Prophet inequalities for optimal stopping rules with probabilistic recall

Abstract

Let X_i, i = 1, ..., n, be independent random variables, and consider an optimal stopping problem where an observation not chosen in the past is still available i steps later with some probability p_i, 1 ≥ p₁ ≥ ... ≥ p_{n -1} ≥ 0. Only one object may be chosen. After formulating the general solution to this optimal stopping problem, we consider `prophet inequalities' for this situation. Let V<sub>\bf p</sub> (X<sub>1</sub>, ..., X<sub>n</sub>) be the optimal value to the statistician. We show that for all non-trivial, non-negative X<sub>i</sub> and all n ≥ 2, the `ratio prophet inequality' \rm E[ \max (X₁, ..., X_n)] < (2 - p_{n -1} ) V_{\bf p} (X₁, ..., X_n) holds, and 2 - p_{n -1} is the `best constant'. This generalizes the classical prophet inequality with no recall, in which the best constant is 2. For any 0 ≤ X<sub>i</sub> ≤ 1, the `difference prophet inequality' \rm E[\max (X₁, ..., X_n)] - V_{\bf p} (X₁, ..., X_n) ≤ (1- p_n-1) [ 1 - (1 - p_{n - 1})^1/2 ]² / p²_n-1 holds. Prophet regions are also discussed.