Normal Form Games with Incomplete Information
Julio Dávila
2009
Julio Dávila
Normal Form Games with Incomplete Information
Normal form games with incomplete information...
1 I : a set of players
Julio Dávila
Normal Form Games with Incomplete Information
Normal form games with incomplete information...
1 I : a set of players
2 for all i ∈ I ,
Si : the sets of actions available to each player
Julio Dávila
Normal Form Games with Incomplete Information
Normal form games with incomplete information...
1 I : a set of players
2 for all i ∈ I ,
Si : the sets of actions available to each player
Σi : the set of probability distributions over Si
Julio Dávila
Normal Form Games with Incomplete Information
Normal form games with incomplete information...
1 I : a set of players
2 for all i ∈ I ,
Si : the sets of actions available to each player
Σi : the set of probability distributions over Si
Θi : the sets of pieces of information about the state of the
world for each player
Julio Dávila
Normal Form Games with Incomplete Information
Normal form games with incomplete information...
1 I : a set of players
2 for all i ∈ I ,
Si : the sets of actions available to each player
Σi : the set of probability distributions over Si
Θi : the sets of pieces of information about the state of the
world for each player
p: a probability distribution over the states of the world
Θ = ×i∈I Θi
Julio Dávila
Normal Form Games with Incomplete Information
Normal form games with incomplete information...
1 I : a set of players
2 for all i ∈ I ,
Si : the sets of actions available to each player
Σi : the set of probability distributions over Si
Θi : the sets of pieces of information about the state of the
world for each player
p: a probability distribution over the states of the world
Θ = ×i∈I Θi
3 for all i ∈ I and all θ ∈ Θ, uiθ ∈ R×j∈I Σi linear in each
σj ∈ Σj , for all j ∈ I , i.e. the payoffs from each profile of
randomizations over actions for each player in each state of
the world.
Julio Dávila
Normal Form Games with Incomplete Information
Normal form games with incomplete information...
1 I : a set of players
2 for all i ∈ I ,
Si : the sets of actions available to each player
Σi : the set of probability distributions over Si
Θi : the sets of pieces of information about the state of the
world for each player
p: a probability distribution over the states of the world
Θ = ×i∈I Θi
3 for all i ∈ I and all θ ∈ Θ, uiθ ∈ R×j∈I Σi linear in each
σj ∈ Σj , for all j ∈ I , i.e. the payoffs from each profile of
randomizations over actions for each player in each state of
the world.
A normal form game with incomplete information, or
Bayesian game, is a collection {Si , (uiθ )θ∈Θ , p}i∈I as above
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
if a player i with information θi is sure to face opponents with
θ
a profile of informations θ−i , then his payoff uiθ ((σj j )j∈I ) is
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
if a player i with information θi is sure to face opponents with
θ
a profile of informations θ−i , then his payoff uiθ ((σj j )j∈I ) is
X
h Y
s∈×i 0 ∈I Si 0
i
θ
σj j (sj ) viθ (s)
j∈I
for some viθ ∈ R×i 0 ∈I Si 0
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
if a player i with information θi is sure to face opponents with
θ
a profile of informations θ−i , then his payoff uiθ ((σj j )j∈I ) is
X
h Y
s∈×i 0 ∈I Si 0
i
θ
σj j (sj ) viθ (s)
j∈I
for some viθ ∈ R×i 0 ∈I Si 0
σiθi (si ) is the probability of a player i with information θi
playing si according to σiθi
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
if a player i with information θi is not sure about the
opponents’ profile of informations θ−i , then his expected
θ
payoff Eθ−i (uiθ ((σj j )j∈I )|θi ) is
X
p(θ−i |θi )



