What is Evolutionary Game Theory?


 Emory Sharp
 1 years ago
 Views:
Transcription
1 What is Evolutionary Game Theory? Origins: Genetics and Biology Explain strategic aspects in evolution of species due to the possibility that individual fitness may depend on population frequency Basic Entities: Genes Basic Example Genes induce characteristics (behaviors) into organisms Fitness of a gene: # of offsprings (organisms) carrying some characteristics induced by that gene Selection in presence of frequencydependent fitness: The number of actual offsprings (organisms) embodying a given gene depends on: frequency of organisms embodying that gene; frequencies of organisms embodying all other genes. Evolutionary Model as a dynamic model whose law of motion reflects three basic forces: 1. Selection: Objects with higher fitness tend to spread 2. Mutation: Introduces new objects (variety) in the system 3. Inheritance: Transfers (successful) behavior across time
2 Why biology metaphor can be applied to social sciences?  Models as As if metaphors  Economic agents as simple, myopic, entities living in complicated environments (trial & error, learning)  Agents carry a set of simple behavioral rules ( genes ) allowing them to cope with the environment  Inheritance of BR through imitation and other learning mechanisms  Social and economic institutions act as selection devices (e.g. firms in markets)  Fitness of a gene depends on: realized outcomes induced by BR (i.e. realized profits) not only on frequency of same BR in the population but also on frequencies of other BR  Selection acts directly on realized outcomes and agents (e.g. firms) and indirectly on genes (only firms carrying genes that generate successful behaviors are reproduced)  Mutations as discoveries which introduce variety and novelty in genetic pool (BR)  Biological evolution understood as cultural evolution
3 Historical Development of EGT Models 1. Original Approach: Evolutionary Stable Strategies No explicit dynamics, no gametheoretic foundations Suppose a strategy is played by all agents in the population We ask: Is it stable to invasion of any mutant strategy? If yes: It is an ESS and can withstand to the evolutionary pressure of mutation, inheritance and selection 2. Gametheoretic approach: Det. Replicator Dynamics Population of agents repeatedly playing bilateral games Each agent is hardwired to play a certain strategy In each time period there are many random bilateral meetings (and payoffs) (Alternatively: Each agent plays against the field ) The number of offsprings each agent (strategy) will have will be higher the larger his payoff (fitness) (Alternatively: Each strategy will be adopted the more its fitness is higher) No mutations and novelty: We assume that mutations are independent and occur relatively infrequently; population is sufficiently large that one can reason in terms of expected values and deterministic processes
4 3. Gametheoretic approach: Stochastic Evolution As before but now mutations are relevant and occur very often Deterministic Force: Inheritance and Selection Stochastic Force: Mutations Variety  Do deterministic predictions change? EGT, Economics and Game Theory EGT vs. Evolutionary Models (EM)  Evolutionary Models as a general framework to describe economies where rationality, selection, heterogeneity, dynamics and endogenous novelty are central  EGT as a particular instance of EM  Why EGT has been so successful in economics and why is it so strongly related to games Models are simple and analytically solvable (cf. EM) Tool for addressing deficiencies in game theory o Hyperrationality o Dynamics o Equilibrium Selection
5 The HyperRationality Problem Standard NonCooperative Game Theory Assumptions  Fully Rational Players with Unbounded Computational Skills  Perfect knowledge of their own and opponent s strategies  Well defined preferences over uncountable sets of lotteries  Perfect knowledge of other preferences  Rationality is common knowledge Example: Prisoner dilemma (1,2) C D C (3,3) (0,5) D (5,0) (1,1) Agents do not play (C,C), (C,D) or (D,C) because everyone knows that the other will deviate as they know each other to be rational. Experimental results tell us that people are not rational If EGT is able to explain social behavior, then maybe rationality is not necessary!
6 The Dynamics Problem Standard NonCooperative Game Theory Assumptions  Games are typically played once  If games are multistage or repeated, rationality assumptions prevents one from appreciate dynamics entirely (all is compressed in a unique decision) EGT is inherently dynamic  Agents are myopic and repeatedly adapt to the environment in a trial & error fashion  The state of the system today (e.g. frequency with which a certain strategy is played in the population) depends on the past states of the system
7 The Equilibrium Selection Problem What if Multiple Equilibria Arise?  Example: Coordination Game  No inbuilt explanation: Everything can happen  Preplay commitment  Some equilibrium may be more appealing than others Perfect NE: A NE which does not involve playing weakly dominated strategies ChainStore Game Subgame Perfect Equilibrium many others  There are so many refinements that in any multiple equilibrium game, each equilibrium may be justified under some refinement!!  From equilibrium selection to refinement selection!!  EGT can be a useful tool to address (and solve) equilibrium selection problems Some questions that we will ask  Are strictly/weakly dominated strategies wiped away by selection (i.e. does EG select equilibria where agents play rationally)?  Can evolutionary processes select a NE?  Do evolutionary processes select among NE? And if yes, how? Do they select efficient equilibria?  What if no equilibria exist?
