Spring 2017 Final Exam
|
|
- Matthew O’Brien’
- 5 years ago
- Views:
Transcription
1 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 notes, books, or phones may be used during the exam. rite your name, answers and work clearly on the answer paper provided. Please ask me if you have any questions. Problem (0 Points) [From Lecture Slides]. Consider the following game between two competing auction houses, Christie s and Sotheby s. Each firm charges its customers a commission on the items sold. Customers view each of the auction houses as essentially identical. For this reason, which ever house charges the lowest commission charge will be the one most of the customers want to use. However, if they can cooperate they might be able to make more money without undercutting each other. The stage-game for the competition between the auction houses is shown below. Sotheby s 7% 5% % 7% 7, 7, 0 -, Christie s 5% 0,,, %, -, 0, 0 (A) (.5 points) List all pure strategy Nash equilibria of the stage-game. (B) (.5 points) List the preferred cooperative outcome of the stage-game and determine the best payoff a player gains by unilaterally deviating from this outcome. (C) (5 points) escribe the Grim-trigger strategy for the infinitely repeated game between Sotheby s and Christie s. () (0 points) If both auction houses have a common discount factor 0 δ, find the condition on this discount factor that will allow the Grim-trigger strategy to be sustained as a Subgame Perfect Nash equilibrium of the infinitely repeated game. Solution: Using ISS it becomes clear that both auction houses have a dominant strategy of 5%. Hence, for part (A) there is a single Nash equilibrium of (5%, 5%). For part (B), the preferred cooperative outcome is (7%, 7%) where both auction houses collude to keep their commission rates high and not undercut each other. hen both auction houses play this
2 strategy, they each receive a payoff of 7. However, each auction house has an incentive to deviate to 5%, getting 0, given the other auction house plays 7%. The grim trigger strategy is for both players to start by playing the cooperative outcome of 7% and then for each following period play 7% if everyone played 7% previously and play 5% forever if anyone cheated the previous round. Given δ, the grim-trigger can be sustained as a SPNE of the infinitely repeated game if 7 δ 0 + δ δ 7 0 0δ + δ 0δ δ 0 7 6δ δ 6 =
3 Problem (0 Points) [From Lecture Slides]. Consider the following simultaneous-move prisoner s dilemma stagegame. Assume that b > a and that d > c. Players will repeat the game an infinite number of times. Player Cooperate efect Player Cooperate a, a, c, b efect b, c d, d (A) (5 points) erive the inequality that must hold to sustain cooperation through a Grimtrigger strategy as a SPNE of the infinitely repeated game. (B) (5 points) erive the condition on the common discount factor δ that allows the Grimtrigger strategy to satisfy the inequality from part (A). (C) (5 points) escribe what happens to the value of δ that will sustain the Grim-trigger strategy as an SPNE as the value (b a) increases. escribe intuitively what the value (b a) is and why it has the relationship with δ that you identified. () (5 points) escribe what happens to the value of δ that will sustain the Grim-trigger strategy as an SPNE as the value (b d) increases. escribe intuitively what the value (b d) is and why it has the relationship with δ that you identified.
4 Problem (0 Points) [From Lecture Slides]. Nature Gunslinger Cowpoke yatt Earp yatt Earp 5 6 There are pure strategy profiles for the simultaneous game of incomplete information above. The stranger s strategies are listed first, then yatt Earps. ( G / C, ) ( G / C, ) ( G / C, ) ( G / C, ) ( G / C, ) ( G / C, ) ( G / C, ) ( G / C, ) (A) (5 points) hat is the Gunslinger type s dominant strategy? hich of the pure strategy profile s above does this eliminate from being possible BNEs? (B) (5 points) Find the pure strategy Bayes-Nash equilibria of the game. Solution: Each player has only two typical strategies, ait = and raw =. An equilibrium will specify a strategy for yatt Earp and a strategy for each type of the stranger (Gunslinger, Cowpoke). Suppose that yatt Earp plays at his single information set. Then what is the best response for each type of the stranger? e can highlight Earp s strategy in blue in the figure. Given this behavior, we can highlight the Gunslinger and Cowpoke s best response. Both types would like to choose if Earp is choosing. Nature Gunslinger Cowpoke yatt Earp yatt Earp 5 6
