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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We study the stable marriage problem in two-sided markets with randomly generated preferences. Agents on each side of the market are divided into a constant number of "soft" tiers, which capture agents' qualities. Specifically, every agent within a tier has the same public score, and agents on each side have preferences independently generated proportionally to the public scores of the other side.
We compute the expected average rank which agents in each tier have for their partners in the man-optimal stable matching, and prove concentration results for the average rank in asymptotically large markets. Furthermore, despite having a significant effect on ranks, public scores do not strongly influence the probability of an agent matching to a given tier of the other side. This generalizes the results by Pittel [Pittel, 1989], which analyzed markets with uniform preferences. The results quantitatively demonstrate the effect of competition due to the heterogeneous attractiveness of agents in the market.

Itai Ashlagi, Mark Braverman, Amin Saberi, Clayton Thomas, and Geng Zhao. Tiered Random Matching Markets: Rank Is Proportional to Popularity. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ashlagi_et_al:LIPIcs.ITCS.2021.46, author = {Ashlagi, Itai and Braverman, Mark and Saberi, Amin and Thomas, Clayton and Zhao, Geng}, title = {{Tiered Random Matching Markets: Rank Is Proportional to Popularity}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {46:1--46:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.46}, URN = {urn:nbn:de:0030-drops-135851}, doi = {10.4230/LIPIcs.ITCS.2021.46}, annote = {Keywords: Stable matching, stable marriage problem, tiered random markets, deferred acceptance} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

In the min-cost bipartite perfect matching with delays (MBPMD) problem, requests arrive online at points of a finite metric space. Each request is either positive or negative and has to be matched to a request of opposite polarity. As opposed to traditional online matching problems, the algorithm does not have to serve requests as they arrive, and may choose to match them later at a cost. Our objective is to minimize the sum of the distances between matched pairs of requests (the connection cost) and the sum of the waiting times of the requests (the delay cost). This objective exhibits a natural tradeoff between minimizing the distances and the cost of waiting for better matches. This tradeoff appears in many real-life scenarios, notably, ride-sharing platforms. MBPMD is related to its non-bipartite variant, min-cost perfect matching with delays (MPMD), in which each request can be matched to any other request. MPMD was introduced by Emek et al. (STOC'16), who showed an O(log^2(n)+log(Delta))-competitive randomized algorithm on n-point metric spaces with aspect ratio Delta.
Our contribution is threefold. First, we present a new lower bound construction for MPMD and MBPMD. We get a lower bound of Omega(sqrt(log(n)/log(log(n)))) on the competitive ratio of any randomized algorithm for MBPMD. For MPMD, we improve the lower bound from Omega(sqrt(log(n))) (shown by Azar et al., SODA'17) to Omega(log(n)/log(log(n))), thus, almost matching their upper bound of O(log(n)). Second, we adapt the algorithm of Emek et al. to the bipartite case, and provide a simplified analysis that improves the competitive ratio to O(log(n)). The key ingredient of the algorithm is an O(h)-competitive randomized algorithm for MBPMD on weighted trees of height h. Third, we provide an O(h)-competitive deterministic algorithm for MBPMD on weighted trees of height h. This algorithm is obtained by adapting the algorithm for MPMD by Azar et al. to the apparently more complicated bipartite setting.

Itai Ashlagi, Yossi Azar, Moses Charikar, Ashish Chiplunkar, Ofir Geri, Haim Kaplan, Rahul Makhijani, Yuyi Wang, and Roger Wattenhofer. Min-Cost Bipartite Perfect Matching with Delays. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ashlagi_et_al:LIPIcs.APPROX-RANDOM.2017.1, author = {Ashlagi, Itai and Azar, Yossi and Charikar, Moses and Chiplunkar, Ashish and Geri, Ofir and Kaplan, Haim and Makhijani, Rahul and Wang, Yuyi and Wattenhofer, Roger}, title = {{Min-Cost Bipartite Perfect Matching with Delays}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {1:1--1:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.1}, URN = {urn:nbn:de:0030-drops-75509}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.1}, annote = {Keywords: online algorithms with delayed service, bipartite matching, competitive analysis} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 5011, Computing and Markets (2005)

Every game in strategic form can be extended by adding a correlation device.
Any Equilibrium in such an extended game is called a correlated equilibrium (Aumann 1974).
Aumann showed that there exist games, where the agents surplus in
a correlated equilibrium is greater than their surplus in every equilibrium.
This suggests the study of two major measures for the value of correlation:
1. The ratio between the maximal surplus obtained in an correlated equilibrium to the maximal surplus obtained in equilibrium.
We refer to this ratio as the mediation value.
2. The ratio between the optimal surplus to the maximal surplus obtained in correlated equilibrium.
We refer to this ratio as the enforcement value.
In this work we initiate the study of the mediation value and of the enforcement value,
providing several general results on the value of correlation as captured by these concepts.
We also present a set of results for the more specialized case of congestion games,
a class of games that received a lot attention in the recent computer science and e-commerce communities.
Indeed, while much work in computer science has been devoted to the study of the ratio between the surplus in optimal strategies to the surplus in the worst Nash equilibrium (the so called "price of anarchy") for congestion games, our work presents and initiates the study of two other complementary measures.

Itai Ashlagi, Dov Monderer, and Moshe Tennenholtz. The Value of Correlation in Strategic Form Games. In Computing and Markets. Dagstuhl Seminar Proceedings, Volume 5011, pp. 1-29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{ashlagi_et_al:DagSemProc.05011.20, author = {Ashlagi, Itai and Monderer, Dov and Tennenholtz, Moshe}, title = {{The Value of Correlation in Strategic Form Games}}, booktitle = {Computing and Markets}, pages = {1--29}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {5011}, editor = {Daniel Lehmann and Rudolf M\"{u}ller and Tuomas Sandholm}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05011.20}, URN = {urn:nbn:de:0030-drops-2317}, doi = {10.4230/DagSemProc.05011.20}, annote = {Keywords: Correlation, mediation, enforcement, equilibrium, mediator} }

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