The Nash equilibrium of the stage game is (D, R), giving payoffs (0, 0). Consider the trigger strategies:
|
|
- Estella Stone
- 5 years ago
- Views:
Transcription
1 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. Show what utilities could be suppted using the Nash threats folk theem. Show that f sufficiently patient players, it is a subgame perfect Nash equilibrium to play (U, L) in every period. What is the minimum δ that achieves cooperation? The Nash equilibrium of the stage game is (D, R), giving payoffs (0, 0). Consider the trigger strategies: If the histy is (U, L), (U, L),..., (U, L), play (U, L) this period. After any other histy, play (D, R) Does either player have a profitable deviation? F the row player, following the proposed strategies gives a payoff of and deviating gives a payoff of so cooperating is better than deviating if 1 + δ1 + δ = δ0 + δ = δ 1 2 F the column player, following the proposed strategies gives a payoff of and deviating gives a payoff of So cooperating is better than deviating if 1 + δ1 + δ = δ0 + δ = δ 4 5 THEREFORE, as long as δ 4/5, the trigger strategies are a Subgame Perfect Nash Equilibrium
2 (b). Consider the infinitely repeated game with discount rate δ, where the strategic fm below is the stage game: B L R U 2, 1 0, 0 A D 0, 0 1, 2 Sketch a graph of the players payoffs. Show what utilities could be suppted using the Nash threats folk theem. Show that f sufficiently patient players, it is a subgame perfect Nash equilibrium to play (U, L) in every even period, and (D, R) in every odd period. What is the minimum δ that achieves cooperation? (Hint: You might need to use mixed strategies) There are three equilibria in the stage game. There are two pure-strategy Nash equilibria, (U, L) and (D, R), and a mixed one with strategies This gives expected payoffs 2/3 to each player. Consider the trigger strategies: σ u = 2 3, σ d = 1 3, σ L = 1 3, σ R = 2 3 If the histy is (D, R), (U, L), (D, R), (U, L),..., (D, R), play (U, L) this period. If the histy is (D, R), (U, L), (D, R), (U, L),..., (U, L), play (D, R) this period. If the histy is anything else, play the mixed strategy equilibrium of the stage game. Does either player have a profitable deviation? When it s your turn to get the 2, the payoff from cooperating is 2 + δ + 2δ 2 + δ 3 + 2δ = (2 + δ)(1 + δ 2 + δ ) = (2 + δ) 2 and when it s your turn to get the 1, the payoff from cooperating is 1 + 2δ + 1δ 2 + 2δ = (1 + 2δ)(1 + δ 2 + δ ) = (1 + 2δ) 2 Well, the row player might deviate after a histy (D, R), (U, L), (D, R), (U, L),..., (D, R) from U to D, after a histy (D, R), (U, L), (D, R), (U, L),..., (U, L) to U, both give a payoff 0 + δ δ = 2 δ 3 Comparing the sums term-by-term, 2, 1 > 0 and 1, 2 > 2/3, and 2, 1 > 2/3, and so on, so the first two sums are always better than the third one. Therefe neither player ever has an incentive to deviate. Or if you look at the sums, (2+δ), (1+δ) > 2/3, and 1/(1 δ 2 ) > 1/(1 δ), so that the discounted sums of the first two payoffs are greater than the discounted sum of the third payoff. Therefe, deviating is never profitable. THEREFORE, as long as δ 0, the trigger strategies are a Subgame Perfect Nash Equilibrium
3 (c). Consider the infinitely repeated game with discount rate δ, where the strategic fm below is the stage game: 2 L C R U 4, 4 0, 5 5, 6 1 M 5, 0 1, 1 6, 0 D 6, 5 0, 6 7, 7 Sketch a graph of the players payoffs. Show what utilities could be suppted using the Nash threats folk theem. Show that f sufficiently patient players, it is a subgame perfect Nash equilibrium to play (U, L) in every period. What is the minimum δ that achieves cooperation? Are there multiple ways to design the punishments f cheating? What is the lowest δ you can achieve? There are three Nash equilibria in the stage game: (M, C), (D, L), and (U, R). Consider the following trigger strategies: If the histy is (U, L), (U, L),..., (U, L), then play (U, L). After any other histy, play (M, C). Cooperating then gives both players a discounted payoff of 4 + δ4 + δ = 4 Then the most profitable deviation f the row player is to choose D and get a payoff of 6, followed by the punishment, and the most profitable deviation f the column player is to choose R and get a payoff of 6, followed by the punishment. This gives a discounted payoff of deviating to either player of 6 + δ1 + δ δ = 6 + δ δ Or 4 6() + δ δ 2 5 THEREFORE, if δ 2/5, the above trigger strategies are a subgame perfect Nash equilibrium Are there other ways to enfce cooperation? Think about these strategies: If the histy is (U, L), (U, L),..., (U, L), then play (U, L). After any histy in which the row player deviated first, play (U, R) After any histy in which the column player deviated first, play (D, L)
