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 Yi=Xi-ic, i=1,2,.... For n=1,2,..., let M(Y1,...,Yn)=E(max1≤ i≤ nYi) and V(Y1,...,Yn)=supτ ∈ CnE(Yτ), where Cn denotes the set of all stopping rules for Y1,...,Yn. Sharp upper bounds for the difference M(Y1,...,Yn)-V(Y1,...,Yn) are given under various restrictions on c and n.
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