How Far Can Nim in Disguise Be Stretched?

A move in the game of nim consists of taking any positive number of tokens from asinglepile. Suppose we add the class of moves of taking a nonnegative number of tokens jointly fromallthe piles. We give a complete answer to the question which moves in the class can be adjoined without changing the winning strategy of nim. The results apply to other combinatorial games with unbounded Sprague–Grundy function values. We formulate two weakened conditions of the notion of nim-sum 0 for proving the results.