Combinatorial 2-player games can be studied from different perspectives. Traditionally the goal has been to acquire a perfect strategy, and to this purpose an efficient procedure (polynomial in succinct input size) is required. However, most combinatorial games are intrinsically hard to analyze; success is limited to a small number of games with predominant “mathematical structure”. The classical games of Nim (Bouton 1901) and Wythoff Nim (Wythoff 1907) are easy to analyze rigorously, but already seemingly modest variants, like (p, q)-GDWN (Larsson 2012, 2014), appear to withstand log-polynomial descriptions. Therefore, development of new methods is highly desirable.

Here, we use methods from physics, such as renormalization, in an attempt to understand the larger geometry of a game's P-positions (safe positions for the Previous player), rather than their exact configurations (Friedman et al. 2007, 2009). By studying evolution diagrams of a general class of linear extensions of Nim, Wythoff Nim and GDWN, we observe that P-positions often distribute uniformly along lines (a.k.a. P-beams), visually separated from the move lines. Given a fundamental hypothesis, a filling property which generalizes directly from Wythoff Nim, we formulate natural equations on the slopes and densities of P-positions along these lines; here, a key innovation, a reorganization model, guides us in selecting the relevant rules (move lines). The exceptional case of the symmetric (p, q)-GDWN is interesting, because of observed quasi-log repetitive fluctuations, and these games have defied all previous analysis.

1. Introduction to the class Linear Nimhoff

In this paper we study a class of combinatorial games using renormalizationbased techniques from physics in combination with computer simulations. This approach leads to a probabilistic geometric analysis of the underlying structure and behavior of a game. A number of interesting features are revealed, including observations of quasi log-periodic fluctuations. Our class of games, dubbed Linear Nimhoff, is a generalization of the classical Wythoff's game [Wythoff 1907] and the more recent GDWN [Larsson 2012a].

The renormalization approach to games involves:

• 1) Identifying broad, overall patterns in games (on a course grained level) by focusing on their scaling, asymptotic, and/or global, probabilistic behavior;

• 2) Analyzing these patterns using scaling/course-graining techniques derived from self-consistency conditions;

• 3) A novel method for this paper, a reorganization model, which filters out game rules that do not contribute to the broad overall patterns.