Find all the NE of this game remember not to forget mixed st

Find all the NE of this game (remember not to forget mixed strategy NE):

Solution

LEts first find all best responses for each player given

Suppose Player 1 has strategies (U,M,D) and PLayer 2 has strategies (L,C,R)

Best Response of player 1 when player 2 chooses L is M & D because we can easily verify that player 1 has higher payoff of 4 rather than 3

Best Response of player 1 when player 2 chooses C is D for same reason mentioned above

Best Response of player 1 when player 2 chooses R is M

Now for Player 2 we have

Best Response of player 2 when player 1 chooses U is R

Best Response of player 2 when player 1 chooses M is L

Best Response of player 2 when player 1 chooses D is L

Hence We have 2 Pure Nash Equilibrium in this game and those are (M,L) & (D,L)

We know that Player 2 would choose only L then player 1 is indifferent becuase he has same payoff therefore Mixed Strategy NAsh Equilibrium is (1/2,1/2)