The Sprague-Grundy Functions of Saturations of Misère Nim

We consider mis\`{e}re Nim as a normal-play game obtained from Nim by removing the terminal position. While explicit formulas are known for the Sprague-Grundy functions of Nim and Welter's game, no explicit formula is known for that of mis\`{e}re Nim. All three of these games can be considered as position restrictions of Nim. What are the differences between them? We point out that Nim and Welter's game are saturated, but mis\`{e}re Nim is not. Moreover, we present explicit formulas for the Sprague-Grundy functions of saturations of mis\`{e}re Nim, which are obtained from mis\`{e}re Nim by adjoining some moves.