Game Theory and Its Practical Applications
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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). -
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. -
Lecture Notes
GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies .................................. -
More Experiments on Game Theory
More Experiments on Game Theory Syngjoo Choi Spring 2010 Experimental Economics (ECON3020) Game theory 2 Spring2010 1/28 Playing Unpredictably In many situations there is a strategic advantage associated with being unpredictable. Experimental Economics (ECON3020) Game theory 2 Spring2010 2/28 Playing Unpredictably In many situations there is a strategic advantage associated with being unpredictable. Experimental Economics (ECON3020) Game theory 2 Spring2010 2/28 Mixed (or Randomized) Strategy In the penalty kick, Left (L) and Right (R) are the pure strategies of the kicker and the goalkeeper. A mixed strategy refers to a probabilistic mixture over pure strategies in which no single pure strategy is played all the time. e.g., kicking left with half of the time and right with the other half of the time. A penalty kick in a soccer game is one example of games of pure con‡icit, essentially called zero-sum games in which one player’s winning is the other’sloss. Experimental Economics (ECON3020) Game theory 2 Spring2010 3/28 The use of pure strategies will result in a loss. What about each player playing heads half of the time? Matching Pennies Games In a matching pennies game, each player uncovers a penny showing either heads or tails. One player takes both coins if the pennies match; otherwise, the other takes both coins. Left Right Top 1, 1, 1 1 Bottom 1, 1, 1 1 Does there exist a Nash equilibrium involving the use of pure strategies? Experimental Economics (ECON3020) Game theory 2 Spring2010 4/28 Matching Pennies Games In a matching pennies game, each player uncovers a penny showing either heads or tails. -
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. -
Repeated Games
REPEATED GAMES 1 Early PD experiments In 1950, Merrill Flood and Melvin Dresher (at RAND) devised an experiment to test Nash’s theory about defection in a two-person prisoners’ dilemma. Experimental Design – They asked two friends to play the PD 100 times. – They measured the success of Nash’s equilibrium concept by counting the number of times the players chose {D;D}. 2 Flood and Dresher’s results Player 1 cooperated in 68 rounds Player 2 cooperated in 78 rounds Both cooperated in 60 of last 89 rounds Flood Dresher Nash 3 Flood and Dresher’s results Player 1 cooperated in 68 rounds Player 2 cooperated in 78 rounds Both cooperated in 60 of last 89 rounds Wait a Ha! That jerk I can’tI’mOh a be Ha! Nash second... Nash was genius...%&@#!wrong! was wrong! wrong! Flood Dresher Nash 4 Nash’s response “If this experiment were conducted with various different players rotating the competition and with no information given to a player of what choices the others have been making until the end of all trials, then the experimental results would have been quite different, for this modification of procedure would remove the interaction between the trials.” 5 Nash’s response “The flaw in this experiment as a test of equilibrium point theory is that the experiment really amounts to having the players play one large multimove game. One cannot...think of the thing as a sequence of independent games...there is too much interaction.” In other words, Nash said that repeating the game changes the game itself. -
8. Maxmin and Minmax Strategies
CMSC 474, Introduction to Game Theory 8. Maxmin and Minmax Strategies Mohammad T. Hajiaghayi University of Maryland Outline Chapter 2 discussed two solution concepts: Pareto optimality and Nash equilibrium Chapter 3 discusses several more: Maxmin and Minmax Dominant strategies Correlated equilibrium Trembling-hand perfect equilibrium e-Nash equilibrium Evolutionarily stable strategies Worst-Case Expected Utility For agent i, the worst-case expected utility of a strategy si is the minimum over all possible Husband Opera Football combinations of strategies for the other agents: Wife min u s ,s Opera 2, 1 0, 0 s-i i ( i -i ) Football 0, 0 1, 2 Example: Battle of the Sexes Wife’s strategy sw = {(p, Opera), (1 – p, Football)} Husband’s strategy sh = {(q, Opera), (1 – q, Football)} uw(p,q) = 2pq + (1 – p)(1 – q) = 3pq – p – q + 1 We can write uw(p,q) For any fixed p, uw(p,q) is linear in q instead of uw(sw , sh ) • e.g., if p = ½, then uw(½,q) = ½ q + ½ 0 ≤ q ≤ 1, so the min must be at q = 0 or q = 1 • e.g., minq (½ q + ½) is at q = 0 minq uw(p,q) = min (uw(p,0), uw(p,1)) = min (1 – p, 2p) Maxmin Strategies Also called maximin A maxmin strategy for agent i A strategy s1 that makes i’s worst-case expected utility as high as possible: argmaxmin ui (si,s-i ) si s-i This isn’t necessarily unique Often it is mixed Agent i’s maxmin value, or security level, is the maxmin strategy’s worst-case expected utility: maxmin ui (si,s-i ) si s-i For 2 players it simplifies to max min u1s1, s2 s1 s2 Example Wife’s and husband’s strategies -
