Econ106G Lecture Note 9
Hong Feng
July 24, 2008
1 Incomplete Information and Bayesian Nash
Equilibrium
A key assumption in most of the games we have discussed so far was that each
player knows the whole structure of the game, i.e. they know their own possi
ble strategies, the strategies of other players, and the payo/s from all possible
situations that we want to explain with game theory. In particular, knowledge
in competition rarely know each others±costs.
A (simpli&ed version of) strategicform game with Incomplete In
formation is given by
A set of players
i
= 1
;
2
Types of each players:
t
i
2
T
i
with distribution
p
i
(
t
i
)
; i
= 1
;
2
:
Actions of each player:
a
i
2
A
i
; i
= 1
;
2
Strategies:
s
i
2
S
i
:
Each pure strategy
s
i
assigns some action
a
i
to every
type
t
i
:
Payo/s:
u
1
(
a
1
;a
2
;t
1
;t
2
)
; u
2
(
a
1
;a
2
;t
1
;t
2
)
:
(
s
1
(
t
1
)
;s
2
(
t
2
))
is a Bayesian Nash equilibrium
if
1
is playing a best reponse to
2
s
1
(
t
1
)
maximizes
E
t
2
[
u
1
(
a
1
;s
2
(
t
2
)
;t
1
;t
2
)]
for all
t
1
i:e: E
t
2
[
u
1
(
s
1
(
t
1
)
;s
2
(
t
2
)
;t
1
;t
2
)]
±
E
t
2
[
u
1
(
a
0
1
;s
2
(
t
2
)
;t
1
;t
2
)]
for all
a
0
1
2
A
1
;
for all
t
1
and
2
is playing a best response to
1
s
2
(
t
2
)
maximizes
E
t
2
[
u
2
(
s
1
(
t
1
)
;a
2
;t
1
;t
2
)]
for all
t
2
in a BNE, for each type
t
i
of player
i
, the assigned action
s
i
(
t
i
)
must be a best
response to
s
j
;
i.e. it must generate the highest
expected payo/
for
t
i
:
1
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stronger one wins. The strength of player 1 is commonly known to be inter
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 Winter '04
 cai
 Game Theory, player, Bayesian Nash equilibrium, Hong Feng July

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