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Does bilevel optimization result in more competitive racing behavior?

Andrew Cinar, Forrest Laine

TL;DR

The paper investigates how the choice of equilibrium concept—Nash versus bilevel (Stackelberg)—affects competitive behavior and safety in two-agent racing. It introduces a nonlinear, model-driven racing framework with drag, drafting, and position-dependent collision responsibility, and develops a solver for nonlinear constrained bilevel optimization. A large-scale empirical study compares 16 strategy configurations (single-player, Nash, bilevel leader, bilevel follower) across random initial conditions, revealing that bilevel leadership can enhance competitiveness but often at the cost of safety, while always-following strategies can improve safety. The results provide guidance on selecting information structures for autonomous multi-agent systems and motivate future work on adaptive strategy switching during interaction.

Abstract

Two-vehicle racing is natural example of a competitive dynamic game. As with most dynamic games, there are many ways in which the underlying solution concept can be structured, resulting in different equilibrium concepts. The assumed solution concept influences the behaviors of two interacting players in racing. For example, blocking behavior emerges naturally in leader-follower play, but to achieve this in Nash play the costs would have to be chosen specifically to trigger this behavior. In this work, we develop a novel model for competitive two-player vehicle racing, represented as an equilibrium problem, complete with simplified aerodynamic drag and drafting effects, as well as position-dependent collision-avoidance responsibility. We use our model to explore how different solution concepts affect competitiveness. We develop a solution for bilevel optimization problems, enabling a large-scale empirical study comparing bilevel strategies (either as leader or follower), Nash equilibrium strategy and a single-player constant velocity baseline. We find the choice of strategies significantly affects competitive performance and safety.

Does bilevel optimization result in more competitive racing behavior?

TL;DR

The paper investigates how the choice of equilibrium concept—Nash versus bilevel (Stackelberg)—affects competitive behavior and safety in two-agent racing. It introduces a nonlinear, model-driven racing framework with drag, drafting, and position-dependent collision responsibility, and develops a solver for nonlinear constrained bilevel optimization. A large-scale empirical study compares 16 strategy configurations (single-player, Nash, bilevel leader, bilevel follower) across random initial conditions, revealing that bilevel leadership can enhance competitiveness but often at the cost of safety, while always-following strategies can improve safety. The results provide guidance on selecting information structures for autonomous multi-agent systems and motivate future work on adaptive strategy switching during interaction.

Abstract

Two-vehicle racing is natural example of a competitive dynamic game. As with most dynamic games, there are many ways in which the underlying solution concept can be structured, resulting in different equilibrium concepts. The assumed solution concept influences the behaviors of two interacting players in racing. For example, blocking behavior emerges naturally in leader-follower play, but to achieve this in Nash play the costs would have to be chosen specifically to trigger this behavior. In this work, we develop a novel model for competitive two-player vehicle racing, represented as an equilibrium problem, complete with simplified aerodynamic drag and drafting effects, as well as position-dependent collision-avoidance responsibility. We use our model to explore how different solution concepts affect competitiveness. We develop a solution for bilevel optimization problems, enabling a large-scale empirical study comparing bilevel strategies (either as leader or follower), Nash equilibrium strategy and a single-player constant velocity baseline. We find the choice of strategies significantly affects competitive performance and safety.
Paper Structure (15 sections, 1 theorem, 26 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 15 sections, 1 theorem, 26 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Game Theory Background
  3. Computing Solutions to Equilibrium Problems
  4. Nash Equilibrium Points
  5. Bilevel Equilibrium Points
  6. Two-player Racing Game
  7. Vehicle Dynamics Model
  8. Objective Functions and Constraints
  9. Track Model and Constraints
  10. Collision Avoidance and Responsibility
  11. Drafting for Acceleration Limits
  12. Simulations and Discussion
  13. Robustness
  14. Performance
  15. Conclusion

Key Result

Lemma 1

Let the sets $D_k$ appearing in eq:simple_union be closed sets. Then a point $x^*$ is a local optimum of eq:simple_union if and only if $x^*$ is a local optimum of all problems:

Figures (3)

  • Figure 1: Red (Player 2) Follower is blocked by Blue (Player 1) Leader while attempting to pass in bilevel L-F competition.
  • Figure 2: $\ell_1$ and $\ell_2$ with respect to relative longitudinal position of the players, bounded above by a small positive value.
  • Figure 3: Triangular drafting region and collision avoidance responsibility based on relative positions of players.

Theorems & Definitions (2)

  • Lemma 1
  • proof