Finding optimal strategies of matrix game

Question

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?

Answer 1

Your solution is correct, since the elements $ a_ {11} = 0 $ and $ a_ {41} = 0 $ are both saddle points of the matrix, so your strategies are optimal solutions.

However, I am not entirely sure of your reasoning. A priori it is not enough that the row player is guaranteed a non-negative payout. He would have to get the biggest payout possible to accept his strategy. Certainly in this case that highest possible payout is zero, since that is the value of the game according to the saddle points. Maybe that's what you had in mind, and in that case everything would be fine. But you should probably justify that this is the highest payout that the row player can guarantee (saddle points argument again).