The problem of finding optimal coordinated signal timing plans for a large number of traffic signals is a challenging problem because of the exponential growth in the number of joint timing plans that need to be explored as the network size grows. In this paper, the game-theoretic paradigm of fictitious play to iteratively search for a coordinated signal timing plan is employed, which improves a system-wide performance criterion for a traffic network. The algorithm is robustly scalable to realistic-size networks modeled with high-fidelity simulations. Results of a case study for the city of Troy, MI, where there are 75 signalized intersections, are reported. Under normal traffic conditions, savings in average travel time of more than 20% are experienced against a static timing plan, and even against an aggressively tuned automatic-signal-retiming algorithm, savings of more than 10% are achieved. The efficiency of the algorithm stems from its parallel nature. With a thousand parallel CPUs available, the algorithm finds the plan above under 10 min, while a version of a hill-climbing algorithm makes virtually no progress in the same amount of wall-clock computational time.