What is Evolutionary Game Theory?

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "What is Evolutionary Game Theory?"

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 frequency-dependent 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 game-theoretic 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. Game-theoretic 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. Game-theoretic 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 Hyper-Rationality Problem Standard Non-Cooperative 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 Non-Cooperative Game Theory Assumptions - Games are typically played once - If games are multi-stage 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 in-built explanation: Everything can happen - Pre-play commitment - Some equilibrium may be more appealing than others Perfect NE: A NE which does not involve playing weakly dominated strategies Chain-Store Game Sub-game 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 Chain-Store 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 real-world 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 hard-wired 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 population-level learning theory: It is the population that learns over time

10 Important Concepts in Game-Theory Focus: Non-cooperative, 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 Best-Replies - 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 Entry-Deterrence)!!

13 NE Refinements: Trembling-Hand Perfection - A (Trembling-Hand) 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 2-player game, any NE which is not WD is a PNE Hence in a 2-player 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 Doubly-Symmetric Games - Two-Player 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. Hawk-Dove Game (Weibull, Ex 1.11, p.27) 4. Rock-Scissor-Paper Game (Weibull, Ex 1.12, p.28) Recalling the invariance results about NE, re-elaborate 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 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 information

ECON 40050 Game Theory Exam 1 - Answer Key. 4) All exams must be turned in by 1:45 pm. No extensions will be granted.

ECON 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 hand-held nonprogrammable calculator, and a ruler. No other materials may be at or near your desk. Books, coats,

More information

6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation

6.207/14.15: 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 Finitely-repeated prisoner s dilemma

More information

ECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY 100-3 90-99 21 80-89 14 70-79 4 0-69 11

ECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY 100-3 90-99 21 80-89 14 70-79 4 0-69 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 100-3 90-99 21 80-89 14 70-79 4 0-69 11 Question 1: 30 points Games

More information

6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games

6.254 : 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 information

Jeux finiment répétés avec signaux semi-standards

Jeux finiment répétés avec signaux semi-standards Jeux finiment répétés avec signaux semi-standards P. Contou-Carrère 1, T. Tomala 2 CEPN-LAGA, Université Paris 13 7 décembre 2010 1 Université Paris 1, Panthéon Sorbonne 2 HEC, Paris Introduction Repeated

More information

Chapter 7. Evolutionary Game Theory

Chapter 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 on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/

More information

6.254 : Game Theory with Engineering Applications Lecture 1: Introduction

6.254 : 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 information

0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text)

0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text) September 2 Exercises: Problem 2 (p. 21) Efficiency: p. 28-29: 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 information

A Game Theoretic Approach to Traffic Flow Control. (Multi-Agent Systems: Paper Project) May 2004

A Game Theoretic Approach to Traffic Flow Control. (Multi-Agent Systems: Paper Project) May 2004 A Game Theoretic Approach to Traffic Flow Control (Multi-Agent Systems: Paper Project) May 2004 Authors Jin Yu Enrico Faldini Programme Master in Artificial Intelligence Master in Artificial Intelligence

More information

Midterm exam PS 30 November 2005

Midterm 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 information

Théorie de la décision et théorie des jeux Stefano Moretti

Théorie 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 Paris-Dauphine email: Stefano.MOREI@dauphine.fr

More information

AN INTRODUCTION TO GAME THEORY

AN INTRODUCTION TO GAME THEORY AN INTRODUCTION TO GAME THEORY 2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. MARTIN J. OSBORNE University of Toronto

More information

Lecture V: Mixed Strategies

Lecture V: Mixed Strategies Lecture V: Mixed Strategies Markus M. Möbius February 26, 2008 Osborne, chapter 4 Gibbons, sections 1.3-1.3.A 1 The Advantage of Mixed Strategies Consider the following Rock-Paper-Scissors game: Note that

More information

Online Appendix to Stochastic Imitative Game Dynamics with Committed Agents

Online 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 information

Computational Learning Theory Spring Semester, 2003/4. Lecture 1: March 2

Computational 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 information

Monopolistic Competition, Oligopoly, and maybe some Game Theory

Monopolistic 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 information

Equilibrium computation: Part 1

Equilibrium 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 information

Games Manipulators Play

Games 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] Gibbard-Satterthwaite Theorem All

More information

Backward Induction and Subgame Perfection

Backward Induction and Subgame Perfection Backward Induction and Subgame Perfection In extensive-form 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 information

Game Theory in Wireless Networks: A Tutorial

Game Theory in Wireless Networks: A Tutorial 1 Game heory in Wireless Networks: A utorial Mark Felegyhazi, Jean-Pierre Hubaux EPFL Switzerland email: {mark.felegyhazi, jean-pierre.hubaux}@epfl.ch EPFL echnical report: LCA-REPOR-2006-002, submitted

