Nim variants

Article type: Research Article

Authors: van den Bergh, Mark^{a; *} | Kosters, Walter^b | Spieksma, Flora^a

Affiliations: [a] Mathematisch instituut, Universiteit Leiden, Netherlands | [b] Leiden Institute of Advanced Computer Science, Universiteit Leiden, Netherlands

Correspondence: [*] Corresponding author. E-mail: m.j.h.van.den.bergh@math.leidenuniv.nl.

Abstract: Combinatorial game theory provides results for the class of two-player, deterministic games with perfect information. With the aim of generalizing this theory to the class of non-perfect information games in mind, we introduce and analyze three variants of the game of Nim. In these variants, the opponent only receives partial information on the move executed by the opponent. We model the variants as games in extensive form and compute Nash equilibria for different starting configurations. For one variant, this provides a full characterization of the game. For the other variants, we prove some partial and structural results, but a full characterization remains elusive.

Keywords: Combinatorial game theory, Nim, game in extensive form, Nash equilibrium, Kriegspiel

DOI: 10.3233/ICG-220206

Journal: ICGA Journal, vol. 44, no. 1, pp. 2-17, 2022