The Shapley Value
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Is Shapley Cost Sharing Optimal?∗ (For the Special Issue in Honor of Lloyd Shapley)
Is Shapley Cost Sharing Optimal?∗ (For the special issue in honor of Lloyd Shapley) Shahar Dobzinskiy Aranyak Mehtaz Tim Roughgardenx Mukund Sundararajan{ August 29, 2016 Abstract A general approach to the design of budget-balanced cost-sharing mechanisms is to use the Shapley value, applied to the given cost function, to define payments from the players to the mechanism. Is the corresponding Shapley value mechanism \optimal" in some sense? We consider the objective of minimizing worst-case inefficiency subject to a revenue constraint, and prove results in three different regimes. 1. For the public excludable good problem, the Shapley value mechanism minimizes the worst-case efficiency loss over all truthful, deterministic, and budget-balanced mechanisms that satisfy equal treatment. This result follows from a characteriza- tion of the Shapley value mechanism as the unique one that satisfies two additional technical conditions. 2. For the same problem, even over a much more general mechanism design space that allows for randomization and approximate budget-balance and does not im- pose equal treatment, the worst-case efficiency loss of the Shapley value mecha- nism is within a constant factor of the minimum possible. ∗The results in Sections 3{5 appeared, in preliminary form, in the extended abstract [8]. The results in Section 6 appeared in the PhD thesis of the fourth author [27]. yDepartment of Applied Math and Computer Science, the Weizmann Institute of Science, Rohovot 76100, Israel. This work was done while the author was visiting Stanford University, and supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities, and by grants from the Israel Science Foundation and the USA-Israel Bi-national Science Foundation. -
Introduction to Computational Social Choice
1 Introduction to Computational Social Choice Felix Brandta, Vincent Conitzerb, Ulle Endrissc, J´er^omeLangd, and Ariel D. Procacciae 1.1 Computational Social Choice at a Glance Social choice theory is the field of scientific inquiry that studies the aggregation of individual preferences towards a collective choice. For example, social choice theorists|who hail from a range of different disciplines, including mathematics, economics, and political science|are interested in the design and theoretical evalu- ation of voting rules. Questions of social choice have stimulated intellectual thought for centuries. Over time the topic has fascinated many a great mind, from the Mar- quis de Condorcet and Pierre-Simon de Laplace, through Charles Dodgson (better known as Lewis Carroll, the author of Alice in Wonderland), to Nobel Laureates such as Kenneth Arrow, Amartya Sen, and Lloyd Shapley. Computational social choice (COMSOC), by comparison, is a very young field that formed only in the early 2000s. There were, however, a few precursors. For instance, David Gale and Lloyd Shapley's algorithm for finding stable matchings between two groups of people with preferences over each other, dating back to 1962, truly had a computational flavor. And in the late 1980s, a series of papers by John Bartholdi, Craig Tovey, and Michael Trick showed that, on the one hand, computational complexity, as studied in theoretical computer science, can serve as a barrier against strategic manipulation in elections, but on the other hand, it can also prevent the efficient use of some voting rules altogether. Around the same time, a research group around Bernard Monjardet and Olivier Hudry also started to study the computational complexity of preference aggregation procedures. -
Stable Allocations and the Practice of Market Design
15 OCTOBER 2012 Scientific Background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2012 STABLE ALLOCATIONS AND THE PRACTICE OF MARKET DESIGN compiled by the Economic Sciences Prize Committee of the Royal Swedish Academy of Sciences THE ROYAL SWEDISH ACADEMY OF SCIENCES has as its aim to promote the sciences and strengthen their influence in society. BOX 50005 (LILLA FRESCATIVÄGEN 4 A), SE-104 05 STOCKHOLM, SWEDEN TEL +46 8 673 95 00, FAX +46 8 15 56 70, [email protected] HTTP://KVA.SE 1 Introduction Economists study how societies allocate resources. Some allocation problems are solved by the price system: high wages attract workers into a particu- lar occupation, and high energy prices induce consumers to conserve energy. In many instances, however, using the price system would encounter legal and ethical