DOCSLIB.ORG

Modeling Strategic Behavior1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modeling Strategic Behavior1 Modeling Strategic Behavior1 A Graduate Introduction to Game Theory and Mechanism Design George J. Mailath Department of Economics, University of Pennsylvania Research School of Economics, Australian National University 1September 16, 2020, copyright by George J. Mailath. World Scientific Press published the November 02, 2018 version. Any significant correc- tions or changes are listed at the back. To Loretta Preface These notes are based on my lecture notes for Economics 703, a first-year graduate course that I have been teaching at the Eco- nomics Department, University of Pennsylvania, for many years. It is impossible to understand modern economics without knowl- edge of the basic tools of game theory and mechanism design. My goal in the course (and this book) is to teach those basic tools so that students can understand and appreciate the corpus of modern economic thought, and so contribute to it. A key theme in the course is the interplay between the formal development of the tools and their use in applications. At the same time, extensions of the results that are beyond the course, but im- portant for context are (briefly) discussed. While I provide more background verbally on many of the exam- ples, I assume that students have seen some undergraduate game theory (such as covered in Osborne, 2004, Tadelis, 2013, and Wat- son, 2013). In addition, some exposure to intermediate microeco- nomics and decision making under uncertainty is helpful. Since these are lecture notes for an introductory course, I have not tried to attribute every result or model described. The result is a somewhat random pattern of citations and references. There is much more here than can be covered in a one semester course. I often do not cover Section 4.2 (Foundations of Nash Equi- librium), selectively cover material in Section 3.3 and Chapter 7 (Re- peated Games), and never both Sections 8.2 (Coase conjecture) and 8.3 (Reputations). I have been very lucky in my coauthors and colleagues (par- ticularly Larry Samuelson and Andrew Postlewaite), from whom I learned a tremendous amount. Thanks to the many generations iii iv Preface of Penn graduate students who were subjected to early versions of these notes, and made many helpful comments. Thanks also to Larry Samuelson and Tilman Börgers for helpful comments. Fi- nally, thanks to Ashwin Kambhampati and Changhwa Lee, who did an outstanding job proofreading these notes. Game Theory in Economics Game theory studies the strategic interactions of agents (often called “players” or decision-makers). An example of a strategic interaction is the pricing behavior of two petrol (gas) stations on the same in- tersection. Each station, in choosing its price, will both respond to the current price of the other station and to how it believes the other station will respond to its price. To study strategic interactions, it is useful to use parlor games such as chess and poker as examples. The first game theorists were mathematicians, and viewed the study of strategic interactions (be- ginning with parlor games) as applied mathematics.1 The goal was to calculate a solution, which was a prediction: How would rational players behave? A solution can also be a recommendation: How should a rational player behave (assuming the other player is ratio- nal)? The perspective of these notes is that of an economist. Economists are social scientists and, as such, want to understand social behav- ior. Any model simplifies the situation being modeled, and models of strategic interactions in an economic or social context are no different. The resulting modeling choices make the use of game theory within economics very different from game theory as ap- plied mathematics (which takes the rules of the game as given). In particular, the modeling choices reflect the modeler’s judgment as to what the players treat as strategically relevant. This judgment determines both the choices of strategy spaces (the actions and in- formation players have), and the choice of solution concept. There is often a subtle interplay between the question under investiga- tion, the modeling choices (including that of the solution concept), 1Leonard (2010) gives a fascinating history of the birth of game theory. Preface v and the resulting analysis. vi Preface Contents Preface iii Game Theory in Economics. iv 1 Normal and Extensive Form Games 1 1.1 Normal Form Games . .1 1.2 Iterated Deletion of Dominated Strategies . .9 1.3 Extensive Form Games . 12 1.3.1 The Reduced Normal Form . 16 1.4 Problems . 18 2 A First Look at Equilibrium 21 2.1 Nash Equilibrium . 