Tacit Collusion in Oligopoly"
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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 .................................. -
The Three Types of Collusion: Fixing Prices, Rivals, and Rules Robert H
University of Baltimore Law ScholarWorks@University of Baltimore School of Law All Faculty Scholarship Faculty Scholarship 2000 The Three Types of Collusion: Fixing Prices, Rivals, and Rules Robert H. Lande University of Baltimore School of Law, [email protected] Howard P. Marvel Ohio State University, [email protected] Follow this and additional works at: http://scholarworks.law.ubalt.edu/all_fac Part of the Antitrust and Trade Regulation Commons, and the Law and Economics Commons Recommended Citation The Three Types of Collusion: Fixing Prices, Rivals, and Rules, 2000 Wis. L. Rev. 941 (2000) This Article is brought to you for free and open access by the Faculty Scholarship at ScholarWorks@University of Baltimore School of Law. It has been accepted for inclusion in All Faculty Scholarship by an authorized administrator of ScholarWorks@University of Baltimore School of Law. For more information, please contact [email protected]. ARTICLES THE THREE TYPES OF COLLUSION: FIXING PRICES, RIVALS, AND RULES ROBERTH. LANDE * & HOWARDP. MARVEL** Antitrust law has long held collusion to be paramount among the offenses that it is charged with prohibiting. The reason for this prohibition is simple----collusion typically leads to monopoly-like outcomes, including monopoly profits that are shared by the colluding parties. Most collusion cases can be classified into two established general categories.) Classic, or "Type I" collusion involves collective action to raise price directly? Firms can also collude to disadvantage rivals in a manner that causes the rivals' output to diminish or causes their behavior to become chastened. This "Type 11" collusion in turn allows the colluding firms to raise prices.3 Many important collusion cases, however, do not fit into either of these categories. -
Collusion Constrained Equilibrium
Theoretical Economics 13 (2018), 307–340 1555-7561/20180307 Collusion constrained equilibrium Rohan Dutta Department of Economics, McGill University David K. Levine Department of Economics, European University Institute and Department of Economics, Washington University in Saint Louis Salvatore Modica Department of Economics, Università di Palermo We study collusion within groups in noncooperative games. The primitives are the preferences of the players, their assignment to nonoverlapping groups, and the goals of the groups. Our notion of collusion is that a group coordinates the play of its members among different incentive compatible plans to best achieve its goals. Unfortunately, equilibria that meet this requirement need not exist. We instead introduce the weaker notion of collusion constrained equilibrium. This al- lows groups to put positive probability on alternatives that are suboptimal for the group in certain razor’s edge cases where the set of incentive compatible plans changes discontinuously. These collusion constrained equilibria exist and are a subset of the correlated equilibria of the underlying game. We examine four per- turbations of the underlying game. In each case,we show that equilibria in which groups choose the best alternative exist and that limits of these equilibria lead to collusion constrained equilibria. We also show that for a sufficiently broad class of perturbations, every collusion constrained equilibrium arises as such a limit. We give an application to a voter participation game that shows how collusion constraints may be socially costly. Keywords. Collusion, organization, group. JEL classification. C72, D70. 1. Introduction As the literature on collective action (for example, Olson 1965) emphasizes, groups often behave collusively while the preferences of individual group members limit the possi- Rohan Dutta: [email protected] David K. -
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 -
Imbalanced Collusive Security Games
Imbalanced Collusive Security Games Han-Ching Ou, Milind Tambe, Bistra Dilkina, and Phebe Vayanos Computer Science Department, University of Southern California {hanchino,tambe,dilkina,phebe.vayanos }@usc.edu Abstract. Colluding adversaries is a crucial challenge for defenders in many real-world applications. Previous literature has provided Collusive Security Games (COSG) to model colluding adversaries, and provided models and algorithms to generate defender strategies to counter colluding adversaries, often by devising strategies that inhibit collusion [6]. Unfortunately, this previous work focused exclusively on situations with perfectly matched adversaries, i.e., where their rewards were symmetrically distributed. In the real world, however, defenders often face adversaries where their rewards are asymmetrically distributed. Such inherent asymmetry raises a question as to whether human adversaries would at- tempt to collude in such situations, and whether defender strategies to counter such collusion should focus on inhibiting collusion. To address these open ques- tions, this paper: (i) explores and theoretically analyzes Imbalanced Collusive Security Games (ICOSG) where defenders face adversaries with asymmetrically distributed rewards; (ii) conducts extensive experiments of three different adver- sary models involving 1800 real human subjects and (iii) derives novel analysis of the reason behind why bounded rational attackers models outperform perfectly rational attackers models. The key principle discovered as the result of our -
Econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Stähler, Frank Working Paper — Digitized Version Markov perfection and cooperation in repeated games Kiel Working Paper, No. 760 Provided in Cooperation with: Kiel Institute for the World Economy (IfW) Suggested Citation: Stähler, Frank (1996) : Markov perfection and cooperation in repeated games, Kiel Working Paper, No. 760, Kiel Institute of World Economics (IfW), Kiel This Version is available at: http://hdl.handle.net/10419/46946 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. www.econstor.eu Kieler Arbeitspapiere Kiel Working Papers Kiel Working Paper No. 760 Markov perfection and cooperation in repeated games by Frank Stahler August 1996 Institut fur Weltwirtschaft an der Universitat Kiel The Kiel Institute of World Economics ISSN 0342 - 0787 The Kiel Institute of World Economics Diisternbrooker Weg 120 D-24105Kiel, FRG Kiel Working Paper No. -
Collusion and Heterogeneity of Firms∗
Collusion and Heterogeneity of Firms∗ Ichiro Obaray Federico Zincenko z June 13, 2016 Abstract We examine the impact of heterogeneous discounting on collusion in dynamic Bertrand competition. We show exactly when collusion can be sustained and how collusion would be organized efficiently with heterogeneous discounting. First we show that collusion is possible if and only if the average discount factor exceeds a certain threshold, with or without capacity constraints. Next we identify a dynamic pattern of market share that characterizes efficient collusion and obtain the unique long-run prediction despite the presence of multiple equilibria. In the long run, the most patient firm and the most impatient firm tend to dominate the market. JEL Classification: C72, C73, D43. Keywords: Bertrand Competition, Capacity Constraint, Collusion, Repeated Game, Subgame Perfect Equilibrium. Unequal Discounting. ∗This article was developed from the second chapter (co-authored with I. Obara) of Zincenko's doctoral dissertation at UCLA. We would like to thank the editor and anonymous referees for their useful comments. We also like to thank the participants at many seminars at various universities, 2011 Asian Meeting of the Econometric Society at Korea University, 2012 Southwest Economic Theory Conference at UC San Diego, and 2013 North American Summer Meeting of the Econometric Society at USC. All remaining errors are ours. yUniversity of California, Los Angeles; [email protected]. zUniversity of Pittsburgh; [email protected]. 1 1 Introduction To understand when and how collusion arises with heterogeneous discounting, we study a dynamic Bertrand competition model, in which firms discount future profits at different discount rates, and examine the impact of heterogeneous discounting on collusion. -
Game Playing
Game Playing Philipp Koehn 29 September 2015 Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Outline 1 ● Games ● Perfect play – minimax decisions – α–β pruning ● Resource limits and approximate evaluation ● Games of chance ● Games of imperfect information Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 2 games Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Games vs. Search Problems 3 ● “Unpredictable” opponent ⇒ solution is a strategy specifying a move for every possible opponent reply ● Time limits ⇒ unlikely to find goal, must approximate ● Plan of attack: – computer considers possible lines of play (Babbage, 1846) – algorithm for perfect play (Zermelo, 1912; Von Neumann, 1944) – finite horizon, approximate evaluation (Zuse, 1945; Wiener, 1948; Shannon, 1950) – first Chess program (Turing, 1951) – machine learning to improve evaluation accuracy (Samuel, 1952–57) – pruning to allow deeper search (McCarthy, 1956) Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Types of Games 4 deterministic chance perfect Chess Backgammon information Checkers Monopoly Go Othello imperfect battleships Bridge information Blind Tic Tac Toe Poker Scrabble Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Game Tree (2-player, Deterministic, Turns) 5 Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Simple Game Tree 6 ● 2 player game ● Each player has one move ● You move first ● Goal: optimize your payoff (utility) Start Your move Opponent -
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).