Negotiation Analysis Supplement NEWS CLEARINGHOUSE by Howard Raiffa
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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. -
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. -
1947–1967 a Conversation with Howard Raiffa
Statistical Science 2008, Vol. 23, No. 1, 136–149 DOI: 10.1214/088342307000000104 c Institute of Mathematical Statistics, 2008 The Early Statistical Years: 1947–1967 A Conversation with Howard Raiffa Stephen E. Fienberg Abstract. Howard Raiffa earned his bachelor’s degree in mathematics, his master’s degree in statistics and his Ph.D. in mathematics at the University of Michigan. Since 1957, Raiffa has been a member of the faculty at Harvard University, where he is now the Frank P. Ramsey Chair in Managerial Economics (Emeritus) in the Graduate School of Business Administration and the Kennedy School of Government. A pioneer in the creation of the field known as decision analysis, his re- search interests span statistical decision theory, game theory, behavioral decision theory, risk analysis and negotiation analysis. Raiffa has su- pervised more than 90 doctoral dissertations and written 11 books. His new book is Negotiation Analysis: The Science and Art of Collaborative Decision Making. Another book, Smart Choices, co-authored with his former doctoral students John Hammond and Ralph Keeney, was the CPR (formerly known as the Center for Public Resources) Institute for Dispute Resolution Book of the Year in 1998. Raiffa helped to create the International Institute for Applied Systems Analysis and he later became its first Director, serving in that capacity from 1972 to 1975. His many honors and awards include the Distinguished Contribution Award from the Society of Risk Analysis; the Frank P. Ramsey Medal for outstanding contributions to the field of decision analysis from the Operations Research Society of America; and the Melamed Prize from the University of Chicago Business School for The Art and Science of Negotiation. -
Minmax Shapley Value and Biform Contests Subgame Perfect Nash Equilibrium
1 סדרת מאמרים לדיון Working paper series No. 14 – 2011 Minmax Shapley Value and Biform Contests Subgame Perfect Nash Equilibrium Mordechai E. Schwarz 142 מערכות מידע חשבונאיות ממוחשבות 1 Minmax Shapley Value and Biform Contests Subgame Perfect Nash Equilibrium Mordechai E. Schwarz1 Abstract A biform contest is a two-stage game. In the first stage, agents exert resources to build up their position towards the second stage. The second stage is either a bargaining game or a cooperative game, but if no cooperative solution is achieved it turns to a rent-seeking contest. It is shown that with a passive regulator, the second stage is a cooperative game and if all contestants are risk-averse, the biform contest non-cooperatively implements the minmax Shapley value, as defined in the paper, in subgame perfect equilibrium. Keywords: biform game, biform contest, rent-seeking, externalities, minmax approach, minmax coalitional worth, minmax Shapley value. JEL classification: C7, D7 1 Department of Economics and Management, The Open University of Israel. Email: [email protected]. I am truly indebted to J. Atsu Amegashie, Shmuel Nitzan and Aviad Heifetz for very helpful comments on earlier versions of this work. Of course, the responsibility for every error is exclusively mine. 2 1. Introduction Cooperation among rent-seekers is usually modeled as a two-stage game. There are at least three main categories of these models. One category which I call wolf-pack games studied a two-stage group rent-seeking without a specific sharing rule of the prize among the winning group members. Contestants cooperate within group rent-seeking against rival groups in the first stage, and then the individual members of the winning group engage in another rent- seeking contest for the prize in the second stage2. -
Comparing the Fairness of Two Popular Solution Concepts of Coalition Games: Shapley Value and Nucleolus1
1 COMPARING THE FAIRNESS OF TWO POPULAR SOLUTION CONCEPTS OF COALITION GAMES: SHAPLEY VALUE AND NUCLEOLUS1 AUTHORS REDACTED TO PRESERVE ANONYMITY ABSTRACT An analytics model is developed to compare two popular solution concepts of coalition games, and to determine which one is fairer and more appropriate to use in real life applications of coalition games. Coalition games have found wide application in economics, finance, politics, and computing. When applications are developed using coalition game theory, designers have the option of choosing from several solutions concepts and the effectiveness of the solution depends on the accuracy of the underlying solution concepts. The algorithm presented sheds light on the effectiveness and fairness of the solution concepts and, in particular, is used to compare the fairness between Shapley Value and Nucleolus solution concepts. When imputations for coalitions are expressed in a barycentric triangle, they create a core, and larger core areas leave room for more competition among the coalitions which can create instability. Smaller core areas are desirable because they leave little room for further collaboration and come to a quick resolution. An example supports the idea of smaller core comparison, and imputations have been calculated using Shapley Value and Nucleolus expressed in a barycentric triangle for comparison. INTRODUCTION Game theory has its foundations in applied mathematics. The mathematical models and built-in consistency of game theory make it a suitable framework and basis for modeling and designing of automated bargaining and decision-making software systems in interactive negotiation. For example, a game theory framework can serve as the most efficient bidding rules for Web Services, e-commerce auction website, or tamper-proof automated negotiations for purchasing communication bandwidth. -