objections. Consider, for instance, the allocation of public-school places to children, or the allocation of human organs to patients who need transplants. Furthermore, there are many markets where the price system operates but the traditional assumption of perfect competition is not even approximately satis…ed. In particular, many goods are indivisible and het- erogeneous, whereby the market for each type of good becomes very thin. How these thin markets allocate resources depends on the institutions that govern transactions. This year’s prizewinning work encompasses a theoretical framework for analyzing resource allocation, as well as empirical studies and actual redesign of real-world institutions such as labor-market clearinghouses and school ad- missions procedures. The foundations for the theoretical framework were laid in 1962, when David Gale and Lloyd Shapley published a mathematical inquiry into a certain class of allocation problems. -
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. -
Stochastic Game Theory Applications for Power Management in Cognitive Networks
STOCHASTIC GAME THEORY APPLICATIONS FOR POWER MANAGEMENT IN COGNITIVE NETWORKS A thesis submitted to Kent State University in partial fulfillment of the requirements for the degree of Master of Digital Science by Sham Fung Feb 2014 Thesis written by Sham Fung M.D.S, Kent State University, 2014 Approved by Advisor , Director, School of Digital Science ii TABLE OF CONTENTS LIST OF FIGURES . vi LIST OF TABLES . vii Acknowledgments . viii Dedication . ix 1 BACKGROUND OF COGNITIVE NETWORK . 1 1.1 Motivation and Requirements . 1 1.2 Definition of Cognitive Networks . 2 1.3 Key Elements of Cognitive Networks . 3 1.3.1 Learning and Reasoning . 3 1.3.2 Cognitive Network as a Multi-agent System . 4 2 POWER ALLOCATION IN COGNITIVE NETWORKS . 5 2.1 Overview . 5 2.2 Two Communication Paradigms . 6 2.3 Power Allocation Scheme . 8 2.3.1 Interference Constraints for Primary Users . 8 2.3.2 QoS Constraints for Secondary Users . 9 2.4 Related Works on Power Management in Cognitive Networks . 10 iii 3 GAME THEORY|PRELIMINARIES AND RELATED WORKS . 12 3.1 Non-cooperative Game . 14 3.1.1 Nash Equilibrium . 14 3.1.2 Other Equilibriums . 16 3.2 Cooperative Game . 16 3.2.1 Bargaining Game . 17 3.2.2 Coalition Game . 17 3.2.3 Solution Concept . 18 3.3 Stochastic Game . 19 3.4 Other Types of Games . 20 4 GAME THEORETICAL APPROACH ON POWER ALLOCATION . 23 4.1 Two Fundamental Types of Utility Functions . 23 4.1.1 QoS-Based Game . 23 4.1.2 Linear Pricing Game . -
The Portfolio Optimization Performance During Malaysia's
Modern Applied Science; Vol. 14, No. 4; 2020 ISSN 1913-1844 E-ISSN 1913-1852 Published by Canadian Center of Science and Education The Portfolio Optimization Performance during Malaysia’s 2018 General Election by Using Noncooperative and Cooperative Game Theory Approach Muhammad Akram Ramadhan bin Ibrahim1, Pah Chin Hee1, Mohd Aminul Islam1 & Hafizah Bahaludin1 1Department of Computational and Theoretical Sciences Kulliyyah of Science International Islamic University Malaysia, 25200 Kuantan, Pahang, Malaysia Correspondence: Pah Chin Hee, Department of Computational and Theoretical Sciences, Kulliyyah of Science, International Islamic University Malaysia, 25200 Kuantan, Pahang, Malaysia. Received: February 10, 2020 Accepted: February 28, 2020 Online Published: March 4, 2020 doi:10.5539/mas.v14n4p1 URL: https://doi.org/10.5539/mas.v14n4p1 Abstract Game theory approach is used in this study that involves two types of games which are noncooperative and cooperative. Noncooperative game is used to get the equilibrium solutions from each payoff matrix. From the solutions, the values then be used as characteristic functions of Shapley value solution concept in cooperative game. In this paper, the sectors are divided into three groups where each sector will have three different stocks for the game. This study used the companies that listed in Bursa Malaysia and the prices of each stock listed in this research obtained from Datastream. The rate of return of stocks are considered as an input to get the payoff from each stock and its coalition sectors. The value of game for each sector is obtained using Shapley value solution concepts formula to find the optimal increase of the returns. The Shapley optimal portfolio, naive diversification portfolio and market portfolio performances have been calculated by using Sharpe ratio. -
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. -
Deterministic and Stochastic Prisoner's Dilemma Games: Experiments in Interdependent Security