5 e now give Earp a chance to deviate from given the s behavior of /. e compute the expected payoffs of choosing each strategy given the specified strategy for the. EU() = (/) + 5(/) = 6/ + 5/ = / EU( ) = (/) + 6(/) = / + 6/ = 9/ () Since EU() = / > 9/ = EU( ) yatt Earp cannot do strictly better by deviating and since both players (and their types) are best responding to the other player s strategy, we have a Bayes-Nash equilibrium of this simultaneous game of incomplete information. But we are not done. Suppose that yatt Earp instead plays wait. Then we highlight the best response for each type of the stranger. Nature Gunslinger Cowpoke yatt Earp yatt Earp 5 6 hen yatt Earp chooses to wait, the Gunslinger best responds with and the Cowpoke best responds with. oes Earp want to deviate given this best response? The expected payoffs are EU( ) = (/) + (/) = / + / = / EU() = (/) + (/) = 6/ + / = 0/ Because Earp cannot do strictly better by deviating to under these probabilities, is a best response to the stranger s strategy /. Hence (/, ) is also a BNE. 5
6 Problem (0 Points) [From Problem Set ]. Suppose you and one other bidder are competing in a private-value auction. The auction format is sealed bit, first price. Let v and b denote your valuation and bid, respectively, and let ˆv and ˆb denote the valuation and bid of your opponent. Your payoff is v b if it is the case that b ˆb. Your payoff is 0 otherwise. Although you do not observe ˆv, you know that ˆv is uniformly distributed over the interval [0, ]. That is, v is the probability that ˆv < v. You also know that your opponent bids according to the function ˆb(ˆv) = ˆv. (A) (5 points) rite your expected payoff function from bidding a value b. (B) (5 points) rite the probability of winning the auction as a function of your bid b. (C) (5 points) erive the optimal (best response) bidding rule (Hint: Take derivative, set equal to zero, etc). () (5 points) Suppose your value is v = /, what is your optimal bid? 6
7 Problem 5 (0 Points) [From Practice H]. Consider the signaling game below. There are players, P and P. Player has types, N and S. Nature chooses N and probability / and S at probability /. After learning their type, P chooses either Left L or right R. Player observes player s strategy and then updates beliefs about player s type before choosing a strategy up U or down. (, 0) L P R P γ U (, ) N (, 0) Nature (, 0) L P S R γ P U (, 0) (, ) The separating pure strategy profiles for this game are (L N /R S, U) (R N /L S, ) (R N /L S, U) (R N /L S, ) The pooling pure strategy profiles for this game are (L N /L S, U) (L N /L S, ) (R N /R S, U) (R N /R S, ) (A) (0 points) Can any of the separating strategy profiles above be sustained as a Perfect Bayes Equilibria (PBE)? If so, fully describe which ones and any required conditions on beliefs. (B) (0 points) Can any of the pooling strategy profiles above be sustained as a Perfect Bayes Equilibria (PBE)? If so, fully describe which ones and any required conditions on beliefs. Solution: For part A we check the separating strategies. Suppose (L N /R S ), then γ = 0 and P chooses. hen P chooses, the N type doesn t deviate, but the S type does ( ). So not a PBE. Suppose (R N /L S ), then γ = and P chooses U. hen P chooses U, the N type won t deviate ( > ) and the S type won t deviate ( > ). So (R N /L S, U) is a PBE. Part (B) we look at the pooling strategies. Suppose (L N /L S ), then P has arbitrary beliefs γ and the expected payoffs are, EU(U) = γ EU() = ( γ) EU(U) EU() γ γ γ γ 7
8 So we have two cases. hen γ / P plays U. The N type wants to deviate in this case ( > ). hen γ < /, P chooses. In this case, the both types do strictly worse by deviating to R. So we have a PBE of (L N /L S, ) γ < The other pooling strategy is (R N /R S ), then γ = /. In this case we have EU(U) = = EU() = So P chooses U. hen P chooses U the N type doesn t want to deviate, but the S type can do strictly better by deviating to L ( > ). So this isn t a PBE (which is obvious since L is strictly dominate for the S type).
