A version of the game of Nim is used to illustrate the minmax strategy in Rosen's Discrete Mathematics and Its Applications, 3ed, chapter 11 p. 768. I understand the +1 and -1 labels here. However, why is there not a [121] node at the second level? (question 1)
However, I had trouble understand this part (question 2):
Say if I start from the rightmost node [21] in the second level. The first player will seek to win, and thus will choose one of the two +1 options in the level below. The second player will be forced to pick up the last stone and thus lose. So shouldn't the value of this node [21] be +1 instead?
Say if I start from the rightmost node [1] of the third level, the first player must pick up the only stone there is. So shouldn't the value of this node be -1 in this case?
I am so confused by the stated Theorem 3. Please help.
As per @JMoravitz's explanation in the comment, theorem 3 should be understood as:
The value of a vertex of a game tree tells us the payoff to the first player if both players follow the minmax strategy and play continues from the position represented by this vertex.
This is supported by the wording of the inductive step of the theorem's proof, which suggests that a continuation of the game and not a reset. A continuation retains the players' turns.
If starts entails maintaining the players' turns, then everything is fine. I am still bothered by the use of starts here, and feel that it is rather misleading in this context.