A Difference Prophet Inequality for Bounded I.I.D. Variables, with Cost for Observations

Abstract

Let $X_i$ be i.i.d. random variables, $0 \leq X_i \leq 1$ and $c \geq 0$, and let $Y_i = X_i - ic$. It is shown that for all $n$, all $c$ and all such $X_i, E(\max_{i \geq 1} Y_i) - \sup_t EY_t < e^{-1}$, where $t$ is a stopping rule and $e^{-1}$ is shown to be the best bound for which the inequality holds. Specific bounds are also obtained for fixed $n$ or fixed $c$. These results are very similar to those obtained by Jones for a similar problem, where $0 \leq X_i \leq 1$ are independent but not necessarily identically distributed. All results are valid and unchanged also when $Y_i$ is replaced by $Y^\ast_i = \max_{1 \leq j \leq i} X_j - ic$.