I have a function f = &#x2212;<!-- − --> 20 p &#x22C5;<!-- ⋅ --> q + 9

Joshua Foley 2022-07-06 Answered
I have a function f = 20 p q + 9 p + 9 q. 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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Answers (1)

Jordin Church
Answered 2022-07-07 Author has 11 answers
Your problem can be formulated as:
  max p [ 0 , 1 ] min q [ 0 , 1 ] f ( p , q ) = max p [ 0 , 1 ] g ( p )
Let's see g ( p ),
  g ( p ) = min q [ 0 , 1 ] 20 p q + 9 p + 9 q = min q [ 0 , 1 ] ( 9 20 p ) q + 9 p
Case-1:
  9 20 p 0 q = 0  Hence  g ( p ) = 9 p  if  p 9 20
Case-2:
  9 20 p < 0 q = 1  Hence  g ( p ) = 9 11 p  if  p > 9 20
You can see p = 9 20 is the best move for the first player. Similar to the previous answer but more mathematical. Note that even if p R we cannot gain any advantage.

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