Abstract: Many processes in the real-world consist in making choices with probabilities in proportion to the total reward accumulated when making those choices in the past. Universal mathematical models for such processes are the interacting urns with reinforcement on graphs. We will discuss a multiple colour generalisation of the model of graph interacting urns studied by Benaim et. al., Random Struct. Alg., 46: 614-634, 2015. We show that for complete graphs and for a broad class of reinforcement functions governing the addition of balls in the urns, the process of colour proportions at each urn converges almost surely to the fixed points of the reinforcement function. The main tools employed are the stochastic approximation method and Lyapunov functions.