Deferred Acceptance Algorithms: History, Theory, Practice, and Open Questions
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Calibrating Determinacy Strength in Levels of the Borel Hierarchy
CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY SHERWOOD J. HACHTMAN Abstract. We analyze the set-theoretic strength of determinacy for levels of the Borel 0 hierarchy of the form Σ1+α+3, for α < !1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α+1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of Π1-reflection principles, Π1-RAPα, whose consistency strength corresponds 0 CK exactly to that of Σ1+α+3-Determinacy, for α < !1 . This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: the least θ so that all winning strategies 0 in Σ4 games belong to Lθ+1 is the least so that Lθ j= \P(!) exists + all wellfounded trees are ranked". x1. Introduction. Given a set A ⊆ !! of sequences of natural numbers, consider a game, G(A), where two players, I and II, take turns picking elements of a sequence hx0; x1; x2;::: i of naturals. Player I wins the game if the sequence obtained belongs to A; otherwise, II wins. For a collection Γ of subsets of !!, Γ determinacy, which we abbreviate Γ-DET, is the statement that for every A 2 Γ, one of the players has a winning strategy in G(A). It is a much-studied phenomenon that Γ -DET has mathematical strength: the bigger the pointclass Γ, the stronger the theory required to prove Γ -DET. -
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. -
Stable Matching: Theory, Evidence, and Practical Design
THE PRIZE IN ECONOMIC SCIENCES 2012 INFORMATION FOR THE PUBLIC Stable matching: Theory, evidence, and practical design This year’s Prize to Lloyd Shapley and Alvin Roth extends from abstract theory developed in the 1960s, over empirical work in the 1980s, to ongoing efforts to fnd practical solutions to real-world prob- lems. Examples include the assignment of new doctors to hospitals, students to schools, and human organs for transplant to recipients. Lloyd Shapley made the early theoretical contributions, which were unexpectedly adopted two decades later when Alvin Roth investigated the market for U.S. doctors. His fndings generated further analytical developments, as well as practical design of market institutions. Traditional economic analysis studies markets where prices adjust so that supply equals demand. Both theory and practice show that markets function well in many cases. But in some situations, the standard market mechanism encounters problems, and there are cases where prices cannot be used at all to allocate resources. For example, many schools and universities are prevented from charging tuition fees and, in the case of human organs for transplants, monetary payments are ruled out on ethical grounds. Yet, in these – and many other – cases, an allocation has to be made. How do such processes actually work, and when is the outcome efcient? Matching theory The Gale-Shapley algorithm Analysis of allocation mechanisms relies on a rather abstract idea. If rational people – who know their best interests and behave accordingly – simply engage in unrestricted mutual trade, then the outcome should be efcient. If it is not, some individuals would devise new trades that made them better of. -
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. -
Equilmrium and DYNAMICS David Gale, 1991 Equilibrium and Dynamics
EQUILmRIUM AND DYNAMICS David Gale, 1991 Equilibrium and Dynamics Essays in Honour of David Gale Edited by Mukul Majumdar H. T. Warshow and Robert Irving Warshow Professor ofEconomics Cornell University Palgrave Macmillan ISBN 978-1-349-11698-0 ISBN 978-1-349-11696-6 (eBook) DOI 10.1007/978-1-349-11696-6 © Mukul Majumdar 1992 Softcover reprint of the hardcover 1st edition 1992 All rights reserved. For information, write: Scholarly and Reference Division, St. Martin's Press, Inc., 175 Fifth Avenue, New York, N.Y. 10010 First published in the United States of America in 1992 ISBN 978-0-312-06810-3 Library of Congress Cataloging-in-Publication Data Equilibrium and dynamics: essays in honour of David Gale I edited by Mukul Majumdar. p. em. Includes bibliographical references (p. ). ISBN 978-0-312-06810-3 1. Equilibrium (Economics) 2. Statics and dynamics (Social sciences) I. Gale, David. II. Majumdar, Mukul, 1944- . HB145.E675 1992 339.5-dc20 91-25354 CIP Contents Preface vii Notes on the Contributors ix 1 Equilibrium in a Matching Market with General Preferences Ahmet Alkan 1 2 General Equilibrium with Infinitely Many Goods: The Case of Separable Utilities Aloisio Araujo and Paulo Klinger Monteiro 17 3 Regular Demand with Several, General Budget Constraints Yves Balasko and David Cass 29 4 Arbitrage Opportunities in Financial Markets are not Inconsistent with Competitive Equilibrium Lawrence M. Benveniste and Juan Ketterer 45 5 Fiscal and Monetary Policy in a General Equilibrium Model Truman Bewley 55 6 Equilibrium in Preemption Games with Complete Information Kenneth Hendricks and Charles Wilson 123 7 Allocation of Aggregate and Individual Risks through Financial Markets Michael J. -
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. -
Determinacy and Large Cardinals
