Extensive games (with perfect information)
|
|
- Cornelia O’Brien’
- 5 years ago
- Views:
Transcription
1 Extensive games (with perfect information) (also referred to as extensive-form games or dynamic games) DEFINITION An extensive game with perfect information has the following components A set N (the set of players). A set H of sequences (finite or infinite) satisfying: The empty sequence is a member of H If (a k ) k=1,...,k H (where K may be infinite) and L < K then (a k ) k=1,...,l H If an infinite sequence (a k ) k=1 H satisfies (ak ) k=1,...,l H for every positive integer L then (a k ) k=1 H (Each member of H is called a history; each component of a history is an action taken by a player.) A history (a k ) k=1,...,k H is called terminal (final) if it is infinite or if there is no a K+1 such that (a k ) k=1,...,k+1 H. The set of terminal histories is denoted Z. A function P that assigns to each nonterminal history (each member of H\Z) a member of N. (P is the player function, P (h) being the player who takes an action after the history h.) For each player i N a preference relation i on Z (the preference relation of player i). Denote a history h followed by action a by (h, a). Interpetation of this definition: After any nonterminal history h Player P (h) chooses a possible action a, i.e. such an action that (h, a) belongs to H. Game tree is a convenient method to represent extensive-form games. Think of a rooted tree in graph-theoretic sense. Actions are represented by branches (or edges) and histories correspond to paths and induce nodes. Non-terminal histories induce decision nodes and terminal histories induce end-nodes or end-points or leaves. Example: mini-ultimatum game 1
2 Strategies Action Strategy! Strategy is a plan of action for every contingency. DEFINITION A strategy of player i N in an extensive game with perfect information N, H, P, i is a function that assigns an admissible action to each nonterminal history h H\Z for which P (h) = i. A combination of strategies induces an outcome of the game - the terminal history that will result when they are implemented. Note that the strategy dictates what to do even at nodes that, under this strategy, will not be visited. Any extensive-form game can now be represented in a matrix form. Example: Entrant game Thus, also the concept of Nash Equilibrium easily applies to dynamic games it is the NE of the appropriate strategic (matrix-form) game. 2
3 Nash is not enough In extensive-form games players may want to revise their equilibrium strategy as the game unfolds. It seems naive to assume that they will not when it is in their best interest to do so it would be tantamount to believing non-credible threats or promises. DEFINITION The subgame of the extensive game with perfect information Γ = N, H, P, ( i ) that follows the history h is the extensive game Γ(h) = N, H h, P h, ( i h ), where H h is the set of sequences h of actions for which (h, h ) H, P h is defined by P h (h ) = P (h, h ) for each h H h, and i h is defined by h i h h if and only if (h, h ) i (h, h ). (examples) Every game is its own subgame (following an empty history). Denote by s i h the strategy induced by s i in subgame Γ(h) and by s h the strategy profile induced by s. DEFINITION Strategy profile s constitutes a subgame perfect equilibrium of a game if s h is a NE of every subgame Γ(h). One problem with the SPNE: shouldn t I give up my belief about rationality of the other player when he makes a dumb choice? (example) 3
4 Centipede game Is common knowledge of rationality so rational? (example) 4
5 Timing matters: Cournot vs Stackelberg n = 2, P = a Q, Q = q 1 + q 2, constant unit cost c 1 = c 2. Profit is given by: Π i = q i (P c i ) = q i (a q i q i c i ) FOC: Π i q i = (a q i q i c i ) q i = (a 2q i q i c i ) = 0 Thus for any q i i s BR is q i = a q i c i 2 If they move simultaneously (Counot), NE is ( a c 3, a c 3 ). If they move sequentially (Stackelberg), there are multiple NE. But the unique SPNE is ( a c ). Total output is different, player 1 is better off (why?), player 2 is worse off., a c 2 4 5
6 Some properties of SPNE Obviously, every SPNE is a NE but not conversely. Existence: every finitie extensive game with perfect information has a subgame perfect equilibrium (can be found by backward induction) (non)uniqueness: SPNE is in general not unique. However, it is unique if no player is indifferent between two end-nodes. (example: the gardening games) 6
7 Two fairly benign extensions Random moves: nature choses at some nodes, following a pre-defined distribution Simultaneous moves: more than one player moves simultaneously at some nodes Note: with simultaneous moves SPNE may fail to exist (example: matching pennies). 7
8 Forward induction Example: forward induction in Battle-of-the-Sexes 8
Extensive Form Games with Perfect Information
Extensive Form Games with Perfect Information Pei-yu Lo 1 Introduction Recap: BoS. Look for all Nash equilibria. show how to nd pure strategy Nash equilibria. Show how to nd mixed strategy Nash equilibria.
