Seeking Nash Equilibrium in Non-cooperative Quadratic Games Under Delayed Information Exchange

Abstract: In this paper, we investigate the seeking of Nash equilibrium (NE) in a non-cooperative quadratic game where all agents exchange their delayed strategy information with their neighbors. To extend best-response algorithms to the delayed information setting, an estimation mechanism for each agent to estimate the current strategy profile is designed. Based on the best-response strategy to the estimations, the strategy profile dynamics of all agents is established, which is revealed to converge asymptotically to the NE when agents exchange multi-step-delay information via the Lyapunov-Krasovskii functional approach. In the scenario where agents exchange one-step-delay information, the exponential convergence of the strategy profile dynamics to the NE can be guaranteed by restricting the learning rate to less than an upper bound. Moreover, a lower bound on the learning rate for instability of the NE is proposed. Numerical simulations are provided for verifying the developed results.

• The paper develops an estimation-based best-response algorithm that converges to the unique Nash equilibrium under multi-step delays when a Lyapunov–Krasovskii LMI is feasible.
• For one-step delays, the method achieves deterministic exponential convergence when the learning rate satisfies ξ < δ₁ and provably diverges when ξ exceeds δ₂.
• Simulations show that stable learning rates generally decrease as delay and network size grow, while the exact LMI-based stability boundary remains topology-dependent and conservative.

Problem setting and motivation

The paper studies distributed Nash equilibrium (NE) seeking in a static non-cooperative quadratic game in which each agent maximizes a private quadratic payoff $$J_i(\mathbf{s})=\frac{1}{2}\mathbf{s}^\top A_i\mathbf{s}+\mathbf{b}_i^\top\mathbf{s}+g_i$$, with $a_{ii}^i<0$. Under a strict diagonal-dominance condition on each $A_i$ (Assumption 1), the NE exists, is unique, and equals $(\mathbf{I}-M)^{-1}\mathbf{c}$ for the best-response matrix $M$ and offset $\mathbf{c}$. The distinguishing feature of the setting is that agents communicate over an undirected connected graph and can only exchange $\tau$-step-delayed strategy and estimation information: at stage $t$, agent $i$ receives only $s_j(t-\tau)$ and $a_{ii}^i<0$0 from neighbors. The authors motivate this both by communication latency in networked systems and by strategic privacy concerns—agents may deliberately disclose only outdated strategies to conceal their current decisions.

The core algorithmic idea is an estimation-based best-response scheme. Each agent maintains estimates $a_{ii}^i<0$1 of every other agent's current strategy and plays the best response to its own estimate vector. Estimates are updated by a consensus-plus-correction rule combining neighbor estimation disagreement $a_{ii}^i<0$2 with the innovation $a_{ii}^i<0$3, scaled by a learning rate $a_{ii}^i<0$4. Stacking all quantities yields a linear delay system of dimension $a_{ii}^i<0$5 driven by matrices built from the Laplacian $a_{ii}^i<0$6, the adjacency-derived diagonal $a_{ii}^i<0$7, and the best-response structure $a_{ii}^i<0$8.

Convergence under multi-step delays ($a_{ii}^i<0$9)

For $A_i$0, the closed-loop dynamics take the form $A_i$1 after differencing. Theorem 1 establishes global asymptotic convergence of the strategy profile to the NE provided there exist positive definite matrices $A_i$2 satisfying a linear-matrix-inequality-type condition $A_i$3 of dimension $A_i$4. The proof constructs a Lyapunov–Krasovskii functional with three terms—a quadratic term on an augmented state, a delay-window sum, and a summation-inequality-based term following Seuret, Gouaisbaut, and Fridman—and shows the forward difference is negative definite. Equilibrium analysis then confirms that the unique equilibrium of the augmented system coincides with the augmented NE state, using irreducible diagonal dominance of $A_i$5.

Two caveats deserve emphasis. First, the Lyapunov–Krasovskii construction is explicitly valid only for $A_i$6: at $A_i$7 the summation limits invert and the analysis breaks down, which is why the one-step case is treated separately. Second, the condition $A_i$8 is sufficient but not constructive—the paper concedes that the structural complexity of $A_i$9 prevents a theoretical characterization of admissible learning rates, and feasibility must be checked numerically via LMI solvers. In the five-agent wheel-graph example, feasible solutions exist up to $(\mathbf{I}-M)^{-1}\mathbf{c}$0 with $(\mathbf{I}-M)^{-1}\mathbf{c}$1 but not for $(\mathbf{I}-M)^{-1}\mathbf{c}$2, where simulations show divergence; the authors note this "may imply" divergence rather than proving it, so the failure of the LMI at $(\mathbf{I}-M)^{-1}\mathbf{c}$3 should not be read as a necessary instability condition.

