REPEATED GAMES. MICROECONOMICS Principles and Analysis Frank Cowell. Frank Cowell: Repeated Games. Almost essential Game Theory: Dynamic.
|
|
- Florence Gregory
- 5 years ago
- Views:
Transcription
1 Prerequisites Almost essential Game Theory: Dynamic REPEATED GAMES MICROECONOMICS Principles and Analysis Frank Cowell April
2 Overview Repeated Games Basic structure Embedding the game in context Equilibrium issues Applications April
3 Introduction Another examination of the role of time Dynamic analysis can be difficult more than a few stages can lead to complicated analysis of equilibrium We need an alternative approach one that preserves basic insights of dynamic games for example, subgame-perfect equilibrium Build on the idea of dynamic games introduce a jump move from the case of comparatively few stages to the case of arbitrarily many April
4 Repeated games The alternative approach take a series of the same game embed it within a time-line structure Basic idea is simple connect multiple instances of an atemporal game model a repeated encounter between the players in the same situation of economic conflict Raises important questions how does this structure differ from an atemporal model? how does the repetition of a game differ from a single play? how does it differ from a collection of unrelated games of identical structure with identical players? April
5 History Why is the time-line different from a collection of unrelated games? The key is history consider history at any point on the timeline contains information about actual play information accumulated up to that point History can affect the nature of the game at any stage all players can know all the accumulated information strategies can be conditioned on this information History can play a role in the equilibrium some interesting outcomes aren t equilibria in a single encounter these may be equilibrium outcomes in the repeated game the game s history is used to support such outcomes April
6 Repeated games: Structure The stage game take an instant in time specify a simultaneous-move game payoffs completely specified by actions within the game Repeat the stage game indefinitely there s an instance of the stage game at time 0,1,2,,t, the possible payoffs are also repeated for each t payoffs at t depends on actions in stage game at t A modified strategic environment all previous actions assumed as common knowledge so agents strategies can be conditioned on this information Modifies equilibrium behaviour and outcome? April
7 Equilibrium Simplified structure has potential advantages whether significant depends on nature of stage game concern nature of equilibrium Possibilities for equilibrium new strategy combinations supportable as equilibria? long-term cooperative outcomes absent from a myopic analysis of a simple game Refinements of subgame perfection simplify the analysis: can rule out empty threats and incredible promises disregard irrelevant might-have-beens April
8 Overview Repeated Games Basic structure Developing the basic concepts Equilibrium issues Applications April
9 Equilibrium: an approach Focus on key question in repeated games: how can rational players use the information from history? need to address this to characterise equilibrium Illustrate a method in an argument by example outline for the Prisoner's Dilemma game same players face same outcomes from their actions that they may choose in periods 1, 2,, t, Prisoner's Dilemma particularly instructive given: its importance in microeconomics pessimistic outcome of an isolated round of the game April
10 * detail on slide can only be seen if you run the slideshow Prisoner s dilemma: Reminder Payoffs in stage game If Alf plays [RIGHT] Bill s best response is [right] Alf [LEFT] [RIGHT] 2,2 3,0 0,3 1,1 If Bill plays [right] Alf s best response is [RIGHT] Nash Equilibrium Outcome that Pareto dominates NE [left] Bill [right] The highlighted NE is inefficient Could the Pareto-efficient outcome be an equilibrium in the repeated game? Look at the structure April
11 * detail on slide can only be seen if you run the slideshow Repeated Prisoner's dilemma 1 [LEFT] Alf [RIGHT] Stage game between (t=1) Stage game (t=2) follows here or here or here Bill or here [left] [right] [left] [right] 2 2 [LEFT] Alf Alf Alf (2,2) (0,3) (3,0) (1,1) 2 2 Alf [LEFT] [RIGHT][LEFT] [RIGHT] [LEFT] [RIGHT] [RIGHT] Bill Bill Bill Bill [left] [right] [left] [left] [right] [left] [right] [left] [left] [right] [left] [right] [left] [right] (2,2) (2,2) (0,3) (2,2) (3,0) (0,3) (2,2) (3,0) (1,1) (0,3) (3,0) (0,3) (1,1) (3,0) (1,1) (1,1) Repeat this structure indefinitely? April
12 Repeated Prisoner's dilemma 1 [LEFT] Alf [RIGHT] The stage game repeated though time Bill [left] [right] [left] [right] (2,2) (0,3) (3,0) (1,1) Alf t [LEFT] [RIGHT] Bill [left] [right] [left] [right] (2,2) (0,3) (3,0) (1,1) Let's look at the detail April
13 Repeated PD: payoffs To represent possibilities in long run: first consider payoffs available in the stage game then those available through mixtures In the one-shot game payoffs simply represented it was enough to denote them as 0,,3 purely ordinal arbitrary monotonic changes of the payoffs have no effect Now we need a generalised notation cardinal values of utility matter we need to sum utilities, compare utility differences Evaluation of a payoff stream: suppose payoff to agent h in period t is υ h (t) value of (υ h (1), υ h (2),, υ h (t) ) is given by [1 δ] δ t 1 υ h (t) t=1 where δ is a discount factor 0 < δ < 1 April
