  We study the dynamics of evolution of points of agents placed in the vertices
of a graph within the rules of two-agent units of competition where an edge is
randomly chosen and the agent with higher points gets a new point with a
probability $p$. The model is closely connected to generalized vertex models
and anti-ferromagnetic Potts models at zero temperature. After studying the
most general properties for generic graphs we confine the study to discrete
d-dimensional tori. We mainly focus on the ring and torus graphs.
