Best response correspondences game theory
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}
&gt; 0 \quad&amp; \text{for } q &gt; 1/4,\\
= 0 &amp; \text{for } q = 1/4,\\
&lt; 0 &amp; \text{for } q &lt; 1/4.
\end{cases}
\end{align}
For $q&gt;1/4$ 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 $q&lt;1/4$ the payoffs are strictly decreasing in $p$ such that $p = 0$ is optimal.
Same logic applies to The Catalan.