So I'm trying to wrap my head around the basics of game theory, and would like to know if I argue correctly:
Say I have the game
$$ A = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & 1 \\ 0 & 1 & -2 & -1 \\ 0 & 0 & 0 & 0 \end{pmatrix} $$ and that I would like to find optimal strategies for this game. Suppose that player $1$ plays the mixed strategy $(p_1,p_2,p_3,p_4)$. If $p_2$ and $p_3$ are greater than $0$, player $1$ is not guaranteed to receive a non-negative value. However, if they are zero, then player $1$ is guaranteed to receive $0$. This means that optimal strategies for player $1$ are of the form $(u, 0, 0, 1-u)$, where $u \in [0,1]$. In the same way one would find that the only optimal strategy for player 2 is $(1,0,0,0)$.
Is this correct?