Mean-Field-Type Game Theory with Rosenblatt Noise

TL;DR

This paper introduces a novel mean-field game framework incorporating Rosenblatt noise, a non-Gaussian process, providing new tools for modeling, control, and equilibrium analysis in systems with complex, long-range dependent stochastic behaviors.

Contribution

It develops stochastic calculus for Rosenblatt processes and extends game theory to non-Gaussian noise environments, revealing new equilibrium conditions and highlighting the limitations of traditional Gaussian-based models.

Findings

Rosenblatt noise captures real data behaviors like skewness and heavy tails.

New stochastic calculus formulas for Rosenblatt processes are derived.

Conditions for saddle-point and Nash equilibria under Rosenblatt noise are established.

Abstract

We study the integration of Rosenblatt noise into stochastic systems, control theory, and mean-field-type game theory, addressing the limitations of traditional Gaussian and Markovian models. Empirical evidence from various domains, including water demand, e-commerce, power grid operations, wireless channels, and agricultural supply chains, demonstrates the prevalence of non-Gaussian characteristics such as skews, heavy tails and strong long-range dependencies. The Rosenblatt process, a non-Gaussian non-Markovian, self-similar process, offers a baseline framework for capturing some the behaviors observed in real data. We develop novel stochastic calculus formulas for Rosenblatt processes, apply these to dynamical systems, and analyze optimal control problems, revealing the suboptimality of traditional noise approximation methods. We extend game-theoretic analysis to environments driven by Rosenblatt noise, establishing conditions for saddle-point equilibria in zero-sum games and identifying state-feedback Nash equilibria in non-zero-sum games. Our findings underscore the importance of incorporating non-Gaussian noise into predictive analytics and control strategies, enhancing the accuracy and robustness of models in real-world applications. These findings represent a significant advancement in mean-field-type game theory with variance-awareness, offering new insights and tools for managing interactive systems influenced by Rosenblatt noise.