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### oldexam3sol

Course: ECON 4349, Fall 2008
School: U. Houston
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4349 Econ Midterm #3 solutions 1. (30 points) a. (10 points) b. (10 points) c. (10 points) 2. (30 points) In the strategic form below, boldface payoffs correspond to pure-strategy best responses. Player 2 Player 1 A B A B 1,1,1 0,9,0 9,0,0 1,1,1 Player 3: A Player 1 A B Player 2 A B 0,0,9 1,1,1 1,1,1 0,0,0 Player 3: B Based on the best responses above, pure-strategy Nash equilibria are (B,B,A), (A,B,B), and...

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4349 Econ Midterm #3 solutions 1. (30 points) a. (10 points) b. (10 points) c. (10 points) 2. (30 points) In the strategic form below, boldface payoffs correspond to pure-strategy best responses. Player 2 Player 1 A B A B 1,1,1 0,9,0 9,0,0 1,1,1 Player 3: A Player 1 A B Player 2 A B 0,0,9 1,1,1 1,1,1 0,0,0 Player 3: B Based on the best responses above, pure-strategy Nash equilibria are (B,B,A), (A,B,B), and (B,A,B). Since the game is symmetric, but none of these equilibria are, there should be a mixed-strategy equilibrium (and it should be symmetric). So, let (p,1-p), (q,1-q), and (r,1-r) be the three players' mixed strategies. For Player 1, the expected payoff to choosing A is 1qr+0(1-q)r+0q(1-r)+1(1-q)(1-r) = 1-q-r+2qr, and the expected payoff to choosing B is 9qr+1(1-q)r+1q(1-r)+0(1-q)(1-r) = q+r+7qr. Player 1 will be willing to choose a mixed strategy only if the payoffs to A and B are equal, which is true if 1-q-r+2qr = q+r+7qr. This equality simplifies to 5qr+2q+2r-1=0. Similar arguments for Player 2 and Player 3 give the conditions for their playing mixed strategies: 5pr+2p+2r-1=0 5qr+2q+2r-1=0 Since we are looking for a symmetric equilibrium, we can set p=q=r, so each of these equations simplifies to 5p2+4p-1=0. This quadratic has two solutions: p=-1 and p=1/5. Since probabilities must be between 0 and 1, we can eliminate p=-1. Then, since p=q=r, we have q=1/5 and r=1/5, so the mixed-strategy Nash equilibrium is ( (1/5,4/5), (1/5,4/5), (1/5,4/5) ). 3. (30 points) a. (8 points) At Player 1's second decision node, she will choose H (which gives her 8) instead of G (which gives her 7). Then, at Player 2's decision nodes, he will choose C (which gives him 3) over D (which gives him 1) and E (which gives him 0) over F (which gives him -1). Then, at Player 1's first decision node, she will choose B (which gives her 7) instead of A (which gives her 0). So, the rollback equilibrium is (BH,CE), and payoffs are (7,0). b. (12 points) A game matrix is shown here. (Other correct are answers possible.) Player 2 CE CF DE DF Player AG 0,3 0,3 7,6 7,6 1 AH 0,3 0,3 8,1 8,1 BG 7,0 -1,-1 7,0 -1,-1 BH 7,0 -1,-1 7,0 -1,-1 c. (10 points) Pure-strategy best responses correspond to boldface payoffs. Pure-strategy Nash equilibria are (AH,CF), (BG,CE), and (BH,CE); note that this last one is the rollback equilibrium. 4. (30 points) a. (10 points) A game tree is shown below. b. (10 points) Player 1's available pure strategies are: Quit, Up, Top Continue, Up, Top Quit, Up, Bottom Continue, Up, Bottom Quit, Down, Top Continue, Down, Top Quit, Down, Bottom Continue, Down, Bottom Player 2's available pure strategies are: First, Left, Links Second, Left, Links First, Left, Rechts Second, Left, Rechts First, Right, Links Second, Right, Links First, Right, Rechts Second, Right, Rechts c. (10 points) This game has four subgames: the entire game, the portion of the game starting from Player 2's first move (after Player 1 has chosen Continue), and the portions starting from after Player 2 has chosen either First or Second. To find the subgame perfect equilibrium, we start from these last two subgames. Both correspond to the simultaneous-move games below. Player 2 Player 2 Lef...