14 PD: stage game A generalised notation for the stage game consider actions and payoffs in each of four fundamental cases Both socially irresponsible: they play [RIGHT], [right] get ( υ a, υ b ) where υ a > 0, υ b > 0 Both socially responsible: they play [LEFT],[left] get (υ *a, υ *b ) where υ *a > υ a, υ *b > υ b Only Alf socially responsible: they play [LEFT], [right] get ( 0, υ b ) where υ b > υ *b Only Bill socially responsible: they play [RIGHT], [left] get ( υ a, 0) where υ a > υ *a A diagrammatic view April
15 Repeated Prisoner s dilemma payoffs _ υ b υ b Space of utility payoffs Payoffs for Prisoner's Dilemma Nash-Equilibrium payoffs Payoffs Pareto-superior to NE Payoffs available through mixing Feasible, superior points "Efficient" outcomes UU * ( υ*a, υ *b ) ( υ a, υ b ) 0 _ υ a υ a April
16 Choosing a strategy: setting Long-run advantage in the Pareto-efficient outcome payoffs (υ *a, υ *b ) in each period clearly better than ( υ a, υ b ) in each period Suppose the agents recognise the advantage what actions would guarantee them this? clearly they need to play [LEFT], [left] every period The problem is lack of trust: they cannot trust each other nor indeed themselves: Alf tempted to be antisocial and get payoff υ a by playing [RIGHT] Bill has a similar temptation April
17 Choosing a strategy: formulation Will a dominated outcome still be inevitable? Suppose each player adopts a strategy that 1. rewards the other party's responsible behaviour by responding with the action [left] 2. punishes antisocial behaviour with the action [right], thus generating the minimax payoffs (υ a, υ b ) Known as a trigger strategy Why the strategy is powerful punishment applies to every period after the one where the antisocial action occurred if punishment invoked offender is minimaxed for ever Look at it in detail April
18 Repeated PD: trigger strategies s T a Bill s action in 0,,t Alf s action at t+1 [left][left],,[left] [LEFT] Anything else [RIGHT] s T b Alf s action in 0,,t Bill s action at t+1 Take situation at t First type of history Response of other player to continue this history Second type of history Punishment response Trigger strategies [s Ta, s Tb ] [LEFT][LEFT],,[LEFT] Anything else [left] [right] Will it work? April
19 Will the trigger strategy work? Utility gain from misbehaving at t: υ a υ *a What is value at t of punishment from t + 1 onwards? Difference in utility per period: υ *a υ a Discounted value of this in period t + 1: V := [υ *a υ a ]/[1 δ ] Value of this in period t: δv = δ[υ *a υ a ]/[1 δ ] So agent chooses not to misbehave if υ a υ *a δ[υ *a υ a ]/[1 δ ] But this is only going to work for specific parameters value of δ relative to υ a, υ a and υ *a What values of discount factor will allow an equilibrium? April
20 Discounting and equilibrium For an equilibrium condition must be satisfied for both a and b Consider the situation of a Rearranging the condition from the previous slide: δ[υ *a υ a ] [1 δ] [ υ a υ *a ] δ[ υ a υ a ] [ υ a υ *a ] Simplifying this the condition must be δ δ a where δ a := [ υ a υ *a ] / [ υ a υ a ] A similar result must also apply to agent b Therefore we must have the condition: δ δ where δ := max {δ a, δ b } April
21 Repeated PD: SPNE Assuming δ δ, take the strategies [s Ta, s Tb ] prescribed by the Table If there were antisocial behaviour at t consider subgame that would start at t + 1 Alf could not increase his payoff by switching from [RIGHT] to [LEFT], given that Bill is playing [left] a similar remark applies to Bill so strategies imply a NE for this subgame likewise for any subgame starting after t + 1 But if [LEFT],[left] has been played in every period up till t: Alf would not wish to switch to [RIGHT] a similar remark applies to Bill again we have a NE So, if δ is large enough, [s Ta, s Tb ] is a Subgame-Perfect Equilibrium yields the payoffs (υ *a, υ *b ) in every period April
22 Folk Theorem The outcome of the repeated PD is instructive illustrates an important result the Folk Theorem Strictly speaking a class of results finite/infinite games different types of equilibrium concepts A standard version of the Theorem: for a two-person infinitely repeated game: suppose discount factor is sufficiently close to 1 any combination of actions observed in any finite number of stages this is the outcome of a subgame-perfect equilibrium April
23 Assessment The Folk Theorem central to repeated games perhaps better described as Folk Theorems a class of results Clearly has considerable attraction Put its significance in context makes relatively modest claims gives a possibility result Only seen one example of the Folk Theorem let s apply it to well known oligopoly examples April
24 Overview Repeated Games Basic structure Some well-known examples Equilibrium issues Applications April
25 Cournot competition: repeated Start by reinterpreting PD as Cournot duopoly two identical firms firms can each choose one of two levels of output [high] or [low] can firms sustain a low-output (i.e. high-profit) equilibrium? Possible actions and outcomes in the stage game: [HIGH], [high]: both firms get Cournot-Nash payoff Π C > 0 [LOW], [low]: both firms get joint-profit maximising payoff Π J > Π C [HIGH], [low]: payoffs are ( Π, 0) where Π > Π J Folk theorem: get SPNE with payoffs (Π J, Π J ) if δ is large enough Critical value for the discount factor δ is Π Π J δ = Π Π C But we should say more Let s review the standard Cournot diagram April
