EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7083 ISSN 1307-5543 – ejpam.com Published by New York Business Global 1 Game-Theoretic Electromagnetic Isolation Framework2 for Wireless Connectivity in Mineral and Phosphate3 Mining Environments4 Mohamed Ayari1,∗, Atef Gharbi2, Zeineb Klai3, Abdelhalim Hasnaoui4,5 Mahmoud Salaheldin Elsayed3, Elsaid Md. Abdelrahim5 6 1 Department of Information Technology, Faculty of Computing and Information7 Technology, Northern Border University, Kingdom of Saudi Arabia8 2 Department of Information Systems, Faculty of Computing and Information Technology,9 Northern Border University, Kingdom of Saudi Arabia10 3 Department of Computer Sciences, Faculty of Computing and Information Technology,11 Northern Border University,12 Kingdom of Saudi Arabia13 4 Mathematics Department, College of Sciences and Arts, Northern Border University,14 Kingdom of Saudi Arabia15 5 Computer Science Department, Science College, Northern Border University (NBU),16 Arar 73213, Saudi Arabia17 18 Abstract. Reliable wireless communication is a critical enabler of digital transformation in the mining industry, particularly in phosphate and mineral extraction environments, where safety, monitoring, and automation depend on robust connectivity. However, underground and semi-enclosed mining environ- ments introduce severe electromagnetic (EM) challenges, including signal attenuation, multipath fading, and interference from dense equipment and layered geological structures. Traditional substrate engineering methods—such as diffused buried layers (DBL), metallized grids, guard rings, and electromagnetic bandgap (EBG) structures—offer partial isolation, but often fail to ensure stable performance in such harsh con- ditions. This paper proposes the application of a game-theoretic electromagnetic isolation framework to wireless communication systems in mining operations. By modeling isolation techniques as strategic players in a non-cooperative game, the framework derives equilibrium solutions that balance isolation, insertion loss, fabrication complexity, and deployment cost. Simulation studies in the 2 to 12 GHz band demonstrate that the proposed method achieves 25–30 dB improvements in coupling reduction compared to conventional approaches while maintaining practical scalability. These results highlight the potential of the framework to enable safe, interference-resilient, and efficient wireless connectivity for real-time monitoring, autonomous equipment control, and worker safety systems in phosphate and mineral mining environments. 2020 Mathematics Subject Classifications: 91A10, 91A35, 90C59, 78M50, 78A4519 Key Words and Phrases: Electromagnetic isolation, game theory, optimization, wireless mining20 communication, phosphate mining, high-frequency systems21 22 ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7083 Email address: Mohamed.ayari@nbu.edu.sa (M. Ayari) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 2 of 17 1. Introduction23 Electromagnetic (EM) coupling and interference remain critical barriers to achieving24 reliable high-frequency communication in industrial environments. As mining operations,25 particularly in the extraction of phosphate and minerals, adopt digital technologies, the26 demand for robust wireless systems has increased. Applications such as real-time safety27 monitoring, autonomous vehicle control, and IoT-based environmental sensing require un-28 interrupted connectivity in environments characterized by metallic machinery, stratified29 geological layers, and confined underground spaces. These conditions intensify EM cou-30 pling, degrade signal integrity, increase bit error rates, and introduce latency in mission-31 critical systems [1–4].32 Traditional substrate engineering techniques have long been explored to mitigate elec-33 tromagnetic interference. Methods such as Buried Diffused Layers (BDL), metallized34 grids, through fences, and guard rings provide partial isolation by controlling capacitive35 and inductive coupling paths [5–7]. More advanced solutions have emerged with Elec-36 tromagnetic Band Gap (EBG) structures and metamaterial inclusions, which suppress37 surface waves and improve isolation in dense multi-antenna systems [8, 9]. Although effec-38 tive in laboratory conditions, these techniques often encounter limitations in scalability,39 cost, and insertion loss, particularly in harsh industrial and mining environments where40 system miniaturization