21 2.1.1 Why Study Nash Equilibrium? . 26 2.2 Credible Threats and Backward Induction . 27 2.2.1 Backward Induction and Iterated Weak Domi- nance . 31 2.3 Subgame Perfection . 33 2.4 Mixing . 38 2.4.1 Mixed Strategies and Security Levels . 38 2.4.2 Domination and Optimality . 40 2.4.3 Equilibrium in Mixed Strategies . 45 2.4.4 Behavior Strategies . 50 2.5 Dealing with Multiplicity . 52 2.5.1 Refinements . 52 2.5.2 Selection . 56 2.6 Problems . 58 vii viii Contents 3 Games with Nature 69 3.1 An Introductory Example . 69 3.2 Purification . 71 3.3 Auctions and Related Games . 75 3.4 Games of Incomplete Information . 91 3.5 Higher Order Beliefs and Global Games . 97 3.6 Problems . 106 4 Nash Equilibrium: Existence and Foundations 113 4.1 Existence . 113 4.2 Learning/Evolutionary Foundations . 117 4.2.1 Social Learning (Evolutionary Game Theory) . 119 4.2.2 Individual learning . 128 4.3 Problems . 129 5 Nash Equilibrium Refinements in Dynamic Games 139 5.1 Sequential Rationality . 139 5.2 Perfect Bayesian Equilibrium . 147 5.3 Sequential Equilibrium . 152 5.4 Problems . 158 6 Signaling 165 6.1 General Theory . 165 6.2 Job Market Signaling . 169 6.2.1 Full Information . 170 6.2.2 Incomplete Information . 171 6.2.3 Refining to Separation . 176 6.2.4 Continuum of Types . 178 6.3 Problems . 180 7 Repeated Games 191 7.1 Perfect Monitoring . 191 7.1.1 The Stage Game . 191 7.1.2 The Repeated Game . 191 7.1.3 Subgame Perfection . 195 7.1.4 Automata . 198 7.1.5 Renegotiation-Proof Equilibria . 208 7.2 Short-Lived Players and Modeling Competitive Agents . 209 Contents ix 7.3 Applications . 216 7.3.1 Efficiency Wages I . 216 7.3.2 Collusion Under Demand Uncertainty . 220 7.4 Enforceability, Decomposability, and a Folk Theorem . 223 7.5 Imperfect Public Monitoring . 230 7.5.1 Efficiency Wages II . 230 7.5.2 Public Perfect Equilibria . 232 7.5.3 Automata . 234 7.6 Problems . 239 8 Topics in Dynamic Games 251 8.1 Dynamic Games and Markov Perfect Equilibria . 251 8.2 Disappearance of Monopoly Power and the Coase Conjecture . 258 8.2.1 One and Two Period Example . 258 8.2.2 Infinite Horizon . 261 8.3 Reputations . 264 8.3.1 Two Periods . 264 8.3.2 Infinite Horizon . 267 8.3.3 Infinite Horizon with Behavioral Types . 270 8.4 Problems . 273 9 Bargaining 281 9.1 Axiomatic Nash Bargaining . 281 9.1.1 The Axioms . 281 9.1.2 Nash’s Theorem . 282 9.2 Rubinstein (1982) Bargaining . 283 9.2.1 The Stationary Equilibrium . 284 9.2.2 All Equilibria . 286 9.2.3 Impatience . 288 9.3 Outside Options . 289 9.3.1 Version I . 289 9.3.2 Version II . 292 9.4 Exogenous Risk of Breakdown . 295 9.5 Problems . 296 x Contents 10 Introduction to Mechanism Design 303 10.1 A Simple Screening Example . 303 10.2 A Less Simple Screening Example . 307 10.3 The Take-It-or-Leave-It Mechanism . 313 10.4 Implementation . 314 10.5 Problems . 318 11 Dominant Strategy Mechanism Design 321 11.1 Social Choice and Arrow’s Impossibility Theorem . 321 11.2 Dominant Strategy Implementation and the Gibbard-Satterthwaite Theorem . 325 11.3 Efficiency in Quasilinear Environments . 330 11.4 Problems . 333 12 Bayesian Mechanism Design 335 12.1 The Bayesian Revelation Principle . 335 12.2 Efficiency in Quasilinear Environments . 337 12.3 Incomplete Information Bargaining . 340 12.3.1 The Impossibility of Ex Post Efficient Trade . 341 12.3.2 Maximizing Ex Ante Gains From Trade . 345 12.4 Independent Private Values Auctions . 352 12.5 Problems . 357 13 Principal Agency 363 13.1 Introduction . 363 13.2 Observable Effort . 364 13.3 Unobservable Effort (Moral Hazard) . 366 13.4 Unobserved Cost of Effort (Adverse Selection) . 372 13.5 A Hybrid Model . 373 13.6 Problems . 377 14 Appendices 379 14.1 Proof of Theorem 2.4.1 . 379 14.2 Trembling Hand Perfection . 382 14.2.1 Existence and Characterization . 382 14.2.2 Extensive Form Trembling Hand Perfection . 384 14.3 Completion of Proof of Theorem 12.3.1 . 386 14.4 Problems . 387 Contents xi References 389 Index 401 Corrections to the 2019 Published version 407 xii Contents Chapter 1 Normal and Extensive Form Games 1.1 Normal Form Games Most introductions to game theory start with the prisoner’s dilemma.1 Two suspects (I and II) are separately interrogated. The prosecu- tors have sufficient evidence to convict each of a minor offence, but wish to convict them of a major offence. The potential results of the interrogation are illustrated in Figure 1.1.1. Clearly, no matter what the other suspect does, it is always better to confess than not confess. This game is often interpreted as a partnership game, in which two partners simultaneously choose between exerting effort and shirking. Effort E produces an output of 6 at a cost of 4, while shirk- ing S yields no output at no cost.
