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Repeated Games
6.254 : Game Theory with Engineering Applications Lecture 15: Repeated Games Asu Ozdaglar MIT April 1, 2010 1 Game Theory: Lecture 15 Introduction Outline Repeated Games (perfect monitoring) The problem of cooperation Finitely-repeated prisoner's dilemma Infinitely-repeated games and cooperation Folk Theorems Reference: Fudenberg and Tirole, Section 5.1. 2 Game Theory: Lecture 15 Introduction Prisoners' Dilemma How to sustain cooperation in the society? Recall the prisoners' dilemma, which is the canonical game for understanding incentives for defecting instead of cooperating. Cooperate Defect Cooperate 1, 1 −1, 2 Defect 2, −1 0, 0 Recall that the strategy profile (D, D) is the unique NE. In fact, D strictly dominates C and thus (D, D) is the dominant equilibrium. In society, we have many situations of this form, but we often observe some amount of cooperation. Why? 3 Game Theory: Lecture 15 Introduction Repeated Games In many strategic situations, players interact repeatedly over time. Perhaps repetition of the same game might foster cooperation. By repeated games, we refer to a situation in which the same stage game (strategic form game) is played at each date for some duration of T periods. Such games are also sometimes called \supergames". We will assume that overall payoff is the sum of discounted payoffs at each stage. Future payoffs are discounted and are thus less valuable (e.g., money and the future is less valuable than money now because of positive interest rates; consumption in the future is less valuable than consumption now because of time preference). We will see in this lecture how repeated play of the same strategic game introduces new (desirable) equilibria by allowing players to condition their actions on the way their opponents played in the previous periods. -
On Dynamic Games with Randomly Arriving Players Pierre Bernhard, Marc Deschamps
On Dynamic Games with Randomly Arriving Players Pierre Bernhard, Marc Deschamps To cite this version: Pierre Bernhard, Marc Deschamps. On Dynamic Games with Randomly Arriving Players. 2015. hal-01377916 HAL Id: hal-01377916 https://hal.archives-ouvertes.fr/hal-01377916 Preprint submitted on 7 Oct 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. O n dynamic games with randomly arriving players Pierre Bernhard et Marc Deschamps September 2015 Working paper No. 2015 – 13 30, avenue de l’Observatoire 25009 Besançon France http://crese.univ-fcomte.fr/ CRESE The views expressed are those of the authors and do not necessarily reflect those of CRESE. On dynamic games with randomly arriving players Pierre Bernhard∗ and Marc Deschamps† September 24, 2015 Abstract We consider a dynamic game where additional players (assumed identi- cal, even if there will be a mild departure from that hypothesis) join the game randomly according to a Bernoulli process. The problem solved here is that of computing their expected payoff as a function of time and the number of players present when they arrive, if the strategies are given. We consider both a finite horizon game and an infinite horizon, discounted game. -
California Institute of Technology
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Caltech Authors - Main DIVISION OF THE HUM ANITIES AND SO CI AL SCIENCES CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 IMPLEMENTATION THEORY Thomas R. Palfrey � < a: 0 1891 u. "')/,.. () SOCIAL SCIENCE WORKING PAPER 912 September 1995 Implementation Theory Thomas R. Palfrey Abstract This surveys the branch of implementation theory initiated by Maskin (1977). Results for both complete and incomplete information environments are covered. JEL classification numbers: 025, 026 Key words: Implementation Theory, Mechanism Design, Game Theory, Social Choice Implementation Theory* Thomas R. Palfrey 1 Introduction Implementation theory is an area of research in economic theory that rigorously investi gates the correspondence between normative goals and institutions designed to achieve {implement) those goals. More precisely, given a normative goal or welfare criterion for a particular class of allocation pro blems (or domain of environments) it formally char acterizes organizational mechanisms that will guarantee outcomes consistent with that goal, assuming the outcomes of any such mechanism arise from some specification of equilibrium behavior. The approaches to this problem to date lie in the general domain of game theory because, as a matter of definition in the implementation theory litera ture, an institution is modelled as a mechanism, which is essentially a non-cooperative game. Moreover, the specific models of equilibrium behavior -