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U. Houston - ECON - 4349
Name:ID#:Econ 4349 Midterm #4Answer all questions as completely as possible. Partial credit may be given for progress toward a solution. True but irrelevant information receives no credit. You may write on the backs of these pages, as long as I
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #1Consider a game with this payoff matrix. Player 2 W 4,6 2,9 6,8Player 1A B CV 7,1 5,5 9,3X 3,2 8,7 1,4a. If Player 1 chooses A and Player 2 chooses X, what will Player 1s payoff be?b. If Player 1 chooses B a
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #10Due at beginning of class Nov. 29Consider the following two-player card game. First, Player 1 and Player 2 are each dealt one card, which can be either an A or a K. There is a chance that both are dealt As, a chance
NMSU - MATH - 683
Spectral SequencesIn this document we gather together facts and calculations about spectral sequences.1Motivation of Spectral Sequences by Double Complexes0 Let T be a rst quadrant double complex. We write Epq for the (p, q) term in T . We ha
U. Houston - ECON - 4349
Econ 4349 Quiz #10 Solutionsa. One acceptable game tree is shown below.b. This game has no proper subgames, so every Nash equilibrium is also a subgame perfect equilibrium. To find the Nash equilibria, we represent the game with a strategic form.
U. Houston - ECON - 4349
Econ 4349 Quiz #1 SolutionsPlayer 1 A B C V 7,1 5,5 9,3 Player 2 W 4,6 2,9 6,8 X 3,2 8,7 1,4a. If Player 1 chooses A and Player 2 chooses X, Player 1s payoff will be 3. b. If Player 1 chooses B and Player 2 chooses V, Player 2s payoff will be 5. c
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #2Consider a game with this payoff matrix. Player 2 Y 1,2 8,7 2,1Player 1A B CX 7,1 5,8 9,3Z 4,3 2,9 6,4a. Does Player 1 have a dominant strategy in this game? If so, which one(s)?b. Does Player 1 have a domin
U. Houston - ECON - 4349
Econ 4349 Quiz #2 SolutionsPlayer 1 A B C X 7,1 5,8 9,3 Player 2 Y 1,2 8,7 2,1 Z 4,3 2,9 6,4a. Player 1 has no dominant strategy. b. Player 1 has a dominated strategy, A (which is dominated by C). c. Player 2 has a dominant strategy, Z. d. Player
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #3Consider a game with this payoff matrix. Player 2 Player 1 A B W 6,1 4,4 X 1,0 2,3 Y 7,5 0,4 Z 2,6 2,5a. Write Player 1s best-response correspondence in the space below.b. Write Player 2s best-response correspondenc
U. Houston - ECON - 4349
Econ 4349 Quiz #3 SolutionsPlayer 2 Player 1 A B W 6,1 4,4 X 1,0 2,3 Y 7,5 0,4 Z 2,6 2,5a. Player 1s best-response correspondence is BR(W)=A BR(X)=B BR(Y)=A BR(Z)={A,B} b. Player 2s best-response correspondence is BR(A)=Z BR(B)=Z c. Nash equilibri
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #4Due at beginning of class Sept. 201. Consider the game below. Player 2 H J 1,3,7 7,1,11 4,5,2 10,6,1 5,12,6 8,2,9 Player 3: X Player 2 H J 9,4,8 12,7,5 11,10,4 3,8,10 2,9,12 6,11,3 Player 3: YPlayer 1A B CPlayer
U. Houston - ECON - 4349
Econ 4349 Quiz #4 Solutions1. Player 1 A B C Player 2 H J 1,3,7 7,1,11 4,5,2 10,6,1 5,12,6 8,2,9 Player 3: X Player 1 A B C Player 2 H J 9,4,8 12,7,5 11,10,4 3,8,10 2,9,12 6,11,3 Player 3: Ya. If Player 1 chooses C, Player 2 chooses J, and Player
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #5Due at beginning of class Oct. 13Consider the game below. Player 2 W X Player A 6,1 2,5 1 B 3,4 4,3 a. Find any pure-strategy Nash equilibria of this game. b. Let (p,1-p) denote Player 1s (mixed) strategy, and (q,1-q)
U. Houston - ECON - 4349
Econ 4349 Quiz #5 SolutionsPlayer 2 Player 1 A B W 6,1 3,4 X 2,5 4,3a. This game has no pure-strategy Nash equilibria. b. Given the players mixed strategies, Player 1s expected payoff to A is 2+4q and her expected payoff to B is 4-q. So, A is her
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #6Consider the game tree below.a. In the space below, list the strategies available to Player 1.b. In the space below, list the strategies available to Player 2.
