Devising a Framework for Efficient and Accurate Matchmaking and Real-Time Web Communication
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On Likely Solutions of the Stable Matching Problem with Unequal
ON LIKELY SOLUTIONS OF THE STABLE MATCHING PROBLEM WITH UNEQUAL NUMBERS OF MEN AND WOMEN BORIS PITTEL Abstract. Following up a recent work by Ashlagi, Kanoria and Leshno, we study a stable matching problem with unequal numbers of men and women, and independent uniform preferences. The asymptotic formulas for the expected number of stable matchings, and for the probabilities of one point–concentration for the range of husbands’ total ranks and for the range of wives’ total ranks are obtained. 1. Introduction and main results Consider the set of n1 men and n2 women facing a problem of selecting a marriage partner. For n1 = n2 = n, a marriage M is a matching (bijection) between the two sets. It is assumed that each man and each woman has his/her preferences for a marriage partner, with no ties allowed. That is, there are given n permutations σj of the men set and n permutations ρj of the women set, each σj (ρj resp.) ordering the women set (the men set resp.) according to the desirability degree of a woman (a man) as a marriage partner for man j (woman j). A marriage is called stable if there is no unmarried pair (a man, a woman) who prefer each other to their respective partners in the marriage. A classic theorem, due to Gale and Shapley [4], asserts that, given any system of preferences {ρj,σj}j∈[n], there exists at least one stable marriage M. arXiv:1701.08900v2 [math.CO] 13 Feb 2017 The proof of this theorem is algorithmic. A bijection is constructed in steps such that at each step every man not currently on hold makes a pro- posal to his best choice among women who haven’t rejected him before, and the chosen woman either provisionally puts the man on hold or rejects him, based on comparison of him to her current suitor if she has one already. -
On the Effi Ciency of Stable Matchings in Large Markets Sangmok Lee
On the Effi ciency of Stable Matchings in Large Markets SangMok Lee Leeat Yarivyz August 22, 2018 Abstract. Stability is often the goal for matching clearinghouses, such as those matching residents to hospitals, students to schools, etc. We study the wedge between stability and utilitarian effi ciency in large one-to-one matching markets. We distinguish between stable matchings’average effi ciency (or, effi ciency per-person), which is maximal asymptotically for a rich preference class, and their aggregate effi ciency, which is not. The speed at which average effi ciency of stable matchings converges to its optimum depends on the underlying preferences. Furthermore, for severely imbalanced markets governed by idiosyncratic preferences, or when preferences are sub-modular, stable outcomes may be average ineffi cient asymptotically. Our results can guide market designers who care about effi ciency as to when standard stable mechanisms are desirable and when new mechanisms, or the availability of transfers, might be useful. Keywords: Matching, Stability, Effi ciency, Market Design. Department of Economics, Washington University in St. Louis, http://www.sangmoklee.com yDepartment of Economics, Princeton University, http://www.princeton.edu/yariv zWe thank Federico Echenique, Matthew Elliott, Aytek Erdil, Mallesh Pai, Rakesh Vohra, and Dan Wal- ton for very useful comments and suggestions. Euncheol Shin provided us with superb research assistance. Financial support from the National Science Foundation (SES 0963583) and the Gordon and Betty Moore Foundation (through grant 1158) is gratefully acknowledged. On the Efficiency of Stable Matchings in Large Markets 1 1. Introduction 1.1. Overview. The design of most matching markets has focused predominantly on mechanisms in which only ordinal preferences are specified: the National Resident Match- ing Program (NRMP), clearinghouses for matching schools and students in New York City and Boston, and many others utilize algorithms that implement a stable matching correspond- ing to reported rank preferences. -
Deferred Acceptance Algorithms: History, Theory, Practice, and Open Questions
