# Best response correspondences game theory

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!

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.