Cheating to Get Better Roommates in a Random Stable Matching
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Lecture 4 Rationalizability & Nash Equilibrium Road
Lecture 4 Rationalizability & Nash Equilibrium 14.12 Game Theory Muhamet Yildiz Road Map 1. Strategies – completed 2. Quiz 3. Dominance 4. Dominant-strategy equilibrium 5. Rationalizability 6. Nash Equilibrium 1 Strategy A strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy). Matching pennies with perfect information 2’s Strategies: HH = Head if 1 plays Head, 1 Head if 1 plays Tail; HT = Head if 1 plays Head, Head Tail Tail if 1 plays Tail; 2 TH = Tail if 1 plays Head, 2 Head if 1 plays Tail; head tail head tail TT = Tail if 1 plays Head, Tail if 1 plays Tail. (-1,1) (1,-1) (1,-1) (-1,1) 2 Matching pennies with perfect information 2 1 HH HT TH TT Head Tail Matching pennies with Imperfect information 1 2 1 Head Tail Head Tail 2 Head (-1,1) (1,-1) head tail head tail Tail (1,-1) (-1,1) (-1,1) (1,-1) (1,-1) (-1,1) 3 A game with nature Left (5, 0) 1 Head 1/2 Right (2, 2) Nature (3, 3) 1/2 Left Tail 2 Right (0, -5) Mixed Strategy Definition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: Si = {si1,si2,…,sik} σ → A mixed strategy: i: S [0,1] s.t. σ σ σ i(si1) + i(si2) + … + i(sik) = 1. If the other players play s-i =(s1,…, si-1,si+1,…,sn), then σ the expected utility of playing i is σ σ σ i(si1)ui(si1,s-i) + i(si2)ui(si2,s-i) + … + i(sik)ui(sik,s-i). -
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. -
A New Approach to Stable Matching Problems
August1 989 Report No. STAN-CS-89- 1275 A New Approach to Stable Matching Problems bY Ashok Subramanian Department of Computer Science Stanford University Stanford, California 94305 3 DISTRIBUTION /AVAILABILITY OF REPORT Unrestricted: Distribution Unlimited GANIZATION 1 1 TITLE (Include Securrty Clamfrcat!on) A New Approach to Stable Matching Problems 12 PERSONAL AUTHOR(S) Ashok Subramanian 13a TYPE OF REPORT 13b TtME COVERED 14 DATE OF REPORT (Year, Month, Day) 15 PAGE COUNT FROM TO August 1989 34 16 SUPPLEMENTARY NOTATION 17 COSATI CODES 18 SUBJECT TERMS (Contrnue on reverse If necessary and jdentrfy by block number) FIELD GROUP SUB-GROUP 19 ABSTRACT (Continue on reverse if necessary and identrfy by block number) Abstract. We show that Stable Matching problems are the same as problems about stable config- urations of X-networks. Consequences include easy proofs of old theorems, a new simple algorithm for finding a stable matching, an understanding of the difference between Stable Marriage and Stable Roommates, NTcompleteness of Three-party Stable Marriage, CC-completeness of several Stable Matching problems, and a fast parallel reduction from the Stable Marriage problem to the ’ Assignment problem. 20 DISTRIBUTION /AVAILABILITY OF ABSTRACT 21 ABSTRACT SECURITY CLASSIFICATION q UNCLASSIFIED/UNLIMITED 0 SAME AS Rf’T 0 DTIC USERS 22a NAME OF RESPONSIBLE INDIVIDUAL 22b TELEPHONE (Include Area Code) 22c OFFICE SYMBOL Ernst Mavr DD Form 1473, JUN 86 Prevrous edrtions are obsolete SECURITY CLASSIFICATION OF TYS PAGt . 1 , S/N 0102-LF-014-6603 \ \ ““*5 - - A New Approach to Stable Matching Problems * Ashok Subramanian Department of Computer Science St anford University Stanford, CA 94305-2140 Abstract. -
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. -
How to Win at Tic-Tac-Toe
More Than Child’s Play How to Get N in a Row Games with Animals Hypercube Tic-Tac-Toe How to Win at Tic-Tac-Toe Norm Do Undoubtably, one of the most popular pencil and paper games in the world is tic-tac-toe, also commonly known as noughts and crosses. In this talk, you will learn how to beat your friends (at tic-tac-toe), discover why snaky is so shaky, and see the amazing tic-tac-toe playing chicken! March 2007 Norm Do How to Win at Tic-Tac-Toe Tic-Tac-Toe is popular: You’ve all played it while sitting at the back of a boring class. In fact, some of you are probably playing it right now! Tic-Tac-Toe is boring: People who are mildly clever should never lose. More Than Child’s Play How to Get N in a Row Some Facts About Tic-Tac-Toe Games with Animals Games to Beat your Friends With Hypercube Tic-Tac-Toe Some facts about tic-tac-toe Tic-Tac-Toe is old: It may have been played under the name of “terni lapilli” in Ancient Rome. Norm Do How to Win at Tic-Tac-Toe Tic-Tac-Toe is boring: People who are mildly clever should never lose. More Than Child’s Play How to Get N in a Row Some Facts About Tic-Tac-Toe Games with Animals Games to Beat your Friends With Hypercube Tic-Tac-Toe Some facts about tic-tac-toe Tic-Tac-Toe is old: It may have been played under the name of “terni lapilli” in Ancient Rome. -
