Non-Normalized Solutions of Generalized Nash Equilibrium in Autonomous Racing

TL;DR

This work addresses non-normalized Generalized Nash Equilibria (GNE) in dynamic games with shared constraints, focusing on autonomous racing. It shows that the standard normalized solution, which enforces identical shared multipliers via $\sigma$, can miss valuable asymmetric equilibria. The authors extend the MCP formulation by introducing player-specific scaling matrices $A_i$ and a common fictitious multiplier $\sigma$ to compute non-normalized GNEs, demonstrating multi-modal strategies and improved overtaking performance in two-car racing simulations. Overall, the approach expands the computational toolbox for distributed decision-making in dynamic games and has practical implications for designing flexible, competitive autonomous systems.

Abstract

In dynamic games with shared constraints, Generalized Nash Equilibria (GNE) are often computed using the normalized solution concept, which assumes identical Lagrange multipliers for shared constraints across all players. While widely used, this approach excludes other potentially valuable GNE. This paper addresses the limitations of normalized solutions in racing scenarios through three key contributions. First, we highlight the shortcomings of normalized solutions with a simple racing example. Second, we propose a novel method based on the Mixed Complementarity Problem (MCP) formulation to compute non-normalized Generalized Nash Equilibria (GNE). Third, we demonstrate that our proposed method overcomes the limitations of normalized GNE solutions and enables richer multi-modal interactions in realistic racing scenarios.

Figures (4)

• Figure 1: Illustration of the 1D racing example.
• Figure 2: First Qualitative Example: Racing Problem Solutions on a Straight Track
• Figure 3: Second Qualitative Example: Racing Problem Solutions on a Curved Track
• Figure 4: Results of 100 Monte Carlo simulations comparing two different strategies on an L-shaped track