The game is as follows:
$$\begin{array}{c|c|c|} &A&B\\ \hline A&2;3&2;3\\ \hline B&-1;2&1;2\\ \hline C&-1;3&4;2\\ \hline \end{array}$$
I've written a program that calculates nash equilibria. Here is one that it calculated with the corresponding payoff vectors:
The greyed-out strategies are the mixed strategies. Is this equilibrium indeed valid? To me it seems to be valid, but I am not sure as other tools, such as Gambit, did not calculate this equilibrium. So is there something wrong with the algorithm I programmed or not?
It looks correct, but maybe the tools didn't calculate this particular one because it lies at the intersection of two families of Nash equilibria. The first family has
$$ P_1(B) = \begin{cases} 0 & p < 2/5 \\ q & p = 2/5 \\ 1 & p > 2/5 \end{cases} $$ $$ P_1(A, F) = \begin{cases} 1 & p < 2/5 \\ 1-q & p = 2/5 \\ 0 & p > 2/5 \end{cases} $$ $$ P_2(C) = p $$ $$ P_2(D) = 1-p $$
and the other family has
$$ P_1(B) = r $$ $$ P_1(A, F) = 1-r $$ $$ P_2(C) = 2/5 $$ $$ P_2(D) = 3/5 $$
where $p, q, r$ are arbitrary in $[0, 1]$.