U. Houston - ECON - 4349
Econ 4349 Quiz #6 Solutionsa. Player 1 has only one decision node, and two choices at that node: A and B. So, her only available strategies are A and B. b. Player 2 has two decision nodes, the left one (after Player 1 has chosen A) and the right on
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #71. Represent the following game tree as a game matrix. (You do not need to solve it!)2. Represent the following game matrix as a game tree. (You do not need to solve it!) Player 2 D 3,2 2,3Player 1A BC 5,0 1,4
U. Houston - ECON - 4349
Econ 4349 Quiz #7 Solutions1. Note that Player 1 has three strategies: A, B, and C. Player 2 has four strategies: WY, WZ, XY, and XZ. Player 2 Player 1 A B C WY 6,2 5,0 7,1 WZ 6,2 5,0 2,3 XY 4,4 5,0 7,1 XZ 4,4 5,0 2,32. Other game trees are accept
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #8Due at beginning of class Nov. 8Consider the two simultaneous-move games below. Player 2 Y Z Player A 7,-7 6,-6 Player 1 1 B 8,-8 9,-9 Game 1 Player 2 W X 2,-1 0,1 -1,2 1,0 Game 2C DNow, consider the following game
U. Houston - ECON - 4349
Econ 4349 Quiz #8 Solutionsa. One acceptable game tree is shown below; others are possible.b. To solve this game using the modified version of backward induction discussed in class, we first find the subgames. Besides the entire game (which is a s
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #9Consider the following game. First, a fair coin is tossed (so that Heads and Tails are equally likely). Player 1 is told the result of the coin toss; Player 2 is not. Then, the two players themselves simultaneously choo
U. Houston - ECON - 4349
Econ 4349 Quiz #9 Solutionsa. One acceptable game tree is shown below.b. Player 1 has two information sets, each with only one node. At either information set, she can choose either H or T. So, her available pure strategies are HH, HT, TH, and TT.
U. Houston - ECON - 4349
Syllabus: Introduction to Game Theory (Econ 4349)The goal of this course is to teach you how to analyze simple strategic situations with the use of gametheoretic techniques. Times of class: Instructor: TTh 1-2:30 pm Nick Feltovich Phone: Fax: Office
U. Houston - ECON - 7342
Name:Econ 7342 Midterm Exam #2(120 points total) Instructions: Answer all questions. Answers that are not correct, but show some progress toward a solution, will receive partial credit. You may use scratch paper, but you will only receive credit f
Allan Hancock College - COMP - 5348
Copyright warningCOMMONWEALTH OF AUSTRALIA Copyright Regulations 1969 WARNING This material has been reproduced and communicated to you by or on behalf of the University of Sydney pursuant to Part VB of the Copyright Act 1968 (the Act). The material
U. Houston - ECON - 7342
Econ 7342 Midterm Exam #2 Solutions(1) We have pA = 5.5 QA , pB = 3.5 QB , and C(Q) = 0.5Q. (a) The customers demand functions are QA = 5.5pA and QB = 3.5pB , so if the monopolist charges a single price p, market demand is Q = QA + QB = 9 2p for
U. Houston - ECON - 7342
Econ 7342 Midterm Exam #3 Solutions3 3 (1) We have Ui = 6ea eb ei , for i = a, b. 1 1 3 3 (a) The sum of Anns and Bills utilities is Ua + Ub = 12ea eb ea eb . This is maximized by setting partial derivatives (with respect to ea and eb ) to zero:
U. Houston - ECON - 7342
Syllabus: Microeconomic Theory II (Econ 7342-04420)Times of Class: Instructor: Office: Office Hours: MW 9-10:30 am Location of Class: McElhinney 104 Nick Feltovich McElhinney 202-C Phone: 743-3815 Email: nfelt@mail.uh.edu Fax: 743-3798 WWW: www.uh.e
NMSU - MATH - 683
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Allan Hancock College - COMP - 5348
CopyrightwarningCOMMONWEALTH OF AUSTRALIA Copyright Regulations 1969 WARNING This material has been reproduced and communicated to you by or on behalf of the University of Sydney pursuant to Part VB of the Copyright Act 1968 (the Act). The material
NMSU - MATH - 683
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NMSU - MATH - 683
Exercises From Section 5.1In this note we ll in some of the details from Chapter 5.1 of Weibel. Let E be a double complex of modules. The rst result we do is part 1 of Exercise 5.1.2, which gives a 2 description of Epq .2 Problem 1 (5.1.2). The mo
Allan Hancock College - COMP - 5348
CopyrightwarningCOMMONWEALTH OF AUSTRALIA Copyright Regulations 1969 WARNING This material has been reproduced and communicated to you by or on behalf of the University of Sydney pursuant to Part VB of the Copyright Act 1968 (the Act). The material
NMSU - MATH - 683
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NMSU - MATH - 683
Mathematics 683 Assignment 2Fall 2006Work three of the following problems. These are due on 8 December. 1. Let n be a positive integer. If n : A ! A is the multiplication by n map on a G-module A, show, by means of using the bar resolution, that th
NMSU - MATH - 683
Handouts for Mathematics 683 Topics in Group TheoryPatrick J. Morandi Fall 20022Contents1 Group Actions 2 Simplicity of An 3 Automorphism Groups of Sn and of An 4 Free Groups 4.1 Construction of Free Groups . . . . . . . . . . . . . . . . . . .