Exponential convergence and an instability threshold for $(\mathbf{I}-M)^{-1}\mathbf{c}$4

For one-step-delay exchange, the estimation update degenerates into a delay-free recursion, and the augmented system becomes $(\mathbf{I}-M)^{-1}\mathbf{c}$5. Theorem 2 proves global exponential convergence when

$(\mathbf{I}-M)^{-1}\mathbf{c}$6

via a Gershgorin disk argument: disks associated with the first block lie inside the unit circle by Assumption 1, while disks of the second block are inscribed in the unit circle whenever $(\mathbf{I}-M)^{-1}\mathbf{c}$7. A contradiction argument rules out $(\mathbf{I}-M)^{-1}\mathbf{c}$8 as an eigenvalue, giving $(\mathbf{I}-M)^{-1}\mathbf{c}$9. This is a stronger result than related work: compared with asynchronous algorithms requiring an auxiliary interference graph that guarantee only almost-sure convergence [Li2025], or sub-linearly convergent delayed seeking schemes [LiuJ2024], this algorithm is synchronous, uses only the communication graph, and achieves deterministic exponential convergence.

Theorem 3 complements this with a lower bound for instability: if $M$0, then the mean eigenvalue of $M$1 falls below $M$2, forcing $M$3 and divergence. Together, Theorems 2 and 3 bracket the admissible learning rate from above and below, though the interval $M$4 may be empty or the bounds conservative. Indeed, the paper acknowledges that $M$5 is conservative—convergence is observed empirically for some $M$6—and numerical sensitivity analysis shows $M$7 tracks the true maximal convergent learning rate $M$8 well for complete graphs (both tending to zero as $M$9 grows) but poorly for ring graphs, where $\mathbf{c}$0 while $\mathbf{c}$1. A notable topological finding is that the guaranteed upper bound for ring graphs is always $\mathbf{c}$2 regardless of agent count, making rings the most robust topology for large games among those analyzed.

Numerical evidence and scalability

Simulations corroborate the theory across three examples. In the five-agent wheel graph with $\mathbf{c}$3 and $\mathbf{c}$4, strategy errors decay exponentially; sweeping $\mathbf{c}$5 from 0.05 to 1 reveals a U-shaped terminal-stage curve, with instability beyond $\mathbf{c}$6 at $\mathbf{c}$7, consistent with $\mathbf{c}$8 being a valid but loose instability bound. A 20-agent example demonstrates scalability: convergence holds in four of five tested configurations, while the divergent case ($\mathbf{c}$9, $\tau$0, ring graph) violates the LMI condition of Theorem 1. The most practically relevant empirical observation is an inverse relationship between the maximal stable learning rate and both the delay step $\tau$1 and the number of agents $\tau$2: longer delays and larger networks require smaller learning rates.

Limitations and open questions

Several limitations are stated or evident. The LMI condition of Theorem 1 provides no analytic recipe for choosing $\tau$3 as a function of $\tau$4 and $\tau$5, and no existence conditions for $\tau$6 are given; the gap between LMI feasibility and actual stability (as at $\tau$7) is unresolved. The bounds $\tau$8 and $\tau$9 are provably conservative for some topologies, and the exact stability boundary $t$0 lacks closed-form characterization. The framework assumes noiseless, lossless, synchronous communication over a fixed undirected connected graph, deterministic quadratic payoffs satisfying strict diagonal dominance, and scalar strategies per agent—all restrictive relative to realistic deployments. The authors list as future work: existence conditions for the LMI variables, more general payoff structures, quantitative analysis of how $t$1 and $t$2 affect convergence speed, and extensions to noise, packet dropout, and asynchronous updates.

Conclusion

This paper delivers a complete convergence picture for estimation-based best-response NE seeking under delayed information exchange in quadratic games: asymptotic convergence via a Lyapunov–Krasovskii LMI condition for multi-step delays, exponential convergence under an explicit learning-rate upper bound for one-step delays, and an explicit learning-rate threshold above which the dynamics provably diverge. The main practical takeaway—that permissible learning rates shrink inversely with delay length and network size—is supported both theoretically and numerically, though the conservatism of the derived bounds and the lack of constructive rate selection remain open problems.