Games with Incomplete Information
Some players have incomplete information about some components of the
game
Microeconomics II
◮
◮
Strategic Form Games with Incomplete Information
Firm does not know rival’s cost
Bidder does not know valuations of other bidders in an auction
We could also say some players have private information
Does incomplete information matter?
Levent Koçkesen
Suppose you make an offer to buy out a company
Koç University
If the value of the company is V it is worth 1.5V to you
The seller accepts only if the offer is at least V
If you know V what do you offer?
You know only that V is uniformly distributed over [0, 100]. What should
you offer?
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Bayesian Games
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Bayesian Games
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Bayesian Games
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Bayesian Games
What is new in a Bayesian game?
Each player has a type: summarizes a player’s private information
◮ Type set for player i: Θi
◮
We will first look at incomplete information games where players move
simultaneously
◮
⋆ A generic type: θi
Set of type profiles: Θ = ×i∈N Θi
⋆ A generic type profile: θ = {θ , θ , . . . , θn }
1 2
Players’ payoffs depend on types
◮ ui : S × Θ → R
◮ ui (s, θ)
Bayesian games
Later on we will study dynamic games of incomplete information
Incomplete information can be anything about the game
◮
◮
◮
Payoff functions
Actions available to others
Beliefs of others; beliefs of others’ beliefs of others’...
Harsanyi (1967-1968) showed that introducing types in payoffs is
adequate
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Bayesian Games
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Bayesian Games
Bayesian Equilibrium
What is new in a Bayesian game?
Since payoffs depend on types different types of the same player may play
different strategies
◮ si : Θi → Si
◮ σi : Θi → ∆(Si )
We assume that players maximize their expected payoffs. Let
Each player has beliefs about others’ types
◮ pi : Θi → ∆ (Θ−i )
◮ pi (θ−i |θi )
Beliefs are consistent
◮
There exists a probability distribution (a common prior) p ∈ ∆(Θ) such that
(Assuming type sets are all finite)
Ui (σ, θ) = ∑
pi (θ−i |θi ) =
p(θ)
∑t−i ∈Θ−i p(θi , t−i )
!
∏ σj(sj |θj )
j∈N
s∈S
ui (s, θ)
or
Ui (σi , σ−i , θi , θ−i ) = ∑
for all θ ∈ Θ and for all i ∈ N .
s∈S
∏
!
σj (sj |θj ) σi (si |θi )ui (s, θi , θ−i )
j∈N\{i}
Expected payoff of player i of type θi is given by
Definition
A Bayesian Game is given by
Ui (σ, θi ) =
∑
pi (θ−i |θi )Ui (σ, θi , θ−i )
θ−i ∈Θ−i
GB = (N, (Si ), (Θi ), (pi ), (ui ))
or
Ui (σi , σ−i , θi ) =
We say that a Bayesian games is finite if N, Si , Θi are finite for all i ∈ N .
Levent Koçkesen (Koç University)
Bayesian Games
∑
pi (θ−i |θi )Ui (σi , σ−i , θi , θ−i )
θ−i ∈Θ−i
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Bayesian Equilibrium
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Bayesian Games
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Bank Runs
You (player 1) and another investor (player 2) have a deposit of $100 each
in a bank
Definition
σ∗
is a Bayesian equilibrium if each type of each player
A strategy profile
maximizes her expected payoff given other players’ strategies, i.e., for all i ∈ N
and θi ∈ Θi
If the bank manager is a good investor you will each get $150 at the end
of the year. If not you loose your money
You can try to withdraw your money now but the bank has only $100 cash
◮
Ui (σ∗i , σ∗−i , θi ) ≥ Ui (σi , σ∗−i , θi ),
for all σi ∈ ∆(Si )Θi
◮
If only one tries to withdraw she gets $100
If both try to withdraw they each can get $50
You believe that the manager is good with probability q
Also known as Bayesian Nash or Bayes Nash equilibrium
Player 2 knows whether the manager is good or bad
You and player 2 simultaneously decide whether to withdraw or not
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Bank Runs
Separating Equilibria
The payoffs can be summarized as follows
W
N
W
50, 50
0, 100
W
N
N
100, 0
150, 150
W
50, 50
0, 100
W
N
N
100, 0
150, 150
Bad (1 − q)
(Good: W, Bad: N)
2
(Good: N, Bad: W)
Player 1’s expected payoffs
Separating Equilibria: Each type plays a different strategy
2
Pooling Equilibria: Each type plays the same strategy
W:
N:
1
Player 2 always withdraws in bad state: σ2 (W|B) = 1 in any Bayesian
equilibrium
Bayesian Games
W
N
2
q < 1/2: Player 1 chooses W. But then player 2 of Good type must play W,
which contradicts our hypothesis that he plays N
q ≥ 1/2: Player 1 chooses N. The best response of Player 2 of Good type
is N, which is the same as our hypothesis
q < 1/2:
q ≥ 1/2:
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No separating equilibrium
σ1 (N) = 1, σ2 (N|G) = 1, σ2 (W|B) = 1
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Bayesian Games
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Mixed Strategy Equilibria?
