Uploaded by Luis Schneider

Game Theory Notes

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Game Theory
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DT
Basics :
→
Representation
Information set
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decision nodes between which
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strategic) form →
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set of strategy profiles
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it F- i
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payoff
fact F is CK between the
common knowledge
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a
:
players
the
if
each
player
others know
player
knows
F
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each
that each
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player
each
player
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that every
knows F , and
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interests
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at
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us
individual
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Prisoner's Dilemma)
,
Pareto dominance
strategy profile
strategy profiles
:
a
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if
s
u
;
(
s
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hits ) for
all
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7
dominates
Pareto
and the
•
s .t
strategy profiles
interests
group
;
i
player
inequality is strict for at least one player
is Pareto efficient
efficiency a strategy profile
Pareto
s
:
if there is
profiles
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player game
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:
strategies
iterated dominance
always lead
uncertainty )
for
player
possible
finite two
a
for
UD2
second
is
sit $ ,
;) for every
of best responses
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in
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sit
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Proposition :
Bz
•
Isi
:
B.
'
strategy
i 's
strictly dominated
the set of
Bi
strategies
responses
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i
the set of
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,
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player
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:
strategies
that
tension : rational izability
a
survive
does not
unique strategy profile
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Big picture
>
Rationality
:
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Rationality
Rationality
strategies
Nash
is
NE
hits ;
,
Strict Nash
hi Isi
,
player
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s,
player
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or
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Duopoly
,
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NE
>
.
.
.
,
a
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That
;) for
player i.
every s.ie $ ; and
( s ;)
;
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s
,
strategy profile
A
-
for each
player
se
i.
$
is
That
a
is
,
and each
every sit %
for
;)
quantity
:
price
:
NE
in )
competition
competition
consider
:
,
,
r
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l M
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ie
,
strategy profile
a
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where
NE
MS
nil si
at D8 ;
✗
each
player
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for
for
each
si
c-
$;
and each
i.
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in
( pure
a
:
proposition F- very
of player 's and a
NE
is
:
Duopoly
rz
$
BR !
only
strategy
,
BR
=
hits ,
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(9
uilri
s
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>
strategy
Profile
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wits ,
7
se
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i.
Bertrand
Mixed
strategy profile
Equilibrium :
Iff
Gourmet
T
.
S
that
so
mutual BRS
A
:
;)
-
Applications
Mixed
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Equilibrium
i.
player
strict NE
•
Nash
izability
single strategy profile
a
iff sit BR ; ( s
s
,
on
are
Equilibrium
each
NE
rational
:
strategy )
•
consistency
+
Equilibrium
Players coordinate
their
'
of rationality
CK
+
Nash
'
and BR
dominance
pure
or
finite
game
(
having
finite strategy
space)
mixed
strategies
a
has
finite number
at
least
one
Extensive Form :
'
Perfect recall :
well
as
-
as
any
is
a
single
:
mode
from
Proposition
Sub
sub
an
of
Sub
-
sub
every info
game
Perfection :
Every
:
game
:
the
finite
BI
it neither
.
that
a
w/
game
identifies
contains
its
info
are
not
a
.
initiate
to
successors
nodes that
a
has
.
equilibrium
is said
any of
war
perfect
an
player
analysing
beginning
,
a
in
are
successors
.
game Perfect
if it
game
NE
specifies
( SPE )
a
NE
:
in
A
strategy profile is called
of the original
sub
every
game
a
.
continuation
a
.
process of
A woolen in the tree
game
info set
a
NE
set at which this
.
The
end to
the
strategy
SPE
-
otherwise
:
a
Backward induction :
pure
•
on
the move
game
'
extensive
the
:
has
'
of
set
.
sequential rationality An optimal strategy for
player should maximise his or her expected payoff
conditional
•
past actions
dashed lines)
two
sequential Rationality and
-
info
Every
Imperfect information
>
own
events that they have observed
other
Perfect information
form
remember their
Players
player's CP
payoff
is
:
the
For
a
given
mode
a
in
payoff that this
on the
path of
a
game
,
player
eventually
get contingent
play passing
through
profile
strategy
mode
a
1 based
on
the
will
)
Incomplete information
Incomplete information refers
:
•
of nature IW1
info
fixed
between the
.
to
having moves
generate asymmetric
prob ) that
.
games
players ( exogenous asymmetry )
Perfect info (
chess)
e.g
→
complete information →
Imperfect info le g.
(
endogenous asymmetry)
Type The different moves of nature that
.
.
>
'
privately
Bayesian
w/
game
>
form :
Normal
-
of
form version
a
nature
NE
and
LBNE )
Bayesian rationalize
ability
:
Second
First
price
-
sealed bid auction
-
dominant
weakly
-
single
observes
normal
Bayesian
Auction
•
a
:
player
'
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.
.
strategy
for each
price sealed bid auction
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bzlvz)
v42
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constitute
a
:
bi.LY)
:
is
a
player
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BNE
vi
=
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I b.dk )
and
12
"
=
YN
WI
N
people)
'
Revenue
expected
Perfect
-
equivalence
revenue
:
the two
Bayesian Equilibrium
Assessment
auctions
for the seller
yield
the
same
:
where ✗ is a strategy
A pair
I ✗ m)
and
set
profile
µ is a belief system ; i. e. , for each into
I MI II) is a
probability measure over nodes in I
:
,
,
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,
•
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Perfect
la , M)
the
Bayesian Equilibrium
is
a
following
weak PBE (WPBE )
are
satisfied
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IPBE )
:
An
assessment
if at each into
.
set I
>
>
sequential rationality Given belief system µ
✗ is
sequentially rational
2) Bayesian updating Given strategy profile ✗
1)
:
:
M is
consistent v1
Bayesian
rule whenever
,
,
possible
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