Chronology of Game Theory
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Learning and Equilibrium
Learning and Equilibrium Drew Fudenberg1 and David K. Levine2 1Department of Economics, Harvard University, Cambridge, Massachusetts; email: [email protected] 2Department of Economics, Washington University of St. Louis, St. Louis, Missouri; email: [email protected] Annu. Rev. Econ. 2009. 1:385–419 Key Words First published online as a Review in Advance on nonequilibrium dynamics, bounded rationality, Nash equilibrium, June 11, 2009 self-confirming equilibrium The Annual Review of Economics is online at by 140.247.212.190 on 09/04/09. For personal use only. econ.annualreviews.org Abstract This article’s doi: The theory of learning in games explores how, which, and what 10.1146/annurev.economics.050708.142930 kind of equilibria might arise as a consequence of a long-run non- Annu. Rev. Econ. 2009.1:385-420. Downloaded from arjournals.annualreviews.org Copyright © 2009 by Annual Reviews. equilibrium process of learning, adaptation, and/or imitation. If All rights reserved agents’ strategies are completely observed at the end of each round 1941-1383/09/0904-0385$20.00 (and agents are randomly matched with a series of anonymous opponents), fairly simple rules perform well in terms of the agent’s worst-case payoffs, and also guarantee that any steady state of the system must correspond to an equilibrium. If players do not ob- serve the strategies chosen by their opponents (as in extensive-form games), then learning is consistent with steady states that are not Nash equilibria because players can maintain incorrect beliefs about off-path play. Beliefs can also be incorrect because of cogni- tive limitations and systematic inferential errors. -
Game Theory Lecture Notes
Game Theory: Penn State Math 486 Lecture Notes Version 2.1.1 Christopher Griffin « 2010-2021 Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License With Major Contributions By: James Fan George Kesidis and Other Contributions By: Arlan Stutler Sarthak Shah Contents List of Figuresv Preface xi 1. Using These Notes xi 2. An Overview of Game Theory xi Chapter 1. Probability Theory and Games Against the House1 1. Probability1 2. Random Variables and Expected Values6 3. Conditional Probability8 4. The Monty Hall Problem 11 Chapter 2. Game Trees and Extensive Form 15 1. Graphs and Trees 15 2. Game Trees with Complete Information and No Chance 18 3. Game Trees with Incomplete Information 22 4. Games of Chance 24 5. Pay-off Functions and Equilibria 26 Chapter 3. Normal and Strategic Form Games and Matrices 37 1. Normal and Strategic Form 37 2. Strategic Form Games 38 3. Review of Basic Matrix Properties 40 4. Special Matrices and Vectors 42 5. Strategy Vectors and Matrix Games 43 Chapter 4. Saddle Points, Mixed Strategies and the Minimax Theorem 45 1. Saddle Points 45 2. Zero-Sum Games without Saddle Points 48 3. Mixed Strategies 50 4. Mixed Strategies in Matrix Games 53 5. Dominated Strategies and Nash Equilibria 54 6. The Minimax Theorem 59 7. Finding Nash Equilibria in Simple Games 64 8. A Note on Nash Equilibria in General 66 Chapter 5. An Introduction to Optimization and the Karush-Kuhn-Tucker Conditions 69 1. A General Maximization Formulation 70 2. Some Geometry for Optimization 72 3. -
Uniqueness and Symmetry in Bargaining Theories of Justice
Philos Stud DOI 10.1007/s11098-013-0121-y Uniqueness and symmetry in bargaining theories of justice John Thrasher Ó Springer Science+Business Media Dordrecht 2013 Abstract For contractarians, justice is the result of a rational bargain. The goal is to show that the rules of justice are consistent with rationality. The two most important bargaining theories of justice are David Gauthier’s and those that use the Nash’s bargaining solution. I argue that both of these approaches are fatally undermined by their reliance on a symmetry condition. Symmetry is a substantive constraint, not an implication of rationality. I argue that using symmetry to generate uniqueness undermines the goal of bargaining theories of justice. Keywords David Gauthier Á John Nash Á John Harsanyi Á Thomas Schelling Á Bargaining Á Symmetry Throughout the last century and into this one, many philosophers modeled justice as a bargaining problem between rational agents. Even those who did not explicitly use a bargaining problem as their model, most notably Rawls, incorporated many of the concepts and techniques from bargaining theories into their understanding of what a theory of justice should look like. This allowed them to use the powerful tools of game theory to justify their various theories of distributive justice. The debates between partisans of different theories of distributive justice has tended to be over the respective benefits of each particular bargaining solution and whether or not the solution to the bargaining problem matches our pre-theoretical intuitions about justice. There is, however, a more serious problem that has effectively been ignored since economists originally J. -
