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Lecture 4 Rationalizability & Nash Equilibrium Road
Lecture 4 Rationalizability & Nash Equilibrium 14.12 Game Theory Muhamet Yildiz Road Map 1. Strategies – completed 2. Quiz 3. Dominance 4. Dominant-strategy equilibrium 5. Rationalizability 6. Nash Equilibrium 1 Strategy A strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy). Matching pennies with perfect information 2’s Strategies: HH = Head if 1 plays Head, 1 Head if 1 plays Tail; HT = Head if 1 plays Head, Head Tail Tail if 1 plays Tail; 2 TH = Tail if 1 plays Head, 2 Head if 1 plays Tail; head tail head tail TT = Tail if 1 plays Head, Tail if 1 plays Tail. (-1,1) (1,-1) (1,-1) (-1,1) 2 Matching pennies with perfect information 2 1 HH HT TH TT Head Tail Matching pennies with Imperfect information 1 2 1 Head Tail Head Tail 2 Head (-1,1) (1,-1) head tail head tail Tail (1,-1) (-1,1) (-1,1) (1,-1) (1,-1) (-1,1) 3 A game with nature Left (5, 0) 1 Head 1/2 Right (2, 2) Nature (3, 3) 1/2 Left Tail 2 Right (0, -5) Mixed Strategy Definition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: Si = {si1,si2,…,sik} σ → A mixed strategy: i: S [0,1] s.t. σ σ σ i(si1) + i(si2) + … + i(sik) = 1. If the other players play s-i =(s1,…, si-1,si+1,…,sn), then σ the expected utility of playing i is σ σ σ i(si1)ui(si1,s-i) + i(si2)ui(si2,s-i) + … + i(sik)ui(sik,s-i). -
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. -
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 .................................. -
Economics 201B Economic Theory (Spring 2021) Strategic Games
Economics 201B Economic Theory (Spring 2021) Strategic Games Topics: terminology and notations (OR 1.7), games and solutions (OR 1.1-1.3), rationality and bounded rationality (OR 1.4-1.6), formalities (OR 2.1), best-response (OR 2.2), Nash equilibrium (OR 2.2), 2 2 examples × (OR 2.3), existence of Nash equilibrium (OR 2.4), mixed strategy Nash equilibrium (OR 3.1, 3.2), strictly competitive games (OR 2.5), evolution- ary stability (OR 3.4), rationalizability (OR 4.1), dominance (OR 4.2, 4.3), trembling hand perfection (OR 12.5). Terminology and notations (OR 1.7) Sets For R, ∈ ≥ ⇐⇒ ≥ for all . and ⇐⇒ ≥ for all and some . ⇐⇒ for all . Preferences is a binary relation on some set of alternatives R. % ⊆ From % we derive two other relations on : — strict performance relation and not  ⇐⇒ % % — indifference relation and ∼ ⇐⇒ % % Utility representation % is said to be — complete if , or . ∀ ∈ % % — transitive if , and then . ∀ ∈ % % % % can be presented by a utility function only if it is complete and transitive (rational). A function : R is a utility function representing if → % ∀ ∈ () () % ⇐⇒ ≥ % is said to be — continuous (preferences cannot jump...) if for any sequence of pairs () with ,and and , . { }∞=1 % → → % — (strictly) quasi-concave if for any the upper counter set ∈ { ∈ : is (strictly) convex. % } These guarantee the existence of continuous well-behaved utility function representation. Profiles Let be a the set of players. — () or simply () is a profile - a collection of values of some variable,∈ one for each player. — () or simply is the list of elements of the profile = ∈ { } − () for all players except . ∈ — ( ) is a list and an element ,whichistheprofile () . -
Interim Correlated Rationalizability
Interim Correlated Rationalizability The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Dekel, Eddie, Drew Fudenberg, and Stephen Morris. 2007. Interim correlated rationalizability. Theoretical Economics 2, no. 1: 15-40. Published Version http://econtheory.org/ojs/index.php/te/article/view/20070015 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:3196333 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Theoretical Economics 2 (2007), 15–40 1555-7561/20070015 Interim correlated rationalizability EDDIE DEKEL Department of Economics, Northwestern University, and School of Economics, Tel Aviv University DREW FUDENBERG Department of Economics, Harvard University STEPHEN MORRIS Department of Economics, Princeton University This paper proposes the solution concept of interim correlated rationalizability, and shows that all types that have the same hierarchies of beliefs have the same set of interim-correlated-rationalizable outcomes. This solution concept charac- terizes common certainty of rationality in the universal type space. KEYWORDS. Rationalizability, incomplete information, common certainty, com- mon knowledge, universal type space. JEL CLASSIFICATION. C70, C72. 1. INTRODUCTION Harsanyi (1967–68) proposes solving games of incomplete -