8 The ChainStore Game (1,2) Acquiesces Retaliates Stays Out (0,4) (0,4) Enters (2,2) ( 4, 4) 1=Potential Entrant; 2=Incumbent Nash Equilibria: (Stays Out, Retaliates), (Enters, Acquiesces) Which one will be played? (Stays Out, Retaliates) seems implausible i. If 2 plays R then it must take into account that if 1 enters he gets 4 ii. If 2 chooses R, it must be because he knows that the threat of retaliation will induce 1 to stay out iii. However, 1 may think that the threat is not credible, because 2 would choose A if he enters If fact, (Stays Out, Retaliates) is not a Perfect NE because involves playing weakly dominated strategies: for 2 strategy A does strictly better than R if 1 enters (2 4) and does at least as well as R if 1 stays out (4 4). Therefore one should eliminate R and concentrate on the game (1,2) Acquiesces Stays Out (0,4) Enters (2,2)
9 Two Important Issues 1. Selection, Rationality and Equilibrium Perfect rationality is not a widespread property of realworld agents (and often cannot be!) As if hypothesis: Equilibrium does not appear because agents are rational, but agents appear rational because only rationality can survive in equilibrium Agents can be assumed to behave as if they were rational But a behavior can be selected in equilibrium only if: (i) (ii) it is feasible and it is present from the start in the head of the agents the environment does not change during selection process. Both conditions are hardly met in reality! 2. Is EGT a Theory of Learning? Players are either hardwired or adaptive EGT is not a model of individual learning Agents do not start from some model of reality and they refine it across time EGT is a populationlevel learning theory: It is the population that learns over time
10 Important Concepts in GameTheory Focus: Noncooperative, finite, games in normal form Set of players: I = {1, 2 } Set of pure strategies: S i = { s 1, s 2,, s k }, S = i S i = S 1 S 2 Payoff Matrices (G is k x k) Mixed Strategies 1. Symmetric Games: G (i) = {g (i) hl}, i=1,2, G (2) = [G (1) ] T 2. Asymmetric Games: G (i) = {g (i) hl},  Probability Distribution over S i : x i = { x i1, x i2,, x ik } k  Subjective vs. frequency interpretation  Pure strategies = vertices of k  Population Profile: x =(x 1,, x n )  Expected payoffs u i when the profile is x Example: Symmetric Game with k=2 strategies Player 1: x 1 = (x 11, x 12 )=(x 11, 1 x 11 ) Player 2: x 2 = (x 21, x 22 )=(x 21, 1 x 21 ) u i (x) = x i G (i) x j, i j  Expected payoffs u i when i plays a pure strategy h and j i plays some other strategy z j u i ( e h i, z j ) = e h i G (i) z j = G (h) (i) z j
11 Weak and strict dominance (for some agent i)  x i WD y i u i ( x i, z ) u i ( y i, z ) all z, > for some z  x i SD y i u i ( x i, z ) > u i ( y i, z ) all z Example for pure strategies (symmetric PD game) (1,2) C D C (3,3) (0,5) D (5,0) (1,1) Iteratively Strictly Dominated Strategies and Strictly Dominance Solvable games  SD pure strategies are never played by rational players (and thus may be eliminated)  Requirement: Everybody knows about each other payoffs (so that they can eliminate each other s SD strategies)  A pure strategy is not iteratively strictly dominated if it is not SD in any reduced game (1) A B C A B C NB: Symmetry!
12 BestReplies  Pure: β i (y) = {h S i : u i ( e h i, y) u i ( e l i, y), all l S i }  Mixed: α i (y) = {x i : u i (x i, y ) u i (z i, y ), all z i } Nash Equilibrium  A mixed strategy profile x is a NE iff x α(x) = i α i (x) 2 players, symm: x is a NE x G x y G x for all y A pure strategy profile (s 1,, s n ) is a NE iff any player will not deviate unilaterally  A NE is Strict iff α(x) = { x }, i.e. there are no alternative BR. A strict NE cannot involve mixed strategies (by convexity)!  Let Θ NE the set of all NE of the game; for any finite game there is at least one NE in mixed strategies, but not necessarily a pure strategy one!  Example. No pure NE, unique mixed NE: (½,½), (½,½) (1,2) A B A (1,1) (1,1) B (1,1) (1,1)  NE is invariant to positive affine transformations of all payoffs and to additions of some constant number to all payoffs to a given player associated to a given pure strategy played by the opponent (i.e. adding a number to a given column)  A NE cannot be SD but it can be WD (see EntryDeterrence)!!
