Two-vehicle racing is natural example of a competitive dynamic game. As with most dynamic games, there are many ways in which the underlying solution concept can be structured, resulting in different equilibrium concepts. The assumed solution concept influences the behaviors of two interacting players in racing. For example, blocking behavior emerges naturally in leader-follower play, but to achieve this in Nash play the costs would have to be chosen specifically to trigger this behavior. In this work, we develop a novel model for competitive two-player vehicle racing, represented as an equilibrium problem, complete with simplified aerodynamic drag and drafting effects, as well as position-dependent collisionavoidance responsibility. We use our model to explore how different solution concepts affect competitiveness. We develop a solution for bilevel optimization problems, enabling a largescale empirical study comparing bilevel strategies (either as leader or follower), Nash equilibrium strategy and a singleplayer constant velocity baseline. We find the choice of strategies significantly affects competitive performance and safety.

Collision detection is a critical functionality for robotics. The degree to which objects collide cannot be represented as a continuously differentiable function for any shapes other than spheres. This paper proposes a framework for handling collision detection between polyhedral shapes. We frame the signed distance between two polyhedral bodies as the optimal value of a convex optimization, and consider constraining the signed distance in a bilevel optimization problem. To avoid relying on specialized bilevel solvers, our method exploits the fact that the signed distance is the minimal point of a convex region related to the two bodies. Our method enumerates the values obtained at all extreme points of this region and lists them as constraints in the higher-level problem. We compare our formulation to existing methods in terms of reliability and speed when solved using the same mixed complementarity problem solver. We demonstrate that our approach more reliably solves difficult collision detection problems with multiple obstacles than other methods, and is faster than existing methods in some cases.