Econ 711 Homework 1 Solutions
|
|
- Eric McBride
- 5 years ago
- Views:
Transcription
1 Econ 711 Homework 1 s January 4, Symmetric, not complete, not transitive. Not a game tree. Asymmetric, not complete, transitive. Game tree. 1
2 Asymmetric, not complete, transitive. Not a game tree. (both and preceed 4, but neither preceeds the other). 4 Asymmetric, not complete, not transitive. Not a game tree. 5 Asymmetric, not complete, vacuously transitive. Not a game tree 1 and both preceed 4, but neither preceeds the other.. 1) WTS Defn 1 Defn and ) Defn Defn 1 1) Suppose Prop A and not Prop B. There are two possibilities: a) no sequence exists or b) two or more sequences exist. First, we ll consider a): t k t by assumption. Then i, j, t i t and t j t implies t i t j or t j t i by Prop A. Asymmetry implies only one of those can hold, and adding transitivity implies there can be no cycles. Thus we can construct a sequence as in Defn. Now let s consider b): Suppose we can order these nodes two different ways. It must be the case, then, that at least one pair of nodes in the set of preceeding nodes switches order between the two sequences, and thus by transitivity that each node in this pair preceeds the other, violating asymmetry. Contradiction. ) Suppose Prop B holds and we have (t,t,t ) such that t t and t t. Prop B tells us that, since t and t preceed t, there is a unique sequence over this set of two nodes. Thus, either t t or t t, proving Prop A.. 1
3 Yes it is. Alice only moves at one node, so she has perfect information, and Bob always knows what node he is at because he knows how much money Alice has offered him the distinguishing feature between the nodes he plays. For the subgame starting with B 1, Bob is indifferent between accepting and rejecting, so either is valid. For all other subgames, Bob strictly prefers accepting. Thus, for Alice, there are two cases: 1) Bob accepts all. In this case, Alice s best response is to offer nothing, so the SPNE for (Alice,Bob) is (0,aaaaa). ) Bob rejects 0 and accepts all others. Then Alice s best response is to offer 1, the least she can give while ensuring Bob s acceptance. Then the SPNE is (1,raaaa). 4
4 Now Bob moves before Alice, but Alice doesn t observe Bob, so she only has one information set despite the fact that she has 5 nodes. Thus, in this case, SPNE doesn t impose any additional constraints beyond NE. There are now many equilibria. If Alice plays 0, all s 1 s s s 4 s 5 are best responses for Bob. If Alice plays i > 0, all strategies where s i = accept are best responses for Bob. For Alice s strategy i to be a best response, we must have i = min{j : s j = a, 0}. Thus the set of SPNE is {0,rrrrr} and {i, s 1 s s s 4 s 5 } where i = min{j : s j = a, 0} Since Bob now cares about all possible actions Alice could chose, rather than just one that she s picked, his only two possibilities are raaaa and aaaaa, as in. However, Alice has no weakly dominant strategies, as her best response is contingent on Bob s strategy. Thus there are no equilibria in weakly dominant strategies. Actions: q 1 [0, ], q (q 1 ) [0, ]. Strategies: firm 1: q 1 [0, ], firm : the set of functions q : [0, ] [0, ] For backwards induction equilibria, first we solve the subgame starting at firm s turn, then we solve firm 1 s problem. Given q 1, firm s best response is to maximize Π (q ) = q ( 1 q 1 q ). Computing the FOC gives us 1 q 1 q = 0 or q = 1 q 1. Thus firm s best response is q (q 1 ) = 1 q 1. Consequently, firm 1 must maximize Π 1 (q 1 ) = q 1 ( 1 q 1 1 q 1 ) = q 1 (1/ q 1 /). Computing the FOC gives us 1/ q 1 = 0 or q 1 = 1. Plugging that into q (q 1 ),we have q = 1/4. Thus the equilibrium outcome is Π 1 = 1/(1 1/ 1/4) = 1/8 and Π = 1/4(1 1/ 1/4) = 1/16. The profit from Cournot is 1/(1/)=1/9, so firm 1 is better off and firm is worse off. 4
5 5 We ll consider 5 first because and 4 are special cases of it. Consider a strategy of q = 1 q1 for firm if q 1 = q 1and q = 1 otherwise. Essentially, we re making deviating from (q1, 1 q 1 ) as costly as possible for firm 1, to give them as strong incentives as we can to produce those quantities. If firm 1 produces anything else, firm will produce a quantity 1, making firm 1 s profit less than or equal to zero. Firm s behavior for contingencies other than q 1 =q 1 seems irrational, and wouldn t survive in a SPNE, but since we only require a Nash Equilibrium it doesn t matter because firm never has to play those irrational moves in the equilibrium. Thus, given that firm 1 s strategy is q 1 =q 1, firm is playing its best response, and so long as firm 1 s profit is nonnegative, playing q 1 is better than the next best response of producing 0 and getting 0 profit. So long as q 1 1, this condition will be satisfied. Thus the range of supportable q 1 s is [0,1] and the range of supportable q s is [0,1/]. Choosing q 1 = 1/ and using the strategies in part 5 gives us the desired equilibrium Choosing q 1 = 1/4 and using the strategies in part 5 gives us the desired equilibrium, as it is the reverse of the SPNE Stackleberg outcome we derived before. u i (x i, x i x i ) = x i ( x i x i ). Notice this has the same functional form as the profit function in the cournot game, with price and quantity replaced with the public and private goods. The best responses are thus the same: x i = x i. Then the equilibrium x is are x i x i = = 1/ + x i /4 and x i = /. & 5
