Question: Suppose three identical, risk-neutral firms must decide simultaneously and irreversibly whether to enter a new market which can accommodate only two of them. If all three firms enter, all get payoff 0; otherwise, entrants get 9 and firms that stay out get 8.
My attempt: Arbitrarily choosing firm A, firm A is deciding whether to enter the market or not. Hence, the Nash equilibrium occurs when $\text{payoff of not entering = entering}$. Let $p$ be the probability of each firm entering the market. Then $$\text{not entering = P(only firm who enters) + P(second firm to enter) + P(third firm to enter)}\\ 8=9(p)(1-p)^2+9(p^2)(1-p)+0(p^3)\\ 8=9(p-p^2) $$
and I get a complex number as the answer. The correct equation should be $8=9(1-p^2)$, but I'm not sure how to get that.
Any help is greatly appreciated!
Edit: Solved it an hour later. I was being silly. I'm leaving my partial answer below for anyone who is curious.