4 After any histy in which both players deviated first at the same time, play (M, C). This punishes the deviat with a 5 payoff fever, and rewards the player who didn t cheat with a 6 fever. The payoff from deviating f either player is 6 5δ 5δ 2 5δ = 6 5 δ δ 4 6() 5δ δ 2 11 THEREFORE, if δ 2/11, the above trigger strategies are a subgame perfect Nash equilibrium This is the lowest you can get, since a payoff of 5 is the wst you can impose on a player, given that the player is trying to maximize his payoff. 2. Consider the N-player prisoners dilemma: If all players choose S, they each get a payoff of N If a single player chooses C while all others choose S, that player gets N 2 and the others get 1 If me than one player confesses, all players who confess get zero while any player who remains silent gets 1 i. What is the Nash equilibrium of the N-player prisoners dilemma? ii. Consider the infinitely repeated game with discount fact δ where the N-player prisoners dilemma is the stage game. Show that f sufficiently patient players, it is a subgame perfect Nash equilibrium to play S in every period. What is the minimum δ that achieves cooperation? iii. What happens to the minimum δ as N increases? i. All players confess is the Nash equilibrium of the game. In fact, it is a strictly dominant strategy to confess. Suppose all other players are silent; by confessing, I get a payoff of N 2, and from silent, I get a payoff of N, so confessing is better. Suppose at least one player is confessing; by remaining silent, I get a payoff of 1, while by confessing I get a payoff of zero, so confessing is better. Therefe, no matter what my opponents are doing, I get a higher payoff by confessing, so it is a strictly dominant strategy. ii. Consider the following trigger strategies: After any histy in which all players chose silent in all previous periods, play silent this period.
5 After any other histy, confess. The discounted payoff of cooperating is Deviating and confessing gives a payoff N + δn + δ 2 N +... = N N 2 + δ0 + δ = N 2 N N 2 δ 1 1 N THEREFORE, if δ 1 1/N, the trigger strategies are a subgame perfect Nash equilibrium iii. As N increases, we get δ(2) = 1 1/2 = 1/1, δ(3) = 2/3, δ(4) = 3/4,..., which tends to 1. Since 1/N 0 as N, when the game is really big, the discount fact will be close to 1, making cooperating me difficult. 3. There s a market with two firms who are contemplating acting collusively. If they cut production so that each is making half the monopoly quantity, they each get π m = 2. If they act as Cournot competits, they both get π c = 1. If one player plays the monopoly quantity, the other player can take advantage of the situation and increase output, getting a larger share of the market f himself at the expense of his partner, who gets 0 if this occurs; call this value π d, f optimal deviation. Here s a strategic fm f the game: Collude F irmb Compete Collude 2, 2 0,π d F irma Compete π d, 0 1, 1 i. F what values of π d does this have the fm of a prisoner s dilemma, i.e., each player has a dominant strategy to play competitively? Assume this inequality holds f the rest of the problem. ii. Assume the game is repeatedly infinitely with discount fact δ. Show that f sufficiently patient players, it is a subgame perfect Nash equilibrium to collude in every period. What is the minimum δ that achieves cooperation? iii. How does the minimum δ change as π d increases? Does this make collusion me less likely to succeed? i. If π d > 2, the players have a dominant strategy to compete. ii. The Nash equilibrium of the stage game is (Compete, Compete), giving payoffs (1, 1). Colluding gives payoffs (2, 2). Consider the trigger strategies
