We study the optimization of ergodic averages for multi-valued dynamical systems, i.e. where points may have multiple different forward orbits. Under upper semi-continuity assumptions, we show that the maximum space average with respect to invariant probability measures for such systems can be characterised in terms of maximum time averages on an auxiliary shift space. For all multi-valued expanding systems that are open mappings, we show that every H\"older continuous real-valued function can be modified by a coboundary, of the same H\"older exponent, such that the resulting function is dominated by its maximum ergodic average.
Jenkinson et al. (Sun,) studied this question.