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Documents authored by Vazirani, Vijay V.


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Vazirani, Vijay V.

Document
Towards a Practical, Budget-Oblivious Algorithm for the Adwords Problem Under Small Bids

Authors: Vijay V. Vazirani

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)


Abstract
Motivated by recent insights into the online bipartite matching problem (OBM), our goal was to extend the optimal algorithm for it, namely Ranking, all the way to the special case of adwords problem, called Small, in which bids are small compared to budgets; the latter has been of considerable practical significance in ad auctions [Mehta et al., 2007]. The attractive feature of our approach was that it would yield a budget-oblivious algorithm, i.e., the algorithm would not need to know budgets of advertisers and therefore could be used in autobidding platforms. We were successful in obtaining an optimal, budget-oblivious algorithm for Single-Valued, under which each advertiser can make bids of one value only. However, our next extension, to Small, failed because of a fundamental reason, namely failure of the No-Surpassing Property. Since the probabilistic ideas underlying our algorithm are quite substantial, we have stated them formally, after assuming the No-Surpassing Property, and we leave the open problem of removing this assumption. With the help of two undergrads, we conducted extensive experiments on our algorithm on randomly generated instances. Our findings are that the No-Surpassing Property fails less than 2% of the time and that the performance of our algorithms for Single-Valued and Small are comparable to that of [Mehta et al., 2007]. If further experiments confirm this, our algorithm may be useful as such in practice, especially because of its budget-obliviousness.

Cite as

Vijay V. Vazirani. Towards a Practical, Budget-Oblivious Algorithm for the Adwords Problem Under Small Bids. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 21:1-21:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{vazirani:LIPIcs.FSTTCS.2023.21,
  author =	{Vazirani, Vijay V.},
  title =	{{Towards a Practical, Budget-Oblivious Algorithm for the Adwords Problem Under Small Bids}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{21:1--21:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.21},
  URN =		{urn:nbn:de:0030-drops-193941},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.21},
  annote =	{Keywords: Adwords problem, ad auctions, online bipartite matching, competitive analysis}
}
Document
A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications

Authors: Rohith Reddy Gangam, Tung Mai, Nitya Raju, and Vijay V. Vazirani

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)


Abstract
Recently [Mai and Vazirani, 2018] identified and initiated work on a new problem, namely understanding structural relationships between the lattices of solutions of two "nearby" instances of stable matching. They also gave an application of their work to finding a robust stable matching. However, the types of changes they allowed in going from instance A to B were very restricted, namely any one agent executes an upward shift. In this paper, we allow any one agent to permute its preference list arbitrarily. Let M_A and M_B be the sets of stable matchings of the resulting pair of instances A and B, and let ℒ_A and ℒ_B be the corresponding lattices of stable matchings. We prove that the matchings in M_A ∩ M_B form a sublattice of both ℒ_A and ℒ_B and those in M_A ⧵ M_B form a join semi-sublattice. These properties enable us to obtain a polynomial time algorithm for not only finding a stable matching in M_A ∩ M_B, but also for obtaining the partial order, as promised by Birkhoff’s Representation Theorem [Birkhoff, 1937]. As a result, we can generate all matchings in this sublattice. Our algorithm also helps solve a version of the robust stable matching problem. We discuss another potential application, namely obtaining new insights into the incentive compatibility properties of the Gale-Shapley Deferred Acceptance Algorithm.

Cite as

Rohith Reddy Gangam, Tung Mai, Nitya Raju, and Vijay V. Vazirani. A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 19:1-19:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{gangam_et_al:LIPIcs.FSTTCS.2022.19,
  author =	{Gangam, Rohith Reddy and Mai, Tung and Raju, Nitya and Vazirani, Vijay V.},
  title =	{{A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{19:1--19:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.19},
  URN =		{urn:nbn:de:0030-drops-174114},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.19},
  annote =	{Keywords: stable matching, robust solutions, finite distributive lattice, Birkhoff’s Representation Theorem}
}
Document
New Characterizations of Core Imputations of Matching and b-Matching Games

Authors: Vijay V. Vazirani

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)


Abstract
We give new characterizations of core imputations for the following games: 1) The assignment game. 2) Concurrent games, i.e., general graph matching games having non-empty core. 3) The unconstrained bipartite b-matching game (edges can be matched multiple times). 4) The constrained bipartite b-matching game (edges can be matched at most once). The classic paper of Shapley and Shubik [Shapley and Shubik, 1971] showed that core imputations of the assignment game are precisely optimal solutions to the dual of the LP-relaxation of the game. Building on this, Deng et al. [Deng et al., 1999] gave a general framework which yields analogous characterizations for several fundamental combinatorial games. Interestingly enough, their framework does not apply to the last two games stated above. In turn, we show that some of the core imputations of these games correspond to optimal dual solutions and others do not. This leads to the tantalizing question of understanding the origins of the latter. We also present new characterizations of the profits accrued by agents and teams in core imputations of the first two games. Our characterization for the first game is stronger than that for the second; the underlying reason is that the characterization of vertices of the Birkhoff polytope is stronger than that of the Balinski polytope.

