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Game Theory- Prisoners Dilemma Vs Battle of the Sexes EXCERPTS
Lesson 14. Game Theory 1 Lesson 14 Game Theory c 2010, 2011 ⃝ Roberto Serrano and Allan M. Feldman All rights reserved Version C 1. Introduction In the last lesson we discussed duopoly markets in which two firms compete to sell a product. In such markets, the firms behave strategically; each firm must think about what the other firm is doing in order to decide what it should do itself. The theory of duopoly was originally developed in the 19th century, but it led to the theory of games in the 20th century. The first major book in game theory, published in 1944, was Theory of Games and Economic Behavior,byJohnvon Neumann (1903-1957) and Oskar Morgenstern (1902-1977). We will return to the contributions of Von Neumann and Morgenstern in Lesson 19, on uncertainty and expected utility. Agroupofpeople(orteams,firms,armies,countries)areinagame if their decision problems are interdependent, in the sense that the actions that all of them take influence the outcomes for everyone. Game theory is the study of games; it can also be called interactive decision theory. Many real-life interactions can be viewed as games. Obviously football, soccer, and baseball games are games.Butsoaretheinteractionsofduopolists,thepoliticalcampaignsbetweenparties before an election, and the interactions of armed forces and countries. Even some interactions between animal or plant species in nature can be modeled as games. In fact, game theory has been used in many different fields in recent decades, including economics, political science, psychology, sociology, computer science, and biology. This brief lesson is not meant to replace a formal course in game theory; it is only an in- troduction. -
Prisoners of Reason Game Theory and Neoliberal Political Economy
C:/ITOOLS/WMS/CUP-NEW/6549131/WORKINGFOLDER/AMADAE/9781107064034PRE.3D iii [1–28] 11.8.2015 9:57PM Prisoners of Reason Game Theory and Neoliberal Political Economy S. M. AMADAE Massachusetts Institute of Technology C:/ITOOLS/WMS/CUP-NEW/6549131/WORKINGFOLDER/AMADAE/9781107064034PRE.3D iv [1–28] 11.8.2015 9:57PM 32 Avenue of the Americas, New York, ny 10013-2473, usa Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107671195 © S. M. Amadae 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication Data Amadae, S. M., author. Prisoners of reason : game theory and neoliberal political economy / S.M. Amadae. pages cm Includes bibliographical references and index. isbn 978-1-107-06403-4 (hbk. : alk. paper) – isbn 978-1-107-67119-5 (pbk. : alk. paper) 1. Game theory – Political aspects. 2. International relations. 3. Neoliberalism. 4. Social choice – Political aspects. 5. Political science – Philosophy. I. Title. hb144.a43 2015 320.01′5193 – dc23 2015020954 isbn 978-1-107-06403-4 Hardback isbn 978-1-107-67119-5 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. -
Paul Milgrom Wins the BBVA Foundation Frontiers of Knowledge Award for His Contributions to Auction Theory and Industrial Organization
Economics, Finance and Management is the seventh category to be decided Paul Milgrom wins the BBVA Foundation Frontiers of Knowledge Award for his contributions to auction theory and industrial organization The jury singled out Milgrom’s work on auction design, which has been taken up with great success by governments and corporations He has also contributed novel insights in industrial organization, with applications in pricing and advertising Madrid, February 19, 2013.- The BBVA Foundation Frontiers of Knowledge Award in the Economics, Finance and Management category goes in this fifth edition to U.S. mathematician Paul Milgrom “for his seminal contributions to an unusually wide range of fields of economics including auctions, market design, contracts and incentives, industrial economics, economics of organizations, finance, and game theory,” in the words of the prize jury. This breadth of vision encompasses a business focus that has led him to apply his theories in advisory work with governments and corporations. Milgrom (Detroit, 1942), a professor of economics at Stanford University, was nominated for the award by Zvika Neeman, Head of The Eitan Berglas School of Economics at Tel Aviv University. “His work on auction theory is probably his best known,” the citation continues. “He has explored issues of design, bidding and outcomes for auctions with different rules. He designed auctions for multiple complementary items, with an eye towards practical applications such as frequency spectrum auctions.” Milgrom made the leap from games theory to the realities of the market in the mid 1990s. He was dong consultancy work for Pacific Bell in California to plan its participation in an auction called by the U.S. -