More informationGames with Perfect Information
Games with Perfect Information Yiling Chen September 7, 2011 Non-Cooperative Game Theory What is it? Mathematical study of interactions between rational and self-interested agents. Non-Cooperative Focus
More informationGame Theory. Wolfgang Frimmel. Perfect Bayesian Equilibrium
Game Theory Wolfgang Frimmel Perfect Bayesian Equilibrium / 22 Bayesian Nash equilibrium and dynamic games L M R 3 2 L R L R 2 2 L R L 2,, M,2, R,3,3 2 NE and 2 SPNE (only subgame!) 2 / 22 Non-credible
More informationSolving Extensive Form Games
Chapter 8 Solving Extensive Form Games 8.1 The Extensive Form of a Game The extensive form of a game contains the following information: (1) the set of players (2) the order of moves (that is, who moves
More informationExtensive Form Games with Perfect Information
Extensive Form Games with Perfect Information Levent Koçkesen 1 Extensive Form Games The strategies in strategic form games are speci ed so that each player chooses an action (or a mixture of actions)
More information4: Dynamic games. Concordia February 6, 2017
INSE6441 Jia Yuan Yu 4: Dynamic games Concordia February 6, 2017 We introduce dynamic game with non-simultaneous moves. Example 0.1 (Ultimatum game). Divide class into two groups at random: Proposers,
More informationMicroeconomics. 2. Game Theory
Microeconomics 2. Game Theory Alex Gershkov http://www.econ2.uni-bonn.de/gershkov/gershkov.htm 18. November 2008 1 / 36 Dynamic games Time permitting we will cover 2.a Describing a game in extensive form
More information1 The General Definition
MS&E 336 Lecture 1: Dynamic games Ramesh Johari April 4, 2007 1 The General Definition A dynamic game (or extensive game, or game in extensive form) consists of: A set of players N; A set H of sequences
More informationBackwards Induction. Extensive-Form Representation. Backwards Induction (cont ) The player 2 s optimization problem in the second stage
Lecture Notes II- Dynamic Games of Complete Information Extensive Form Representation (Game tree Subgame Perfect Nash Equilibrium Repeated Games Trigger Strategy Dynamic Games of Complete Information Dynamic
More informationEconomics 201A Economic Theory (Fall 2009) Extensive Games with Perfect and Imperfect Information
Economics 201A Economic Theory (Fall 2009) Extensive Games with Perfect and Imperfect Information Topics: perfect information (OR 6.1), subgame perfection (OR 6.2), forward induction (OR 6.6), imperfect
More informationUC Berkeley Haas School of Business Game Theory (EMBA 296 & EWMBA 211) Summer 2016
UC Berkeley Haas School of Business Game Theory (EMBA 296 & EWMBA 211) Summer 2016 More on strategic games and extensive games with perfect information Block 2 Jun 12, 2016 Food for thought LUPI Many players
More informationIndustrial Organization Lecture 3: Game Theory
Industrial Organization Lecture 3: Game Theory Nicolas Schutz Nicolas Schutz Game Theory 1 / 43 Introduction Why game theory? In the introductory lecture, we defined Industrial Organization as the economics
More information6.207/14.15: Networks Lecture 11: Introduction to Game Theory 3
6.207/14.15: Networks Lecture 11: Introduction to Game Theory 3 Daron Acemoglu and Asu Ozdaglar MIT October 19, 2009 1 Introduction Outline Existence of Nash Equilibrium in Infinite Games Extensive Form
More informationExtensive Form Games I
Extensive Form Games I Definition of Extensive Form Game a finite game tree X with nodes x X nodes are partially ordered and have a single root (minimal element) terminal nodes are z Z (maximal elements)
More informationNegotiation: Strategic Approach
Negotiation: Strategic pproach (September 3, 007) How to divide a pie / find a compromise among several possible allocations? Wage negotiations Price negotiation between a seller and a buyer Bargaining
