Nisan comments on Why you should minimax in two-player zero-sum games

[Nisan]

A proof of the lemma $V(a,b^{\ast })\le v^{\ast }$:

$$V(a,b^{\ast })=V(a,{\mathrm{argmin}}_b\underset{a^{\prime }}{max}V(a^{\prime },b\left)\right)$$

$$\le \underset{a^{\prime }}{max}V(a^{\prime },{\mathrm{argmin}}_b\underset{a^{\prime }}{max}V(a^{\prime },b\left)\right)$$

$$=\underset{b}{min}\underset{a^{\prime }}{max}V(a^{\prime },b)$$

$$=v^{\ast }$$