Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized open-loop Nash equilibria in linear-quadratic difference games with coupled-affine inequality constraints

Partha Sarathi Mohapatra, Puduru Viswanadha Reddy

TL;DR

The paper addresses deterministic finite-horizon linear-quadratic difference games with coupled affine inequality constraints and derives both necessary and sufficient conditions for generalized open-loop Nash equilibria. It develops two coupled discrete-time linear complementarity systems (LCS1 and LCS2), proves their equivalence, and Reformulates the problem as a large-scale linear complementarity problem (LCP) to enable numerical computation. By introducing an auxiliary static constrained game (SGCC) and a backward Riccati framework, it extends prior results to fully coupled control variables and provides a concrete LCP-based solution pathway. The network-flow numerical example demonstrates the method's ability to compute GOLNE under capacity and battery constraints, highlighting practical applicability to networked engineering systems.

Abstract

In this note, we study a class of deterministic finite-horizon linear-quadratic difference games with coupled affine inequality constraints involving both state and control variables. We show that the necessary conditions for the existence of generalized open-loop Nash equilibria in this game class lead to two strongly coupled discrete-time linear complementarity systems. Subsequently, we derive sufficient conditions by establishing an equivalence between the solutions of these systems and convexity of the players' objective functions. These conditions are then reformulated as a solution to a linear complementarity problem, providing a numerical method to compute these equilibria. We illustrate our results using a network flow game with constraints.

Generalized open-loop Nash equilibria in linear-quadratic difference games with coupled-affine inequality constraints

TL;DR

The paper addresses deterministic finite-horizon linear-quadratic difference games with coupled affine inequality constraints and derives both necessary and sufficient conditions for generalized open-loop Nash equilibria. It develops two coupled discrete-time linear complementarity systems (LCS1 and LCS2), proves their equivalence, and Reformulates the problem as a large-scale linear complementarity problem (LCP) to enable numerical computation. By introducing an auxiliary static constrained game (SGCC) and a backward Riccati framework, it extends prior results to fully coupled control variables and provides a concrete LCP-based solution pathway. The network-flow numerical example demonstrates the method's ability to compute GOLNE under capacity and battery constraints, highlighting practical applicability to networked engineering systems.

Abstract

In this note, we study a class of deterministic finite-horizon linear-quadratic difference games with coupled affine inequality constraints involving both state and control variables. We show that the necessary conditions for the existence of generalized open-loop Nash equilibria in this game class lead to two strongly coupled discrete-time linear complementarity systems. Subsequently, we derive sufficient conditions by establishing an equivalence between the solutions of these systems and convexity of the players' objective functions. These conditions are then reformulated as a solution to a linear complementarity problem, providing a numerical method to compute these equilibria. We illustrate our results using a network flow game with constraints.
Paper Structure (10 sections, 9 theorems, 70 equations, 3 figures)

This paper contains 10 sections, 9 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Dynamic game with coupled constraints
  3. Generalized Open-loop Nash Equilibrium
  4. Necessary conditions
  5. Sufficient conditions
  6. Related static game with coupled constraints
  7. Main result
  8. Reformulation of LCS1 as a linear complementarity problem
  9. Numerical Illustration
  10. Conclusions

Key Result

Theorem 1

Let Assumption ass:Yassumption hold. Let $\{\bm{x}^\star,\bm{\lambda},\bm{\mu}^\star\}^1$ be a solution of LCS1. Then, the objective function eq:objective1 of each player $i\in \mathsf{N}$ can be expressed as: where $y_k^i:=(Y_k^i)^{-1}{B_k^i}^\prime E_{k+1}^i A_k x_k+b_k^i$, and ${x_k,~k\in \mathsf{K}}$ satisfies eq:state.

Figures (3)

  • Figure 1: N.C. (S.C.) refer to necessary (sufficient) conditions. Replace $\dagger$ with $\star$ for Theorem \ref{['th:TPBlink']} and $\dagger$ with $-$ for Theorem \ref{['th:TPBlink2']}.
  • Figure 2: Network flow game with two players and two relay nodes.
  • Figure 3: Flow-rates/controls of the players (panel (a)), battery levels (panel (b)), aggregate flow-rates at the relay nodes (panel (c)) and destination nodes (panel (d)). Arrow with equation number pointing to a dashed dark line indicates that one of the constraints \ref{['eq:NFconstraints']} is active.

Theorems & Definitions (28)

  • Remark 1
  • Definition 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Remark 4
  • Remark 5
  • Definition 2
  • Remark 6
  • ...and 18 more