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?

Question

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&#039;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&#039;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?

Jordin Church · Accepted Answer

Your problem can be formulated as:         max          p      ∈      [      0      ,      1      ]            min          q      ∈      [      0      ,      1      ]        f  (  p  ,  q  )  =      max          p      ∈      [      0      ,      1      ]        g  (  p  )Let&#039;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  pCase-1:     9  −  20  p  ≥  0    ⟹    q  =  0     Hence   g  (  p  )  =  9  p   if   p  ≤      9    20  Case-2:     9  −  20  p  &amp;lt;  0    ⟹    q  =  1     Hence   g  (  p  )  =  9  −  11  p   if   p  &amp;gt;      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.

I have a function f = − 20 p ⋅ q + 9

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