More information6.254 : Game Theory with Engineering Applications Lecture 13: Extensive Form Games
6.254 : Game Theory with Engineering Lecture 13: Extensive Form Games Asu Ozdaglar MIT March 18, 2010 1 Introduction Outline Extensive Form Games with Perfect Information One-stage Deviation Principle
More information6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2
6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2 Daron Acemoglu and Asu Ozdaglar MIT October 14, 2009 1 Introduction Outline Mixed Strategies Existence of Mixed Strategy Nash Equilibrium
More informationBasics of Game Theory
Basics of Game Theory Giacomo Bacci and Luca Sanguinetti Department of Information Engineering University of Pisa, Pisa, Italy {giacomo.bacci,luca.sanguinetti}@iet.unipi.it April - May, 2010 G. Bacci and
More information1 Extensive Form Games
1 Extensive Form Games De nition 1 A nite extensive form game is am object K = fn; (T ) ; P; A; H; u; g where: N = f0; 1; :::; ng is the set of agents (player 0 is nature ) (T ) is the game tree P is the
More information: Cryptography and Game Theory Ran Canetti and Alon Rosen. Lecture 8
0368.4170: Cryptography and Game Theory Ran Canetti and Alon Rosen Lecture 8 December 9, 2009 Scribe: Naama Ben-Aroya Last Week 2 player zero-sum games (min-max) Mixed NE (existence, complexity) ɛ-ne Correlated
More information3.3.3 Illustration: Infinitely repeated Cournot duopoly.
will begin next period less effective in deterring a deviation this period. Nonetheless, players can do better than just repeat the Nash equilibrium of the constituent game. 3.3.3 Illustration: Infinitely
More informationEconomics 3012 Strategic Behavior Andy McLennan October 20, 2006
Economics 301 Strategic Behavior Andy McLennan October 0, 006 Lecture 11 Topics Problem Set 9 Extensive Games of Imperfect Information An Example General Description Strategies and Nash Equilibrium Beliefs
More informationON FORWARD INDUCTION
Econometrica, Submission #6956, revised ON FORWARD INDUCTION SRIHARI GOVINDAN AND ROBERT WILSON Abstract. A player s pure strategy is called relevant for an outcome of a game in extensive form with perfect
More informationSF2972 Game Theory Exam with Solutions March 15, 2013
SF2972 Game Theory Exam with s March 5, 203 Part A Classical Game Theory Jörgen Weibull and Mark Voorneveld. (a) What are N, S and u in the definition of a finite normal-form (or, equivalently, strategic-form)
More informationEconS Advanced Microeconomics II Handout on Subgame Perfect Equilibrium (SPNE)
EconS 3 - Advanced Microeconomics II Handout on Subgame Perfect Equilibrium (SPNE). Based on MWG 9.B.3 Consider the three-player nite game of perfect information depicted in gure. L R Player 3 l r a b
More informationSF2972 Game Theory Written Exam with Solutions June 10, 2011
SF97 Game Theory Written Exam with Solutions June 10, 011 Part A Classical Game Theory Jörgen Weibull and Mark Voorneveld 1. Finite normal-form games. (a) What are N, S and u in the definition of a finite
More informationRefinements - change set of equilibria to find "better" set of equilibria by eliminating some that are less plausible
efinements efinements - change set of equilibria to find "better" set of equilibria by eliminating some that are less plausible Strategic Form Eliminate Weakly Dominated Strategies - Purpose - throwing
More informationEquilibrium Refinements
Equilibrium Refinements Mihai Manea MIT Sequential Equilibrium In many games information is imperfect and the only subgame is the original game... subgame perfect equilibrium = Nash equilibrium Play starting
More informationGame Theory. Kuhn s Theorem. Bernhard Nebel, Robert Mattmüller, Stefan Wölfl, Christian Becker-Asano
Game Theory Albert-Ludwigs-Universität Freiburg Bernhard Nebel, Robert Mattmüller, Stefan Wölfl, Christian Becker-Asano Research Group Foundations of Artificial Intelligence June 17, 2013 June 17, 2013