6 This is equivalent to Stackleberg, which was solved earlier. 4-1 As we saw in the last homework, then the set of possible Nash Equilibria is x such that x 1 x = 1/4. and x i Player 1 s strategies are simply x 1 [0, 1]. Player s strategies are the set of functions x : [0, 1] [0, 1]. 4- First, we ll solve player s problem. Given x 1, u (x, x 1 ) = x + x 1 x F OC : u (x, x 1 ) x = 1 1 x 1 x = 0 x = 7/4 x 1 This is our best response for player. Then player 1 solves u 1 (x 1, x ) = x 1 + 1/4,which has a corner solution of x 1 = 1. Essentially, since expenditures on the public good are the same no matter what player 1 does, since player has an incentive to make up for player 1 s shortchanging, player 1 might as well not contribute anything and leave it all to player to finance the public good. Thus we have a NE of (1, 7/4 x 1 ) and an equilibrium outcome of / for player 1 and 5/4 for player. Note that te SPNE is the element of the set of NE in the simultaneous move game that is most favorable to player 1 and least favorable to player
7 Notice that at the last node of the game tree (the last subgame), stopping is the unique best response. Given that player will stop in the last period, player 1 s best response at the second to last period (the second to last subgame) is to stop. Continuing inductively, we see that, given optimal play in all later periods, the best response at every node is to stop. Thus the only SPNE strategy profile is s 1 s...s 99 s 100 for both players. As in part, the game must stop in the first period. Suppose we had a NE where the first n actions were continue and the n+1th action was stop (any strategy profile with continue as the first move will be of this form, since the game stops after the last period). Then whoever plays at period n will have an incentive to deviate, as stopping n the nth period rather than the n+1th period will net them 1 additional dollar. Then the only possible NE strategy profiles have player 1 stopping in the first period, and player stopping in the second period. However, all other elements of the strategy that is, actions at every node 7
8 after the second can vary freely, since they are never reached and thus have no effect on the payoffs. Player must play stop at the second period so that player 1 doesn t have an incentive to deviate to continue. Thus the NE strategy profiles are s 1 a...a 99 a 100 for player 1 and s 1 a...a 99 a 100, where a can be either stop or continue. 4 There are no strictly dominated strategies. Every node after the first will never be reached under certain strategies specifically where player 1 stops in the first period, so there are situations where the actions at those nodes don t matter. Thus, we only need consider player 1 s first action. Continue is strictly better if player s first action is continue, and strictly worse otherwise, so there is no action in period 1 that is strictly better for player 1 under all contingencies
9 Strategies for player 1 are of the form a 1 a a a 4...a 8, where a 1 {(A, B), (B, C), (C, D)}and all other actions are chosen from the two options that are being voted for at that node. We end up with 16 information sets because player 1 chooses the first voting pair (1 info set), votes on the first pair ( info sets), and votes on the second pair (*8=4 info sets). Strategies for player are of the form a 1 a a a 4...a 54, where all actions are chosen from the two options that are being voted for at that node. We end up with 54 information sets because player votes on the first pair (*=6 info sets) and votes on the second pair (*8*=48 info sets). Strategies for player are of the form a 1 a a a 4...a 108, where all actions are chosen from the 9
10 two options that are being voted for at that node. We end up with 108 information sets because player votes on the first pair (*4=1 info sets) and votes on the second pair (*4*8=96 info sets). As you can see in the extensive form of the game tree, the second rounds of voting yield outcome A for (A,B), C for (A,C), and B for (B,C). Then, once we ve finished the first round of voting, we know the payoff, and can plug it in as in the figure. Now we see that, if player 1 chooses (A,C), the voting will proceed to (A,B), which is optimal for player 1. Thus player 1 will choose (A,C) and the final vote will be for A. There are many SPNE strategy profiles there are many information sets where players are indifferent, and each of those doubles the number of SPNE strategy profiles since either choice is consistent with best responding (a quick count gives me ˆ50 strategies, but I m obviously not going to write them out). As before, there are many, many SPNE strategy profiles, and they can be defined analogously to 7., though the reordering of moves will change the number of information sets each player has and the way their strategies must be written down. The set of SPNE outcomes will remain the same, however notice that, in a voting game with options and voters, a given voter is either not pivotal, in which case their vote doesn t matter, or they are pivotal, in which case they choose their most preferred option. Later voters observe the votes of earlier voters, but the only change this can cause in their strategy is to make them vote for a less preferred option if they know their vote isn t pivotal, which only happens in this case if the same outcome obtains regardless of their vote. Thus, changing the order of voting will not change the equilibrium payoffs. 10
CUR 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 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 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 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 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 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 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 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 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 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 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 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 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 informationDynamic Games. Econ 400. University of Notre Dame. Econ 400 (ND) Dynamic Games 1 / 18
