307–326. https://doi.org/10.1504/IJCSM.2022.128650