Cite as

Vijay V. Vazirani. New Characterizations of Core Imputations of Matching and b-Matching Games. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 28:1-28:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{vazirani:LIPIcs.FSTTCS.2022.28,
  author =	{Vazirani, Vijay V.},
  title =	{{New Characterizations of Core Imputations of Matching and b-Matching Games}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{28:1--28:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.28},
  URN =		{urn:nbn:de:0030-drops-174207},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.28},
  annote =	{Keywords: LP-duality theory, cooperative game theory, core of a game, assignment game, general graph matching game, bipartite b-matching game}
}
Document
Invited Talk
Online Bipartite Matching and Adwords (Invited Talk)

Authors: Vijay V. Vazirani

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)


Abstract
The purpose of this paper is to give a "textbook quality" proof of the optimal algorithm, called Ranking, for the online bipartite matching problem (OBM) and to highlight its role in matching-based market design. In particular, we discuss a generalization of OBM, called the adwords problem, which has had a significant impact in the ad auctions marketplace.

Cite as

Vijay V. Vazirani. Online Bipartite Matching and Adwords (Invited Talk). In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 5:1-5:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{vazirani:LIPIcs.MFCS.2022.5,
  author =	{Vazirani, Vijay V.},
  title =	{{Online Bipartite Matching and Adwords}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{5:1--5:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.5},
  URN =		{urn:nbn:de:0030-drops-168031},
  doi =		{10.4230/LIPIcs.MFCS.2022.5},
  annote =	{Keywords: matching-based market design, online algorithms, ad auctions, competitive analysis}
}
Document
Nash-Bargaining-Based Models for Matching Markets: One-Sided and Two-Sided; Fisher and Arrow-Debreu

Authors: Mojtaba Hosseini and Vijay V. Vazirani

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)


Abstract
This paper addresses two deficiencies of models in the area of matching-based market design. The first arises from the recent realization that the most prominent solution that uses cardinal utilities, namely the Hylland-Zeckhauser (HZ) mechanism [Hylland and Zeckhauser, 1979], is intractable; computation of even an approximate equilibrium is PPAD-complete [Vazirani and Yannakakis, 2021; Chen et al., 2021]. The second is the extreme paucity of models that use cardinal utilities, in sharp contrast with general equilibrium theory. Our paper addresses both these issues by proposing Nash-bargaining-based matching market models. Since the Nash bargaining solution is captured by a convex program, efficiency follow; in addition, it possesses a number of desirable game-theoretic properties. Our approach yields a rich collection of models: for one-sided as well as two-sided markets, for Fisher as well as Arrow-Debreu settings, and for a wide range of utility functions, all the way from linear to Leontief. We also give very fast implementations for these models which solve large instances, with n = 2000, in one hour on a PC, even for a two-sided matching market. A number of new ideas were needed, beyond the standard methods, to obtain these implementations.

Cite as

Mojtaba Hosseini and Vijay V. Vazirani. Nash-Bargaining-Based Models for Matching Markets: One-Sided and Two-Sided; Fisher and Arrow-Debreu. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 86:1-86:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{hosseini_et_al:LIPIcs.ITCS.2022.86,
  author =	{Hosseini, Mojtaba and Vazirani, Vijay V.},
  title =	{{Nash-Bargaining-Based Models for Matching Markets: One-Sided and Two-Sided; Fisher and Arrow-Debreu}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{86:1--86:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.86},
  URN =		{urn:nbn:de:0030-drops-156821},
  doi =		{10.4230/LIPIcs.ITCS.2022.86},
  annote =	{Keywords: Matching-based market design, Nash bargaining, convex optimization, Frank-Wolfe algorithm, cutting planes, general equilibrium theory, one-sided markets, two-sided markets}
}
Document
Computational Complexity of the Hylland-Zeckhauser Scheme for One-Sided Matching Markets

