On the Rationality of Escalation Are Advised to Read Sections 6, 7 and 9
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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? -
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. -
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 .................................. -
Paper and Pencils for Everyone
(CM^2) Math Circle Lesson: Game Theory of Gomuku and (m,n,k-games) Overview: Learning Objectives/Goals: to expose students to (m,n,k-games) and learn the general history of the games through out Asian cultures. SWBAT… play variations of m,n,k-games of varying degrees of difficulty and complexity as well as identify various strategies of play for each of the variations as identified by pattern recognition through experience. Materials: Paper and pencils for everyone Vocabulary: Game – we will create a working definition for this…. Objective – the goal or point of the game, how to win Win – to do (achieve) what a certain game requires, beat an opponent Diplomacy – working with other players in a game Luck/Chance – using dice or cards or something else “random” Strategy – techniques for winning a game Agenda: Check in (10-15min.) Warm-up (10-15min.) Lesson and game (30-45min) Wrap-up and chill time (10min) Lesson: Warm up questions: Ask these questions after warm up to the youth in small groups. They may discuss the answers in the groups and report back to you as the instructor. Write down the answers to these questions and compile a working definition. Try to lead the youth so that they do not name a specific game but keep in mind various games that they know and use specific attributes of them to make generalizations. · What is a game? · Are there different types of games? · What make something a game and something else not a game? · What is a board game? · How is it different from other types of games? · Do you always know what your opponent (other player) is doing during the game, can they be sneaky? · Do all of games have the same qualities as the games definition that we just made? Why or why not? Game history: The earliest known board games are thought of to be either ‘Go’ from China (which we are about to learn a variation of), or Senet and Mehen from Egypt (a country in Africa) or Mancala. -
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. -
Understanding Emotions in Electronic Auctions: Insights from Neurophysiology
Understanding Emotions in Electronic Auctions: Insights from Neurophysiology Marc T. P. Adam and Jan Krämer Abstract The design of electronic auction platforms is an important field of electronic commerce research. It requires not only a profound understanding of the role of human cognition in human bidding behavior but also of the role of human affect. In this chapter, we focus specifically on the emotional aspects of human bidding behavior and the results of empirical studies that have employed neurophysiological measurements in this regard. By synthesizing the results of these studies, we are able to provide a coherent picture of the role of affective processes in human bidding behavior along four distinct theoretical pathways. 1 Introduction Electronic auctions (i.e., electronic auction marketplaces) are information systems that facilitate and structure the exchange of products and services between several potential buyers and sellers. In practice, electronic auctions have been employed for a wide range of goods and services, such as commodities (e.g., Internet consumer auction markets, such as eBay.com), perishable goods (e.g., Dutch flower markets, such as by Royal FloraHolland), consumer services (e.g., procurement auction markets, such as myhammer.com), or property rights (e.g., financial double auction markets such as NASDAQ). In this vein, a large share of our private and commercial economic activity is being organized through electronic auctions today (Adam et al. 2017;Kuetal.2005). Previous research in information systems, economics, and psychology has shown that the design of electronic auctions has a tremendous impact on the auction M. T. P. Adam College of Engineering, Science and Environment, The University of Newcastle, Callaghan, Australia e-mail: [email protected] J. -
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]). -
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 -
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. -
CS711: Introduction to Game Theory and Mechanism Design
