AbstractWe treat situations in which independent structurally identical decision problems are to be faced either simultaneously or serially. Recent work on such compound decision problems has centered on finding procedures that satisfy the strengthened asymptotic optimality property of Gilliland and Hannan (Ann. Math. Statist.40 (1969), 1536–1541) and on providing decision rules that at once satisfy an admissibility property and the classical asymptotic optimality property. We suggest a simplified method of accomplishing the first of these goals in general situations and then provide decision rules for finite state components simultaneously admissible and satisfying the strengthened optimality property.