Ratio prophet inequalities when the mortal has several choices

Let Xi be nonnegative, 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,.,kXti]. For n.k the optimal return is Vnk(X1,.,Xn)=supE[maxi=1,.,kXti] 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<gkVnk(X1,.,Xn) holds for all such X1,.,Xn and all n.k; \hbox{g1=2}, g2=1+e.1=1.3678.,g3=1+e1.e=1.1793.,\breakg4=1.0979. and g5=1.0567.. Similar results hold for infinite sequences X1,X2,..