6 If the histy is (Collude, Collude), (Collude, Collude),... Collude this period., (Collude, Collude), play If the histy is anything else, Compete this period. Then the payoff to cooperating is and the payoff to deviating is Solving f δ gives 2 + δ2 + δ δ = 2 π d + δ + δ 2 + δ = π d + 2 π d + δ 2 ()π d + δ δ π d 2 π d 1 δ THEREFORE, as long as δ π d 2, the above trigger strategies are a subgame perfect Nash π d 1 equilibrium iii. The minimum δ is δ = π d 2 π d 1 Differentiating with respect to π d gives so the minimum δ is increasing in π d. δ π d = (π d 1) (π d 2) (π d 1) 2 = 1 (π d 1) 2 > 0
Repeated 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 informationS 2,2-1, x c C x r, 1 0,0
Problem Set 5 1. There are two players facing each other in the following random prisoners dilemma: S C S, -1, x c C x r, 1 0,0 With probability p, x c = y, and with probability 1 p, x c = 0. With probability
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 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 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 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 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 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 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 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 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 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 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 informationGame 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 informationEcon 414 Midterm Exam
Econ 44 Midterm Exam Name: There are three questions taken from the material covered so far in the course. All questions are equally weighted. If you have a question, please raise your hand and I will
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 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 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 informationSolution Problem Set 2
ECON 282, Intro Game Theory, (Fall 2008) Christoph Luelfesmann, SFU Solution Problem Set 2 Due at the beginning of class on Tuesday, Oct. 7. Please let me know if you have problems to understand one of
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 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 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 informationREPEATED GAMES. MICROECONOMICS Principles and Analysis Frank Cowell. Frank Cowell: Repeated Games. Almost essential Game Theory: Dynamic.
Prerequisites Almost essential Game Theory: Dynamic REPEATED GAMES MICROECONOMICS Principles and Analysis Frank Cowell April 2018 1 Overview Repeated Games Basic structure Embedding the game in context
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 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 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 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 informationNoncooperative Oligopoly
Noncooperative Oligopoly Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j s actions affect firm i s profits Example: price war
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 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 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 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 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 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 informationEcon 302 Assignment 3 Solution. a 2bQ c = 0, which is the monopolist s optimal quantity; the associated price is. P (Q) = a b
Econ 302 Assignment 3 Solution. (a) The monopolist solves: The first order condition is max Π(Q) = Q(a bq) cq. Q a Q c = 0, or equivalently, Q = a c, which is the monopolist s optimal quantity; the associated
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 informationEco AS , J. Sandford, spring 2019 March 9, Midterm answers
Midterm answers Instructions: You may use a calculator and scratch paper, but no other resources. In particular, you may not discuss the exam with anyone other than the instructor, and you may not access
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 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 informationAS/ECON 2350 S2 N Answers to Mid term Exam July time : 1 hour. Do all 4 questions. All count equally.
