Modeling the multi-traffic signal-control synchronization: A Markov chains game theory approach

This paper presents a new paradigm for modeling the multi-traffic signal-control synchronization problem using game theory based on the extraproximal method. The objective in a crossing is to minimize the queuing delay and the problem for a signal controller is to find an optimal signal timing strategy, i.e. establishing green timing for each signal phase. Signal controllers are considered the players of the game. Each intersection aims at finding the green time that minimizes its signal and queuing delay. The problem presents natural restrictions: (a) the number of entering cars and exiting cars is different for each street of the intersection and (b) the interval of time given for vehicles to the green light is equal to the red light in the respective opposite direction. The first restriction determines a leader-follower Stackelberg game model: streets having more traffic require more green time. The last restriction establishes a simultaneous-solution of the game as a better consideration of the real situation. Then, to take advantage of the structure of the game, the paper shows that the solution is given by a Nash equilibrium. We introduce the c-variable method for finding the optimal signal timing distribution and making the problem computationally tractable. The extraproximal method is a two-step iterated procedure: (a) the first step calculates a preliminary approximation to the equilibrium point, and (b) the second step is designed to find an adjustment of the previous step. The formulation of the game is given in terms of coupled nonlinear programming problems implementing the Lagrange principle. Tikhonov's regularization method is employed to ensure the convergence of the costfunctions to a Nash equilibrium point. In addition, the extraproximal method is developed in terms of Markov chains. The usefulness of the method is demonstrated by a three-way intersection example. The contributions have major implications for real-world applications.