Combinatorial games on Galton–Watson trees involving several-generation-jump moves

Abstract

We study the k-jump normal and k-jump misère games on rooted Galton–Watson trees, expressing the probabilities of various possible outcomes of these games as specific fixed points of functions that depend on k and the offspring distribution. We discuss phase transition results pertaining to draw probabilities when the offspring distribution is Poisson(λ). We compare the probabilities of various outcomes of the 2-jump normal game with those of the 2-jump misère game, and a similar comparison is drawn between the 2-jump normal game and the 1-jump normal game, under the Poisson regime. We describe the rate of decay of the probability that the first player loses the 2-jump normal game as λ→∞. We also discuss a sufficient condition for the average duration of the k-jump normal game to be finite.