Before revision:  Informally, we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: R^d \times R^d \rightarrow R if y^\star is a second-order local maximum for f(x^\star,\cdot) in a sense of Nesterov 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: R^d \times R^d \rightarrow 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.