# Multiple sequences Prophet Inequality Under Observation Constraints

CoRR（2024）

摘要

In our problem, we are given access to a number of sequences of nonnegative
i.i.d. random variables, whose realizations are observed sequentially. All
sequences are of the same finite length. The goal is to pick one element from
each sequence in order to maximize a reward equal to the expected value of the
sum of the selections from all sequences. The decision on which element to pick
is irrevocable, i.e., rejected observations cannot be revisited. Furthermore,
the procedure terminates upon having a single selection from each sequence. Our
observation constraint is that we cannot observe the current realization of all
sequences at each time instant. Instead, we can observe only a smaller, yet
arbitrary, subset of them. Thus, together with a stopping rule that determines
whether we choose or reject the sample, the solution requires a sampling rule
that determines which sequence to observe at each instant. The problem can be
solved via dynamic programming, but with an exponential complexity in the
length of the sequences. In order to make the solution computationally
tractable, we introduce a decoupling approach and determine each stopping time
using either a single-sequence dynamic programming, or a Prophet Inequality
inspired threshold method, with polynomial complexity in the length of the
sequences. We prove that the decoupling approach guarantees at least 0.745 of
the optimal expected reward of the joint problem. In addition, we describe how
to efficiently compute the optimal number of samples for each sequence, and
its' dependence on the variances.

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