How do I find mixed-strategy Nash equilibria for the following 3 by 2 game?

Question

------- L' ----R'

L-----3,0----0,1

M----0,1----3,0

R----2,2----2,2

I am trying eliminate one of the strategies for player 1 so that I can have a 2x2 game. However, mixing L(1/2) and M(1/2) gives me equivalent payoff as playing R. Is there some other trick to solve for a mixed strategy equilibrium for this game?

Answer 1

Suppose the column player uses mixed strategy $L'(p), R'(1-p)$. The row player's average payoffs are then $3p$ for $L$, $3-3p$ for $M$, $2$ for $R$. Thus the row player should play $M$ if $p < 1/3$, $R$ if $1/3 < p < 2/3$, $L$ if $p > 2/3$.

Now it's always better for the column player if the row player plays R. Thus the Nash equilibria have the row player always playing R and the column player using the mixed strategy $L'(p), R'(1-p)$ with $1/3 \le p \le 2/3$.