Percolation games on rooted, edge-weighted random trees

TL;DR

The paper studies a two-player bond percolation game on rooted Galton–Watson trees with edges labeled by $+1$, $0$, or $-1$ and analyzes the probabilities of three outcomes: P1 win, P1 loss, and draw. It develops a fixed-point framework that expresses outcom e probabilities as fixed points of multivariate maps on matrices, with draw probabilities vanishing iff the fixed point is unique. The authors derive recurrence relations and establish contraction-type criteria, enabling sharp results for κ=2 and substantial partial results for κ=3, including phase-transition-like behavior of draw probabilities under varying edge-weight probabilities and offspring distributions. Simulations complement theory, illustrating phase transitions and guiding conjectures about draw probabilities across parameter regimes. The work connects to broader percolation, random-tree processes, and two-player combinatorial game literature, and provides both exact and numerically-supported insights into when draws occur and when game duration remains finite.

Abstract

Consider a rooted Galton-Watson tree $T$, to each of whose edges we assign, independently, a weight that equals $+1$ with probability $p_{1}$, $0$ with probability $p_{0}$ and $-1$ with probability $p_{-1}=1-p_{1}-p_{0}$. We play a game on a realization of this tree, involving two players and a token that is allowed to be moved from where it is currently located, say a vertex $u$ of $T$, to any child $v$ of $u$. The players begin with initial capitals that amount to $i$ and $j$ units respectively, and a player wins if either she is the first to amass a capital worth $κ$ units, where $κ\in \mathbb{N}$ is prespecified, or she is able to move the token to a leaf vertex, from where her opponent cannot move it any farther, or her opponent's capital is the first to dwindle to $0$. This paper is concerned with analyzing the probabilities of the three possible outcomes such a game may culminate in, as well as with finding conditions under which the expected duration of the game is finite. Of particular interest to us is the exploration of criteria that guarantee the probability of draw in such a game to be $0$. The theory we develop is further supported by observations obtained via computer simulations, providing a deeper insight into how the above-mentioned probabilities behave as the underlying parameters and / or offspring distributions are allowed to vary. We include in this paper conjectures pertaining to the behaviour of the probability of draw in our game (including a phase transition phenomenon, in which the probability of draw goes from being $0$ to being strictly positive) as the parameter-pair $(p_{0},p_{1})$ is varied suitably while keeping the underlying offspring distribution of $T$ fixed.

Theorems & Definitions (19)

• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Remark 1
• Theorem 4.4
• Theorem 4.5
• Theorem 4.6
• Corollary 1
• Lemma 1
• Lemma 2