Contemporaneous Perfect Epsilon-Equilibria
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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. -
Equilibrium Refinements
Equilibrium Refinements Mihai Manea MIT Sequential Equilibrium I In many games information is imperfect and the only subgame is the original game. subgame perfect equilibrium = Nash equilibrium I Play starting at an information set can be analyzed as a separate subgame if we specify players’ beliefs about at which node they are. I Based on the beliefs, we can test whether continuation strategies form a Nash equilibrium. I Sequential equilibrium (Kreps and Wilson 1982): way to derive plausible beliefs at every information set. Mihai Manea (MIT) Equilibrium Refinements April 13, 2016 2 / 38 An Example with Incomplete Information Spence’s (1973) job market signaling game I The worker knows her ability (productivity) and chooses a level of education. I Education is more costly for low ability types. I Firm observes the worker’s education, but not her ability. I The firm decides what wage to offer her. In the spirit of subgame perfection, the optimal wage should depend on the firm’s beliefs about the worker’s ability given the observed education. An equilibrium needs to specify contingent actions and beliefs. Beliefs should follow Bayes’ rule on the equilibrium path. What about off-path beliefs? Mihai Manea (MIT) Equilibrium Refinements April 13, 2016 3 / 38 An Example with Imperfect Information Courtesy of The MIT Press. Used with permission. Figure: (L; A) is a subgame perfect equilibrium. Is it plausible that 2 plays A? Mihai Manea (MIT) Equilibrium Refinements April 13, 2016 4 / 38 Assessments and Sequential Rationality Focus on extensive-form games of perfect recall with finitely many nodes. An assessment is a pair (σ; µ) I σ: (behavior) strategy profile I µ = (µ(h) 2 ∆(h))h2H: system of beliefs ui(σjh; µ(h)): i’s payoff when play begins at a node in h randomly selected according to µ(h), and subsequent play specified by σ. -
Algorithms and Collusion - Background Note by the Secretariat
Unclassified DAF/COMP(2017)4 Organisation for Economic Co-operation and Development 9 June 2017 English - Or. English DIRECTORATE FOR FINANCIAL AND ENTERPRISE AFFAIRS COMPETITION COMMITTEE Cancels & replaces the same document of 17 May 2017. Algorithms and Collusion - Background Note by the Secretariat 21-23 June 2017 This document was prepared by the OECD Secretariat to serve as a background note for Item 10 at the 127th Meeting of the Competition Committee on 21-23 June 2017. The opinions expressed and arguments employed herein do not necessarily reflect the official views of the Organisation or of the governments of its member countries. More documentation related to this discussion can be found at http://www.oecd.org/daf/competition/algorithms-and-collusion.htm Please contact Mr. Antonio Capobianco if you have any questions about this document [E-mail: [email protected]] JT03415748 This document, as well as any data and map included herein, are without prejudice to the status of or sovereignty over any territory, to the delimitation of international frontiers and boundaries and to the name of any territory, city or area. 2 │ DAF/COMP(2017)4 Algorithms and Collusion Background note by the Secretariat Abstract The combination of big data with technologically advanced tools, such as pricing algorithms, is increasingly diffused in today everyone’s life, and it is changing the competitive landscape in which many companies operate and the way in which they make commercial and strategic decisions. While the size of this phenomenon is to a large extent unknown, there are a growing number of firms using computer algorithms to improve their pricing models, customise services and predict market trends. -
Evolutionary Game Theory: ESS, Convergence Stability, and NIS
Evolutionary Ecology Research, 2009, 11: 489–515 Evolutionary game theory: ESS, convergence stability, and NIS Joseph Apaloo1, Joel S. Brown2 and Thomas L. Vincent3 1Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada, 2Department of Biological Sciences, University of Illinois, Chicago, Illinois, USA and 3Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona, USA ABSTRACT Question: How are the three main stability concepts from evolutionary game theory – evolutionarily stable strategy (ESS), convergence stability, and neighbourhood invader strategy (NIS) – related to each other? Do they form a basis for the many other definitions proposed in the literature? Mathematical methods: Ecological and evolutionary dynamics of population sizes and heritable strategies respectively, and adaptive and NIS landscapes. Results: Only six of the eight combinations of ESS, convergence stability, and NIS are possible. An ESS that is NIS must also be convergence stable; and a non-ESS, non-NIS cannot be convergence stable. A simple example shows how a single model can easily generate solutions with all six combinations of stability properties and explains in part the proliferation of jargon, terminology, and apparent complexity that has appeared in the literature. A tabulation of most of the evolutionary stability acronyms, definitions, and terminologies is provided for comparison. Key conclusions: The tabulated list of definitions related to evolutionary stability are variants or combinations of the three main stability concepts. Keywords: adaptive landscape, convergence stability, Darwinian dynamics, evolutionary game stabilities, evolutionarily stable strategy, neighbourhood invader strategy, strategy dynamics. INTRODUCTION Evolutionary game theory has and continues to make great strides. -
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 .................................. -
Evolutionary Stable Strategy Application of Nash Equilibrium in Biology
