<p>Let <span class="math-container">$\pi_L(p,q) = 3pq + p(1-q) + 2(1-p)(1-q)$</span> denote the expected payoff of the lecturer. The best response is defined as <span class="math-container">$BR_L(q) = \arg\max_{p \in [0,1]}\pi_L(p,q)$</span>.
Now, note that 
<span class="math-container">\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}</span>
For <span class="math-container">$q&gt;1/4$</span> the payoffs are strictly increasing in <span class="math-container">$p$</span> such that <span class="math-container">$p = 1$</span> is optimal.
For <span class="math-container">$q=1/4$</span> the payoffs are constant in <span class="math-container">$p$</span> such that any <span class="math-container">$p \in [0,1]$</span> is optimal.
For <span class="math-container">$q&lt;1/4$</span> the payoffs are strictly decreasing in <span class="math-container">$p$</span> such that <span class="math-container">$p = 0$</span> is optimal.
Same logic applies to The Catalan.</p>

CC BY-SA 4.0
2020-05-04T12:39:49.927Z
2020-05-04T12:39:49.927Z
clueless
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<p>I'm studying game theory and there is a slide where I don't quite understand how exactly each value was deduced particularly:</p>

<p>For <span class="math-container">$BR_L(q)$</span> and <span class="math-container">$BR_c(q)$</span>. I understand why they are functions of <span class="math-container">$q$</span> and <span class="math-container">$p$</span>, 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 <span class="math-container">$1$</span> and <span class="math-container">$[0,1]$</span> and <span class="math-container">$0$</span> for the specific values of <span class="math-container">$q$</span> and <span class="math-container">$p$</span>. If someone could explain the derivation of these, that would help a lot!</p>

<game-theory>
Best response correspondences game theory
Slim Shady
# Best response correspondences game theory

(Asked by: username_0 on 2020-05-04)

[![enter image description here](https://i.sstatic.net/BM0LA.png)](https://i.sstatic.net/BM0LA.png)

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!

**Tags:** game-theory

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## 1 answers

### Answer by username_1 on 2020-05-04. Score: 3 | Accepted answer

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.


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