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?
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 ..................................
    [Show full text]
  • Comparing Auction Designs Where Suppliers Have Uncertain Costs and Uncertain Pivotal Status
    RAND Journal of Economics Vol. 00, No. 0, Winter 2018 pp. 1–33 Comparing auction designs where suppliers have uncertain costs and uncertain pivotal status ∗ Par¨ Holmberg and ∗∗ Frank A. Wolak We analyze how market design influences bidding in multiunit procurement auctions where suppliers have asymmetric information about production costs. Our analysis is particularly relevant to wholesale electricity markets, because it accounts for the risk that a supplier is pivotal; market demand is larger than the total production capacity of its competitors. With constant marginal costs, expected welfare improves if the auctioneer restricts offers to be flat. We identify circumstances where the competitiveness of market outcomes improves with increased market transparency. We also find that, for buyers, uniform pricing is preferable to discriminatory pricing when producers’ private signals are affiliated. 1. Introduction Multiunit auctions are used to trade commodities, securities, emission permits, and other divisible goods. This article focuses on electricity markets, where producers submit offers before the level of demand and amount of available production capacity are fully known. Due to demand shocks, unexpected outages, transmission-constraints, and intermittent output from renewable energy sources, it often arises that an electricity producer is pivotal, that is, that realized demand is larger than the realized total production capacity of its competitors. A producer that is certain to be pivotal possess a substantial ability to exercise market power because it can withhold output ∗ Research Institute of Industrial Economics (IFN), Associate Researcher of the Energy Policy Research Group (EPRG), University of Cambridge; [email protected]. ∗∗ Program on Energy and Sustainable Development (PESD) and Stanford University; [email protected].
    [Show full text]
  • Natural Games
    Ntur J A a A A ,,,* , a Department of Biosciences, FI-00014 University of Helsinki, Finland b Institute of Biotechnology, FI-00014 University of Helsinki, Finland c Department of Physics, FI-00014 University of Helsinki, Finland ABSTRACT Behavior in the context of game theory is described as a natural process that follows the 2nd law of thermodynamics. The rate of entropy increase as the payoff function is derived from statistical physics of open systems. The thermodynamic formalism relates everything in terms of energy and describes various ways to consume free energy. This allows us to associate game theoretical models of behavior to physical reality. Ultimately behavior is viewed as a physical process where flows of energy naturally select ways to consume free energy as soon as possible. This natural process is, according to the profound thermodynamic principle, equivalent to entropy increase in the least time. However, the physical portrayal of behavior does not imply determinism. On the contrary, evolutionary equation for open systems reveals that when there are three or more degrees of freedom for behavior, the course of a game is inherently unpredictable in detail because each move affects motives of moves in the future. Eventually, when no moves are found to consume more free energy, the extensive- form game has arrived at a solution concept that satisfies the minimax theorem. The equilibrium is Lyapunov-stable against variation in behavior within strategies but will be perturbed by a new strategy that will draw even more surrounding resources to the game. Entropy as the payoff function also clarifies motives of collaboration and subjective nature of decision making.
