Suppose I am given the following matrix:
I would like to find all MSNE
I started by doing the double underline method to find any PSNE. I discovered that none exist. I then looked at which strategies are strictly dominated. I noticed that Player 1 will never play M, because player 1 can always get a better payoff by playing L or R. So I eliminated the middle row "m".
After doing this, I get the following reduced matrix:
I assigned Player 2 a probability of $p_1$ for strategy A, $p_2$ for strategy B, and $1-p_1-p_2$ for strategy C. Similarly I assigned Player 2 a probability of $q$ for strategy L and $1-q$ for strategy R.
So now, we can see that Player 1's expected payoff of choosing L, that is $E(L)$, is $(1/2)(p_1)+(2/3)(p_2)+(1)(1-p_1-p_2)$
= Thus, $p_1=(6-2p_2)/3)$ and $p_2=(6-3p_1)/2$
Player 1's expected payoff of choosing R, that is $E(R)$, is $(1)(p_1)+(2/3)(p_2)+(1/3)(1-p_1-p_2)$
Did I do this correctly?
Similarly, Player 2's expected payoff for choosing strategy A, $E(A)$, is $(1/2)(q)+(1)(1-q)$
Player 2's expected payoff for strategy B, $E(B)$, is $(2/3)(q)+(2/3)(1-q)$
=$2q/3+(2-2q)/3$ =$2q+2-2q$ =$2$
Player 2's expected payoff for strategy C is $(1)(q)+(1/3)(1-q)$
I am not sure if I'm doing this correctly.