[PPT] - Game Theory Extensive Form Games: Applications Levent Ko ckesen PowerPoint Presentation

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Game Theory

Extensive Form Games: Applications Levent Ko¸ ckesen

Ko¸ c University

Levent Ko¸ ckesen (Ko¸ c 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

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 2 / 23

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Ultimatum Bargaining

Two players, A and B, bargain

ver a cake of size 1

Player A makes an offer x ∈ [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

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 3 / 23

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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 Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 4 / 23

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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 strategic considerations

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 5 / 23

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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?

uA uB 1 1 dA dB

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 6 / 23

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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?

uA uB 1 1 (dA, dB) 45◦ 0.5 0.5

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 7 / 23

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Bargaining

How 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

r x = 0.45

So A gets 0.45 and B gets 0.55

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 8 / 23

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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 Solution
2. Non-cooperative: Alternating offers bargaining game

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 9 / 23

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Bargaining: Axiomatic Approach

John Nash (1950): The Bargaining Problem, Econometrica

1. 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

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 10 / 23

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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

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 11 / 23

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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 a share xi

◮ player i’s payoff is δt

ixi

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

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

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 12 / 23

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A B xA, 1 − xA B xA Y N A δA(1 − xB), δBxB A xB Y N B δ2

AxA, δ2 B(1 − xA)

A xA Y N Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 13 / 23

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Alternating Offers Bargaining

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

Similarly 1 − x∗

B = δAx∗ A

A B xA, 1 − xA B xA Y N A δA(1 − xB), δBxB A xB Y N B δ2

AxA, δ2 B(1 − xA)

A xA Y N

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 14 / 23

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Alternating Offers Bargaining

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

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 15 / 23

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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

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 16 / 23

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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 lim

δ→1 πi = lim δ→1 πB = 1

2

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 17 / 23

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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

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 18 / 23

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Nash Equilibrium of Cournot Duopoly

Best response correspondences: Q1 = a − c − bQ2 2b Q2 = a − c − bQ1 2b Nash equilibrium: (Qc

1, Qc 2) =

a − c 3b , a − c 3b

In equilibrium each firm’s profit is

πc

1 = πc 2 = (a − c)2

9b

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 19 / 23

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Cournot Best Response Functions

Q1 Q2

a−c b a−c b a b a b a−c 2b a−c 2b

b

a−c 3b a−c 3b

B1 B2

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 20 / 23

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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 2 u1, u2 Q1 Q2

ui(Q1, Q2) = (a − b(Q1 + Q2))Qi − cQi

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 21 / 23

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Backward Induction Equilibrium of Stackelberg Game

Sequential rationality of Firm 2 implies that for any Q1 it must play a best 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

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 22 / 23

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Backward Induction Equilibrium of Stackelberg Game

Backward Induction Equilibrium Qs

1 = a − c

2b Qs

2(Q1) = a − c − bQ1

2b Backward Induction Outcome Qs

1 = a − c

2b > a − c 3b = Qc

1

Qs

2 = a − c

4b < a − c 3b = Qc

2

Firm 1 commits to an aggressive strategy Equilibrium Profits πs

1 = (a − c)2

8b > (a − c)2 9b = πc

1

πs

2 = (a − c)2

16b < (a − c)2 9b = πc

2

There is first mover advantage

Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 23 / 23