Let Xi be non-negative, independent random variables with finite expectation, and X∗ n = max{X1, . . . , Xn}. The value EX∗ n is what can be obtained by a “prophet”. A “mortal” on the other hand, may use k ≥ 1 stopping rules t1, . . . , tk, yielding a return of E[maxi=1,...,k Xti ]. For n ≥ k the optimal return is V n k (X1, . . . , Xn) = supE[maxi=1,...,k Xti ] where the supremum is over all s...