An Introductory Course on Mathematical Game Theory
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Game Theory 2: Extensive-Form Games and Subgame Perfection
Game Theory 2: Extensive-Form Games and Subgame Perfection 1 / 26 Dynamics in Games How should we think of strategic interactions that occur in sequence? Who moves when? And what can they do at different points in time? How do people react to different histories? 2 / 26 Modeling Games with Dynamics Players Player function I Who moves when Terminal histories I Possible paths through the game Preferences over terminal histories 3 / 26 Strategies A strategy is a complete contingent plan Player i's strategy specifies her action choice at each point at which she could be called on to make a choice 4 / 26 An Example: International Crises Two countries (A and B) are competing over a piece of land that B occupies Country A decides whether to make a demand If Country A makes a demand, B can either acquiesce or fight a war If A does not make a demand, B keeps land (game ends) A's best outcome is Demand followed by Acquiesce, worst outcome is Demand and War B's best outcome is No Demand and worst outcome is Demand and War 5 / 26 An Example: International Crises A can choose: Demand (D) or No Demand (ND) B can choose: Fight a war (W ) or Acquiesce (A) Preferences uA(D; A) = 3 > uA(ND; A) = uA(ND; W ) = 2 > uA(D; W ) = 1 uB(ND; A) = uB(ND; W ) = 3 > uB(D; A) = 2 > uB(D; W ) = 1 How can we represent this scenario as a game (in strategic form)? 6 / 26 International Crisis Game: NE Country B WA D 1; 1 3X; 2X Country A ND 2X; 3X 2; 3X I Is there something funny here? I Is there something funny here? I Specifically, (ND; W )? I Is there something funny here? -
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. -
Best Experienced Payoff Dynamics and Cooperation in the Centipede Game
Theoretical Economics 14 (2019), 1347–1385 1555-7561/20191347 Best experienced payoff dynamics and cooperation in the centipede game William H. Sandholm Department of Economics, University of Wisconsin Segismundo S. Izquierdo BioEcoUva, Department of Industrial Organization, Universidad de Valladolid Luis R. Izquierdo Department of Civil Engineering, Universidad de Burgos We study population game dynamics under which each revising agent tests each of his strategies a fixed number of times, with each play of each strategy being against a newly drawn opponent, and chooses the strategy whose total payoff was highest. In the centipede game, these best experienced payoff dynamics lead to co- operative play. When strategies are tested once, play at the almost globally stable state is concentrated on the last few nodes of the game, with the proportions of agents playing each strategy being largely independent of the length of the game. Testing strategies many times leads to cyclical play. Keywords. Evolutionary game theory, backward induction, centipede game, computational algebra. JEL classification. C72, C73. 1. Introduction The discrepancy between the conclusions of backward induction reasoning and ob- served behavior in certain canonical extensive form games is a basic puzzle of game the- ory. The centipede game (Rosenthal (1981)), the finitely repeated prisoner’s dilemma, and related examples can be viewed as models of relationships in which each partic- ipant has repeated opportunities to take costly actions that benefit his partner and in which there is a commonly known date at which the interaction will end. Experimen- tal and anecdotal evidence suggests that cooperative behavior may persist until close to William H. -
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. -
Draft Abstract
SN 1327-8231 ECONOMICS, ECOLOGY AND THE ENVIRONMENT WORKING PAPER NO. 38 Neglected Features of the Safe Minimum Standard: Socio-economic and Institutional Dimensions by IRMI SEIDL AND CLEM TISDELL March 2000 THE UNIVERSITY OF QUEENSLAND ISSN 1327-8231 WORKING PAPERS ON ECONOMICS, ECOLOGY AND THE ENVIRONMENT Working Paper No. 38 Neglected Features of the Safe Minimum Standard: Socio-Economic and Institutional Dimensions by Irmi Seidl1 and Clem Tisdell2 March 2000 © All rights reserved 1 Irmi Seidl, Institut fuer Umweltwissenschaften, University of Zuerich, Winterthurerstr. 190, CH-8057, Zuerich, Switzerland. Email: [email protected] 2 School of Economics, The University of Queensland, Brisbane QLD 4072, Australia Email: [email protected] WORKING PAPERS IN THE SERIES, Economics, Ecology and the Environment are published by the School of Economics, University of Queensland, 4072, Australia, as follow up to the Australian Centre for International Agricultural Research Project 40 of which Professor Clem Tisdell was the Project Leader. Views expressed in these working papers are those of their authors and not necessarily of any of the organisations associated with the Project. They should not be reproduced in whole or in part without the written permission of the Project Leader. It is planned to publish contributions to this series over the next few years. Research for ACIAR project 40, Economic impact and rural adjustments to nature conservation (biodiversity) programmes: A case study of Xishuangbanna Dai Autonomous Prefecture, Yunnan, China was sponsored by the Australian Centre for International Agricultural Research (ACIAR), GPO Box 1571, Canberra, ACT, 2601, Australia. The research for ACIAR project 40 has led in part, to the research being carried out in this current series. -
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 .................................. -
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. -
OR-M5-Ktunotes.In .Pdf
