# Game theory : winning positions ?

Can we apply the Sprague-Grundy theorem/algorithm if the final position is a winning position ?

If we have a finite two-person impartial game where the final position is a loosing position, we can apply the Sprague-Grundy method to find which player has a winning strategy by assigning each position a nonnegative integer, called the nimber :

• The nimber of the final position is $0$
• The nimber of a position is the mex (minimal excluded number) of all the nimbers of all the next possible positions

We have the very useful property :

• The nimber of the sum of two games is the XOR of the numbers of the two games

Now, from a position with non-zero nimber, the winning-strategy is to go to a position with zero nimber (it is proven to be always possible). The (nimber=0)- positions correspond to the loosing positions.

How to do the same if the final position is winning ?

• Is "nimber" a nickname for "number" or is it a distinct notion?
– mlc
Aug 24, 2017 at 18:07
• It is a nonnegative number associated with a position in a (generalised) game of Nim. This is all linked with the Sprague-Grundy theorem :) Aug 24, 2017 at 18:11
• Nim games where the last person to move loses are called misère games. See this previous math exchange question"Nimbers for misère games" and the website miseregames.org for details. Aug 25, 2017 at 4:28