Tacit Collusion in Oligopoly"
Tacit Collusion in Oligopoly Edward J. Green,y Robert C. Marshall,z and Leslie M. Marxx February 12, 2013 Abstract We examine the economics literature on tacit collusion in oligopoly markets and take steps toward clarifying the relation between econo- mists’analysis of tacit collusion and those in the legal literature. We provide examples of when the economic environment might be such that collusive pro…ts can be achieved without communication and, thus, when tacit coordination is su¢ cient to elevate pro…ts versus when com- munication would be required. The authors thank the Human Capital Foundation (http://www.hcfoundation.ru/en/), and especially Andrey Vavilov, for …nancial support. The paper bene…tted from discussions with Roger Blair, Louis Kaplow, Bill Kovacic, Vijay Krishna, Steven Schulenberg, and Danny Sokol. We thank Gustavo Gudiño for valuable research assistance. [email protected], Department of Economics, Penn State University [email protected], Department of Economics, Penn State University [email protected], Fuqua School of Business, Duke University 1 Introduction In this chapter, we examine the economics literature on tacit collusion in oligopoly markets and take steps toward clarifying the relation between tacit collusion in the economics and legal literature. Economists distinguish between tacit and explicit collusion. Lawyers, using a slightly di¤erent vocabulary, distinguish between tacit coordination, tacit agreement, and explicit collusion. In hopes of facilitating clearer communication between economists and lawyers, in this chapter, we attempt to provide a coherent resolution of the vernaculars used in the economics and legal literature regarding collusion.1 Perhaps the easiest place to begin is to de…ne explicit collusion. -
Revelation Principles in Multistage Games∗
Revelation Principles in Multistage Games Takuo Sugaya and Alexander Wolitzky Stanford GSB and MIT September 22, 2017 Abstract We study the revelation principle in multistage games for perfect Bayesian equi- libium and sequential equilibrium. The question of whether or not the revelation principle holds is subtle, as expanding players’ opportunities for communication ex- pands the set of consistent beliefs at off-path information sets. In settings with adverse selection but no moral hazard, Nash, perfect Bayesian, and sequential equilibrium are essentially equivalent, and a virtual-implementation version of the revelation principle holds. In general, we show that the revelation principle holds for weak perfect Bayesian equilibrium and for a stronger notion of perfect Bayesian equilibrium introduced by Myerson (1986). Our main result is that the latter notion of perfect Bayesian equilib- rium is outcome-equivalent to sequential equilibrium when the mediator can tremble; this provides a revelation principle for sequential equilibrium. When the mediator cannot tremble, the revelation principle for sequential equilibrium can fail. Preliminary. Comments Welcome. For helpful comments, we thank Drew Fudenberg, Dino Gerardi, Shengwu Li, Roger Myerson, Alessandro Pavan, Harry Pei, Juuso Toikka, and Bob Wilson. 1 Introduction The revelation principle states that any social choice function that can be implemented by any mechanism can also be implemented by a canonical mechanism where communication between players and the mechanism designer or mediator takes a circumscribed form: players communicate only their private information to the mediator, and the mediator communi- cates only recommended actions to the players. This cornerstone of modern microeconomics was developed by several authors in the 1970s and early 1980s, reaching its most general formulation in the principal-agent model of Myerson (1982), which treats one-shot games with both adverse selection and moral hazard. -
The Revelation Principle for Mechanism Design with Reporting Costs