More informationECO421: Reputation. Marcin P ski. March 29, 2018
ECO421: Reputation Marcin P ski March 29, 2018 Plan Chain store game Model of reputation Reputations in innite games Applications Model No pure strategy equilibria Mixed strategy equilibrium Basic model
More informationGame Theory and Social Psychology
Game Theory and Social Psychology cf. Osborne, ch 4.8 Kitty Genovese: attacked in NY in front of 38 witnesses no one intervened or called the police Why not? \Indierence to one's neighbour and his troubles
More informationMicroeconomics for Business Practice Session 3 - Solutions
Microeconomics for Business Practice Session - Solutions Instructor: Eloisa Campioni TA: Ugo Zannini University of Rome Tor Vergata April 8, 016 Exercise 1 Show that there are no mixed-strategy Nash equilibria
More informationPerfect Bayesian Equilibrium
Perfect Bayesian Equilibrium For an important class of extensive games, a solution concept is available that is simpler than sequential equilibrium, but with similar properties. In a Bayesian extensive
More informationSeptember 5 Exercises: Nash equilibrium and dominant strategy equilibrium: p. 44: 1, 3, 4
September 5 Exercises: Nash equilibrium and dominant strategy equilibrium: p. 44: 1, 3, 4 Example 12 (Continued) Now assume that in addition to payingc v i when the project is provided, each division is
More informationWEAKLY DOMINATED STRATEGIES: A MYSTERY CRACKED
WEAKLY DOMINATED STRATEGIES: A MYSTERY CRACKED DOV SAMET Abstract. An informal argument shows that common knowledge of rationality implies the iterative elimination of strongly dominated strategies. Rationality
More informationLecture 7. Simple Dynamic Games
Lecture 7. Simple Dynamic Games 1. Two-Stage Games of Complete and Perfect Information Two-Stages dynamic game with two players: player 1 chooses action a 1 from the set of his feasible actions A 1 player
More informationMS&E 246: Lecture 12 Static games of incomplete information. Ramesh Johari
MS&E 246: Lecture 12 Static games of incomplete information Ramesh Johari Incomplete information Complete information means the entire structure of the game is common knowledge Incomplete information means
More informationEconS Sequential Competition
EconS 425 - Sequential Competition Eric Dunaway Washington State University eric.dunaway@wsu.edu Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 47 A Warmup 1 x i x j (x
More informationEC3224 Autumn Lecture #04 Mixed-Strategy Equilibrium
Reading EC3224 Autumn Lecture #04 Mixed-Strategy Equilibrium Osborne Chapter 4.1 to 4.10 By the end of this week you should be able to: find a mixed strategy Nash Equilibrium of a game explain why mixed
More informationCopyright (C) 2013 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of the Creative
Copyright (C) 2013 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of the Creative Commons attribution license http://creativecommons.org/licenses/by/2.0/
More informationNotes on Coursera s Game Theory
Notes on Coursera s Game Theory Manoel Horta Ribeiro Week 01: Introduction and Overview Game theory is about self interested agents interacting within a specific set of rules. Self-Interested Agents have
More informationAdvanced Microeconomics
Advanced Microeconomics ECON5200 - Fall 2012 Introduction What you have done: - consumers maximize their utility subject to budget constraints and firms maximize their profits given technology and market
More informationGame Theory. Professor Peter Cramton Economics 300
Game Theory Professor Peter Cramton Economics 300 Definition Game theory is the study of mathematical models of conflict and cooperation between intelligent and rational decision makers. Rational: each
More informationLong-Run versus Short-Run Player