13 NE Refinements: TremblingHand Perfection  A (TremblingHand) Perfect NE is a NE that is robust to trembles or mistakes in strategy plays  Consider for each player i some numbers µ i = (µ i1,, µ ik ) where µ ih (0,1) is the probability with which i plays h by mistake  Consider the perturbed game associated to the original game and compute the new set of NE: Θ NE (µ)  Then x is a PNE iff, for some sequence µ 0, there exists a sequence of x(µ) Θ NE (µ) such that x(µ) x  Important If x is a PNE then it is not WD In a 2player game, any NE which is not WD is a PNE Hence in a 2player game, x is a PNE iff it is not WD
14 Two Player Symmetric Games  I = {1,2}, S 1 = S 2  Payoff matrices are s.t. G (1) = (G (2) ) T  Example: Two Strategies, S 1 = S 2 = {1,2} G (1) 1 2 G (2) g' 11 g' 12 1 g" 11 g" 12 2 g' 21 g' 22 2 g" 21 g" 22 g' 11 = g" 11, g' 22 = g" 22, g' 12 = g" 21 and g' 21 = g" 12 PD 1 2 MP 1 2 ED (4,4) (0,5) 1 (1,1) (1,1) 1 (2,2) (0,0) 2 (5,0) (3,3) 2 (1,1) (1,1) 2 (1,4) (1,4) Two Player DoublySymmetric Games  TwoPlayer Symmetric Game where G (i) = [G (i) ] T, i =1,2  Example: Two Strategies g' 12 = g' 21, g 12 = g (2,2) (0,0) 2 (0,0) (1,1) 2 player Symmetric NE: Is a NE (x,y) where x=y i.e. where both players use the same (mixed or pure) strategy
15 Exercises Find NE in mixed and pure strategies for the games: 1. Prisoner Dilemma Game (Weibull, Ex 1.1, p.2) 2. Coordination Game (Weibull, Ex 1.10, p.26) 3. HawkDove Game (Weibull, Ex 1.11, p.27) 4. RockScissorPaper Game (Weibull, Ex 1.12, p.28) Recalling the invariance results about NE, reelaborate in all details the classification in Weibull, par , p Find for any of the 4 categories the NE of the game. 2. Find why games 1,2 and 3 of the exercise above belong to one of the 4 categories 3. Which is the difference between Pareto efficient and risk dominant equilibria in a coordination game (p.31)?
10 Evolutionarily Stable Strategies
10 Evolutionarily Stable Strategies There is but a step between the sublime and the ridiculous. Leo Tolstoy In 1973 the biologist John Maynard Smith and the mathematician G. R. Price wrote an article in
More informationECON 40050 Game Theory Exam 1  Answer Key. 4) All exams must be turned in by 1:45 pm. No extensions will be granted.
1 ECON 40050 Game Theory Exam 1  Answer Key Instructions: 1) You may use a pen or pencil, a handheld nonprogrammable calculator, and a ruler. No other materials may be at or near your desk. Books, coats,
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitelyrepeated prisoner s dilemma
More informationECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY 1003 9099 21 8089 14 7079 4 069 11
The distribution of grades was as follows. ECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY Range Numbers 1003 9099 21 8089 14 7079 4 069 11 Question 1: 30 points Games
More information6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games
6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1 Introduction Outline Decisions, utility maximization Strategic form games Best responses
More informationJeux finiment répétés avec signaux semistandards
Jeux finiment répétés avec signaux semistandards P. ContouCarrère 1, T. Tomala 2 CEPNLAGA, Université Paris 13 7 décembre 2010 1 Université Paris 1, Panthéon Sorbonne 2 HEC, Paris Introduction Repeated
More informationChapter 7. Evolutionary Game Theory
From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2010. Complete preprint online at http://www.cs.cornell.edu/home/kleinber/networksbook/
More information6.254 : Game Theory with Engineering Applications Lecture 1: Introduction
6.254 : Game Theory with Engineering Applications Lecture 1: Introduction Asu Ozdaglar MIT February 2, 2010 1 Introduction Optimization Theory: Optimize a single objective over a decision variable x R
