A Non-Cooperative Game Approach to Autonomous Racing

We consider autonomous racing of two cars and present an approach to formulate the decision making as a non-cooperative non-zero-sum game. The game is formulated by restricting both players to fulfill static track constraints as well as collision constraints which depend on the combined actions of the two players. At the same time the players try to maximize their own progress. In the case where the action space of the players is finite, the racing game can be reformulated as a bimatrix game. For this bimatrix game, we show that the actions obtained by a sequential maximization approach where only the follower considers the action of the leader are identical to a Stackelberg and a Nash equilibrium in pure strategies. Furthermore, we propose a game promoting blocking, by additionally rewarding the leading car 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 an online implementation, we propose to play the games in a moving horizon fashion, and we present 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 Zürich. The simulation study shows that the presented games can successfully model different racing behaviors and generate interesting racing situations.