    [Show full text]
  • Beyond Truth-Telling: Preference Estimation with Centralized School Choice and College Admissions∗
    Beyond Truth-Telling: Preference Estimation with Centralized School Choice and College Admissions∗ Gabrielle Fack Julien Grenet Yinghua He October 2018 (Forthcoming: The American Economic Review) Abstract We propose novel approaches to estimating student preferences with data from matching mechanisms, especially the Gale-Shapley Deferred Acceptance. Even if the mechanism is strategy-proof, assuming that students truthfully rank schools in applications may be restrictive. We show that when students are ranked strictly by some ex-ante known priority index (e.g., test scores), stability is a plausible and weaker assumption, implying that every student is matched with her favorite school/college among those she qualifies for ex post. The methods are illustrated in simulations and applied to school choice in Paris. We discuss when each approach is more appropriate in real-life settings. JEL Codes: C78, D47, D50, D61, I21 Keywords: Gale-Shapley Deferred Acceptance Mechanism, School Choice, College Admissions, Stable Matching, Student Preferences ∗Fack: Universit´eParis-Dauphine, PSL Research University, LEDa and Paris School of Economics, Paris, France (e-mail: [email protected]); Grenet: CNRS and Paris School of Economics, 48 boulevard Jourdan, 75014 Paris, France (e-mail: [email protected]); He: Rice University and Toulouse School of Economics, Baker Hall 246, Department of Economcis MS-22, Houston, TX 77251 (e-mail: [email protected]). For constructive comments, we thank Nikhil Agarwal, Peter Arcidi- acono, Eduardo Azevedo,
    [Show full text]
  • Dynamic Games Under Bounded Rationality
    Munich Personal RePEc Archive Dynamic Games under Bounded Rationality Zhao, Guo Southwest University for Nationalities 8 March 2015 Online at https://mpra.ub.uni-muenchen.de/62688/ MPRA Paper No. 62688, posted 09 Mar 2015 08:52 UTC Dynamic Games under Bounded Rationality By ZHAO GUO I propose a dynamic game model that is consistent with the paradigm of bounded rationality. Its main advantages over the traditional approach based on perfect rationality are that: (1) the strategy space is a chain-complete partially ordered set; (2) the response function is certain order-preserving map on strategy space; (3) the evolution of economic system can be described by the Dynamical System defined by the response function under iteration; (4) the existence of pure-strategy Nash equilibria can be guaranteed by fixed point theorems for ordered structures, rather than topological structures. This preference-response framework liberates economics from the utility concept, and constitutes a marriage of normal-form and extensive-form games. Among the common assumptions of classical existence theorems for competitive equilibrium, one is central. That is, individuals are assumed to have perfect rationality, so as to maximize their utilities (payoffs in game theoretic usage). With perfect rationality and perfect competition, the competitive equilibrium is completely determined, and the equilibrium depends only on their goals and their environments. With perfect rationality and perfect competition, the classical economic theory turns out to be deductive theory that requires almost no contact with empirical data once its assumptions are accepted as axioms (see Simon 1959). Zhao: Southwest University for Nationalities, Chengdu 610041, China (e-mail: [email protected]).