Int J Game Theory (2008) 36:537–569 DOI 10.1007/s00182-008-0117-6 ORIGINAL PAPER Deferred acceptance algorithms: history, theory, practice, and open questions Alvin E. Roth Accepted: 19 July 2007 / Published online: 29 January 2008 © Springer-Verlag 2008 Abstract The deferred acceptance algorithm proposed by Gale and Shapley (1962) has had a profound influence on market design, both directly, by being adapted into practical matching mechanisms, and, indirectly, by raising new theoretical questions. Deferred acceptance algorithms are at the basis of a number of labor market clea- ringhouses around the world, and have recently been implemented in school choice systems in Boston and New York City. In addition, the study of markets that have failed in ways that can be fixed with centralized mechanisms has led to a deeper understanding of some of the tasks a marketplace needs to accomplish to perform well. In particular, marketplaces work well when they provide thickness to the mar- ket, help it deal with the congestion that thickness can bring, and make it safe for participants to act effectively on their preferences. Centralized clearinghouses organi- zed around the deferred acceptance algorithm can have these properties, and this has sometimes allowed failed markets to be reorganized. Keywords Matching · Market design · Gale-shapley · Deferred acceptance Introduction Matching is one of the important functions of markets. Who gets which jobs, which school places, who marries whom, these help shape lives and careers. Partly for this reason, a substantial literature has grown out of the remarkable paper “College Admissions and the Stability of Marriage” that David Gale and Lloyd Prepared for Gale’s Feast: a Day in honor of the 85th birthday of David Gale, July 2007, Stony Brook. -
Matchings and Games on Networks
Matchings and games on networks by Linda Farczadi A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics and Optimization Waterloo, Ontario, Canada, 2015 c Linda Farczadi 2015 Author's Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract We investigate computational aspects of popular solution concepts for different models of network games. In chapter 3 we study balanced solutions for network bargaining games with general capacities, where agents can participate in a fixed but arbitrary number of contracts. We fully characterize the existence of balanced solutions and provide the first polynomial time algorithm for their computation. Our methods use a new idea of reducing an instance with general capacities to an instance with unit capacities defined on an auxiliary graph. This chapter is an extended version of the conference paper [32]. In chapter 4 we propose a generalization of the classical stable marriage problem. In our model the preferences on one side of the partition are given in terms of arbitrary bi- nary relations, that need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable mar- riage where preferences are allowed to have ties and to be incomplete. Our main result shows that deciding the existence of a stable matching in our model is NP-complete. -
Matchingmarkets
Package ‘matchingMarkets’ February 20, 2015 Version 0.1-2 Depends R (>= 2.15.2) Imports Rcpp (>= 0.11.2), lpSolve (>= 5.6.6), partitions LinkingTo Rcpp, RcppArmadillo Date 2014-08-14 Title An R package for the analysis of stable matchings Author Thilo Klein Maintainer Thilo Klein <[email protected]> Description This package implements a structural estimator to correct for the sample selection bias from observed outcomes in matching markets. It also contains R code for matching algorithms such as the deferred-acceptance algorithm for college admissions, the top-trading-cycles algorithm for house allocation and a partitioning linear program for the roommates problem. URL https://github.com/thiloklein/matchingMarkets License GPL (>= 2) | file LICENSE Repository CRAN Repository/R-Forge/Project matchingmarkets Repository/R-Forge/Revision 16 Repository/R-Forge/DateTimeStamp 2014-11-22 23:10:11 Date/Publication 2014-11-23 09:35:22 NeedsCompilation yes R topics documented: matchingMarkets-package . .2 baac00 . .4 daa..............................................6 khb .............................................8 mfx .............................................9 1 2 matchingMarkets-package plp..............................................9 stabit . 