Repeated Games 1 Long-Run versus Short-Run Player a fixed simultaneous move stage game Player 1 is long-run with discount factor δ actions a A a finite set 1 1 1 1 2 utility u ( a, a ) Player 2 is short-run
More informationOpen Sequential Equilibria of Multi-Stage Games with Infinite Sets of Types and Actions
Open Sequential Equilibria of Multi-Stage Games with Infinite Sets of Types and Actions By Roger B. Myerson and Philip J. Reny Department of Economics University of Chicago Paper can be found at https://sites.google.com/site/philipjreny/home/research
More informationDynamic Games, II: Bargaining
Dynamic Games, II: Bargaining Econ 400 University of Notre Dame Econ 400 (ND) Dynamic Games, II: Bargaining 1 / 16 Bargaining In bargaining games, we are generally trying to model agents haggling back-and-forth
More informationReview of topics since what was covered in the midterm: Topics that we covered before the midterm (also may be included in final):
Review of topics since what was covered in the midterm: Subgame-perfect eqms in extensive games with perfect information where players choose a number (first-order conditions, boundary conditions, favoring
More informationANSWER KEY 4 GAME THEORY, ECON 395
ANSWER KEY 4 GAME THEORY, ECON 395 PROFESSOR A. JOSEPH GUSE (1) (Gibbons 2.1) Suppose a parent and child play the following game. First, the child takes an action A, that produces net income for the child,
More informationBasic Game Theory. Kate Larson. January 7, University of Waterloo. Kate Larson. What is Game Theory? Normal Form Games. Computing Equilibria
Basic Game Theory University of Waterloo January 7, 2013 Outline 1 2 3 What is game theory? The study of games! Bluffing in poker What move to make in chess How to play Rock-Scissors-Paper Also study of
More informationA New and Robust Subgame Perfect Equilibrium in a model of Triadic Power Relations *
A New and Robust Subgame Perfect Equilibrium in a model of Triadic Power Relations * by Magnus Hatlebakk ** Department of Economics, University of Bergen Abstract: We present a new subgame perfect equilibrium
More informationLecture Notes for Economics 200C: Games and Information Vincent Crawford, revised March 2000; do not reproduce except for personal use
Lecture Notes for Economics C: Games and Information Vincent Crawford, revised March ; do not reproduce except for personal use. Introduction MWG 7-33; Kreps 355-384; Varian 59-65; McMillan 3-4 Robert
More information6 The Principle of Optimality
6 The Principle of Optimality De nition A T shot deviation from a strategy s i is a strategy bs i such that there exists T such that bs i (h t ) = s i (h t ) for all h t 2 H with t T De nition 2 A one-shot
More informationOligopoly. Molly W. Dahl Georgetown University Econ 101 Spring 2009
Oligopoly Molly W. Dahl Georgetown University Econ 101 Spring 2009 1 Oligopoly A monopoly is an industry consisting a single firm. A duopoly is an industry consisting of two firms. An oligopoly is an industry
More informationEconomics 209B Behavioral / Experimental Game Theory (Spring 2008) Lecture 3: Equilibrium refinements and selection
Economics 209B Behavioral / Experimental Game Theory (Spring 2008) Lecture 3: Equilibrium refinements and selection Theory cannot provide clear guesses about with equilibrium will occur in games with multiple
More informationEcon 618: Correlated Equilibrium
Econ 618: Correlated Equilibrium Sunanda Roy 1 Basic Concept of a Correlated Equilibrium MSNE assumes players use a random device privately and independently, that tells them which strategy to choose for
More informationEvolutionary Dynamics and Extensive Form Games by Ross Cressman. Reviewed by William H. Sandholm *
Evolutionary Dynamics and Extensive Form Games by Ross Cressman Reviewed by William H. Sandholm * Noncooperative game theory is one of a handful of fundamental frameworks used for economic modeling. It
More informationProblem Set 6 Solutions
