Partizan Serial Nim

A combinatorial game is a two-player game without hidden information or chance elements. The main object of combinatorial game theory is to determine the outcome (i.e., which player has a winning strategy) of a given position in combinatorial games. NIM is a well-known and fundamental ruleset in combinatorial game theory. This paper proposes a novel partizan variant of NIM called PARTIZAN-SERIAL-NIM, defined as follows: there are $n$ piles of stones indexed by $1,2,\dots ,n$; the two players have permutations ${\mathbf{\sigma }}^L$ and ${\mathbf{\sigma }}^R$ of $(1,2,\dots ,n)$, respectively; a move is to remove any positive number of stones from the non-empty pile with the minimum value in the player’s permutation; the player who cannot make a move loses. This ruleset is a generalization of SERIAL-NIM and PARTIZAN-END-NIM. We give an algorithm to compute the outcome of a given position in PARTIZAN-SERIAL-NIM in $O(n^2)$ time, provided that each arithmetic and comparison operation is performed in $O(1)$ time. Also, for the case where all non-empty piles have the same number $m$ of stones, we prove that the outcome does not depend on $m$ for $m\ge 2$ and present an algorithm to compute the outcome in $O(n)$ time. Further, we prove that the atomic weight of every position in PARTIZAN-SERIAL-NIM is an integer.