Game Theory
Problem Set 9
Levent Kokesen
1. Consider the following prisoners dilemma game.
C
D
C
2; 2
3; 0
D
0; 3
1; 1
For what values of ; if any, the following strategies constitute subgame perfect equilibria?
(a) Tit-For-Tat: Choose C in period 1 and t
Columbia University Department of Economics
W4415 Game Theory
Midterm Solutions
Prajit Dutta
October 16th, 2013, in class.
Notes:
This exam is worth 100 points, and will last 75 minutes. Answer all the questions. You
are advised to divide your time among
Economics 379
Game Theory
Department of Economics
Colby College
Problem Set 1
Due in class Wednesday, February 23
From Dutta
CH 3: Exercises 1, 2, 8 - 10, 11 - 14
CH 5: Exercises 5.7 - 5.13
Additional Exercises
1. Determine the set of strategies that surv
Columbia University Department of Economics
W4415 Game Theory
Problem Set 4
Prajit Dutta
Due Wednesday, October 7th, 2014 in class
1
Mixed Strategies
Consider the game represented in the following payo matrix.
1\2
H
D
H
0, 0
2, 8
D
8,2
4, 4
(a) List all p
Game Theory
Problem Set 6
Levent Kokesen
1. Two people are involved in a dispute. Person 1 does not know whether person 2 is strong
or weak; she assigns probability to person 2 being strong. Person 2 is fully informed.
Each person can either ght or yield.
Economics 379
Game Theory
Department of Economics
Colby College
Problem Set 5
Due Wednesday, April 14, 2011
From Dutta
Chapter 15: Exercises 11, 12, 15, 17, 20 23 (Note: For question 22, use a discount rate of 0.9)
Additional Exercises
1. Find conditions
Economics 379
Game Theory
Department of Economics
Colby College
Problem Set 2
Solutions
Chapter 6
6.8
Firm 1s profit maximization problem is:
max( a bq1 dq2 ) q1 cq1
q1
FOC : a 2bq1 dq2 = c q1 =
a c dq2
2b
ac
a c dq2
is nonnegative so long as q2
. Thus f
Game Theory
Problem Set 5
Levent Kokesen
1. Find all the pure and mixed strategy equilibria of the following games by constructing
the best response correspondences of the players:
(a) Matching Pennies:
H
T
1; 1 1; 1
1; 1 1; 1
H
T
(b) Hawk-Dove:
H
D
H
D
0
Game Theory
Problem Set 5 Solutions
Levent Kokesen
1. Find all the pure and mixed strategy equilibria of the following games by constructing
the best response correspondences of the players:
(a) Matching Pennies:
H
T
1; 1 1; 1
1; 1 1; 1
H
T
Let 1 (H ) = p
Game Theory (S4415): Answers to Problem Set 4
Prajit K. Dutta
June 22, 2001
Chapter 13
1.4 Write down the extensive form of the game above. How many subgames are there in this game? How many strategies does Coke have?
Answer:
Coke
Pepsi
-2,-1
A
T
-3,1
A
E
Columbia University Department of Economics
W4415 Game Theory
Problem Set 2
Prajit Dutta
Due September 23th, 2015 in class
1
Dominant and Dominated Strategies
Consider the game represented in the following payo matrix.
T
M
B
L
8, 12
6, 14
16, 10
C
14, 2
2
Game Theory
Problem Set 1
Levent Kokesen
1. Consider the following game of divide the dollar. There is a dollar to be split between
two players. Player 1 makes an oer (an oer by Player 1 species how much he would
like Player 2 to have). Without observing
Game Theory
Problem Set 4
Levent Kokesen
1. Find all the Nash equilibria of a rst-price sealed bid auction with two bidders by
constructing the players best response correspondences.
2. Find all the Nash equilibria of a second-price sealed bid auction wit
Game Theory
Problem Set 7
Levent Kokesen
1. (Centipede Game) Find the backward induction equilibrium of the game in the following gure.
1 C 2 C 1 C 2 C 1 C 2 C r 6; 5
b
r
r
r
r
r
S
S
S
S
S
S
r
r
r
r
r
r
1; 0 0; 2 3; 1 2; 4 5; 3 4; 6
Figure 1: A six-period
Game Theory
Problem Set 8
Levent Kokesen
1. (Burning the Bridge) Army 1, of country 1, must decide whether to attack Army
2, of country 2, which is occupying an island between the two countries. In the event
of an attack, army 2 may ght or retreat over a
Columbia University Department of Economics
S4415Q Game Theory, Problem Set 3
Due Monday August 1st in class.
