Lets say I have 3 players, call them $p_1, p_2, p_3$. All players $p_i$ can choose between either play $A$ or $B$. Denote the probability that player $i$ will play $B$ with $q_i$. How do I find the mixed strategy for this game?
I was thinking something in the line of, assume $p_1$ plays $B$ then with probability $q_2$ $p_2$ plays $B$ and with probability $q_3$ $p_3$ plays $B$ so the expected payoff should be (where $u_1(B)$ denotes the "reward for $p_1$ playing $B$)
$$ q_3(q_2u_1(B) + (1-q_2)u_1(B)) + (1-q_3)(q_2u_i(B) + (1-q_2)u_i(B)) $$
On the other hand if $p_1$ plays $A$ the expected payoff is
$$ q_3(q_2u_1(A) + (1-q_2)u_i(A)) + (1-q_3)(q_2u_1(A) + (1-q_2)u_1(A)) $$
In order for $p_1$ to mix between strategies, this two payoffs have to be equal. Am I on the right track?