Let $f:[0,1]\times[0,1]\to\mathbb{R}$ be a continuous function. Define $$ M_x:=\max\limits_{0\leq y\leq 1} f(x,y), \qquad m_y:=\min\limits_{0\leq x\leq 1} f(x,y). $$ Is there a useful set of assumptions under which one can conclude that $$ \inf\limits_{0\leq x\leq 1}M_x = \sup\limits_{0\leq y\leq 1} m_y ? $$ As the example $f(x,y)=(xy)^2$ shows, this is not true in general. On the other hand, I think (though I haven't checked the details carefully) that this is true provided that for any given $x$, the maximum of $f(x,y)$ is attained at a $unique$ $y$, and that for any given $y$, the minimum of $f(x,y)$ is attained at a $unique$ $x$. I would be grateful for any reference that discusses this question in detail.
There are many results of this type, and in general they go under the name of minimax theorems. Often some sort of convexity / concavity assumption is made, as in Sion's Minimax Theorem. You are correct in stating that uniqueness of optimizers gives you equality when $f$ is continuous with domain $[0,1]\times [0,1]$. However, this can fail if the domain of $f$ is different. For example, let $f$ be the arc length metric on the circle. Then $M_x = \pi$ for all $x$, $m_y = 0$ for all $y$. The optima are achieved uniquely at $y = x$ and $x=y$, respectively, but $\pi = \inf_x M_x \neq \sup_y m_y = 0$. 