Fact-Checking Glen Weyl's and Stefano Feltri's
The Market Design Community and the Broadcast Incentive Auction: Fact-Checking Glen Weyl’s and Stefano Feltri’s False Claims By Paul Milgrom* June 3, 2020 TV Station Interference Constraints in the US and Canada In a recent Twitter rant and a pair of subsequent articles in Promarket, Glen Weyl1 and Stefano Feltri2 invent a conspiratorial narrative according to which the academic market design community is secretive and corrupt, my own actions benefitted my former business associates and the hedge funds they advised in the 2017 broadcast incentive auction, and the result was that far too little TV spectrum was reassigned for broadband at far too little value for taxpayers. The facts bear out none of these allegations. In fact, there were: • No secrets: all of Auctionomics’ communications are on the public record, • No benefits for hedge funds: the funds vigorously opposed Auctionomics’ proposals, which reduced their auction profits, • No spectrum shortfalls: the number of TV channels reassigned was unaffected by the hedge funds’ bidding, and • No taxpayer losses: the money value created for the public by the broadband spectrum auction was more than one hundred times larger than the alleged revenue shortfall. * Paul Milgrom, the co-founder and Chairman of Auctionomics, is the Shirley and Leonard Ely Professor of Economics at Stanford University. According to his 2020 Distinguished Fellow citation from the American Economic Association, Milgrom “is the world’s leading auction designer, having helped design many of the auctions for radio spectrum conducted around the world in the last thirty years.” 1 “It Is Such a Small World: The Market-Design Academic Community Evolved in a Business Network.” Stefano Feltri, Promarket, May 28, 2020. -
Putting Auction Theory to Work
Putting Auction Theory to Work Paul Milgrom With a Foreword by Evan Kwerel © 2003 “In Paul Milgrom's hands, auction theory has become the great culmination of game theory and economics of information. Here elegant mathematics meets practical applications and yields deep insights into the general theory of markets. Milgrom's book will be the definitive reference in auction theory for decades to come.” —Roger Myerson, W.C.Norby Professor of Economics, University of Chicago “Market design is one of the most exciting developments in contemporary economics and game theory, and who can resist a master class from one of the giants of the field?” —Alvin Roth, George Gund Professor of Economics and Business, Harvard University “Paul Milgrom has had an enormous influence on the most important recent application of auction theory for the same reason you will want to read this book – clarity of thought and expression.” —Evan Kwerel, Federal Communications Commission, from the Foreword For Robert Wilson Foreword to Putting Auction Theory to Work Paul Milgrom has had an enormous influence on the most important recent application of auction theory for the same reason you will want to read this book – clarity of thought and expression. In August 1993, President Clinton signed legislation granting the Federal Communications Commission the authority to auction spectrum licenses and requiring it to begin the first auction within a year. With no prior auction experience and a tight deadline, the normal bureaucratic behavior would have been to adopt a “tried and true” auction design. But in 1993 there was no tried and true method appropriate for the circumstances – multiple licenses with potentially highly interdependent values. -
Chapter 16 Oligopoly and Game Theory Oligopoly Oligopoly
Chapter 16 “Game theory is the study of how people Oligopoly behave in strategic situations. By ‘strategic’ we mean a situation in which each person, when deciding what actions to take, must and consider how others might respond to that action.” Game Theory Oligopoly Oligopoly • “Oligopoly is a market structure in which only a few • “Figuring out the environment” when there are sellers offer similar or identical products.” rival firms in your market, means guessing (or • As we saw last time, oligopoly differs from the two ‘ideal’ inferring) what the rivals are doing and then cases, perfect competition and monopoly. choosing a “best response” • In the ‘ideal’ cases, the firm just has to figure out the environment (prices for the perfectly competitive firm, • This means that