Emperor nim and emperor sum: a new sum of impartial games

The emperor sum of combinatorial games is discussed in this study. In this sum, a player moves arbitrarily many times in one component. For every other component, the player moves once at most. The $$\mathcal {P}$$ P -positions of emperor sums are characterized using a parameter referred to as $$\mathcal {P}$$ P -position length. An emperor sum is a $$\mathcal {P}$$ P -position if and only if every component is a $$\mathcal {P}$$ P -position and the nim-sum of the $$\mathcal {P}$$ P -position lengths of all components is 0. This is similar to using the nim-sum of $$\mathcal {G}$$ G -values to characterize the $$\mathcal {P}$$ P -positions of the disjunctive sum of games.