Question: Suppose that f : [0, 1] × [0, 1] → R is strictly concave in x ∈ [0, 1] and strictly con vex in y ∈ [0, 1] and continuous. Then there is a point (x ∗, y ∗) so that
In fact, define y = ?(x ) as the function so that f (x , ?(x )) = miny f (x , y). This function is well defined and continuous by the assumptions. Also define the function x = ψ(y) by f (ψ(y), y) = maxx f (x , y). The new function g(x ) = ψ(?(x )) is then a continuous function taking points in [0, 1] and resulting in points in [0, 1]. There is a theorem, called the Brouwer fixed-point theorem, which now guarantees that there is a point x ∗ ∈ [0, 1] so that g(x ∗) = x ∗. Set y ∗ = ?(x ∗). Verify that (x ∗, y ∗) satisfies the requirements of a saddle point for f.