Incomplete information:
Bayesian Nash equilibrium
Lectures in Game Theory
Fall 2011, Part 6
24.07.2011
G.B. Asheim, ECON3/4200-6
1

Incomplete information: At least one player
does not know who his opponents are.

An illustrating example

Definition of
Bayesian games,
B
Bayesian
i normall fform,
Bayesian Nash equilibrium

First-price sealed-bid auction as an example.

Cournot competition as an example
24.07.2011
2
G.B. Asheim, ECON3/4200-6
A Bayesian game
One type of
player 1 U
1
A ( 12 )
N t
Nature
B ( 12 )
D
U
1
Another D
type of player 1
24.07.2011
2 L
2, 6
R
2 L
R
2 L
2, 0
0, 4
0,, 8
0, 6
0, 0
2, 4
R
2, 8
R
2 L
Either
1
2
R
L
U 2, 6 2, 0
D 0, 4 0, 8
2
or
R
L
1
U 0, 6 0, 0
D 2, 4 2, 8
G.B. Asheim, ECON3/4200-6
is played, but only
1 knows which.
3
1
The Bayesian normal form (an ex ante
perspective)
2
U
1
A ( 12 )
N t
Nature
B ( 12 )
D
U
1
D
2 L
2, 6
R
2 L
R
2 L
2, 0
0, 4
0,, 8
0, 6
0, 0
2, 4
R
2, 8
R
2 L
24.07.2011
R
L
1
UU 1, 6 1, 0
UD 2, 5 2, 4
DU 0,
0 5 0,
0 4
DD 1, 4 1, 8
Bayesian rationalizability and Bayesian
Nash equilibrium
4
G.B. Asheim, ECON3/4200-6
Treating different types as different players
(an ex post perspective) 1B 2 L R
U
1
A ( 12 )
N t
Nature
B ( 12 )
D
U
1
D
24.07.2011
2 L
2, 6 1A
U 2, 0, 6 2, 0, 0
U
D 2, 2, 5 2, 2, 4
R
2 L
2, 0
0, 4
0,, 8
0, 6
0, 0
2, 4
R
2, 8
R
2 L
R
2 L
2
R
L
1B
0,
0,
5
0,
0, 4
U
D
D 0, 2, 4 0, 2, 8
Equivalent way to
determine Bayesian
Nash equilibrium
5
G.B. Asheim, ECON3/4200-6
A Bayesian game specifies
Example of p:
t2
t 2
t1 1 6 512
Players: {1, ... , i, ... , n}
For each player, an action set: Ai t  13 112
 For each player, a type set: Ti
 A probability distribution over type profiles: p
 For each player, a payoff function: ui


1
Bayesian game : G  ( A1 , , An ; T1 , , Tn ; p; u1 , , un )
Player i' s type ti  Ti is private information.
Player i' s payoff ui (a1 , , an ; t1 , , tn )
depends on the action and type profiles.
24.07.2011
G.B. Asheim, ECON3/4200-6
6
2
Strategy
Definition : In the Bayesian game
G  ( A1 , , An ; T1 , , Tn ; p; u1 , , un )
a strategy for player i is a function si () that,
for each type ti  Ti , specifies a feasible action si (ti ).
The Bayesian normal form specifies
Players: {1, ... , i, ... , n}
 For each player, the strategy set: Si
 For each player, the expected payoff function

Definition: A Bayesian Nash equilibrium of a Bayesian
game is a Nash equilibrium of the Bayesian normal form.
24.07.2011
G.B. Asheim, ECON3/4200-6
7
1st price sealed bid auction w/private values
Bid : bi  Ai  [0, 1000], i  1, 2
Valuations : vi  Ti  [0, 1000], i  1, 2, are distribute d
x
independen tly and uniformly : For each i, Pr(v j  x)  1000
.
ui (bi , b j ; vi ) 
vi  bi if bi is bigger than the other bid.
otherwise.
th i
0
One must bid less than true value in order to earn if one
wins. This must be traded off against the fact that a lower
bid reduces the probability for having a winning bid.
Consider strategies of the following form: bi (vi )  12 vi
24.07.2011
G.B. Asheim, ECON3/4200-6
8
A Bayesian Nash equil. in 1st price auction
We must show that bi (vi )  12 vi maximizes
density
2
1000
(vi  bi )  Pr (Other bids less than bi )
2
bi
Probability that other bids less than bi: 1000
1
1000
2
max (vi  bi )  1000
bi
bi
bid
FOC yields: bi  12 vi
This shows that it is a Bayesian
Nash equil. that both bid
n bidder auction
half of true value.
500
1000
Cournot competition
24.07.2011
G.B. Asheim, ECON3/4200-6
9
3