More information0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text)
September 2 Exercises: Problem 2 (p. 21) Efficiency: p. 2829: 1, 4, 5, 6 0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text) We discuss here a notion of efficiency that is rooted in the individual preferences
More informationA Game Theoretic Approach to Traffic Flow Control. (MultiAgent Systems: Paper Project) May 2004
A Game Theoretic Approach to Traffic Flow Control (MultiAgent Systems: Paper Project) May 2004 Authors Jin Yu Enrico Faldini Programme Master in Artificial Intelligence Master in Artificial Intelligence
More informationMidterm exam PS 30 November 2005
Midterm exam PS 30 November 2005 Name: TA: Section number: This is a closed book exam. The only thing you can take into this exam is yourself and writing instruments. Everything you write should be your
More informationThéorie de la décision et théorie des jeux Stefano Moretti
héorie de la décision et théorie des jeux Stefano Moretti UMR 7243 CNRS Laboratoire d'analyse et Modélisation de Systèmes pour l'aide à la décision (LAMSADE) Université ParisDauphine email: Stefano.MOREI@dauphine.fr
More informationAN INTRODUCTION TO GAME THEORY
AN INTRODUCTION TO GAME THEORY 2008 AGIInformation Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. MARTIN J. OSBORNE University of Toronto
More informationLecture V: Mixed Strategies
Lecture V: Mixed Strategies Markus M. Möbius February 26, 2008 Osborne, chapter 4 Gibbons, sections 1.31.3.A 1 The Advantage of Mixed Strategies Consider the following RockPaperScissors game: Note that
More informationOnline Appendix to Stochastic Imitative Game Dynamics with Committed Agents
Online Appendix to Stochastic Imitative Game Dynamics with Committed Agents William H. Sandholm January 6, 22 O.. Imitative protocols, mean dynamics, and equilibrium selection In this section, we consider
More informationComputational Learning Theory Spring Semester, 2003/4. Lecture 1: March 2
Computational Learning Theory Spring Semester, 2003/4 Lecture 1: March 2 Lecturer: Yishay Mansour Scribe: Gur Yaari, Idan Szpektor 1.1 Introduction Several fields in computer science and economics are
More informationMonopolistic Competition, Oligopoly, and maybe some Game Theory
Monopolistic Competition, Oligopoly, and maybe some Game Theory Now that we have considered the extremes in market structure in the form of perfect competition and monopoly, we turn to market structures
More informationEquilibrium computation: Part 1
Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium
More informationGames Manipulators Play
Games Manipulators Play Umberto Grandi Department of Mathematics University of Padova 23 January 2014 [Joint work with Edith Elkind, Francesca Rossi and Arkadii Slinko] GibbardSatterthwaite Theorem All
More informationBackward Induction and Subgame Perfection
Backward Induction and Subgame Perfection In extensiveform games, we can have a Nash equilibrium profile of strategies where player 2 s strategy is a best response to player 1 s strategy, but where she
More informationGame Theory in Wireless Networks: A Tutorial
1 Game heory in Wireless Networks: A utorial Mark Felegyhazi, JeanPierre Hubaux EPFL Switzerland email: {mark.felegyhazi, jeanpierre.hubaux}@epfl.ch EPFL echnical report: LCAREPOR2006002, submitted
More informationNash Equilibrium. Ichiro Obara. January 11, 2012 UCLA. Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31
Nash Equilibrium Ichiro Obara UCLA January 11, 2012 Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31 Best Response and Nash Equilibrium In many games, there is no obvious choice (i.e. dominant action).