The Revelation Principle for Mechanism Design with Reporting Costs ANDREW KEPHART, Duke University VINCENT CONITZER, Duke University The revelation principle is a key tool in mechanism design. It allows the designer to restrict attention to the class of truthful mechanisms, greatly facilitating analysis. This is also borne out in an algorithmic sense, al- lowing certain computational problems in mechanism design to be solved in polynomial time. Unfortunately, when not every type can misreport every other type (the partial verification model), or—more generally— misreporting can be costly, the revelation principle can fail to hold. This also leads to NP-hardness results. The primary contribution of this paper consists of characterizations of conditions under which the revelation principle still holds when misreporting can be costly. (These are generalizations of conditions given earlier for the partial verification case [Green and Laffont 1986; Yu 2011].) We also study associated computational problems. Additional Key Words and Phrases: automated mechanism design, signaling costs, partial verification, rev- elation principle 1. INTRODUCTION Mechanism design concerns making decisions based on the private information of one or more agents, who will not report this information truthfully if they do not see this to be in their interest. The goal is to define a mechanism—a game to be played by the agent(s)—that defines actions for the agents to take and maps these actions to outcomes, such that in equilibrium, the agents’ true types map to outcomes as de- sired. This may, at first, appear to leave us in the unmanageable situation of having to search through all possible games we could define. -
Theory of Mechanism Design
Theory of Mechanism Design Debasis Mishra1 April 14, 2020 1Economics and Planning Unit, Indian Statistical Institute, 7 Shahid Jit Singh Marg, New Delhi 110016, India, E-mail: [email protected] 2 Contents 1 Introduction to Mechanism Design7 1.0.1 Private Information and Utility Transfers................8 1.0.2 Examples in Practice........................... 10 1.1 A General Model of Preferences......................... 11 1.2 Social Choice Functions and Mechanisms.................... 14 1.3 Dominant Strategy Incentive Compatibility................... 16 1.4 Bayesian Incentive Compatibility........................ 18 1.4.1 Failure of revelation principle...................... 21 2 Mechanism Design with Transfers and Quasilinearity 23 2.1 A General Model................................. 24 2.1.1 Allocation Rules............................. 24 2.1.2 Payment Functions............................ 26 2.1.3 Incentive Compatibility.......................... 27 2.1.4 An Example................................ 27 2.1.5 Two Properties of Payments....................... 28 2.1.6 Efficient Allocation Rule is Implementable............... 29 2.2 The Vickrey-Clarke-Groves Mechanism..................... 32 2.2.1 Illustration of the VCG (Pivotal) Mechanism.............. 33 2.2.2 The VCG Mechanism in the Combinatorial Auctions......... 35 2.2.3 The Sponsored Search Auctions..................... 37 2.3 Affine Maximizer Allocation Rules are Implementable............. 38 2.3.1 Public Good Provision.......................... 40 2.3.2 Restricted and Unrestricted Type Spaces................ 41 3 3 Mechanism Design for Selling a Single Object 45 3.1 The Single Object Auction Model........................ 45 3.1.1 The Vickrey Auction........................... 45 3.1.2 Facts from Convex Analysis....................... 46 3.1.3 Monotonicity and Revenue Equivalence................. 49 3.1.4 The Efficient Allocation Rule and the Vickrey Auction....... -
Mean Field Equilibrium in Dynamic Games with Complementarities
Mean Field Equilibrium in Dynamic Games with Complementarities Sachin Adlakha Department of Electrical Engineering, Stanford University, Stanford, CA, 94305, [email protected] Ramesh Johari Department of Management Science and Engineering, Stanford University, Stanford, CA, 94305, [email protected] We study a class of stochastic dynamic games that exhibit strategic complementarities between players; formally, in the games we consider, the payoff of a player has increasing differences between her own state and the empirical distribution of the states of other players. Such games can be used to model a diverse set of applications, including network security models, recommender systems, and dynamic search in markets. Stochastic games are generally difficult to analyze, and these difficulties are only exacerbated when the number of players is large (as might be the case in the preceding examples). We consider an approximation methodology called mean field equilibrium to study these games. In such an equilibrium, each player reacts to only the long run average state of other players. We find necessary conditions for the existence of a mean field equilibrium in such games. Furthermore, as a simple consequence of this existence theorem, we obtain several natural monotonicity properties. We show that there exist a “largest” and a “smallest” equilibrium among all those where the equilibrium strategy used by a player is nondecreasing, and we also show that players converge to each of these equilibria via natural myopic learning dynamics; as we argue, these dynamics are more reasonable than the standard best response dynamics. We also provide sensitivity results, where we quantify how the equilibria of such games move in response to changes in parameters of the game (e.g., the introduction of incentives to players). -
