In a game theory textbook there is something similar to the table below where there is one pure strategy nash equilibrium and multiple mixed strategy nash equilibria. It is a simultaneous game with the payoffs presented below.
If we assume that this game is played twice, How do I identify all subgame perfect equilibria for this game, as well as nash equilibrium that is not a subgame perfect equilibrium?
For a game with multiple pure strategy nash equilibrium I think I can find a solution by using backward induction, but for a game like this with just one pure strategy nash equilibrium and multiple mixed strategy nash equilibria, I have no idea how to identify the subgame perfect equilibria and possibly a nash equilibrium that is not subgame perfect equilibrium, especially when there are mixed strategy equilibria included.
Any help in this would be appreciated.
\begin{array}{|c|c|c|c|} \hline & A & B & C \\\hline A & (1,1) & (0,0) & (0,0)\\\hline B & (0,0) & (2,1) & (1,2)\\\hline C & (0,0) & (1,2) & (2,1)\\\hline \end{array}