θ−i ∈×j6=i Θj
X
s∈×i 0 ∈I Si 0
h Y
j∈I


i
θ
σj j (sj ) viθ (s)

for some viθ ∈ R×i 0 ∈I Si 0
σiθi (si ) is the probability of a player i with information θi
playing si according to σiθi
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
where
p(θ−i |θi ) = P
0
θ−i
p(θi , θ−i )
0
∈×j6=i Θj p(θi , θ−i )
is the probability of his opponents’ informations to be θ−i if i’s
information is θi
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
if a player i with information θi is not sure about the
opponents’ profile of informations θ−i , then his expected
θ
payoff Eθ−i (uiθ ((σj j )j∈I )|θi ) is
X
p(θ−i |θi )



θ−i ∈×j6=i Θj
X
s∈×i 0 ∈I Si 0
h Y
j∈I


i
θ
σj j (sj ) viθ (s)

for some viθ ∈ R×i 0 ∈I Si 0
σiθi (si ) is the probability of a player i of type θi playing si
according to σiθi
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
then player i chooses, for each information θi he may receive,
σiθi to maximize
X
p(θ−i |θi )



θ−i ∈×j6=i Θj
X
s∈×i 0 ∈I Si 0
h Y
j∈I


i
θ
σj j (sj ) viθ (s)

θ
given σj j , for all j 6= i
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
then player i chooses (σiθi )θi ∈Θi to maximize
X
θi ∈Θi
pi (θi )


X
p(θ−i |θi )




θ−i ∈×j6=i Θj
X
s∈×i 0 ∈I Si 0
h Y
j∈I


i
θ
σj j (sj ) viθ (s)

θ
given σj j , for all j 6= i
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
then player i chooses (σiθi )θi ∈Θi to maximize, for each
information θi he may receive,
X


θ∈×j∈I Θj

X
h Y
s∈×i 0 ∈I Si 0
j∈I


i
θ
σj j (sj ) p(θ)viθ (s)

