NIM with Cash

TL;DR

This paper extends the classic NIM game by incorporating monetary resources, analyzing how the interplay of stones and dollars affects winning strategies for various finite sets A.

Contribution

It introduces a new variant of NIM with monetary costs and provides general theorems to determine winning conditions for different sets A.

Findings

Derived win conditions for A={1,L} and A={1,L,L+1}

Extended classical NIM analysis to include monetary constraints

Provided a framework for analyzing complex win scenarios in resource-based NIM variants

Abstract

Let A be a finite subset of $\nat$. Then NIM(A;n) is the following 2-player game: initially there are $n$ stones on the board and the players alternate removing $a\in A$ stones. The first player who cannot move loses. This game has been well studied. We investigate an extension of the game where Player I starts out with d dollars, Player II starts out with e dollars, and when a player removes a\in A he loses a dollars. The first player who cannot move loses; however, note this can happen for two different reasons: (1) the number of stones is less than min(A), (2) the player has less than $\min(A)$ dollars. This game leads to more complex win conditions then standard NIM. We prove some general theorems from which we can obtain win conditions for a large variety of finite sets A. We then apply them to the sets A={1,L}, and A={1,L,L+1}.