Economics 201B Economic Theory (Spring 2021) Strategic Games
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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. -
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. -
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 .................................. -
SEQUENTIAL GAMES with PERFECT INFORMATION Example
SEQUENTIAL GAMES WITH PERFECT INFORMATION Example 4.9 (page 105) Consider the sequential game given in Figure 4.9. We want to apply backward induction to the tree. 0 Vertex B is owned by player two, P2. The payoffs for P2 are 1 and 3, with 3 > 1, so the player picks R . Thus, the payoffs at B become (0, 3). 00 Next, vertex C is also owned by P2 with payoffs 1 and 0. Since 1 > 0, P2 picks L , and the payoffs are (4, 1). Player one, P1, owns A; the choice of L gives a payoff of 0 and R gives a payoff of 4; 4 > 0, so P1 chooses R. The final payoffs are (4, 1). 0 00 We claim that this strategy profile, { R } for P1 and { R ,L } is a Nash equilibrium. Notice that the 0 00 strategy profile gives a choice at each vertex. For the strategy { R ,L } fixed for P2, P1 has a maximal payoff by choosing { R }, ( 0 00 0 00 π1(R, { R ,L }) = 4 π1(R, { R ,L }) = 4 ≥ 0 00 π1(L, { R ,L }) = 0. 0 00 In the same way, for the strategy { R } fixed for P1, P2 has a maximal payoff by choosing { R ,L }, ( 00 0 00 π2(R, {∗,L }) = 1 π2(R, { R ,L }) = 1 ≥ 00 π2(R, {∗,R }) = 0, where ∗ means choose either L0 or R0. Since no change of choice by a player can increase that players own payoff, the strategy profile is called a Nash equilibrium. Notice that the above strategy profile is also a Nash equilibrium on each branch of the game tree, mainly starting at either B or starting at C. -
Strong Stackelberg Reasoning in Symmetric Games: an Experimental
Strong Stackelberg reasoning in symmetric games: An experimental ANGOR UNIVERSITY replication and extension Pulford, B.D.; Colman, A.M.; Lawrence, C.L. PeerJ DOI: 10.7717/peerj.263 PRIFYSGOL BANGOR / B Published: 25/02/2014 Publisher's PDF, also known as Version of record Cyswllt i'r cyhoeddiad / Link to publication Dyfyniad o'r fersiwn a gyhoeddwyd / Citation for published version (APA): Pulford, B. D., Colman, A. M., & Lawrence, C. L. (2014). Strong Stackelberg reasoning in symmetric games: An experimental replication and extension. PeerJ, 263. https://doi.org/10.7717/peerj.263 Hawliau Cyffredinol / General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. 23. Sep. 2021 Strong Stackelberg reasoning in symmetric games: An experimental replication and extension Briony D. Pulford1, Andrew M. Colman1 and Catherine L. Lawrence2 1 School of Psychology, University of Leicester, Leicester, UK 2 School of Psychology, Bangor University, Bangor, UK ABSTRACT In common interest games in which players are motivated to coordinate their strate- gies to achieve a jointly optimal outcome, orthodox game theory provides no general reason or justification for choosing the required strategies. -
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. -
Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction
Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction Renato Paes Leme Eva´ Tardos Department of Computer Science Department of Computer Science Cornell University, Ithaca, NY Cornell University, Ithaca, NY [email protected] [email protected] Abstract—The Generalized Second Price Auction has for advertisements and slots higher on the page are been the main mechanism used by search companies more valuable (clicked on by more users). The bids to auction positions for advertisements on search pages. are used to determine both the assignment of bidders In this paper we study the social welfare of the Nash equilibria of this game in various models. In the full to slots, and the fees charged. In the simplest model, information setting, socially optimal Nash equilibria are the bidders are assigned to slots in order of bids, and known to exist (i.e., the Price of Stability is 1). This paper the fee for each click is the bid occupying the next is the first to prove bounds on the price of anarchy, and slot. This auction is called the Generalized Second Price to give any bounds in the Bayesian setting. Auction (GSP). More generally, positions and payments Our main result is to show that the price of anarchy is small assuming that all bidders play un-dominated in the Generalized Second Price Auction depend also on strategies. In the full information setting we prove a bound the click-through rates associated with the bidders, the of 1.618 for the price of anarchy for pure Nash equilibria, probability that the advertisement will get clicked on by and a bound of 4 for mixed Nash equilibria. -
ECONS 424 – STRATEGY and GAME THEORY MIDTERM EXAM #2 – Answer Key
