Matching-Updating Mechanism: A Solution for the Stable Marriage Problem with Dynamic Preferences
Abstract
:1. Introduction
2. Preliminaries
2.1. The Stable Marriage Problem
2.2. Dynamic Preferences in the SMP
3. Updating the Stable Matching
- Determining the impacts of modifying an agent’s preferences: We identify the effects of changing an agent’s preference and determine whether it leads to the occurrence of a blocking pair in a matching.
- Initiating an update of the matching if a blocking pair exists.
- If there is no blocking pair, the previous instance’s matching will likely be stable for the new instance.
3.1. Identifying the Preference Changes
Algorithm 1 Finding a potential blocking pair. |
Input: -Current SMP Instance: mPref, wPref -Previous SMP Instance: prevMPref, prevWPref, Stable Matching (SM)
|
3.2. Initiating the Matching Update
- Imagine that there is one room with one entrance; randomly select a pair from each matching process. Let the selected pair enter the room. The selected pair can be confirmed as a stable matching in this room because no other choice can break the pair. Meanwhile, the rest of the agents form a queue outside the room to enter the room one by one.
- Ask an agent who is in front of the room to enter the room. There will be a matching process inside the room. The door of the room will remain closed before a stable matching is formed in the room.
- Repeat the second step until there are no remaining queues and a stable matching is obtained. As a result, a stable matching will be obtained without any blocking pairs.
- Imagine that there is one room with one entrance; select some pairs that have been confirmed as a stable matching. Let of all the selected pairs enter the room together. Meanwhile, the rest of the agents form a queue outside the room to enter the room one by one.
- Ask an agent who is in front of the room to enter the room. There will be a matching process inside the room. The door of the room will remain closed before a stable matching is formed in the room.
- Repeat the second step until there is no remaining queue and a stable matching is obtained. As a result, a stable matching will be obtained without any blocking pairs.
3.3. Reducing the Previous Matching
3.4. Controlling the Matching Orientation
Algorithm 2 Updating stable matching. |
Input: -Current SMP Instance: mPref, wPref -Previous SMP Instance: prevMPref, prevWPref, Stable Matching (SM)
|
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Algorithm A1 Procedure of the path to stability. |
Input: -SMP Instance (manPrefers and womanPrefers)
|
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Alimudin, A.; Ishida, Y. Matching-Updating Mechanism: A Solution for the Stable Marriage Problem with Dynamic Preferences. Entropy 2022, 24, 263. https://doi.org/10.3390/e24020263
Alimudin A, Ishida Y. Matching-Updating Mechanism: A Solution for the Stable Marriage Problem with Dynamic Preferences. Entropy. 2022; 24(2):263. https://doi.org/10.3390/e24020263
Chicago/Turabian StyleAlimudin, Akhmad, and Yoshiteru Ishida. 2022. "Matching-Updating Mechanism: A Solution for the Stable Marriage Problem with Dynamic Preferences" Entropy 24, no. 2: 263. https://doi.org/10.3390/e24020263
APA StyleAlimudin, A., & Ishida, Y. (2022). Matching-Updating Mechanism: A Solution for the Stable Marriage Problem with Dynamic Preferences. Entropy, 24(2), 263. https://doi.org/10.3390/e24020263