More information

Nash Equilibrium. Ichiro Obara. January 11, 2012 UCLA. Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31

Nash 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 information

Avoiding the Consolation Prize: The Mathematics of Game Shows

Avoiding 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 information

Game Theory in Wireless Networks: A Tutorial

Game Theory in Wireless Networks: A Tutorial 1 Game Theory in ireless Networks: Tutorial Mark Felegyhazi, Jean-Pierre Hubaux EPFL Switzerland email: {mark.felegyhazi, jean-pierre.hubaux}@epfl.ch EPFL Technical report: LC-REPORT-2006-002, submitted

More information

Chapter 7. Sealed-bid Auctions

Chapter 7. Sealed-bid Auctions Chapter 7 Sealed-bid 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 information

FINAL EXAM, Econ 171, March, 2015, with answers

FINAL 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 simultaneous-move game, then

More information

WEAK DOMINANCE: A MYSTERY CRACKED

WEAK 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 information

How to Solve Strategic Games? Dominant Strategies

How 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 information

Sociobiology and Altruism

Sociobiology 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 information

Summary 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 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 information

Competition and Regulation. Lecture 2: Background on imperfect competition

Competition 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 information

Game Theory and Algorithms Lecture 10: Extensive Games: Critiques and Extensions

Game 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 information

Moral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania

Moral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically

More information

Games Played in a Contracting Environment

Games 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 information

Games of Incomplete Information

Games 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 information

Repeated Games and Reputations

Repeated 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 information

ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015

ECON 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 information

A Note on Best Response Dynamics

A 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 information

Oligopoly 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.

Oligopoly 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 8-9. 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 information

On Stability Properties of Economic Solution Concepts

On 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 information

Evolutionary dynamics and backward induction

Evolutionary 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 information

Mistakes Allow Evolutionary Stability in the Repeated Prisoner's Dilemma Game. I. Introduction

Mistakes Allow Evolutionary Stability in the Repeated Prisoner's Dilemma Game. I. Introduction J. theor. Biol. (1989) 136, 47-56 Mistakes Allow Evolutionary Stability in the Repeated Prisoner's Dilemma Game ROBERT BOYD Department of Anthropology, University of California, Los Angeles, California

More information

Internet Advertising and the Generalized Second Price Auction:

Internet 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 information

Learning to Play 33 Games: Neural Networks as Bounded-Rational Players Technical Appendix

Learning to Play 33 Games: Neural Networks as Bounded-Rational Players Technical Appendix Learning to Play 33 Games: Neural Networks as Bounded-Rational Players Technical Appendix Daniel Sgroi 1 Daniel J. Zizzo 2 Department of Economics School of Economics, University of Warwick University

More information

Towards a compliance audit of SLAs for data replication in Cloud storage

Towards 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 information

Genetic 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)? 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 information

Solution Concepts. Jonathan Levin. April 2006

Solution 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 information

Evolution, Natural Selection, and Adaptation

Evolution, Natural Selection, and Adaptation Evolution, Natural Selection, and Adaptation Nothing in biology makes sense except in the light of evolution. (Theodosius Dobzhansky) Charles Darwin (1809-1882) Voyage of HMS Beagle (1831-1836) Thinking

More information

The Basics of Game Theory

The 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 information

6.1 What is a Game? 156 CHAPTER 6. GAMES

6.1 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 on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/

More information

GAMES FOR BUSINESS AND ECONOMICS

GAMES 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 information

Game Theory 1. Introduction

Game Theory 1. Introduction Game Theory 1. Introduction Dmitry Potapov CERN What is Game Theory? Game theory is about interactions among agents that are self-interested I ll use agent and player synonymously Self-interested: Each

More information

Price competition with homogenous products: The Bertrand duopoly model [Simultaneous move price setting duopoly]

Price 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 information

Game Theory: Supermodular Games 1

Game 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 information

Modules 5: Behavior Genetics and Evolutionary Psychology

Modules 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 information

Nash and game theory

Nash 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 information

Midterm Advanced Economic Theory, ECO326F1H Marcin Pęski February, 2015

Midterm 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 information

Do not open this exam until told to do so.