9 Problem 6 (0 Points) [From Practice H]. Consider the scene from The Princess Pride where Prince Humperdinck discovers esley alive in a bedroom with Princess Buttercup after having killed him earlier that day. Prince Humperdinck suspects that esley has no strength (is eak) but isn t sure. He also knows that if esley is strong there is no way he could take him. esley therefore has two types: Strong and eak. Prince Humperdinck believes the probability that he is strong is only / (after all esley has been mostly-dead all day). esley has two strategies, he can get out of bed O (act tough, bluff, etc.) or he can stay in bed, B. After observing esley s action, Prince Humperdinck can choose to surrender S or fight F. esley s payoffs are listed first. (, 0) S F µ B esley O γ S F (, 0) (0, ) (, 0) S Humperdinck µ F B esley Strong Nature eak O Humperdinck γ S F (0, ) (, 0) (, ) (, ) Find all Perfect Bayes Nash equilibria (PBE) of the game. Solution: Consider the separating strategy (B, O ). Then µ = and γ = 0. H chooses S if B and F if O. Given H s choices, the strong type doesn t want to deviate, but the weak type wants to deviate to B, so (B, O ) cannot be part of a PBE. Consider the separating strategy (O, B ). Then µ = 0 and γ =. H chooses F if B and S if O. Under this policy the strong type doesn t want to deviate and the weak type does want to deviate - going from to /. Consider the pooling strategy (B, B ), then µ = / and γ is arbitrary. H s choice of B is EU(S) = 0 EU(F ) = + = = So H will choose F if B. On the other side, we have EU(S ) = 0 EU(F ) = γ + ( γ) = γ 0 Two cases γ 9
10 . γ / choose F.. γ > / choose S. In case, the strong type cannot do strictly better by deviating. The weak type doesn t want to deviate either since Humperdinck fights no matter what. In case, the strong type wants to deviate to O since they will get instead of 0. So we have a PBE (B/B, F/F ) when γ The last strategy is (O, O ) which makes γ = / and µ arbitrary. EU(S) = 0 EU(F ) = µ + µ = µ µ hen µ / then H chooses F if B. On the other information set, EU(S ) = 0 EU(F ) = + = > 0 So when both types choose O, H chooses F. In this case, the strong type is getting 0 and by deviating gets 0, so no strict benefit. The weak type is getting / and by deviating gets so the weak type wants to deviate. In the other case when µ > / then H chooses S if B. ere this the case, the strong type gets by switching and so not sustainable as PBE. 0
Simon 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 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 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 informationECONS 424 STRATEGY AND GAME THEORY HOMEWORK #7 ANSWER KEY
ECONS 424 STRATEGY AND GAME THEORY HOMEWORK #7 ANSWER KEY Exercise 3 Chapter 28 Watson (Checking the presence of separating and pooling equilibria) Consider the following game of incomplete information:
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 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 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: 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 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 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 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 informationECONS 424 STRATEGY AND GAME THEORY HANDOUT ON PERFECT BAYESIAN EQUILIBRIUM- III Semi-Separating equilibrium
ECONS 424 STRATEGY AND GAME THEORY HANDOUT ON PERFECT BAYESIAN EQUILIBRIUM- III Semi-Separating equilibrium Let us consider the following sequential game with incomplete information. Two players are playing
More informationHW Consider the following game:
HW 1 1. Consider the following game: 2. HW 2 Suppose a parent and child play the following game, first analyzed by Becker (1974). First child takes the action, A 0, that produces income for the child,
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 informationExtensive-Form Games with Imperfect Information
May 6, 2015 Example 2, 2 A 3, 3 C Player 1 Player 1 Up B Player 2 D 0, 0 1 0, 0 Down C Player 1 D 3, 3 Extensive-Form Games With Imperfect Information Finite No simultaneous moves: each node belongs to
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 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 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 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 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 2 1. Consider a zero-sum game, where
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 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 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 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 informationExtensive form games - contd
Extensive form games - contd Proposition: Every finite game of perfect information Γ E has a pure-strategy SPNE. Moreover, if no player has the same payoffs in any two terminal nodes, then there is a unique
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 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 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 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 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 informationEconS Games with Incomplete Information II and Auction Theory
EconS 424 - Games with Incomplete Information II and Auction Theory Félix Muñoz-García Washington State University fmunoz@wsu.edu April 28, 2014 Félix Muñoz-García (WSU) EconS 424 - Recitation 9 April
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 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 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 informationMicroeconomic Theory II Spring 2016 Final Exam Solutions