More informationAvoiding the Consolation Prize: The Mathematics of Game Shows
Avoiding the Consolation Prize: The Mathematics of Game Shows STUART GLUCK, Ph.D. CENTER FOR TALENTED YOUTH JOHNS HOPKINS UNIVERSITY STU@JHU.EDU CARLOS RODRIGUEZ CENTER FOR TALENTED YOUTH JOHNS HOPKINS
More informationGame Theory in Wireless Networks: A Tutorial
1 Game Theory in ireless Networks: Tutorial Mark Felegyhazi, JeanPierre Hubaux EPFL Switzerland email: {mark.felegyhazi, jeanpierre.hubaux}@epfl.ch EPFL Technical report: LCREPORT2006002, submitted
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationFINAL EXAM, Econ 171, March, 2015, with answers
FINAL EXAM, Econ 171, March, 2015, with answers There are 9 questions. Answer any 8 of them. Good luck! Problem 1. (True or False) If a player has a dominant strategy in a simultaneousmove game, then
More informationWEAK DOMINANCE: A MYSTERY CRACKED
WEAK DOMINANCE: A MYSTERY CRACKED JOHN HILLAS AND DOV SAMET Abstract. What strategy profiles can be played when it is common knowledge that weakly dominated strategies are not played? A comparison to the
More informationHow to Solve Strategic Games? Dominant Strategies
How to Solve Strategic Games? There are three main concepts to solve strategic games: 1. Dominant Strategies & Dominant Strategy Equilibrium 2. Dominated Strategies & Iterative Elimination of Dominated
More informationSociobiology and Altruism
Sociobiology and Altruism E. O. Wilson: Sociobiology: The new synthesis (1975) Most of the book deals with ants and ant social behavior Last chapter: Human Sociobiology Human behavioral traits are adaptations
More informationSummary of Doctoral Dissertation: Voluntary Participation Games in Public Good Mechanisms: Coalitional Deviations and Efficiency
Summary of Doctoral Dissertation: Voluntary Participation Games in Public Good Mechanisms: Coalitional Deviations and Efficiency Ryusuke Shinohara 1. Motivation The purpose of this dissertation is to examine
More informationCompetition and Regulation. Lecture 2: Background on imperfect competition
Competition and Regulation Lecture 2: Background on imperfect competition Monopoly A monopolist maximizes its profits, choosing simultaneously quantity and prices, taking the Demand as a contraint; The
More informationGame Theory and Algorithms Lecture 10: Extensive Games: Critiques and Extensions
Game Theory and Algorithms Lecture 0: Extensive Games: Critiques and Extensions March 3, 0 Summary: We discuss a game called the centipede game, a simple extensive game where the prediction made by backwards
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationGames Played in a Contracting Environment
Games Played in a Contracting Environment V. Bhaskar Department of Economics University College London Gower Street London WC1 6BT February 2008 Abstract We analyze normal form games where a player has
More informationGames of Incomplete Information
Games of Incomplete Information Jonathan Levin February 00 Introduction We now start to explore models of incomplete information. Informally, a game of incomplete information is a game where the players
More informationRepeated Games and Reputations
1 / 1 Repeated Games and Reputations George J Mailath UPenn June 26, 2014 9th Tinbergen Institute Conference: 70 Years of Theory of Games and Economic Behaviour The slides and associated bibliography are
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1
More informationA Note on Best Response Dynamics
Games and Economic Behavior 29, 138 150 (1999) Article ID game.1997.0636, available online at http://www.idealibrary.com on A Note on Best Response Dynamics Ed Hopkins Department of Economics, University
More informationOligopoly markets: The price or quantity decisions by one rm has to directly in uence pro ts by other rms if rms are competing for customers.
15 Game Theory Varian: Chapters 89. The key novelty compared to the competitive (Walrasian) equilibrium analysis is that game theoretic analysis allows for the possibility that utility/pro t/payo s depend
More informationOn Stability Properties of Economic Solution Concepts
On Stability Properties of Economic Solution Concepts Richard J. Lipton Vangelis Markakis Aranyak Mehta Abstract In this note we investigate the stability of game theoretic and economic solution concepts
More informationEvolutionary dynamics and backward induction
Games and Economic Behavior 41 (2002) 227 264 www.elsevier.com/locate/geb Evolutionary dynamics and backward induction Sergiu Hart Center for Rationality and Interactive Decision Theory, Department of
More informationMistakes Allow Evolutionary Stability in the Repeated Prisoner's Dilemma Game. I. Introduction
J. theor. Biol. (1989) 136, 4756 Mistakes Allow Evolutionary Stability in the Repeated Prisoner's Dilemma Game ROBERT BOYD Department of Anthropology, University of California, Los Angeles, California
More informationInternet Advertising and the Generalized Second Price Auction:
Internet Advertising and the Generalized Second Price Auction: Selling Billions of Dollars Worth of Keywords Ben Edelman, Harvard Michael Ostrovsky, Stanford GSB Michael Schwarz, Yahoo! Research A Few
More informationLearning to Play 33 Games: Neural Networks as BoundedRational Players Technical Appendix
Learning to Play 33 Games: Neural Networks as BoundedRational Players Technical Appendix Daniel Sgroi 1 Daniel J. Zizzo 2 Department of Economics School of Economics, University of Warwick University
More informationTowards a compliance audit of SLAs for data replication in Cloud storage
Towards a compliance audit of SLAs for data replication in Cloud storage J. Leneutre B. Djebaili, C. Kiennert, J. Leneutre, L. Chen, Data Integrity and Availability Verification Game in Untrusted Cloud
More informationGenetic Drift Simulation. Experimental Question: How do random events cause evolution (a change in the gene pool)?