26 Cournot stage game q 2 q 2 χ 1 ( ) Firm 1 s Iso-profit curves Firm 2 s Iso-profit curves Firm 1 s reaction function Firm 2 s reaction function Cournot-Nash equilibrium Outputs with higher profits for both firms Joint profit-maximising solution Output that forces other firm s profit to 0 (q 1 C, q 2 C ) χ 2 ( ) (q1 J, q2 J ) 0 q 1 q 1 April
27 Repeated Cournot game: Punishment Standard Cournot model is richer than simple PD: action space for PD stage game just has the two output levels continuum of output levels introduces further possibilities Minimax profit level for firm 1 in a Cournot duopoly is zero, not the NE outcome Π C arises where firm 2 sets output to q 2 such that 1 makes no profit Imagine a deviation by firm 1 at time t raises q 1 above joint profit-max level Would minimax be used as punishment from t + 1 to? clearly (0, q 2 ) is not on firm 2's reaction function so cannot be best response by firm 2 to an action by firm 1 so it cannot belong to the NE of the subgame everlasting minimax punishment is not credible in this case April
28 Repeated Cournot game: Payoffs Π Π 2 Space of profits for the two firms Cournot-Nash outcome Joint-profit maximisation Minimax outcomes Payoffs available in repeated game (Π J,Π J ) (Π C,Π C ) 0 Π Π 1 Now review Bertrand competition April
29 Bertrand stage game p 2 Marginal cost pricing Monopoly pricing Firm 1 s reaction function Firm 2 s reaction function Nash equilibrium p M c c p M p 1 April
30 Bertrand competition: repeated NE of the stage game: set price equal to marginal cost c results in zero profits NE outcome is the minimax outcome minimax outcome is implementable as a Nash equilibrium in all the subgames following a defection from cooperation In repeated Bertrand competition firms set p M if acting cooperatively split profits between them if one firm deviates from this others then set price to c Repeated Bertrand: result can enforce joint profit maximisation through trigger strategy provided discount factor is large enough April
31 Repeated Bertrand game: Payoffs Π M Π 2 Space of profits for the two firms Bertrand-Nash outcome Firm 1 as a monopoly Firm 2 as a monopoly Payoffs available in repeated game 0 Π M Π 1 April
32 Repeated games: summary New concepts: Stage game History The Folk Theorem Trigger strategy What next? Games under uncertainty April
Game Theory. Wolfgang Frimmel. Repeated Games
Game Theory Wolfgang Frimmel Repeated Games 1 / 41 Recap: SPNE The solution concept for dynamic games with complete information is the subgame perfect Nash Equilibrium (SPNE) Selten (1965): A strategy
More informationGAME THEORY: DYNAMIC. MICROECONOMICS Principles and Analysis Frank Cowell. Frank Cowell: Dynamic Game Theory
Prerequisites Almost essential Game Theory: Strategy and Equilibrium GAME THEORY: DYNAMIC MICROECONOMICS Principles and Analysis Frank Cowell April 2018 1 Overview Game Theory: Dynamic Mapping the temporal
More informationIn reality; some cases of prisoner s dilemma end in cooperation. Game Theory Dr. F. Fatemi Page 219
Repeated Games Basic lesson of prisoner s dilemma: In one-shot interaction, individual s have incentive to behave opportunistically Leads to socially inefficient outcomes In reality; some cases of prisoner
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Bargaining We will now apply the concept of SPNE to bargaining A bit of background Bargaining is hugely interesting but complicated to model It turns out that the
More informationIntroduction to Game Theory Lecture Note 5: Repeated Games
Introduction to Game Theory Lecture Note 5: Repeated Games Haifeng Huang University of California, Merced Repeated games Repeated games: given a simultaneous-move game G, a repeated game of G is an extensive
More informationDuopoly models Multistage games with observed actions Subgame perfect equilibrium Extensive form of a game Two-stage prisoner s dilemma
Recap Last class (September 20, 2016) Duopoly models Multistage games with observed actions Subgame perfect equilibrium Extensive form of a game Two-stage prisoner s dilemma Today (October 13, 2016) Finitely
More informationDUOPOLY. MICROECONOMICS Principles and Analysis Frank Cowell. July 2017 Frank Cowell: Duopoly. Almost essential Monopoly
Prerequisites Almost essential Monopoly Useful, but optional Game Theory: Strategy and Equilibrium DUOPOLY MICROECONOMICS Principles and Analysis Frank Cowell 1 Overview Duopoly Background How the basic
More informationRepeated Games. September 3, Definitions: Discounting, Individual Rationality. Finitely Repeated Games. Infinitely Repeated Games
Repeated Games Frédéric KOESSLER September 3, 2007 1/ Definitions: Discounting, Individual Rationality Finitely Repeated Games Infinitely Repeated Games Automaton Representation of Strategies The One-Shot
More informationWarm Up Finitely Repeated Games Infinitely Repeated Games Bayesian Games. Repeated Games
Repeated Games Warm up: bargaining Suppose you and your Qatz.com partner have a falling-out. You agree set up two meetings to negotiate a way to split the value of your assets, which amount to $1 million
More informationThe Nash equilibrium of the stage game is (D, R), giving payoffs (0, 0). Consider the trigger strategies:
Problem Set 4 1. (a). Consider the infinitely repeated game with discount rate δ, where the strategic fm below is the stage game: B L R U 1, 1 2, 5 A D 2, 0 0, 0 Sketch a graph of the players payoffs.