Problem 1
Suppose two players play an infinitely repeated version of the stage game represented in the
following payoff matrix:
1\2
A
B
A
2, 2
0,
Perfect Bayesian Equilibrium
Game Theory, Summer 2016
1
Sequential Games
In 4.1 we looked a simultaneous game with incomplete
information. The players played simultaneously, but one (or
more) had private information that made the other uncertain
which pay
Koc University
Department of Economics
ECON/MGEC 333
Game Theory And Strategy
Midterm Examination II Solutions
Levent Kokesen
c
January 7, 2010
1. (30pts.) Find the set of pure strategy Nash equilibria and subgame perfect equilibria of the following
game:
Game Theory
Problem Set 3
Levent Kokesen
1. Consider the Cournot duopoly game introduced in the class (see page 6 of Lecture Notes
1) with linear demands and constant unit costs with the exception that each rm has a
dierent unit cost, i.e. the cost functi
Koc University
Department of Economics
Name:
ECON/MGEC 333
Game Theory And Strategy
Midterm Examination II
Levent Kokesen
c
January 6, 2011
Instructions
Please write your name in the space provided at the top.
Answer all questions.
Write your answers i
Koc University
Department of Economics
ECON/MGEC 333
Game Theory And Strategy
Midterm Examination II Solutions
Levent Kokesen
c
January 6, 2011
1. (30pts.) Consider the following extensive form game:
1
C
c
2
2, 1
s
S
a, 4
1, 0
(a) (10pts.) Assume that a =
Koc University
Department of Economics
Name:
ECON/MGEC 333
Game Theory And Strategy
Midterm Examination II
Levent Kokesen
c
December 28, 2011
Instructions
Please write your name in the space provided at the top.
Answer all questions.
Write your answers
Koc University
Department of Economics
ECON/MGEC 333
Game Theory And Strategy
Midterm Examination II Solutions
Levent Kokesen
c
December 28, 2011
1. (30pts.) Consider the following extensive form game:
1
L
R
2
x, 4
u
d
1
2, 1
l
0, 0
r
1, 0
(a) (10pts.) As
Koc University
Department of Economics
ECON/MGEC 333
Game Theory And Strategy
Problem Set 9 Solutions
Levent Kokesen
c
January 6, 2011
1. (a) Tit-For-Tat: The behavior of a player who adopts this strategy depends only on the last periods
outcome. Therefor
Game Theory - W4415
Final Examination
Total time allowed is 120 minutes. Please show all your work. Good luck!
Levent Kokesen
December 20, 2000
1. Consider the following extensive form game. Player 1 either goes hunting with player 2
in which case they en
Game Theory - W4415
Final Examination
Total time allowed is 150 minutes. Please show all your work. Good luck!
Levent Kokesen
December 17, 2001
1. Consider the following extensive form game. Player 1 either decides to go to a concert
with player 2 in whic
Game Theory - W4415
Final Examination
Total time allowed is 150 minutes. Please show all your work. Good luck!
Levent Kokesen
December 16, 2000
1. (30 pts.) Consider the following extensive form game.
1 C 2 C 1 C r 2; 4
b
r
r
S
r
S
r
S
r
1; 1 0; 2 3; 1
(a
Game Theory - W4415
Final Examination
Total time allowed is 180 minutes. Please show all your work. Good luck!
Levent Kokesen
December 20, 1999
1. (25 pts) Consider the following extensive form game. Player 1 either goes hunting with
player 2 in which cas
Game Theory - W4415
Solutions to the Final Examination
Levent Kokesen
December 20, 2000
1. (a) The strategy sets are S1 = fLS; LH; RS; RH g ; S2 = fS; H g
LS
LH
RS
RH
S
3; 3
3; 3
4; 4
3; 0
H
3; 3
3; 3
0; 3
2; 2
The set of pure strategy Nash equilibria is
Game Theory - W4415
Solutions to Final Examination
December 17, 2001
Levent Kokesen
1. (a) The strategy sets are S1 = fHB; HS; CB; CS g ; S2 = fB; S g
B
2; 2
2; 2
3; 1
0; 0
HB
HS
CB
CS
S
2; 2
2; 2
0; 0
1; 3
The set of pure strategy Nash equilibria is f(HB
Columbia University Department of Economics
W4415 Game Theory
Midterm Solutions
Prajit Dutta
Fall 2015
Problem 1 (25 Points)
For each of the following claims, first state whether it is true or false. If it is true, provide a
short proof. If it is false, p
2(ii)
True. I will prove that a mixed strategy cannot be dominant, and thus cannot be part of a DSS,
and therefore a game can only have a DSS in pure strategies.
To see this, assume that mixed strategy i is dominant for player i. Let Si Si be the set
of p
W4415 Game Theory
2008 Midterm Solutions
Question #1
a) si is a (weakly) dominated strategy of player i if there exists an0
other strategy si that gives player i a payoff at least as high as strategy si
no matter what the other players do, and a strictly