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Quantized distributed Nash equilibrium seeking under DoS attacks

Shuai Feng, Maojiao Ye, Lihua Xie, Shengyuan Xu

TL;DR

The paper addresses distributed Nash equilibrium seeking in networks where communications are both quantized and intermittently disrupted by Denial-of-Service (DoS) attacks. It introduces a dynamic quantization scheme with zooming-in and holding, paired with a consensus-based estimator, and proves a DoS resilience bound $1/T + \Delta/\tau_D < 1$ along with explicit lower bounds on quantizer levels that guarantee non-saturation and convergence to the NE. The analysis handles nonlinear payoff coupling via the nonlinear term $P(\eta)$ and derives four DoS-case dynamics to establish convergence. Simulations demonstrate robust convergence to the NE under strong DoS, with the scaling parameter $\theta(k)$ shrinking to zero while quantized signals stay within permitted ranges. These results enable resilient, bandwidth-efficient distributed NE seeking in cyber-threat environments.

Abstract

This paper studies distributed Nash equilibrium (NE) seeking under Denial-of-Service (DoS) attacks and quantization. The players can only exchange information with their own direct neighbors. The transmitted information is subject to quantization and packet losses induced by malicious DoS attacks. We propose a quantized distributed NE seeking strategy based on the approach of dynamic quantized consensus. To solve the quantizer saturation problem caused by DoS attacks, the quantization mechanism is equipped to have zooming-in and holding capabilities, in which the holding capability is consistent with the results in quantized consensus under DoS. A sufficient condition on the number of quantizer levels is provided, under which the quantizers are free from saturation under DoS attacks. The proposed distributed quantized NE seeking strategy is shown to have the so-called maximum resilience to DoS attacks. Namely, if the bound characterizing the maximum resilience is violated, an attacker can deny all the transmissions and hence distributed NE seeking is impossible.

Quantized distributed Nash equilibrium seeking under DoS attacks

TL;DR

The paper addresses distributed Nash equilibrium seeking in networks where communications are both quantized and intermittently disrupted by Denial-of-Service (DoS) attacks. It introduces a dynamic quantization scheme with zooming-in and holding, paired with a consensus-based estimator, and proves a DoS resilience bound along with explicit lower bounds on quantizer levels that guarantee non-saturation and convergence to the NE. The analysis handles nonlinear payoff coupling via the nonlinear term and derives four DoS-case dynamics to establish convergence. Simulations demonstrate robust convergence to the NE under strong DoS, with the scaling parameter shrinking to zero while quantized signals stay within permitted ranges. These results enable resilient, bandwidth-efficient distributed NE seeking in cyber-threat environments.

Abstract

This paper studies distributed Nash equilibrium (NE) seeking under Denial-of-Service (DoS) attacks and quantization. The players can only exchange information with their own direct neighbors. The transmitted information is subject to quantization and packet losses induced by malicious DoS attacks. We propose a quantized distributed NE seeking strategy based on the approach of dynamic quantized consensus. To solve the quantizer saturation problem caused by DoS attacks, the quantization mechanism is equipped to have zooming-in and holding capabilities, in which the holding capability is consistent with the results in quantized consensus under DoS. A sufficient condition on the number of quantizer levels is provided, under which the quantizers are free from saturation under DoS attacks. The proposed distributed quantized NE seeking strategy is shown to have the so-called maximum resilience to DoS attacks. Namely, if the bound characterizing the maximum resilience is violated, an attacker can deny all the transmissions and hence distributed NE seeking is impossible.
Paper Structure (17 sections, 4 theorems, 47 equations, 3 figures, 1 algorithm)

This paper contains 17 sections, 4 theorems, 47 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Framework
  3. System description
  4. Time-constrained DoS
  5. Quantizer
  6. Control objectives
  7. quantized distributed NE seeking controller design
  8. Control architecture
  9. Quantized system dynamics
  10. Main results
  11. Analysis of zooming-in and holding quantization
  12. Quantizer unsaturation and convergence to the NE
  13. Simulation
  14. Conclusions and future research
  15. Proof of $\|\xi^x(s_r)\|_\infty \le 1/2\gamma_1$ and $\|\xi^y(s_r)\|_\infty \le 1/2\gamma_1$
  16. ...and 2 more sections

Key Result

Lemma 1

feng2023tcns Consider the DoS attacks characterized by Assumptions ass 3 and ass 4 and the network sampling period $\Delta$. If $1/T+ \Delta /\tau_D <1$, then $T_S(0, k\Delta)) \ge \left(1- 1/T - \Delta/\tau_D \right) k - \frac{\kappa+\eta\Delta}{\Delta}.$$\blacksquare$

Figures (3)

  • Figure 1: Time responses of $x(k)$ (top) and $\theta(k)$ (bottom).
  • Figure 2: The values of $Q_R(\cdot)$ for $x$ and $y$. The legend for the bottom plot is similar to that in the top plot and hence is omitted.
  • Figure 3: Dynamics of $\|x(k)-x^*\|$ without and under quantization.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 1 more