and ruggedness are required.41 Recent research has expanded to algorithmic and optimization-driven methods. Ap-42 proaches such as Particle Swarm Optimization (PSO), Genetic Algorithms (GA), and43 Differential Evolution (DE) have been applied to fine-tune substrate geometries, antenna44 layouts, and material parameters, thus balancing isolation performance with practical45 trade-offs [10–12]. In addition, machine learning techniques—including neural networks46 and support vector machines—are increasingly used to predict EM isolation metrics from47 design parameters, accelerating design space exploration, and reducing reliance on re-48 peated simulations [13–15]. However, these computational strategies have not yet been49 fully exploited in mining communication systems, where adaptive and interference-resilient50 solutions are urgently needed.51 Game theory provides a promising but underutilized approach in this field. Widely52 used in energy systems [16], transport networks [17, 18], and wireless power control [19],53 game-theoretic frameworks capture competitive and cooperative dynamics among subsys-54 tems. When applied to EM isolation, techniques such as BDL, metallized grids, guard55 rings, and EBG surfaces can be modeled as strategic players, each optimizing its perfor-56 mance under constraints of fabrication cost, insertion loss, and size. Equilibrium solu-57 tions—whether Nash or Stackelberg—yield stable trade-offs that conventional optimiza-58 tion often overlooks [20–22].59 This integration is particularly relevant for mining applications, where communication60 systems must balance isolation performance with cost-effective deployment in challenging61 underground conditions. By unifying physical and algorithmic methods under a game-62 theoretic optimization framework, this study addresses both the theoretical gap in EM63 modeling and the practical need for resilient wireless infrastructure in phosphate and64 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 3 of 17 mineral mining environments.65 The paper is structured as follows: Section 2 discusses substrate isolation methods,66 meta-material-based techniques, and optimization approaches relevant to mining commu-67 nication. Section 3 presents the proposed game-theoretic framework, including definitions68 of players, utility functions, and equilibrium formulations. Section 4 details the simulation69 setup and results across the GHz ranges. Section 5 discusses the practical implications in70 mining environments. Finally, Section 6 concludes the paper and outlines directions for71 future research.72 2. Substrate Engineering and Electromagnetic Isolation Methods73 Electromagnetic isolation has traditionally relied on substrate engineering to suppress74 unwanted coupling in high-frequency circuits and antenna systems. In the context of min-75 eral and phosphate mining, where metallic machinery, underground tunnels, and stratified76 substrates exacerbate EM interference, these methods provide a critical foundation for re-77 liable wireless connectivity.78 2.1. Buried Diffused Layers and Substrate Doping79 Continuous BDL layers have long been employed to mitigate capacitive coupling be-80 tween substrate traces. By creating a buried conductive layer beneath the active substrate,81 capacitive leakage paths are reduced, improving signal integrity across GHz frequencies.82 Substrate doping techniques further enhance conductivity control, but trade-offs emerge in83 the form of increased fabrication complexity and potential degradation of high-frequency84 response [23–25].85 2.2. Guard Rings and Isolation Pockets86 Guard rings—conductive loops embedded around critical lines or antenna elements—re-87 main a popular choice for localized isolation. Recent studies show that optimized guard-88 ring geometries and embedded isolation pockets can achieve substantial improvements89 across wide frequency bands [26–28]. For ruggedized mining devices, guard rings offer a90 compact, low-overhead solution, though they provide limited performance against broad-91 band interference.92 2.3. Electromagnetic Band Gap and Metamaterial Inclusions93 EBG structures and metamaterials represent advanced substrate engineering tech-94 niques designed to block or absorb surface waves. By embedding periodic structures or95 resonant inclusions, these methods disrupt coupling paths and deliver isolation enhance-96 ments exceeding 20–30 dB in compact antenna arrays [29, 30]. Their tunability