Genetic Drift Simulation Experimental Question: How do random events cause evolution (a change in the gene pool)? Hypothesis: Introduction: What is Genetic Drift? Let's examine a simple model of a population
More informationSolution Concepts. Jonathan Levin. April 2006
Solution Concepts Jonathan Levin April 2006 These notes discuss some of the central solution concepts for normalform games: Nash and correlated equilibrium, iterated deletion of strictly dominated strategies,
More informationEvolution, Natural Selection, and Adaptation
Evolution, Natural Selection, and Adaptation Nothing in biology makes sense except in the light of evolution. (Theodosius Dobzhansky) Charles Darwin (18091882) Voyage of HMS Beagle (18311836) Thinking
More informationThe Basics of Game Theory
Sloan School of Management 15.010/15.011 Massachusetts Institute of Technology RECITATION NOTES #7 The Basics of Game Theory Friday  November 5, 2004 OUTLINE OF TODAY S RECITATION 1. Game theory definitions:
More information6.1 What is a Game? 156 CHAPTER 6. GAMES
From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2010. Complete preprint online at http://www.cs.cornell.edu/home/kleinber/networksbook/
More informationGAMES FOR BUSINESS AND ECONOMICS
GAMES FOR BUSINESS AND ECONOMICS ROY/GARDNER Indiana University Nachrichtentechnische BibliotHek TUD Inv.Nr.: /S.JOtUM John Wiley & Sons, Inc. 5" New York Chichester Brisbane Toronto Singapore Contents
More informationGame Theory 1. Introduction
Game Theory 1. Introduction Dmitry Potapov CERN What is Game Theory? Game theory is about interactions among agents that are selfinterested I ll use agent and player synonymously Selfinterested: Each
More informationPrice competition with homogenous products: The Bertrand duopoly model [Simultaneous move price setting duopoly]
ECON9 (Spring 0) & 350 (Tutorial ) Chapter Monopolistic Competition and Oligopoly (Part ) Price competition with homogenous products: The Bertrand duopoly model [Simultaneous move price setting duopoly]
More informationGame Theory: Supermodular Games 1
Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution
More informationModules 5: Behavior Genetics and Evolutionary Psychology
Modules 5: Behavior Genetics and Evolutionary Psychology Source of similarities and differences Similarities with other people such as developing a languag, showing similar emotions, following similar
More informationNash and game theory
Nash and game theory Antonio Cabrales 1 I am asked to give my view on the contribution of John Nash to the development of game theory. Since I have received most of my early influence through textbooks,
More informationMidterm Advanced Economic Theory, ECO326F1H Marcin Pęski February, 2015
Midterm Advanced Economic Theory, ECO326F1H Marcin Pęski February, 2015 There are three questions with total worth of 100 points. Read the questions carefully. You must give a supporting argument and an
More informationDo not open this exam until told to do so.
Do not open this exam until told to do so. Department of Economics College of Social and Applied Human Sciences K. Annen, Winter 004 Final (Version ): Intermediate Microeconomics (ECON30) Solutions Final
More informationGame Theory. An introduction to the concepts of dominant strategies, Nash equilibrium and strategic commitment
Game Theory An introduction to the concepts of dominant strategies, Nash equilibrium and strategic commitment Introduction The theory of games is a theory of economic behaviour in multiperson decision
More informationComputational Game Theory and Clustering
Computational Game Theory and Clustering Martin Hoefer mhoefer@mpiinf.mpg.de 1 Computational Game Theory? 2 Complexity and Computation of Equilibrium 3 Bounding Inefficiencies 4 Conclusion Computational
More informationUCLA. Department of Economics Ph. D. Preliminary Exam MicroEconomic Theory
UCLA Department of Economics Ph. D. Preliminary Exam MicroEconomic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.
More informationBiform Games: Additional Online Material
Biform Games: Additional Online Material Adam Brandenburger Harborne Stuart July 2006 These appendices supplement Brandenburger and Stuart [1, 2006], [2, 2006] ( Biform Games ). Appendix C uses the efficiency
More informationIntroduction. Bargaining  whether over arms control, the terms of a peace settlement, exchange rate
Bargaining in International Relations Introduction Bargaining  whether over arms control, the terms of a peace settlement, exchange rate coordination, alliances, or trade agreements  is a central feature
More informationLecture 11: Oligopoly and Strategic Behavior
Lecture 11: Oligopoly and Strategic Behavior Few Firms in the Market: Each aware of others actions Each firm in the industry has market power Entry is Feasible, although incumbent(s) may try to deter it.
