Solutions to Assignment 10
Stat 155: Game Theory
Question 1
a) When the mixed strategies of PI and PII are x and y respectively and the
payo is A,then the expected payo to PI is formally written as
i1
aij xi yj =
i
j
yj
xi (
i
j=1
yj ) =:
j=i+1
ci (y)xi

Solutions to Assignment 3
Stat 155: Game Theory
Question 1
a)
12 = (01100)2
21 = (10101)2
N imsum = (11001)2 = 25
15 = (1111)2
10 = (1010)2
5 = (0101)2
N imsum = (0000)2 = 0
Question 2
This game is a game of Staircase Nim under disguise. To see this, cons

Solutions to Assignment 1
Stat 155: Game Theory
September 15, 2013
Question 1
Here we are trying to enumerate the P-positions and N-positions of a subtraction
game with subtraction set cfw_1, 2, 4, 5. As we are playing under normal winning
rule, 0 is a P-

Solutions to Assignment 6
Stat 155: Game Theory
Question 1
We will write ei for a vector with 1 at the ith position and 0 everywhere else. p
will be a mixed strategy for Player I and q will be a mixed strategy for Player
II. Then, if you choose p = ei and

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 bottom-left to top-right as a Nim pile. The size of the pile depends on the
number of moves the bishop in that

Solutions to Assignment 2
Stat 155: Game Theory
Question 1
This is a simple restatement of the subtraction game we have discussed in Assignment #1. The answer is also exactly the same.
Question 2
The game of Chomp can also be solved using backward inducti

UNIVERSITY OF CALIFORNIA, BERKELEY
DEPARTMENT OF STATISTICS
STAT-155: Game Theory
Fall 2013
Instructor: Antar Bandyopadhyay
GSI: Sujayam Saha
Practice Midterm
Date Given: October 16, 2013
Total Points: 30
Total Time: 50 minutes
1. Consider the following C

Solutions to Assignment 8
Stat 155: Game Theory
Question 1
Firstly, observe that Column 3 dominates Column 2 for Player II. The new
payo matrix is
8 0 5
0 4 1
Then, assigning probabilities p and 1 p to rows 1 and 2 for Player I,
we have that the expected

Solutions to Assignment 12
Stat 155: Game Theory
Question 1
First we try to simplify the denition of symmetric game for two-player games.
The only permutations of cfw_1, 2 are cfw_1, 2 and cfw_2, 1. Then, the condition given
in the question simplies to U2

Solutions to Assignment 5
Stat 155: Game Theory
Question 1
For this problem, we will use induction. 0 is a terminal position and we have
h(0) = 0. From 1, the only possible move is to 0, implies h(1) = 1 = g(0) + 1.
So the base step holds.
For any other s

Solutions to Assignment 11
Stat 155: Game Theory
Question 1
This follow from the denition of safety values and Nash Equilibrium. Indeed,
xT Ay min xT Ay for all x
y
T
T
taking max on both sides, max x Ay max min xT Ay
x
by denition of NE, x
x
y
Ay vI by d

Solutions to Assignment 9
Stat 155: Game Theory
Question 1
Using translation invariance of optimal strategies, we solve the following modied payo matrix made by adding 3 to each payo. Recall that the optimal
strategies of these modied game is same as that

UNIVERSITY OF CALIFORNIA, BERKELEY
DEPARTMENT OF STATISTICS
STAT-155: Game Theory
Fall 2013
Instructor: Antar Bandyopadhyay
GSI: Sujayam Saha
Assignment # 12
Date Given: December 02, 2013 (Monday)
Date Due: December 09, 2013 (Monday)
Total Points: 20
1. R

Solutions to Assignment 7
Stat 155: Game Theory
Question 1
This is very similar to the second question in the midterm paper. It is easy to
guess that the uniform mixed strategy is going to be a Nash equilibrium and
as such an optimal strategy. To prove th

Solutions to Assignment 4
Stat 155: Game Theory
Question 1
This game has a mind-boggling number of symmetries which can be exploited to
analyse the game. The simplest one is detailed here. Observe that, at each step
a move is to select a rook and then mov