makes97 them attractive for mining applications where communication spans multiple frequency98 bands, though fabrication tolerances and environmental variability (e.g., temperature, hu-99 midity, dust) can reduce reliability.100 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 4 of 17 2.4. Algorithmic and Computational Methods101 Beyond structural techniques, computational approaches have gained traction. Meta-102 heuristic algorithms such as Particle Swarm Optimization (PSO), Genetic Algorithms103 (GA), and Differential Evolution (DE) have been applied to optimize geometry and ma-104 terial parameters, striking a balance between isolation, insertion loss, and cost [31–33].105 Similarly, machine learning methods—including neural networks and support vector ma-106 chines (SVMs)—are increasingly used to predict isolation metrics from design parameters,107 thereby accelerating design exploration and reducing reliance on extensive prototyping108 [34, 35]. These methods are particularly relevant for mining, where harsh deployment109 conditions demand adaptive and predictive design strategies.110 2.5. Research Gap111 Although substrate engineering and algorithmic optimization have each produced no-112 table improvements, they remain fragmented approaches. Structural methods maximize113 isolation but often ignore fabrication cost, while algorithmic models improve design speed114 but lack physical trade-off modeling. To address these gaps, this paper introduces a115 game-theoretic optimization framework, where different isolation techniques are modeled116 as strategic players competing or cooperating under real-world constraints. This unified117 approach provides a stable equilibrium solution, ensuring scalability and resilience for118 wireless mining communication systems.119 3. Proposed Game-Theoretic Optimization Framework120 3.1. Framework Overview121 The proposed framework formulates electromagnetic isolation design as a non-cooperative122 game where each isolation technique—such as Buried Diffused Layers (BDL), metallized123 grids, guard rings, and EBG structures—is modeled as a strategic player. Each player124 seeks to optimize its contribution to isolation performance while minimizing associated125 costs (e.g., insertion loss, fabrication complexity, or deployment footprint).126 The outcome of this interaction is defined by a Nash Equilibrium (NE), where no127 player can unilaterally improve its utility without degrading system-level performance.128 This formulation provides a balanced design strategy suitable for harsh environments129 such as mineral and phosphate mines, where robustness, scalability, and cost-efficiency130 are essential.131 3.2. Game Formulation132 Let:133 • P = {1, 2, . . . , N} denote the set of isolation techniques (players).134 • Si denote the strategy space of player i, representing tunable design variables (e.g.,135 grid spacing, doping concentration, guard ring width).136 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 5 of 17 • Ui(Si, S−i) denote the utility function of player i, which depends on its own strategy137 Si and the strategies of all other players S−i.138 Each player aims to maximize its utility:139 Ui(Si, S−i) = α · I(S)− β · L(Si)− γ · C(Si), (1) where:140 • I(S) = achieved isolation (in dB, higher is better),141 • L(Si) = insertion loss penalty introduced by player i,142 • C(Si) = normalized fabrication complexity index,143 • α, β, γ = weighting coefficients representing system priorities.144 3.3. Nash Equilibrium Condition145 A strategy profile S∗ = (S∗ 1 , S ∗ 2 , . . . , S ∗ N ) is a Nash Equilibrium if:146 Ui(S ∗ i , S ∗ −i) ≥ Ui(Ŝi, S ∗ −i), ∀i ∈ P, ∀Ŝi ∈ Si. (2) Existence and Uniqueness of Nash Equilibrium: The existence of a Nash Equi-147 librium in this framework is guaranteed under standard game-theoretic assumptions,148 namely that each strategy space Si is compact and convex, and that each utility func-149 tion Ui(Si, S−i) is continuous in all strategy profiles and quasi-concave in Si. Under150 these conditions, the game admits at least one pure-strategy equilibrium according to the151 Glicksberg–Debreu fixed-point theorem. Uniqueness can be ensured if each Ui is strictly152 concave in Si or if best responses are contraction mappings, which holds approximately for153 small coupling between isolation techniques. These assumptions are reasonable in electro-154 magnetic design problems, where the strategy spaces (geometric or material parameters)155 are