NBER TECHNICAL WORKING PAPER SERIES DETERMINISTIC AND STOCHASTIC PRISONER'S DILEMMA GAMES: EXPERIMENTS IN INTERDEPENDENT SECURITY Howard Kunreuther Gabriel Silvasi Eric T. Bradlow Dylan Small Technical Working Paper 341 http://www.nber.org/papers/t0341 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 August 2007 We appreciate helpful discussions in designing the experiments and comments on earlier drafts of this paper by Colin Camerer, Vince Crawford, Rachel Croson, Robyn Dawes, Aureo DePaula, Geoff Heal, Charles Holt, David Krantz, Jack Ochs, Al Roth and Christian Schade. We also benefited from helpful comments from participants at the Workshop on Interdependent Security at the University of Pennsylvania (May 31-June 1 2006) and at the Workshop on Innovation and Coordination at Humboldt University (December 18-20, 2006). We thank George Abraham and Usmann Hassan for their help in organizing the data from the experiments. Support from NSF Grant CMS-0527598 and the Wharton Risk Management and Decision Processes Center is gratefully acknowledged. The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research. © 2007 by Howard Kunreuther, Gabriel Silvasi, Eric T. Bradlow, and Dylan Small. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. Deterministic and Stochastic Prisoner's Dilemma Games: Experiments in Interdependent Security Howard Kunreuther, Gabriel Silvasi, Eric T. Bradlow, and Dylan Small NBER Technical Working Paper No. 341 August 2007 JEL No. C11,C12,C22,C23,C73,C91 ABSTRACT This paper examines experiments on interdependent security prisoner's dilemma games with repeated play. -
ECONOMICS: the NEXT PHYSICAL SCIENCE? by J. Doyne Farmer
ECONOMICS: THE NEXT PHYSICAL SCIENCE? By J. Doyne Farmer, Martin Shubik and Eric Smith June 2005 COWLES FOUNDATION DISCUSSION PAPER NO. 1520 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/ Economics: the next physical science? J. Doyne Farmer,1 Martin Shubik,2 and Eric Smith1 1Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe NM 87501 2Economics Department, Yale University, New Haven CT (Dated: June 9, 2005) We review an emerging body of work by physicists addressing questions of economic organization and function. We suggest that, beyond simply employing models familiar from physics to economic observables, remarkable regularities in economic data may suggest parts of social order that can usefully be incorporated into, and in turn can broaden, the conceptual structure of physics. Contents physics departments specializing in econophysics. There is even a new annual research prize, titled the “Young I. Physics and economics 1 Scientist Award for Social and Econophysics”. Is this just a fad, or is there something more substantial here? II. Data analysis and the search for empirical regularitiesIf physicists2 want to do research in economics, why don’t they just get degrees in economics in the first place? Why don’t the econophysicists retool, find jobs in eco- III. Modeling the behavior of agents 4 nomics departments and publish in traditional economics journals? Perhaps this is just a temporary phenomenon, IV. The quest for simple models of non-rational choicedriven4 by a generation of physicists who made a bad ca- reer choice. Is there any reason why research in eco- nomics should be done in physics departments as an on- V. -
Stochastic Games with Hidden States∗
Stochastic Games with Hidden States¤ Yuichi Yamamoto† First Draft: March 29, 2014 This Version: January 14, 2015 Abstract This paper studies infinite-horizon stochastic games in which players ob- serve noisy public information about a hidden state each period. We find that if the game is connected, the limit feasible payoff set exists and is invariant to the initial prior about the state. Building on this invariance result, we pro- vide a recursive characterization of the equilibrium payoff set and establish the folk theorem. We also show that connectedness can be replaced with an even weaker condition, called asymptotic connectedness. Asymptotic con- nectedness is satisfied for generic signal distributions, if the state evolution is irreducible. Journal of Economic Literature Classification Numbers: C72, C73. Keywords: stochastic game, hidden state, connectedness, stochastic self- generation, folk theorem. ¤The author thanks Naoki Aizawa, Drew Fudenberg, Johannes Horner,¨ Atsushi Iwasaki, Michi- hiro Kandori, George Mailath, Takeaki Sunada, and Masatoshi Tsumagari for helpful conversa- tions, and seminar participants at various places. †Department of Economics, University of Pennsylvania. Email: [email protected] 1 1 Introduction When agents have a long-run relationship, underlying economic conditions may change over time. A leading example is a repeated Bertrand competition with stochastic demand shocks. Rotemberg and Saloner (1986) explore optimal col- lusive pricing when random demand shocks are i.i.d. each period. Haltiwanger and Harrington (1991), Kandori (1991), and Bagwell and Staiger (1997) further extend the analysis to the case in which demand fluctuations are cyclic or persis- tent. One of the crucial assumptions of these papers is that demand shocks are publicly observable before firms make their decisions in each period. -