N
100, 0
150, 150
W
N
Good (q)
1
(Good: N, Bad: N)
2
(Good: W, Bad: W)
Player 1’s expected payoffs
◮
q × 100 + (1 − q) × 50
q × 150 + (1 − q) × 0
Separating Equilibrium
Pooling Equilibria
W
50, 50
0, 100
N
100, 0
0, 0
Two possibilities
How would you play if you were Player 2 who knew the banker was bad?
Levent Koçkesen (Koç University)
W
50, 50
0, 100
Not possible since W is a dominant strategy for Bad
Two Possible Types of Bayesian Equilibria
1
W
N
Bad (1 − q)
1
◮
Good (q)
W
50, 50
0, 100
Good (q)
N
100, 0
0, 0
W
50, 50
0, 100
N
100, 0
0, 0
W
N
N
100, 0
150, 150
W
N
Good (q)
Bad (1 − q)
W
50, 50
0, 100
N
100, 0
0, 0
Bad (1 − q)
Is there an equilibrium in which σ2 (W|G) ∈ (0, 1)?
This requires the expected payoffs to W and N to be the same for player 2
of Good type:
Not possible since W is a dominant strategy for Bad
W:
N:
W
50, 50
0, 100
σ1 (W) × 50 + (1 − σ1 (W)) × 100 = (1 − σ1 (W)) × 150
or σ1 (W) = 1/2
This, in turn, implies that the expected payoffs to W and N must be same
for player 1:
q × 50 + (1 − q) × 50
q × 0 + (1 − q) × 0
Player 1 chooses W. Player 2 of Good type’s best response is W.
Therefore, for any value of q the following is the unique
q × [σ2 (W|G) × 50 + (1 − σ2 (W|G)) × 100] + (1 − q) × 50 =
Pooling Equilibrium
q × [σ2 (W|G) × 0 + (1 − σ2(W|G)) × 150] + (1 − q) × 0
σ1 (W) = 1, σ2 (W|G) = 1, σ2 (W|B) = 1
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Bayesian Games
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Levent Koçkesen (Koç University)
Bayesian Games
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Mixed Strategy Equilibrium
Cournot Duopoly with Incomplete Information about Costs
This is solved as
σ2 (W|G) = 1 −
We have
50
100q
Two firms. They choose how much to produce qi ∈ R+
Firm 1 has high cost: cH
Firm 2 has either low or high cost: cL or cH
1
σ2 (W|G) ∈ (0, 1) ⇔ q >
2
Firm 1 believes that Firm 2 has low cost with probability µ ∈ [0, 1]
payoff function of player i with cost cj
Mixed Strategy Equilibrium
ui (q1 , q2 , cj ) = (a − (q1 + q2 )) qi − cj qi
The following is a Bayesian equilibrium for any q > 1/2:
Strategies:
q1 ∈ R+
1
50
σ1 (W) = , σ2 (W|G) = 1 −
, σ2 (W|B) = 1
2
100q
q2 : {cL , cH } → R+
If q < 1/2 the only equilibrium is a bank run
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Bayesian Games
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Complete Information
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Complete Information
Firm 1
max (a − (q1 + q2 )) q1 − cH q1
q1
Nash Equilibrium
Best response correspondence
If Firm 2’s cost is cH
a − q2 − cH
B1 (q2 ) =
2
q1 = q2 =
Firm 2
max (a − (q1 + q2 )) q2 − cj q2
q1 =
q2
Best response correspondences
B2 (q1 , cL ) =
B2 (q1 , cH ) =
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a − cH
3
If Firm 2’s cost is cL
q2 =
a − q1 − cL
2
a − q1 − cH
2
Bayesian Games
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Levent Koçkesen (Koç University)
