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

TL;DR

This paper analyzes combinatorial $k$-jump games on Galton-Watson trees, deriving probabilities of outcomes, phase transitions, and decay rates, with a focus on Poisson offspring distributions and game duration conditions.

Contribution

It introduces fixed point characterizations of game outcomes on Galton-Watson trees and compares different $k$-jump game variants under Poisson offspring distributions.

Findings

Probabilities of game outcomes expressed as fixed points.

Phase transition conditions for draw probabilities in Poisson trees.

Decay rate of losing probability as offspring mean increases.

Abstract

We study the $k$-jump normal and $k$-jump mis\`{e}re games on rooted Galton-Watson trees, expressing the probabilities of various outcomes of these games as specific fixed points of certain functions that depend on $k$ and the offspring distribution. We discuss results on phase transitions pertaining to draw probabilities when the offspring distribution is Poisson$(\lambda)$ (i.e. for which values of $\lambda$, the draw probability is strictly positive). We compare the probabilities of the various outcomes of the $2$-jump normal game with those of the $2$-jump mis\`{e}re 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 $\lambda \rightarrow \infty$. Finally, we discuss a sufficient condition for the average duration of the $k$-jump normal game to be finite.