The Dynamical Landscape of Beggar-My-Neighbour: Ultra-long Matches, Loops, and Infinite Matches

TL;DR

An automated `Infinite Loop Factory'algorithm is introduced which, by implementing adaptive insertion strategies, proves effective in identifying non-terminating cycles with balanced initial deck configurations, thereby confirming the existence of non-terminating dynamics in standard and generalised settings of the game.

Abstract

We present a rigorous mathematical and computational analysis of the deterministic card game \emph{Beggar-My-Neighbour}. By establishing a formal state-space framework, we investigate the game's dynamical landscape, focussing on the dichotomy between terminating and non-terminating matches. Extensive numerical simulations reveal that the distribution of finite match durations \emph{approximates} an exponential decay, with relevant deviations, confirming an emergent memory-less dynamics. This statistical behaviour is further analysed in the context of ultra-long matches, where we identify characteristic multi-scale oscillatory patterns and entropic regimes. Theoretically, we address the problem of backwards determinism, formalising the lack of injectivity of the trick function even within the set of reachable states. Crucially, we contribute to the recent resolution of the long-standing question regarding the existence of infinite games. We introduce an automated `Infinite Loop Factory' algorithm which, by implementing adaptive insertion strategies, proves effective in identifying non-terminating cycles with balanced initial deck configurations, thereby confirming the existence of non-terminating dynamics in standard and generalised settings of the game.

Figures (15)

• Figure 1: Example game with $N=40$ cards, and special cards up to maximum rank $\mathfrak{R}=3$, which is won in 5 tricks.
• Figure 2: Left: Output of a numerical simulation of $10^7$ independent matches for the game with $N=40$ cards, and special cards' highest rank $\mathfrak{R}=3$. Right: frequency distribution of the matches versus their length in number of tricks per match.
• Figure 3: Frequency distribution of the number of tricks per match for each $(N,\mathfrak{R})$-setting game, with $N=40$ and $\mathfrak{R}\in\{2,3,4,5,6,7\}$.
• Figure 4: Semi-logarithmic plot of game duration frequency for the $(40,3)$-setting based on the matches simulation reported in Table \ref{['tab:numanalrecap']} and \ref{['tab:numanalrecapN']}. The approximately linear decay in the range $n \in [10, 100]$ is consistent with exponential distribution $\rho(k) \propto e^{-\lambda n}$ with $\lambda \approx 0.0394$. Strong deviation from linearity occurs for $n \leqslant 9$ (short games).
• Figure 5: Explicit detection of lack of injectivity, in the space of reachable configurations, of the trick rule function $\mathfrak{F}$ in the $(52,4)$-setting: two distinct matches display distinct decks after the third trick, and yet the same decks in the following fourth trick.