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. -
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. -
Improving Fictitious Play Reinforcement Learning with Expanding Models
Improving Fictitious Play Reinforcement Learning with Expanding Models Rong-Jun Qin1;2, Jing-Cheng Pang1, Yang Yu1;y 1National Key Laboratory for Novel Software Technology, Nanjing University, China 2Polixir emails: [email protected], [email protected], [email protected]. yTo whom correspondence should be addressed Abstract Fictitious play with reinforcement learning is a general and effective framework for zero- sum games. However, using the current deep neural network models, the implementation of fictitious play faces crucial challenges. Neural network model training employs gradi- ent descent approaches to update all connection weights, and thus is easy to forget the old opponents after training to beat the new opponents. Existing approaches often maintain a pool of historical policy models to avoid the forgetting. However, learning to beat a pool in stochastic games, i.e., a wide distribution over policy models, is either sample-consuming or insufficient to exploit all models with limited amount of samples. In this paper, we pro- pose a learning process with neural fictitious play to alleviate the above issues. We train a single model as our policy model, which consists of sub-models and a selector. Everytime facing a new opponent, the model is expanded by adding a new sub-model, where only the new sub-model is updated instead of the whole model. At the same time, the selector is also updated to mix up the new sub-model with the previous ones at the state-level, so that the model is maintained as a behavior strategy instead of a wide distribution over policy models. -
Approximation Guarantees for Fictitious Play
Approximation Guarantees for Fictitious Play Vincent Conitzer Abstract— Fictitious play is a simple, well-known, and often- significant progress has been made in the computation of used algorithm for playing (and, especially, learning to play) approximate Nash equilibria. It has been shown that for any games. However, in general it does not converge to equilibrium; ǫ, there is an ǫ-approximate equilibrium with support size even when it does, we may not be able to run it to convergence. 2 Still, we may obtain an approximate equilibrium. In this O((log n)/ǫ ) [1], [22], so that we can find approximate paper, we study the approximation properties that fictitious equilibria by searching over all of these supports. More play obtains when it is run for a limited number of rounds. recently, Daskalakis et al. gave a very simple algorithm We show that if both players randomize uniformly over their for finding a 1/2-approximate Nash equilibrium [12]; we actions in the first r rounds of fictitious play, then the result will discuss this algorithm shortly. Feder et al. then showed is an ǫ-equilibrium, where ǫ = (r + 1)/(2r). (Since we are examining only a constant number of pure strategies, we know a lower bound on the size of the supports that must be that ǫ < 1/2 is impossible, due to a result of Feder et al.) We considered to be guaranteed to find an approximate equi- show that this bound is tight in the worst case; however, with an librium [16]; in particular, supports of constant sizes can experiment on random games, we illustrate that fictitious play give at best a 1/2-approximate Nash equilibrium. -
Game Theoretic Interaction and Decision: a Quantum Analysis
games Article Game Theoretic Interaction and Decision: A Quantum Analysis Ulrich Faigle 1 and Michel Grabisch 2,* 1 Mathematisches Institut, Universität zu Köln, Weyertal 80, 50931 Köln, Germany; [email protected] 2 Paris School of Economics, University of Paris I, 106-112, Bd. de l’Hôpital, 75013 Paris, France * Correspondence: [email protected]; Tel.: +33-144-07-8744 Received: 12 July 2017; Accepted: 21 October 2017; Published: 6 November 2017 Abstract: An interaction system has a finite set of agents that interact pairwise, depending on the current state of the system. Symmetric decomposition of the matrix of interaction coefficients yields the representation of states by self-adjoint matrices and hence a spectral representation. As a result, cooperation systems, decision systems and quantum systems all become visible as manifestations of special interaction systems. The treatment of the theory is purely mathematical and does not require any special knowledge of physics. It is shown