This paper considers the differential game approach to the leaderless formation control problem of a linear dynamical multi-robot system with directed graph topology. The formation requirement is formulated in terms of a finite-horizon quadratic cost function for each robot through the use of graph Laplacian and optimal control theory. The motion equations of robots and formation cost functions are the state equations and cost functions of the differential game. In the differential game approach, each robot can select its neighbours and implement distributed controllers; however, there is no guarantee that the formation control exists. The main contribution of this paper is a differential game formulation to formation control admitting a unique Nash equilibrium that ensures the existence of the formation control.