Summary: Multiple-pursuer multiple-evader games raise fundamental and novel problems in control theory and robotics. In this paper, we propose a distributed solution to this problem that simultaneously addresses the discrete assignment of pursuers to evaders as well as the continuous control strategies for capturing individual evaders. The resulting hybrid control framework guarantees the mutual exclusion property of the final assignment for all initial conditions as well as capturing all evaders after exploring at most a polynomial number of assignments, dramatically reducing the combinatorial nature of purely discrete assignment problems.