More informationPrisoner s dilemma with T = 1
REPEATED GAMES Overview Context: players (e.g., firms) interact with each other on an ongoing basis Concepts: repeated games, grim strategies Economic principle: repetition helps enforcing otherwise unenforceable
More informationTable 10.1: Elimination and equilibrium. 1. Is there a dominant strategy for either of the two agents?
Chapter 10 Strategic Behaviour Exercise 10.1 Table 10.1 is the strategic form representation of a simultaneous move game in which strategies are actions. s b 1 s b 2 s b 3 s a 1 0, 2 3, 1 4, 3 s a 2 2,
More informationEconomics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Repeated games OR 8 and 9, and FT 5
Economics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Repeated games OR 8 and 9, and FT 5 The basic idea prisoner s dilemma The prisoner s dilemma game with one-shot payoffs 2 2 0
More informationCUR 412: Game Theory and its Applications Final Exam Ronaldo Carpio Jan. 13, 2015
CUR 41: Game Theory and its Applications Final Exam Ronaldo Carpio Jan. 13, 015 Instructions: Please write your name in English. This exam is closed-book. Total time: 10 minutes. There are 4 questions,
More informationIn the Name of God. Sharif University of Technology. Graduate School of Management and Economics
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics (for MBA students) 44111 (1393-94 1 st term) - Group 2 Dr. S. Farshad Fatemi Game Theory Game:
More informationRepeated Games. Econ 400. University of Notre Dame. Econ 400 (ND) Repeated Games 1 / 48
Repeated Games Econ 400 University of Notre Dame Econ 400 (ND) Repeated Games 1 / 48 Relationships and Long-Lived Institutions Business (and personal) relationships: Being caught cheating leads to punishment
More informationEconomics 171: Final Exam
Question 1: Basic Concepts (20 points) Economics 171: Final Exam 1. Is it true that every strategy is either strictly dominated or is a dominant strategy? Explain. (5) No, some strategies are neither dominated
More informationChapter 8. Repeated Games. Strategies and payoffs for games played twice
Chapter 8 epeated Games 1 Strategies and payoffs for games played twice Finitely repeated games Discounted utility and normalized utility Complete plans of play for 2 2 games played twice Trigger strategies
More informationCHAPTER 14: REPEATED PRISONER S DILEMMA
CHAPTER 4: REPEATED PRISONER S DILEMMA In this chapter, we consider infinitely repeated play of the Prisoner s Dilemma game. We denote the possible actions for P i by C i for cooperating with the other
More informationRepeated Games. EC202 Lectures IX & X. Francesco Nava. January London School of Economics. Nava (LSE) EC202 Lectures IX & X Jan / 16
Repeated Games EC202 Lectures IX & X Francesco Nava London School of Economics January 2011 Nava (LSE) EC202 Lectures IX & X Jan 2011 1 / 16 Summary Repeated Games: Definitions: Feasible Payoffs Minmax
More informationIntroduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4)
Introduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4) Outline: Modeling by means of games Normal form games Dominant strategies; dominated strategies,
More informationECE 586BH: Problem Set 5: Problems and Solutions Multistage games, including repeated games, with observed moves
University of Illinois Spring 01 ECE 586BH: Problem Set 5: Problems and Solutions Multistage games, including repeated games, with observed moves Due: Reading: Thursday, April 11 at beginning of class
More informationIntroductory Microeconomics
Prof. Wolfram Elsner Faculty of Business Studies and Economics iino Institute of Institutional and Innovation Economics Introductory Microeconomics More Formal Concepts of Game Theory and Evolutionary
More informationRepeated Games with Perfect Monitoring
Repeated Games with Perfect Monitoring Mihai Manea MIT Repeated Games normal-form stage game G = (N, A, u) players simultaneously play game G at time t = 0, 1,... at each date t, players observe all past
More informationM.Phil. Game theory: Problem set II. These problems are designed for discussions in the classes of Week 8 of Michaelmas term. 1
