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Game Theory
Extensive Form Games: Applications
Levent Koçkesen
Koç University
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
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A Simple Game
You have 10 TL to share
A makes an offer
◮
x for me and 10 − x for you
If B accepts
◮
A’s offer is implemented
If B rejects
◮
Both get zero
Half the class will play A (proposer) and half B (responder)
◮
◮
Proposers should write how much they offer to give responders
I will distribute them randomly to responders
⋆
They should write Yes or No
Click here for the EXCEL file
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
2 / 23
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Ultimatum Bargaining
A
Two players, A and B, bargain
over a cake of size 1
x
Player A makes an offer
x ∈ [0, 1] to player B
If player B accepts the offer
(Y ), agreement is reached
◮
◮
B
Y
A receives x
B receives 1 − x
If player B rejects the offer (N )
both receive zero
Levent Koçkesen (Koç University)
x, 1 − x
Extensive Form Games: Applications
N
0, 0
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Subgame Perfect Equilibrium of Ultimatum Bargaining
We can use backward induction
B’s optimal action
◮
◮
x < 1 → accept
x = 1 → accept or reject
1. Suppose in equilibrium B accepts any offer x ∈ [0, 1]
◮
◮
What is the optimal offer by A? x = 1
The following is a SPE
x∗ = 1
s∗B (x) = Y for all x ∈ [0, 1]
2. Now suppose that B accepts if and only if x < 1
◮
What is A’s optimal offer?
⋆
⋆
x = 1?
x < 1?
Unique SPE
x∗ = 1, s∗B (x) = Y for all x ∈ [0, 1]
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
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page.5
Bargaining
Bargaining outcomes depend on many factors
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Social, historical, political, psychological, etc.
Early economists thought the outcome to be indeterminate
John Nash introduced a brilliant alternative approach
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Axiomatic approach: A solution to a bargaining problem must satisfy
certain “reasonable” conditions
⋆
◮
◮
These are the axioms
How would such a solution look like?
This approach is also known as cooperative game theory
Later non-cooperative game theory helped us identify critical strategic
considerations
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
5 / 23
page.6
Bargaining
Two individuals, A and B, are trying to share a cake of size 1
If A gets x and B gets y,utilities are uA (x) and uB (y)
If they do not agree, A gets utility dA and B gets dB
What is the most likely outcome?
uB
1
dB
dA
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
1
uA
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Bargaining
Let’s simplify the problem
uA (x) = x, and uB (x) = x
dA = dB = 0
A and B are the same in every other respect
What is the most likely outcome?
uB
1
45◦
0.5
(dA , dB )
Levent Koçkesen (Koç University)
0.5
Extensive Form Games: Applications
1
uA
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page.8
Bargaining
How about now? dA = 0.3, dB = 0.4
uB
1
45◦
0.55
0.4
0.3 0.45
1
uA
Let x be A’s share. Then
Slope = 1 =
1 − x − 0.4
x − 0.3
or x = 0.45
So A gets 0.45 and B gets 0.55
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
8 / 23
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Bargaining
In general A gets
1
dA + (1 − dA − dB )
2
B gets
1
dB + (1 − dA − dB )
2
But why is this reasonable?
Two answers:
1. Axiomatic: Nash Bargaining Solution
2. Non-cooperative: Alternating offers bargaining game
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
9 / 23
page.10
Bargaining: Axiomatic Approach
John Nash (1950): The Bargaining Problem, Econometrica
1. Efficiency
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No waste
2. Symmetry
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If bargaining problem is symmetric, shares must be equal
3. Scale Invariance
⋆
Outcome is invariant to linear changes in the payoff scale
4. Independence of Irrelevant Alternatives
⋆
If you remove alternatives that would not have been chosen, the
solution does not change
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
10 / 23
page.11
Nash Bargaining Solution
What if parties have different bargaining powers?
Remove symmetry axiom
Then A gets
xA = dA + α(1 − dA − dB )
B gets
xB = dB + β(1 − dA − dB )
α, β > 0 and α + β = 1 represent bargaining powers
If dA = dB = 0
xA = α
Levent Koçkesen (Koç University)
and
xB = β
Extensive Form Games: Applications
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page.12
Alternating Offers Bargaining
Two players, A and B, bargain over a cake of size 1
At time 0, A makes an offer xA ∈ [0, 1] to B
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◮
If B accepts, A receives xA and B receives 1 − xA
If B rejects, then
at time 1, B makes a counteroffer xB ∈ [0, 1]
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◮
If A accepts, B receives xB and A receives 1 − xB
If A rejects, he makes another offer at time 2
This process continues indefinitely until a player accepts an offer
If agreement is reached at time t on a partition that gives player i a
share xi
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◮
player i’s payoff is δit xi
δi ∈ (0, 1) is player i’s discount factor
If players never reach an agreement, then each player’s payoff is zero
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
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page.13
A
xA
B
Y
N
B
xA , 1 − xA
xB
A
Y
N
A
δA (1 − xB ), δB xB
xA
B
Y
N
A
2 x , δ 2 (1 − x )
δA
A B
A
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
13 / 23
page.14
Alternating Offers Bargaining
Stationary No-delay Equilibrium
A
1. No Delay: All equilibrium offers are accepted
xA
B
2. Stationarity: Equilibrium offers do not depend on time
Y
Let equilibrium offers be (x∗A , x∗B )
What does B expect to get if she rejects x∗A ?