Valuing Inputs Under Supply Uncertainty : the Bayesian Shapley Value∗
Valuing Inputs Under Supply Uncertainty : The Bayesian Shapley Value∗ Roland Pongouy and Jean-Baptiste Tondjiz April 12, 2017 Abstract We consider the problem of valuing inputs in a production environment in which input supply is uncertain. Inputs can be workers in a firm, risk factors for a disease, securities in a financial market, or nodes in a networked economy. Each input takes its values from a finite set and uncertainty is modeled as a probability distribution over this set. First, we provide an axiomatic solution to our valuation problem, defining three intuitive axioms which we use to uniquely characterize a valuation scheme that we call the a priori Shapley value. Second, we solve the problem of valuing inputs a posteriori { that is, after observing output. This leads to the Bayesian Shapley value. Third, we consider the problem of rationalizing uncertainty when the inputs are ra- tional workers supplying labor in a non-cooperative production game in which payoffs are given by the Shapley wage function. We find that probability distributions over labor supply that can be supported as mixed strategy Nash equilibria always exist. We also provide an intuitive condition under which we prove the existence of a pure strategy Nash equilibrium. We present several applications of our theory to real-life situations. JEL classification codes: C70, D20, D80, J30. Keywords: Input valuation, uncertainty, a priori Shapley value, Bayesian Shapley value, rationalizability. ∗We would like to thank Pierre Evariste Nguimkeu for his valuable comments. yDepartment of Economics, University of Ottawa, [email protected] zDepartment of Economics, University of Ottawa, [email protected] 1 Introduction In this paper, we consider the problem of valuing inputs in a production environment where input supply is uncertain. -
Howard Raiffa, 92 a Leader in the Study of How Decisions Are Made
The Washington Post July 24, 2016 Howard Raiffa, 92 A leader in the study of how decisions are made Dr. Raiffa in a Harvard classroom in 1975. (Richard A. Chase/Harvard Business School) by Emily Langer Howard Raiffa, a professor of economics who was credited with leading an intellectual revolution in business and other arenas by teaching decision-making not as an art, but as a science grounded in mathematics and statistics, died July 8 at his home in Oro Valley, Ariz. He was 92. His death was announced by the Harvard Business School, where Dr. Raiffa joined the faculty in 1957, and by the Harvard Kennedy School of government and public affairs, which cited him as one of the founding fathers of its modern incarnation. The cause was Parkinson’s disease, said his daughter Judith Raiffa. During a nearly four-decade career at Harvard, Dr. Raiffa became known as one of the most creative thinkers in his field: the study of how people make decisions — good, bad, wise or unwise — in situations of uncertainty. He was the quintessential interdisciplinary academician. Trained in mathematics and statistics, he was a joint chair at Harvard’s business and government schools and did work that was applied to commerce, law, foreign affairs, public policy, medicine and sports. In an interview, Max H. Bazerman, a professor of business administration at the Harvard Business School and co-director of the Center for Public Leadership at the Harvard Kennedy School, said Dr. Raiffa was someone who “wanted to use his brilliance in applied contexts to make the world better.” In the early years of his career, Dr. -
The Art and Science of Negotiation
tl t ,I The Art and Science of Negotiation HOWARD RAIFFA The BelknapPress of Harvard University Press Cambridge, Massachusetts and London, England I I i i I I I Acknowledgments Ideas are incestuous. They commingle and refuse to sort thenr- selves out so that one can say, "These ideas are his or hers arrd those mine." I know, however, that many of the ideas in the chap- ters that follow are the ideas of others, and some of these others can be identified. To no one am I more indebted than to John Ham- mond. This book would not have been written if I had not chosen to teach a course in competitive decision making at the Harvard Busi- Copyright O f 982 by the President and Fellows of Harvard College All rights reserved ness School, a course that evolved over more than a decade. In the Printed in the United States of America mid-1960s I taught a doctoral seminar in individual, group, and irr- teractive decisions, and in the early seventies John Hammond irr- corporated some of the material from that seminar in a pioneering 'he "Competitive Art of Negotiating" is a registered trademark of Gerard I. Nierenberg. This M.B.A. course entitled Decision Nlaking." I later in- :,The Art and Science of Negotiation, is not connected with Mr. Nierenberg's herited that course from John and built on his materials. Although i or programs. my course evolved into one that was substantially different frorn John's, he had set the tone; and even when I departed from his 87 6543 work, I had a very comfortable launching pad. -
