This paper presents a new model for computing optimal randomized security policies in non-cooperative Stackelberg Security Games (SSGs) for multiple players. Our framework rests upon the extraproximal method and its extension to Markov chains, within which we explicitly compute the unique Stackelberg/Nash equilibrium of the game by employing the Lagrange method and introducing the Tikhonov regularization method. We also consider a game-theory realization of the problem that involves defenders and attackers performing a discrete-time random walk over a finite state space. Following the Kullback-Leibler divergence the players' actions are fixed and, then the next-state distribution is computed. The player's goal at each time step is to specify the probability distribution for the next state. We present an explicit construction of a computationally efficient strategy under mild defenders and attackers conditions and demonstrate the performance of the proposed method on a simulated target tracking problem.