AS/ECON 2350 S2 N Answers to Mid term Exam July 2017 time : 1 hour Do all 4 questions. All count equally. Q1. Monopoly is inefficient because the monopoly s owner makes high profits, and the monopoly s
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 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 informationSpring 2017 Final Exam
Spring 07 Final Exam ECONS : Strategy and Game Theory Tuesday May, :0 PM - 5:0 PM irections : Complete 5 of the 6 questions on the exam. You will have a minimum of hours to complete this final exam. No
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 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 informationAnswers to Problem Set 4
Answers to Problem Set 4 Economics 703 Spring 016 1. a) The monopolist facing no threat of entry will pick the first cost function. To see this, calculate profits with each one. With the first cost function,
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 information14.12 Game Theory Midterm II 11/15/ Compute all the subgame perfect equilibria in pure strategies for the following game:
4. Game Theory Midterm II /5/7 Prof. Muhamet Yildiz Instructions. This is an open book exam; you can use any written material. You have one hour and minutes. Each question is 5 points. Good luck!. Compute
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 informationB w x y z a 4,4 3,3 5,1 2,2 b 3,6 2,5 6,-3 1,4 A c -2,0 2,-1 0,0 2,1 d 1,4 1,2 1,1 3,5
Econ 414, Exam 1 Name: There are three questions taken from the material covered so far in the course. All questions are equally weighted. If you have a question, please raise your hand and I will come
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 informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2017
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2017 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
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 informationECON106P: Pricing and Strategy
ECON106P: Pricing and Strategy Yangbo Song Economics Department, UCLA June 30, 2014 Yangbo Song UCLA June 30, 2014 1 / 31 Game theory Game theory is a methodology used to analyze strategic situations 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 informationMicroeconomics II. CIDE, MsC Economics. List of Problems
Microeconomics II CIDE, MsC Economics List of Problems 1. There are three people, Amy (A), Bart (B) and Chris (C): A and B have hats. These three people are arranged in a room so that B can see everything
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 informationEcon 323 Microeconomic Theory. Chapter 10, Question 1
Econ 323 Microeconomic Theory Practice Exam 2 with Solutions Chapter 10, Question 1 Which of the following is not a condition for perfect competition? Firms a. take prices as given b. sell a standardized
More informationSimon Fraser University Spring 2014
Simon Fraser University Spring 2014 Econ 302 D200 Final Exam Solution This brief solution guide does not have the explanations necessary for full marks. NE = Nash equilibrium, SPE = subgame perfect equilibrium,
More informationIterated Dominance and Nash Equilibrium
Chapter 11 Iterated Dominance and Nash Equilibrium In the previous chapter we examined simultaneous move games in which each player had a dominant strategy; the Prisoner s Dilemma game was one example.
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 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. # 03 Illustrations of Nash Equilibrium Lecture No. # 04
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 informationStrategies and Nash Equilibrium. A Whirlwind Tour of Game Theory
Strategies and Nash Equilibrium A Whirlwind Tour of Game Theory (Mostly from Fudenberg & Tirole) Players choose actions, receive rewards based on their own actions and those of the other players. Example,
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 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 informationMicroeconomics I - Seminar #9, April 17, Suggested Solution
Microeconomics I - Seminar #9, April 17, 009 - Suggested Solution Problem 1: (Bertrand competition). Total cost function of two firms selling computers is T C 1 = T C = 15q. If these two firms compete
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 informationElements of Economic Analysis II Lecture X: Introduction to Game Theory
Elements of Economic Analysis II Lecture X: Introduction to Game Theory Kai Hao Yang 11/14/2017 1 Introduction and Basic Definition of Game So far we have been studying environments where the economic
More informationProblem 3,a. ds 1 (s 2 ) ds 2 < 0. = (1+t)
Problem Set 3. Pay-off functions are given for the following continuous games, where the players simultaneously choose strategies s and s. Find the players best-response functions and graph them. Find
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 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 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 informationEcon 711 Homework 1 Solutions
Econ 711 Homework 1 s January 4, 014 1. 1 Symmetric, not complete, not transitive. Not a game tree. Asymmetric, not complete, transitive. Game tree. 1 Asymmetric, not complete, transitive. Not a game tree.