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 .................................. -
Finitely Repeated Games
Repeated games 1: Finite repetition Universidad Carlos III de Madrid 1 Finitely repeated games • A finitely repeated game is a dynamic game in which a simultaneous game (the stage game) is played finitely many times, and the result of each stage is observed before the next one is played. • Example: Play the prisoners’ dilemma several times. The stage game is the simultaneous prisoners’ dilemma game. 2 Results • If the stage game (the simultaneous game) has only one NE the repeated game has only one SPNE: In the SPNE players’ play the strategies in the NE in each stage. • If the stage game has 2 or more NE, one can find a SPNE where, at some stage, players play a strategy that is not part of a NE of the stage game. 3 The prisoners’ dilemma repeated twice • Two players play the same simultaneous game twice, at ! = 1 and at ! = 2. • After the first time the game is played (after ! = 1) the result is observed before playing the second time. • The payoff in the repeated game is the sum of the payoffs in each stage (! = 1, ! = 2) • Which is the SPNE? Player 2 D C D 1 , 1 5 , 0 Player 1 C 0 , 5 4 , 4 4 The prisoners’ dilemma repeated twice Information sets? Strategies? 1 .1 5 for each player 2" for each player D C E.g.: (C, D, D, C, C) Subgames? 2.1 5 D C D C .2 1.3 1.5 1 1.4 D C D C D C D C 2.2 2.3 2 .4 2.5 D C D C D C D C D C D C D C D C 1+1 1+5 1+0 1+4 5+1 5+5 5+0 5+4 0+1 0+5 0+0 0+4 4+1 4+5 4+0 4+4 1+1 1+0 1+5 1+4 0+1 0+0 0+5 0+4 5+1 5+0 5+5 5+4 4+1 4+0 4+5 4+4 The prisoners’ dilemma repeated twice Let’s find the NE in the subgames. -
A Game Theory Interpretation for Multiple Access in Cognitive Radio
1 A Game Theory Interpretation for Multiple Access in Cognitive Radio Networks with Random Number of Secondary Users Oscar Filio Rodriguez, Student Member, IEEE, Serguei Primak, Member, IEEE,Valeri Kontorovich, Fellow, IEEE, Abdallah Shami, Member, IEEE Abstract—In this paper a new multiple access algorithm for (MAC) problem has been analyzed using game theory tools cognitive radio networks based on game theory is presented. in previous literature such as [1] [2] [8] [9]. In [9] a game We address the problem of a multiple access system where the theory model is presented as a starting point for the Distributed number of users and their types are unknown. In order to do this, the framework is modelled as a non-cooperative Poisson game in Coordination Function (DCF) mechanism in IEEE 802.11. In which all the players are unaware of the total number of devices the DCF there are no base stations or access points which participating (population uncertainty). We propose a scheme control the access to the channel, hence all nodes transmit their where failed attempts to transmit (collisions) are penalized. In data frames in a competitive manner. However, as pointed out terms of this, we calculate the optimum penalization in mixed in [9], the proposed DCF game does not consider how the strategies. The proposed scheme conveys to a Nash equilibrium where a maximum in the possible throughput is achieved. different types of traffic can affect the sum throughput of the system. Moreover, in [10] it is assumed that the total number of players is known in order to evaluate the performance I. -
Norms, Repeated Games, and the Role of Law
Norms, Repeated Games, and the Role of Law Paul G. Mahoneyt & Chris William Sanchiricot TABLE OF CONTENTS Introduction ............................................................................................ 1283 I. Repeated Games, Norms, and the Third-Party Enforcement P rob lem ........................................................................................... 12 88 II. B eyond T it-for-Tat .......................................................................... 1291 A. Tit-for-Tat for More Than Two ................................................ 1291 B. The Trouble with Tit-for-Tat, However Defined ...................... 1292 1. Tw o-Player Tit-for-Tat ....................................................... 1293 2. M any-Player Tit-for-Tat ..................................................... 1294 III. An Improved Model of Third-Party Enforcement: "D ef-for-D ev". ................................................................................ 1295 A . D ef-for-D ev's Sim plicity .......................................................... 1297 B. Def-for-Dev's Credible Enforceability ..................................... 1297 C. Other Attractive Properties of Def-for-Dev .............................. 1298 IV. The Self-Contradictory Nature of Self-Enforcement ....................... 1299 A. The Counterfactual Problem ..................................................... 1300 B. Implications for the Self-Enforceability of Norms ................... 1301 C. Game-Theoretic Workarounds ................................................ -