Problem Set 6 Solutions Exercise. Prom Example : (Alex Alice, Bob Betty), College Admission Example: (A (β,γ), B α); (A (α,β), B γ), Prom Example 2: (Alex Alice, Bob Betty); (Alex Betty, Bob Alice). Exercise
More informationAppendix B for The Evolution of Strategic Sophistication (Intended for Online Publication)
Appendix B for The Evolution of Strategic Sophistication (Intended for Online Publication) Nikolaus Robalino and Arthur Robson Appendix B: Proof of Theorem 2 This appendix contains the proof of Theorem
More informationRobust Knowledge and Rationality
Robust Knowledge and Rationality Sergei Artemov The CUNY Graduate Center 365 Fifth Avenue, 4319 New York City, NY 10016, USA sartemov@gc.cuny.edu November 22, 2010 Abstract In 1995, Aumann proved that
More informationOligopoly. Oligopoly. Xiang Sun. Wuhan University. March 23 April 6, /149
Oligopoly Xiang Sun Wuhan University March 23 April 6, 2016 1/149 Outline 1 Introduction 2 Game theory 3 Oligopoly models 4 Cournot competition Two symmetric firms Two asymmetric firms Many symmetric firms
More informationConfronting Theory with Experimental Data and vice versa. European University Institute. May-Jun Lectures 7-8: Equilibrium
Confronting Theory with Experimental Data and vice versa European University Institute May-Jun 2008 Lectures 7-8: Equilibrium Theory cannot provide clear guesses about with equilibrium will occur in games
More informationDynamic Games and Bargaining. Johan Stennek
Dynamic Games and Bargaining Johan Stennek 1 Dynamic Games Logic of cartels Idea: We agree to both charge high prices and share the market Problem: Both have incentive to cheat Solution: Threat to punish
More information1 Oligopoly: Bertrand Model
1 Oligopoly: Bertrand Model Bertrand model: There are two rms and no entry is possible. Homogeneity of product. Single period. Consumers always purchase from the cheapest seller. If the two selllers charge
More informationSF2972 Game Theory Problem set on extensive form games
SF2972 Game Theor Problem set on etensive form games Mark Voorneveld There are five eercises, to be handed in at the final lecture (March 0). For a bonus point, please answer all questions; at least half
More informationGovernment 2005: Formal Political Theory I
Government 2005: Formal Political Theory I Lecture 11 Instructor: Tommaso Nannicini Teaching Fellow: Jeremy Bowles Harvard University November 9, 2017 Overview * Today s lecture Dynamic games of incomplete
More informationEvery Choice Correspondence is Backwards-Induction Rationalizable
Every Choice Correspondence is Backwards-Induction Rationalizable John Rehbeck Department of Economics University of California-San Diego jrehbeck@ucsd.edu Abstract We extend the result from Bossert and
More informationBargaining, Contracts, and Theories of the Firm. Dr. Margaret Meyer Nuffield College
Bargaining, Contracts, and Theories of the Firm Dr. Margaret Meyer Nuffield College 2015 Course Overview 1. Bargaining 2. Hidden information and self-selection Optimal contracting with hidden information
More informationFirst Prev Next Last Go Back Full Screen Close Quit. Game Theory. Giorgio Fagiolo
Game Theory Giorgio Fagiolo giorgio.fagiolo@univr.it https://mail.sssup.it/ fagiolo/welcome.html Academic Year 2005-2006 University of Verona Summary 1. Why Game Theory? 2. Cooperative vs. Noncooperative
More informationMixed Strategies. Krzysztof R. Apt. CWI, Amsterdam, the Netherlands, University of Amsterdam. (so not Krzystof and definitely not Krystof)
Mixed Strategies Krzysztof R. Apt (so not Krzystof and definitely not Krystof) CWI, Amsterdam, the Netherlands, University of Amsterdam Mixed Strategies p. 1/1 Mixed Extension of a Finite Game Probability
More informationOn Acyclicity of Games with Cycles