Determinacy and Large Cardinals Itay Neeman∗ Abstract. The principle of determinacy has been crucial to the study of definable sets of real numbers. This paper surveys some of the uses of determinacy, concentrating specifically on the connection between determinacy and large cardinals, and takes this connection further, to the level of games of length ω1. Mathematics Subject Classification (2000). 03E55; 03E60; 03E45; 03E15. Keywords. Determinacy, iteration trees, large cardinals, long games, Woodin cardinals. 1. Determinacy Let ωω denote the set of infinite sequences of natural numbers. For A ⊂ ωω let Gω(A) denote the length ω game with payoff A. The format of Gω(A) is displayed in Diagram 1. Two players, denoted I and II, alternate playing natural numbers forming together a sequence x = hx(n) | n < ωi in ωω called a run of the game. The run is won by player I if x ∈ A, and otherwise the run is won by player II. I x(0) x(2) ...... II x(1) x(3) ...... Diagram 1. The game Gω(A). A game is determined if one of the players has a winning strategy. The set A is ω determined if Gω(A) is determined. For Γ ⊂ P(ω ), det(Γ) denotes the statement that all sets in Γ are determined. Using the axiom of choice, or more specifically using a wellordering of the reals, it is easy to construct a non-determined set A. det(P(ωω)) is therefore false. On the other hand it has become clear through research over the years that det(Γ) is true if all the sets in Γ are definable by some concrete means. -
A FURTHER NOTE on the STABLE MATCHING PROBLEM Gabrielle DEMANGE David GALE Marilda SOTOMAYOR* in Recent Work on the Matching
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Applied Mathematics 16 (1987) 217-222 217 North-Holland A FURTHER NOTE ON THE STABLE MATCHING PROBLEM Gabrielle DEMANGE Laboratoire d'Econom~trie de l'Ecole Polytechnique, Paris, France David GALE Dept. of Mathematics, University of California, Berkeley, CA 94720, USA Marilda SOTOMAYOR* Pontifica Universidade Catolica do Rio de Janeiro, Brasil Received 15 October 1985 Revised 15 July 1986 Using a lcmma of J.S. Hwang we obtain a generalization of a theorem of Dubins and Freed- man. It is shown that the core of the matching game is non-manipulable in a suitable sense by coalitions consisting of both men and women. A further strong stability property of the core is derived. Introduction In recent work on the matching model a lemma of Hwang was used to give a short proof of the non-manipu!ability theorem of Dubins and Freedman [2], [5]. In this note we give two other applications of this lemma; one is a stronger version of the Dubins-Freedman's result; the other one concerns a strong stability property of the core of the matching model. In the next section we recall briefly the matching model; in Section 2 we give the relevant earlier results before presenting our non-manipulability theorem; the strong stability result is stated in Section 3 and proofs are gathered in the final section. 1. The model As is usual, we phrase our result in terms of a marriage problem in which there is a set M of men and W of women. -
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 -
Devising a Framework for Efficient and Accurate Matchmaking and Real-Time Web Communication
Dartmouth College Dartmouth Digital Commons Dartmouth College Undergraduate Theses Theses and Dissertations 6-1-2016 Devising a Framework for Efficient and Accurate Matchmaking and Real-time Web Communication Charles Li Dartmouth College Follow this and additional works at: https://digitalcommons.dartmouth.edu/senior_theses Part of the Computer Sciences Commons Recommended Citation Li, Charles, "Devising a Framework for Efficient and Accurate Matchmaking and Real-time Web Communication" (2016). Dartmouth College Undergraduate Theses. 106. https://digitalcommons.dartmouth.edu/senior_theses/106 This Thesis (Undergraduate) is brought to you for free and open access by the Theses and Dissertations at Dartmouth Digital Commons. It has been accepted for inclusion in Dartmouth College Undergraduate Theses by an authorized administrator of Dartmouth Digital Commons. For more information, please contact [email protected]. Devising a framework for efficient and accurate matchmaking and real-time web communication Charles Li Department of Computer Science Dartmouth College Hanover, NH [email protected] Abstract—Many modern applications have a great need for propose and evaluate original matchmaking algorithms, and matchmaking and real-time web communication. This paper first also discuss and evaluate implementations of real-time explores and details the specifics of original algorithms for communication using existing web protocols. effective matchmaking, and then proceeds to dive into implementations of real-time communication between different clients across the web. Finally, it discusses how to apply the II. PRIOR WORK AND RELEVANT TECHNOLOGIES techniques discussed in the paper in practice, and provides samples based on the framework. A. Matchmaking: No Stable Matching Exists Consider a pool of users. What does matchmaking entail? It Keywords—matchmaking; real-time; web could be the task of finding, for each user, the best compatriot or opponent in the pool to match with, given the user’s set of I. -
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.