Multi-agent motion planning using differential games with lexicographic preferences

Multi-player games with lexicographic cost functions can capture a variety of driving and racing scenarios and are known to have pure-strategy Nash Equilibria (NE) under certain conditions. The standard Iterated Best Response (IBR) procedure for finding such equilibria can be slow because computing the best response for each agent generally involves solving a non-convex optimization problem. In this paper, we introduce a type of game which uses a lexicographic cost function. We show that for this class of games, the best responses can be effectively computed through piece-wise linear approximations. This enables us to approximate the NE using a linearized version of IBR. We show the gap between the linear approximations returned by our linearized IBR and the true best response drops asymptotically. We implement the algorithm and show that it can find approximate NE for a handful of agents driving in realistic scenarios in under 10 seconds.