2
1. Find the N ash bargaining model NTU solution to the bimatrix game
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with threat point (3 1).
3:31;) LL1-3>( WWW/Wu)
_ jaw/Ww'W/z
I ' ' , 5m +14 : O U 3 H75
f 3
2. Given the bimatrix game below, nd all real numbers 3 and t such that the ga
167 Midterm 2 Solutions1
1. Question 1
Prisoner I
Recall the prisoners dilemma, which has the following payoffs.
Prisoner II
silent
confess
silent (1, 1) (10, 0)
confess (0, 10) (8, 8)
Find all Nash equilibria for this game.
Solution. See Example 4.6 in t
Game Theory
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due April 5, in the discussion section.
Homework 1
Exercise 1. Let n be a positive integer. Consider the game Chomp played on an n n
board. Explicitl
Game Theory
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due February 2nd, in the discussion section.
Homework 3
Exercise 1. This exercise deals with subsets of the real line. Show that [0, 1] is closed, bu
Game Theory
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due February 9th, in the discussion section.
Homework 4
Exercise 1. Find the value of the two-person zero-sum game described by the payoff matrix
0 9
Game Theory
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due February 16th, in the discussion section.
Homework 5
Exercise 1. Suppose we have a two-person zero-sum game with (n + 1) (n + 1) payoff
matrix A
Game Theory
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due February 23rd, in the discussion section.
Homework 6
Player I
Exercise 1. Recall the Game of Chicken is defined as follows. Each player chooses t
Game Theory
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due March 1st, in the discussion section.
Homework 7
Exercise 1. Let n be a positive integer. Let v : 2cfw_1,.,n cfw_0, 1 be a characteristic functio
Game Theory
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due March 8th, in the discussion section.
Homework 8
Exercise 1. Prove the following Lemma from the notes: The set of functions cfw_WS Scfw_1,.,n
is
Math 167, Winter 2016, UCLA
Name:
Instructor: Steven Heilman
UCLA ID:
Date:
Signature:
.
(By signing here, I certify that I have taken this test while refraining from cheating.)
Mid-Term 1
This exam contains 7 pages (including this cover page) and 5 probl
167 Final Solutions1
1. Question 1
(a) Every two-player general sum game has a Nash equilibrium such that this Nash equilibrium is evolutionarily stable.
False. Rock paper scissors has only one nash equilibrium, and it is not evolutionarily
stable. (We sh
Final Examination
Mathematics 167, Game Theory
Ferguson
Tues. June 14, 2005
1. (a) Consider a game of nim with 3 piles of sizes 9, 17 and 21. Is this a P-position
or an N-position? If an N-position, what is a winning first move?
(b) Consider the take or b
167 Final Solutions1
1. Question 1
(a) There is a two-person zero sum game with two Nash equilibria and one optimal strategy.
FALSE. In zero sum games, Nash equilibria are equivalent to optimal strategies, i.e. the
number of each must be the same. (We sho
Solutions to Exercises from Homework 4, Math 432 Spring 2012
4.7#1 Consider the game with matrix
0
A = 1
9
7
4
3
2
8
1
4
2 .
6
(a) A Bayes strategy against (1/5, 1/5, 1/5, 2/5) is the row which gives the best payoff for Player I.
Row 1 gives 7/5 + 2/5 + 8
Solutions to Homework 2 Exercises from Part I of Fergusons Game Theory
4.5#4 Kayles problem of Dudeney and Loyd
Of 13 pins in a row the second has been knocked down.
(a) In order to show that this is an N-position, we have to prove that the SG value at th
Solutions to Exercises from HW5, Math 432, Spring 2012
Section 1.6
1.6#2 Find the safety levels and maxmin strategies for the players in the bimatrix games
(a)
A=
B =
(5, 0)
(4, 4)
5
, saddle point: VI = 1, p = (1, 0).
4
1
0
T
(1, 1)
(0, 5)
5
, saddle poi
Solutions to Exercises from Part I of Fergusons Game Theory
Section 1.5
1.5#1 Mis`ere version of the take-away game. There are 21 chips, we can remove 1, 2, or 3.
Last player to move loses, hence position 1 is a P-position, from positions 2,3, and 4 we ca
Solutions to Homework 3 Exercises from Part I I of Fergusons Game Theory
Section 1.5
1.5#2 Player I holds a black Ace and a red 8. Player II holds a red 2 and a black 7. If the chosen cards
are of the same color Player I wins, if they differ Player II doe
Math 167, Winter 2016, UCLA
Name:
Instructor: Steven Heilman
UCLA ID:
Date:
Signature:
.
(By signing here, I certify that I have taken this test while refraining from cheating.)
Final Exam
This exam contains 16 pages (including this cover page) and 11 pro
167 Spring Midterm 2 Solutions1
1. Question 1
Prisoner I
Recall the prisoners dilemma, which has the following payoffs.
Prisoner II
silent
confess
silent (1, 1) (10, 0)
confess (0, 10) (8, 8)
Find all Correlated equilibria for this game.
Solution. Suppose
167 Midterm 1 Spring Quarter Solutions1
1. Question 1
TRUE/FALSE
(a) In the game of chess, it is known that both players can force at least a draw.
FALSE. It could be the case that the white player (or the black player) has a winning
strategy. We just don
Math 167
Homework 1 Solutions
1
1. Use induction to prove that for any n N,
n
X
1
i2 = n(n + 1)(2n + 1).
6
i=1
First check the base case n = 1:
n
X
i=1
1
i2 = 1 = n(n + 1)(2n + 1).
6
Now assume inductively that for some given n N,
n
X
1
i2 = n(n + 1)(2n +
167 Midterm 1 Solutions1
1. Question 1
(i) Give an example of a game mentioned in class, or the notes, or any course textbook,
such that: the first player has a winning strategy. (You only need to mention the game; you
do not need to prove anything.)
Solu