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 $\mathrm{Poisson}<!--FUNCTION\; APPLICATION-->(\lambda )$. 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 $\lambda \to \infty$. We also discuss a sufficient condition for the average duration of the $k$-jump normal game to be finite.

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