Explanatory Notes for the Tectonic Map of the Circum-Pacific Region Southwest Quadrant
U.S. DEPARTMENT OF THE INTERIOR TO ACCOMPANY MAP CP-37 U.S. GEOLOGICAL SURVEY Explanatory Notes for the Tectonic Map of the Circum-Pacific Region Southwest Quadrant 1:10,000,000 ICIRCUM-PACIFIC i • \ COUNCIL AND MINERAL RESOURCES 1991 CIRCUM-PACIFIC COUNCIL FOR ENERGY AND MINERAL RESOURCES Michel T. Halbouty, Chairman CIRCUM-PACIFIC MAP PROJECT John A. Reinemund, Director George Gryc, General Chairman Erwin Scheibner, Advisor, Tectonic Map Series EXPLANATORY NOTES FOR THE TECTONIC MAP OF THE CIRCUM-PACIFIC REGION SOUTHWEST QUADRANT 1:10,000,000 By Erwin Scheibner, Geological Survey of New South Wales, Sydney, 2001 N.S.W., Australia Tadashi Sato, Institute of Geoscience, University of Tsukuba, Ibaraki 305, Japan H. Frederick Doutch, Bureau of Mineral Resources, Canberra, A.C.T. 2601, Australia Warren O. Addicott, U.S. Geological Survey, Menlo Park, California 94025, U.S.A. M. J. Terman, U.S. Geological Survey, Reston, Virginia 22092, U.S.A. George W. Moore, Department of Geosciences, Oregon State University, Corvallis, Oregon 97331, U.S.A. 1991 Explanatory Notes to Supplement the TECTONIC MAP OF THE CIRCUM-PACIFTC REGION SOUTHWEST QUADRANT W. D. Palfreyman, Chairman Southwest Quadrant Panel CHIEF COMPILERS AND TECTONIC INTERPRETATIONS E. Scheibner, Geological Survey of New South Wales, Sydney, N.S.W. 2001 Australia T. Sato, Institute of Geosciences, University of Tsukuba, Ibaraki 305, Japan C. Craddock, Department of Geology and Geophysics, University of Wisconsin-Madison, Madison, Wisconsin 53706, U.S.A. TECTONIC ELEMENTS AND STRUCTURAL DATA AND INTERPRETATIONS J.-M. Auzende et al, Institut Francais de Recherche pour 1'Exploitacion de la Mer (IFREMER), Centre de Brest, B. -
Oligopolistic Competition
Lecture 3: Oligopolistic competition EC 105. Industrial Organization Mattt Shum HSS, California Institute of Technology EC 105. Industrial Organization (Mattt Shum HSS,Lecture California 3: Oligopolistic Institute of competition Technology) 1 / 38 Oligopoly Models Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j's actions affect firm i's profits PC: firms are small, so no single firm’s actions affect other firms’ profits Monopoly: only one firm EC 105. Industrial Organization (Mattt Shum HSS,Lecture California 3: Oligopolistic Institute of competition Technology) 2 / 38 Oligopoly Models Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j's actions affect firm i's profits PC: firms are small, so no single firm’s actions affect other firms’ profits Monopoly: only one firm EC 105. Industrial Organization (Mattt Shum HSS,Lecture California 3: Oligopolistic Institute of competition Technology) 2 / 38 Oligopoly Models Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j's actions affect firm i's profits PC: firms are small, so no single firm’s actions affect other firms’ profits Monopoly: only one firm EC 105. Industrial Organization (Mattt Shum HSS,Lecture California 3: Oligopolistic Institute of competition Technology) 2 / 38 Oligopoly Models Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j's actions affect firm i's profits PC: firms are small, so no single firm’s actions affect other firms’ profits Monopoly: only one firm EC 105. -
BAYESIAN EQUILIBRIUM the Games Like Matching Pennies And
