Combinatorial games on multi-type Galton-Watson trees

TL;DR

This paper investigates combinatorial games played on multi-type Galton-Watson trees, analyzing how game outcomes depend on tree structure, vertex types, and offspring distributions, revealing conditions for draws and asymptotic behaviors.

Contribution

It introduces and analyzes novel versions of normal, misère, and escape games on multi-type Galton-Watson trees with disjoint move sets for players, exploring outcome probabilities and asymptotic properties.

Findings

Draw probabilities are zero unless each vertex has one blue and one red child.

On Poisson trees with increasing λ, game outcome probabilities approach 1.

Outcome behaviors depend on offspring distribution and edge move restrictions.

Abstract

When normal and mis\`{e}re games are played on bi-type binary Galton-Watson trees (with vertices coloured blue or red and each having either no child or precisely $2$ children), with one player allowed to move along monochromatic edges and the other along non-monochromatic edges, the draw probabilities equal $0$ unless every vertex gives birth to one blue and one red child. On bi-type Poisson trees where each vertex gives birth to Poisson$(\lambda)$ offspring in total, the draw probabilities approach $1$ as $\lambda \rightarrow \infty$. We study such \emph{novel} versions of normal, mis\`{e}re and escape games on rooted multi-type Galton-Watson trees, with the "permissible" edges for one player being disjoint from those of her opponent. The probabilities of the games' outcomes are analyzed, compared with each other, and their behaviours as functions of the underlying law explored.