CHAPTER – 12 Decision Theory 12.1. INTRODUCTION The decisions are classified according to the degree of certainty as deterministic models, where the manager assumes complete certainty and each strategy results in a unique payoff, and Probabilistic models, where each strategy leads to more than one payofs and the manager attaches a probability measure to these payoffs. The scale of assumed certainty can range from complete certainty to complete uncertainty hence one can think of decision making under certainty (DMUC) and decision making under uncertainty (DMUU) on the two extreme points on a scale. The region that falls between these extreme points corresponds to the concept of probabilistic models, and referred as decision-making under risk (DMUR). Hence we can say that most of the decision making problems fall in the category of decision making under risk and the assumed degree of certainty is only one aspect of a decision problem. The other way of classifying is: Linear or non-linear behaviour, static or dynamic conditions, single or multiple objectives.KTUNOTES.IN One has to consider all these aspects before building a model. Decision theory deals with decision making under conditions of risk and uncertainty. For our purpose, we shall consider all types of decision models including deterministic models to be under the domain of decision theory. In management literature, we have several quantitative decision models that help managers identify optima or best courses of action. Complete uncertainty Degree of uncertainty Complete certainty Decision making Decision making Decision-making Under uncertainty Under risk Under certainty. Before we go to decision theory, let us just discuss the issues, such as (i) What is a decision? (ii) Why must decisions be made? (iii) What is involved in the process of decision-making? (iv) What are some of the ways of classifying decisions? This will help us to have clear concept of decision models. -
The Shapley Value for Airport and Irrigation Games
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Márkus, Judit; Pintér, Péter Miklós; Radványi, Anna Working Paper The Shapley Value for Airport and Irrigation Games IEHAS Discussion Papers, No. MT-DP - 2012/7 Provided in Cooperation with: Institute of Economics, Centre for Economic and Regional Studies, Hungarian Academy of Sciences Suggested Citation: Márkus, Judit; Pintér, Péter Miklós; Radványi, Anna (2012) : The Shapley Value for Airport and Irrigation Games, IEHAS Discussion Papers, No. MT-DP - 2012/7, ISBN 978-615-5243-00-4, Hungarian Academy of Sciences, Institute of Economics, Centre for Economic and Regional Studies, Budapest This Version is available at: http://hdl.handle.net/10419/108259 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. -
Implementation Theory*
Chapter 5 IMPLEMENTATION THEORY* ERIC MASKIN Institute for Advanced Study, Princeton, NJ, USA TOMAS SJOSTROM Department of Economics, Pennsylvania State University, University Park, PA, USA Contents Abstract 238 Keywords 238 1. Introduction 239 2. Definitions 245 3. Nash implementation 247 3.1. Definitions 248 3.2. Monotonicity and no veto power 248 3.3. Necessary and sufficient conditions 250 3.4. Weak implementation 254 3.5. Strategy-proofness and rich domains of preferences 254 3.6. Unrestricted domain of strict preferences 256 3.7. Economic environments 257 3.8. Two agent implementation 259 4. Implementation with complete information: further topics 260 4.1. Refinements of Nash equilibrium 260 4.2. Virtual implementation 264 4.3. Mixed strategies 265 4.4. Extensive form mechanisms 267 4.5. Renegotiation 269 4.6. The planner as a player 275 5. Bayesian implementation 276 5.1. Definitions 276 5.2. Closure 277 5.3. Incentive compatibility 278 5.4. Bayesian monotonicity 279 * We are grateful to Sandeep Baliga, Luis Corch6n, Matt Jackson, Byungchae Rhee, Ariel Rubinstein, Ilya Segal, Hannu Vartiainen, Masahiro Watabe, and two referees, for helpful comments. Handbook of Social Choice and Welfare, Volume 1, Edited by K.J Arrow, A.K. Sen and K. Suzumura ( 2002 Elsevier Science B. V All rights reserved 238 E. Maskin and T: Sj'str6m 5.5. Non-parametric, robust and fault tolerant implementation 281 6. Concluding remarks 281 References 282 Abstract The implementation problem is the problem of designing a mechanism (game form) such that the equilibrium outcomes satisfy a criterion of social optimality embodied in a social choice rule. -
Scenario Analysis Normal Form Game
Overview Economics 3030 I. Introduction to Game Theory Chapter 10 II. Simultaneous-Move, One-Shot Games Game Theory: III. Infinitely Repeated Games Inside Oligopoly IV. Finitely Repeated Games V. Multistage Games 1 2 Normal Form Game A Normal Form Game • A Normal Form Game consists of: n Players Player 2 n Strategies or feasible actions n Payoffs Strategy A B C a 12,11 11,12 14,13 b 11,10 10,11 12,12 Player 1 c 10,15 10,13 13,14 3 4 Normal Form Game: Normal Form Game: Scenario Analysis Scenario Analysis • Suppose 1 thinks 2 will choose “A”. • Then 1 should choose “a”. n Player 1’s best response to “A” is “a”. Player 2 Player 2 Strategy A B C a 12,11 11,12 14,13 Strategy A B C b 11,10 10,11 12,12 a 12,11 11,12 14,13 Player 1 11,10 10,11 12,12 c 10,15 10,13 13,14 b Player 1 c 10,15 10,13 13,14 5 6 1 Normal Form Game: Normal Form Game: Scenario Analysis Scenario Analysis • Suppose 1 thinks 2 will choose “B”. • Then 1 should choose “a”. n Player 1’s best response to “B” is “a”. Player 2 Player 2 Strategy A B C Strategy A B C a 12,11 11,12 14,13 a 12,11 11,12 14,13 b 11,10 10,11 12,12 11,10 10,11 12,12 Player 1 b c 10,15 10,13 13,14 Player 1 c 10,15 10,13 13,14 7 8 Normal Form Game Dominant Strategy • Regardless of whether Player 2 chooses A, B, or C, Scenario Analysis Player 1 is better off choosing “a”! • “a” is Player 1’s Dominant Strategy (i.e., the • Similarly, if 1 thinks 2 will choose C… strategy that results in the highest payoff regardless n Player 1’s best response to “C” is “a”.