War Gaming in the Information Age—Theory and Purpose Paul Bracken
Naval War College Review Volume 54 Article 6 Number 2 Spring 2001 War Gaming in the Information Age—Theory and Purpose Paul Bracken Martin Shubik Follow this and additional works at: https://digital-commons.usnwc.edu/nwc-review Recommended Citation Bracken, Paul and Shubik, Martin (2001) "War Gaming in the Information Age—Theory and Purpose," Naval War College Review: Vol. 54 : No. 2 , Article 6. Available at: https://digital-commons.usnwc.edu/nwc-review/vol54/iss2/6 This Article is brought to you for free and open access by the Journals at U.S. Naval War College Digital Commons. It has been accepted for inclusion in Naval War College Review by an authorized editor of U.S. Naval War College Digital Commons. For more information, please contact [email protected]. Bracken and Shubik: War Gaming in the Information Age—Theory and Purpose WAR GAMING IN THE INFORMATION AGE Theory and Purpose Paul Bracken and Martin Shubik ver twenty years ago, a study was carried out under the sponsorship of the ODefense Advanced Research Projects Agency and in collaboration with the General Accounting Office to survey and critique the models, simulations, and war games then in use by the Department of Defense.1 From some points of view, twenty years ago means ancient history; changes in communication tech- nology and computers since then can be measured Paul Bracken, professor of management and of political science at Yale University, specializes in national security only in terms of orders of magnitudes. The new world and management issues. He is the author of Command of the networked battlefield, super-accurate weapons, and Control of Nuclear Forces (1983) and of Fire in the and the information technology (IT) revolution, with its East: The Rise of Asian Military Power and the Second Nuclear Age (HarperCollins, 1999). -
2Majors and Minors.Qxd.KA
APPLIED MATHEMATICS AND STATISTICS Spring 2007: updates since Fall 2006 are in red Applied Mathematics and Statistics (AMS) Major and Minor in Applied Mathematics and Statistics Department of Applied Mathematics and Statistics, College of Engineering and Applied Sciences CHAIRPERSON: James Glimm UNDERGRADUATE PROGRAM DIRECTOR: Alan C. Tucker UNDERGRADUATE SECRETARY: Elizabeth Ahmad OFFICE: P-139B Math Tower PHONE: (631) 632-8370 E-MAIL: [email protected] WEB ADDRESS: http://naples.cc.sunysb.edu/CEAS/amsweb.nsf Students majoring in Applied Mathematics and Statistics often double major in one of the following: Computer Science (CSE), Economics (ECO), Information Systems (ISE) Faculty Bradley Plohr, Adjunct Professor, Ph.D., he undergraduate program in Princeton University: Conservation laws; com- Hongshik Ahn, Associate Professor, Ph.D., Applied Mathematics and Statistics putational fluid dynamics. University of Wisconsin: Biostatistics; survival Taims to give mathematically ori- analysis. John Reinitz, Professor, Ph.D., Yale University: ented students a liberal education in quan- Mathematical biology. Esther Arkin, Professor, Ph.D., Stanford titative problem solving. The courses in University: Computational geometry; combina- Robert Rizzo, Assistant Professor, Ph.D., Yale this program survey a wide variety of torial optimization. University: Bioinformatics; drug design. mathematical theories and techniques Edward J. Beltrami, Professor Emeritus, Ph.D., David Sharp, Adjunct Professor, Ph.D., that are currently used by analysts and Adelphi University: Optimization; stochastic California Institute of Technology: Mathematical researchers in government, industry, and models. physics. science. Many of the applied mathematics Yung Ming Chen, Professor Emeritus, Ph.D., Ram P. Srivastav, Professor, D.Sc., University of courses give students the opportunity to New York University: Partial differential equa- Glasgow; Ph.D., University of Lucknow: Integral develop problem-solving techniques using equations; numerical solutions.