The Roommate Problem Revisited
Department of Economics- FEA/USP The Roommate Problem Revisited MARILDA SOTOMAYOR WORKING PAPER SERIES Nº 2016-04 DEPARTMENT OF ECONOMICS, FEA-USP WORKING PAPER Nº 2016-04 The Roommate Problem Revisited Marilda Sotomayor ([email protected]) Abstract: We approach the roommate problem by focusing on simple matchings, which are those individually rational matchings whose blocking pairs, if any, are formed with unmatched agents. We show that the core is non-empty if and only if no simple and unstable matching is Pareto optimal among all simple matchings. The economic intuition underlying this condition is that blocking can be done so that the transactions at any simple and unstable matching need not be undone, as agents reach the core. New properties of economic interest are proved. Keywords: Core; stable matching . JEL Codes: C78; D78. THE ROOMMATE PROBLEM REVISITED by MARILDA SOTOMAYOR1 Department of Economics Universidade de São Paulo, Cidade Universitária, Av. Prof. Luciano Gualberto 908 05508-900, São Paulo, SP, Brazil e-mail: [email protected] 7/2005 ABSTRACT We approach the roommate problem by focusing on simple matchings, which are those individually rational matchings whose blocking pairs, if any, are formed with unmatched agents. We show that the core is non-empty if and only if no simple and unstable matching is Pareto optimal among all simple matchings. The economic intuition underlying this condition is that blocking can be done so that the transactions at any simple and unstable matching need not be undone, as agents reach the core. New properties of economic interest are proved. Keywords: core, stable matching JEL numbers: C78, D78 1 This paper is partially supported by CNPq-Brazil. -
Fractional Hedonic Games
Fractional Hedonic Games HARIS AZIZ, Data61, CSIRO and UNSW Australia FLORIAN BRANDL, Technical University of Munich FELIX BRANDT, Technical University of Munich PAUL HARRENSTEIN, University of Oxford MARTIN OLSEN, Aarhus University DOMINIK PETERS, University of Oxford The work we present in this paper initiated the formal study of fractional hedonic games, coalition formation games in which the utility of a player is the average value he ascribes to the members of his coalition. Among other settings, this covers situations in which players only distinguish between friends and non-friends and desire to be in a coalition in which the fraction of friends is maximal. Fractional hedonic games thus not only constitute a natural class of succinctly representable coalition formation games, but also provide an interesting framework for network clustering. We propose a number of conditions under which the core of fractional hedonic games is non-empty and provide algorithms for computing a core stable outcome. By contrast, we show that the core may be empty in other cases, and that it is computationally hard in general to decide non-emptiness of the core. 1 INTRODUCTION Hedonic games present a natural and versatile framework to study the formal aspects of coalition formation which has received much attention from both an economic and an algorithmic perspective. This work was initiated by Drèze and Greenberg[1980], Banerjee et al . [2001], Cechlárová and Romero-Medina[2001], and Bogomolnaia and Jackson[2002] and has sparked a lot of follow- up work. A recent survey was provided by Aziz and Savani[2016]. In hedonic games, coalition formation is approached from a game-theoretic angle. -
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.