More informationInfinitely Repeated Games with Discounting Ù
Infinitely Repeated Games with Discounting Page 1 Infinitely Repeated Games with Discounting Ù Introduction 1 Discounting the future 2 Interpreting the discount factor 3 The average discounted payoff 4
More informationBiology 1406  Notes for exam 5  Population genetics Ch 13, 14, 15
Biology 1406  Notes for exam 5  Population genetics Ch 13, 14, 15 Species  group of individuals that are capable of interbreeding and producing fertile offspring; genetically similar 13.7, 14.2 Population
More informationGame Theory. Themes. 1. Introduction to Game Theory 2. Sequential Games 3. Simultaneous Games 4. Conclusion. Introduction to Game Theory
Game Theory Themes 1. Introduction to Game Theory 2. Sequential Games 3. Simultaneous Games 4. Conclusion Introduction to Game Theory Game theory is the branch of decision theory concerned with interdependent
More informationSequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium
Sequential lmove Games Using Backward Induction (Rollback) to Find Equilibrium Sequential Move Class Game: Century Mark Played by fixed pairs of players taking turns. At each turn, each player chooses
More informationGame Theory. CDAM Research Report LSECDAM200109 October 8, 2001. 1 What is game theory? 4. 2 Definitions of games 6.
Game Theory Theodore L. Turocy Texas A&M University Bernhard von Stengel London School of Economics CDAM Research Report LSECDAM29 October 8, 2 Contents What is game theory? 4 2 Definitions of games
More informationCongestion Games with PlayerSpecific Payoff Functions
GAMES AND ECONOMIC BEHAVIOR 13, 111 124 (1996) ARTICLE NO. 0027 Congestion Games with PlayerSpecific Payoff Functions Igal Milchtaich Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem
More informationSimon Fraser University Spring 2015. Econ 302 D200 Final Exam Solution Instructor: Songzi Du Tuesday April 21, 2015, 12 3 PM
Simon Fraser University Spring 2015 Econ 302 D200 Final Exam Solution Instructor: Songzi Du Tuesday April 21, 2015, 12 3 PM The brief solutions suggested here may not have the complete explanations necessary
More informationOligopoly is a market structure more susceptible to gametheoretic analysis, because of apparent strategic interdependence among a few producers.
1 Market structure from a gametheoretic perspective: Oligopoly After our more theoretical analysis of different zerosum and variablesum games, let us return to the more familiar territory of economicsespecially
More informationHistory Independent Prediction in Evolutionary Game Theory *
History Independent Prediction in Evolutionary Game Theory * William H. Sandholm MES KGSM Northwestern niversity Evanston, IL 60208,.S.A. email: whs@nwu.edu January 28, 1998 Forthcoming in Rationality
More information6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium
6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline PricingCongestion Game Example Existence of a Mixed
More informationOn the Evolution of Behavioral Heterogeneity in Individuals and Populations
Biology and Philosophy 13: 205 231, 1998. c 1998 Kluwer Academic Publishers. Printed in the Netherlands. On the Evolution of Behavioral Heterogeneity in Individuals and Populations CARL T. BERGSTROM Department
More informationHeterogeneous Expectations, Adaptive Learning, and Evolutionary Dynamics
Heterogeneous Expectations, Adaptive Learning, and Evolutionary Dynamics Eran A. Guse West Virginia University November 006 (Revised December 008) Abstract This paper presents a linear selfreferential
More informationMicroeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012. (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3
Microeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012 1. Subgame Perfect Equilibrium and Dominance (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3 Highlighting
More informationNetwork Security Validation Using Game Theory
Network Security Validation Using Game Theory Vicky Papadopoulou and Andreas Gregoriades Computer Science and Engineering Dep., European University Cyprus, Cyprus {v.papadopoulou,a.gregoriades}@euc.ac.cy
More informationGame Mining: How to Make Money from those about to Play a Game
Game Mining: How to Make Money from those about to Play a Game David H. Wolpert NASA Ames Research Center MailStop 2691 Moffett Field, CA 940351000 david.h.wolpert@nasa.gov James W. Bono Department of
More informationApplication of Game Theory in Inventory Management
Application of Game Theory in Inventory Management Rodrigo TranamilVidal Universidad de Chile, Santiago de Chile, Chile Rodrigo.tranamil@ug.udechile.cl Abstract. Game theory has been successfully applied
More informationEvolution and Information in a GiftGiving Game *
Evolution and nformation in a GiftGiving Game * By Phillip Johnson, (johnsonp@yahoo.com) Centro de nvestigacion Economía, nstituto echnologico Autonomo de Mexico, David K. Levine, (dlevine@ucla.edu) Department
