  Multi-agent differential games are important and useful tools for analyzing
many practical problems. With the recent surge of interest in using UAVs for
civil purposes, the importance and urgency of developing tractable multi-agent
analysis techniques that provide safety and performance guarantees is at an
all-time high. Hamilton-Jacobi (HJ) reachability has successfully provided
safety guarantees to small-scale systems and is flexible in terms of system
dynamics. However, the exponential complexity scaling of HJ reachability
prevents its direct application to large scale problems when the number of
vehicles is greater than two. In this paper, we overcome the scalability
limitations of HJ reachability by using a mixed integer program that exploits
the properties of HJ solutions to provide higher-level control logic. Our
proposed method provides safety guarantee for three-vehicle systems -- a
previously intractable task for HJ reachability -- without incurring
significant additional computation cost. Furthermore, our method is scalable
beyond three vehicles and performs significantly better by several metrics than
an extension of pairwise collision avoidance to multi-vehicle collision
avoidance. We demonstrate our proposed method in simulations.
