Game Theory - Extensive Form Games: Applicationshome.ku.edu.tr/.../8...Games-Applications-Handout.pdf · page.12 Alternating Oﬀers Bargaining Two players, A and B, bargain over

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Game Theory

Extensive Form Games: Applications

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Extensive Form Games: Applications 1 / 23

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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


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Ultimatum Bargaining

Two players, A and B, bargainover a cake of size 1

Player A makes an offerx ∈ [0, 1] to player B

If player B accepts the offer(Y ), agreement is reached

◮ A receives x◮ B receives 1− x

If player B rejects the offer (N)both receive zero

A

B

x, 1− x 0, 0

x

Y N


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Subgame Perfect Equilibrium of Ultimatum BargainingWe 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]


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Bargaining

Bargaining outcomes depend on many factors◮ Social, historical, political, psychological, etc.

Early economists thought the outcome to be indeterminate

John Nash introduced a brilliant alternative approach◮ 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 strategicconsiderations


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Bargaining

Two individuals, A and B, are trying to share a cake of size 1If 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 dBWhat is the most likely outcome?

uA

uB

1

1

dA

dB


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BargainingLet’s simplify the problem

uA(x) = x, and uB(x) = x

dA = dB = 0A and B are the same in every other respectWhat is the most likely outcome?

uA

uB

1

1

(dA, dB)

45◦

0.5

0.5


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BargainingHow about now? dA = 0.3, dB = 0.4

uA

uB

1

1

0.3

0.4

45◦

0.45

0.55

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


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Bargaining

In general A gets

dA +1

2(1− dA − dB)

B gets

dB +1

2(1− dA − dB)

But why is this reasonable?

Two answers:

1. Axiomatic: Nash Bargaining Solution2. Non-cooperative: Alternating offers bargaining game


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Bargaining: Axiomatic Approach

John Nash (1950): The Bargaining Problem, Econometrica1. Efficiency

⋆ No waste

2. Symmetry⋆ 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


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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 and xB = β


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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◮ 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]◮ 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 ashare xi

◮ player i’s payoff is δtixi

◮ δi ∈ (0, 1) is player i’s discount factor

If players never reach an agreement, then each player’s payoff is zero


page.13 A

B

xA, 1− xA

B

xA

Y N

A

δA(1− xB), δBxB

A

xB

Y N

B

δ2AxA, δ2

B(1− xA)

A

xA

Y N


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Stationary No-delay Equilibrium

1. No Delay: All equilibrium offers are accepted

2. Stationarity: Equilibrium offers do not depend on time

Let equilibrium offers be (x∗A, x∗

B)

What does B expect to get if she rejects x∗A?◮ δBx

∗

B

Therefore, we must have

1− x∗A = δBx∗

B

Similarly1− x∗B = δAx

∗

A

A

B

xA, 1− xA

B

xA

Y N

A

δA(1 − xB), δBxB

A

xB

Y N

B

δ2AxA, δ2

B(1− xA)

A

xA

Y N


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There is a unique solution

x∗A =1− δB

1− δAδB

x∗B =1− δA

1− δAδB

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 ≥ δAx∗

A

Player B: Always offer x∗B, accept any xA with 1− xA ≥ δBx∗

B


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Properties of the Equilibrium

Bargaining Power

Player A’s share

πA = x∗A =1− δB

1− δAδB

Player B’s share

πB = 1− x∗A =δB(1− δA)

1− δAδB

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


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Properties of the Equilibrium

First mover advantage

If players are equally patient: δA = δB = δ

πA =1

1 + δ>

δ

1 + δ= πB

First mover advantage disappears as δ → 1

limδ→1

πi = limδ→1

πB =1

2


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Capacity Commitment: Stackelberg Duopoly

Remember Cournot Duopoly model?◮ 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◮ 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


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Nash Equilibrium of Cournot Duopoly

Best response correspondences:

Q1 =a− c− bQ2

2b

Q2 =a− c− bQ1

2b

Nash equilibrium:

(Qc1, Q

c2) =

(

a− c

3b,a− c

3b

)

In equilibrium each firm’s profit is

πc1 = πc

2 =(a− c)2

9b


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Cournot Best Response Functions

Q1

Q2

a−cb

a−cb

ab

ab

a−c2b

a−c2b

b

a−c3b

a−c3b

B1

B2


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Stackelberg ModelThe 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

2

u1, u2

Q1

Q2

ui(Q1, Q2) = (a− b(Q1 +Q2))Qi − cQi


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Backward Induction Equilibrium of Stackelberg Game

Sequential rationality of Firm 2 implies that for any Q1 it must play abest response:

Q2(Q1) =a− c− bQ1

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

2b)]Q1 − cQ1

This is solved as

Q1 =a− c

2b


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Backward Induction Equilibrium of Stackelberg GameBackward Induction Equilibrium

Qs1 =

a− c

2b

Qs2(Q1) =

a− c− bQ1

2b

Backward Induction Outcome

Qs1 =

a− c

2b>

a− c

3b= Qc

1

Qs2 =

a− c

4b<

a− c

3b= Qc

2

Firm 1 commits to an aggressivestrategy

Equilibrium Profits

πs1 =

(a− c)2

8b>

(a− c)2

9b= πc

1

πs2 =

(a− c)2

16b<

(a− c)2

9b= πc

2

There is first mover advantage


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Game Theory - Extensive Form Games: Applicationshome.ku.edu.tr/.../8...Games-Applications-Handout.pdf · page.12 Alternating Oﬀers Bargaining Two players, A and B, bargain over