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arxiv: 2606.23011 · v1 · pith:R7SRHEX4new · submitted 2026-06-22 · 📡 eess.SY · cs.SY

Robust Data-Driven Nash Equilibrium Seeking under Partial-Decision Information

Linqi Wang , Yifei Li , Wenjie Liu , Yuzhou Wei , Gang Wang , Lihua Xie This is my paper

Pith reviewed 2026-06-26 07:17 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords data-driven controlNash equilibrium seekingmulti-agent systemspartial-decision informationoutput regulationsemi-definite programmingdisturbance rejection
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The pith

Nash equilibrium controllers for multi-agent systems with unknown dynamics are synthesized directly from noisy data by recasting the problem as cooperative output regulation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a data-driven approach for agents to reach a Nash equilibrium when their linear dynamics are unknown, they receive only partial information about others' decisions, and external disturbances and equality constraints are present. The central move is to rewrite the equilibrium-seeking task as a cooperative output regulation problem that incorporates an internal model for disturbance rejection. Controllers are then obtained by solving semi-definite programs on collected noisy input-state measurements, with proofs of closed-loop stability and asymptotic convergence. This matters because many real multi-agent applications, such as vehicle formations, operate without accurate models and with limited communication. The method is further extended to certain nonlinear systems with constant disturbances through integral control and quadratic constraints on the nonlinearities.

Core claim

By reformulating decentralized Nash equilibrium seeking under partial-decision information and equality constraints as a cooperative output regulation problem, controllers can be synthesized directly from noisy input-state data via semi-definite programs, providing formal guarantees of closed-loop stability and asymptotic convergence to the Nash equilibrium for linear systems subject to exogenous disturbances, with an extension to a class of nonlinear systems via integral control and quadratic constraints.

What carries the argument

Reformulation of the Nash equilibrium seeking problem as a cooperative output regulation problem, followed by data-driven semi-definite program synthesis of controllers from noisy input-state measurements

If this is right

  • Agents reach the Nash equilibrium with formal stability guarantees even when exact dynamics are unavailable.
  • Partial-decision information and equality constraints are handled without requiring full communication among agents.
  • Exogenous disturbances are rejected through the internal model while convergence is maintained.
  • The same data-driven synthesis extends to nonlinear systems with constant disturbances using integral action and quadratic constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could reduce the need for system identification in environments where dynamics change over time.
  • Similar reformulations might apply to other distributed game problems or optimization tasks in control.
  • Validation on physical multi-agent hardware would test whether the noisy-data guarantees hold beyond simulation.

Load-bearing premise

The multi-agent system with unknown linear dynamics, exogenous disturbances, partial-decision information, and equality constraints can be validly recast as a cooperative output regulation problem for which data-driven SDP synthesis yields the required stability and convergence properties.

What would settle it

Apply the semi-definite program to noisy data collected from the system; if the resulting controller produces a closed-loop trajectory that does not converge asymptotically to the Nash equilibrium or loses stability under the modeled disturbances, the claim is falsified.

Figures

Figures reproduced from arXiv: 2606.23011 by Gang Wang, Lihua Xie, Linqi Wang, Wenjie Liu, Yifei Li, Yuzhou Wei.