Authors: Vijay V. Vazirani and Mihalis Yannakakis

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


Abstract
In 1979, Hylland and Zeckhauser [Hylland and Zeckhauser, 1979] gave a simple and general scheme for implementing a one-sided matching market using the power of a pricing mechanism. Their method has nice properties - it is incentive compatible in the large and produces an allocation that is Pareto optimal - and hence it provides an attractive, off-the-shelf method for running an application involving such a market. With matching markets becoming ever more prevalent and impactful, it is imperative to finally settle the computational complexity of this scheme. We present the following partial resolution: 1) A combinatorial, strongly polynomial time algorithm for the dichotomous case, i.e., 0/1 utilities, and more generally, when each agent’s utilities come from a bi-valued set. 2) An example that has only irrational equilibria, hence proving that this problem is not in PPAD. 3) A proof of membership of the problem in the class FIXP. 4) A proof of membership of the problem of computing an approximate HZ equilibrium in the class PPAD. We leave open the (difficult) questions of determining if computing an exact HZ equilibrium is FIXP-hard and an approximate HZ equilibrium is PPAD-hard.

Cite as

Vijay V. Vazirani and Mihalis Yannakakis. Computational Complexity of the Hylland-Zeckhauser Scheme for One-Sided Matching Markets. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 59:1-59:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{vazirani_et_al:LIPIcs.ITCS.2021.59,
  author =	{Vazirani, Vijay V. and Yannakakis, Mihalis},
  title =	{{Computational Complexity of the Hylland-Zeckhauser Scheme for One-Sided Matching Markets}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{59:1--59:19},
  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.2021.59},
  URN =		{urn:nbn:de:0030-drops-135987},
  doi =		{10.4230/LIPIcs.ITCS.2021.59},
  annote =	{Keywords: Hyland-Zeckhauser scheme, one-sided matching markets, mechanism design, dichotomous utilities, PPAD, FIXP}
}
Document
Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds

Authors: Karthik Gajulapalli, James A. Liu, Tung Mai, and Vijay V. Vazirani

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
We address the following dynamic version of the school choice question: a city, named City, admits students in two temporally-separated rounds, denoted R₁ and R₂. In round R₁, the capacity of each school is fixed and mechanism M₁ finds a student optimal stable matching. In round R₂, certain parameters change, e.g., new students move into the City or the City is happy to allocate extra seats to specific schools. We study a number of Settings of this kind and give polynomial time algorithms for obtaining a stable matching for the new situations. It is well established that switching the school of a student midway, unsynchronized with her classmates, can cause traumatic effects. This fact guides us to two types of results: the first simply disallows any re-allocations in round R₂, and the second asks for a stable matching that minimizes the number of re-allocations. For the latter, we prove that the stable matchings which minimize the number of re-allocations form a sublattice of the lattice of stable matchings. Observations about incentive compatibility are woven into these results. We also give a third type of results, namely proofs of NP-hardness for a mechanism for round R₂ under certain settings.

Cite as

Karthik Gajulapalli, James A. Liu, Tung Mai, and Vijay V. Vazirani. Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 21:1-21:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{gajulapalli_et_al:LIPIcs.FSTTCS.2020.21,
  author =	{Gajulapalli, Karthik and Liu, James A. and Mai, Tung and Vazirani, Vijay V.},
  title =	{{Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.21},
  URN =		{urn:nbn:de:0030-drops-132626},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.21},
  annote =	{Keywords: stable matching, mechanism design, NP-Hardness}
}
Document
Matching Is as Easy as the Decision Problem, in the NC Model

Authors: Nima Anari and Vijay V. Vazirani

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)


Abstract
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of randomized NC matching algorithms [Karp et al., 1985; Mulmuley et al., 1987]. Over the last five years, the theoretical computer science community has launched a relentless attack on this question, leading to the discovery of several powerful ideas. We give what appears to be the culmination of this line of work: An NC algorithm for finding a minimum-weight perfect matching in a general graph with polynomially bounded edge weights, provided it is given an oracle for the decision problem. Consequently, for settling the main open problem, it suffices to obtain an NC algorithm for the decision problem. We believe this new fact has qualitatively changed the nature of this open problem. All known efficient matching algorithms for general graphs follow one of two approaches: given by [Edmonds, 1965] and [Lovász, 1979]. Our oracle-based algorithm follows a new approach and uses many of ideas discovered in the last five years. The difficulty of obtaining an NC perfect matching algorithm led researchers to study matching vis-a-vis clever relaxations of the class NC. In this vein, recently [Goldwasser and Grossman, 2015] gave a pseudo-deterministic RNC algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC algorithm with the additional requirement that on the same graph, it should return the same (i.e., unique) perfect matching for almost all choices of random bits. A corollary of our reduction is an analogous algorithm for general graphs.