M.Phil. Game theory: Problem set II These problems are designed for discussions in the classes of Week 8 of Michaelmas term.. Private Provision of Public Good. Consider the following public good game:
More information1 Solutions to Homework 4
1 Solutions to Homework 4 1.1 Q1 Let A be the event that the contestant chooses the door holding the car, and B be the event that the host opens a door holding a goat. A is the event that the contestant
More informationFebruary 23, An Application in Industrial Organization
An Application in Industrial Organization February 23, 2015 One form of collusive behavior among firms is to restrict output in order to keep the price of the product high. This is a goal of the OPEC oil
More informationOutline for Dynamic Games of Complete Information
Outline for Dynamic Games of Complete Information I. Examples of dynamic games of complete info: A. equential version of attle of the exes. equential version of Matching Pennies II. Definition of subgame-perfect
More informationAlmost essential MICROECONOMICS
Prerequisites Almost essential Games: Mixed Strategies GAMES: UNCERTAINTY MICROECONOMICS Principles and Analysis Frank Cowell April 2018 1 Overview Games: Uncertainty Basic structure Introduction to the
More informationMA300.2 Game Theory 2005, LSE
MA300.2 Game Theory 2005, LSE Answers to Problem Set 2 [1] (a) This is standard (we have even done it in class). The one-shot Cournot outputs can be computed to be A/3, while the payoff to each firm can
More informationIPR Protection in the High-Tech Industries: A Model of Piracy. Thierry Rayna University of Bristol
IPR Protection in the High-Tech Industries: A Model of Piracy Thierry Rayna University of Bristol thierry.rayna@bris.ac.uk Digital Goods Are Public, Aren t They? For digital goods to be non-rival, copy
More informationAnswer Key: Problem Set 4
Answer Key: Problem Set 4 Econ 409 018 Fall A reminder: An equilibrium is characterized by a set of strategies. As emphasized in the class, a strategy is a complete contingency plan (for every hypothetical
More informationNot 0,4 2,1. i. Show there is a perfect Bayesian equilibrium where player A chooses to play, player A chooses L, and player B chooses L.
Econ 400, Final Exam Name: There are three questions taken from the material covered so far in the course. ll questions are equally weighted. If you have a question, please raise your hand and I will come
More informationInfinitely Repeated Games
February 10 Infinitely Repeated Games Recall the following theorem Theorem 72 If a game has a unique Nash equilibrium, then its finite repetition has a unique SPNE. Our intuition, however, is that long-term
More information1 Solutions to Homework 3
1 Solutions to Homework 3 1.1 163.1 (Nash equilibria of extensive games) 1. 164. (Subgames) Karl R E B H B H B H B H B H B H There are 6 proper subgames, beginning at every node where or chooses an action.
More informationFDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.
FDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.) Hints for Problem Set 3 1. Consider the following strategic
More informationLecture 5 Leadership and Reputation
Lecture 5 Leadership and Reputation Reputations arise in situations where there is an element of repetition, and also where coordination between players is possible. One definition of leadership is that
More informationCUR 412: Game Theory and its Applications, Lecture 9
CUR 412: Game Theory and its Applications, Lecture 9 Prof. Ronaldo CARPIO May 22, 2015 Announcements HW #3 is due next week. Ch. 6.1: Ultimatum Game This is a simple game that can model a very simplified
More informationGame Theory Fall 2003
Game Theory Fall 2003 Problem Set 5 [1] Consider an infinitely repeated game with a finite number of actions for each player and a common discount factor δ. Prove that if δ is close enough to zero then
More informationpreferences of the individual players over these possible outcomes, typically measured by a utility or payoff function.
