Difference prophet inequalities for [0,1]-valued i.i.d. random variables with cost for observations

Abstract

Let X₁,X₂,… be a sequence of [0,1]-valued i.i.d. random variables, let c≥0 be a sampling cost for each observation and let Y_i=X_i−ic, i=1,2,…. For n=1,2,…, let M(Y₁,…,Y_n)=E(max _1≤i≤nY_i) and V(Y₁,…,Y_n)=sup _τ∈CⁿE(Y_τ), where Cⁿ denotes the set of all stopping rules for Y₁,…,Y_n. Sharp upper bounds for the difference M(Y₁,…,Y_n)−V(Y₁,…,Y_n) are given under various restrictions on c and n.