Cite as

Nima Anari and Vijay V. Vazirani. Matching Is as Easy as the Decision Problem, in the NC Model. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 54:1-54:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{anari_et_al:LIPIcs.ITCS.2020.54,
  author =	{Anari, Nima and Vazirani, Vijay V.},
  title =	{{Matching Is as Easy as the Decision Problem, in the NC Model}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{54:1--54:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.54},
  URN =		{urn:nbn:de:0030-drops-117399},
  doi =		{10.4230/LIPIcs.ITCS.2020.54},
  annote =	{Keywords: Parallel Algorithm, Pseudo-Deterministic, Perfect Matching, Tutte Matrix}
}
Document
Finding Stable Matchings That Are Robust to Errors in the Input

Authors: Tung Mai and Vijay V. Vazirani

Published in: LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)


Abstract
In this paper, we introduce the issue of finding solutions to the stable matching problem that are robust to errors in the input and we obtain the first algorithmic results on this topic. In the process, we also initiate work on a new structural question concerning the stable matching problem, namely finding relationships between the lattices of solutions of two "nearby" instances. Our main algorithmic result is the following: We identify a polynomially large class of errors, D, that can be introduced in a stable matching instance. Given an instance A of stable matching, let B be the instance that results after introducing one error from D, chosen via a discrete probability distribution. The problem is to find a stable matching for A that maximizes the probability of being stable for B as well. Via new structural properties of the type described in the question stated above, we give a polynomial time algorithm for this problem.

Cite as

Tung Mai and Vijay V. Vazirani. Finding Stable Matchings That Are Robust to Errors in the Input. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 60:1-60:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{mai_et_al:LIPIcs.ESA.2018.60,
  author =	{Mai, Tung and Vazirani, Vijay V.},
  title =	{{Finding Stable Matchings That Are Robust to Errors in the Input}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{60:1--60:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Azar, Yossi and Bast, Hannah and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.60},
  URN =		{urn:nbn:de:0030-drops-95238},
  doi =		{10.4230/LIPIcs.ESA.2018.60},
  annote =	{Keywords: Stable Matching, Robust}
}
Document
Mutation, Sexual Reproduction and Survival in Dynamic Environments

Authors: Ruta Mehta, Ioannis Panageas, Georgios Piliouras, Prasad Tetali, and Vijay V. Vazirani

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)


Abstract
A new approach to understanding evolution [Valiant, JACM 2009], namely viewing it through the lens of computation, has already started yielding new insights, e.g., natural selection under sexual reproduction can be interpreted as the Multiplicative Weight Update (MWU) Algorithm in coordination games played among genes [Chastain, Livnat, Papadimitriou, Vazirani, PNAS 2014]. Using this machinery, we study the role of mutation in changing environments in the presence of sexual reproduction. Following [Wolf, Vazirani, Arkin, J. Theor. Biology], we model changing environments via a Markov chain, with the states representing environments, each with its own fitness matrix. In this setting, we show that in the absence of mutation, the population goes extinct, but in the presence of mutation, the population survives with positive probability. On the way to proving the above theorem, we need to establish some facts about dynamics in games. We provide the first, to our knowledge, polynomial convergence bound for noisy MWU in a coordination game. Finally, we also show that in static environments, sexual evolution with mutation converges, for any level of mutation.

Cite as

Ruta Mehta, Ioannis Panageas, Georgios Piliouras, Prasad Tetali, and Vijay V. Vazirani. Mutation, Sexual Reproduction and Survival in Dynamic Environments. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 16:1-16:29, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{mehta_et_al:LIPIcs.ITCS.2017.16,
  author =	{Mehta, Ruta and Panageas, Ioannis and Piliouras, Georgios and Tetali, Prasad and Vazirani, Vijay V.},
  title =	{{Mutation, Sexual Reproduction and Survival in Dynamic Environments}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{16:1--16:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.16},
  URN =		{urn:nbn:de:0030-drops-81655},
  doi =		{10.4230/LIPIcs.ITCS.2017.16},
  annote =	{Keywords: Evolution, Non-linear dynamics, Mutation}
}
Document
Opinion Dynamics in Networks: Convergence, Stability and Lack of Explosion

