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Game-Theoretic Autonomous Driving: A Graphs of Convex Sets Approach

Nikolaj Käfer, Ahmed Khalil, Edward Huynh, Efstathios Bakolas, David Fridovich-Keil

TL;DR

The paper tackles multi-vehicle highway planning with strategic, hybrid decision making by framing the problem as a generalized Nash game and solving it with an Iterative Best Response in Graphs of Convex Sets (IBR-GCS). Each vehicle constructs a strategy-dependent GCS capturing lane-specific safe regions over time, turning the best-response into a shortest-path problem on that graph, with a convex relaxation often yielding tight solutions. The authors prove that, in a generalized potential game, exact best-response updates descend a global potential, and they quantify how inexact (relaxed) updates lead to convergence to an approximate generalized Nash equilibrium. Simulations in multi-lane scenarios demonstrate safe, strategically coherent trajectories and overtaking maneuvers, with analyses of convergence and robustness to relaxation looseness. The framework effectively separates discrete maneuver selection from continuous trajectory optimization while exploiting convexity to enable scalable planning under interaction constraints.

Abstract

Multi-vehicle autonomous driving couples strategic interaction with hybrid (discrete-continuous) maneuver planning under shared safety constraints. We introduce IBR-GCS, an Iterative Best Response (IBR) planning approach based on the Graphs of Convex Sets (GCS) framework that models highway driving as a generalized noncooperative game. IBR-GCS integrates combinatorial maneuver reasoning, trajectory planning, and game-theoretic interaction within a unified framework. The key novelty is a vehicle-specific, strategy-dependent GCS construction. Specifically, at each best-response update, each vehicle builds its own graph conditioned on the current strategies of the other vehicles, with vertices representing lane-specific, time-varying, convex, collision-free regions and edges encoding dynamically feasible transitions. This yields a shortest-path problem in GCS for each best-response step, which admits an efficient convex relaxation that can be solved using convex optimization tools without exhaustive discrete tree search. We then apply an iterative best-response scheme in which vehicles update their trajectories sequentially and provide conditions under which the resulting inexact updates converge to an approximate generalized Nash equilibrium. Simulation results across multi-lane, multi-vehicle scenarios demonstrate that IBR-GCS produces safe trajectories and strategically consistent interactive behaviors.

Game-Theoretic Autonomous Driving: A Graphs of Convex Sets Approach

TL;DR

The paper tackles multi-vehicle highway planning with strategic, hybrid decision making by framing the problem as a generalized Nash game and solving it with an Iterative Best Response in Graphs of Convex Sets (IBR-GCS). Each vehicle constructs a strategy-dependent GCS capturing lane-specific safe regions over time, turning the best-response into a shortest-path problem on that graph, with a convex relaxation often yielding tight solutions. The authors prove that, in a generalized potential game, exact best-response updates descend a global potential, and they quantify how inexact (relaxed) updates lead to convergence to an approximate generalized Nash equilibrium. Simulations in multi-lane scenarios demonstrate safe, strategically coherent trajectories and overtaking maneuvers, with analyses of convergence and robustness to relaxation looseness. The framework effectively separates discrete maneuver selection from continuous trajectory optimization while exploiting convexity to enable scalable planning under interaction constraints.

Abstract

Multi-vehicle autonomous driving couples strategic interaction with hybrid (discrete-continuous) maneuver planning under shared safety constraints. We introduce IBR-GCS, an Iterative Best Response (IBR) planning approach based on the Graphs of Convex Sets (GCS) framework that models highway driving as a generalized noncooperative game. IBR-GCS integrates combinatorial maneuver reasoning, trajectory planning, and game-theoretic interaction within a unified framework. The key novelty is a vehicle-specific, strategy-dependent GCS construction. Specifically, at each best-response update, each vehicle builds its own graph conditioned on the current strategies of the other vehicles, with vertices representing lane-specific, time-varying, convex, collision-free regions and edges encoding dynamically feasible transitions. This yields a shortest-path problem in GCS for each best-response step, which admits an efficient convex relaxation that can be solved using convex optimization tools without exhaustive discrete tree search. We then apply an iterative best-response scheme in which vehicles update their trajectories sequentially and provide conditions under which the resulting inexact updates converge to an approximate generalized Nash equilibrium. Simulation results across multi-lane, multi-vehicle scenarios demonstrate that IBR-GCS produces safe trajectories and strategically consistent interactive behaviors.
Paper Structure (15 sections, 1 theorem, 22 equations, 1 figure)

This paper contains 15 sections, 1 theorem, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shortest Path Problem in Graphs of Convex Sets
  4. Convex Relaxation
  5. Generalized Potential Games
  6. Problem Formulation
  7. Dynamics and bounds:
  8. Lane evolution:
  9. Relative coordinates:
  10. Safety rules:
  11. Rule 1 (Longitudinal safety in the same lane).
  12. Rule 2 (Lateral safety across adjacent lanes).
  13. Preferences and objectives:
  14. Individual vehicle strategy and optimization problem
  15. Example objective function.

Key Result

proposition thmcounterproposition

Suppose $G$ is a generalized potential game with generalized potential $\Phi$ (Definition def_GPG). If, at some intermediate profile $\theta^{k,i-1}$, vehicle $i \in \mathcal{N}$ performs an exact best-response update eq_best_response, then Consequently, if all vehicles update with exact best responses, the sequence $\{\Phi(\theta^k)\}_{k\ge 0}$ is monotonically non-increasing along IBR, and ther

Figures (1)

  • Figure 1: Vehicles operating on a highway with vehicle-specific variables (left) and longitudinal distance (right).

Theorems & Definitions (6)

  • definition thmcounterdefinition: Graph of Convex Sets
  • definition thmcounterdefinition: Path
  • definition thmcounterdefinition: Game
  • definition thmcounterdefinition: Generalized Nash Equilibrium, GNE
  • definition thmcounterdefinition: Generalized potential game
  • proposition thmcounterproposition: Potential descent under exact IBR