Summary: The article concerns game problems for controlled systems with arbitrary Riemann-Liouville fractional derivatives [S. G. Samko, A. A. Kilbas and O. I. Marichev, “Integrals and derivatives of fractional orders and some of their applications” (1987, Zbl 0617.26004; English translation 1993, Zbl 0818.26003)]. Under fixed controls of the players, solution of such systems is presented in the form of analog of Cauchy formula. On the basis of the method of resolving functions [A. A. Chikrij, “Conflict-controlled processes”, Kluwer, Dordrecht (1997; Zbl 0868.93001)], we derive sufficient conditions for the finite-time game termination from given initial states. At the heart of these conditions lies the modified Pontryagin’s condition consisting in the nonemptiness of certain set-valued mappings. These mappings are expressed through the control domains of the players and the generalized Mittag-Leffler functions for the matrix of the system. To evaluate the latter, the apparatus of interpolating polynomials of Lagrange-Sylvester is used [F.R. Gantmakher, Theory of Matrices, Nauka, Moscow, 1967)]. The results are illustrated in the example with ’simple’ matrix and fractional generalized controlling Pontryagin’s example [L. S. Pontryagin, “Selected scientific works. Vol. II: Differential equations. Operator theory. Optimal control. Differential games”, Nauka, Moscow (1988; Zbl 0656.01015)]. In so doing various specific cases are analyzed, including known model examples ‘The Boy and the Crocodile’ and ‘Isotropic Rockets’ [R. Isaacs, “Differential games” (1965; Zbl 0125.38001)].