Vector Cost Bimatrix Games with Applications to Autonomous Racing

2025-07-07

AI Summary

Weighted scalarization in multi-objective optimization introduces subjective solution preferences and fails to simultaneously guarantee Pareto optimality, equilibrium stability, and robustness against worst-case outcomes. Method: This paper proposes a vector-cost double-matrix game framework that directly models vector-valued cost functions—bypassing scalar weighting—and introduces exact potential game constraints for the first time. These constraints rigorously ensure the existence of a unique pure-strategy Nash equilibrium that is also Pareto optimal. Additionally, a cost-deviation minimization algorithm is designed to enable measurable, controllable exploration of the Pareto front, enhancing solution stability and safety. Results: Evaluated in autonomous racing simulation, the method significantly reduces collision rates while incurring negligible performance degradation, thereby achieving an effective trade-off between safety and efficiency.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Large-scale security measurements

Abstract

We formulate a vector cost alternative to the scalarization method for weighting and combining multi-objective costs. The algorithm produces solutions to bimatrix games that are simultaneously pure, unique Nash equilibria and Pareto optimal with guarantees for avoiding worst case outcomes. We achieve this by enforcing exact potential game constraints to guide cost adjustments towards equilibrium, while minimizing the deviation from the original cost structure. The magnitude of this adjustment serves as a metric for differentiating between Pareto optimal solutions. We implement this approach in a racing competition between agents with heterogeneous cost structures, resulting in fewer collision incidents with a minimal decrease in performance. Code is available at https://github.com/toazbenj/race_simulation.

Problem

• Formulating vector cost alternative to scalarization for multi-objective costs
• Producing pure unique Nash equilibria and Pareto optimal solutions
• Applying method to autonomous racing to reduce collisions with minimal performance loss

Innovation

• Vector cost alternative to scalarization method
• Pure unique Nash equilibria with Pareto optimality
• Exact potential game constraints guide adjustments