Stat 155 Homework # 6 Solutions
Problems:
Q 1 Ferguson Chapter III Section 2.5 Q 5
By marking the optimal value in each column for player one and the optimal value in each
row for player two we can nd the pure strategies.
(a) The only pure Nash equilibriu
Solutions to Homework 1
Stat 155: Game Theory
Soumendu Sundar Mukherjee
Note: Only solutions to the problems from PeresKarlin are given here. Solutions to the problems from Fergusons text are available online at http:
/www.math.ucla.edu/~tom/Game_Theory/
Game Theory, Alive
Anna R. Karlin and Yuval Peres
Draft January 20, 2013
Please send comments and corrections to
karlin@cs.washington.edu and peres@microsoft.com
i
We are grateful to Alan Hammond, Yun Long, Gbor Pete, and Peter
a
Ralph for scribing early
Statistics 155 Homework Assignment 4 (due Tuesday, October 11, 2016)
1. (Low Regret Learning algorithm)
Consider a zerosum game with the following payoff matrix;
12 7 4 3 2 1
1 2 3 4 6 11
(a) Find optimal strategies x and y for the first player and the s
Solutions to Midtermguide Problems
Stat 155: Game Theory
Soumendu Sundar Mukherjee
Note: If you have comments or questions about the solutions, feel free to email
me at soumendu@berkeley.edu.
Problem 1
This is of course a progressively bounded impartial g
Solutions to Homework 4
Stat 155: Game Theory
Soumendu Sundar Mukherjee
Note: If you have comments or questions about the solutions, feel free to email
me at soumendu@berkeley.edu.
Problem 3.3 from KP
The payo matrix is given by
W
D
W
(8, 8)
(10, 3)
D
(3,
Solutions to Homework 3
Stat 155: Game Theory
Soumendu Sundar Mukherjee
Note: If you have comments or questions about the solutions, feel free to email
me at soumendu@berkeley.edu.
Problem 1
The payo matrix is given by
Rock
Scissor
Paper
Rock
0
1
1
Sciss
Solutions to Homework 2
Stat 155: Game Theory
Soumendu Sundar Mukherjee
Note: If you have comments or questions about the solutions, feel free to email
me at soumendu@berkeley.edu.
Problem 9.2
That there always is a winning strategy for one of the players
Stat 155 Fall 2014
Applications of Game theory to Biology: Evolutionarily Stable
Strategies
The following example is taken from Philip Strans book Game Theory
and Strategy.
Hawks and Doves: (Introduced by John Maynard Smith and G.R. Price
in 1973)
Individ
Stat 155 Fall 2014
S. Stoyanov
Homework 1, due Friday, September 19
Note that homework turned in after 10:20 will be marked late, and 10%
of the possible points will be deducted. (The assignment is 36 points, so
3.5 points will be deducted.)
Please make
Solutions to Midterm Problems
Stat 155: Game Theory
Soumendu Sundar Mukherjee
Note: If you have comments or questions about the solutions, feel free to email
me at soumendu@berkeley.edu.
Problem 1
Consider the takeaway (subtraction) game with subtraction
Solutions to Homework 6
Stat 155: Game Theory
Soumendu Sundar Mukherjee
Note: If you have comments or questions about the solutions, feel free to email
me at soumendu@berkeley.edu.
Problem 1
Recall the example in class, where we found the BayesNash equil
Social Choice
Stat 155, Fall 2014
Main ques8on
How do we aggregate the preferences of individuals
in a society?
Two op8ons: majority rule
If more than two op8ons, might have inconsistent
results, assuming voters
STAT 135, Problem Set 1, due 2/2/15
January 28, 2015
Problem 1 [20%]
Let X1 , X2 be iid continuous random variables, with cumulative probability
distribution F (x) = P (Xi x) .
1. Show that P (X2 > X1 ) = E (F (X1 ).
2. Show that E (F (X1 ) =
this?
1
2
.
STAT 135, Problem Set 2, due 2/9/15
February 4, 2015
Problem 1 [33%]
Suppose X1 , X2 are independent samples from a Bernoulli distribution B(p),
and let T = X1 + X2 .
1. Show that T Bin(2, p).
2. If = 0 + 1 p + 2 p2 , show that
0
1
= 2 (1 + 20 )
0 + 1 +
STAT 135, Problem Set 3, due 2/18/15
February 9, 2015
The objective of this Problem Set is to study the Stein Phenomenon (1955).
Suppose that = (1 , 2 , . . . , n ) consists of n unknown parameters, with n 3.