how standard notions in cooperative game theory arise naturally in this context. In particular, states of general interaction systems are seen to arise as linear superpositions of pure quantum states and Fourier transformation to become meaningful. Moreover, quantum games fall into this framework. Finally, a theory of Markov evolution of interaction states is presented that generalizes classical homogeneous Markov chains to the present context. Keywords: cooperative game; decision system; evolution; Fourier transform; interaction system; measurement; quantum game 1. Introduction In an interaction system, economic (or general physical) agents interact pairwise, but do not necessarily cooperate towards a common goal. However, this model arises as a natural generalization of the model of cooperative TU games, for which already Owen [1] introduced the co-value as an assessment of the pairwise interaction of two cooperating players1. -
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. -
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. -
Viable Nash Equilibria: Formation and Defection (Feb 2020)
VIABLE NASH EQUILIBRIA: FORMATION AND DEFECTION (FEB 2020) EHUD KALAI In memory of John Nash Abstract. To be credible, economic analysis should restrict itself to the use of only those Nash equilibria that are viable. To assess the viability of an equilibrium , I study simple dual indices: a formation index, F (), that speci…es the number of loyalists needed to form ; and a defection index, D(), that speci…es the number of defectors that can sustain. Surprisingly, these simple indices (1) predict the performance of Nash equilibria in social systems and lab experiments, and (2) uncover new prop- erties of Nash equilibria and stability issues that have so far eluded game theory re…nements. JEL Classi…cation Codes: C0, C7, D5, D9. 1. Overview Current economic analysis often relies on the notion of a Nash equilibrium. Yet there are mixed opinions about the viability of this notion. On the one hand, many equilibria, referred to as viable in this paper, play critical roles in functioning social systems and perform well in lab and …eld experiments. Date: March 9, 2018, this version Feb 24, 2020. Key words and phrases. Normal form games, Nash equilibrium, Stability, Fault tolerance, Behavioral Economics. The author thanks the following people for helpful conversations: Nemanja Antic, Sunil Chopra, Vince Crawford, K…r Eliaz, Drew Fudenberg, Ronen Gradwohl, Yingni Guo, Adam Kalai, Fern Kalai, Martin Lariviere, Eric Maskin, Rosemarie Nagel, Andy Postlewaite, Larry Samuelson, David Schmeidler, James Schummer, Eran Shmaya, Joel Sobel, and Peyton Young; and seminar participants at the universities of the Basque Country, Oxford, Tel Aviv, Yale, Stanford, Berkeley, Stony Brook, Bar Ilan, the Technion, and the Hebrew University. -
Updating Beliefs When Evidence Is Open to Interpretation: Implications for Bias and Polarization∗
Updating Beliefs when Evidence is Open to Interpretation: Implications for Bias and Polarization∗ Roland G. Fryer, Jr., Philipp Harms, and Matthew O. Jacksony This Version: October 2017 Forthcoming in Journal of the European Economic Association Abstract We introduce a model in which agents observe signals about the state of the world, some of which are open to interpretation. Our decision makers first interpret each signal and then form a posterior on the sequence of interpreted signals. This `double updating' leads to confirmation bias and can lead agents who observe the same information to polarize. We explore the model's predictions in an on-line experiment in which individuals interpret research summaries about climate change and the death penalty. Consistent with the model, there is a significant relationship between an individual's prior and their interpretation of the summaries; and - even more striking - over half of the subjects exhibit polarizing behavior. JEL classification numbers: D10, D80, J15, J71, I30 Keywords: beliefs, polarization, learning, updating, Bayesian updating, biases, dis- crimination, decision making ∗We are grateful to Sandro Ambuehl, Ken Arrow, Tanaya Devi, Juan Dubra, Jon Eguia, Ben Golub, Richard Holden, Lawrence Katz, Peter Klibanoff, Scott Kominers, Franziska Michor, Giovanni Parmigiani, Matthew Rabin, Andrei Shleifer, and Joshua Schwartzein for helpful comments and suggestions. Adriano Fernandes provided exceptional research assistance. yFryer is at the Department of Economics, Harvard University, and the NBER, ([email protected]); Jackson is at the Department of Economics, Stanford University, the Santa Fe Institute, and CIFAR (jack- [email protected]); and Harms is at the Department of Mathematical Stochastics, University of Freiburg, ([email protected]).