Microeconomic Theory II Spring 206 Final Exam Solutions Warning: Brief, incomplete, and quite possibly incorrect. Mikhael Shor Question. Consider the following game. First, nature (player 0) selects t
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 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 informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2017
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2017 These notes have been used and commented on before. If you can still spot any errors or have any suggestions for improvement, please
More informationSI 563 Homework 3 Oct 5, Determine the set of rationalizable strategies for each of the following games. a) X Y X Y Z
SI 563 Homework 3 Oct 5, 06 Chapter 7 Exercise : ( points) Determine the set of rationalizable strategies for each of the following games. a) U (0,4) (4,0) M (3,3) (3,3) D (4,0) (0,4) X Y U (0,4) (4,0)
More informationGames of Incomplete Information ( 資訊不全賽局 ) Games of Incomplete Information
1 Games of Incomplete Information ( 資訊不全賽局 ) Wang 2012/12/13 (Lecture 9, Micro Theory I) Simultaneous Move Games An Example One or more players know preferences only probabilistically (cf. Harsanyi, 1976-77)
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 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 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 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 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 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 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 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 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 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 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 informationToday. Applications of NE and SPNE Auctions English Auction Second-Price Sealed-Bid Auction First-Price Sealed-Bid Auction
Today Applications of NE and SPNE Auctions English Auction Second-Price Sealed-Bid Auction First-Price Sealed-Bid Auction 2 / 26 Auctions Used to allocate: Art Government bonds Radio spectrum Forms: Sequential
More informationThe Intuitive and Divinity Criterion: Explanation and Step-by-step examples
: Explanation and Step-by-step examples EconS 491 - Felix Munoz-Garcia School of Economic Sciences - Washington State University Reading materials Slides; and Link on the course website: http://www.bepress.com/jioe/vol5/iss1/art7/
More informationGame Theory Lecture #16
Game Theory Lecture #16 Outline: Auctions Mechanism Design Vickrey-Clarke-Groves Mechanism Optimizing Social Welfare Goal: Entice players to select outcome which optimizes social welfare Examples: Traffic
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 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 informationMIDTERM ANSWER KEY GAME THEORY, ECON 395
MIDTERM ANSWER KEY GAME THEORY, ECON 95 SPRING, 006 PROFESSOR A. JOSEPH GUSE () There are positions available with wages w and w. Greta and Mary each simultaneously apply to one of them. If they apply
More informationProblem Set 5 Answers
Problem Set 5 Answers ECON 66, Game Theory and Experiments March 8, 13 Directions: Answer each question completely. If you cannot determine the answer, explaining how you would arrive at the answer might
More informationNotes for Section: Week 7
Economics 160 Professor Steven Tadelis Stanford University Spring Quarter, 004 Notes for Section: Week 7 Notes prepared by Paul Riskind (pnr@stanford.edu). spot errors or have questions about these notes.
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 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 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 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 informationOut of equilibrium beliefs and Refinements of PBE
Refinements of PBE Out of equilibrium beliefs and Refinements of PBE Requirement 1 and 2 of the PBE say that no player s strategy can be strictly dominated beginning at any information set. The problem
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 informationEcon 101A Final exam May 14, 2013.
Econ 101A Final exam May 14, 2013. Do not turn the page until instructed to. Do not forget to write Problems 1 in the first Blue Book and Problems 2, 3 and 4 in the second Blue Book. 1 Econ 101A Final
More informationGames of Incomplete Information
Games of Incomplete Information EC202 Lectures V & VI Francesco Nava London School of Economics January 2011 Nava (LSE) EC202 Lectures V & VI Jan 2011 1 / 22 Summary Games of Incomplete Information: Definitions:
More information(a) (5 points) Suppose p = 1. Calculate all the Nash Equilibria of the game. Do/es the equilibrium/a that you have found maximize social utility?
GAME THEORY EXAM (with SOLUTIONS) January 20 P P2 P3 P4 INSTRUCTIONS: Write your answers in the space provided immediately after each question. You may use the back of each page. The duration of this exam
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 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 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 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 informationEcon 101A Final exam May 14, 2013.
Econ 101A Final exam May 14, 2013. Do not turn the page until instructed to. Do not forget to write Problems 1 in the first Blue Book and Problems 2, 3 and 4 in the second Blue Book. 1 Econ 101A Final
More informationTHE PENNSYLVANIA STATE UNIVERSITY. Department of Economics. January Written Portion of the Comprehensive Examination for
THE PENNSYLVANIA STATE UNIVERSITY Department of Economics January 2014 Written Portion of the Comprehensive Examination for the Degree of Doctor of Philosophy MICROECONOMIC THEORY Instructions: This examination
