  Computing Nash equilibrium (NE) of multi-player games has witnessed renewed
interest due to recent advances in generative adversarial networks. However,
computing equilibrium efficiently is challenging. To this end, we introduce the
Gradient-based Nikaido-Isoda (GNI) function which serves: (i) as a merit
function, vanishing only at the first-order stationary points of each player's
optimization problem, and (ii) provides error bounds to a stationary Nash
point. Gradient descent is shown to converge sublinearly to a first-order
stationary point of the GNI function. For the particular case of bilinear
min-max games and multi-player quadratic games, the GNI function is convex.
Hence, the application of gradient descent in this case yields linear
convergence to an NE (when one exists). In our numerical experiments, we
observe that the GNI formulation always converges to the first-order stationary
point of each player's optimization problem.
