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Fractal Fract
Volume 9
Issue 12
10.3390/fractalfract9120756
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Article

Nabla Fractional Distributed Nash Seeking for Non-Cooperative Games

by
Yao Xiao
1,
Sunming Ge
1,
Yihao Qiao
2,
Tieqiang Gang
1,* and
Lijie Chen
1,*
1
School of Aerospace Engineering, Xiamen University, Xiamen 361000, China
2
H. Milton School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
*
Authors to whom correspondence should be addressed.
Fractal Fract. 2025, 9(12), 756; https://doi.org/10.3390/fractalfract9120756
Submission received: 22 October 2025 / Revised: 14 November 2025 / Accepted: 19 November 2025 / Published: 21 November 2025
(This article belongs to the Special Issue Distributed Control and Optimization of Fractional-Order Systems: Theory and Applications)
Versions Notes

Abstract

This paper pioneers the introduction of nabla fractional calculus into distributed Nash equilibrium (NE) seeking for non-cooperative games (NGs), proposing several novel discrete-time fractional-order algorithms. We first develop a gradient play-based algorithm under perfect information and subsequently extend it to partial-information settings. Two types of communication network topologies among agents, namely connected undirected graphs and strongly connected unbalanced directed graphs, are explicitly considered. When the pseudo-gradient mapping of the NG is Lipschitz continuous and strongly monotone, the proposed algorithms are proven to achieve asymptotic convergence to the NE with at least a Mittag–Leffler convergence rate. Both the step size and the fractional order act as tunable parameters that jointly influence the convergence performance. Numerical experiments on potential games and Nash–Cournot games demonstrate the effectiveness of the proposed algorithms.
Keywords: non-cooperative games; distributed Nash equilibrium seeking; fractional-order dynamics; nabla fractional calculus; network topologies non-cooperative games; distributed Nash equilibrium seeking; fractional-order dynamics; nabla fractional calculus; network topologies

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MDPI and ACS Style

Xiao, Y.; Ge, S.; Qiao, Y.; Gang, T.; Chen, L. Nabla Fractional Distributed Nash Seeking for Non-Cooperative Games. Fractal Fract. 2025, 9, 756. https://doi.org/10.3390/fractalfract9120756

AMA Style

Xiao Y, Ge S, Qiao Y, Gang T, Chen L. Nabla Fractional Distributed Nash Seeking for Non-Cooperative Games. Fractal and Fractional. 2025; 9(12):756. https://doi.org/10.3390/fractalfract9120756

Chicago/Turabian Style

Xiao, Yao, Sunming Ge, Yihao Qiao, Tieqiang Gang, and Lijie Chen. 2025. "Nabla Fractional Distributed Nash Seeking for Non-Cooperative Games" Fractal and Fractional 9, no. 12: 756. https://doi.org/10.3390/fractalfract9120756

APA Style

Xiao, Y., Ge, S., Qiao, Y., Gang, T., & Chen, L. (2025). Nabla Fractional Distributed Nash Seeking for Non-Cooperative Games. Fractal and Fractional, 9(12), 756. https://doi.org/10.3390/fractalfract9120756

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