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This letter formulates the leader-following cluster consensus problem of a homogeneous linear time-invariant multi-agent system over a directed graph as an player graphical differential game. The agents utilize distributed state feedback control laws based on the available information to calculate their best response. In general, the absence of centralized information prevents the best responses of agents from constituting a Nash equilibrium solution. We show that the best responses depending on the unique solution of an algebraic Riccati equation constitute a Nash equilibrium for the proposed graphical differential game with a modified cost function. A numerical example illustrates the Nash equilibrium solution by showing that the agents can not improve their performance by altering their responses while solving the leader-following cluster consensus problem.