Finding mixed Nash equilibrium

Question

I have following simultaneous move game table

Game table

Player 1 can take one of three actions A,B,C and player 2’s possible actions are D,E,F.

First of all I found pure Nash equilibrium

And (B,E) and (C,D) are pure Nash equilibrium.

Secondly I want to find mixed Nash equilibrium

For that I calculate it

$$u(A,p_2)=1p_d+4p_e+2(1-p_d-p_e)=2p_e-p_d+2$$ $$u(B,p_2)=2p_d+5p_e+3(1-p_d-p_e)=2p_e-p_d+3$$ $$u(c,p_2)=3p_d+2p_e-6(1-p_d-p_e)=-4p_e-3p_d+6$$

And I know that $$u(A,p_2)=u(B,p_2)=u(C,p_2)$$

But after this calculation, I cannot reach any result.

I’m stuck at this point.

Please help me to find mixed Nash equilibrium

Answer 1

The problem here is that in any mixed strategy Nash equilibrium players assign a positive probability only to strategies that aren't dominated. Note that strategy $B$ dominates $A$ and because of this last fact, it can be shown that there is a mixed strategy with $D$ and $E$ that dominates $F$.

The mixed strategy Nash equilibrium can be found from the equations:

$3p+4(1-p)=4p+(1-p)$

$2q+5(1-q)=3q+2(1-q)$

Where $p$ is used for the probability distribution over the first player.