Before revision:  Roughly, we say  a point (x^\star, y^\star) is an \varepsilon-greedy adversarial equilibrium if y^\star is an \varepsilon-approximate local maximum for f(x^\star,\cdot), and x^\star is an \varepsilon-approximate local minimum for a "greedy approximation" to the function \max_z f(x, z) which can be efficiently estimated using second-order optimization algorithms.
After revision:  Roughly, we say that a point (x^\star, y^\star) is an \varepsilon-greedy adversarial equilibrium if y^\star is an \varepsilon-approximate local maximum for f(x^\star,\cdot), and x^\star is an \varepsilon-approximate local minimum for a "greedy approximation" to the function \max_z f(x, z) which can be efficiently estimated using second-order optimization algorithms.