Mixed-Strategy Nash Equilibrium Resolution in Multi-target–Attacker–Defender Games via an Improved Particle Swarm Optimization Algorithm

This paper proposes a unified game-theoretic framework that integrates geometric modeling based on Apollonius circles with Mixed-Strategy Nash Equilibrium (MSNE) analysis to formulate the Multi-Target–Attacker–Defender (MTAD) confrontation problem. A novel payoff function is designed to quantify the relative superiority of agents in attack, defense, and evasion based on geometric overlaps. The engagement is modeled as a finite nonzero-sum matrix game, where each player’s strategy corresponds to a dynamic task allocation scheme.Such strategic interactions among multiple agents naturally lead to a complex decision-making landscape, giving rise to a high-dimensional, nonconvex optimization problem. To address the resulting optimization problem, a Strong–Weak Adaptive Particle Swarm Optimization (SW-APSO) algorithm is developed, incorporating a strength-guided evaluation mechanism and adaptive inertia to balance exploration and exploitation. Simulation results and UAV-based experiments indicate that the proposed framework achieves stable Nash equilibria with accurate convergence and strong adaptability in dynamic adversarial scenarios.