a − cH − (cH − cL )
3
a − cH + (cH − cL )
3
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Incomplete Information
Bayesian Equilibrium
Firm 2
max (a − (q1 + q2 )) q2 − cj q2
q2
Best response correspondences
B2 (q1 , cL ) =
B2 (q1 , cH ) =
a − cH − µ(cH − cL )
3
a − cL + (cH − cL )
cH − cL
− (1 − µ)
3
6
a − cH
cH − cL
q2 (cH ) =
+µ
3
6
a − q1 − cL
2
a − q1 − cH
2
q1 =
q2 (cL ) =
Firm 1 maximizes
µ{[a − (q1 + q2 (cL ))] q1 − cH q1 }
+ (1 − µ){[a − (q1 + q2 (cH ))] q1 − cH q1 }
Is information good or bad for Firm 1?
Does Firm 2 want Firm 1 to know its costs?
Best response correspondence
B1 (q2 (cL ), q2 (cH )) =
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a − [µq2 (cL ) + (1 − µ)q2 (cH )] − cH
2
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Complete vs. Incomplete Information
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Purification of Mixed Strategy Equilibria
Interpreting equilibria in mixed strategies is difficult
individuals flipping coins to determine their actions is counter-intuitive
Complete Information
Incomplete Information
q2
a − cH
why should individuals choose the exact equilibrium probabilities when
they are indifferent between actions?
q2
a − cH
BR1 (q2 )
Two responses:
a−cL
2
1
BR2 (q1 , cL )
q2 (cL )
E[q2 ]
a−cH
2
q2 (cH )
q1
a−cH
2
a − cH
q1
q1
a − cL
Interpret the mixed strategy of player i as the (common) conjecture of
other players about player i’s play
Mixed strategy equilibrium as an equilibrium in conjectures
BR2 (q1 , cH )
a−cH
2
a − cH
2
Harsanyi’s (1973) purification idea
a − cL
Mixed strategy equilibria as the limit of Bayesian equilibria of perturbed games
in which each player is almost always choosing his uniquely optimal action.
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Bayesian Games
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Levent Koçkesen (Koç University)
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Purification of Mixed Strategy Equilibria
Purification of Mixed Strategy Equilibria
Consider the Battle of the Sexes game
Slightly perturb the payoffs
Calculate the pure strategy Bayesian equilibria of the games parametrized
on the perturbations
Limit of the equilibria as perturbations go to zero is the mixed strategy
equilibria of the original game
◮
B
S
B
2, 1
0, 0
S
0, 0
1, 2
Three Nash equilibria: (σ1 (B), σ2 (B)) ∈ {(1, 1), (0, 0), (2/3, 1/3)}
Perturbed game:
A single sequence of perturbed games can be used to purify all the mixed
strategy equilibria
B
2 + εθ1 , 1
0, 0
B
S
We will demonstrate the idea using an example
S
εθ1 , εθ2
1, 2 + εθ2
where ε ∈ (0, 1) and θ1 and θ2 are two independently distributed random
variables with uniform distribution over [0, 1]. This defines a Bayesian game.
We will next find its pure strategy Bayesian equilibria.
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Proof (cont.)