ECONS 424 – STRATEGY AND GAME THEORY MIDTERM EXAM #2 – Answer key Exercise #1. Hawk-Dove game. Consider the following payoff matrix representing the Hawk-Dove game. Intuitively, Players 1 and 2 compete for a resource, each of them choosing to display an aggressive posture (hawk) or a passive attitude (dove). Assume that payoff > 0 denotes the value that both players assign to the resource, and > 0 is the cost of fighting, which only occurs if they are both aggressive by playing hawk in the top left-hand cell of the matrix. Player 2 Hawk Dove Hawk , , 0 2 2 − − Player 1 Dove 0, , 2 2 a) Show that if < , the game is strategically equivalent to a Prisoner’s Dilemma game. b) The Hawk-Dove game commonly assumes that the value of the resource is less than the cost of a fight, i.e., > > 0. Find the set of pure strategy Nash equilibria. Answer: Part (a) • When Player 2 (in columns) chooses Hawk (in the left-hand column), Player 1 (in rows) receives a positive payoff of by paying Hawk, which is higher than his payoff of zero from playing Dove. − Therefore, for Player2 1 Hawk is a best response to Player 2 playing Hawk. Similarly, when Player 2 chooses Dove (in the right-hand column), Player 1 receives a payoff of by playing Hawk, which is higher than his payoff from choosing Dove, ; entailing that Hawk is Player 1’s best response to Player 2 choosing Dove. Therefore, Player 1 chooses2 Hawk as his best response to all of Player 2’s strategies, implying that Hawk is a strictly dominant strategy for Player 1. -
Extensive Form Games with Perfect Information
Notes Strategy and Politics: Extensive Form Games with Perfect Information Matt Golder Pennsylvania State University Sequential Decision-Making Notes The model of a strategic (normal form) game suppresses the sequential structure of decision-making. When applying the model to situations in which players move sequentially, we assume that each player chooses her plan of action once and for all. She is committed to this plan, which she cannot modify as events unfold. The model of an extensive form game, by contrast, describes the sequential structure of decision-making explicitly, allowing us to study situations in which each player is free to change her mind as events unfold. Perfect Information Notes Perfect information describes a situation in which players are always fully informed about all of the previous actions taken by all players. This assumption is used in all of the following lecture notes that use \perfect information" in the title. Later we will also study more general cases where players may only be imperfectly informed about previous actions when choosing an action. Extensive Form Games Notes To describe an extensive form game with perfect information we need to specify the set of players and their preferences, just as for a strategic game. In addition, we also need to specify the order of the players' moves and the actions each player may take at each point (or decision node). We do so by specifying the set of all sequences of actions that can possibly occur, together with the player who moves at each point in each sequence. We refer to each possible sequence of actions (a1; a2; : : : ak) as a terminal history and to the function that denotes the player who moves at each point in each terminal history as the player function. -
Introduction to Game Theory I
Introduction to Game Theory I Presented To: 2014 Summer REU Penn State Department of Mathematics Presented By: Christopher Griffin [email protected] 1/34 Types of Game Theory GAME THEORY Classical Game Combinatorial Dynamic Game Evolutionary Other Topics in Theory Game Theory Theory Game Theory Game Theory No time, doesn't contain differential equations Notion of time, may contain differential equations Games with finite or Games with no Games with time. Modeling population Experimental / infinite strategy space, chance. dynamics under Behavioral Game but no time. Games with motion or competition Theory Generally two player a dynamic component. Games with probability strategic games Modeling the evolution Voting Theory (either induced by the played on boards. Examples: of strategies in a player or the game). Optimal play in a dog population. Examples: Moves change the fight. Chasing your Determining why Games with coalitions structure of a game brother across a room. Examples: altruism is present in or negotiations. board. The evolution of human society. behaviors in a group of Examples: Examples: animals. Determining if there's Poker, Strategic Chess, Checkers, Go, a fair voting system. Military Decision Nim. Equilibrium of human Making, Negotiations. behavior in social networks. 2/34 Games Against the House The games we often see on television fall into this category. TV Game Shows (that do not pit players against each other in knowledge tests) often require a single player (who is, in a sense, playing against The House) to make a decision that will affect only his life. Prize is Behind: 1 2 3 Choose Door: 1 2 3 1 2 3 1 2 3 Switch: Y N Y N Y N Y N Y N Y N Y N Y N Y N Win/Lose: L W W L W L W L L W W L W L W L L W The Monty Hall Problem Other games against the house: • Blackjack • The Price is Right 3/34 Assumptions of Multi-Player Games 1.