  The Finite Selection Model (FSM) was developed by Carl Morris in the 1970s
for the design of the RAND Health Insurance Experiment (HIE) (Morris 1979,
Newhouse et al. 1993), one of the largest and most comprehensive social science
experiments conducted in the U.S. The idea behind the FSM is that each
treatment group takes its turns selecting units in a fair and random order to
optimize a common assignment criterion. At each of its turns, a treatment group
selects the available unit that maximally improves the combined quality of its
resulting group of units in terms of the criterion. In the HIE and beyond, we
revisit, formalize, and extend the FSM as a general tool for experimental
design.
  Leveraging the idea of D-optimality, we propose and analyze a new selection
criterion in the FSM. The FSM using the D-optimal selection function has no
tuning parameters, is affine invariant, and when appropriate, retrieves several
classical designs such as randomized block and matched-pair designs. For
multi-arm experiments, we propose algorithms to generate a fair and random
selection order of treatments. We demonstrate FSM's performance in a case study
based on the HIE and in ten randomized studies from the health and social
sciences. On average, the FSM achieves 68% better covariate balance than
complete randomization and 56% better covariate balance than rerandomization in
a typical study. We recommend the FSM be considered in experimental design for
its conceptual simplicity, efficiency, and robustness.