BAYESIAN EQUILIBRIUM MICHAEL PETERS The games like matching pennies and prisoner’s dilemma that form the core of most undergrad game theory courses are games in which players know each others’ preferences. Notions like iterated deletion of dominated strategies, and rationalizability actually go further in that they exploit the idea that each player puts him or herself in the shoes of other players and imagines that the others do the same thing. Games and reasoning like this apply to situations in which the preferences of the players are common knowledge. When we want to refer to situations like this, we usually say that we are interested in games of complete information. Most of the situations we study in economics aren’t really like this since we are never really sure of the motivation of the players we are dealing with. A bidder on eBay, for example, doesn’t know whether other bidders are interested in a good they would like to bid on. Once a bidder sees that another has submit a bid, he isn’t sure exactly how much the good is worth to the other bidder. When players in a game don’t know the preferences of other players, we say a game has incomplete information. If you have dealt with these things before, you may have learned the term asymmetric information. This term is less descriptive of the problem and should probably be avoided. The way we deal with incomplete information in economics is to use the ap- proach originally described by Savage in 1954. If you have taken a decision theory course, you probably learned this approach by studying Anscombe and Aumann (last session), while the approach we use in game theory is usually attributed to Harsanyi. -
A Beautiful Math : John Nash, Game Theory, and the Modern Quest for a Code of Nature / Tom Siegfried
A BEAUTIFULA BEAUTIFUL MATH MATH JOHN NASH, GAME THEORY, AND THE MODERN QUEST FOR A CODE OF NATURE TOM SIEGFRIED JOSEPH HENRY PRESS Washington, D.C. Joseph Henry Press • 500 Fifth Street, NW • Washington, DC 20001 The Joseph Henry Press, an imprint of the National Academies Press, was created with the goal of making books on science, technology, and health more widely available to professionals and the public. Joseph Henry was one of the founders of the National Academy of Sciences and a leader in early Ameri- can science. Any opinions, findings, conclusions, or recommendations expressed in this volume are those of the author and do not necessarily reflect the views of the National Academy of Sciences or its affiliated institutions. Library of Congress Cataloging-in-Publication Data Siegfried, Tom, 1950- A beautiful math : John Nash, game theory, and the modern quest for a code of nature / Tom Siegfried. — 1st ed. p. cm. Includes bibliographical references and index. ISBN 0-309-10192-1 (hardback) — ISBN 0-309-65928-0 (pdfs) 1. Game theory. I. Title. QA269.S574 2006 519.3—dc22 2006012394 Copyright 2006 by Tom Siegfried. All rights reserved. Printed in the United States of America. Preface Shortly after 9/11, a Russian scientist named Dmitri Gusev pro- posed an explanation for the origin of the name Al Qaeda. He suggested that the terrorist organization took its name from Isaac Asimov’s famous 1950s science fiction novels known as the Foun- dation Trilogy. After all, he reasoned, the Arabic word “qaeda” means something like “base” or “foundation.” And the first novel in Asimov’s trilogy, Foundation, apparently was titled “al-Qaida” in an Arabic translation. -
A Companion for Aspirant Air Warriors a Handbook for Personal Professional Study
A Companion for Aspirant Air Warriors A Handbook for Personal Professional Study DAVID R. METS, PHD Air University Press Air Force Research Institute Maxwell Air Force Base, Alabama May 2010 Muir S. Fairchild Research Information Center Cataloging Data Mets, David R. A companion for aspirant air warriors : a handbook for personal professional study / David R. Mets. p. ; cm. Includes bibliographical references. ISBN 978-1-58566-206-7 1. Air power—History. 2. Aeronautics, Military—History. 3. Aeronautics, Military—Biography. 4. Military art and science—History. I. Title. 358.4—dc22 Disclaimer Opinions, conclusions, and recommendations expressed or implied within are solely those of the author and do not necessarily represent the views of Air University, the Air Force Research Institute, the United States Air Force, the Department of Defense, or any other US government agency. Cleared for public release: distribution unlimited. Air University Press 155 N. Twining Street Maxwell AFB, AL 36112-6026 http://aupress.au.af.mil ii Dedicated to Maj Lilburn Stow, USAF, and his C-130 crew, who lost their lives over the A Shau Valley, Vietnam, 26 April 1968, while supporting their Army countrymen on the ground Contents Chapter Page DISCLAIMER . ii DEDICATION . iii FOREWORD . vii ABOUT THE AUTHOR . ix ACKNOWLEDGMENTS . xi INTRODUCTION . 1 1 THE INFANCY OF AIRPOWER. 3 2 NAVAL AVIATION . 7 3 AIRPOWER IN WORLD WAR I . 11 4 LAYING THE INTELLECTUAL FOUNDATIONS, 1919–1931 . 15 5 AN AGE OF INNOVATION, 1931–1941 . 19 6 NAVAL AVIATION BETWEEN THE WARS . 23 7 WORLD WAR II: THE RISE OF THE LUFTWAFFE . 29 8 WORLD WAR II: EUROPE—THE STRATEGIC BOMBING DIMENSION .