Economics 3M03, Spring 2016
Practice Problems 1: Answer key
1. Consider the following game between players 1 and 2.
2
M
5; 2
3; 4
6 ;5
L
U 3 ;3
1 I 2; 9
D 0; 5
R
4 ;3
4 ;5
1; 4
Find all Nash equilibria in the game by:
(a) cell-by-cell inspection;
(b) best
Introduction to Game Theory
Part 2: Examples of games
Maxim Ivanov
McMaster University
January 11, 2016
Maxim Ivanov (McMaster University)
Introduction to Game Theory
January 11, 2016
1/4
Prisoners dilemma
Consider a 2x2 game, i.e., a game between 2 playe
Introduction to Game Theory
Part 6: Applications of Nash equilibrium (Bertrand Oligopoly)
Maxim Ivanov
McMaster University
February 1, 2016
Maxim Ivanov (McMaster University)
Introduction to Game Theory
February 1, 2016
1/6
NE: Bertrand Oligopoly
2 rms, i
Introduction to Game Theory
Part 5: Dominance
Maxim Ivanov
McMaster University
January 18, 2016
Maxim Ivanov (McMaster University)
Introduction to Game Theory
January 18, 2016
1/8
Actions versus strategies
In normal-form games, the set of actions coincide
Introduction to Game Theory
Part 1: Introduction
Maxim Ivanov
McMaster University
January 11, 2016
Maxim Ivanov (McMaster University)
Introduction to Game Theory
January 11, 2016
1/9
What is game theory?
Game theory formalizes strategic interaction betwee
Introduction to Game Theory
Part 4: Best Response
Maxim Ivanov
McMaster University
January 17, 2016
Maxim Ivanov (McMaster University)
Introduction to Game Theory
January 17, 2016
1/6
Finding Nash equilibria: cell-by-cell inspection
There are several meth
Part I: Representing Games
(Chapter 1-5)
1
Introduction
(Chapter 1-3)
Game Theory: Definitions and overview
Representation: Extensive forms
Strategies
Representation: Normal forms
2
Its Your Move
3
What is a game?
A game is being played whenever peop
Nash Equilibrium
(Watson Chapters 9, 10, 11)
1
Pure Strategy Nash Equilibrium
A set of strategies forms a Nash equilibrium if the
strategies are best replies to each other
Recall: A strategy is a best reply to a particular strategy of another player
if i
Part III:
Analysis of Dynamic Settings
1
Outline
Backward induction, subgame perfection (Ch.14-16)
Bargaining* (Ch.18-19)
Repeated games, and applications (Ch. 22-23)
2
Details of the ExtensiveForm
Chapter 14
3
Game Trees: Basic Language
Recall: Trees
Strictly Competitive Games
and Security Strategies
(Chapter 12)
1
Strictly Competitive Game
In a strictly competitive game, the two players have exactly opposite
rankings over the outcomes. Wherever one players payoff
increases, the other ones payoff dec
Repeated Games and
Applications
Chapter 22, 23
Dynamic game
People often interact in ongoing relationships
Employment relationships
Countries competing over tariff levels
Players condition their decisions on the history
of their relationship
An employee
Part IV:
Information
1
Outline
Random Events and Incomplete Information (Ch.24)
Bayesian Nash Equilibrium, and applications
(Ch.26-27)
Perfect Bayesian Nash Equilibrium, and applications
(Ch. 28-29)
2
Random Events and
Incomplete Information
Chapter 24
ECON 3M03 Spring 2013 Quiz 3
Version 1
In a variation of the Bertrand model, there are two firms each with costs
TCi = 3qi i = 1, 2 . They sell in a market where demand for their output is QD = 20 P .
Each firm simultaneously chooses a price which can be
Game Theory - Econ 3M03 Fall 2014
Quiz 3
Version 1
There are 100 firms producing doohickeys. Each firm can produce either blue doohickeys or red
doohickeys but not both, and they must decide independently and simultaneously which colour to produce.
Given
ECON 3M03 Fall 2014 Quiz 5
Version 1
There is a parent (P) and a kid (K). The parent has $90 and has to choose some quantity
to give to the kid-the remainder will be kept for the parent's own use. The kid then
spends any money received from the parent on
Introduction to Game Theory
Part 3: Nash Equilibrium
Maxim Ivanov
McMaster University
January 11, 2016
Maxim Ivanov (McMaster University)
Introduction to Game Theory
January 11, 2016
1/7
Simultaneous-move games of complete information
Simultaneous-move ga
Introduction to Game Theory
Part 10: Extensive games of perfect information
Maxim Ivanov
McMaster University
April 19, 2016
Maxim Ivanov (McMaster University)
Introduction to Game Theory
April 19, 2016
1 / 27
Extensive Form Games of perfect information
In
Economics 3M03, Spring 2016
Mid-term Exam 1 Answer Key
1. In each game below, nd all Nash equilibria (just circle the equilibria, you dont
need to prove that the circled action proles are Nash equilibria). If you believe that the
game does not have Nash e
Economics 3M03, Spring 2016
Solutions to Practice Problems 2
1. Each of two rms has one job opening. The rms oer dierent wages: rm 1 oers the
wage $10 per hour, and rm 2 oers the wage $12 per hour. There are two workers, each of
