How to define Bayesian
rationalizability?
Incomplete information:
Examples
U
1
A ( 12 )
N t
Nature
B ( 12 )
Lectures in Game Theory
Fall 2011, Collection of examples 3
09.11.2011
G.B. Asheim, ECON3/4200-ex3
1
A ( 12 )
N t
Nature
B ( 12 )
D
U
1
D
09.11.2011
2 L
2, 2
R
2 L
R
2 L
0, 0
0, 0
4,, 4
0, 2
2, 0
4, 0
R
0, 4
R
2 L
R
0, 4
R
L
U 2, 2 0, 0
D 0, 0 4, 4
2
or
R
L
1
U 0, 2 2, 0
D 4, 0 0, 4
is played, but only
1 knows which.
2
G.B. Asheim, ECON3/4200-ex3
1
A ( 12 )
N t
Nature
B ( 12 )
UD 3, 1 0, 2
DU 0,
0 1 3,
3 2
DD 2, 0 2, 4
2 L
D
R
2 L
U
D
3
09.11.2011
2
R
L
1B
0, 0 1A U 2, 0, 2 0, 2, 0
0, 0 U
D 2, 4, 1 0, 0, 2
4,, 4
2
0, 2
R
L
1B
2, 0
U 0, 0, 1 4, 2, 2
4, 0 D
D 0, 4, 0 4, 0, 4
0, 4
2, 2
R
2 L
1
Bid : bi  Ai  [0, 1000], i  1,, n
R
2 L
R
4
G.B. Asheim, ECON3/4200-ex3
A Bayesian Nash equil. in 1st price auction
We must show that bi ( vi )  nn1 vi maximizes
Valuations : vi  Ti  [0, 1000], i  1, , n, are distribute d
x
independen tly and uniformly : For each i, Pr(v j  x)  1000
.
v  b if bi is bigger than other bids.
ui (bi , bi ; vi )  i i
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 )  nn1 vi
G.B. Asheim, ECON3/4200-ex3
R
2 L
0, 0
0, 0
4,, 4
0, 2
2, 0
4, 0
09.11.2011
U
1st price sealed bid auction w/private values
09.11.2011
R
2 L
1
Rationalizability has no bite in this game when treating
different types as different players
2
R
L
1
UU 1, 2 1, 0
G.B. Asheim, ECON3/4200-ex3
U
D
1
2, 2
R
2 L
1
Only one strategy profile is rationalizable in
the Bayesian normal form
U
D
2 L
Either
2
5
(vi  bi )  Pr (Others bid less than bi )
density
n
( n 1)1000
1
1000
Probability that one bids less than bi:
n bi
n 1 1000
Prob.
ob. that all o
others
e s bid
b d less
ess than bi:
n 1
n bi
n 1 1000
bid

b

max (vi  bi )  nn1 1000
i
( n 1)1000 1000
n

n 1
bi
FOC yields: bi  nn1 vi
This shows that it is a Bayesian Nash equil. that
all bid (n-1)/n of true value. Other equilibria?
09.11.2011
G.B. Asheim, ECON3/4200-ex3
6
1
Cournot
comp. w/
incompl.
info.
price
Bayesian Nash equilibrium
Firm 1' s unit cost : c
Firm 2' s unit costs : cH or cL
 2 (q1 , q2 ; cHL ))   PP((Q
Q))  ccLHqq22
 a  ( q1  q2 )  cLH qq22
Invers
demand fn.:
P (Q )  a  Q
Firm 2 knows
its own cost.
Firm 1: 2 high
cost w/prob .
cH
cL
q1
09.11.2011
quantity
Q
q2
Firm 2 if high cost : q2 (cH ) max a  ( q1  q2 )  cH q2
FOC : q2 (cH )  12 a  q1  cH 
Firm 2 if low cost : q2 (cL ) max a  ( q1  q2 )  cL q2
FOC : q2 (cL )  12 a  q1  cL 

Firm 1 : q1 max  a  (q1  q2 (cH ))  c q1
 (1   )a  ( q1  q2 (cL ))  c q1
FOC : q  12  a  q2 (cH )  c   (1   )a  q2 (cL )  c 

1
7
G.B. Asheim, ECON3/4200-ex3
Bayesian
Nash equil.:
q2 (cH )  13 a  2cH  c   16 cH  cL 
Comparison w/compl. info.
q2
Are there quantitites for firm 1, firm 2 if high cost and
firm 2 if low cost so that no firm would regret its own
choice if it were informed of the choices of the others?
09.11.2011
The model of Cournot comp. with
incomplete information includes
q2 (cL )  13 a  2cL  c   6 cH  cL 
1’s best resp fn
1
3
2’s best resp fn
if low cost
q2 (cL )
Fi 2 w/high
Firm
/hi h cost
produces more than
under compl info.
2’s best resp fn
if high cost
q2 (cH )
Firm 2 w/low cost
produces less than
under compl info.
q1
q1
09.11.2011
Action sets : A1  [0, ) and A2  [0, )
q  a  2c  cH  (1   )cL 

1
9
G.B. Asheim, ECON3/4200-ex3
Type sets : T1  {c} and T2  {cH , cL }
Nature' s choice : p (cH )   and p (cL )  1  
 1 (q1 , q2 ; c )  a  (q1  q2 )  c q1
Profit functions :
(Payoff functions)  2 ( q1 , q2 ; cH )  a  ( q1  q2 )  cH q2
 2 (q1 , q2 ; cL )  a  (q1  q2 )  cL q2
09.11.2011
Beer – Quiche game
Player 1 has four
pure strategies.
Player 1 has four
pure strategies.
3, 2 U 2 L 1
R 2 U 1, 0
(r )
(q ) D 0, 1
PBE w/(LL)? YES 2, 0 D
A ( 12 )
PBE w/(RR)? NO
Nature
B ( 12 )
1 0 U (1  r )
1,
21
(1  q ) U 2,

R
L
1
D 0, 0
1, 1 D
PBE w/(LR)? YES
[(LR ), (UU), q  0, r  1]
PBE w/(RL)? NO
09.11.2011
Choosing R is dominated for 1A.
[(LL), (DU), q , r  1 / 2] is an unreasonable equilibrium, because
it requires 2 to have q  1 / 2.
G.B. Asheim, ECON3/4200-ex3
11
10
G.B. Asheim, ECON3/4200-ex3
Are all perfect Bayesian equilibria reasonable?
[(LL), (DU), q , r  1 / 2]
where q  1 / 2.
8
G.B. Asheim, ECON3/4200-ex3
2, 0 U 2 Q 1
PBE w/(QQ)? YES 0, - 1 D
[(QQ), (DU), q , r  9 / 10 ]
where q  1 / 2.
(r )
B 2 U 3, 0
(q ) D 1, - 1
S ( 109 )
Nature
W ( 101 )
3 0 U (1  r )
3,
2 0
(1  q ) U 2,
PBE w/(BB)? YES

Q
B
1
D 0, 1
[(BB), (UD), q  9 / 10, r ] 1, 1 D
where r  1 / 2.
PBE w/(BQ)? NO
PBE w/(QB)? NO
09.11.2011
Is [(QQ), (DU), q , r  9 / 10 ] a
reasonable equilibrium?
Only 1S has possibly something to
gain by choosing B. But q  1 / 2.
G.B. Asheim, ECON3/4200-ex3
12
2