Claim
There are three pure strategy Bayesian equilibria of the perturbed Battle of the
Sexes Game:
1
σ1 (B|θ1 ) = σ2 (B|θ2 ) = 1, for all θ1 , θ2 ∈ [0, 1]
2
σ1 (B|θ1 ) = σ2 (B|θ2 ) = 0, for all θ1 , θ2 ∈ [0, 1]
Last, suppose σ1 (B|θ1 ) = 1 for some θ1 and σ1 (B|θ1 ) = 0 for some other θ1 .
The payoff matrix makes it clear that then the equilibrium strategy must be
such that there exists a θ∗1 ∈ (0, 1) such that
σ1 (B|θ1 ) =
3
(
1, θ1 >
σ1 (B|θ1 ) =
0, θ1 ≤
1
3+ε
1
3+ε
(
1, θ2 <
σ2 (B|θ2 ) =
0, θ2 ≥
1
3+ε
1
3+ε
Suppose first that σ1 (B|θ1 ) = 1, for all θ1 ∈ [0, 1]. Then, it must be that
σ2 (B|θ2 ) = 1, for all θ2 ∈ [0, 1]. This is indeed an equilibrium.Second, if
σ1 (B|θ1 ) = 0, for all θ1 ∈ [0, 1]. Then, it must be that σ2 (B|θ2 ) = 0, for all
θ2 ∈ [0, 1]. This is also an equilibrium.
U2 (σ1 , B, θ2 ) = 1 ×
Z 1
dθ1 = (1 − θ∗1 )
U2 (σ1 , S, θ2 ) = εθ2
Z 1
dθ1 + (2 + εθ2 )
Define
θ∗2
by
U2 (σ1 , B, θ∗2 )
=
θ∗1
θ∗1
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Levent Koçkesen (Koç University)
Z θ∗
1
dθ1 = εθ2 + 2θ∗1
0
U2 (σ1 , S, θ∗2 ),
θ∗2 =
Bayesian Games
1, θ1 > θ∗1
0, θ1 ≤ θ∗1
We then have
Proof
Levent Koçkesen (Koç University)
(
i.e.,
1 − 3θ∗1
ε
Bayesian Games
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Proof (cont.)
As ε → 0, we have the following three pure strategy Bayesian equilibria:
Therefore, we must have
σ2 (B|θ2 ) =
(
1, θ2 <
0, θ2 >
θ∗2
θ∗2
σ1 (B|θ1 ) = σ2 (B|θ2 ) = 0, for all θ1 , θ2 ∈ [0, 1]
σ1 (B|θ1 ) =
U1 (B, σ2 , θ1 ) = (2 + εθ1 )
Z 1
θ∗2
Z θ∗
2
dθ2 + εθ1
0
dθ2 =
Z 1
θ∗2
θ∗1
Bayesian Games
1, θ1 >
0, θ1 ≤
1
3
1
3
σ2 (B|θ2 ) =
(
1, θ2 <
0, θ2 ≥
1
3
1
3
Harsanyi (1973)
“Accordingly, mixed-strategy equilibrium points are stable even though the
players may make no deliberate effort to use their pure strategies with the
probability weights prescribed by their mixed equilibrium strategies – because
the random fluctuations in their payoffs will make them use their pure strategies
approximately with the prescribed probabilities.”
1 − 3θ∗2
=
ε
1
and this completes the proof.
Solving for θ∗1 and θ∗2 we get θ∗1 = θ∗2 = 3+ε
(
Therefore, any equilibrium of the BoS game can be obtained as a limit of a
pure-strategy equilibrium in the perturbed game as the perturbation goes to
zero.
dθ2 = εθ1 + 2θ∗2
(1 − θ∗2 )
We must have U1 (B, σ2 , θ∗1 ) = U1 (S, σ2 , θ∗1 ), or
Levent Koçkesen (Koç University)
σ1 (B|θ1 ) = σ2 (B|θ2 ) = 1, for all θ1 , θ2 ∈ [0, 1]
2
3
We now have
U1 (S, σ2 , θ1 ) = 1 ×
1
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