Adversarial ergodic optimization

In this article, we introduce an ergodic optimization problem inspired by information theory, which can be presented informally as follows: given a factor map $\pi : (X, T) \to (Y, S)$ of topological dynamical systems, and a continuous function $f \in C(X)$, what can be said about the extrema $$\sup_{y \in Y} \inf_{x \in \pi^{-1} \{y\}} \lim_{k \to \infty} \frac{1}{k} \sum_{j = 0}^{k - 1} f \left( T^j x \right) ?$$