Figure 1
Figure 1. Figure 1: Scheme of the closed-loop system. where ηi is the internal model state driven by the local error ei . Remark 3.3 Assuming the exosystem matrix Ei to be known is standard in output regulation theory (Huang, 2004). In the present direct data-driven setting, this knowledge is required to construct the internal model in (14). By contrast, observer-based disturbance-rejection methods typically require the plant… view at source ↗
Figure 2
Figure 2. Figure 2: Communication graph of the UAV network. where yi , vi , ui ∈ R 3 denote the position, veloc￾ity, and control input, respectively. Defining xi = col(yi , vi), the above dynamics can be written in the form of (1). The agent parameters are m = [1.2, 1.5, 1.8, 1.35, 1.65, 1.4] kg, and k = [0.24, 0.30, 0.40, 0.29, 0.27, 0.32] N·s/m. The wind disturbance di ∈ R 3 is generated by the exosystem ˙di =     0 π 1… view at source ↗
Figure 5
Figure 5. Figure 5: NE seeking error of each agent under data-driven control and model-based control. the communication protocol (13) and one without the KKT dynamics (12a) [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 3
Figure 3. Figure 3: Trajectories of the UAV network. 0 5 10 -200 -100 0 100 0 5 10 -200 -100 0 100 0 5 10 -200 -100 0 100 0 5 10 -200 -100 0 100 0 5 10 -200 -100 0 100 0 5 10 -200 -100 0 100 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Regulation errors of the UAV network. 1) Comparison with the model-based approach. For com￾parison, we implement a model-based controller in which the exact system matrices, available only in simulation, are used to compute the decentralized feedback gains. The NE model, communication protocol, and internal model are kept the same. As shown in [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: Equality-constraint residual with and without the KKT dynamics (12a). 13 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: NE seeking error under the proposed and reduced designs. u LR I E M mg T [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Schematic diagram of the wing load. 6.2 Rotary-Wing Aerial Vehicle Formation To evaluate the proposed nonlinear design on heteroge￾neous agents, we consider a network of five rotary-wing aerial vehicles communicating over the connected graph in [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Communication graph of rotary-wing aerial vehicles [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Trajectories of rotary-wing aerial vehicles. 0 10 20 30 40 50 60 -10 -5 0 5 0 10 20 30 40 50 60 -10 -5 0 5 0 10 20 30 40 50 60 -10 -5 0 5 0 10 20 30 40 50 60 -10 -5 0 5 0 10 20 30 40 50 60 -10 -5 0 5 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Regulation errors of rotary-wing aerial vehicles. under unknown dynamics and exogenous disturbances. By recasting the problem as a cooperative output regu￾lation problem, the proposed approach integrates an NE model, a communication protocol, an internal model, and a direct data-driven controller, without explicit system identification. For linear systems subject to ex￾ogenous disturbances generated by kn… view at source ↗
read the original abstract

This paper presents a data-driven framework for decentralized Nash equilibrium (NE) seeking in multi-agent systems with unknown linear dynamics subject to exogenous disturbances, operating under partial-decision information (where agents lack direct access to the decisions of all others) and equality constraints. The proposed framework integrates an NE model, a distributed communication protocol, an internal model for disturbance rejection, and a data-driven stabilization strategy. By reformulating the problem as a cooperative output regulation problem, we synthesize controllers directly from noisy input-state data via semi-definite programs (SDPs), providing formal guarantees for closed-loop stability and asymptotic convergence to the NE. The approach is further extended to a class of nonlinear systems with constant disturbances by leveraging integral control and describing nonlinearities via quadratic constraints. Numerical simulations involving unmanned aerial vehicle networks and a rotary-wing aerial vehicle formation validate the efficacy and robustness of the proposed method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims a data-driven framework for decentralized Nash equilibrium seeking in multi-agent systems with unknown linear dynamics, exogenous disturbances, partial-decision information, and equality constraints. It integrates an NE model, distributed protocol, internal model, and data-driven stabilization; reformulates the problem as cooperative output regulation; and synthesizes controllers from noisy input-state data via SDPs to guarantee closed-loop stability and asymptotic NE convergence. The approach extends to nonlinear systems with constant disturbances via integral control and quadratic constraints, with validation on UAV networks and rotary-wing formations.