Leigh Tesfatsion 26 January 2009 Game Theory: Basic Concepts and Terminology A GAME consists of: a collection of decision-makers, called players; the possible information states of each player at each
More informationMicroeconomics of Banking: Lecture 5
Microeconomics of Banking: Lecture 5 Prof. Ronaldo CARPIO Oct. 23, 2015 Administrative Stuff Homework 2 is due next week. Due to the change in material covered, I have decided to change the grading system
More information13.1 Infinitely Repeated Cournot Oligopoly
Chapter 13 Application: Implicit Cartels This chapter discusses many important subgame-perfect equilibrium strategies in optimal cartel, using the linear Cournot oligopoly as the stage game. For game theory
More informationProblem 3 Solutions. l 3 r, 1
. Economic Applications of Game Theory Fall 00 TA: Youngjin Hwang Problem 3 Solutions. (a) There are three subgames: [A] the subgame starting from Player s decision node after Player s choice of P; [B]
More informationEarly PD experiments
REPEATED GAMES 1 Early PD experiments In 1950, Merrill Flood and Melvin Dresher (at RAND) devised an experiment to test Nash s theory about defection in a two-person prisoners dilemma. Experimental Design
More informationIMPERFECT COMPETITION AND TRADE POLICY
IMPERFECT COMPETITION AND TRADE POLICY Once there is imperfect competition in trade models, what happens if trade policies are introduced? A literature has grown up around this, often described as strategic
More informationSI Game Theory, Fall 2008
University of Michigan Deep Blue deepblue.lib.umich.edu 2008-09 SI 563 - Game Theory, Fall 2008 Chen, Yan Chen, Y. (2008, November 12). Game Theory. Retrieved from Open.Michigan - Educational Resources
More informationEconS 424 Strategy and Game Theory. Homework #5 Answer Key
EconS 44 Strategy and Game Theory Homework #5 Answer Key Exercise #1 Collusion among N doctors Consider an infinitely repeated game, in which there are nn 3 doctors, who have created a partnership. In
More informationGame Theory: Additional Exercises
Game Theory: Additional Exercises Problem 1. Consider the following scenario. Players 1 and 2 compete in an auction for a valuable object, for example a painting. Each player writes a bid in a sealed envelope,
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532L Lecture 10 Stochastic Games and Bayesian Games CPSC 532L Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games Stochastic Games
More informationStatic Games and Cournot. Competition
Static Games and Cournot Introduction In the majority of markets firms interact with few competitors oligopoly market Each firm has to consider rival s actions strategic interaction in prices, outputs,
More informationRepeated games. Felix Munoz-Garcia. Strategy and Game Theory - Washington State University
Repeated games Felix Munoz-Garcia Strategy and Game Theory - Washington State University Repeated games are very usual in real life: 1 Treasury bill auctions (some of them are organized monthly, but some
More informationGame Theory. Important Instructions
Prof. Dr. Anke Gerber Game Theory 2. Exam Summer Term 2012 Important Instructions 1. There are 90 points on this 90 minutes exam. 2. You are not allowed to use any material (books, lecture notes etc.).
More informationCMSC 474, Introduction to Game Theory 16. Behavioral vs. Mixed Strategies
CMSC 474, Introduction to Game Theory 16. Behavioral vs. Mixed Strategies Mohammad T. Hajiaghayi University of Maryland Behavioral Strategies In imperfect-information extensive-form games, we can define
More informationMicroeconomic Theory II Preliminary Examination Solutions
Microeconomic Theory II Preliminary Examination Solutions 1. (45 points) Consider the following normal form game played by Bruce and Sheila: L Sheila R T 1, 0 3, 3 Bruce M 1, x 0, 0 B 0, 0 4, 1 (a) Suppose
More informationAnswers to Microeconomics Prelim of August 24, In practice, firms often price their products by marking up a fixed percentage over (average)
Answers to Microeconomics Prelim of August 24, 2016 1. In practice, firms often price their products by marking up a fixed percentage over (average) cost. To investigate the consequences of markup pricing,
More informationSF2972 GAME THEORY Infinite games
SF2972 GAME THEORY Infinite games Jörgen Weibull February 2017 1 Introduction Sofar,thecoursehasbeenfocusedonfinite games: Normal-form games with a finite number of players, where each player has a finite
More informationMKTG 555: Marketing Models
MKTG 555: Marketing Models A Brief Introduction to Game Theory for Marketing February 14-21, 2017 1 Basic Definitions Game: A situation or context in which players (e.g., consumers, firms) make strategic
More informationAdvanced Microeconomic Theory EC104
Advanced Microeconomic Theory EC104 Problem Set 1 1. Each of n farmers can costlessly produce as much wheat as she chooses. Suppose that the kth farmer produces W k, so that the total amount of what produced
More informationCUR 412: Game Theory and its Applications, Lecture 12
CUR 412: Game Theory and its Applications, Lecture 12 Prof. Ronaldo CARPIO May 24, 2016 Announcements Homework #4 is due next week. Review of Last Lecture In extensive games with imperfect information,
More informationPRISONER S DILEMMA. Example from P-R p. 455; also 476-7, Price-setting (Bertrand) duopoly Demand functions
ECO 300 Fall 2005 November 22 OLIGOPOLY PART 2 PRISONER S DILEMMA Example from P-R p. 455; also 476-7, 481-2 Price-setting (Bertrand) duopoly Demand functions X = 12 2 P + P, X = 12 2 P + P 1 1 2 2 2 1
More informationEconomics 431 Infinitely repeated games
Economics 431 Infinitely repeated games Letuscomparetheprofit incentives to defect from the cartel in the short run (when the firm is the only defector) versus the long run (when the game is repeated)
More informationIntroduction to Game Theory
Introduction to Game Theory Part 2. Dynamic games of complete information Chapter 1. Dynamic games of complete and perfect information Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas
More informationHE+ Economics Nash Equilibrium
HE+ Economics Nash Equilibrium Nash equilibrium Nash equilibrium is a fundamental concept in game theory, the study of interdependent decision making (i.e. making decisions where your decision affects
More informationNotes for Section: Week 4
Economics 160 Professor Steven Tadelis Stanford University Spring Quarter, 2004 Notes for Section: Week 4 Notes prepared by Paul Riskind (pnr@stanford.edu). spot errors or have questions about these notes.
