Ratio Prophet Inequalities when the Mortal has Several Choices

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 stopping rules t1, . . . , tk such that P (ti ≤ n) = 1. We show that for a sequence of constants gk which can be evaluated recursively, the inequality EX∗ n < gkV n k (X1, . . . , Xn) holds for all such X1, . . . , Xn and all n ≥ k; g1 = 2, g2 = 1+e−1 = 1.3678 . . . , g3 = 1+e1−e = 1.1793 . . . , g4 = 1.0979 . . . and g5 = 1.0567 . . .. Similar results hold for infinite sequences X1, X2, . . .. Since with five choices the mortal is thus guaranteed over 94% of the prophet’s value, more than five choices may not be practical.