Strategic Approval Voting in a Large Electorate
Strategic approval voting in a large electorate Jean-François Laslier∗ Laboratoire d’Économétrie, École Polytechnique 1 rue Descartes, 75005 Paris [email protected] June 16, 2006 Abstract The paper considers approval voting for a large population of vot- ers. It is proven that, based on statistical information about candidate scores, rational voters vote sincerly and according to a simple behav- ioral rule. It is also proven that if a Condorcet-winner exists, this can- didate is elected. ∗Thanks to Steve Brams, Nicolas Gravel, François Maniquet, Remzi Sanver and Karine Van der Straeten for their remarks. Errors are mine. 1 1Introduction Approval Voting (AV) is the method of election according to which a voter can vote for as many candidates as she wishes, the elected candidate being the one who receives the most votes. In this paper two results are established about AV in the case of a large electorate when voters behave strategically: the sincerity of individual behavior (rational voters choose sincere ballots) and the Condorcet-consistency of the choice function defined by approval voting (whenever a Condorcet winner exists, it is the outcome of the vote). Under AV, a ballot is a subset of the set candidates. A ballot is said to be sincere, for a voter, if it shows no “hole” with respect to the voter’s preference ranking; if the voter sincerely approves of a candidate x she also approves of any candidate she prefers to x. Therefore, under AV, a voter has several sincere ballots at her disposal: she can vote for her most preferred candidate, or for her two, or three, or more most preferred candidates.1 It has been found by Brams and Fishburn (1983) that a voter should always vote for her most-preferred candidate and never vote for her least- preferred one. -
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]). -
DP Cover1512 Kalai.Cdr
Discussion Paper #1512 June 16, 2010 A A Cooperative Value for Bayesian Games Key words: cooperative game theory, non- cooperative game theory, bargaining, min-max value JEL classification: C70, C71, C72, C78 Adam Kalai Microsoft Research Ehud Kalai Northwestern University www.kellogg.northwestern.edu/research/math everone Hall Evanston, IL 60208-2014 US CMS-EMS oad 580 L The Center for Mathematical Studies in Economics & Management Sciences Northwestern University 2001 Sheridan R A COOPERATIVE VALUE FOR BAYESIAN GAMES ADAM TAUMAN KALAI∗ AND EHUD KALAIy;x Abstract. Selfish, strategic players may benefit from cooperation, provided they reach agreement. It is therefore important to construct mechanisms that facilitate such cooperation, especially in the case of asymmetric private information. The two major issues are: (1) singling out a fair and efficient outcome among the many individually rational possibilities in a strategic game, and (2) establishing a play protocol under which strategic players may achieve this outcome. The paper presents a general solution for two-person Bayesian games with monetary payoffs, under a strong revealed-payoff assumption. The proposed solution builds upon earlier concepts in game theory. It coincides with the von Neumann minmax value on the class of zero sum games and with the major solution concepts to the Nash Bargaining Problem. Moreover, the solution is based on a simple decomposition of every game into cooperative and competitive components, which is easy to compute. 1. Introduction Selfish players in strategic games benefit from cooperation, provided they come to mutually beneficial agreements. In the case of asymmetric private information, the benefits may be even greater, but avoiding strategic manipulations is more subtle. -
(Eds.) Foundations in Microeconomic Theory Hugo F
Matthew O. Jackson • Andrew McLennan (Eds.) Foundations in Microeconomic Theory Hugo F. Sonnenschein Matthew O. Jackson • Andrew McLennan (Eds.) Foundations in Microeconomic Theory A Volume in Honor of Hugo F. Sonnenschein ^ Springer Professor Matthew O. Jackson Professor Andrew McLennan Stanford University School of Economics Department of Economics University of Queensland Stanford, CA 94305-6072 Rm. 520, Colin Clark Building USA The University of Queensland [email protected] NSW 4072 Australia [email protected] ISBN: 978-3-540-74056-8 e-ISBN: 978-3"540-74057-5 Library of Congress Control Number: 2007941253 © 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper 987654321 spnnger.com Contents Introduction 1 A Brief Biographical Sketch of Hugo F. Sonnenschein 5 1 Kevin C. Sontheimer on Hugo F.