Strategic Coloring of a Graph

We study a strategic game where every node of a graph is owned by a player who has to choose a color. A player’s payoff is 0 if at least one neighbor selected the same color, otherwise it is the number of players who selected the same color. The social cost of a state is defined as the number of distinct colors that the players use. It is ideally equal to the chromatic number of the graph but it can substantially deviate because every player cares about his own payoff, whatever how bad the social cost is. Following a previous work done by Panagopoulou and Spirakis [1] on the Nash equilibria of the coloring game, we give worst case bounds on the social cost of stable states. Our main contribution is an improved (tight) bound for the worst case social cost of a Nash equilibrium, and the study of strong equilibria, their existence and how far they are from social optima.ou