This paper studies the likelihood of the existence of a pure Nash equilibrium (PNE) in random payoff graphical games. Here, players are represented by vertices, they choose a strategy in finite discrete sets of strategies, and the scope of a player’s utility function is only local. In this setting, the probability of existence of a PNE has been deeply studied for various graphical structures when the number of players tends to infinity, but only in the two strategies-per-player case: this paper extends these studies to an arbitrary number of strategies-per-player. We prove theoretically how more strategies-per-player makes the distribution of the number of equilibria get closer to a Poisson distribution. We apply these results to various graph structures and conclude with numerical experiments.