The Question
We consider the following zero-sum strategic game in matrix form
\begin{array}{c|lcr} & \text{A} & \text{B} & \text{C} \\ \hline A & 0 & +\epsilon & -\delta \\ B & -\epsilon & 0 & 0 \\ C & \delta & 0 & 0 \end{array}
where $\epsilon$ and $\delta$ are nonnegative real numbers.
(a) Find all equilibria for this game (show your working) assuming $\epsilon \gt 0$ and $\delta \gt 0$.
(b) Assume some probabilities for playing $A$ or playing $B$, both for focal player and opponent. Calculate the expected payoffs for each strategy.
My Understanding
I am completely stuck as I seem to calculate that the strategy $(C,C)$ is a saddle point but then if we reduce it to the $2 \times 2$ game $(A,B)$ then $(A,A)$ is a saddle point therefore there can't be a mixed strategy correct?
The game is symmetric (i.e. the payoff matrix is skew-symmetric) so you know its value must be $\ 0\ $. Therefore any optimal mixed strategy $\ \big(p_1,p_2,p_3\big)\ $ for the second player must guarantee that the expected payoff to the first player be non-positive. It must therefore satisfy the inequalities \begin{align} &\epsilon p_2-\delta p_3&\le0\\ -\epsilon p_1&&\le 0\\ \delta p_1&&\le0\\ &p_i\ge0&\text{for }\ i=1,2,3, \end{align} and the equation $$ p_1+p_2+p_3=1\ . $$ The second and third inequalities imply that $\ p_1=0\ $, while the equation and the first inequality give $$ 0\ge\epsilon p_2-\delta p_3=(\epsilon+\delta)p_2-\delta\ \ \text{, or}\\ 0\le p_2\le\frac{\delta}{\epsilon+\delta}\ . $$ Conversely, if $\ p_2\ $ satisfies this final pair of inequalities, $\ p_3=1-p_2\ $, and $\ p_1=0\ $, then all six inequalities and the equation are satisfied, so $\ \big(0,p_2,p_3\big)\ $ is an optimal strategy.
Therefore, a mixed strategy $\ \big(p_1,p_2,p_3\big)\ $ is optimal if and only if: