Generalized Nash Equilibrium Solutions in Dynamic Games With Shared Constraints

TL;DR

This paper addresses the limitation of the standard normalized Generalized Nash Equilibrium (GNE) in dynamic games with shared constraints by introducing a mixed complementarity problem (MCP) formulation that computes non-normalized GNEs. It introduces diagonal scaling matrices $A_i$ to differentially scale shared multipliers, proving that such scaling preserves GNE validity and provides a tunable attack on the solution set, with normalization recaptured when all $A_i$ coincide. A bi-level equilibrium-selection mechanism is proposed to pick among many GNEs by optimizing a higher-level objective $J_0$ over the chosen $A_i$ configurations, enabling more desirable outcomes in applications like racing games. The framework is validated analytically through examples and demonstrated numerically on dynamic racing problems, showing that non-normalized equilibria can yield more intuitive, aggressive strategies and improved performance (e.g., higher win rates in simulated races). Overall, the work expands the computational toolbox for dynamic games by enabling non-normalized GNE search, interpretation via relative aggressiveness, and principled equilibrium selection with existing MCP solvers such as PATH.

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 presents a novel method based on the Mixed Complementarity Problem (MCP) formulation to compute non-normalized GNE, expanding the solution space. We also propose a systematic approach for selecting the optimal GNE based on predefined criteria, enhancing practical flexibility. Numerical examples illustrate the methods effectiveness, offering an alternative to traditional normalized solutions.

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

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• proof
• Corollary 2.1
• proof : Proof of Corollary \ref{['cor:num1']}
• Corollary 2.2
• proof : Proof of Corollary \ref{['cor:num2']}