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 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 informationGame theory and applications: Lecture 1
Game theory and applications: Lecture 1 Adam Szeidl September 20, 2018 Outline for today 1 Some applications of game theory 2 Games in strategic form 3 Dominance 4 Nash equilibrium 1 / 8 1. Some applications
More informationPreliminary Notions in Game Theory
Chapter 7 Preliminary Notions in Game Theory I assume that you recall the basic solution concepts, namely Nash Equilibrium, Bayesian Nash Equilibrium, Subgame-Perfect Equilibrium, and Perfect Bayesian
More informationECONS STRATEGY AND GAME THEORY QUIZ #3 (SIGNALING GAMES) ANSWER KEY
ECONS - STRATEGY AND GAME THEORY QUIZ #3 (SIGNALING GAMES) ANSWER KEY Exercise Mike vs. Buster Consider the following sequential move game with incomplete information. The first player to move is Mike,
More informationThe Ohio State University Department of Economics Econ 601 Prof. James Peck Extra Practice Problems Answers (for final)
The Ohio State University Department of Economics Econ 601 Prof. James Peck Extra Practice Problems Answers (for final) Watson, Chapter 15, Exercise 1(part a). Looking at the final subgame, player 1 must
More informationGame Theory: Global Games. Christoph Schottmüller
Game Theory: Global Games Christoph Schottmüller 1 / 20 Outline 1 Global Games: Stag Hunt 2 An investment example 3 Revision questions and exercises 2 / 20 Stag Hunt Example H2 S2 H1 3,3 3,0 S1 0,3 4,4
More informationCS711 Game Theory and Mechanism Design
CS711 Game Theory and Mechanism Design Problem Set 1 August 13, 2018 Que 1. [Easy] William and Henry are participants in a televised game show, seated in separate booths with no possibility of communicating
More information1. (15 points) P1 P2 P3 P4. GAME THEORY EXAM (with SOLUTIONS) January 2010
GAME THEORY EXAM (with SOLUTIONS) January 2010 P1 P2 P3 P4 INSTRUCTIONS: Write your answers in the space provided immediately after each question. You may use the back of each page. The duration of this
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 information6 Dynamic Games with Incomplete Information
February 24, 2014, Eric Rasmusen, Erasmuse@indiana.edu. Http://www.rasmusen.org. 6 Dynamic Games with Incomplete Information Entry Deterrence II: Fighting Is Never Profitable: X=1 Subgame perfectness does
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 informationName. FINAL EXAM, Econ 171, March, 2015
Name FINAL EXAM, Econ 171, March, 2015 There are 9 questions. Answer any 8 of them. Good luck! Remember, you only need to answer 8 questions Problem 1. (True or False) If a player has a dominant strategy
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 informationCSE 316A: Homework 5
CSE 316A: Homework 5 Due on December 2, 2015 Total: 160 points Notes There are 8 problems on 5 pages below, worth 20 points each (amounting to a total of 160. However, this homework will be graded out
More informationBayesian Nash Equilibrium
Bayesian Nash Equilibrium We have already seen that a strategy for a player in a game of incomplete information is a function that specifies what action or actions to take in the game, for every possibletypeofthatplayer.
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 information4. Beliefs at all info sets off the equilibrium path are determined by Bayes' Rule & the players' equilibrium strategies where possible.
A. Perfect Bayesian Equilibrium B. PBE Examples C. Signaling Examples Context: A. PBE for dynamic games of incomplete information (refines BE & SPE) *PBE requires strategies to be BE for the entire game
More informationRecap First-Price Revenue Equivalence Optimal Auctions. Auction Theory II. Lecture 19. Auction Theory II Lecture 19, Slide 1
Auction Theory II Lecture 19 Auction Theory II Lecture 19, Slide 1 Lecture Overview 1 Recap 2 First-Price Auctions 3 Revenue Equivalence 4 Optimal Auctions Auction Theory II Lecture 19, Slide 2 Motivation
More informationIn the Name of God. Sharif University of Technology. Microeconomics 2. Graduate School of Management and Economics. Dr. S.
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 2 44706 (1394-95 2 nd term) - Group 2 Dr. S. Farshad Fatemi Chapter 8: Simultaneous-Move Games
More informationRepeated, Stochastic and Bayesian Games
Decision Making in Robots and Autonomous Agents Repeated, Stochastic and Bayesian Games Subramanian Ramamoorthy School of Informatics 26 February, 2013 Repeated Game 26/02/2013 2 Repeated Game - Strategies
More informationBeliefs and Sequential Rationality
Beliefs and Sequential Rationality A system of beliefs µ in extensive form game Γ E is a specification of a probability µ(x) [0,1] for each decision node x in Γ E such that x H µ(x) = 1 for all information
More informationEcon 711 Final Solutions
Econ 711 Final Solutions April 24, 2015 1.1 For all periods, play Cc if history is Cc for all prior periods. If not, play Dd. Payoffs for 2 cooperating on the equilibrium path are optimal for and deviating
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 informationGame Theory Problem Set 4 Solutions
Game Theory Problem Set 4 Solutions 1. Assuming that in the case of a tie, the object goes to person 1, the best response correspondences for a two person first price auction are: { }, < v1 undefined,
More informationIn Class Exercises. Problem 1
In Class Exercises Problem 1 A group of n students go to a restaurant. Each person will simultaneously choose his own meal but the total bill will be shared amongst all the students. If a student chooses
More information