bounded and the utilities are smooth and continuous.156 This condition ensures that no isolation technique (player) can improve its utility or157 the overall system outcome by deviating unilaterally from its equilibrium strategy.158 3.4. Algorithmic Implementation159 To compute the equilibrium, an iterative best-response algorithm is used. Each player160 updates its strategy sequentially based on the current strategies of others until conver-161 gence. The pseudocode is presented in Algorithm 1.162 Complexity Analysis: The computational complexity of the algorithm isO(N ·T ), where163 N is the number of players (isolation techniques) and T is the number of iterations until164 convergence. Since each update requires evaluating the utility function, the framework165 scales efficiently even for larger sets of isolation techniques.166 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 6 of 17 Algorithm 1 Iterative Best-Response Algorithm Require: Players P = {1, . . . , N}, initial strategies S(0), tolerance ε Ensure: Equilibrium strategies S∗ 1: Initialize t← 0 2: repeat 3: for each player i ∈ P do 4: Compute best response: S (t+1) i = arg max Si∈Si Ui(Si, S (t) −i ) 5: end for 6: if ‖S(t+1) − S(t)‖ < ε then 7: Convergence achieved; set S∗ ← S(t+1) 8: break 9: else 10: t← t+ 1 11: end if 12: until maximum iterations reached 3.5. Strategy Space and Design Variables167 The effectiveness of the proposed game-theoretic framework depends on the strategy168 space defined for each isolation technique. Every player (isolation method) is assigned a set169 of tunable design variables, with specific ranges reflecting realistic fabrication constraints.170 Table 1 summarizes the strategy variables considered for Buried Diffused Layers (BDL),171 Metallized Grids, Guard Rings, and Electromagnetic Band Gap (EBG) structures.172 Table 1: Example Strategy Spaces for Isolation Techniques Player (Technique) Strategy Variables Range Notes BDL Doping concentration 0.1–0.9 Higher doping improves isolation but also increases insertion loss. Metallized Grid Grid pitch (µm) 5–50 Smaller pitch enhances isolation but raises fabrication complexity. Guard Ring Ring width (µm) 1–20 Wider rings reduce edge coupling but increase footprint. EBG Structures Unit-cell period (mm) 1–5 Determines stopband frequencies and overall bandwidth. The ranges selected represent practical design trade-offs commonly encountered in high-173 frequency circuits. By modeling these choices as strategies within the proposed framework,174 the optimizer identifies equilibrium configurations that balance isolation performance, in-175 sertion loss, and fabrication cost.176 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 7 of 17 3.6. Expected Outcomes177 At equilibrium, the system achieves:178 • Superior Isolation: Coupling reductions of 25–30 dB across GHz ranges.179 • Balanced Trade-offs: Controlled insertion loss and fabrication costs compared to180 brute-force hybrid methods.181 • Scalability: Applicability to sub-6 GHz, mmWave, and mining-relevant wireless182 bands.183 • Robustness: Stability under varying substrate and environmental conditions.184 4. Simulation Setup, System Architecture, and Results185 4.1. System Architecture in Mining Context186 To apply the proposed framework in mineral and phosphate mining environments,187 a reference system architecture was designed (Figure 1). Mining operations typically188 deploy wireless sensor nodes, mobile equipment, and communication relays in underground189 tunnels and surface plants. These nodes are exposed to severe electromagnetic interference190 from metallic machinery, stratified rock formations, and confined tunnel geometries.191 The architecture integrates substrate isolation techniques—Buried Diffused Layers192 (BDL), Metallized Grids, Guard Rings, and Electromagnetic Band Gap (EBG) structures—193 treated as strategic players in the game-theoretic optimizer. Through equilibrium analysis,194 the framework derives balanced strategies that minimize coupling while constraining inser-195 tion loss and fabrication cost. The output is a stable equilibrium solution, enabling robust196 and cost-effective wireless connectivity for applications such as worker safety monitoring,197 autonomous vehicle guidance, and environmental sensing in phosphate mines.198 4.2. Simulation Environment199 All simulations were performed in Ansys HFSS, with post-processing and optimization200 executed in MATLAB.201 • Frequency Range: 2–12 GHz (capturing sub-6 GHz and higher mining communi-202 cation bands).203 • Substrate Material: High-resistivity silicon, thickness 500 µm, relative permittiv-204 ity εr = 11.9.205 • Isolation Methods: BDL, Interrupted BDL, Metallized Grid, Guard Ring, Hybrid206 (BDL+Grid), Game-Theoretic framework.207 • Boundary Conditions: Perfectly Matched Layers (PML).208 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 8 of 17 Figure 1: System architecture for game-theoretic optimization of electromagnetic isolation in mining environ- ments. • Metrics: Coupling parameter (S21), insertion loss (S11), shielding effectiveness209 (SE), fabrication complexity index (FCI), Pareto efficiency.210 Solver Settings and Reproducibility: All simulations were conducted using the Ansys211 HFSS finite element solver. A driven modal solution type was used with adaptive meshing.212 The maximum mesh refinement was set to achieve a convergence criterion of ∆S < 0.01 dB213 between successive passes. The initial mesh density was configured automatically based214 on the smallest feature size, typically yielding 100,000–150,000 tetrahedral elements. Ra-215 diation boundaries were modeled using Perfectly Matched Layers (PML), and symmetry216 planes were applied when applicable to reduce computational load. Frequency sweeps217 were performed using an interpolating sweep from 2 to 12 GHz with a step resolution of218 0.1 GHz. These settings ensure numerical stability, high accuracy, and reproducibility of219 all reported results.220 4.3. Benchmark Configurations221 Before presenting the simulation outcomes, it is essential to define the reference config-222 urations that serve as benchmarks for comparison. Each configuration represents a distinct223 isolation approach, ranging from conventional buried layers to the proposed equilibrium-224 based framework. Table 2 provides a structured overview of the six cases studied, which225 form the basis for the performance evaluations in the following subsections.226 4.4. Coupling Parameter Results (S21)227 The coupling parameter (S21) quantifies the electromagnetic interference between ad-228 jacent structures and is a critical metric for assessing isolation performance. Strong sup-229 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 9 of 17 Table 2: Simulated Configurations Case Configuration Description C1 Uninterrupted BDL Baseline reference. C2 Interrupted BDL Periodic substrate gaps. C3 Metallized Grid Shielding mesh structure. C4 Guard Ring Via-based isolation ring. C5 Hybrid BDL+Grid Structural hybrid approach. C6 Game-Theoretic Framework Proposed equilibrium solution. pression of S21 across the operating frequency band is essential to ensure reliable wireless230 communication in mining environments. Table 3 and Figure 2 summarize the comparative231 results across all configurations. Table 3: Coupling Parameter (S21) in dB Freq (GHz) C1 C2 C3 C4 C5 C6 2 -30 -40 -38 -36 -50 -55 6 -25 -35 -34 -33 -45 -50 10 -20 -30 -28 -27 -40 -48 12 -18 -28 -26 -25 -38 -45 232 2 4 6 8 10 12 Frequency (GHz) −55 −50 −45 −40 −35 −30 −25 −20 Co u( lin g Pa ra m t r S 21 (d B) C1: Unint rru(t d BDL C2: Int rru(t d BDL C3: M talliz d Grid C4: Guard Ring C5: H−brid BDL+Grid C6: Game-Theoretic Figure 2: Frequency response of coupling parameter S21 across all configurations. Interpretation: Baseline BDL (C1) degrades severely above 8 GHz. Guard Ring (C4)233 improves isolation moderately, while the hybrid design (C5) enhances performance, but234 at a higher cost. The proposed framework (C6) sustains > −45 dB isolation at 12 GHz,235 suitable for mining links.236 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 10 of 17 4.5. Insertion Loss Results (S11)237 Insertion loss (S11) measures the additional attenuation introduced by each isolation238 technique, reflecting its impact on overall signal integrity. While improved isolation is239 desirable, excessive insertion loss can undermine power efficiency—particularly critical in240 energy-constrained mining devices. Table 4 and Figure 3 present the simulated insertion241 loss for the six benchmark cases. Table 4: Insertion Loss (S11) in dB Freq (GHz) C1 C2 C3 C4 C5 C6 2 -2.0 -2.5 -2.8 -2.6 -3.2 -3.0 6 -3.5 -4.0 -4.2 -4.0 -4.8 -4.5 10 -5.0 -5.5 -5.8 -5.4 -6.5 -6.2 12 -6.0 -6.5 -6.8 -6.5 -7.2 -6.9 242 2 4 6 8 10 12 Frequency (GHz) −7 −6 −5 −4 −3 −2 In s rti on L os s S 11 (d B) C1: Unint rr−(t d BDL C2: Int rr−(t d BDL C3: M talliz d Grid C4: G−ard Ring C5: Hybrid BDL+Grid C6: Game-Theoretic Figure 3: Frequency response of insertion loss S11 across all configurations. Interpretation: Insertion loss increases with frequency for all methods. The proposed243 method (C6) introduces only ≈ 0.5–1 dB overhead, acceptable given its ≈ 30 dB isolation244 gain.245 4.6. Shielding Effectiveness and Fabrication Complexity246 Beyond isolation and insertion loss, the practicality of each design must be evaluated in247 terms of manufacturability and cost. Shielding effectiveness (SE) reflects the structure’s248 ability to block unwanted radiation, while the fabrication complexity index (FCI) pro-249 vides a normalized estimate of design and manufacturing overhead. Table 5 and Figure 4250 illustrate this trade-off across the six configurations.251 Interpretation: Hybrid C5 achieves strong shielding but at high complexity. C6 balances252 both, yielding the best shielding (32 dB) with moderate complexity.253 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 11 of 17 Table 5: Shielding Effectiveness (SE) and Fabrication Complexity Index (FCI) Case SE (dB) FCI (0–1) C1 15 0.2 C2 22 0.4 C3 20 0.5 C4 19 0.3 C5 28 0.7 C6 32 0.6 C1 C2 C3 C4 C5 C6 Configuration 0 5 10 15 20 25 30 Sh ie ld in g Ef fe ct iv en es s ( dB ) 0.2 0.3 0.4 0.5 0.6 0.7 Fa br ica tio n Co m pl ex ity In de x (0 –1 ) Figure 4: Comparison of shielding effectiveness (SE) and fabrication complexity index (FCI). 4.7. Pareto Front Analysis254 Since isolation, insertion loss, and complexity are interdependent objectives, it is in-255 sufficient to evaluate them in isolation. Pareto front analysis provides a multi-objective256 perspective, highlighting trade-offs and identifying configurations that cannot be improved257 in one metric without compromising another. Figures 5 and 6 depict the two- and three-258 dimensional Pareto fronts, respectively, demonstrating the efficiency of the proposed equi-259 librium framework.260 Interpretation: C1–C4 lie inside the Pareto front (sub-optimal). C5 performs well in261 isolation but poorly in complexity. C6 lies on the Pareto frontier, demonstrating multi-262 objective optimality.263 4.8. Practical Implications for Mining Applications264 The proposed equilibrium framework provides practical advantages for mining opera-265 tions:266 • Underground Phosphate Mines: Ensures stable communication near drilling267 and crushing machinery, even under severe EM interference.268 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 12 of 17 25 30 35 40 45 50 Isolation Deficit |S21| (lower is better) 4.2 4.4 4.6 4.8 5.0 5.2 5.4 In se rti on L os s D ef ici t | S 1 1| (lo we r i s b et te r) C1 C2 C3 C4 C5 C6 Figure 5: 2D Pareto front: isolation vs. insertion loss. Isolation Deficit |S21| (lower better) 25 30 35 40 45 50 Ins ert ion Lo ss De fici t |S 11 | (l ow er be tte r) 4.2 4.4 4.6 4.8 5.0 5.2 5.4 Fa br ica tio n Co m pl ex ity In de x (lo we r b et te r) 0.2 0.3 0.4 0.5 0.6 0.7 C1 C2 C3 C4 C5 C6 Figure 6: 3D Pareto front: isolation, insertion loss, and fabrication complexity. • Underground Mineral Mines: Supports reliable MIMO links for safety sensors269 and IoT-based control systems in confined tunnels.270 • Surface Extraction Sites: Enables scalable deployment of wireless nodes for au-271 tonomous vehicles and UAV-based inspection systems.272 5. Discussion and Practical Implications273 5.1. Interpretation of Results274 The simulation results presented in Section IV demonstrate the superiority of the275 proposed game-theoretic framework (C6) across multiple performance dimensions. The276 coupling parameter analysis (S21) confirmed that baseline approaches such as BDL (C1)277 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 13 of 17 and Guard Rings (C4) fail to maintain isolation above 8 GHz, while the equilibrium-278 driven design sustains values below −45 dB up to 12 GHz. This resilience is critical in279 underground mining environments, where metallic structures and stratified rock surfaces280 exacerbate high-frequency coupling.281 The insertion loss analysis (S11) showed that although the proposed framework in-282 troduces a small penalty (≈ 0.5–1 dB), this is negligible compared to the 25–30 dB im-283 provements in isolation. Moreover, the shielding effectiveness vs fabrication complexity284 comparison revealed that the equilibrium solution achieves the highest SE (32 dB) while285 avoiding the steep complexity of brute-force hybrid designs. The Pareto analysis further286 established that C6 lies directly