Dynamic Games Econ 400 University of Notre Dame Econ 400 (ND) Dynamic Games 1 / 18 Dynamic Games A dynamic game of complete information is: A set of players, i = 1,2,...,N A payoff function for each player
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 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 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 informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Modelling Dynamics Up until now, our games have lacked any sort of dynamic aspect We have assumed that all players make decisions at the same time Or at least no
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 informationExercises Solutions: Game Theory
Exercises Solutions: Game Theory Exercise. (U, R).. (U, L) and (D, R). 3. (D, R). 4. (U, L) and (D, R). 5. First, eliminate R as it is strictly dominated by M for player. Second, eliminate M as it is strictly
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 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 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 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 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 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 informationECON 803: MICROECONOMIC THEORY II Arthur J. Robson Fall 2016 Assignment 9 (due in class on November 22)
ECON 803: MICROECONOMIC THEORY II Arthur J. Robson all 2016 Assignment 9 (due in class on November 22) 1. Critique of subgame perfection. 1 Consider the following three-player sequential game. In the first
More informationSimon Fraser University Fall Econ 302 D200 Final Exam Solution Instructor: Songzi Du Wednesday December 16, 2015, 8:30 11:30 AM
Simon Fraser University Fall 2015 Econ 302 D200 Final Exam Solution Instructor: Songzi Du Wednesday December 16, 2015, 8:30 11:30 AM NE = Nash equilibrium, SPE = subgame perfect equilibrium, PBE = perfect
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 informationExercise Chapter 10
Exercise 10.8.1 Where the isoprofit curves touch the gradients of the profits of Alice and Bob point in the opposite directions. Thus, increasing one agent s profit will necessarily decrease the other
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 informationSo we turn now to many-to-one matching with money, which is generally seen as a model of firms hiring workers
Econ 805 Advanced Micro Theory I Dan Quint Fall 2009 Lecture 20 November 13 2008 So far, we ve considered matching markets in settings where there is no money you can t necessarily pay someone to marry
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 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 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 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 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 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 informationECO 5341 (Section 2) Spring 2016 Midterm March 24th 2016 Total Points: 100
Name:... ECO 5341 (Section 2) Spring 2016 Midterm March 24th 2016 Total Points: 100 For full credit, please be formal, precise, concise and tidy. If your answer is illegible and not well organized, if
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 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 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 informationMA200.2 Game Theory II, LSE
MA200.2 Game Theory II, LSE Answers to Problem Set [] In part (i), proceed as follows. Suppose that we are doing 2 s best response to. Let p be probability that player plays U. Now if player 2 chooses
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 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 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
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 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 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 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 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 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 informationPlayer 2 H T T -1,1 1, -1
1 1 Question 1 Answer 1.1 Q1.a In a two-player matrix game, the process of iterated elimination of strictly dominated strategies will always lead to a pure-strategy Nash equilibrium. Answer: False, In
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 informationLECTURE NOTES ON GAME THEORY. Player 2 Cooperate Defect Cooperate (10,10) (-1,11) Defect (11,-1) (0,0)
LECTURE NOTES ON GAME THEORY September 11, 01 Introduction: So far we have considered models of perfect competition and monopoly which are the two polar extreme cases of market outcome. In models of monopoly,
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 informationChapter 11: Dynamic Games and First and Second Movers
Chapter : Dynamic Games and First and Second Movers Learning Objectives Students should learn to:. Extend the reaction function ideas developed in the Cournot duopoly model to a model of sequential behavior
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 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 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 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 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 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 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 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 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 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 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 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 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 informationEconomic Management Strategy: Hwrk 1. 1 Simultaneous-Move Game Theory Questions.
