Abstract: We explore the generic properties of Two-Agent games on graphs. The basic unit of competition depends on a very simple rule: Given two agents connected by a link in a graph the one with the largest point is granted a point with probability p and the one with smallest point is granted a point with probability 1-p. Only one agent is granted a point. If we successively iterate this, the problem can be mapped to a constrained graph coloring problem which is directly connected to constrained anti-ferroagnetic Potts models at zero temperature. This model and its simple extensions were studied previously on the complete graph where every agent is connected to another -a mean field situation-. In this talk we discuss the time evolution of the model in n-dimensional tori, after generalizing the framework to arbitrary graphs.