Partizan Subtraction Games

TL;DR

This paper explores various asymptotic behaviors in partizan subtraction games, extending the concept of dominance to include weak dominance, fairness, and ultimate impartiality, with computational insights linked to the Frobenius coin problem.

Contribution

It introduces new behavioral classifications for partizan subtraction games and provides arithmetic and geometric characterizations for small move sets.

Findings

Identified conditions for dominance, weak dominance, fairness, and impartiality.

Connected outcome behaviors to the Frobenius coin problem.

Provided characterizations for sets of size at most 2.

Abstract

Partizan subtraction games are combinatorial games where two players, say Left and Right, alternately remove a number n of tokens from a heap of tokens, with $n \in S_L$ (resp. $n \in S_R$) when it is Left's (resp. Right's) turn. The first player unable to move loses. These games were introduced by Fraenkel and Kotzig in 1987, where they introduced the notion of dominance, i.e. an asymptotic behavior of the outcome sequence where Left always wins if the heap is sufficiently large. In the current paper, we investigate the other kinds of behaviors for the outcome sequence. In addition to dominance, three other disjoint behaviors are defined, namely weak dominance, fairness and ultimate impartiality. We consider the problem of computing this behavior with respect to $S_L$ and $S_R$, which is connected to the well-known Frobenius coin problem. General results are given, together with arithmetic and geometric characterizations when the sets $S_L$ and $S_R$ have size at most 2.