Before revision:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} }  if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
After revision:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} } if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using second-order optimization algorithms.