◮
N
B
xA , 1 − xA
xB
δB x∗B
A
Therefore, we must have
Y
N
A
1 − x∗A = δB x∗B
δA (1 − xB ), δB xB
xA
B
Similarly
1 − x∗B = δA x∗A
Y
N
A
2 x , δ 2 (1 − x )
δA
A B
A
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Extensive Form Games: Applications
14 / 23
page.15
Alternating Offers Bargaining
There is a unique solution
1 − δB
1 − δA δB
1 − δA
x∗B =
1 − δA δB
x∗A =
There is at most one stationary no-delay SPE
Still have to verify there exists such an equilibrium
The following strategy profile is a SPE
Player A: Always offer x∗A , accept any xB with 1 − xB ≥ δA x∗A
Player B: Always offer x∗B , accept any xA with 1 − xA ≥ δB x∗B
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
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page.16
Properties of the Equilibrium
Bargaining Power
Player A’s share
πA = x∗A =
1 − δB
1 − δA δB
Player B’s share
πB = 1 − x∗A =
δB (1 − δA )
1 − δA δB
Share of player i is increasing in δi and decreasing in δj
Bargaining power comes from patience
Example
δA = 0.9, δB = 0.95 ⇒ πA = 0.35, πB = 0.65
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
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page.17
Properties of the Equilibrium
First mover advantage
If players are equally patient: δA = δB = δ
πA =
δ
1
>
= πB
1+δ
1+δ
First mover advantage disappears as δ → 1
lim πi = lim πB =
δ→1
Levent Koçkesen (Koç University)
δ→1
1
2
Extensive Form Games: Applications
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page.18
Capacity Commitment: Stackelberg Duopoly
Remember Cournot Duopoly model?
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Two firms simultaneously choose output (or capacity) levels
What happens if one of them moves first?
⋆
or can commit to a capacity level?
The resulting model is known as Stackelberg oligopoly
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After the German economist Heinrich von Stackelberg in Marktform
und Gleichgewicht (1934)
The model is the same except that, now, Firm 1 moves first
Profit function of each firm is given by
ui (Q1 , Q2 ) = (a − b(Q1 + Q2 ))Qi − cQi
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
18 / 23
page.19
Nash Equilibrium of Cournot Duopoly
Best response correspondences:
Q1 =
a − c − bQ2
2b
Q2 =
a − c − bQ1
2b
Nash equilibrium:
(Qc1 , Qc2 )
=
a−c a−c
,
3b
3b
In equilibrium each firm’s profit is
π1c = π2c =
Levent Koçkesen (Koç University)
(a − c)2
9b
Extensive Form Games: Applications
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page.20
Cournot Best Response Functions
Q2
a
b
a−c
b
a−c
2b
a−c
3b
B1
b
B2
a−c a−c
3b 2b
Levent Koçkesen (Koç University)
a−c
b
Extensive Form Games: Applications
a
b
Q1
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Stackelberg Model
The game has two stages:
1. Firm 1 chooses a capacity level Q1 ≥ 0
2. Firm 2 observes Firm 1’s choice and chooses a capacity Q2 ≥ 0
1
Q1
2
Q2
u1 , u2
ui (Q1 , Q2 ) = (a − b(Q1 + Q2 ))Qi − cQi
Levent Koçkesen (Koç University)
Extensive Form Games: Applications
21 / 23
page.22
Backward Induction Equilibrium of Stackelberg Game
Sequential rationality of Firm 2 implies that for any Q1 it must play a
best response:
a − c − bQ1
Q2 (Q1 ) =
2b
Firms 1’s problem is to choose Q1 to maximize:
[a − b(Q1 + Q2 (Q1 ))]Q1 − cQ1
given that Firm 2 will best respond.
Therefore, Firm 1 will choose Q1 to maximize
[a − b(Q1 +
a − c − bQ1
)]Q1 − cQ1
2b
This is solved as
Q1 =
Levent Koçkesen (Koç University)
a−c
2b
Extensive Form Games: Applications
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page.23
Backward Induction Equilibrium of Stackelberg Game
Backward Induction Equilibrium
a−c
2b
a
− c − bQ1
Qs2 (Q1 ) =
2b
Qs1 =
Backward Induction Outcome
a−c
a−c
>
= Qc1
2b
3b
a−c
a−c
Qs2 =
<
= Qc2
4b
3b
Equilibrium Profits
Qs1 =
(a − c)2
(a − c)2
>
= π1c
8b
9b
(a − c)2
(a − c)2
<
= π2c
π2s =
16b
9b
π1s =
Levent Koçkesen (Koç University)
Firm 1 commits to an aggressive
strategy
There is first mover advantage
Extensive Form Games: Applications
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