On Acyclicity of Games with Cycles Daniel Andersson, Vladimir Gurvich, and Thomas Dueholm Hansen Dept. of Computer Science, Aarhus University, {koda,tdh}@cs.au.dk RUTCOR, Rutgers University, gurvich@rutcor.rutgers.edu
More informationGame theory lecture 4. September 24, 2012
September 24, 2012 Finding Nash equilibrium Best-response or best-reply functions. We introduced Nash-equilibrium as a profile of actions (an action for each player) such that no player has an incentive
More informationRationalization of Collective Choice Functions by Games with Perfect Information. Yongsheng Xu
Rationalization of Collective Choice Functions by Games with Perfect Information by Yongsheng Xu Department of Economics, Andrew Young School of Policy Studies Georgia State University, Atlanta, GA 30303
More informationEquivalences of Extensive Forms with Perfect Recall
Equivalences of Extensive Forms with Perfect Recall Carlos Alós-Ferrer and Klaus Ritzberger University of Cologne and Royal Holloway, University of London, 1 and VGSF 1 as of Aug. 1, 2016 1 Introduction
More informationEconomics 703 Advanced Microeconomics. Professor Peter Cramton Fall 2017
Economics 703 Advanced Microeconomics Professor Peter Cramton Fall 2017 1 Outline Introduction Syllabus Web demonstration Examples 2 About Me: Peter Cramton B.S. Engineering, Cornell University Ph.D. Business
More informationGame Theory Lecture 10+11: Knowledge
Game Theory Lecture 10+11: Knowledge Christoph Schottmüller University of Copenhagen November 13 and 20, 2014 1 / 36 Outline 1 (Common) Knowledge The hat game A model of knowledge Common knowledge Agree
More informationAlpha-Beta Pruning: Algorithm and Analysis
Alpha-Beta Pruning: Algorithm and Analysis Tsan-sheng Hsu tshsu@iis.sinica.edu.tw http://www.iis.sinica.edu.tw/~tshsu 1 Introduction Alpha-beta pruning is the standard searching procedure used for solving
More informationBertrand Model of Price Competition. Advanced Microeconomic Theory 1
Bertrand Model of Price Competition Advanced Microeconomic Theory 1 ҧ Bertrand Model of Price Competition Consider: An industry with two firms, 1 and 2, selling a homogeneous product Firms face market
More informationTheory Field Examination Game Theory (209A) Jan Question 1 (duopoly games with imperfect information)
Theory Field Examination Game Theory (209A) Jan 200 Good luck!!! Question (duopoly games with imperfect information) Consider a duopoly game in which the inverse demand function is linear where it is positive
More informationDynamic stochastic game and macroeconomic equilibrium
Dynamic stochastic game and macroeconomic equilibrium Tianxiao Zheng SAIF 1. Introduction We have studied single agent problems. However, macro-economy consists of a large number of agents including individuals/households,
More informationAlpha-Beta Pruning: Algorithm and Analysis
Alpha-Beta Pruning: Algorithm and Analysis Tsan-sheng Hsu tshsu@iis.sinica.edu.tw http://www.iis.sinica.edu.tw/~tshsu 1 Introduction Alpha-beta pruning is the standard searching procedure used for 2-person
More informationAlgorithms for cautious reasoning in games
Algorithms for cautious reasoning in games Geir B. Asheim a Andrés Perea b October 16, 2017 Abstract We provide comparable algorithms for the Dekel-Fudenberg procedure, iterated admissibility, proper rationalizability
More informationIndustrial Organization, Fall 2011: Midterm Exam Solutions and Comments Date: Wednesday October
Industrial Organization, Fall 2011: Midterm Exam Solutions and Comments Date: Wednesday October 23 2011 1 Scores The exam was long. I know this. Final grades will definitely be curved. Here is a rough
More informationCommon Knowledge of Rationality is Self-Contradictory. Herbert Gintis
Common Knowledge of Rationality is Self-Contradictory Herbert Gintis February 25, 2012 Abstract The conditions under which rational agents play a Nash equilibrium are extremely demanding and often implausible.