Authors: Tung Mai, Ioannis Panageas, and Vijay V. Vazirani

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)


Abstract
Inspired by the work of Kempe et al. [Kempe, Kleinberg, Oren, Slivkins, EC 2013], we introduce and analyze a model on opinion formation; the update rule of our dynamics is a simplified version of that of [Kempe, Kleinberg, Oren, Slivkins, EC 2013]. We assume that the population is partitioned into types whose interaction pattern is specified by a graph. Interaction leads to population mass moving from types of smaller mass to those of bigger mass. We show that starting uniformly at random over all population vectors on the simplex, our dynamics converges point-wise with probability one to an independent set. This settles an open problem of [Kempe, Kleinberg, Oren, Slivkins, EC 2013], as applicable to our dynamics. We believe that our techniques can be used to settle the open problem for the Kempe et al. dynamics as well. Next, we extend the model of Kempe et al. by introducing the notion of birth and death of types, with the interaction graph evolving appropriately. Birth of types is determined by a Bernoulli process and types die when their population mass is less than epsilon (a parameter). We show that if the births are infrequent, then there are long periods of "stability" in which there is no population mass that moves. Finally we show that even if births are frequent and "stability" is not attained, the total number of types does not explode: it remains logarithmic in 1/epsilon.

Cite as

Tung Mai, Ioannis Panageas, and Vijay V. Vazirani. Opinion Dynamics in Networks: Convergence, Stability and Lack of Explosion. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 140:1-140:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{mai_et_al:LIPIcs.ICALP.2017.140,
  author =	{Mai, Tung and Panageas, Ioannis and Vazirani, Vijay V.},
  title =	{{Opinion Dynamics in Networks: Convergence, Stability and Lack of Explosion}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{140:1--140:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.140},
  URN =		{urn:nbn:de:0030-drops-74440},
  doi =		{10.4230/LIPIcs.ICALP.2017.140},
  annote =	{Keywords: Opinion Dynamics, Convergence, Jacobian, Center-stable Manifold}
}
Document
Allocation of Divisible Goods Under Lexicographic Preferences

Authors: Leonard J. Schulman and Vijay V. Vazirani

Published in: LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)


Abstract
We present a simple and natural non-pricing mechanism for allocating divisible goods among strategic agents having lexicographic preferences. Our mechanism has favorable properties of strategy-proofness (incentive compatibility). In addition (and even when extended to the case of Leontief bundles) it enjoys Pareto efficiency, envy-freeness, and time efficiency.

Cite as

Leonard J. Schulman and Vijay V. Vazirani. Allocation of Divisible Goods Under Lexicographic Preferences. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 543-559, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{schulman_et_al:LIPIcs.FSTTCS.2015.543,
  author =	{Schulman, Leonard J. and Vazirani, Vijay V.},
  title =	{{Allocation of Divisible Goods Under Lexicographic Preferences}},
  booktitle =	{35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)},
  pages =	{543--559},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-97-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{45},
  editor =	{Harsha, Prahladh and Ramalingam, G.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.543},
  URN =		{urn:nbn:de:0030-drops-56279},
  doi =		{10.4230/LIPIcs.FSTTCS.2015.543},
  annote =	{Keywords: Mechanism design, lexicographic preferences, strategyproof, Pareto optimal, incentive compatible}
}
Document
Equilibrium Computation (Dagstuhl Seminar 14342)

Authors: Nimrod Megiddo, Kurt Mehlhorn, Rahul Savani, and Vijay V. Vazirani

Published in: Dagstuhl Reports, Volume 4, Issue 8 (2015)


Abstract
This report documents the program and outcomes of Dagstuhl Seminar 14342 "Equilibrium Computation". The seminar was at the leading edge of current topics related to equilibrium computation for games and markets. We summarize these topics, give the talk abstracts, and give brief summaries of the problems that were discussed in the open problem sessions.

