Vector Cost Bimatrix Games with Applications to Autonomous Racing

TL;DR

This paper introduces a vector cost approach for bimatrix games that ensures solutions are pure, unique Nash equilibria and Pareto optimal, with practical application to autonomous racing to reduce collisions.

Contribution

It proposes a novel vector cost formulation with an algorithm that guarantees equilibrium solutions while minimizing deviation from original costs.

Findings

Fewer collision incidents in autonomous racing simulations.

Solutions are both pure Nash equilibria and Pareto optimal.

Minimal performance decrease achieved in practical application.

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.