How do I solve this game by mixed strategy Nash equilibrium

Question

     L        C        R
T  (7,2)    (2,7)    (3,6)

B  (2,7)    (7,2)    (4,5)

I can not find any strategy dominated by mixed strategy. So it is impossible to decrease game size to 2*2 game. Is it possible that no Nash equilibrium exists in this game?

Answer 1

There is always a mixed strategy Nash equilibrium for any bimatrix game, which can be found with the Lemke–Howson algorithm. For a $\ 2\times 3\ $ matrix game, however, it's probably easier to find a Nash equilibrium by finding the mixed max-min strategies of the player with only two strategies for all three $\ 2\times 2\ $ subgames. For at least one of these, expected payoffs of the omitted strategy of the other player will be dominated by those of some mixture of the other two strategies.

To save you work, I suggest you start by Finding a mixed strategy Nash equilibrium for the $\ 2\times2\ $ matrix game $$ \pmatrix{(7,2)&(3,6)\\(2,7)&(4,5)}\ , $$ obtained by omitting the second strategy of the second player.