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On finite-horizon approximation of a feedback Nash equilibrium in LQ games

Shengyuan Huang, Xiaoguang Yang, Yifen Mu, Wenjun Mei

Abstract

Dynamic games provide a fundamental framework for multi-agent decision-making over time, yet computing feedback Nash equilibria (FNEs) in infinite-horizon discrete-time linear-quadratic (LQ) settings remains computationally challenging. Motivated by the need for tractable and implementable strategies, this paper studies a finite-horizon strategy for approximating a certain infinite-horizon equilibrium. Specifically, at each stage, each player solves a T-stage game and implements only the first-stage control, thereby avoiding the direct solution of coupled infinite-horizon Riccati equations. We first analyze the finite-horizon game and characterize the structure of the associated coupled generalized discrete Riccati difference equations. Based on this analysis, we establish a sufficient condition for uniqueness of the FNE and propose an efficient algorithm that computes it via a sequence of linear equations. We then consider the infinite-horizon game in which players adopt the finite-horizon strategies with heterogeneous prediction horizons and show that, under suitable conditions, the total cost under the finite-horizon strategies converges to the cost under the limiting infinite-horizon FNE. Moreover, we derive an explicit upper bound on this cost gap in terms of the distance between the corresponding strategy matrices. These results provide theoretical justification and quantitative performance guarantees for finite-horizon strategies in infinite-horizon LQ dynamic games. A nonscalar numerical example illustrates the effectiveness of the proposed framework.

On finite-horizon approximation of a feedback Nash equilibrium in LQ games

Abstract

Dynamic games provide a fundamental framework for multi-agent decision-making over time, yet computing feedback Nash equilibria (FNEs) in infinite-horizon discrete-time linear-quadratic (LQ) settings remains computationally challenging. Motivated by the need for tractable and implementable strategies, this paper studies a finite-horizon strategy for approximating a certain infinite-horizon equilibrium. Specifically, at each stage, each player solves a T-stage game and implements only the first-stage control, thereby avoiding the direct solution of coupled infinite-horizon Riccati equations. We first analyze the finite-horizon game and characterize the structure of the associated coupled generalized discrete Riccati difference equations. Based on this analysis, we establish a sufficient condition for uniqueness of the FNE and propose an efficient algorithm that computes it via a sequence of linear equations. We then consider the infinite-horizon game in which players adopt the finite-horizon strategies with heterogeneous prediction horizons and show that, under suitable conditions, the total cost under the finite-horizon strategies converges to the cost under the limiting infinite-horizon FNE. Moreover, we derive an explicit upper bound on this cost gap in terms of the distance between the corresponding strategy matrices. These results provide theoretical justification and quantitative performance guarantees for finite-horizon strategies in infinite-horizon LQ dynamic games. A nonscalar numerical example illustrates the effectiveness of the proposed framework.

Paper Structure

This paper contains 10 sections, 4 theorems, 59 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Finite-horizon LQ game with i/o/s dynamics
  3. Basic Notations
  4. Game Setup
  5. Analysis of feedback Nash equilibrium
  6. Infinite-horizon discrete-time LQ game with i/o/s dynamics and finite-horizon strategy
  7. Numerical simulations
  8. Conclusions
  9. Proof of Lemma \ref{['lm:convergence']}
  10. Proof of Theorem \ref{['theo:finite-horizon strategy']}

Key Result

Lemma 1

The $N$-person $T$-stage finite-horizon LQ game eq:isosystem, eq:costfun has a unique FNE $(\gamma_{1:T}^{1*},\dots, \gamma_{1:T}^{N*})$ if and only if the following coupled generalized discrete Riccati difference equations admits a unique solution $\{P^{i*}_t, S^{i*}_t, w^{i*}_t, K^{i*}_t, L^{i*}_ with $P^i_{T+1}=\mathbf{0}, S^i_{T+1}=\mathbf{0}, w^i_{T+1}=0$ for any $i \in \mathcal{N}$. Here $

Figures (2)

  • Figure 1: Player $i$'s first-stage strategy matrices ${K}^{i*}_1(T) = [{K}^{i*}_1(T)[1],\ {K}^{i*}_1(T)[2],\ {K}^{i*}_1(T)[3]]$ and ${L}^{i*}_1(T)$ of the unique FNE in the $T$-stage game \ref{['eq:isosystem_simulation']}, \ref{['eq:costfun_simulation']}, for $i \in \mathcal{N}$ and $T = 1, \dots, 20$, where $T$ denotes the length of the game. The horizontal black dashed lines represent the strategy matrices of the limiting FNE derived from the coupled equations \ref{['eq:K']}$\sim$\ref{['eq:w']} in the infinite-horizon game.
  • Figure 2: Player $i$'s total cost $\tilde{J}^i(x_1)(T )$ under the finite-horizon strategy "looking $T$ steps ahead and moving one step" in the infinite-horizon game with \ref{['eq:isosystem_simulation']} and \ref{['eq:costfun_simulation']}, for $i=1,2$ and $T=2,\dots,50$, where $T$ denotes each player’s prediction horizon. The horizontal black dashed lines represent the total cost $J^i(x_1)$ under the limiting FNE derived from the coupled equations \ref{['eq:K']}$\sim$\ref{['eq:w']} in the infinite-horizon game for players $i=1,2$.

Theorems & Definitions (8)

  • Definition 1: Feedback Nash Equilibrium (FNE), 1998basarNoncooperativeGame
  • Lemma 1
  • Proposition 1
  • proof
  • Lemma 2
  • Theorem 3
  • proof
  • proof