    [Show full text]
  • Chemical Game Theory
    Chemical Game Theory Jacob Kautzky Group Meeting February 26th, 2020 What is game theory? Game theory is the study of the ways in which interacting choices of rational agents produce outcomes with respect to the utilities of those agents Why do we care about game theory? 11 nobel prizes in economics 1994 – “for their pioneering analysis of equilibria in the thoery of non-cooperative games” John Nash Reinhard Selten John Harsanyi 2005 – “for having enhanced our understanding of conflict and cooperation through game-theory” Robert Aumann Thomas Schelling Why do we care about game theory? 2007 – “for having laid the foundatiouns of mechanism design theory” Leonid Hurwicz Eric Maskin Roger Myerson 2012 – “for the theory of stable allocations and the practice of market design” Alvin Roth Lloyd Shapley 2014 – “for his analysis of market power and regulation” Jean Tirole Why do we care about game theory? Mathematics Business Biology Engineering Sociology Philosophy Computer Science Political Science Chemistry Why do we care about game theory? Plato, 5th Century BCE Initial insights into game theory can be seen in Plato’s work Theories on prisoner desertions Cortez, 1517 Shakespeare’s Henry V, 1599 Hobbes’ Leviathan, 1651 Henry orders the French prisoners “burn the ships” executed infront of the French army First mathematical theory of games was published in 1944 by John von Neumann and Oskar Morgenstern Chemical Game Theory Basics of Game Theory Prisoners Dilemma Battle of the Sexes Rock Paper Scissors Centipede Game Iterated Prisoners Dilemma
    [Show full text]
  • Disambiguation of Ellsberg Equilibria in 2X2 Normal Form Games
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Decerf, Benoit; Riedel, Frank Working Paper Disambiguation of Ellsberg equilibria in 2x2 normal form games Center for Mathematical Economics Working Papers, No. 554 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Decerf, Benoit; Riedel, Frank (2016) : Disambiguation of Ellsberg equilibria in 2x2 normal form games, Center for Mathematical Economics Working Papers, No. 554, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, http://nbn-resolving.de/urn:nbn:de:0070-pub-29013618 This Version is available at: http://hdl.handle.net/10419/149011 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
    [Show full text]
  • Contemporaneous Perfect Epsilon-Equilibria
    Games and Economic Behavior 53 (2005) 126–140 www.elsevier.com/locate/geb Contemporaneous perfect epsilon-equilibria George J. Mailath a, Andrew Postlewaite a,∗, Larry Samuelson b a University of Pennsylvania b University of Wisconsin Received 8 January 2003 Available online 18 July 2005 Abstract We examine contemporaneous perfect ε-equilibria, in which a player’s actions after every history, evaluated at the point of deviation from the equilibrium, must be within ε of a best response. This concept implies, but is stronger than, Radner’s ex ante perfect ε-equilibrium. A strategy profile is a contemporaneous perfect ε-equilibrium of a game if it is a subgame perfect equilibrium in a perturbed game with nearly the same payoffs, with the converse holding for pure equilibria. 2005 Elsevier Inc. All rights reserved. JEL classification: C70; C72; C73 Keywords: Epsilon equilibrium; Ex ante payoff; Multistage game; Subgame perfect equilibrium 1. Introduction Analyzing a game begins with the construction of a model specifying the strategies of the players and the resulting payoffs. For many games, one cannot be positive that the specified payoffs are precisely correct. For the model to be useful, one must hope that its equilibria are close to those of the real game whenever the payoff misspecification is small. To ensure that an equilibrium of the model is close to a Nash equilibrium of every possible game with nearly the same payoffs, the appropriate solution concept in the model * Corresponding author. E-mail addresses: [email protected] (G.J. Mailath), [email protected] (A. Postlewaite), [email protected] (L.
    [Show full text]
  • A Pigouvian Approach to Congestion in Matching Markets
    DISCUSSION PAPER SERIES IZA DP No. 11967 A Pigouvian Approach to Congestion in Matching Markets Yinghua He Thierry Magnac NOVEMBER 2018 DISCUSSION PAPER SERIES IZA DP No. 11967 A Pigouvian Approach to Congestion in Matching Markets Yinghua He Rice University and Toulouse School of Economics Thierry Magnac Toulouse School of Economics and IZA NOVEMBER 2018 Any opinions expressed in this paper are those of the author(s) and not those of IZA. Research published in this series may include views on policy, but IZA takes no institutional policy positions. The IZA research network is committed to the IZA Guiding Principles of Research Integrity. The IZA Institute of Labor Economics is an independent economic research institute that conducts research in labor economics and offers evidence-based policy advice on labor market issues. Supported by the Deutsche Post Foundation, IZA runs the world’s largest network of economists, whose research aims to provide answers to the global labor market challenges of our time. Our key objective is to build bridges between academic research, policymakers and society. IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available directly from the author. IZA – Institute of Labor Economics Schaumburg-Lippe-Straße 5–9 Phone: +49-228-3894-0 53113 Bonn, Germany Email: [email protected] www.iza.org IZA DP No. 11967 NOVEMBER 2018 ABSTRACT A Pigouvian Approach to Congestion in Matching Markets* Recruiting agents, or “programs” costly screen “applicants” in matching processes, and congestion in a market increases with the number of applicants to be screened.
    [Show full text]