  Optimizing the allocation of units into treatment groups can help researchers
improve the precision of causal estimators and decrease costs when running
factorial experiments. However, existing optimal allocation results typically
assume a super-population model and that the outcome data comes from a known
family of distributions. Instead, we focus on randomization-based causal
inference for the finite-population setting, which does not require model
specifications for the data or sampling assumptions. We propose exact
theoretical solutions for optimal allocation in $2^K$ factorial experiments
under complete randomization with A-, D- and E-optimality criteria. We then
extend this work to factorial designs with block randomization. We also derive
results for optimal allocations when using cost-based constraints. To connect
our theory to practice, we provide convenient integer-constrained programming
solutions using a greedy optimization approach to find integer optimal
allocation solutions for both complete and block randomization. The proposed
methods are demonstrated using two real-life factorial experiments conducted by
social scientists.