GENERAL ARTICLE Evolutionary Stable Strategy Application of Nash Equilibrium in Biology Jayanti Ray-Mukherjee and Shomen Mukherjee Every behaviourally responsive animal (including us) make decisions. These can be simple behavioural decisions such as where to feed, what to feed, how long to feed, decisions related to finding, choosing and competing for mates, or simply maintaining ones territory. All these are conflict situations between competing individuals, hence can be best understood Jayanti Ray-Mukherjee is using a game theory approach. Using some examples of clas- a faculty member in the School of Liberal Studies sical games, we show how evolutionary game theory can help at Azim Premji University, understand behavioural decisions of animals. Game theory Bengaluru. Jayanti is an (along with its cousin, optimality theory) continues to provide experimental ecologist who a strong conceptual and theoretical framework to ecologists studies mechanisms of species coexistence among for understanding the mechanisms by which species coexist. plants. Her research interests also inlcude plant Most of you, at some point, might have seen two cats fighting. It invasion ecology and is often accompanied with the cats facing each other with puffed habitat restoration. up fur, arched back, ears back, tail twitching, with snarls, growls, Shomen Mukherjee is a and howls aimed at each other. But, if you notice closely, they faculty member in the often try to avoid physical contact, and spend most of their time School of Liberal Studies in the above-mentioned behavioural displays. Biologists refer to at Azim Premji University, this as a ‘limited war’ or conventional (ritualistic) strategy (not Bengaluru. He uses field experiments to study causing serious injury), as opposed to dangerous (escalated) animal behaviour and strategy (Figure 1) [1]. -
Game Theory]: Basics of Game Theory
Artificial Intelligence Methods for Social Good M2-1 [Game Theory]: Basics of Game Theory 08-537 (9-unit) and 08-737 (12-unit) Instructor: Fei Fang [email protected] Wean Hall 4126 1 5/8/2018 Quiz 1: Recap: Optimization Problem Given coordinates of 푛 residential areas in a city (assuming 2-D plane), denoted as 푥1, … , 푥푛, the government wants to find a location that minimizes the sum of (Euclidean) distances to all residential areas to build a hospital. The optimization problem can be written as A: min |푥푖 − 푥| 푥 푖 B: min 푥푖 − 푥 푥 푖 2 2 C: min 푥푖 − 푥 푥 푖 D: none of above 2 Fei Fang 5/8/2018 From Games to Game Theory The study of mathematical models of conflict and cooperation between intelligent decision makers Used in economics, political science etc 3/72 Fei Fang 5/8/2018 Outline Basic Concepts in Games Basic Solution Concepts Compute Nash Equilibrium Compute Strong Stackelberg Equilibrium 4 Fei Fang 5/8/2018 Learning Objectives Understand the concept of Game, Player, Action, Strategy, Payoff, Expected utility, Best response Dominant Strategy, Maxmin Strategy, Minmax Strategy Nash Equilibrium Stackelberg Equilibrium, Strong Stackelberg Equilibrium Describe Minimax Theory Formulate the following problem as an optimization problem Find NE in zero-sum games (LP) Find SSE in two-player general-sum games (multiple LP and MILP) Know how to find the method/algorithm/solver/package you can use for solving the games Compute NE/SSE by hand or by calling a solver for small games 5 Fei Fang 5/8/2018 Let’s Play! Classical Games -
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. -
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. -
Comparative Statics and Perfect Foresight in Infinite Horizon Economies
Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/comparativestatiOOkeho working paper department of economics Comparative Statics And Perfect Foresight in Infinite Horizon Economies Timothy J. Kehoe David K. Levine* Number 312 December 1982 massachusetts institute of technology 50 memorial drive Cambridge, mass. 02139 Comparative Statics And Perfect Foresight in Infinite Horizon Economies Timothy J. Kehoe David K. Levine* Number 312 December 1982 Department of Economics, M.I.T. and Department of Economics, U.C.L.A., respectively. Abstract This paper considers whether infinite horizon economies have determinate perfect foresight equilibria. ¥e consider stationary pure exchange economies with both finite and infinite numbers of agents. When there is a finite number of infinitely lived agents, we argue that equilibria are generically determinate. This is because the number of equations determining the equilibria is not infinite, but is equal to the number of agents minus one and must determine the marginal utility of income for all but one agent. In an overlapping generations model with infinitely many finitely lived agents, this reasoning breaks down. ¥e ask whether the initial conditions together with the requirement of convergence to a steady state locally determine an equilibrium price path. In this framework there are many economies with isolated equilibria, many with continue of equilibria, and many with no equilibria at all. With two or more goods in every period not only can the price level be indeterminate but relative prices as well. Furthermore, such indeterminacy can occur whether or not there is fiat money in the economy. -
Kranton Duke University
The Devil is in the Details – Implications of Samuel Bowles’ The Moral Economy for economics and policy research October 13 2017 Rachel Kranton Duke University The Moral Economy by Samuel Bowles should be required reading by all graduate students in economics. Indeed, all economists should buy a copy and read it. The book is a stunning, critical discussion of the interplay between economic incentives and preferences. It challenges basic premises of economic theory and questions policy recommendations based on these theories. The book proposes the path forward: designing policy that combines incentives and moral appeals. And, therefore, like such as book should, The Moral Economy leaves us with much work to do. The Moral Economy concerns individual choices and economic policy, particularly microeconomic policies with goals to enhance the collective good. The book takes aim at laws, policies, and business practices that are based on the classic Homo economicus model of individual choice. The book first argues in great detail that policies that follow from the Homo economicus paradigm can backfire. While most economists would now recognize that people are not purely selfish and self-interested, The Moral Economy goes one step further. Incentives can amplify the selfishness of individuals. People might act in more self-interested ways in a system based on incentives and rewards than they would in the absence of such inducements. The Moral Economy warns economists to be especially wary of incentives because social norms, like norms of trust and honesty, are critical to economic activity. The danger is not only of incentives backfiring in a single instance; monetary incentives can generally erode ethical and moral codes and social motivations people can have towards each other.