Cite as

Nimrod Megiddo, Kurt Mehlhorn, Rahul Savani, and Vijay V. Vazirani. Equilibrium Computation (Dagstuhl Seminar 14342). In Dagstuhl Reports, Volume 4, Issue 8, pp. 73-88, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)


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@Article{megiddo_et_al:DagRep.4.8.73,
  author =	{Megiddo, Nimrod and Mehlhorn, Kurt and Savani, Rahul and Vazirani, Vijay V.},
  title =	{{Equilibrium Computation (Dagstuhl Seminar 14342)}},
  pages =	{73--88},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2014},
  volume =	{4},
  number =	{8},
  editor =	{Megiddo, Nimrod and Mehlhorn, Kurt and Savani, Rahul and Vazirani, Vijay V.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.8.73},
  URN =		{urn:nbn:de:0030-drops-47990},
  doi =		{10.4230/DagRep.4.8.73},
  annote =	{Keywords: Algorithms, Computational Complexity, Equilibrium Computation, Game Theory, Market Equilibrium, Nash Equilibrium}
}
Document
10171 Abstracts Collection – Equilibrium Computation

Authors: Edith Elkind, Nimrod Megiddo, Peter Bro Miltersen, Bernhard von Stengel, and Vijay V. Vazirani

Published in: Dagstuhl Seminar Proceedings, Volume 10171, Equilibrium Computation (2010)


Abstract
From April 25 to April 30, 2010, the Dagstuhl Seminar 10171 ``Equilibrium Computation'' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.

Cite as

Edith Elkind, Nimrod Megiddo, Peter Bro Miltersen, Bernhard von Stengel, and Vijay V. Vazirani. 10171 Abstracts Collection – Equilibrium Computation. In Equilibrium Computation. Dagstuhl Seminar Proceedings, Volume 10171, pp. 1-18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)


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@InProceedings{elkind_et_al:DagSemProc.10171.1,
  author =	{Elkind, Edith and Megiddo, Nimrod and Miltersen, Peter Bro and von Stengel, Bernhard and Vazirani, Vijay V.},
  title =	{{10171 Abstracts Collection – Equilibrium Computation}},
  booktitle =	{Equilibrium Computation},
  pages =	{1--18},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{10171},
  editor =	{Edith Elkind and Nimrod Megiddo and Peter Bro Miltersen and Vijay V. Vazirani and Bernahrd von Stengel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10171.1},
  URN =		{urn:nbn:de:0030-drops-26738},
  doi =		{10.4230/DagSemProc.10171.1},
  annote =	{Keywords: Equilibrium computation, algorithmic game theory}
}
Document
Algorithmic Game Theory and the Internet (Dagstuhl Seminar 03291)

Authors: Marek Karpinski, Christos H. Papadimitriou, and Vijay V. Vazirani

Published in: Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)


Abstract

Cite as

Marek Karpinski, Christos H. Papadimitriou, and Vijay V. Vazirani. Algorithmic Game Theory and the Internet (Dagstuhl Seminar 03291). Dagstuhl Seminar Report 386, pp. 1-9, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2003)


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@TechReport{karpinski_et_al:DagSemRep.386,
  author =	{Karpinski, Marek and Papadimitriou, Christos H. and Vazirani, Vijay V.},
  title =	{{Algorithmic Game Theory and the Internet (Dagstuhl Seminar 03291)}},
  pages =	{1--9},
  ISSN =	{1619-0203},
  year =	{2003},
  type = 	{Dagstuhl Seminar Report},
  number =	{386},
  institution =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.386},
  URN =		{urn:nbn:de:0030-drops-152660},
  doi =		{10.4230/DagSemRep.386},
}

Vazirani, Vijay

Document
Solvency Games

Authors: Noam Berger, Nevin Kapur, Leonard Schulman, and Vijay Vazirani

Published in: LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)


Abstract
We study the decision theory of a maximally risk-averse investor --- one whose objective, in the face of stochastic uncertainties, is to minimize the probability of ever going broke. With a view to developing the mathematical basics of such a theory, we start with a very simple model and obtain the following results: a characterization of best play by investors; an explanation of why poor and rich players may have different best strategies; an explanation of why expectation-maximization is not necessarily the best strategy even for rich players. For computation of optimal play, we show how to apply the Value Iteration method, and prove a bound on its convergence rate.

Cite as

Noam Berger, Nevin Kapur, Leonard Schulman, and Vijay Vazirani. Solvency Games. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 61-72, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{berger_et_al:LIPIcs.FSTTCS.2008.1741,
  author =	{Berger, Noam and Kapur, Nevin and Schulman, Leonard and Vazirani, Vijay},
  title =	{{Solvency Games}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{61--72},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-08-8},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{2},
  editor =	{Hariharan, Ramesh and Mukund, Madhavan and Vinay, V},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1741},
  URN =		{urn:nbn:de:0030-drops-17419},
  doi =		{10.4230/LIPIcs.FSTTCS.2008.1741},
  annote =	{Keywords: Decision making under uncertainity, multi-arm bandit problems, game theory}
}
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