Truthful Revelation Mechanisms for Simultaneous Common Agency Games† 132 Truthful Revelation Mechanisms for Simultaneous Common A
American Economic Journal: Microeconomics 2 (May 2010): 132–190 http://www.aeaweb.org/articles.php?doi 10.1257/mic.2.2.132 = Contents Truthful Revelation Mechanisms for Simultaneous Common Agency Games† 132 Truthful Revelation Mechanisms for Simultaneous Common A. Simple Menu-Auction Example 136 Agency Games† I. The Environment 140 II. Simple Revelation Mechanisms 143 By Alessandro Pavan and Giacomo Calzolari* III. Using Revelation Mechanisms in Applications 150 A. Competition in Nonlinear Tariffs 150 B. Menu Auctions 156 We introduce new revelation mechanisms for simultaneous common C. Moral Hazard 159 agency games which, although they do not always permit a complete IV. Enriched Mechanisms 161 equilibrium characterization, do facilitate the characterization of A. Non-Markov Strategies 161 the equilibrium outcomes that are typically of interest in applica- tions. We then show how these mechanisms can be used in applica- B. Mixed Strategies 163 tions such as menu auctions, competition in nonlinear tariffs, and V. Conclusions 166 moral hazard settings. Lastly, we show how one can enrich the rev- Appendix 1: Take-It-or-Leave-It-Offer Equilibria in the Menu-Auction Example in the Introduction elation mechanisms, albeit at a cost of an increase in complexity, 167 to characterize all possible equilibrium outcomes, including those Appendix 2: Omitted Proofs 168 sustained by non-Markov strategies and or mixed-strategy profiles. REFERENCES 189 / JEL C72, D82, D86 ( ) any economic environments can be modelled as common agency games—that M is, games where multiple principals contract simultaneously and noncoopera- tively with the same agent.1 Despite their relevance for applications, the analysis of these games has been made difficult by the fact that one cannot safely assume that the agent selects a contract with each principal by simply reporting his “type” i.e., his ( exogenous payoff-relevant information . -
Lecture 1: Ascending and Ex Post Incentive Compatible Mechanisms
CS364B: Frontiers in Mechanism Design Lecture #1: Ascending and Ex Post Incentive Compatible Mechanisms∗ Tim Roughgardeny January 8, 2014 1 Introduction These twenty lectures cover advanced topics in mechanism design. They assume familiarity with some of the material covered in the instructor's CS364A course | specifically, lectures 2{4 and 7{9. Recall that mechanism design is the \science of rule-making." The goal is to understand how to design systems with strategic participants | autonomous decision-makers whose objectives are generally different from the the designer's | that have good performance guarantees. For example, a mechanism designer might want to compute a socially efficient allocation of scarce resources or raise significant revenue, while a mechanism participant only cares about its own utility. 2 Course Outline The plan is to cover the following five topics. 1. (6 lectures.) Welfare-maximization in combinatorial auctions: tractable special cases and ascending implementations. 2. (4 lectures.) Welfare-maximization in combinatorial auctions: dominant-strategy ap- proximation mechanisms for NP-hard cases. 3. (3 lectures.) Welfare-maximization in combinatorial auctions: better guarantees for weaker solutions concepts (undominated strategies and Bayes-Nash equilibria). ∗ c 2014, Tim Roughgarden. yDepartment of Computer Science, Stanford University, 462 Gates Building, 353 Serra Mall, Stanford, CA 94305. Email: [email protected]. 1 4. (4 lectures.) The price of anarchy in simple auctions. 5. (3 lectures.) Revenue-maximization in multi-parameter settings. 3 Review: The k-Vickrey Auction Let's recall a basic example from last quarter. Scenario #1: • k identical items (even k = 1 is interesting); • each bidder is unit demand, meaning it only wants one item; • each bidder has a private valuation vi for a item.