Chapter 3
Representation of Games
We are now ready to formally introduce games and some fundamental concepts, such as
a strategy. In order to analyze a strategic situations, one needs to know
who the players are,
which actions are available to them,
Lecture 3
Representation of Games
14.12 Game Theory
Muhamet Yildiz
1
Game: Ingredients
Who are the players (decision makers)?
What moves are available to each player
and when?
What does each player know at the time of
each of his decisions?
What are t
Lecture 12
Finitely Repeated Games
14.12 Game Theory
Muhamet Yildiz
1
Road Map
1.
2.
3.
4.
Entry-Deterrence/Chain-store paradox
Finitely repeated Prisoners Dilemma
A general result
Repeated games with multiple equilibria
2
Prisoners' Dilemma, repeated twi
Lecture 13
Infinitely Repeated Games
14.12 Game Theory
Muhamet Yildiz
1
Road Map
1. Definitions
2. Single-deviation principle
3. Examples
2
Infinitely repeated Games with
observable actions
T= cfw_O,1,2, . .,t, .
G = "stage game" = a finite game
At ea
Lecture 8
Backward Induction
14.12 Game Theory
Muhamet Yildiz
1
Road Map
1.
2.
3.
4.
Backward Induction
Examples
Application: Stackelberg Duopoly
[Next Application: Negotiation]
2
Definitions
Perfect-Information game is a game in which all
the information
14.12 Economic Applications of
Game Theory
Professor: Muhamet Yildiz
1
Swansea
Merthyr
Banbury Luton
Oxford
Tydfil
Swindon
London
Prime Meridian
WALES
Chelmsford
Gillingham
Cardiff
Bristol
Newbury
Barnstaple
Maidstone
UNITED KINGDOM
Tiverton
Hastings 2o
Chapter 4
Dominance
The previous lectures focused on how to formally describe a strategic situation. We now
start analyzing strategic situations in order to nd which outcomes are more reasonable
and likely to realize. In order to do that, we consider cert
14.12 Game Theory Final (Answers)
12/21/2007
Prof. Muhamet Yildiz
Instructions. This is an open book exam; you can use any written material. You have two
hour and 50 minutes. Each question is 25 points. Good luck!
1. There are two siblings, who have inher
14.12 Game Theory - Midterm I
10/13/2011
Prof. Muhamet Yildiz
Instructions. This is a closed book exam. You have 90 minutes. You need to show your
work when it is needed. All questions have equal weights. You may be able to receive partial
credit for stat
14.12 Game Theory Midterm I
10/19/2010
Prof. Muhamet Yildiz
Instructions. This is an open book exam; you can use any written material. You have one
hour and 20 minutes. Each question is 25 points. Good luck!
1. Consider the following game.
(a) Using backw
14.12 Midterm Exam #1
October 8, 2009
Answer all questions. You have 85 minutes in which to complete the exam. Please
show your calculations and provide rough explanations where you cant give formal state
ments so I can give you partial credit.
1. (15 Min
Answers to Selected Problems from Past Final Exams
Maksim Pinkovskiy
December 17, 2011
1
Problem 1, Final 2004
Denote the type with probability 0.4 as strong, and the type with probability 0.6 as weak.
It is clear that if player 1 is strong, he will play
14.12 Game Theory
MIDTERM 1 SOLUTIONS
10/16/2008
Prof. Casey Rothschild
Instructions. This is an open book exam; you can use any written material. You may use a
calculator. You may not use a computer or any electronic device with wireless communication
ca
Chapter 5
Rationalizability
A player is said to be rational if he maximizes expected value of his utility function, as
described in the game. The previous lecture explored the implications of rationality. This
was captured by dominance. In natural strateg
Lectures 16
Incomplete Information
Static Case
14.12 Game Theory
Muhamet Yildiz
1
Road Map
1.
2.
3.
4.
5.
Example
Bayesian Games
Bayesian Nash Equilibrium
More Examples
Bayes' Rule
2
Incomplete information
one player knows something (relevant)
that some o
Lecture 9
Negotiation
14.12 Game Theory
Muhamet Yildiz
1
Examples of Bargaining
Buying a car, house, or shopping at a bazaar
Wage Negotiations
International Agreements
Legislative Bargaining
Litigation
2
Road Map
1. Congressional Bargaining & Agenda
Sett
Lecture 10
Subgame-perfect Equilibrium
14.12 Game Theory
Muhamet Yildiz
1
Road Map
1. Subgame-perfect Equilibrium
1.
2.
3.
4.
Motivation
What is a subgame?
Definition
Example
2. Applications
1. BankRun
2. Infinite-horizon Bargaining
2
A game
1
l~
(2,6)
T
Lecture 14
Infinitely Repeated Games II
14.12 Game Theory
Muhamet Yildiz
1
Road Map
1. Folk Theorem
2. Applications (Problems)
2
Folk Theorem
Definition: v = (v 1 ,v2 , . . ,vn) is feasible iff v is a convex
combination of pure-strategy payoff-vectors:
v
Lecture 5
Rationalizability
14.12 Game Theory
Muhamet Yildiz
1
vM = 0
V T = 2p-(I-p) = 3p-1
Recall: A Game
VB = -p+2 (I-p) = 2-3p
V
L
R
T
(2,0)
(-1,1)
M
(0,10) (0,0)
B
(-1,-6)
p
(2,0)
2
o ~-~-~-
-I
L -_~
o
I-p
p
1
2
Recap: Rationality & Dominance
Belief:
Chapter 12
Repeated Games
In real life, most games are played within a larger context, and actions in a given situation
aect not only the present situation but also the future situations that may arise. When
a player acts in a given situation, he takes in
Chapter 7
Application: Imperfect Competition
Some of the earliest applications of game theory is the analyses of imperfect competition
by Cournot (1838) and Bertrand (1883), a century before Nash (1950). This chapter
applies the solution concepts of ratio
Chapter 8
Further Applications
This chapter is devoted to exercises that apply the ideas developed in previous chapters
to various real-world problems.All of the exercises come from past exams and homework
problems. The reader is recommended to solve them
Chapter 6
Nash Equilibrium
6.1
Introduction and Denition
Both dominant-strategy equilibrium and rationalizability are well-founded solution concepts. If players are rational and they are cautious in the sense that they assign positive
probability to each
14.12 Game Theory Midterm II
11/15/2007
Prof. Muhamet Yildiz
Instructions. This is an open book exam; you can use any written material. You have one hour
and 20 minutes. Each question is 25 points. Good luck!
1. Compute all the subgameperfect equilibria i