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Predictive Compensation in Finite-Horizon LQ Games under Gauss-Markov Deviations

Navid Mojahed, Mahdis Rabbani, Shima Nazari

TL;DR

A predictive compensation framework for finite-horizon, discrete-time linear quadratic dynamic games subject to Gauss-Markov execution deviations from feedback Nash strategies is developed and the reduction in expected cost achieved by optimal predictive compensation is characterized.

Abstract

This paper develops a predictive compensation framework for finite-horizon, discrete-time linear quadratic dynamic games subject to Gauss-Markov execution deviations from feedback Nash strategies. One player's control is corrupted by temporally correlated stochastic perturbations modeled as a first-order autoregressive (AR(1)) process, while the opposing player has causal access to past deviations and employs a predictive feedforward strategy that anticipates their future effect. We derive closed-form recursions for mean and covariance propagation under the resulting perturbed closed loop, establish boundedness and sensitivity properties of the equilibrium trajectory, and characterize the reduction in expected cost achieved by optimal predictive compensation. Numerical experiments corroborate the theoretical results and demonstrate performance gains relative to nominal Nash feedback across a range of disturbance persistence levels.

Predictive Compensation in Finite-Horizon LQ Games under Gauss-Markov Deviations

TL;DR

A predictive compensation framework for finite-horizon, discrete-time linear quadratic dynamic games subject to Gauss-Markov execution deviations from feedback Nash strategies is developed and the reduction in expected cost achieved by optimal predictive compensation is characterized.

Abstract

This paper develops a predictive compensation framework for finite-horizon, discrete-time linear quadratic dynamic games subject to Gauss-Markov execution deviations from feedback Nash strategies. One player's control is corrupted by temporally correlated stochastic perturbations modeled as a first-order autoregressive (AR(1)) process, while the opposing player has causal access to past deviations and employs a predictive feedforward strategy that anticipates their future effect. We derive closed-form recursions for mean and covariance propagation under the resulting perturbed closed loop, establish boundedness and sensitivity properties of the equilibrium trajectory, and characterize the reduction in expected cost achieved by optimal predictive compensation. Numerical experiments corroborate the theoretical results and demonstrate performance gains relative to nominal Nash feedback across a range of disturbance persistence levels.

Paper Structure

This paper contains 6 sections, 3 theorems, 56 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation & Background
  3. Structured Execution Deviations & Moment--Based Sensitivity Analysis
  4. Predictive Compensator Design
  5. Numerical Example
  6. Conclusion

Key Result

Lemma 1

Under Assumption assump:ar1, define Then, for all $k\ge \ell\ge0$ In particular, the marginal covariance admits a closed form and satisfies $\Phi_{k,\ell}^\top=\Phi_{\ell,k}$. Hence $\Phi_k \to \sigma_0^2 I_{m_2}$ as $k\to\infty$.

Figures (4)

  • Figure 1: Nominal feedback–Nash trajectories for both players. Top: state components $x_k^{(i)}$, $i=1,2,3$. Bottom: corresponding control inputs $u_{1,k}^{(i)}$ (left) and $u_{2,k}^{(i)}$ (right) under the equilibrium policies ${K_{1,k}^*,K_{2,k}^*}$. The stable closed-loop evolution serves as the reference for all subsequent deviation experiments.
  • Figure 2: Validation of the second-moment recursion in Theorem \ref{['thm:mean-covariance']}. Comparison between theoretical and Monte Carlo (MC) traces $\mathrm{tr}(\Sigma_k)$ and $\mathrm{tr}(\widehat{\Sigma}_k)$ across the horizon. The near-perfect overlap confirms the accuracy of the analytical covariance propagation and its boundedness under the Schur-stable closed loop.
  • Figure 3: Monte Carlo realizations of the deviation state $\Delta x_k$ at $(\sigma_0,\rho)=(0.06,0.5)$. Faded light gray lines show individual runs; bold curves denote componentwise means. Trajectories remain zero-mean and bounded, illustrating the empirical validity of Theorem \ref{['thm:mean-covariance']} and the anisotropic spread induced by $(A,B_2)$ and $A_{\mathrm{cl},k}$.
  • Figure 4: Effect of disturbance persistence and predictive compensation on Player 1’s cost. Left: expected cost $\mathbb{E}[J_1]$ versus persistence factor $\rho$ across various steady-state per-channel standard deviations $\sigma_0$. Right: expected cost reduction $\mathbb{E}[\Delta J_1]$ due to the predictive compensator. Dashed curves (predictive compensator) consistently lie below solid ones (no compensation).

Theorems & Definitions (5)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Definition 2
  • Theorem 2