 # I have a function f = &#x2212;<!-- − --> 20 p &#x22C5;<!-- ⋅ --> q + 9 Joshua Foley 2022-07-06 Answered
I have a function $f=-20p\cdot q+9p+9q$. Player 1 chooses $p$ and player 2 chooses $q$. Both $p$ and $q$ are in the inclusive interval [0,1]. Player 1 wants to maximize $f$ while player 2 wants to minimize $f$.

Player 1 goes first, what is the most optimal value of $p$ he should choose knowing that player 2 will choose a $q$ in response to player 1's choice of $p$?

This seems to be some sort of minimization-maximization problem, but I am unsure how to solve it. I was thinking about approaching this from a calculus perspective by taking the partial derivative of $f$ with respect to $p$, but it doesn't seem I get an intuition by doing this, and it seems that $p$ and $q$ are a function of each other. How should I approach solving this problem analytically?
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Your problem can be formulated as:

Let's see $g\left(p\right)$,

Case-1:

Case-2:

You can see $p=\frac{9}{20}$ is the best move for the first player. Similar to the previous answer but more mathematical. Note that even if $p\in \mathbb{R}$ we cannot gain any advantage.

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