Semi-Explicit Solution of Some Discrete-Time Mean-Field-Type Games with Higher-Order Costs

Traditional solvable game theory and mean-field-type game theory (risk-aware games) predominantly focus on quadratic costs due to their analytical tractability. Nevertheless, they often fail to capture critical non-linearities inherent in real-world systems. In this work, we present a unified framework for solving discrete-time game problems with higher-order state and strategy costs involving power-law terms. We derive semi-explicit expressions for equilibrium strategies, cost-to-go functions, and recursive coefficient dynamics across deterministic, stochastic, and multi-agent system settings by convex-completion techniques. The contributions include variance-aware solutions under additive and multiplicative noise, extensions to mean-field-type-dependent dynamics, and conditions that ensure the positivity of recursive coefficients. Our results provide a foundational methodology for analyzing non linear multi-agent systems under higher-order penalization, bridging classical game theory and mean-field-type game theory with modern computational tools for engineering applications.