More informationm 11 m 12 Non-Zero Sum Games Matrix Form of Zero-Sum Games R&N Section 17.6
Non-Zero Sum Games R&N Section 17.6 Matrix Form of Zero-Sum Games m 11 m 12 m 21 m 22 m ij = Player A s payoff if Player A follows pure strategy i and Player B follows pure strategy j 1 Results so far
More informationIntroduction to Game Theory
Introduction to Game Theory What is a Game? A game is a formal representation of a situation in which a number of individuals interact in a setting of strategic interdependence. By that, we mean that each
More informationGame Theory Notes: Examples of Games with Dominant Strategy Equilibrium or Nash Equilibrium
Game Theory Notes: Examples of Games with Dominant Strategy Equilibrium or Nash Equilibrium Below are two different games. The first game has a dominant strategy equilibrium. The second game has two Nash
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 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 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 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 informationMicroeconomic Theory II Preliminary Examination Solutions Exam date: August 7, 2017
Microeconomic Theory II Preliminary Examination Solutions Exam date: August 7, 017 1. Sheila moves first and chooses either H or L. Bruce receives a signal, h or l, about Sheila s behavior. The distribution
More informationEconomics 109 Practice Problems 1, Vincent Crawford, Spring 2002
Economics 109 Practice Problems 1, Vincent Crawford, Spring 2002 P1. Consider the following game. There are two piles of matches and two players. The game starts with Player 1 and thereafter the players
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 informationOligopoly (contd.) Chapter 27
Oligopoly (contd.) Chapter 7 February 11, 010 Oligopoly Considerations: Do firms compete on price or quantity? Do firms act sequentially (leader/followers) or simultaneously (equilibrium) Stackelberg models:
More informationEconomics 502 April 3, 2008
Second Midterm Answers Prof. Steven Williams Economics 502 April 3, 2008 A full answer is expected: show your work and your reasoning. You can assume that "equilibrium" refers to pure strategies unless
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 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 informationECON/MGEC 333 Game Theory And Strategy Problem Set 9 Solutions. Levent Koçkesen January 6, 2011
Koç University Department of Economics ECON/MGEC 333 Game Theory And Strategy Problem Set Solutions Levent Koçkesen January 6, 2011 1. (a) Tit-For-Tat: The behavior of a player who adopts this strategy
More informationEcon 323 Microeconomic Theory. Practice Exam 2 with Solutions
Econ 323 Microeconomic Theory Practice Exam 2 with Solutions Chapter 10, Question 1 Which of the following is not a condition for perfect competition? Firms a. take prices as given b. sell a standardized
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 informationECO303: Intermediate Microeconomic Theory Benjamin Balak, Spring 2008
ECO303: Intermediate Microeconomic Theory Benjamin Balak, Spring 2008 Game Theory: FINAL EXAMINATION 1. Under a mixed strategy, A) players move sequentially. B) a player chooses among two or more pure
More informationGame Theory. VK Room: M1.30 Last updated: October 22, 2012.
Game Theory VK Room: M1.30 knightva@cf.ac.uk www.vincent-knight.com Last updated: October 22, 2012. 1 / 33 Overview Normal Form Games Pure Nash Equilibrium Mixed Nash Equilibrium 2 / 33 Normal Form Games
More informationOptimal selling rules for repeated transactions.
Optimal selling rules for repeated transactions. Ilan Kremer and Andrzej Skrzypacz March 21, 2002 1 Introduction In many papers considering the sale of many objects in a sequence of auctions the seller
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 informationSolutions to Homework 3
Solutions to Homework 3 AEC 504 - Summer 2007 Fundamentals of Economics c 2007 Alexander Barinov 1 Price Discrimination Consider a firm with MC = AC = 2, which serves two markets with demand functions
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 information6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts
6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts Asu Ozdaglar MIT February 9, 2010 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria
More informationFinal Examination December 14, Economics 5010 AF3.0 : Applied Microeconomics. time=2.5 hours
YORK UNIVERSITY Faculty of Graduate Studies Final Examination December 14, 2010 Economics 5010 AF3.0 : Applied Microeconomics S. Bucovetsky time=2.5 hours Do any 6 of the following 10 questions. All count
More informationAgenda. Game Theory Matrix Form of a Game Dominant Strategy and Dominated Strategy Nash Equilibrium Game Trees Subgame Perfection
Game Theory 1 Agenda Game Theory Matrix Form of a Game Dominant Strategy and Dominated Strategy Nash Equilibrium Game Trees Subgame Perfection 2 Game Theory Game theory is the study of a set of tools that
More information