on the multi-objective frontier, demonstrating that it is287 the most rational choice when balancing performance and cost.288 5.2. Strategic Advantages of the Game-Theoretic Approach289 Conventional optimization methods often focus on maximizing one metric while ne-290 glecting trade-offs. The game-theoretic model naturally integrates these competing ob-291 jectives by treating isolation techniques as strategic players. Key strategic advantages292 include:293 • Equilibrium Stability: Once equilibrium is reached, no player can unilaterally294 improve performance, leading to robust and balanced outcomes.295 • Scalability: New techniques, such as metamaterials and intelligent surfaces, can be296 incorporated as new players without disrupting the framework.297 • Flexibility: Utility weights (α, β, γ) allow mining system designers to prioritize298 isolation, cost, or energy efficiency depending on operational needs.299 • Efficiency: The iterative best-response algorithm converges rapidly, reducing com-300 putational burden compared to exhaustive design searches.301 5.3. Practical Implications for Mining Systems302 The results have direct implications for real-world mining operations:303 • Phosphate Mining Operations: Wireless nodes deployed near drilling and crush-304 ing equipment face extreme EM interference. The proposed framework ensures reli-305 able communication links for safe worker coordination and process monitoring.306 • Underground Mineral Mines: Confined tunnels amplify multipath interference.307 The equilibrium solution stabilizes MIMO links for safety sensors and IoT-based308 control systems.309 • Surface Extraction Sites: Open-pit mines increasingly use autonomous haul310 trucks and UAV inspection. The proposed framework provides scalable, interference-311 resilient designs that can be manufactured and deployed at scale.312 M. Ayari et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7083 14 of 17 5.4. Limitations of the Current Study313 While promising, the present study has limitations that must be acknowledged:314 (i) Frequency Range Constraint: Simulations were limited to 2–12 GHz. Validation315 at mmWave and terahertz frequencies is necessary for broader applicability.316 (ii) Simplified Fabrication Cost Modeling: The fabrication complexity index was317 normalized and does not yet reflect detailed industrial cost metrics, such as lithog-318 raphy tolerances or ruggedization.319 (iii) Lack of Experimental Prototyping: Current results are simulation-based. Fab-320 ricated prototypes and field trials in mining environments are required to validate321 robustness under real conditions.322 6. Conclusion323 This paper has introduced a game-theoretic optimization framework for electromag-324 netic isolation in high-frequency communication systems, applied to the context of mineral325 and phosphate mining environments. By modeling isolation techniques—including BDL,326 metallized grids, guard rings, and EBG structures—as strategic players, the framework327 achieves equilibrium solutions that balance isolation performance, insertion loss, and fab-328 rication complexity.329 Simulation results across 2–12 GHz demonstrated superior performance, with coupling330 reductions of up to 30 dB beyond conventional methods, while maintaining acceptable in-331 sertion loss. The equilibrium design also achieved Pareto efficiency, delivering the highest332 shielding effectiveness with moderate fabrication complexity. These findings underscore333 the ability of the framework to ensure robust, cost-effective and interference-resistant334 wireless connectivity, supporting safety, automation, and IoT applications in mining op-335 erations.336 Future research will extend the framework to include prototype fabrication and experi-337 mental validation in real mining sites, as well as integration with AI/ML prediction models338 to accelerate equilibrium computation. Dynamic and cooperative game formulations will339 also be explored to enhance adaptability under changing conditions, paving the way for340 deployment at higher frequency ranges (28–60 GHz and beyond) and in large-scale mining341 infrastructures.342 Acknowledgements343 The authors extend their appreciation to the Deanship of Scientific Research at North-344 ern Border University, Arar, KSA for funding this research work through the project345 number “NBU-FFMRA-2025-2443-01”.346 M. Ayari et al. / Eur. J. Pure Appl. 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