I am tasked with finding all of the Nash Equilibria (pure or mixed) in the of the following game:
$$\begin{array} \\&L&C&R\\ T&2,2&2,3&1,2\\ M&0,3&3,2&1,1\\ B&3,1&0,2&0,3\end{array}$$
I have already deduced that there are no pure strategy equilibria here by using the "underline the best response" method. In order to find mixed strategies, I know that one must be indifferent to all of his own options given a probability distribution from the other player, i.e. the expected utility from any of his choices is the same given what his opponent might play.
To do this, I started by finding the expected utility of each of the pure strategies by assigning probabilities to each of their actions. For Player $1$, $T$, $M$, and $B$ got probabilities of $x$, $y$, and $1-x-y$, respectively; I did a similar thing for Player $2$, with $p$, $q$, and $1-p-q$. Then, I calculated each players' utility given that his opponent picked a pure strategy; for Player $1$:
$$U_1((x,y,1-x-y),L)=3-x-3y$$ $$U_1((x,y,1-x-y), C)=2x+3y$$ $$U_1((x,y,1-x-y), R)=x+y$$
and for Player $2$:
$$U_2(T, (p,q,1-p-q))=q+2$$ $$U_2(M, (p,q,1-p-q))=2p+q+1$$ $$U_2(B, (p,q,1-p-q))=3-2p-q$$
Using these, I hoped to be able to see whether or not any of the pure strategies was strictly dominated given some other mixed strategy (e.g. $M$ is strictly dominated by some combination of $t$ and $B$). However, I'm not confident that this is the best way of actually doing this problem. Any suggestions as to where to go/what method to employ would be sincerely appreciated. Cheers.