10 stabsim . 15 ttc.............................................. 17 Index 18 matchingMarkets-package An R package for the analysis of stable matchings. Description The matchingMarkets package contains R and C++ code for the estimation of structural models that correct for the sample selection bias of observed outcomes in matching markets. Matching is concerned with who transacts with whom, and how. For example, who works at which job, which students go to which school, who forms a workgroup with whom, and so on. The empirical analysis of matching markets is naturally subject to sample selection problems. -
15 Hedonic Games Haris Azizaand Rahul Savanib
Draft { December 5, 2014 15 Hedonic Games Haris Azizaand Rahul Savanib 15.1 Introduction Coalitions are a central part of economic, political, and social life, and coalition formation has been studied extensively within the mathematical social sciences. Agents (be they humans, robots, or software agents) have preferences over coalitions and, based on these preferences, it is natural to ask which coalitions are expected to form, and which coalition structures are better social outcomes. In this chapter, we consider coalition formation games with hedonic preferences, or simply hedonic games. The outcome of a coalition formation game is a partitioning of the agents into disjoint coalitions, which we will refer to synonymously as a partition or coalition structure. The defining feature of hedonic preferences is that every agent only cares about which agents are in its coalition, but does not care how agents in other coali- tions are grouped together (Dr`ezeand Greenberg, 1980). Thus, hedonic preferences completely ignore inter-coalitional dependencies. Despite their relative simplicity, hedonic games have been used to model many interesting settings, such as research team formation (Alcalde and Revilla, 2004), scheduling group activities (Darmann et al., 2012), formation of coalition governments (Le Breton et al., 2008), cluster- ings in social networks (see e.g., Aziz et al., 2014b; McSweeney et al., 2014; Olsen, 2009), and distributed task allocation for wireless agents (Saad et al., 2011). Before we give a formal definition of a hedonic game, we give a standard hedonic game from the literature that we will use as a running example (see e.g., Banerjee et al. -
Stable Matching with PCF Version 2, an Étude in Secure Computation
Stable Matching with PCF Version 2, an Etude´ in Secure Computation A Thesis Presented to the Faculty of the School of Engineering and Applied Science University of Virginia In Partial Fulfillment of the requirements for the Degree Master of Science (Computer Science) by Benjamin Terner August 2015 To Mom Everything I am is a reflection of the woman who raised me, and everything I do is an honor to her memory. iii Abstract The classic stable-matching algorithm of Gale and Shapley and subsequent variants by, e.g., Abdulkadiroglu et al. [APR05], have been used successfully in a number of real-world scenarios, including the assignment of US medical school graduates to residency programs and students in New York, Norway, and Singapore to high schools and universities. One shortcoming of the Gale-Shapley family of matching algorithms is susceptibility to strategic manipulation by the participants. The commonly used paradigm to mitigate this shortcoming, employing a trusted third party to compute matchings explicitly, is outdated, expensive, and in some scenarios, impossible. This makes stable matching a natural problem for secure, multiparty computation (SMPC). Secure multiparty computation allows two or more mutually distrustful parties to evaluate a joint function on their inputs without revealing more information about the inputs than each player can derive from his own input and output. We use Portable Circuit Format (PCF), a compiler and interpreter for Yao’s garbled circuit protocols, to produce the first feasible, privacy-preserving protocol for stable matching. In doing so, we improve the theoretical bounds for stable matching constructions, develop global optimizations for PCF circuits, and improve garbling techniques used by PCF. -
Class Descriptions—Week 5, Mathcamp 2016
