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Noncooperative Game in Multi-controller System under Delayed and Asymmetric Information

Xin Li, Qingyuan Qi, Kemi Ding

Abstract

We address a noncooperative game problem in multi-controller system under delayed and asymmetric information structure. Under these conditions, the classical separation principle fails as estimation and control design become strongly coupled, complicating the derivation of an explicit Nash equilibrium. To resolve this, we employ a common-private information decomposition approach, effectively decoupling control inputs and state estimation to obtain a closed-form Nash equilibrium. By applying a forward iterative method, we establish the convergence of the coupled Riccati and estimation error covariance recursions, yielding both the steady-state Kalman filter and the Nash equilibrium. Furthermore, we quantify the impact of asymmetric information, proving that a richer information set reduces the costs for the corresponding player. Finally, numerical examples are provided to demonstrate the effectiveness of the results.

Noncooperative Game in Multi-controller System under Delayed and Asymmetric Information

Abstract

We address a noncooperative game problem in multi-controller system under delayed and asymmetric information structure. Under these conditions, the classical separation principle fails as estimation and control design become strongly coupled, complicating the derivation of an explicit Nash equilibrium. To resolve this, we employ a common-private information decomposition approach, effectively decoupling control inputs and state estimation to obtain a closed-form Nash equilibrium. By applying a forward iterative method, we establish the convergence of the coupled Riccati and estimation error covariance recursions, yielding both the steady-state Kalman filter and the Nash equilibrium. Furthermore, we quantify the impact of asymmetric information, proving that a richer information set reduces the costs for the corresponding player. Finally, numerical examples are provided to demonstrate the effectiveness of the results.

Paper Structure

This paper contains 17 sections, 7 theorems, 56 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related works
  3. Contribution
  4. Problem Formulation
  5. System Model
  6. Asymmetric Information Structure
  7. The Game-theoretic Formulation
  8. Main Results
  9. Common-private Information Decomposition
  10. The state estimate via Kalman filter
  11. The Nash equilibrium
  12. Comparison of equilibrium costs under symmetric and asymmetric information
  13. Steady-State Kalman filter and Nash Equilibrium
  14. Numerical Example
  15. Conclusion
  16. ...and 2 more sections

Key Result

Lemma 1

For all $k \in \{0, \dots, N\}$ and $e_k^2=x_k-\hat{x}_{k|k-1}^{\,2}$, the following orthogonality relations hold:

Figures (6)

  • Figure 1: System model under the delayed and asymmetric information structure
  • Figure 2: The Riccati matrices $P_k^1$, $\Phi_k^1$
  • Figure 3: The Riccati matrices $P_k^2$, $\Phi_k^2$
  • Figure 4: The feedback gains $K_k^1$, $K_k^2$
  • Figure 5: Trajectory of $x_k$, $\hat{x}_{k|k-1}^1$ and $\hat{x}_{k|k-1}^2$
  • ...and 1 more figures

Theorems & Definitions (21)

  • Example 1
  • Definition 1
  • Remark 1
  • Lemma 1
  • Remark 2
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 3
  • ...and 11 more