Consider the setting where treatment is a limited resource so that a fixed proportion of the population can receive treatment. We wish to assign the treatment resource to subgroups of the population (based on their covariate information) to maximize the population mean outcome. Consider the greedy treatment assignment strategy which prioritizes treating the groups that will benefit most from treatment. In the unconstrained case this strategy is optimal, but in the constrained case it can be suboptimal. This ties to the 0-1 knapsack problem from combinatorial optimization, which is a surprisingly difficult problem (NP-hard) given how naturally it arises.
The greedy deterministic solution is suboptimal in the population in the above image. On the ensuing slides, we show that there exists a greedy stochastic treatment strategy (treatment can be random) that outperforms the greedy and the optimal deterministic strategies. We prove a general version of this result, and develop inference for the mean outcome under the optimal resource-constrained treatment strategy, in
A. R. Luedtke and M. J. van der Laan, “Optimal indvidualized treatments in resource-limited settings,” International Journal of Biostatistics (to appear), 2015. [tech rep]