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
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
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
Columbia University Department of Economics
W4415 Game Theory
Midterm Solutions
Prajit Dutta
October 15th
Problem 1: Definitions (10 Points)
Provide formal definitions of the following concepts:
(a) Strategy si weakly dominates strategy s0i . [5p]
Strateg
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,
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 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 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
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
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 (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
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
W4415: Problem Set #2
Qingmin Liu
Columbia University
October 17, 2016
Abstract
Due in class on Monday, October 24, 2016. See the syllabus for problem set
policies.
1. Consider the following two-player game:
A
B
C
D
L
4; 2
1; 4
0; 0
0; 0
M
0; 0
1; 4
2; 4
Econ GU4415: Problem Set #1
Qingmin Liu
Columbia University
September 19, 2016
Abstract
Due in class on September 26, 2016. See the syllabus for problem set
policies.
1. Consider the following two-player game:
A
B
C
D
L
4; 2
1; 4
0; 0
0; 0
M
0; 0
1; 4
2;
Game Theory: Midterm Exam 1
Columbia University
Fall, 2016
Instructions
Please make sure to follow the instructions below. Failure to follow them can
result in getting a score of zero, so please follow them very carefully.
1. This is a closed book, closed
1/19/17
GU4415: Game Theory
Lecture 2
Normal Form Games, Dominance,
Iterated Dominance
Game Theory
Recall from last time:
Game Theory is multi-person decision theory
To understand how a (rational) player should act,
we need to understand what the struct
2/20/17
GU4415: Game Theory
Lecture 4
Weak Dominance, Mixed Strategies, Best
Responses, Rationalizability
Last Time
1. IDSDS & Applications
2. Weak Dominance
1
2/20/17
Weak Dominance Reminder
A strategy weakly dominates another if the former does at leas
GU4415: Game Theory
Lecture 5
Rationalizability,
Pure Strategy Nash Equilibrium
Last Time
1. Weak Dominance: Application, Pitfalls of Iteration
2. Mixed Strategies and (Iterated) Strict Dominance
3. (Never) Best Responses
4. Rationalizability
Rationalizab
Econ GU4415, Spring 2017
Problem Set #1
Due in class on Jan 31, 2017. Be extremely legible! Your answers should give
some justification, but be succinct. See the syllabus for other policies.
1. What is the definition of a strictly dominated strategy? What
Game Theory Midterm 1 Solutions
October 3, 2016
1. Three game theorists work on a team project. Assume they work independently, and it is not possible
for them to monitor each other. No one likes working, and the cost of effort is measured in dollars.
Eac
2/7/17
GU4415: Game Theory
Lecture 7
Market Games,
Mixed Strategy Nash Equilibrium
Last Time
1. Pure Strategy Nash Equilibrium & Examples
2. Introduced Cournot Oligopoly
1
2/7/17
Cournot Oligopoly
N1 firms in a market (Monopoly if N=1; Duopoly if N=2)
S
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
Game Theory
Problem Set 2
Levent Kokesen
1. Find the Nash equilibria of the games that were given in Problem Set 1 (There are ve
games).
2. (Altruistic players in the Prisoners Dilemma) Each of two players has two possible
actions, cooperate (C) and defec
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
Assorted Extras
Game Theory, Summer 2016
1
Strategies
Throughout the week weve used relatively simple strategies:
History-independent strategies.
This includes strategies which are time independent - i.e.
always play C in all periods and for all histori
Econ GU4415, Spring 2017
Problem Set #2 Solutions
1. Consider the 3 player game shown below, in which player 3s strategy set is cfw_M1 , M2 , M3 , M4 .
Each cell only lists one payoff, which represents all three players payoffs from that
strategy profile.
Econ GU4415, Spring 2017
Midterm Exam 1 Solutions
February 17, 2017
1. Provide formal definitions for the following, allowing for mixed strategies:
(a) (6 points.) A strictly dominated pure strategy.
Solution: A pure strategy si is strictly dominated if t
Econ GU4415, Spring 2017
Problem Set #3
Due in class on March 7, 2017. Be extremely legible! Your answers should give
some justification, but be succinct. See the syllabus for other policies.
1. Suppose Anne and Bob must simultaneously name a number from
Econ GU4415, Spring 2017
Problem Set #1 Solutions
1. What is the definition of a strictly dominated strategy? What is the definition of a
weakly dominated strategy? Give an example of a two-player game (in matrix form)
in which one player has three pure s