1
$\begingroup$

enter image description here

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!

$\endgroup$
3
$\begingroup$

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.

| improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.