CS711: Introduction to Game Theory and Mechanism Design Teacher: Swaprava Nath Extensive Form Games Extensive Form Games Chocolate Division Game: Suppose a mother gives his elder son two (indivisible) chocolates to share between him and his younger sister. She also warns that if there is any dispute in the sharing, she will take the chocolates back and nobody will get anything. The brother can propose the following sharing options: (2-0): brother gets two, sister gets nothing, or (1-1): both gets one each, or (0-2): both chocolates to the sister. After the brother proposes the sharing, his sister may \Accept" the division or \Reject" it. Brother 2-0 1-1 0-2 Sister ARARAR (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 1 / 16 Game Theory and Mechanism Design Extensive Form Games S1 = f2 − 0; 1 − 1; 0 − 2g S2 = fA; Rg × fA; Rg × fA; Rg = fAAA; AAR; ARA; ARR; RAA; RAR; RRA; RRRg u1(2 − 0;A) = 2; u1(1 − 1;A) = 1; u2(1 − 1;A) = 1; u2(0 − 2;A) = 2 u1(0 − 2;A) = u1(0 − 2;R) = u1(1 − 1;R) = u1(2 − 0;R) = 0 u2(0 − 2;R) = u2(1 − 1;R) = u2(2 − 0;R) = u2(2 − 0;A) = 0 P (?) = 1;P (2 − 0) = P (1 − 1) = P (0 − 2) = 2 X (2 − 0) = X (1 − 1) = X (0 − 2) = fA; Rg X (?) = f(2 − 0); (1 − 1); (0 − 2)g Z = f(2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g H = f?; (2 − 0); (1 − 1); (0 − 2); (2 − 0;A); (2 − 0;R); (1 − 1;A); (1 − 1;R); (0 − 2;A); (0 − 2;R)g N = f1 (brother); 2 (sister)g;A = f2 − 0; 1 − 1; 0 − 2; A; Rg Representing the Chocolate Division Game Brother 2-0 1-1 0-2 Sister ARARAR (2; 0) (0; 0) (1; 1)(0; 0) (0; 2) (0; 0) 2 / 16 Game Theory -
A Note on Stabilizing Cooperation in the Centipede Game
games Article A Note on Stabilizing Cooperation in the Centipede Game Steven J. Brams 1,* and D. Marc Kilgour 2 1 Department of Politics, New York University, 19 W. 4th Street, New York, NY 10012, USA 2 Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada; [email protected] * Correspondence: [email protected] Received: 14 June 2020; Accepted: 18 August 2020; Published: 20 August 2020 Abstract: In the much-studied Centipede Game, which resembles the Iterated Prisoners’ Dilemma, two players successively choose between (1) cooperating, by continuing play, or (2) defecting and terminating play. The subgame-perfect Nash equilibrium implies that play terminates on the first move, even though continuing play can benefit both players—but not if the rival defects immediately, which it has an incentive to do. We show that, without changing the structure of the game, interchanging the payoffs of the two players provides each with an incentive to cooperate whenever its turn comes up. The Nash equilibrium in the transformed Centipede Game, called the Reciprocity Game, is unique—unlike the Centipede Game, wherein there are several Nash equilibria. The Reciprocity Game can be implemented noncooperatively by adding, at the start of the Centipede Game, a move to exchange payoffs, which it is rational for the players to choose. What this interchange signifies, and its application to transforming an arms race into an arms-control treaty, are discussed. Keywords: centipede game; Prisoners’ Dilemma; subgame-perfect equilibrium; payoff exchange 1. Introduction Since Rosenthal [1] introduced what has come to be called the Centipede Game, there has been controversy about what constitutes rational play in it, which we discuss in the next section. -
Second-Price All-Pay Auctions and Best-Reply Matching Equilibria Gisèle Umbhauer
Second-Price All-Pay Auctions and Best-Reply Matching Equilibria Gisèle Umbhauer To cite this version: Gisèle Umbhauer. Second-Price All-Pay Auctions and Best-Reply Matching Equilibria. International Game Theory Review, World Scientific Publishing, 2019, 21 (02), 10.1142/S0219198919400097. hal- 03164468 HAL Id: hal-03164468 https://hal.archives-ouvertes.fr/hal-03164468 Submitted on 9 Mar 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. SECOND-PRICE ALL-PAY AUCTIONS AND BEST-REPLY MATCHING EQUILIBRIA Gisèle UMBHAUER. Bureau d’Economie Théorique et Appliquée, University of Strasbourg, Strasbourg, France [email protected] Accepted October 2018 Published in International Game Theory Review, Vol 21, Issue 2, 40 pages, 2019 DOI 10.1142/S0219198919400097 The paper studies second-price all-pay auctions - wars of attrition - in a new way, based on classroom experiments and Kosfeld et al.’s best-reply matching equilibrium. Two players fight over a prize of value V, and submit bids not exceeding a budget M; both pay the lowest bid and the prize goes to the highest bidder. The behavior probability distributions in the classroom experiments are strikingly different from the mixed Nash equilibrium.