Best response correspondences game theory

Question

I'm studying game theory and there is a slide where I don't quite understand how exactly each value was deduced particularly:

For $BR_L(q)$ and $BR_c(q)$. I understand why they are functions of $q$ and $p$, since the best response of one player is dependent on what strategy the other player will choose, but i'm not sure exactly why they take values of $1$ and $[0,1]$ and $0$ for the specific values of $q$ and $p$. If someone could explain the derivation of these, that would help a lot!

Answer 1

Let $\pi_L(p,q) = 3pq + p(1-q) + 2(1-p)(1-q)$ denote the expected payoff of the lecturer. The best response is defined as $BR_L(q) = \arg\max_{p \in [0,1]}\pi_L(p,q)$. Now, note that \begin{align} \frac{\partial \pi_L(p,q)}{\partial p} \begin{cases} > 0 \quad& \text{for } q > 1/4,\\ = 0 & \text{for } q = 1/4,\\ < 0 & \text{for } q < 1/4. \end{cases} \end{align} For $q>1/4$ the payoffs are strictly increasing in $p$ such that $p = 1$ is optimal. For $q=1/4$ the payoffs are constant in $p$ such that any $p \in [0,1]$ is optimal. For $q<1/4$ the payoffs are strictly decreasing in $p$ such that $p = 0$ is optimal. Same logic applies to The Catalan.