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Game Theory Notes
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Table of Contents
1.
Background Stuff
3
2.
Overview
5
3.
Extensive Form
6
4.
Strategies
10
5.
Preferences
12
6.
Normal Form
13
7.
Efficiency
16
8.
Beliefs
17
9.
Mixed Strategies
18
10.
Expected Utility
19
11.
Dominance
20
12.
Best Response
22
13.
Nash Equilibrium
23
14.
MixedStrategy Nash Equilibrium
27
15.
Existence of Nash Equilibrium
30
16.
Cournot Model
32
17.
Hotelling Model
34
18.
Bertrand Model
35
19.
Weaknesses of Nash Equilibrium
40
20.
Subgames
41
21.
SubgamePerfect Nash Equilibrium
42
22.
Stackelberg Model
43
23.
Limit Capacity
44
24.
Advertising
47
25.
Price Guarantees
48
26.
Strictly Competitive Games
50
27.
Equivalent Nash Equilibria
51
28.
Security Strategies
52
29.
Parlour Games
54
30.
OneCard Poker
55
3
Background Material for Introductory Game Theory
Calculus
As with most standard microeconomic theory, game theory typically assumes that an
agent behaves optimally (from its own point of view).
In formal modelling, this
frequently involves taking a derivative of some objective function (the thing the agent
cares about) with respect to some choice variable(s) (the thing(s) the agent directly
affects) and setting that derivative equal to zero (in hopes of characterizing the maximum
of the objective function).
The bottom line is that you want to have a solid grounding in
introductory calculus for this course.
Economics
You need suprisingly little background in economics for this course.
It helps to be
familiar with utility and the way economists think about efficiency.
Set Theory
Set theory is used in game theory largely for compact notation.
A set is an unordered collection of things. The things that make up the set are the
elements of the set, sometimes called the members of the set.
A set can consist of a
collection of anything, even other sets.
For the purpose of this course, sets will normally
consist of strategies.
Notation:
A = {1, 5, 7} means the set A has elements 1, 5, and 7.
Curly brackets are normally used
to enclose the elements of a set and the elements are separated by commas.
(Round
brackets are used for vectors for which the order of the elements matters.)
The order of elements doesn't matter:
If B = {5, 1, 7} then A = B and we can also write B = {1, 5, 7}.
Repeated elements shouldn't be there:
Writing A = {1, 5, 5, 7} is bad.
Not really that it's wrongits most likely effect is to
confuse the reader.
Sets can be members of sets:
If C = {2, 3} and D = {A, C} then D = {{1, 5, 7}, {2, 3}}.
The 'nesting' of elements within sets within sets matters:
If E = {1, 5, 7, 2, 3} then D
E. {{1}}
≠
≠
{1}.
The empty set:
There is something known as the 'empty set', denoted
φ
(the Greek letter phi, sometimes
written as
ϕ
) which has no members.
It's also sometimes called the 'null set'.
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Additional notation:
∈
means 'is a member of'.
e.g. 1
∈
A, A
∈
D
∉
means 'is not a member of.
e.g. 1
∉
C
∪
is an operator denoting the union of two sets.
e.g. E=A
∪
C, {1, 5}
∪
{5, 7}=A
∩
is an operator denoting the intersection of two sets.
e.g. C
∩
{2, 4}={2}, A
∩
C=
φ
⊆
means 'is a subset of'.
e.g. A
⊆
E, A
⊆
A
⊂
means 'is a proper subset of'. e.g. A
E
⊂
×
is an operator denoting the cross product of two or more sets.
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This note was uploaded on 09/05/2008 for the course ECONOMICS ECON 3M03 taught by Professor Brucejames during the Fall '08 term at McMaster University.
 Fall '08
 BRUCEJAMES
 Game Theory

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