NMSU - MATH - 683
Free GroupsIn this note we give an alternative construction of free groups by working with semigroups. We also give some basic properties of free groups. A semigroup is a nonempty set with an associative binary operation. If S is a semigroup, we de
U. Houston - ECON - 4349
Econ 4349 Homework #11. Consider a sequential-move game with this game tree. (Notice that if Player 1 chooses D, the game ends without Player 2 making a move.)a. List all strategies available to Player 1.b. List all strategies available to Playe
U. Houston - ECON - 4349
Econ 4349 Homework #21. Three people simultaneously decide whether to contribute to a public project. Contributing costs \$20. If at least one person contributes, the project is successful and all three people get a benefit: Player 1 gets \$50, Player
NMSU - MATH - 683
Some Groups of Order 27Let G = x, y | x9 = y 9 = 1, xy 4 = y 7 x, yx4 = x7 y . In this note we determine the structure of G. For our rst observation, we note that, by the symmetry of the relations, there is an automorphism of G with (x) = y and (y
U. Houston - ECON - 4349
Econ 4349 Homework #2 solutions1. We abbreviate the strategies as C=contribute and N=not contribute. a. One possible payoff table is shown here; others are possible. Player 2 C N 30,10,-10 30,30,-10 50,10,-10 50,30,-10 Player 3: C Player 2 C N 30,10
U. Houston - ECON - 4349
Econ 4349 Homework #3 solutions1. Players are abbreviated: P1=Player 1, P2=Player 2, P3=Player 3. Strategies are abbreviated: 0=dont contribute, 1=contribute one unit, and 2=contribute two units. a. One possible payoff table is shown here; others ar
NMSU - MATH - 683
Changing GeneratorsIf a group G = X | R is given by generators and relations, and if Y is another generating set, can one use the given presentation to present G with the new generating set? There is a natural guess of how to do this, which we desc
U. Houston - ECON - 4349
Econ 4349 Homework #5For each game below, find all Nash equilibria (pure- and mixed-strategy). 1. Player 2 Player 1 2. Player 2 Player 1 3. Player 2 Player 1 4. Player 2 Player 1 C D C 0,0 2,6 D 9,3 0,0 S R S 70,70 60,10 R 10,60 60,60 H T H 4,0 0,2
NMSU - MATH - 683
Extensions of Groups1IntroductionTo better understand groups it is often useful to see how a group can be built from smaller groups. For example, given two groups G and H, we can form the direct product G H. This new group can be studied in te
U. Houston - ECON - 4349
Name:ID#:Econ 4349 Quiz #11. Consider a sequential-move game with this game tree. (Notice that if Player 1 chooses D, the game ends without Player 2 making a move.)a. List all strategies available to Player 1.b. List all strategies available
U. Houston - ECON - 4349
Name:Econ 4349 Quiz #101. Consider the following game tree.a. List all of Player 1s available strategies in the space below.b. List all of Player 2s available strategies in the space below.c. Write the subgame perfect equilibrium in the spac
U. Houston - ECON - 4349
Econ 4349 Quiz #10 Solutions1.a. (2 points) Player 1 has only one information set (in fact, only one node), and has three options at this information set: C, D, and E. So, her available strategies are C, D, and E. b. (4 points) Player 2 has two in
U. Houston - ECON - 4349
Econ 4349 Quiz #3 Solutions1. X 7,1 5,8 9,6 Player 2 Y 1,2 8,7 2,1 Z 4,3 2,9 6,4Player 1A B Ca. (1 point) For Player 1, strategy C strictly dominates strategy A. b. (1 point) For Player 2, strategy Z strictly dominates strategy Y. 2. S 4,6 3,0
