The Dependent Doors Problem: An Investigation into Sequential Decisions without Feedback
This work addresses a theoretical problem in sequential decision-making without feedback, which is incremental as it builds on existing abstractions but provides new analytical insights and algorithms.
The paper tackles the dependent doors problem, an abstraction for sequential decisions without feedback, by analyzing scenarios with positively correlated and acyclic dependencies, showing that configurations with the same fundamental distributions have optimal running times up to a universal constant and providing algorithms optimal up to constant factors, with specific results like logarithmic price for memoryless doors and near-optimal algorithms for two cascading memoryless doors.
We introduce the dependent doors problem as an abstraction for situations in which one must perform a sequence of possibly dependent decisions, without receiving feedback information on the effectiveness of previously made actions. Informally, the problem considers a set of $d$ doors that are initially closed, and the aim is to open all of them as fast as possible. To open a door, the algorithm knocks on it and it might open or not according to some probability distribution. This distribution may depend on which other doors are currently open, as well as on which other doors were open during each of the previous knocks on that door. The algorithm aims to minimize the expected time until all doors open. Crucially, it must act at any time without knowing whether or which other doors have already opened. In this work, we focus on scenarios where dependencies between doors are both positively correlated and acyclic.The fundamental distribution of a door describes the probability it opens in the best of conditions (with respect to other doors being open or closed). We show that if in two configurations of $d$ doors corresponding doors share the same fundamental distribution, then these configurations have the same optimal running time up to a universal constant, no matter what are the dependencies between doors and what are the distributions. We also identify algorithms that are optimal up to a universal constant factor. For the case in which all doors share the same fundamental distribution we additionally provide a simpler algorithm, and a formula to calculate its running time. We furthermore analyse the price of lacking feedback for several configurations governed by standard fundamental distributions. In particular, we show that the price is logarithmic in $d$ for memoryless doors, but can potentially grow to be linear in $d$ for other distributions.We then turn our attention to investigate precise bounds. Even for the case of two doors, identifying the optimal sequence is an intriguing combinatorial question. Here, we study the case of two cascading memoryless doors. That is, the first door opens on each knock independently with probability $p\_1$. The second door can only open if the first door is open, in which case it will open on each knock independently with probability $p\_2$. We solve this problem almost completely by identifying algorithms that are optimal up to an additive term of 1.