In generative adversarial models (GANs), why should we solve min-max problem and not max-min?
You can use either a min-max or a max-min game, it doesn't matter. Look, this is the &quot;usual&quot; loss function (i. e. if you use the original loss proposed by Goodfellow et al. in their paper on GANs, though I use a slightly different notation):
$$\min_{G}\max_{D}\mathcal L_{\text{GAN}}\left(D, G \right) = \mathbb E_{x \sim p_{\text{data}\left( x\right) }}\left[ \log \left\{ D\left( x \right) \right\} \right] + \mathbb E_{z\sim p_{z}\left( z\right)}\left[ \log\left\{ 1 - D\left( G\left( z\right) \right) \right\}\right].$$
Now, you could also write this as:
$$\max_{G}\min_{D}\mathcal L_{\text{GAN}}\left(D, G \right) = \mathbb E_{x \sim p_{\text{data}\left( x\right) }}\left[ \log \left\{ 1 - D\left( x \right) \right\} \right] + \mathbb E_{z\sim p_{z}\left( z\right)}\left[ \log\left\{ D\left( G\left( z\right) \right) \right\}\right].$$
I hope that clarifies your doubt.