CLASS DESCRIPTIONS|WEEK 5, MATHCAMP 2016 Contents 9:10 Classes 2 Building Groups out of Other Groups 2 More Problem Solving: Polynomials 2 The Hairy and Still Lonely Torus 2 Turning Points in the History of Mathematics 4 Why Aren't We Learning This: Fun Stuff in Statistics 4 10:10 Classes 4 A Very Difficult Definite Integral 4 Homotopy (Co)limits 5 Scandalous Curves 5 Special Relativity 5 String Theory 6 The Magic of Determinants 6 The Mathematical ABCs 7 The Stable Marriage Problem 7 Quadratic Reciprocity 7 11:10 Classes 8 Benford's Law 8 Egyptian Fractions 8 Gambling with Nathan Ramesh 8 Hyper-Dimensional Geometry 8 IOL 2016 8 Many Cells Separating Points 9 Period Three Implies Chaos 9 Problem Solving: Tetrahedra 9 The Brauer Group 9 The BSD conjecture 10 The Cayley{Hamilton Theorem 10 The \Free Will" Theorem 10 The Yoneda Lemma 10 Twisty Puzzles are Easy 11 Wedderburn's Theorem 11 Wythoff's Game 11 1:10 Classes 11 Big Numbers 11 Graph Polynomials 12 Hard Problems That Are Almost Easy 12 Hyperbolic Geometry 12 Latin Squares and Finite Geometries 13 1 MC2016 ◦ W5 ◦ Classes 2 Multiplicative Functions 13 Permuting Conditionally Convergent Series 13 The Cap Set Problem 13 2:10 Classes 14 Computer-Aided Design 14 Factoring with Elliptic Curves 14 Math and Literature 14 More Group Theory! 15 Paradoxes in Probability 15 The Hales{Jewett Theorem 15 9:10 Classes Building Groups out of Other Groups. ( , Don, Wednesday{Friday) In group theory, everybody learns how to take a group, and shrink it down by taking a quotient by a normal subgroup|but what they don't tell you is that you can totally go the other direction. -
WEEK 5 CLASS PROPOSALS, MATHCAMP 2018 Contents
WEEK 5 CLASS PROPOSALS, MATHCAMP 2018 Abstract. We have a lot of fun class proposals for Week 5! Take a read through these proposals, and then let us know which ones you would like us to schedule for Week 5 by submitting your preferences on the appsys appsys.mathcamp.org. We'll keep voting open until sign-in on Wednesday evening. Please only vote for classes that you would attend if offered. Thanks for helping us create the schedule! Contents Aaron's Classes 3 Braid groups and the fundamental group of configuration space 3 Counting points over finite fields 4 Dimensional Analysis 4 Finite Fields 4 Math is Applied Physics 4 Riemann-Roching out on the projective line 4 The Art of Live-TeXing 4 The dimension formula for modular forms via the moduli stack of elliptic curves 5 Aaron, Vivian's Classes 5 Galois Theory and Number Theory 5 Ania, Jessica's Classes 5 Cycles in permutations 5 Apurva's Classes 6 Galois Correspondence of Covering Spaces 6 MP3 6= JPEG 6 The Quantum Spring 6 There isn't enough Homological Algebra in the world 6 Would I ever Lie Group to you? 6 Ben Dees's Classes 6 Bairely Sensible 6 Finite Field Fourier Fun 7 What is a \Sylow" anyways? 7 Yes, you can trisect angles 7 Brian Reinhart's Classes 8 A Slice of PIE 8 Dyusha, Michelle's Classes 8 Curse of Dimensionality 8 J-Lo's Classes 9 Axiomatic Music Theory part 2: Rhythm 9 Lattices 9 Minus Choice, Still Paradoxes 9 The Borsuk Problem 9 The Music of Zeta 10 Yes, you can solve the Quintic 10 Jeff's Classes 10 1 MC2018 ◦ W5 ◦ Classes 2 Combinatorial Topology 10 Jeff's Favorite Pictures -
Cheating to Get Better Roommates in a Random Stable Matching
Dartmouth College Dartmouth Digital Commons Computer Science Technical Reports Computer Science 12-1-2006 Cheating to Get Better Roommates in a Random Stable Matching Chien-Chung Huang Dartmouth College Follow this and additional works at: https://digitalcommons.dartmouth.edu/cs_tr Part of the Computer Sciences Commons Dartmouth Digital Commons Citation Huang, Chien-Chung, "Cheating to Get Better Roommates in a Random Stable Matching" (2006). Computer Science Technical Report TR2006-582. https://digitalcommons.dartmouth.edu/cs_tr/290 This Technical Report is brought to you for free and open access by the Computer Science at Dartmouth Digital Commons. It has been accepted for inclusion in Computer Science Technical Reports by an authorized administrator of Dartmouth Digital Commons. For more information, please contact [email protected]. Cheating to Get Better Roommates in a Random Stable Matching Chien-Chung Huang Technical Report 2006-582 Dartmouth College