Planning Against a Prophet: A Graph-Theoretic Framework for Making Sequential Decisions
Andr\'es Cristi, Sigal Oren

TL;DR
This paper introduces a graph-theoretic framework for prophet inequalities, analyzing sequential decision-making under uncertainty and providing algorithms with provable performance guarantees.
Contribution
It presents a novel graph-based model for prophet inequalities, offering tight bounds and generalizations that encompass many existing problems in the field.
Findings
Algorithm guarantees a 1/2k prophet inequality ratio.
Upper bound of 1/(k+1) for the ratio.
Tight bound of 1/2 for the leasing variant.
Abstract
We devise a general graph-theoretic framework for studying prophet inequalities. In this framework, an agent traverses a directed acyclic graph from a starting node to a target node . Each edge has a value that is sampled from a known distribution. When the agent reaches a node it observes the realized values of all the outgoing edges from . The agent's objective is to maximize the expected total value of the path it takes. As in prophet inequalities, we compare the agent's performance against a prophet who observes all the realizations of the edges' values ahead of time. Our analysis reveals that this ratio highly depends on the number of paths required to cover all the nodes in the graph. In particular, we provide an algorithm that guarantees a prophet inequality ratio of and show an upper bound of . Our framework captures planning…
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Taxonomy
TopicsComplex Systems and Decision Making
