Symmetric mixed-strategy equilibrium: Entering markets [Solved]
I solved it, haha. Turns out I was being silly. I had completely neglected the fact that this was a 3-player game, and $\text{not entering}$ still had probability values attached to it. Let $A$, $B$, and $C$ be the three firms. So the game looks like this:
$$
\text{P(A enter;B,C enter)+P(A enter;only B enter)+P(A enter;only C enter)+}\\
\text{P(A enter;neither enters) = P(A dne;B,C enter)+P(A dne;only B enter)+}\\
\text{P(A dne;only C enter)+P(A dne;neither enters)}
$$
It's a giant mess of $p$'s and $(1-p)$'s, but at the end of the day, the final version $8=9(1-p^2)$ is derived.