Significance. If the recasting and SDP-based synthesis hold with the claimed guarantees, the work would offer a notable advance in combining output regulation, data-driven control, and game-theoretic NE seeking under partial information and disturbances, enabling robust decentralized protocols for applications like UAV formations without requiring full model knowledge.

major comments (2)
  1. [Main body (reformulation and SDP synthesis sections)] No section supplies an explicit state-space embedding that simultaneously encodes the NE variational inequality, equality constraints, and partial-information graph while preserving the distributed information structure. This embedding is required for the reformulation to cooperative output regulation and for the subsequent noisy-data SDP to certify both internal stability and regulation to the NE; without it the central claim cannot be verified.
  2. [Abstract and § on data-driven synthesis] The abstract asserts formal guarantees for stability and asymptotic convergence via SDPs but provides no derivation steps, error bounds on the data, or conditions under which the noisy input-state pairs yield a valid controller. These details are load-bearing for the data-driven claim and must be supplied with explicit assumptions on the data set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address each major comment below, indicating where revisions will be made to strengthen the presentation of the reformulation and data-driven guarantees.

read point-by-point responses

  1. Referee: [Main body (reformulation and SDP synthesis sections)] No section supplies an explicit state-space embedding that simultaneously encodes the NE variational inequality, equality constraints, and partial-information graph while preserving the distributed information structure. This embedding is required for the reformulation to cooperative output regulation and for the subsequent noisy-data SDP to certify both internal stability and regulation to the NE; without it the central claim cannot be verified.

    Authors: We agree that a single consolidated state-space embedding would improve clarity. The current manuscript encodes the variational inequality via the pseudo-gradient and the equality constraints via an auxiliary variable in the internal model, while the partial-information graph appears through the Laplacian in the distributed protocol. However, these elements are presented piecewise. In the revision we will insert an explicit augmented state vector x_aug = [x; λ; e] (where λ are the multipliers for the constraints and e the regulation errors) together with the block-structured system matrix that preserves the block-diagonal structure required by the distributed protocol. This will be placed in a new subsection of the reformulation section and will be used to state the cooperative output-regulation problem before the SDP synthesis. revision: yes

  2. Referee: [Abstract and § on data-driven synthesis] The abstract asserts formal guarantees for stability and asymptotic convergence via SDPs but provides no derivation steps, error bounds on the data, or conditions under which the noisy input-state pairs yield a valid controller. These details are load-bearing for the data-driven claim and must be supplied with explicit assumptions on the data set.

    Authors: The derivation of the SDP (based on the data-driven matrix inequality obtained via the matrix S-lemma applied to the noise bound ||Δ|| ≤ ε) is contained in the data-driven synthesis section, together with the persistence-of-excitation assumption on the collected trajectories. The abstract, however, is too terse. We will revise the abstract to include the sentence “under the assumption of bounded measurement noise and sufficiently rich data satisfying a persistence-of-excitation condition.” In the main text we will add an explicit theorem stating the LMI feasibility condition, the resulting closed-loop stability margin, and the bound on the residual regulation error as a function of the noise level ε. A short remark will also list the precise data-set assumptions required for the SDP to certify internal stability and asymptotic NE convergence. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external reformulation and data

full rationale

The provided abstract and description contain no quoted equations or steps that reduce by construction to fitted inputs, self-definitions, or self-citation chains. The central move (reformulating NE seeking under partial information and constraints as cooperative output regulation, then applying data-driven SDP synthesis) is presented as a modeling choice drawing on noisy input-state data rather than a tautological renaming or parameter fit. No load-bearing uniqueness theorem or ansatz is shown to originate solely from the authors' prior work within the given text. This is the expected honest non-finding for a paper whose core synthesis step uses external measurements.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes several domain assumptions whose validity cannot be inspected without the full manuscript: that the system admits a reformulation as cooperative output regulation, that noisy input-state data suffice for SDP synthesis of stabilizing controllers, and that the extension to nonlinear systems via quadratic constraints preserves the guarantees.

axioms (2)
  • domain assumption The multi-agent dynamics admit a reformulation as a cooperative output regulation problem under partial-decision information and equality constraints.
    Stated in the abstract as the basis for controller synthesis.
  • domain assumption Noisy input-state data collected from the unknown linear system are sufficient for SDP-based data-driven stabilization with formal closed-loop guarantees.
    Central to the data-driven part of the method.

pith-pipeline@v0.9.1-grok · 5689 in / 1480 out tokens · 14263 ms · 2026-06-26T07:17:20.116549+00:00 · methodology

discussion (0)

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