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

Moumanti Podder, Dhruv Bhasin

Abstract

We study the $k$-jump normal and $k$-jump misère 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$(λ)$ (i.e. for which values of $λ$, 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 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 $λ\rightarrow \infty$. Finally, we discuss a sufficient condition for the average duration of the $k$-jump normal game to be finite.

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

Abstract

We study the -jump normal and -jump misère 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 and the offspring distribution. We discuss results on phase transitions pertaining to draw probabilities when the offspring distribution is Poisson (i.e. for which values of , the draw probability is strictly positive). We compare the probabilities of the various outcomes of the -jump normal game with those of the -jump misère game, and a similar comparison is drawn between the -jump normal game and the -jump normal game, under the Poisson regime. We describe the rate of decay of the probability that the first player loses the -jump normal game as . Finally, we discuss a sufficient condition for the average duration of the -jump normal game to be finite.
Paper Structure (39 sections, 23 theorems, 197 equations, 6 figures)

This paper contains 39 sections, 23 theorems, 197 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Introduction to the games
  3. Motivations for studying these games
  4. Notations and some definitions
  5. Main results
  6. A brief discussion of the literature on combinatorial games
  7. Organization of the paper
  8. Proof of Theorem \ref{['thm:main_1_normal']}
  9. Showing that $\mathop{\mathrm{n\ell}}\nolimits_{k}$ is a fixed point of $H_{k}$
  10. Showing that $\mathop{\mathrm{n\ell}}\nolimits_{k}$ is the minimum positive fixed point of $H_{k}$
  11. Proof of Theorem \ref{['thm:main_1_misere']}
  12. Proof of Theorem \ref{['main:bounds_on_nl_k_ml_k']}
  13. Proof of Theorem \ref{['main:normal_draw_probab_limit_Poisson']}
  14. Understanding the behaviour of $c_{k,\lambda}$ as a function of $\lambda$
  15. Understanding the behaviour of $F_{j,\lambda}(c_{k,\lambda})$ for all $1 \leqslant j \leqslant k-1$
  16. ...and 24 more sections

Key Result

Theorem 1.1

Consider the $k$-jump normal game for $k \in \mathbb{N}$. Define the function $H_{k}: [0,c_{k-1}] \rightarrow [0,1]$ as Then $\mathop{\mathrm{n\ell}}\nolimits_{k}$ is the minimum positive fixed point of $H_{k}$. Moreover, $\mathop{\mathrm{nw}}\nolimits_{k} = 1 - F_{k}(\mathop{\mathrm{n\ell}}\nolimits_{k})$.

Figures (6)

  • Figure 1: Comparing probabilities $\mathop{\mathrm{n\ell}}\nolimits_{k}$ of P1 losing as functions of $\lambda$, for $k = 1, 2, 3$
  • Figure 2: Comparing probabilities $\mathop{\mathrm{nw}}\nolimits_{k}$ of P1 winning as functions of $\lambda$, for $k = 1, 2, 3$
  • Figure 3: Comparing probabilities $\mathop{\mathrm{nd}}\nolimits_{k}$ of draw as functions of $\lambda$, for $k = 1, 2, 3$
  • Figure 4: The function $H_{2}$ is not convex between $c_{2} \approx 0.095$ and $c_{1} \approx 0.265$, when $\chi$ is Poisson$(5)$
  • Figure 5: Here, $k = 2$ and $u_{0} \in \mathop{\mathrm{NL}}\nolimits^{(n+2)}$ (indicated in red), so that $u_{i} \in \mathop{\mathrm{NW}}\nolimits^{(n+1)}$ for all $1 \leqslant i \leqslant 6$ (indicated in blue). Since $u_{1} \in \mathop{\mathrm{NW}}\nolimits^{(n+1)}$ and its children $u_{3}$ and $u_{4}$ are in $\mathop{\mathrm{NW}}\nolimits^{(n+1)}$, $u_{1}$ must have at least one grandchild, say $v_{1}$ (indicated in red) in $\mathop{\mathrm{NL}}\nolimits^{(n)}$. Note that $v_{1} \in \mathop{\mathrm{NL}}\nolimits^{(n)}$ ensures $u_{4} \in \mathop{\mathrm{NW}}\nolimits^{(n+1)}$. Likewise, $u_{2}$ and its children $u_{5}$ and $u_{6}$ are in $\mathop{\mathrm{NW}}\nolimits^{(n+1)}$, hence $u_{2}$ must have at least one grandchild, say $v_{4}$, in $\mathop{\mathrm{NL}}\nolimits^{(n)}$. That $v_{4} \in \mathop{\mathrm{NL}}\nolimits^{(n)}$ ensures $u_{6} \in \mathop{\mathrm{NW}}\nolimits^{(n+1)}$. Let $u_{3}$ and $u_{5}$ have grandchildren $v_{2}$ and $v_{3}$ (respectively) in $\mathop{\mathrm{NL}}\nolimits^{(n)}$, but no child in $\mathop{\mathrm{NL}}\nolimits^{(n)}$. Then $u_{4}$ and $u_{6}$ are in $\mathcal{C}_{0,1,n}$, $u_{3}$ and $u_{5}$ are in $\mathcal{C}_{0,2,n}$, and $u_{1}$ and $u_{2}$ are in $\mathcal{C}_{1,2,n}$.
  • ...and 1 more figures

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 30 more