Why does this paper say that the Nash-equilibrium of GAN is given by a discriminator which is 0 everywhere on the data distribution?

Question

I am facing difficulty in understanding the bolded portion of the following statement from this paper

GANs are defined by a min-max two-player game between a discriminative network $D_\Psi(x)$ and generative network $G_\theta(z)$. While the discriminator tries to distinguish between real data point and data points produced by the generator, the generator tries to fool the discriminator. It can be shown that if both the generator and discriminator are powerful enough to approximate any real-valued function, the unique Nash-equilibrium of this two-player game is given by a generator that produces the true data distribution and a discriminator which is 0 everywhere on the data distribution.

My understanding is that discriminator gives $\dfrac{1}{2}$ for any further inputs after training. But, what is the $0$ mentioned?

Answer 1

What this paper is not saying is that the discriminator, $D_{\phi}$, always returns a scalar value of zero. What they are saying is that the generator, $G_{\theta}(z)$, has accurately learned the distribution of the input data, and that the discriminator produces the correct answer for each input from the generator. It's a mathematical description of the global optimum for a GAN. This Medium article by Jonathan Hui talks more about Nash equilibria, the Kullback-Leibler, and more.