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On the Upper Bound of Near Potential Differential Games

Balint Varga

TL;DR

The paper addresses how close a Linear Quadratic Near Potential Differential Game (LQ NPDG) is to an exact potential differential game (EPDG) by introducing a differential distance measure $\sigma^{(i)}_d(t)$ and an upper bound $\Delta$. It derives a linear relationship between this distance and the resulting trajectory error, yielding a bound $\|\boldsymbol{x}^{(p)}(t)-\boldsymbol{x}^*(t)\|_2 \le C_{\mathrm{NPDG}}(t) \Delta$ with an explicit expression for $C_{\mathrm{NPDG}}(t)$. The core contributions include a sufficient condition ensuring the NPDG property via a bound on the Riccati solution differences and a closed-form bound on closed-loop dynamics. This work broadens the applicability of potential-game concepts to practical engineering scenarios, such as human-machine interactions, by providing a tractable way to quantify and bound deviations from NE.

Abstract

This letter presents an extended analysis and a novel upper bound of the subclass of Linear Quadratic Near Potential Differential Games (LQ NPDG). LQ NPDGs are a subclass of potential differential games, for which a distance between an LQ exact potential differential game and the LQ NPDG. LQ NPDGs exhibit a unique characteristic: the smaller the distance from an LQ exact potential differential game, the closer their dynamic trajectories. This letter introduces a novel upper bound for this distance. Moreover, a linear relation between this distance and the resulting trajectory errors is established, opening the possibility for further application of LQ NPDGs.

On the Upper Bound of Near Potential Differential Games

TL;DR

The paper addresses how close a Linear Quadratic Near Potential Differential Game (LQ NPDG) is to an exact potential differential game (EPDG) by introducing a differential distance measure and an upper bound . It derives a linear relationship between this distance and the resulting trajectory error, yielding a bound with an explicit expression for . The core contributions include a sufficient condition ensuring the NPDG property via a bound on the Riccati solution differences and a closed-form bound on closed-loop dynamics. This work broadens the applicability of potential-game concepts to practical engineering scenarios, such as human-machine interactions, by providing a tractable way to quantify and bound deviations from NE.

Abstract

This letter presents an extended analysis and a novel upper bound of the subclass of Linear Quadratic Near Potential Differential Games (LQ NPDG). LQ NPDGs are a subclass of potential differential games, for which a distance between an LQ exact potential differential game and the LQ NPDG. LQ NPDGs exhibit a unique characteristic: the smaller the distance from an LQ exact potential differential game, the closer their dynamic trajectories. This letter introduces a novel upper bound for this distance. Moreover, a linear relation between this distance and the resulting trajectory errors is established, opening the possibility for further application of LQ NPDGs.
Paper Structure (9 sections, 2 theorems, 50 equations)

This paper contains 9 sections, 2 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exact Potential Differential Games
  4. Distance between two Potential Differential Games
  5. Upper Bound of NPDGs
  6. Properties of an NPDG
  7. Dynamics of LQ NPDGs
  8. Discussion
  9. Summary and Outlook

Key Result

Theorem 1

Let an LQ exact potential differential game $\Gamma^{(p)}_{\text{ed}}$ with its state trajectories $\boldsymbol{x}^{(p)}(t)$ in its NE be given. Furthermore, let an arbitrary LQ differential game $\Gamma_\mathrm{npd}$ according to Definition def:diff_game with its state trajectories $\boldsymbol{x}^ hold $\forall t \in [0,\tau_{\mathrm{end}}]$. If holds, where $\Delta$ is defined in def:NPDG_dist

Theorems & Definitions (10)

  • Definition 1: LQ Differential Game 2016_NonzeroSumDifferentialGames_basar
  • Definition 2: Nash Equilibrium 2016_NonzeroSumDifferentialGames_basar
  • Definition 3: LQ Exact Potential Differential Games 2016_SurveyStaticDynamic_gonzalez-sanchez
  • Definition 4: Differential Distance 2023_LimitedInformationShared_varga
  • Definition 5: Near Potential Differential Game 2023_LimitedInformationShared_varga
  • Theorem 1: LQ NPDG
  • proof
  • Definition 6: Closed-Loop System Matrix Error
  • Theorem 2: Boundedness of NPDGs
  • proof