Stat 155 Fall 2009: Homework 5
Due October 8, 2009
• Please show all your steps. No credit will be given for just giving the
answer, without any supporting work.
• Grading: 3 points for a complete sol
Homework 1
Stat 155, summer 2012
Due in section Thursday 28th June or before
Please give explanations or show working for all answers. You may, and should, discuss problems
with classmates and others,
Homework 2
Stat 155, summer 2012
Due in section Thursday 5th July or before
Please give explanations or show working for all answers. You may, and should, discuss problems
with classmates and others,
Homework 3
Stat 155, summer 2012
Due in section Thursday 12th July or before
Please give explanations or show working for all answers. You may, and should, discuss problems
with classmates and others,
Homework 4
Stat 155, summer 2012
Due in section Thursday 19th July or before
Please give explanations or show working for all answers. You may, and should, discuss problems
with classmates and others,
Homework 5
Stat 155, summer 2012
Due in section Thursday 26th July or before
Please give explanations or show working for all answers. You may, and should, discuss problems
with classmates and others,
Homework 6
Stat 155, summer 2012
Due in section Thursday 2nd August or before
Please give explanations or show working for all answers. You may, and should, discuss problems
with classmates and others
Homework 7
Stat 155, summer 2012
Due in class Wednesday 8th August or before
Please give explanations or show working for all answers. You may, and should, discuss problems
with classmates and others,
Some miscellaneous questions
Stat 135 summer 2012
1. In this zero-sum game, Player I ips a fair coin and looks at the result, while hiding the result
from Player II. This is followed by Player I choos
Practice problems: General-sum games
Stat 155, summer 2012
1. The following are payo matrices for general-sum games.
(a) Another prisoners dilemma:
Silent
Confess
Silent (1, 2)
(2, 0)
Confess (0, 4) (
Practice problems: Sprague-Grundy
Stat 155, summer 2012
For the following positions in the following impartial games, nd the Sprague-Grundy function
and state whether the position is N or P.
1. A pile
HW 5 Solutions
1) First, note the following typo in the text (page 103):
General fact: In a symmetric game, if bii > bi,j for all j = i, then pure strategy i is an evolutionarily
stable strategy. This
Stat 155 Fall 2009: Solutions to Homework 3
(was due October 1, 2009)
1. The payoﬀ matrix is shown below, with player I’s choices along the rows,
and player II’s choices along the columns.
Black
Red
B
Stat 155 Fall 2009: Solutions to Homework 3
(was due September 24, 2009)
1. This is a sum of two subtraction games. Using the notation from class, we
can call the subtraction sets S4 and S5 . Then we
Stat 155 Fall 2009: Homework 3
Due September 24, 2009
• Please show all your steps. No credit will be given for just giving the
answer, without any supporting work.
• Grading: 3 points for a complete
Stat 155 Fall 2009: Solutions to Homework 2
(was due September 17, 2009)
1. The Sprague-Grundy function of the 2 × 3 rectangular piece of chocolate
for the game of Chomp is enumerated below. It was ob
Stat 155 Fall 2009: Homework 2
Due September 17, 2009
• Please show all your steps. No credit will be given for just giving the
answer, without any supporting work.
• Grading: 3 points for a complete
Stat 155 Fall 2009: Solutions to Homework 1
(was due September 10, 2009)
1. Chomp the square at (3,1), gobbling 9 pieces:
∈P
z
z
2. Using the binary expressions of the heap sizes,
9 = 1001, 10 = 1010,
HW 1 Solutions
1) Observe the following table:
Number
42
x
14
32
1
?
0
16
0
?
0
8
1
?
1
4
0
?
1
2
1
?
1
1
0
?
0
To make the Nim-sum of 42 and x equal to 14, we need x to have a one in the 32 column, a
HW 1 Solutions
1) First, consider the conguration
P
P.
(1)
When a player places any letter in a blank between the P s, the next player will be able to spell P OP .
P congurations within the board, the
HW 3 Solutions
1) Writing down the possible plays:
Player 2
1
2
3
1 3
2
0
2 3
2
Player 1 2
3
0
2 3
Thus, the payo matrix A is
3
A= 2
0
2
0
3
2
2 3
We rst try to nd an optimal strategy for player 1 by
HW 4 Solutions
1) Without the rule change, the payo matrix is as follows:
Player 1
CO
IW
Player 2
CO
IW
(1, 1)
(1, 2)
(2, 1) (a, a)
We can safely assume that a > 1.
When players play (CO, CO), there i
Practice problems: Zero-sum games
Stat 155, summer 2012
For the following positions in the following impartial games, nd the Sprague-Grundy function
and state whether the position is N or P.
1. The fo