14.12 Game Theory Lecture Notes
∗
Lectures 36
Muhamet Yildiz
†
We will formally de
f
ne the games and some solution concepts, such as Nash Equi
librium, and discuss the assumptions behind these solution concepts.
In order to analyze a game, we need to know
•
who the players are,
•
which actions are available to them,
•
how much each player values each outcome,
•
what each player knows.
Notice that we need to specify not only what each player knows about external
parameters, such as the payo
f
s, but also about what they know about the other players’
knowledge and beliefs about these parameters, etc. In the
f
rst half of this course, we
will con
f
ne ourselves to the games of complete information, where everything that is
known by a player is common knowledge.
1
(We say that X is common knowledge if
∗
These notes are somewhat incomplete – they do not include some of the topics covered in the
class.
†
Some parts of these notes are based on the notes by Professor Daron Acemoglu, who taught this
course before.
1
Knowledge is de
f
ned as an operator on the propositions satisfying the following properties:
1. if I know X, X must be true;
2. if I know X, I know that I know X;
3. if I don’t know X, I know that I don’t know X;
4. if I know something, I know all its logical implications.
1
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View Full Documenteveryone knows X, and everyone knows that everyone knows X, and everyone knows
that everyone knows that everyone knows X, ad in
f
nitum.) In the second half, we will
relax this assumption and allow players to have asymmetric information, focusing on
informational issues.
1R
e
p
r
e
s
e
n
t
a
t
i
o
n
s
o
f
g
a
m
e
s
The games can be represented in two forms:
1. The normal (strategic) form,
2. The extensive form.
1.1
Normal form
De
f
nition 1
(Normal form) An nplayer game is any list
G
=(
S
1
,...,S
n
;
u
1
,...,u
n
)
,
where, for each
i
∈
N
=
{
1
,...,n
}
,
S
i
is the set of all strategies that are available to
player
i
,and
u
i
:
S
1
×
...
×
S
n
→
R
is player
i
’s von NeumannMorgenstern utility
function.
Notice that a player’s utility depends not only on his own strategy but also on the
strategies played by other players. Moreover, each player
i
tries to maximize the expected
value of
u
i
(where the expected values are computed with respect to his own beliefs); in
other words,
u
i
is a von NeumannMorgenstern utility function. We will say that player
i
is rational i
f
hetr
iestomax
im
izetheexpectedva
lueo
f
u
i
(given his beliefs).
2
It is also assumed that it is common knowledge that the players are
N
=
{
1
}
,
that the set of strategies available to each player
i
is
S
i
,andthateach
i
tries to maximize
expected value of
u
i
given his beliefs.
When there are only two players, we can represent the (normal form) game by a
bimatrix (i.e., by two matrices):
1
\
2l
e
f
t
r
i
g
h
t
up
0,2
1,1
down
4,1
3,2
2
We have also made another very strong “rationality” assumption in de
f
ning knowledge, by assuming
that, if I know something, then I know all its logical consequences.
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 Spring '05
 MuhammadYildiz
 Game Theory, player, Head Tail

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