A note on the network coloring game

The network coloring game has been proposed in the literature of social sciences as a model for conflict-resolution circumstances. The players of the game are the vertices of a graph with $n$ vertices and maximum degree $\Delta$. The game is played over rounds, and in each round all players simultaneously choose a color from a set of available colors. Players have local information of the graph: they only observe the colors chosen by their neighbors and do not communicate or cooperate with one another. A player is happy when she has chosen a color that is different from the colors chosen by her neighbors, otherwise she is unhappy, and a configuration of colors for which all players are happy is a proper coloring of the graph. It has been shown in the literature that, when the players adopt a particular greedy randomized strategy, the game reaches a proper coloring of the graph within $O(\log(n))$ rounds, with high probability, provided the number of colors available to each player is at least $\Delta+2$. In this note we show that a modification of the aforementioned greedy strategy yields likewise a proper coloring of the graph, provided the number of colors available to each player is at least $\Delta+1$.