θ
given σj j , for all j 6= i
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
then player i chooses (σiθi )θi ∈Θi to maximize, for each
information θi he may receive,
h Y
X
(s,θ)∈(×j∈I Sj )×(×j∈I Θj )
θ
σj j (sj )
i
θ
p(θ)vi (s)
j∈I
θ
given σj j , for all j 6= i
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
then player i chooses (σiθi )θi ∈Θi to maximize, for each
information θi he may receive,
h Y
X
(s,θ)0 ∈×j∈I (Sj ×Θj )
i
θ
σj j (sj )
p(θ)viθ (s)
j∈I
θ
given σj j , for all j 6= i
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
then player i chooses (σiθi )θi ∈Θi to maximize, for each
information θi he may receive,
θ
σj j (sj )
Y
X
(s,θ)0 ∈×j∈I (Sj ×Θj )
j∈I
P
(sj0 ,θj0 )
θ0
σj j (sj0 )
θ
p(θ)vi (s)
θ
given σj j , for all j 6= i
Julio Dávila
Normal Form Games with Incomplete Information
... or Bayesian games
then player i chooses (σiθi )θi ∈Θi to maximize, for each
information θi he may receive,
Y
X
(s,θ)0 ∈×j∈I (Sj ×Θj )
0
σ̃j (sj , θj ) ṽi ((s, θ) )
j∈I
θ
given σj j , for all j 6= i
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
Julio Dávila
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
A
A
u1 (σ) =.7[8pUA pL + 5pUA pR + 7pD
pL + 0pD
pR ]
B
B
+ .3[8pUB pL + 5pUB pR + 9pD
pL + 2pD
pR ]
A
A
u2 (σ) =.7[5pUA pL + 4pUA pR + 0pD
pL + 3pD
pR ]
B
B
+ .3[5pUB pL + 4pUB pR + 0pD
pL + 3pD
pR ]
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
A
u1 (σ) =.7(8pL + 5pR )pUA + .7(7pL + 0pR )pD
B
+ .3(8pL + 5pR )pUB + .3(9pL + 2pR )pD
A
B
+ .3pD
)]pL
u2 (σ) =[5(.7pUA + .3pUB ) + 0(.7pD
A
B
+ [4(.7pUA + .3pUB ) + 3(.7pD
+ .3pD
)]pR
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
1 mixes in A only if 8pL + 5pR = 7pL + 0pR
1 mixes in B only if 8pL + 5pR = 9pL + 2pR
A
B
u2 (σ) =[5(.7pUA + .3pUB ) + 0(.7pD
+ .3pD
)]pL
A
B
+ [4(.7pUA + .3pUB ) + 3(.7pD
+ .3pD
)]pR
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
1 mixes in A only if pL + 5pR = 0!!
1 mixes in B only if pL − 3pR = 0
A
B
u2 (σ) =[5(.7pUA + .3pUB ) + 0(.7pD
+ .3pD
)]pL
A
B
+ [4(.7pUA + .3pUB ) + 3(.7pD
+ .3pD
)]pR
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
A
1 does not mix in A: (pUA , pD
) = (1, 0)
3 1
1 mixes in B only if (pL , pR ) = ( , )
4 4
A
B
u2 (σ) =[5(.7pUA + .3pUB ) + 0(.7pD
+ .3pD
)]pL
A
B
+ [4(.7pUA + .3pUB ) + 3(.7pD
+ .3pD
)]pR
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
A
1 does not mix in A: (pUA , pD
) = (1, 0)
3 1
1 mixes in B only if (pL , pR ) = ( , )
4 4
A
B
2 mixes only if .7pUA + .3pUB = 3(.7pD
+ .3pD
)
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
A
1 does not mix in A: (pUA , pD
) = (1, 0)
3 1
1 mixes in B only if (pL , pR ) = ( , )
4 4
2 mixes only if .7pUA + .3pUB = 2.1(1 − pUA ) + .9(1 − pUB )
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
A
1 does not mix in A: (pUA , pD
) = (1, 0)
3 1
1 mixes in B only if (pL , pR ) = ( , )
4 4
2 mixes only if 2.8pUA + 1.2pUB = 3
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
A
1 does not mix in A: (pUA , pD
) = (1, 0)
3 1
1 mixes in B only if (pL , pR ) = ( , )
4 4
2 mixes only if 1.2pUB = .2
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
A
) = (1, 0)
1 does not mix in A: (pUA , pD
3 1
1 mixes in B only if (pL , pR ) = ( , )
4 4
5
1
B
2 mixes only if (pUB , pD
)=( , )
6 6
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
Bayesian Nash equilibrium:
1 5
A
B
{(pUA , pD
), (pUB , pD
)} = {(1, 0), ( , )}
6 6
3 1
(pL , pR ) = ( , )
4 4
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
Bayesian Nash equilibrium:
1 5
A
B
{(pUA , pD
), (pUB , pD
)} = {(1, 0), ( , )}
6 6
3 1
(pL , pR ) = ( , )
4 4
A
u1 (σ) =.7(8pL + 5pR )pUA + .7(7pL + 0pR )pD
B
+ .3(8pL + 5pR )pUB + .3(9pL + 2pR )pD
Julio Dávila
Normal Form Games with Incomplete Information
example
U
D
[.7]
L
R
8, 5 5, 4
7, 0 0, 3
U
D
[.3]
L
R
8, 5 5, 4
9, 0 2, 3
Bayesian Nash equilibrium:
1 5
A
B
{(pUA , pD
), (pUB , pD
)} = {(1, 0), ( , )}
6 6
3 1
(pL , pR ) = ( , )
4 4
A
u1 (σ) =.7(8pL + 5pR )pUA + .7(7pL + 0pR )pD
B
+ .3(8pL + 5pR )pUB + .3(9pL + 2pR )pD
A
B
u2 (σ) =[5(.7pUA + .3pUB ) + 0(.7pD
+ .3pD
)]pL
A
B
+ [4(.7pUA + .3pUB ) + 3(.7pD
+ .3pD
)]pR
Julio Dávila
Normal Form Games with Incomplete Information
...
Julio Dávila
Normal Form Games with Incomplete Information