Minimax theorem

In the mathematical area of <a href="/facts/Game_theory/zkEIj1ya">game theory</a> and of <a href="/facts/Convex_optimization/7D6U4MTN">convex optimization</a>, a minimax theorem is a theorem that claims that

max
          
            x
            ∈
            X
          
        
        
          min
          
            y
            ∈
            Y
          
        
        f
        (
        x
        ,
        y
        )
        =
        
          min
          
            y
            ∈
            Y
          
        
        
          max
          
            x
            ∈
            X
          
        
        f
        (
        x
        ,
        y
        )
      
    
    {\displaystyle \max _{x\in X}\min _{y\in Y}f(x,y)=\min _{y\in Y}\max _{x\in X}f(x,y)}

under certain conditions on the sets 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 and on the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
. It is always true that the left-hand side is at most the right-hand side (<a href="/facts/Max%25E2%2580%2593min_inequality/48pXX4sN">max–min inequality</a>) but equality only holds under certain conditions identified by minimax theorems. The first theorem in this sense is <a href="/facts/John_von_Neumann/aUk6urof">von Neumann</a>'s minimax theorem about two-player <a href="/facts/Zero-sum_game/ccouScgK">zero-sum games</a> published in 1928, which is considered the starting point of <a href="/facts/Game_theory/zkEIj1ya">game theory</a>. Von Neumann is quoted as saying "As far as I can see, there could be no theory of games ... without that theorem ... I thought there was nothing worth publishing until the Minimax Theorem was proved". Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature.

Minimax theorem open-in-new

Minimax theorem