A Non-Cooperative Game Approach to Autonomous Racing (1712.03913v2)

Abstract: We consider autonomous racing of two cars and present an approach to formulate racing decisions as a non-cooperative non-zero-sum game. We design three different games where the players aim to fulfill static track constraints as well as avoid collision with each other; the latter constraint depends on the combined actions of the two players. The difference between the games are the collision constraints and the payoff. In the first game collision avoidance is only considered by the follower, and each player maximizes their own progress towards the finish line. We show that, thanks to the sequential structure of this game, equilibria can be computed through an efficient sequential maximization approach. Further, we show these actions, if feasible, are also a Stackelberg and Nash equilibrium in pure strategies of our second game where both players consider the collision constraints. The payoff of our third game is designed to promote blocking, by additionally rewarding the cars for staying ahead at the end of the horizon. We show that this changes the Stackelberg equilibrium, but has a minor influence on the Nash equilibria. For online implementation, we propose to play the games in a moving horizon fashion, and discuss two methods for guaranteeing feasibility of the resulting coupled repeated games. Finally, we study the performance of the proposed approaches in simulation for a set-up that replicates the miniature race car tested at the Automatic Control Laboratory of ETH Zurich. The simulation study shows that the presented games can successfully model different racing behaviors and generate interesting racing situations.

• The paper’s main contribution is the formulation of autonomous racing as a non-cooperative, non-zero-sum game with three distinct game models.
• It employs sequential maximization and a moving horizon approach to compute equilibria, achieving an empirical collision probability of approximately 0.005.
• The study advances game theory and MPC-based control for autonomous vehicles, offering practical insights for competitive driving scenarios.

A Non-Cooperative Game Approach to Autonomous Racing: An In-Depth Analysis

The paper "A Non-Cooperative Game Approach to Autonomous Racing" by Alexander Liniger and John Lygeros investigates the interactions of autonomous cars in a racing context. This research explores the challenging dynamics of competitive driving through a game theoretic perspective, formulating racing decisions as a non-cooperative non-zero-sum game. The authors introduce a series of three distinct games to model these interactions, each varying in complexity of constraints and payoff structures.

Key Contributions

The key contribution of this paper is the formulation of autonomous racing as a non-cooperative non-zero-sum game, a distinct approach compared to many existing models which assume a degree of cooperation from all vehicles involved. The proposed games are strategically designed to address track constraints and vehicle interactions under competitive conditions.

1. Sequential Game: In this game, collision avoidance is considered only by the following vehicle. Players aim to maximize their respective progress towards the finish line without mutual consideration of both players. The authors show that equilibria for this game type can be computed efficiently through sequential maximization.
2. Cooperative Game: Both players account for collision constraints, aiming to derive solutions that ensure avoiding collisions, resulting in a mutual understanding of the game's dynamics. It incorporates a more synchronous consideration of potential interactions.
3. Blocking Game: This game introduces a strategic component of blocking, rewarding players for hindering the opponent's progress. This is achieved via a modified payoff structure rewarding cars that remain ahead at the horizon's end, influencing Stackelberg equilibria while minimally impacting Nash equilibria.

Methodology and Results

The researchers analyze equilibria in terms of Stackelberg and Nash concepts, revealing cases where equilibria are feasible under certain assumptions. Importantly, the paper employs a moving horizon approach to implement these games, akin to Model Predictive Control (MPC), allowing adaptive decision-making in dynamic scenarios.

Simulations conducted mimic realistic racing conditions with small-scale race cars. The paper reveals intriguing outcomes: the sequential game is effective for blocking but carries a higher collision risk, whereas the cooperative game optimizes for collision avoidance; the blocking game strikes a balance by encouraging strategic blocking while minimizing collisions. A notable observation is the empirical collision probability of approximately 0.005, underscoring the robust modeling in competitive environments.

Practical and Theoretical Implications

Practically, these models enhance our understanding of autonomous vehicle interactions in competitive settings where traditional cooperative assumptions may not hold. Theoretically, the work advances game theory applications in autonomous systems, suggesting pathways to model complex agent interactions without cooperative assumptions.

Conclusion and Future Directions

This research marks a significant step towards more sophisticated modeling of autonomous vehicle behavior in competitive racing contexts. The potential for real-time implementation and consideration of varying vehicle parameters and dynamics could greatly benefit future applications. Continued exploration in this domain could extend to multi-vehicle racing scenarios or incorporate more nuanced vehicle dynamics and decision-making processes, further bridging the gap between theoretical models and real-world autonomous vehicle interactions.

The paper's findings and methodologies represent a notable contribution to both the fields of game theory and autonomous vehicle control systems, paving the way for more nuanced and adaptable models in rapidly evolving and competitive vehicular environments.