Math 131a Lecture 2 Spring 2009
Midterm 1
Name:
Instructions: There are 4 problems. Make sure you are not missing any pages. Unless stated otherwise (or unless it trivializes the problem), you may use without proof anything proven in the sections of the b
Solutions to homework 1 1.5#1 Mis`re version of the take-away game. There are 21 chips, we can remove 1, 2, or 3. e Last player to move loses, hence position 1 is a P-position, from positions 2,3, and 4 we can move to 1, hence these are N-positions. Now,
Basic Concepts
In a strategic setting, a person may not always have an choice of what is best that is independent of the actions of others A game is an abstract model of a strategic situation Three elements to a game
players strategies payoffs
Players
Selected Solutions from Thomas S. Ferguson's Game Theory
Jeffrey Lee Hellrung, Jr. April 03, 2009
1.1.3 (a) The strategy of the previous exercise will not work. You would begin by choosing a 3, and each time your opponent chooses a 4, your response is to
Game Theory Solutions to Exercises The Strategic Form of a Game, Matrix Games & Domination
Jan-Jaap Oosterwijk Fall 2007
2
Matrix Games Domination
Contrary to Ferguson's notation, we prefer to use the horizontal notation (, , . . . , ) to represent a row
Game Theory Solutions to Exercises: Noncooperative Games
Jan-Jaap Oosterwijk Fall 2007
2
Noncooperative Games
Strategic Equilibria Are Individually Rational.
2.5.1
A payoff vector is said to be individually rational if each player receives at least his sa
Game Theory Solutions to Exercises: Cooperative Games
Jan-Jaap Oosterwijk Winter 2007-2008
4
Cooperative Games
4.5.1 For the following bimatrix games, draw the NTU and TU feasible sets. What are the Pareto optimal outcomes? (a) (0, 4) (3, 2) (4, 0) (2, 3)
Game Theory Solutions to Exercises The Principle of Indifference
Jan-Jaap Oosterwijk Fall 2007
3
3.1
The Principle of Indifference
-2 2 -1 1 . Consider the game with matrix 1 1 3 0 1 (a) Note that this game has a saddle point. (b) Show that the inverse o
Game Theory Solutions to Exercises: Bimatrix Games Safety Levels
Jan-Jaap Oosterwijk Fall 2007
1
Bimatrix Games Safety Levels
1.6.1 Convert the following extensive form game to strategic form. Solution: (See graphical appendix at the end of this document
Game Theory Solutions to Exercises: Models of Duopoly
Jan-Jaap Oosterwijk Fall 2007
3
Models of Duopoly
3.5.1 (a) Suppose in the Cournot model that the firms have different production costs. Let c1 and c2 be the costs of production per unit for firms 1 an
Game Theory Solutions to Exercises: The Extensive Form of a Game
Jan-Jaap Oosterwijk Fall 2007
5
The Extensive Form of a Game
The Silver Dollar
5.9.1
Player II chooses one of two rooms in which to hide a silver dollar. Then, Player I, not knowing which ro
Math 167 Homework 8
December 9, 2008
Game Theory Thomas Ferguson
Section II.3.7 Problem 15. Battleship. The game of Battleship, sometimes called Salvo, is played on two square boards, usually 10 by 10. Each player hides a fleet of ships on his own board a
1
Part I: Multiple choice
Choose exactly one response for each 1. A natural monopoly (a) is a monopoly in the production of raw materials. (b) occurs when one .rm can supply the entire market more cheaply than can a number of .rms. (c) is one result of a
Monopoly
You will not be responsible for monopoly and quality
Monopoly
A monopoly is a single supplier to a market Barriers to entry are the source of all monopoly power There are two general types of barriers to entry technical barriers legal barriers
Te
Short-Run Decisions: Pricing & Output
When there are only a few firms in a market, predicting output and price can be difficult
how aggressively do firms compete? how much information do firms have about rivals? how often do firms interact?
The difficu
Repeated Games
In many real-world settings, players play the same game over and over again
the simple constituent game that is played repeatedly is called the stage game
Repeated play opens up the possibility of cooperation in equilibrium
players can adop
Solutions for Final Exam
1. State the field axioms of the real number system: See the book or my notes. Several students made the minor error of forgetting that an element must be nonzero to have an inverse. 2. State the Completeness Axiom of the real num
M413 Test 1 Solutions 1. Use induction to prove that 5 + 11 + 17 + + (6n - 1) = 3n2 + 2n for all n N. First, this is true for n = 1: 6(1) - 1 = 3(12 ) + 2(1). Now assume the statement is true for n. So we assume that 5 + 11 + 17 + + (6n - 1) = 3n2 + 2n. O
Solutions to 131A Midterm 1, Spring 09. Note: Proofs must be rigorous. Drawings will not be credited, though allowed. Q1. Let A and B be two subsets of R, both bounded below. Let S = A + B, the set of elements of the form a + b with a A and b B. a). Show
1. Show that n!> 2 n for all n 4 by induction. 2. Show that n!> 2 n +1 for all n > 5 by induction. n!> k 2 n 3. Use induction to show that n straight lines in the plane, with no two parallel and no 2 three going through a common point, divide the plane in