Howard Raiffa Jeffrey Keisler
University of Massachusetts Boston From the SelectedWorks of Jeffrey Keisler September, 2016 In Memoriam: Howard Raiffa Jeffrey Keisler Available at: https://works.bepress.com/jeffrey_keisler/72/ In Memoriam: Howard Raiffa Decision Analysis Today 35(2):4-7, September 2016 Howard Raiffa died peacefully on Friday, July 8, 2016 at the age of 92. Raiffa was the Frank Plumpton Ramsey Professor of Managerial Economics Emeritus at the Business School and the Kennedy School at Harvard University. He was both a towering intellect and a kind and good man, who inspired colleagues across many disciplines and organizations with his keen mind, ecumenical approach to academic research, and his generous spirit. Raiffa’s intellectual and personal qualities persist in the many PhD students (I was fortunate to be one) he advised over his 40 years as a professor. Without him there would be no field of decision analysis as we know it. His work spanning over six decades continues to shape the field, through his mentorship of many leading researchers, and his own seminal publications. Along with his many honors in other fields, Raiffa was the first recipient of the Decision Analysis Society’s Ramsey Medal, and was recently honored along with Ronald Howard at the celebration of the 50th Anniversary of the field of Decision Analysis, with the creation of the annual Raiffa-Howard Award. Much has been written about his life and career. A list of tributes, biographical essays and Raiffa’s own writings is provided at the end of this essay. Within decision analysis, his impact is incalculable. Raiffa had the mind of the mathematician and the heart of a coach. -
Fairness Through the Lens of Cooperative Game Theory: an Experimental Approach∗
Fairness through the Lens of Cooperative Game Theory: An Experimental Approach∗ Geoffroy de Clippely Kareen Rozenz May 2019 Abstract We experimentally investigate how impartial observers allocate money to agents whose complementarity and substitutability determine the surplus each group can achieve. Analyzing the data through the lens of axioms and solutions from cooperative-game theory, a one-parameter model (mixing equal split and Shapley value) arises as a parsimonious description of the data. Three treatments establish the robustness of this qualitative conclusion, while also illustrating how context may im- pact the parameter estimate. ∗First version October 2013. We are grateful to the editor, three referees, Pedro Dal B´o,Mark Dean, Kfir Eliaz, Louis Putterman, Larry Samuelson, Roberto Serrano, Rajiv Vohra, and various seminar audiences for valuable suggestions, and also thank Roberto Serrano for arranging part of the funding for this project. The paper benefited from the excellent research assistance of Camilo Dominguez Gonzalez and Emily Nix in the initial stages, and Zeky Murra in later stages. We also thank Eli Zvuluny of Possible Worlds Ltd. for programming the experiment. yBrown University, Department of Economics. zBrown University, Department of Economics. 1 Introduction In many economic settings, including trading and joint production, the surplus to be shared is created through collaboration. Complementarity and substi- tutability among agents determine how much there is to share when they all cooperate, and what surplus would be created should smaller groups form. Consider, for instance, three musicians who can play together as a duo or a trio for an event (but not as soloists). They will collect $900 for performing as a trio. -
John Von Neumann's Work in the Theory of Games and Mathematical Economics
JOHN VON NEUMANN'S WORK IN THE THEORY OF GAMES AND MATHEMATICAL ECONOMICS H. W. KUHN AND A. W. TUCKER Of the many areas of mathematics shaped by his genius, none shows more clearly the influence of John von Neumann than the Theory of Games. This modern approach to problems of competition and cooperation was given a broad foundation in his superlative paper of 1928 [A].1 In scope and youthful vigor this work can be compared only to his papers of the same period on the axioms of set theory and the mathematical foundations of quantum mechanics. A decade later, when the Austrian economist Oskar Morgenstern came to Princeton, von Neumann's interest in the theory was reawakened. The result of their active and intensive collaboration during the early years of World War II was the treatise Theory of games and economic behavior [D], in which the basic structure of the 1928 paper is elaborated and extended. Together, the paper and treatise contain a remarkably complete outline of the subject as we know it today, and every writer in the field draws in some measure upon concepts which were there united into a coherent theory. The crucial innovation of von Neumann, which was to be both the keystone of his Theory of Games and the central theme of his later research in the area, was the assertion and proof of the Minimax Theorem. Ideas of pure and randomized strategies had been intro duced earlier, especially by Êmile Borel [3]. However, these efforts were restricted either to individual examples or, at best, to zero-sum two-person games with skew-symmetric payoff matrices.