More informationPrice Dispersion. Ed Hopkins Economics University of Edinburgh Edinburgh EH8 9JY, UK. November, 2006. Abstract
Price Dispersion Ed Hopkins Economics University of Edinburgh Edinburgh EH8 9JY, UK November, 2006 Abstract A brief survey of the economics of price dispersion, written for the New Palgrave Dictionary
More informationIndividual security and network design
Individual security and network design Diego Cerdeiro Marcin Dziubiński Sanjeev Goyal FIT 2015 Motivation Networks often face external threats in form of strategic or random attacks The attacks can be
More informationFairness with an Honest Minority and a Rational Majority
Fairness with an Honest Minority and a Rational Majority The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Published
More informationSummary. 16 1 Genes and Variation. 16 2 Evolution as Genetic Change. Name Class Date
Chapter 16 Summary Evolution of Populations 16 1 Genes and Variation Darwin s original ideas can now be understood in genetic terms. Beginning with variation, we now know that traits are controlled by
More informationBayesian Nash Equilibrium
. Bayesian Nash Equilibrium . In the final two weeks: Goals Understand what a game of incomplete information (Bayesian game) is Understand how to model static Bayesian games Be able to apply Bayes Nash
More informationA Game Theoretical Framework for Adversarial Learning
A Game Theoretical Framework for Adversarial Learning Murat Kantarcioglu University of Texas at Dallas Richardson, TX 75083, USA muratk@utdallas Chris Clifton Purdue University West Lafayette, IN 47907,
More informationWorking Paper Series
RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos HervésBeloso, Emma Moreno García and
More informationSimple ChannelChange Games for Spectrum Agile Wireless Networks
Simple ChannelChange Games for Spectrum Agile Wireless Networks Roli G. Wendorf and Howard Blum Seidenberg School of Computer Science and Information Systems Pace University White Plains, New York, USA
More informationSealed Bid Second Price Auctions with Discrete Bidding
Sealed Bid Second Price Auctions with Discrete Bidding Timothy Mathews and Abhijit Sengupta August 16, 2006 Abstract A single item is sold to two bidders by way of a sealed bid second price auction in
More informationComplementarities in information acquisition with shortterm trades
Theoretical Economics (007), 441 467 15557561/0070441 Complementarities in information acquisition with shortterm trades CHRISTOPHE CHAMLEY ParisJourdan Sciences Économiques and Department of Economics,
More informationMicroevolution is a change in population s gene pool [1]
GUIDED READING  Ch. 14  Section 4 NAME: Please print out these pages and HANDWRITE the answers directly on the printouts. Typed work or answers on separate sheets of paper will not be accepted. Importantly,
More informationPatent Litigation with Endogenous Disputes
Patent itigation with Endogenous Disputes Presentation at the AEA Annual Meeting January 6, 006 Session: Intellectual Property, itigation, and Innovation By James Bessen and Michael J. Meurer James Bessen
More informationPerfect Bayesian Equilibrium
Perfect Bayesian Equilibrium When players move sequentially and have private information, some of the Bayesian Nash equilibria may involve strategies that are not sequentially rational. The problem is
More informationMidterm March 2015. (a) Consumer i s budget constraint is. c i 0 12 + b i c i H 12 (1 + r)b i c i L 12 (1 + r)b i ;
Masters in EconomicsUC3M Microeconomics II Midterm March 015 Exercise 1. In an economy that extends over two periods, today and tomorrow, there are two consumers, A and B; and a single perishable good,
More informationUnderstanding by Design. Title: BIOLOGY/LAB. Established Goal(s) / Content Standard(s): Essential Question(s) Understanding(s):
Understanding by Design Title: BIOLOGY/LAB Standard: EVOLUTION and BIODIVERSITY Grade(s):9/10/11/12 Established Goal(s) / Content Standard(s): 5. Evolution and Biodiversity Central Concepts: Evolution
More informationOligopoly: Cournot/Bertrand/Stackelberg
Outline Alternative Market Models Wirtschaftswissenschaften Humboldt Universität zu Berlin March 5, 2006 Outline 1 Introduction Introduction Alternative Market Models 2 Game, Reaction Functions, Solution
More informationIntroduction To Genetic Algorithms
1 Introduction To Genetic Algorithms Dr. Rajib Kumar Bhattacharjya Department of Civil Engineering IIT Guwahati Email: rkbc@iitg.ernet.in References 2 D. E. Goldberg, Genetic Algorithm In Search, Optimization
More informationWeek 7  Game Theory and Industrial Organisation
Week 7  Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number
More information9 Repeated Games. Tomorrow, and tomorrow, and tomorrow, Creeps in this petty pace from day to day To the last syllable of recorded time Shakespeare
9 Repeated Games Tomorrow, and tomorrow, and tomorrow, Creeps in this petty pace from day to day To the last syllable of recorded time Shakespeare When a game G is repeated an indefinite number of times
More information