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

Kaichen Jiang, Yuyue Yan, Mingda Yue, Yuhu Wu

TL;DR

This paper investigates the seeking of Nash equilibrium in a non-cooperative quadratic game where all agents exchange their delayed strategy information with their neighbors and proposes a lower bound on the learning rate for instability of the NE.

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.

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

TL;DR

This paper investigates the seeking of Nash equilibrium in a non-cooperative quadratic game where all agents exchange their delayed strategy information with their neighbors and proposes a lower bound on the learning rate for instability of the NE.

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.
Paper Structure (9 sections, 3 theorems, 58 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 9 sections, 3 theorems, 58 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Static Quadratic Game
  4. Myopic Best-Response-Based Seeking Dynamics
  5. Design of Seeking Algorithm for NE
  6. Design of Seeking Algorithm
  7. Convergence Analysis of Algorithm \ref{['SeekingAlgo']} With $\tau\geq2$
  8. Convergence analysis of Algorithm \ref{['SeekingAlgo']} With $\tau=1$
  9. Conclusion

Key Result

Theorem 1

Suppose that Assumption strctilydominant is satisfied and all agents exchange $\tau$-step-delay ($\tau\geq2$) information at current stage $t$. Under Algorithm SeekingAlgo with learning rate $\xi>0$, if there exist matrices $Q_1\in\mathbb{S}_+^{4n^2}$ and $Q_2,Q_3\in\mathbb{S}_+^{2n^2}$ such that where $F(\tau,\xi)=E_2^\top(\xi) Q_1E_2(\xi)-E_1^\top Q_1E_1+\mathop{\mathrm{diag}}\nolimits(Q_2,-Q_2

Figures (8)

  • Figure 1: The wheel graph for interaction among agents in Example \ref{['ex1']}.
  • Figure 2: The evolutions of agents' strategies with $\tau=3$ and $\xi=0.08$
  • Figure 3: The evolutions of agents' strategies with $\tau=4$ and $\xi=0.08$
  • Figure 4: The illustrations of unit circle and the (yellow) Geršgorin disk whose center is $(h_i,0)^\top$ and radius is $1-h_i$.
  • Figure 5: The evolutions of agents' errors under the case of $\tau=1$
  • ...and 3 more figures

Theorems & Definitions (12)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Example 1
  • Theorem 2
  • Remark 4
  • Theorem 3
  • Remark 5
  • ...and 2 more