More informationFinitely repeated simultaneous move game.
Finitely repeated simultaneous move game. Consider a normal form game (simultaneous move game) Γ N which is played repeatedly for a finite (T )number of times. The normal form game which is played repeatedly
More informationRepeated Games. Debraj Ray, October 2006
Repeated Games Debraj Ray, October 2006 1. PRELIMINARIES A repeated game with common discount factor is characterized by the following additional constraints on the infinite extensive form introduced earlier:
More informationPlayer 2 L R M H a,a 7,1 5,0 T 0,5 5,3 6,6
Question 1 : Backward Induction L R M H a,a 7,1 5,0 T 0,5 5,3 6,6 a R a) Give a definition of the notion of a Nash-Equilibrium! Give all Nash-Equilibria of the game (as a function of a)! (6 points) b)
More informationTopics in Contract Theory Lecture 1
Leonardo Felli 7 January, 2002 Topics in Contract Theory Lecture 1 Contract Theory has become only recently a subfield of Economics. As the name suggest the main object of the analysis is a contract. Therefore
More informationProblem Set 2 Answers
Problem Set 2 Answers BPH8- February, 27. Note that the unique Nash Equilibrium of the simultaneous Bertrand duopoly model with a continuous price space has each rm playing a wealy dominated strategy.
More informationCUR 412: Game Theory and its Applications, Lecture 4
CUR 412: Game Theory and its Applications, Lecture 4 Prof. Ronaldo CARPIO March 22, 2015 Homework #1 Homework #1 will be due at the end of class today. Please check the website later today for the solutions
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532l Lecture 10 Stochastic Games and Bayesian Games CPSC 532l Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games 4 Analyzing Bayesian
More informationRenegotiation in Repeated Games with Side-Payments 1
Games and Economic Behavior 33, 159 176 (2000) doi:10.1006/game.1999.0769, available online at http://www.idealibrary.com on Renegotiation in Repeated Games with Side-Payments 1 Sandeep Baliga Kellogg
More informationEC 202. Lecture notes 14 Oligopoly I. George Symeonidis
EC 202 Lecture notes 14 Oligopoly I George Symeonidis Oligopoly When only a small number of firms compete in the same market, each firm has some market power. Moreover, their interactions cannot be ignored.
More informationMicroeconomics I. Undergraduate Programs in Business Administration and Economics
Microeconomics I Undergraduate Programs in Business Administration and Economics Academic year 2011-2012 Second test 1st Semester January 11, 2012 Fernando Branco (fbranco@ucp.pt) Fernando Machado (fsm@ucp.pt)
More informationSequential Rationality and Weak Perfect Bayesian Equilibrium
Sequential Rationality and Weak Perfect Bayesian Equilibrium Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu June 16th, 2016 C. Hurtado (UIUC - Economics)
More informationECONS 424 STRATEGY AND GAME THEORY MIDTERM EXAM #2 ANSWER KEY
ECONS 44 STRATEGY AND GAE THEORY IDTER EXA # ANSWER KEY Exercise #1. Hawk-Dove game. Consider the following payoff matrix representing the Hawk-Dove game. Intuitively, Players 1 and compete for a resource,
More informationBasic Game-Theoretic Concepts. Game in strategic form has following elements. Player set N. (Pure) strategy set for player i, S i.
Basic Game-Theoretic Concepts Game in strategic form has following elements Player set N (Pure) strategy set for player i, S i. Payoff function f i for player i f i : S R, where S is product of S i s.
More informationSequential-move games with Nature s moves.
Econ 221 Fall, 2018 Li, Hao UBC CHAPTER 3. GAMES WITH SEQUENTIAL MOVES Game trees. Sequential-move games with finite number of decision notes. Sequential-move games with Nature s moves. 1 Strategies in
More informationMicroeconomic Theory II Preliminary Examination Solutions Exam date: June 5, 2017
Microeconomic Theory II Preliminary Examination Solutions Exam date: June 5, 07. (40 points) Consider a Cournot duopoly. The market price is given by q q, where q and q are the quantities of output produced
More informationGame Theory with Applications to Finance and Marketing, I
Game Theory with Applications to Finance and Marketing, I Homework 1, due in recitation on 10/18/2018. 1. Consider the following strategic game: player 1/player 2 L R U 1,1 0,0 D 0,0 3,2 Any NE can be
More informationIntroduction to Multi-Agent Programming
Introduction to Multi-Agent Programming 10. Game Theory Strategic Reasoning and Acting Alexander Kleiner and Bernhard Nebel Strategic Game A strategic game G consists of a finite set N (the set of players)
More informationA folk theorem for one-shot Bertrand games
Economics Letters 6 (999) 9 6 A folk theorem for one-shot Bertrand games Michael R. Baye *, John Morgan a, b a Indiana University, Kelley School of Business, 309 East Tenth St., Bloomington, IN 4740-70,
More informationCUR 412: Game Theory and its Applications, Lecture 4
CUR 412: Game Theory and its Applications, Lecture 4 Prof. Ronaldo CARPIO March 27, 2015 Homework #1 Homework #1 will be due at the end of class today. Please check the website later today for the solutions
More informationThe Ohio State University Department of Economics Second Midterm Examination Answers
Econ 5001 Spring 2018 Prof. James Peck The Ohio State University Department of Economics Second Midterm Examination Answers Note: There were 4 versions of the test: A, B, C, and D, based on player 1 s
More informationGame Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati.