Sudikoff Lab 6211 for Computer Science Hanover, NH 03755, USA [email protected] Abstract This paper addresses strategies for the stable roommates problem, assuming that a stable matching is chosen at random. We investigate how a cheating man should permute his preference list so that he has a higher-ranking roommate probabilistically. In the first part of the paper, we identify a necessary condition for creating a new stable roommate for the cheating man. This condition precludes any possibility of his getting a new roommate ranking higher than all his stable room- mates when everyone is truthful. Generalizing to the case that multiple men collude, we derive another impossibility result: given any stable matching in which a subset of men get their best possible roommates, they cannot cheat to create a new stable matching in which they all get strictly better roommates than in the given matching. -
Seen As Stable Marriages
This paper was presented as part of the Mini-Conference at IEEE INFOCOM 2011 Seen As Stable Marriages Hong Xu, Baochun Li Department of Electrical and Computer Engineering University of Toronto Abstract—In this paper, we advocate the use of stable matching agents from both sides to find the optimal solution, which is framework in solving networking problems, which are tradi- computationally expensive. tionally solved using utility-based optimization or game theory. Born in economics, stable matching efficiently resolves conflicts of interest among selfish agents in the market, with a simple and elegant procedure of deferred acceptance. We illustrate through (w1,w3,w2) (m1,m2,m3) one technical case study how it can be applied in practical m1 w1 scenarios where the impeding complexity of idiosyncratic factors makes defining a utility function difficult. Due to its use of (w2,w3,w1) (m1,m3,m2) generic preferences, stable matching has the potential to offer efficient and practical solutions to networking problems, while its m2 w2 mathematical structure and rich literature in economics provide many opportunities for theoretical studies. In closing, we discuss (w2,w1,w3) open questions when applying the stable matching framework. (m3,m2,m1) m3 w3 I. INTRODUCTION Matching is perhaps one of the most important functions of Fig. 1. A simple example of the marriage market. The matching shown is a stable matching. markets. The stable marriage problem, introduced by Gale and Shapley in their seminal work [1], is arguably one of the most Many large-scale networked systems, such as peer-to-peer interesting and successful abstractions of such markets. -
Carrier Aggregation Between Operators in Next
1 Carrier Aggregation between Operators in Next Generation Cellular Networks: A Stable Roommate Market Yong Xiao, Senior Member, IEEE, Zhu Han, Fellow, IEEE, Chau Yuen, Senior Member, IEEE, Luiz A. DaSilva, Senior Member, IEEE Abstract—This paper studies carrier aggregation between next generation of mobile technology will rely on CA to multiple mobile network operators (MNO), referred to as achieve its promised peak data rates. In a typical network, inter-operator carrier aggregation (IO-CA). In IO-CA, each a mobile network operator (MNO) may aggregate MNO can transmit on its own licensed spectrum and aggregate the spectrum licensed to other MNOs. We focus frequency resource blocks contiguously or on the case that MNOs are distributedly partitioned into non-contiguously within a single frequency band, i.e. small groups, called IO-CA pairs, each of which consists of intra-band CA, or it may aggregate resources which are two MNOs that mutually agree to share their spectrum with located in separate frequency bands. While much of the each other. We model the IO-CA pairing problem between current work on CA investigates the aggregation of MNOs as a stable roommate market and derive a condition for which a stable matching structure among all MNOs exclusive blocks of spectrum from the perspective of exists. We propose an algorithm that achieves a stable typical macrocell topologies [3]–[5], we study carrier matching if it exists. Otherwise, the algorithm results in a aggregation (CA) for cellular networks from a cognitive stable partition. For each IO-CA pair, we derive the optimal radio (CR) network perspective.