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 solution, 2 points for an almost correct
solution, 1 point
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, or consult books. However, you must write up your own
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, or consult books. However, you must write up your own w
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, or consult books. However, you must write up your own
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, or consult books. However, you must write up your own
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, or consult books. However, you must write up your own
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, or consult books. However, you must write up your own
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, or consult books. However, you must write up your own
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 choosing a hand, then Player II choosing a hand
(their choic
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) (3, 10)
Find all Nash equilibria.
(b) Football vs. socce
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 of size 6 in a subtraction game with subtraction set:
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 is clear, since if Player 1 plays i, then Player 2s be
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
Black
1
−7
Red
−2
8
Using equalizing strategies, as disc
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 know that if the two piles
have n and m chips, then g4
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 solution, 2 points for an almost correct
solution, 1 po
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 obtained using the
graph on page 11 of the text, starting
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 solution, 2 points for an almost correct
solution, 1 po
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, 11 = 1011, 12 = 1100
, and so the nim-sum of the heap
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 one in the
4 column, and a zero in all other columns.
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, there
Regardless of the size of the board and the number o
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 equating the components of
3
2
0
2 .
(p1 , p2 , 1 p1 p2
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 is a (1 p)2 probability that the result is (CO, CO), a p
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 following are payo matrices for two-player zero sum games