Game Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati. Module No. # 06 Illustrations of Extensive Games and Nash Equilibrium
More informationAn introduction on game theory for wireless networking [1]
An introduction on game theory for wireless networking [1] Ning Zhang 14 May, 2012 [1] Game Theory in Wireless Networks: A Tutorial 1 Roadmap 1 Introduction 2 Static games 3 Extensive-form games 4 Summary
More informationMA200.2 Game Theory II, LSE
MA200.2 Game Theory II, LSE Problem Set 1 These questions will go over basic game-theoretic concepts and some applications. homework is due during class on week 4. This [1] In this problem (see Fudenberg-Tirole
More informationLecture 6 Dynamic games with imperfect information
Lecture 6 Dynamic games with imperfect information Backward Induction in dynamic games of imperfect information We start at the end of the trees first find the Nash equilibrium (NE) of the last subgame
More informationWhen one firm considers changing its price or output level, it must make assumptions about the reactions of its rivals.
Chapter 3 Oligopoly Oligopoly is an industry where there are relatively few sellers. The product may be standardized (steel) or differentiated (automobiles). The firms have a high degree of interdependence.
More informationTopics in Contract Theory Lecture 3
Leonardo Felli 9 January, 2002 Topics in Contract Theory Lecture 3 Consider now a different cause for the failure of the Coase Theorem: the presence of transaction costs. Of course for this to be an interesting
More informationA brief introduction to evolutionary game theory
A brief introduction to evolutionary game theory Thomas Brihaye UMONS 27 October 2015 Outline 1 An example, three points of view 2 A brief review of strategic games Nash equilibrium et al Symmetric two-player
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 Review Examples of Pure Strategy Nash Equilibria Mixed Strategies
More informationName. Answers Discussion Final Exam, Econ 171, March, 2012
Name Answers Discussion Final Exam, Econ 171, March, 2012 1) Consider the following strategic form game in which Player 1 chooses the row and Player 2 chooses the column. Both players know that this is
More informationis the best response of firm 1 to the quantity chosen by firm 2. Firm 2 s problem: Max Π 2 = q 2 (a b(q 1 + q 2 )) cq 2
Econ 37 Solution: Problem Set # Fall 00 Page Oligopoly Market demand is p a bq Q q + q.. Cournot General description of this game: Players: firm and firm. Firm and firm are identical. Firm s strategies:
More informationDiscounted Stochastic Games with Voluntary Transfers
Discounted Stochastic Games with Voluntary Transfers Sebastian Kranz University of Cologne Slides Discounted Stochastic Games Natural generalization of infinitely repeated games n players infinitely many
More informationPrerequisites. Almost essential Risk MORAL HAZARD. MICROECONOMICS Principles and Analysis Frank Cowell. April 2018 Frank Cowell: Moral Hazard 1
Prerequisites Almost essential Risk MORAL HAZARD MICROECONOMICS Principles and Analysis Frank Cowell April 2018 Frank Cowell: Moral Hazard 1 The moral hazard problem A key aspect of hidden information
More informationLecture 9: Basic Oligopoly Models
Lecture 9: Basic Oligopoly Models Managerial Economics November 16, 2012 Prof. Dr. Sebastian Rausch Centre for Energy Policy and Economics Department of Management, Technology and Economics ETH Zürich
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 22 COOPERATIVE GAME THEORY Correlated Strategies and Correlated
More informationEconS 424 Strategy and Game Theory. Homework #5 Answer Key
EconS 44 Strategy and Game Theory Homework #5 Answer Key Exercise #1 Collusion among N doctors Consider an infinitely repeated game, in which there are nn 3 doctors, who have created a partnership. In
More informationUniversité du Maine Théorie des Jeux Yves Zenou Correction de l examen du 16 décembre 2013 (1 heure 30)
Université du Maine Théorie des Jeux Yves Zenou Correction de l examen du 16 décembre 2013 (1 heure 30) Problem (1) (8 points) Consider the following lobbying game between two firms. Each firm may lobby
More information