Do 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 information

Game 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 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 multi-person decision

More information

Computational Game Theory and Clustering

Computational Game Theory and Clustering Computational Game Theory and Clustering Martin Hoefer mhoefer@mpi-inf.mpg.de 1 Computational Game Theory? 2 Complexity and Computation of Equilibrium 3 Bounding Inefficiencies 4 Conclusion Computational

More information

UCLA. Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory

UCLA. Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory UCLA Department of Economics Ph. D. Preliminary Exam Micro-Economic 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 information

Biform Games: Additional Online Material

Biform 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 information

Introduction. Bargaining - whether over arms control, the terms of a peace settlement, exchange rate

Introduction. 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 information

Lecture 11: Oligopoly and Strategic Behavior

Lecture 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 information

Infinitely Repeated Games with Discounting Ù

Infinitely 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 information

Biology 1406 - Notes for exam 5 - Population genetics Ch 13, 14, 15

Biology 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 information

Game 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 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 information

Sequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium

Sequential 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 information

Game Theory. CDAM Research Report LSE-CDAM-2001-09 October 8, 2001. 1 What is game theory? 4. 2 Definitions of games 6.

Game Theory. CDAM Research Report LSE-CDAM-2001-09 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 LSE-CDAM-2-9 October 8, 2 Contents What is game theory? 4 2 Definitions of games

More information

Congestion Games with Player-Specific Payoff Functions

Congestion Games with Player-Specific Payoff Functions GAMES AND ECONOMIC BEHAVIOR 13, 111 124 (1996) ARTICLE NO. 0027 Congestion Games with Player-Specific Payoff Functions Igal Milchtaich Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem

More information

Simon 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 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 information

Oligopoly is a market structure more susceptible to game-theoretic analysis, because of apparent strategic interdependence among a few producers.

Oligopoly is a market structure more susceptible to game-theoretic analysis, because of apparent strategic interdependence among a few producers. 1 Market structure from a game-theoretic perspective: Oligopoly After our more theoretical analysis of different zero-sum and variable-sum games, let us return to the more familiar territory of economics---especially

More information

History Independent Prediction in Evolutionary Game Theory *

History 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. e-mail: whs@nwu.edu January 28, 1998 Forthcoming in Rationality

More information

6.254 : 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 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed

More information

On the Evolution of Behavioral Heterogeneity in Individuals and Populations

On 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 information

Heterogeneous Expectations, Adaptive Learning, and Evolutionary Dynamics

Heterogeneous 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 self-referential

More information

Microeconomic 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. (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 information

Network Security Validation Using Game Theory

Network 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 information

Game 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 Game Mining: How to Make Money from those about to Play a Game David H. Wolpert NASA Ames Research Center MailStop 269-1 Moffett Field, CA 94035-1000 david.h.wolpert@nasa.gov James W. Bono Department of

More information

Application of Game Theory in Inventory Management

Application of Game Theory in Inventory Management Application of Game Theory in Inventory Management Rodrigo Tranamil-Vidal Universidad de Chile, Santiago de Chile, Chile Rodrigo.tranamil@ug.udechile.cl Abstract. Game theory has been successfully applied

More information

Evolution and Information in a Gift-Giving Game *

Evolution and Information in a Gift-Giving Game * Evolution and nformation in a Gift-Giving Game * By Phillip Johnson, (johnsonp@yahoo.com) Centro de nvestigacion Economía, nstituto echnologico Autonomo de Mexico, David K. Levine, (dlevine@ucla.edu) Department

More information

Price 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 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 information

Individual security and network design

Individual 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 information

Fairness with an Honest Minority and a Rational Majority

Fairness 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 information

Summary. 16 1 Genes and Variation. 16 2 Evolution as Genetic Change. Name Class Date

Summary. 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 information

Bayesian Nash Equilibrium

Bayesian 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 information

A Game Theoretical Framework for Adversarial Learning

A 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 information

Working Paper Series

Working 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 Hervés-Beloso, Emma Moreno- García and

More information

Simple Channel-Change Games for Spectrum- Agile Wireless Networks

Simple Channel-Change Games for Spectrum- Agile Wireless Networks Simple Channel-Change 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 information

Sealed Bid Second Price Auctions with Discrete Bidding

Sealed 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 information

Complementarities in information acquisition with short-term trades

Complementarities in information acquisition with short-term trades Theoretical Economics (007), 441 467 1555-7561/0070441 Complementarities in information acquisition with short-term trades CHRISTOPHE CHAMLEY Paris-Jourdan Sciences Économiques and Department of Economics,

More information

Microevolution is a change in population s gene pool [1]

Microevolution 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 information

Patent Litigation with Endogenous Disputes

Patent 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 information

Perfect Bayesian Equilibrium

Perfect 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 information

Midterm 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 ;

Midterm 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 Economics-UC3M 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 information

Understanding by Design. Title: BIOLOGY/LAB. Established Goal(s) / Content Standard(s): Essential Question(s) Understanding(s):

Understanding 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 information

Oligopoly: Cournot/Bertrand/Stackelberg

Oligopoly: 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 information

Introduction To Genetic Algorithms

Introduction 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 information

Week 7 - Game Theory and Industrial Organisation

Week 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 information

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

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 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