firms in oligopoly markets are demand curve for the monopolist) and select output to playing a ‘game’ against each other. maximize profits • To understand how they might act, we need to • An oligopolist, on the other hand, also has to figure out the understand how players play games. environment before computing the best output. • This is the role of Game Theory. Some Concepts We Will Use Strategies • Strategies • Strategies are the choices that a player is allowed • Payoffs to make. • Sequential Games •Examples: • Simultaneous Games – In game trees (sequential games), the players choose paths or branches from roots or nodes. • Best Responses – In matrix games players choose rows or columns • Equilibrium – In market games, players choose prices, or quantities, • Dominated strategies or R and D levels. • Dominant Strategies. – In Blackjack, players choose whether to stay or draw. -
Nash Equilibrium
Lecture 3: Nash equilibrium Nash equilibrium: The mathematician John Nash introduced the concept of an equi- librium for a game, and equilibrium is often called a Nash equilibrium. They provide a way to identify reasonable outcomes when an easy argument based on domination (like in the prisoner's dilemma, see lecture 2) is not available. We formulate the concept of an equilibrium for a two player game with respective 0 payoff matrices PR and PC . We write PR(s; s ) for the payoff for player R when R plays 0 s and C plays s, this is simply the (s; s ) entry the matrix PR. Definition 1. A pair of strategies (^sR; s^C ) is an Nash equilbrium for a two player game if no player can improve his payoff by changing his strategy from his equilibrium strategy to another strategy provided his opponent keeps his equilibrium strategy. In terms of the payoffs matrices this means that PR(sR; s^C ) ≤ P (^sR; s^C ) for all sR ; and PC (^sR; sC ) ≤ P (^sR; s^C ) for all sc : The idea at work in the definition of Nash equilibrium deserves a name: Definition 2. A strategy s^R is a best-response to a strategy sc if PR(sR; sC ) ≤ P (^sR; sC ) for all sR ; i.e. s^R is such that max PR(sR; sC ) = P (^sR; sC ) sR We can now reformulate the idea of a Nash equilibrium as The pair (^sR; s^C ) is a Nash equilibrium if and only ifs ^R is a best-response tos ^C and s^C is a best-response tos ^R. -
NBER WORKING PAPER SERIES COMPLEMENTARITIES and COLLUSION in an FCC SPECTRUM AUCTION Patrick Bajari Jeremy T. Fox Working Paper
NBER WORKING PAPER SERIES COMPLEMENTARITIES AND COLLUSION IN AN FCC SPECTRUM AUCTION Patrick Bajari Jeremy T. Fox Working Paper 11671 http://www.nber.org/papers/w11671 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 September 2005 Bajari thanks the National Science Foundation, grants SES-0112106 and SES-0122747 for financial support. Fox would like to thank the NET Institute for financial support. Thanks to helpful comments from Lawrence Ausubel, Timothy Conley, Nicholas Economides, Philippe Fevrier, Ali Hortacsu, Paul Milgrom, Robert Porter, Gregory Rosston and Andrew Sweeting. Thanks to Todd Schuble for help with GIS software, to Peter Cramton for sharing his data on license characteristics, and to Chad Syverson for sharing data on airline travel. Excellent research assistance has been provided by Luis Andres, Stephanie Houghton, Dionysios Kaltis, Ali Manning, Denis Nekipelov and Wai-Ping Chim. Our email addresses are [email protected] and [email protected]. The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research. ©2005 by Patrick Bajari and Jeremy T. Fox. 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. Complementarities and Collusion in an FCC Spectrum Auction Patrick Bajari and Jeremy T. Fox NBER Working Paper No. 11671 September 2005 JEL No. L0, L5, C1 ABSTRACT We empirically study bidding in the C Block of the US mobile phone spectrum auctions. Spectrum auctions are conducted using a simultaneous ascending auction design that allows bidders to assemble packages of licenses with geographic complementarities. -
The Folk Theorem in Repeated Games with Discounting Or with Incomplete Information
The Folk Theorem in Repeated Games with Discounting or with Incomplete Information Drew Fudenberg; Eric Maskin Econometrica, Vol. 54, No. 3. (May, 1986), pp. 533-554. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198605%2954%3A3%3C533%3ATFTIRG%3E2.0.CO%3B2-U Econometrica is currently published by The Econometric Society. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/econosoc.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Tue Jul 24 13:17:33 2007 Econornetrica, Vol. -
What Is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?
