Pure-Strategy Bayesian Nash equilibrium with general common prior I'm doing a problem set on the subject of Bayesian Nash equilibrium. I'm asked to find the pure-strategy BNE of the following. I've calculated to matrix shown below. My first concern is if I've calculated the expected payoff matrix correctly, and second how do I find all of the pure-strategy BNE when the common prior is not concrete.
Matrix looks correct. To final all pure strategy BNEs, you'll have to discuss cases based on the value of $p$.
For example, if $p\in(0,1)$, then $FT$ is player 2's unique best response to $F$. Thus, to have a BNE, you'd want $F$ to be player 1's best response to $FT$ as well, meaning that you'd require $3p&gt;1-p$, or $p&gt;\frac14$. Hence, $(F,FT)$ is a BNE if $p\in(\frac14,1)$.
You should be able to find other BNEs following a similar line of reasoning. (Hint: don't forget the edge cases where $p=0$ and $p=1$.)