U. Houston - ECON - 4349
Name:Econ 4349 Quiz #41. Consider the simultaneous-move game with the payoff table below. Player 2 W X Y Z A 20,20 25,10 10,15 15,15 Player B 10,0 10,5 20,25 35,10 1 C 15,15 25,20 30,10 5,15 D 30,10 0,20 15,5 10,20 a. (5 points) Write Player 1s en
U. Houston - ECON - 4349
Econ 4349 Quiz #4 Solutions1. Player 2 A B C D W 20,20 10,0 15,15 30,10 X 25,10 10,5 25,20 0,20 Y 10,15 20,25 30,10 15,5 Z 15,15 35,10 5,15 10,20Player 1a. (5 points) Player 1s best-response correspondence is BR(W)=D BR(X)={A,C} BR(Y)=C BR(Z)=B
U. Houston - ECON - 4349
Name:Econ 4349 Quiz #5Consider the game below. Player 2 H J 10,5,1 6,6,3 3,6,7 5,3,9 4,1,10 7,2,5 Player 3: X Player 2 H J 4,7,2 5,3,9 0,10,5 8,6,2 3,4,2 1,7,3 Player 3: YPlayer 1A B CPlayer 1A B Ca. Write Player 1s best-response corresp
U. Houston - ECON - 4349
Name:Econ 4349 Quiz #7For the game below, let Player 1s mixed strategy be (p,1-p) and Player 2s be (q,1-q). Player 2 Y Z Player A 3,0 0,1 1 B 0,1 1,0 a. Write Player 1s best-response correspondence (for both pure and mixed strategies) in the space
U. Houston - ECON - 4349
Econ 4349 Quiz #8 SolutionsPlayer 2 Player 1 A B A 6,4 0,0 B 0,0 1,1There are two pure-strategy Nash equilibria: (A,A) and (B,B). Since almost all two-player games have an odd number of Nash equilibria, we expect there to be one more, and for it t
U. Houston - ECON - 2304
Econ 2304-04308 Midterm Exam #1 Fall 2003Name_ ID#_Choose the one alternative that best completes the statement or answers the question. 1. The term &quot;ceteris paribus&quot; is a synonym for A) small changes. B) other things equal. C) quantity demanded.
U. Houston - ECON - 2304
Econ 2304-04308 Midterm Exam #2 Fall 2003Name_ ID#_Choose the one alternative that best completes the statement or answers the question. 1. The market for milk is in equilibrium, with an equilibrium price of \$2 per gallon. The dairy producers ass
U. Houston - ECON - 2304
Econ 2304-04308 Midterm Exam #3 Fall 2003Name_ ID#_Choose the one alternative that best completes the statement or answers the question. 1. A perfectly competitive firm has chosen its profit-maximizing level of output. The good's price is \$10, an
U. Houston - ECON - 2304
Econ 2304 Homework #1 Solutions:2. a. Maria's production possibilities frontier is shown below.b. It takes her 2 hours to read 100 pages of sociology. In 2 hours, she could read 40 pages of economics. So, her opportunity cost of reading 100 pages
U. Houston - ECON - 2304
Econ 2304 Homework #2 Solutions:2. This statement is false. An increase in the demand for notebooks will (ceteris paribus) raise the equilibrium quantity, so the quantity supplied and quantity demanded will increase. The only case in which quantity
U. Houston - ECON - 2304
Econ 2304 Homework #3 Solutions:3. The demand and supply schedules are: Price (\$) 11 10 9 8 7 6 Quantity demanded (millions) 1 2 4 6 8 10 Quantity supplied (millions) 15 12 9 6 3 1a. The equilibrium price and quantity are \$8 and 6 million. b. If a
U. Houston - ECON - 2304
Econ 2304 Homework #4 Solutions:1. a. Mystery novels have many substitutes (other novels, mystery videos), so they should have more elastic demand than required textbooks, which have no close substitutes. b. Beethoven recordings belong to a narrowly