We show that for several variations of partially observable Markov decision processes, polynomial-time algorithms for finding control policies are unlikely to or simply don't have guarantees of finding policies within a constant factor or a constant summand of optimal. Here ``unlikely'' means ``unless some complexity classes collapse,'' where the collapses considered are P=NP, P=PSPACE, or P=EXP. Until or unless these collapses are shown to hold, any control-policy designer must choose between such performance guarantees and efficient computation.