What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization? Chi Jin Praneeth Netrapalli University of California, Berkeley Microsoft Research, India [email protected] [email protected] Michael I. Jordan University of California, Berkeley [email protected] August 18, 2020 Abstract Minimax optimization has found extensive application in modern machine learning, in settings such as generative adversarial networks (GANs), adversarial training and multi-agent reinforcement learning. As most of these applications involve continuous nonconvex-nonconcave formulations, a very basic question arises—“what is a proper definition of local optima?” Most previous work answers this question using classical notions of equilibria from simultaneous games, where the min-player and the max-player act simultaneously. In contrast, most applications in machine learning, including GANs and adversarial training, correspond to sequential games, where the order of which player acts first is crucial (since minimax is in general not equal to maximin due to the nonconvex-nonconcave nature of the problems). The main contribution of this paper is to propose a proper mathematical definition of local optimality for this sequential setting—local minimax—as well as to present its properties and existence results. Finally, we establish a strong connection to a basic local search algorithm—gradient descent ascent (GDA)—under mild conditions, all stable limit points of GDA are exactly local minimax points up to some degenerate points. 1 Introduction arXiv:1902.00618v3 [cs.LG] 15 Aug 2020 Minimax optimization refers to problems of two agents—one agent tries to minimize the payoff function f : X × Y ! R while the other agent tries to maximize it. -
Game Theory: Basic Concepts and Terminology
Leigh Tesfatsion 29 December 2019 Game Theory: Basic Concepts and Terminology A GAME consists of: • a collection of decision-makers, called players; • the possible information states of each player at each decision time; • the collection of feasible moves (decisions, actions, plays,...) that each player can choose to make in each of his possible information states; • a procedure for determining how the move choices of all the players collectively determine the possible outcomes of the game; • preferences of the individual players over these possible out- comes, typically measured by a utility or payoff function. 1 EMPLOYER CD C (40,40) (10,60) WORKER D (60,10) (20,20) Illustrative Modeling of a Work-Site Interaction as a \Prisoner's Dilemma Game" D = Defect (Shirk) C = Cooperate (Work Hard), (P1,P2) = (Worker Payoff, Employer Payoff) 2 A PURE STRATEGY for a player in a particular game is a complete contingency plan, i.e., a plan describing what move that player should take in each of his possible information states. A MIXED STRATEGY for a player i in a particular game is a probability distribution defined over the collection Si of player i's feasible pure strategy choices. That is, a mixed strategy assigns a nonnegative probability Prob(s) to each pure strategy s in Si, with X Prob(s) = 1 : (1) s2Si EXPOSITIONAL NOTE: For simplicity, the remainder of these brief notes will develop defi- nitions in terms of pure strategies; the unqualified use of \strategy" will always refer to pure strategy. Extension to mixed strategies is conceptually straightforward. 3 ONE-STAGE SIMULTANEOUS-MOVE N-PLAYER GAME: • The game is played just once among N players. -
Nash Equilibrium and Mechanism Design ✩ ∗ Eric Maskin A,B, a Institute for Advanced Study, United States B Princeton University, United States Article Info Abstract
Games and Economic Behavior 71 (2011) 9–11 Contents lists available at ScienceDirect Games and Economic Behavior www.elsevier.com/locate/geb Commentary: Nash equilibrium and mechanism design ✩ ∗ Eric Maskin a,b, a Institute for Advanced Study, United States b Princeton University, United States article info abstract Article history: I argue that the principal theoretical and practical drawbacks of Nash equilibrium as a Received 23 December 2008 solution concept are far less troublesome in problems of mechanism design than in most Available online 18 January 2009 other applications of game theory. © 2009 Elsevier Inc. All rights reserved. JEL classification: C70 Keywords: Nash equilibrium Mechanism design Solution concept A Nash equilibrium (called an “equilibrium point” by John Nash himself; see Nash, 1950) of a game occurs when players choose strategies from which unilateral deviations do not pay. The concept of Nash equilibrium is far and away Nash’s most important legacy to economics and the other behavioral sciences. This is because it remains the central solution concept—i.e., prediction of behavior—in applications of game theory to these fields. As I shall review below, Nash equilibrium has some important shortcomings, both theoretical and practical. I will argue, however, that these drawbacks are far less troublesome in problems of mechanism design than in most other applications of game theory. 1. Solution concepts Game-theoretic solution concepts divide into those that are noncooperative—where the basic unit of analysis is the in- dividual player—and those that are cooperative, where the focus is on coalitions of players. John von Neumann and Oskar Morgenstern themselves viewed the cooperative part of game theory as more important, and their seminal treatise, von Neumann and Morgenstern (1944), devoted fully three quarters of its space to cooperative matters.