Economic Management Strategy: Hwrk 1 1 Simultaneous-Move Game Theory Questions. 1.1 Chicken Lee and Spike want to see who is the bravest. To do so, they play a game called chicken. (Readers, don t try
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 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 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 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 informationMIDTERM 1 SOLUTIONS 10/16/2008
4. Game Theory MIDTERM SOLUTIONS 0/6/008 Prof. Casey Rothschild Instructions. Thisisanopenbookexam; you canuse anywritten material. You mayuse a calculator. You may not use a computer or any electronic
More informationMath 135: Answers to Practice Problems
Math 35: Answers to Practice Problems Answers to problems from the textbook: Many of the problems from the textbook have answers in the back of the book. Here are the answers to the problems that don t
More informationMATH 4321 Game Theory Solution to Homework Two
MATH 321 Game Theory Solution to Homework Two Course Instructor: Prof. Y.K. Kwok 1. (a) Suppose that an iterated dominance equilibrium s is not a Nash equilibrium, then there exists s i of some player
More informationFrancesco Nava Microeconomic Principles II EC202 Lent Term 2010
Answer Key Problem Set 1 Francesco Nava Microeconomic Principles II EC202 Lent Term 2010 Please give your answers to your class teacher by Friday of week 6 LT. If you not to hand in at your class, make
More informationAnswers to Odd-Numbered Problems, 4th Edition of Games and Information, Rasmusen
ODD Answers to Odd-Numbered Problems, 4th Edition of Games and Information, Rasmusen Eric Rasmusen, Indiana University School of Business, Rm. 456, 1309 E 10th Street, Bloomington, Indiana, 47405-1701.
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 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 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 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 informationPAULI MURTO, ANDREY ZHUKOV
GAME THEORY SOLUTION SET 1 WINTER 018 PAULI MURTO, ANDREY ZHUKOV Introduction For suggested solution to problem 4, last year s suggested solutions by Tsz-Ning Wong were used who I think used suggested
More informationStrategic Pre-Commitment
Strategic Pre-Commitment Felix Munoz-Garcia EconS 424 - Strategy and Game Theory Washington State University Strategic Commitment Limiting our own future options does not seem like a good idea. However,
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 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 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 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 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 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 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 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 informationAdvanced Microeconomics
Advanced Microeconomics ECON5200 - Fall 2014 Introduction What you have done: - consumers maximize their utility subject to budget constraints and firms maximize their profits given technology and market
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 informationMohammad Hossein Manshaei 1394
Mohammad Hossein Manshaei manshaei@gmail.com 1394 Let s play sequentially! 1. Sequential vs Simultaneous Moves. Extensive Forms (Trees) 3. Analyzing Dynamic Games: Backward Induction 4. Moral Hazard 5.
More information1 x i c i if x 1 +x 2 > 0 u i (x 1,x 2 ) = 0 if x 1 +x 2 = 0
Game Theory - Midterm Examination, Date: ctober 14, 017 Total marks: 30 Duration: 10:00 AM to 1:00 PM Note: Answer all questions clearly using pen. Please avoid unnecessary discussions. In all questions,
More informationDynamic games with incomplete information
Dynamic games with incomplete information Perfect Bayesian Equilibrium (PBE) We have now covered static and dynamic games of complete information and static games of incomplete information. The next step
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 information