More informationConsistent Beliefs in Extensive Form Games
Games 2010, 1, 415-421; doi:10.3390/g1040415 OPEN ACCESS games ISSN 2073-4336 www.mdpi.com/journal/games Article Consistent Beliefs in Extensive Form Games Paulo Barelli 1,2 1 Department of Economics,
More informationA Model of Modeling. Itzhak Gilboa, Andy Postelwaite, Larry Samuelson, and David Schmeidler. March 2, 2015
A Model of Modeling Itzhak Gilboa, Andy Postelwaite, Larry Samuelson, and David Schmeidler March 2, 2015 GPSS () Model of Modeling March 2, 2015 1 / 26 Outline Four distinctions: Theories and paradigms
More informationA Dynamic Level-k Model in Games
Dynamic Level-k Model in Games Teck Ho and Xuanming Su UC erkeley March, 2010 Teck Hua Ho 1 4-stage Centipede Game 4 2 16 8 1 8 4 32 1 2 3 4 64 16 5 Outcome Round 1 2 3 4 5 1 5 6.2% 30.3% 35.9% 20.0% 7.6%
More informationIterated Strict Dominance in Pure Strategies
Iterated Strict Dominance in Pure Strategies We know that no rational player ever plays strictly dominated strategies. As each player knows that each player is rational, each player knows that his opponents
More informationAlpha-Beta Pruning: Algorithm and Analysis
Alpha-Beta Pruning: Algorithm and Analysis Tsan-sheng Hsu tshsu@iis.sinica.edu.tw http://www.iis.sinica.edu.tw/~tshsu 1 Introduction Alpha-beta pruning is the standard searching procedure used for 2-person
More informationDiscrete Mathematics. On Nash equilibria and improvement cycles in pure positional strategies for Chess-like and Backgammon-like n-person games
Discrete Mathematics 312 (2012) 772 788 Contents lists available at SciVerse ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On Nash equilibria and improvement cycles
More informationUnique Nash Implementation for a Class of Bargaining Solutions
Unique Nash Implementation for a Class of Bargaining Solutions Walter Trockel University of California, Los Angeles and Bielefeld University Mai 1999 Abstract The paper presents a method of supporting
More informationGame Theory Review Questions
Game Theory Review Questions Sérgio O. Parreiras All Rights Reserved 2014 0.1 Repeated Games What is the difference between a sequence of actions and a strategy in a twicerepeated game? Express a strategy
More informationEconomics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Extensive games with perfect information OR6and7,FT3,4and11
Economics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Extensive games with perfect information OR6and7,FT3,4and11 Perfect information A finite extensive game with perfect information
More informationIndividual Rationality in Collective Choice
Individual Rationality in Collective Choice Hiroki Nishimura February 21, 2014 Abstract This paper studies the rationality of an individual player in sequential games of perfect information played with
More informationStatic (or Simultaneous- Move) Games of Complete Information
Static (or Simultaneous- Move) Games of Complete Information Introduction to Games Normal (or Strategic) Form Representation Teoria dos Jogos - Filomena Garcia 1 Outline of Static Games of Complete Information
More informationIntroduction to Game Theory
Introduction to Game Theory Part 2. Dynamic games of complete information Chapter 2. Two-stage games of complete but imperfect information Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas
More information4. CONTINUOUS VARIABLES AND ECONOMIC APPLICATIONS
STATIC GAMES 4. CONTINUOUS VARIABLES AND ECONOMIC APPLICATIONS Universidad Carlos III de Madrid CONTINUOUS VARIABLES In many games, ure strategies that layers can choose are not only, 3 or any other finite
More informationThe Local Best Response Criterion: An Epistemic Approach to Equilibrium Refinement. Herbert Gintis
The Local est Response Criterion: An Epistemic Approach to Equilibrium Refinement Herbert Gintis February 6, 2009 Abstract The standard refinement criteria for extensive form games, including subgame perfect,
More informationEVOLUTIONARY STABILITY FOR TWO-STAGE HAWK-DOVE GAMES
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS olume 25, Number 1, Winter 1995 EOLUTIONARY STABILITY FOR TWO-STAGE HAWK-DOE GAMES R. CRESSMAN ABSTRACT. Although two individuals in a biological species often interact
More informationAmbiguity and the Centipede Game
Ambiguity and the Centipede Game Jürgen Eichberger, Simon Grant and David Kelsey Heidelberg University, Australian National University, University of Exeter. University of Exeter. June 2018 David Kelsey
More informationAxiomatic Equilibrium Selection for Generic Two-Player Games
Stanford University From the SelectedWorks of Robert B Wilson May, 2009 Axiomatic Equilibrium Selection for Generic Two-Player Games Srihari Govindan Robert B Wilson Available at: https://works.bepress.com/wilson_robert/16/
More information