I've just started following a game theory course. I'm still getting used to the concepts so I hope I can get some comment on my thoughts. This is a homework exercise.
Consider a four square board. There are two players, players X and O. The game consists of four rounds. In round 1 and 3 player X writes a 'X' in one of the squares. In rounds 2 and 4 player Y writes a 'Y' in one of the squares. It is not allowed to write something in a square in which something has been written.
Determine the total number of possible pure strategies for each player.
I think I can calculate the answer by using a more general statement.
Suppose player $i$ has $N$ information sets. Denote by $M_n$ the number of possible actions player $i$ can take at information set $n$, $n = 1,\ldots,N$. Then the total number of possible pure strategies of player $i$ is $\prod_{n=1}^{N} M_n$.
My attempt at a proof: creating a pure strategy boils down to picking from each information set a possible action. Therefore the number of possible pure strategies is equal to the number of ways you can pick an action from information set 1 times the number of ways you can pick an action from information set 2, etcetera, up to information set N. In otherwords, it is equal to $\prod_{n=1}^{N} M_n$.
If this is correct, then the number of possible pure strategies for player X are $4\cdot 2^{12}$. For player Y, this would then be $3^4\cdot 1^{24}$.
Is this right? If not, where do I go wrong? Thanks in advance.