We wish to estimate these parameters with n me
STAT 135, Problem Set 5, due 3/4/15
February 25, 2015
Problem 1 [50%]
Let X1 , . . . , Xn be iid Poisson random variables with rate . We want to test
the hypothesis H0 : = 1 against H1 : = 1.21. Let denote the TypeI
error probability and denote the Type
STAT 135, Problem Set 6, due 3/11/15
March 4, 2015
Problem 1 [33%]
Suppose that under H0 a measurement X N (0, 2 ) and that under H1 , X N (1, 2 ), and that
the prior probabilities of H0 and H1 are equal: P (H0 ) = P (H1 ).
For = 1 and x [0, 3], plot and
Stat 155 Fall 2014
Auctions (Chapter 15)
Shobhana Stoyanov
Dept of Statistics
November 21, 2014
Shobhana Stoyanov
Stat 155 Fall 2014 Auctions (Chapter 15)
1
Auctions
Mechanism for buying and selling goods
Use auction if not sure of what price to x for ite
Stat 155: fuller solution to III5 # 5
Michael Lugo
November 24, 2010
1
III5 # 5
(a) Firm 2, knowing Firm 1s production, will produce q2 in order to maximize q2 (a q1
q2 )+ c2 q2 . This gives q2 (q1 ) = (a q1 c2 )/2)+ .
Then firm 1 will produce q1 [0, a
Solutions to Practice Midterm
Stat 155: Game Theory
Question 1
a) This game is same as a game of Nim if you think of each diagonal leading
from bottomleft to topright as a Nim pile. The size of the pile depends on the
number of moves the bishop in that
Solutions to Practice Midterm
Stat 155: Game Theory
Questions
1. Consider a modified game of Nim with 3 piles containing x1 , x2 and x3
chips. In a turn a player is allowed to move at most 3 chips from pile one,
or at most 4 chips from pile two or at most
Stat 155 Midterm Practice Solutions
Problems:
Attempt all questions and show your working  solutions without explanation will not receive
full credit. One double sided sheets of notes are permitted.
Q 1 Find the value and optimal strategy of the followin
Stat 155 Midterm Practice Solutions
Problems:
Attempt all questions and show your working  solutions without explanation will not receive
full credit. One double sided sheets of notes are permitted.
Q 1 Find the value and optimal strategy of the followin
Practice Final
Stat 155: Game Theory
TrueFalse
(2030 mins)
For each of these statements state whether they are true of false. With
reason.
a) If a m n matrix has 2 saddle points, they have the same value.
b) For any matrix Amn = (aij ), maxi minj aij =
Statistics 155 Homework Assignment 2 (due Thursday, September 15, 2016)
1. (Upanddown Rooks) Karlin and Peres, Exercise 1.7, p33.
Solution:
As the picture suggests, this game is a variant of Nim where initially there are eight piles, and each
pile conta
Stat 155: solutions to midterm exam
Michael Lugo
October 21, 2010
1. We have a board consisting of infinitely many squares labeled 0, 1, 2, 3, . . . from left
to right. Finitely many counters are placed on these squares. A game is played, in which a
move
Stat 155 HW 1 Solution
Fall 2016
1
Subtraction game
Consider a subtraction game in which a player can remove from 0 to 4 chips, but removing 0 chips is only
allowed if the previous player removed at least 1 chip. We can model this game as the following ch
Midterm Solution for Game Theory
Soren Reinhold
SPRING
2015Kunzel
March 30, 2015
1. Consider the takeaway (subtraction) game with subtraction set S = cfw_1, 3, 9. That is, this is a
two player game, with n piles of xn chips each. At any players turn, the
ZeroSum Game

(ZeroSum Game) Symmetry in twoplayer zerosum game

skew symmetric
We notice that the remaining matrix is skew symmetric. Thus the value of the game is 0 and for the reduced game an optimal
strategy for one of the players is optimal for
Stat 155 Midterm 1 Review
1
Chomp
Theorem: Every nonterminal rectangle is in N
General Questions:
 Who has a winning strategy? Is either player able to control the game? If so, which
player can assure he/she wins?
o The game is finite. Someone ha
Stat 155 Midterm Spring 2014
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Name:
SID:
This exam has 5 problems and a total of 75 points. Attempt all questions and show your
working  solutions without explanation will not receive full cr
Stat 155 Midterm 1 Review
1
Chomp
Theorem: Every nonterminal rectangle is in N
General Questions:
 Who has a winning strategy? Is either player able to control the game? If so, which
player can assure he/she wins?
o The game is finite. Someone has to wi