whom can apply to only one
Economics 3M03, Spring 2016
Practice Problems 1
1. Consider the following game between players 1 and 2.
2
L M R
U 3; 3 5; 2 4; 3
1 I 2; 9 3; 4 4; 5
D 0; 5 6; 5 1; 4
Find all Nash equilibria in the game by:
(a) cell-by-cell inspection;
(b) best response fu
Economics 3M03, Spring 2016
Mid-term Exam 2 Answer Key
1. Consider a game between two players with the following payos of player 1. Suppose
that player 2 randomizes between L and R with probabilities q and 1 q, respectively.
(a) Determines the pure strate
Economics 3M03, Spring 2016
Problem-Solving Session 1: Answer key
1. Exercise 17.1 from the textbook.
X Y
X 3; 3 1; 5
Y 5; 1 0; 0
The game above is NOT the Prisoners Dilemma as Player 1 prefers (X; X) to (Y; Y ).
Also, Y does not strictly dominate X for e
Economics 3M03, Spring 2016
Practice problems for the nal exam
Note 1: the nal exam is comprehensive.
Note 2: I will keep my regular o ce hours on Monday at 11am-12pm and 1pm-2pm. Please see me to discuss
solutions.
Extensive-form games of perfect informa
Department of Economics
McMaster University
Class: Econ 3M03
Winter 2016
Introduction to Game Theory
Class location/time: DSB B105/Monday 2:30PM3:20PM, Wednesday 2:30PM4:20PM
Instructor: Maxim Ivanov
Office: 408 KTH
Office hours: Monday, 1:00PM2:00PM
Emai
Econ 3M03
Problem-solving Session 1 (Chapters 1-3)
1. Solve Exercise 17.1 from the textbook.
2. Show that the Rock-paper-scissors game does not have a Nash equilibrium in pure
strategies. Put the payoff of each player 1 in the case of winning, -1 in the c
Introduction to Game Theory
Part 9: Mixed-Strategy Nash Equilibria
Maxim Ivanov
McMaster University
February 29, 2016
Maxim Ivanov (McMaster University)
Introduction to Game Theory
February 29, 2016
1 / 17
Set of mixed strategies
Let i be the set of all m
Economics 3M03, Spring 2016
Practice Problems 2
1. Each of two rms has one job opening. The rms oer dierent wages: rm 1 oers the
wage $10 per hour, and rm 2 oers the wage $12 per hour. There are two workers, each of
whom can apply to only one rm. The work
Introduction to Game Theory
Part 6: Applications of Nash equilibrium (Cournot Oligopoly)
Maxim Ivanov
McMaster University
January 27, 2016
Maxim Ivanov (McMaster University)
Introduction to Game Theory
January 27, 2016
1/9
Application of NE: Cournot Oligo
Econ 3M03, 2016
Problem-solving Session 2
1. Consider a 2-player game with the following payoffs to Player 1.
Player 2
Player 1
Actions
L (q)
R (1 q)
T
1
1
M
4
0
B
0
3
Find and draw all mixed strategies of player 1 that strictly dominates T.
2. Consider a
Introduction to Game Theory
Part 11: Bayesian Games (Games of Incomplete Information)
Maxim Ivanov
April 20, 2016
Maxim Ivanov ()
Introduction to Game Theory
April 20, 2016
1/8
Bayesian Games (Games with Incomplete Information)
In games with complete info
ECON 3M03 Fall 2014
Answers to Quiz 1
For the game described in each version:
1. Draw the extensive form of this game.
2. What are the Nash equilibria (both pure- and mixed-strategy) for this game?
Note that all versions of the quiz involve Chicken (a.k.a
Game Theory - Econ 3M03 Fall 2014
Answers to Quiz 2
For all versions, we have as a starting point something of the form:
P= a bQ
Q= q1 + q2
=
TCi c=
i 1, 2
i qi
i =
Pqi TCi
1, 2
i=
Our first move is to express profits/payoff as a function of quantities/st
Game Theory - Econ 3M03 Fall 2014
Quiz 2
Version 1
1. A market has two firms, Firm One and Firm Two. The firms each simultaneously and
independently choose a quantity. Their output is sold at the market clearing price
P
= 10 16 Q where Q is the total outp
Game Theory - Econ 3M03 Fall 2014
Quiz 3
Version 1
There are 100 firms producing doohickeys. Each firm can produce either blue doohickeys or red
doohickeys but not both, and they must decide independently and simultaneously which colour to produce.
Given