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

Julian Barreiro-Gomez, Tyrone E. Duncan, Bozenna Pasik-Duncan, Hamidou Tembine

TL;DR

This work addresses discrete-time mean-field-type games with higher-order costs by developing a convex-completion-based, semi-explicit Solution framework that yields equilibrium strategies, cost-to-go functions, and recursive coefficient dynamics. It extends the method to variance-aware and risk-aware settings, including multiplicative noise and mean-field dependent disturbances, with rigorous conditions ensuring positivity and well-posedness. A measure-space verification approach is provided to handle non-Markovian noise, and numerical examples demonstrate the practicality of the framework for non-quadratic, non-linear multi-agent problems. The results bridge classical game theory and modern MFTG techniques, offering a tractable pathway for analyzing and controlling nonlinear multi-agent systems in engineering contexts.

Abstract

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.

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

TL;DR

This work addresses discrete-time mean-field-type games with higher-order costs by developing a convex-completion-based, semi-explicit Solution framework that yields equilibrium strategies, cost-to-go functions, and recursive coefficient dynamics. It extends the method to variance-aware and risk-aware settings, including multiplicative noise and mean-field dependent disturbances, with rigorous conditions ensuring positivity and well-posedness. A measure-space verification approach is provided to handle non-Markovian noise, and numerical examples demonstrate the practicality of the framework for non-quadratic, non-linear multi-agent problems. The results bridge classical game theory and modern MFTG techniques, offering a tractable pathway for analyzing and controlling nonlinear multi-agent systems in engineering contexts.

Abstract

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.
Paper Structure (23 sections, 8 theorems, 111 equations, 3 figures)

This paper contains 23 sections, 8 theorems, 111 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Previous Works
  3. Contributions
  4. How does the Convex Completion Works?
  5. Convexity of Higher-Order Terms
  6. Structure of the Paper
  7. Selfish Agents with Higher Order Costs
  8. Selfish Agents with Quartic Costs
  9. Extension to $\bar{x}^{2p}$ Costs
  10. Selfish Agents with Variance-Aware Higher-Order Costs
  11. Selfish Agents with Variance-aware Quartic Costs
  12. Extension to $\bar{x}^{2p}$ Costs
  13. Risk-Aware Games under Multiplicative Noise
  14. Variance-aware Quartic Cost under Multiplicative Noise
  15. Extension to $\bar{x}^{2p}$ Costs
  16. ...and 8 more sections

Key Result

Lemma 1

Let $p\geq 1, a\neq 0, b\neq 0.$ Then the mapping $z \mapsto f(z) = z^{2p} + (az+b)^{2p}$ is strictly convex. $\square$

Figures (3)

  • Figure 1: Results example corresponding to Proposition \ref{['propos:proposition_8']} and Proposition \ref{['propos:proposition_9']}.
  • Figure 2: Results example corresponding to Proposition \ref{['propos:proposition_10']} and Proposition \ref{['propos:proposition_11']}.
  • Figure 3: Results example corresponding to Proposition \ref{['propos:proposition_12']} and Proposition \ref{['propos:proposition_13']}.

Theorems & Definitions (19)

  • Lemma 1
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Proposition 3
  • Remark 2
  • Proposition 4
  • Proposition 5
  • Remark 3
  • Proposition 6
  • ...and 9 more