Lecture 10: Normal Form Games with
Incomplete Information
Incomplete information: private information that is not common
knowlegde, and that is relevant for play (e.g. information about payo¤s,
information about strategy spaces etc.)
To solve problem: game transformed
Example: Market entrance game (…gure 6.2., 6.3)
1. Normal form game of incomplete information (Bayesian
Game)
I : …nite set of players; typical element i
each player i is of type θ i ; Θi …nite set of possible types of player i.
θ i summarizes all private information
()
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if jΘi j = 1 for all i, then game with complete information
objective probability distribution of type-pro…les
p:
∏ Θi ! ∆I
i
p (θ 1 , ...θ I ) : probability, that type-pro…le θ 1 , ...θ I realized
Assumed that for each θ i there exists a θ
i
with
p (θ i , θ i ) > 0
Every player i knows own type, but not that of other players
p (θ i jθ i ): conditional probability that all but i are of type pro…le θ i , if i
is θ i .
()
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ei
set of pure actions S
ei
payo¤ function u
ei : ∏ S
i
∏ Θi ! R
i
Game with incomplete information transformed (expanded):
ei with si (θ i ) being the action choosen by i if he
pure strategy si : Θi ! S
is of type θ i
e
mixed strategy σi : Θi ! ∆jSi j
1
payo¤s function:
ui (σ1 , ...σI ) =
∑ p (θ )uei (σ1 (θ 1 ), ...σI (θ I ), θ i )
θ 2Θ
This expanded game is a standard normal form game.
()
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2. Baysian equilibrium
De…nition: The Bayesian equilibrium of a game with incomplete
information is a Nash-equilibrium of the expanded game.
Existence guaranteed by the Nash’s equilibrium existence theorem.
Example: Cournot game with unknown cost structure
2 …rms with constant returns to scale technology choose simultaneously
quantities q1 and q2
Firm 1: marginal costs c1 , common knowlegde
Firm 2: marginal costs c2l with probability 21 and marginal costs c2h with
probability 21 . 20s cost type only known to 2.
Market demand: p = (A
()
q1
q2 )
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One type of …rm 1; two types of …rm 2, Θ2 = fh, l g
Sets of pure strategies: S1 = R+ ; S2 = R+
R+
pro…t functions:
π1 =
1
(q1 (A
2
q1
π2 =
()
q2 (c2l )
1
c1 )) + (q1 (A
2
q1
q2 (c2h )
c1 ))
1
(q2 (c2l )(A q1 q2 (c2l ) c2l ))
2
1
+ (q2 (c2h )(A q1 q2 (c2h ) c2h ))
2
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Pro…t maximization of 2:
Max
π2
0 = A
q1
2q2 (c2l )
c2l
0 = A
q1
2q2 (c2h )
c2h
q 2 (c2l ),q 2 (c2h )
FOC’s
=)
()
q2 (c2l ) =
A
q2 (c2h ) =
A
q1
2
q1
2
c2l
c2h
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Pro…t maximization of …rm 1:
1
Max (q1 (A
q1 2
q1
1
c1 )) + (q1 (A
2
q2 (c2l )
q1
q2 (c2h )
c1 ))
=)
0=
1
(A
2
q2 (c2l )
2q1
=)
q1 =
1
c1 ) + ( A
2
q2 (c2l )
2A
q2 (c2h )
2q1
q2 (c2h )
c1 )
2c1
4
Equilibrium:
q1
=
q2 (c2l ) =
q2 (c2h ) =
()
2A + c2l + c2h 4c1
6
4A 7c2l c2h + 4c1
12
4A c2l 7c2h + 4c1
12
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