Sequential Bargaining
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Sequential Bargaining (Rubinstein Bargaining Model). Two players divide a cake S Each in his turn makes an offer, which the other accepts or rejects. The game ends when someone accepts The players alternate in making offers There is a discount rate of δ. Y. N. t = 2. 1.

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Sequential Bargaining (Rubinstein Bargaining Model)

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Sequential bargaining rubinstein bargaining model

Sequential Bargaining

(Rubinstein Bargaining Model)

  • Two players divide a cake S

  • Each in his turn makes an offer, which the other accepts or rejects.

  • The game ends when someone accepts

  • The players alternate in making offers

  • There is a discount rate of δ


Sequential bargaining rubinstein bargaining model

Y

N

t = 2

1

(x,y) ε S

Sequential Bargaining

(Rubinstein Bargaining Model)

1

denote these stages by 1/2

t = 1

(x,y) ε S

2

Y

i.e. 1 makes an offer,

2 accepts or rejects

(x,y)

N

2

(x,y) ε S

1

(δx, δy)

etc.


Sequential bargaining rubinstein bargaining model

histories:

2/1

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

Strategies

δ

δ2

δ3

δ4


Sequential bargaining rubinstein bargaining model

histories:

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

Strategies

t = 1

δ

t = 2

δ2

t = 3


Sequential bargaining rubinstein bargaining model

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

payoffs

t = 1

δ

t = 2

δ2

t = 3


Sequential bargaining rubinstein bargaining model

2/1

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

Nash Equilibria

δ

δ2

δ3

δ4


Sequential bargaining rubinstein bargaining model

2/1

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

Subgame Perfect Equilibria

δ

δ2

δ3

δ4


Sequential bargaining rubinstein bargaining model

2/1

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

δ

δ2

δ3

δ4


Sequential bargaining rubinstein bargaining model

2/1

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

δ

δ2

δ3

δ4


Sequential bargaining rubinstein bargaining model

2/1

1/2

2 can ensure this payoff

by making this offer

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

?

Can be supported as

an equilibrium payoff

Can be supported as

an equilibrium payoff


Sequential bargaining rubinstein bargaining model

2/1

1/2

2 will not agree to less

1 cannot take more

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2


Sequential bargaining rubinstein bargaining model

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2


Sequential bargaining rubinstein bargaining model

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

using similar arguments


Sequential bargaining rubinstein bargaining model

2/1

1/2

Similarly the only possible

(SPE) payoff for 2 in 2/1 is

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2


Sequential bargaining rubinstein bargaining model

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

Check that it is a SPE !!


Sequential bargaining rubinstein bargaining model

2/1

1/2

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

Graphically

1/2


Sequential bargaining rubinstein bargaining model

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

Show that there is a unique SPE, and that it’s payoff is:


Sequential bargaining rubinstein bargaining model

2/1

(a,b)

(a,b)

1/2

2

2

2/1

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

Bargaining with an Outside Option

a+b < 1

δ

δ2

δ3


Sequential bargaining rubinstein bargaining model

2/1

(a,b)

(a,b)

1/2

2

2

2/1

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with

an Outside Option

1/2

δ

δ2

δ3


Sequential bargaining rubinstein bargaining model

2/1

(a,b)

(a,b)

1/2

2

2

2/1

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with

an Outside Option

1/2

δ

δ2

δ3


Sequential bargaining rubinstein bargaining model

2/1

(a,b)

(a,b)

1/2

2

2

2/1

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with

an Outside Option

1/2

δ

δ2

δ3


Sequential bargaining rubinstein bargaining model

2/1

(a,b)

(a,b)

1/2

2

2

2/1

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with

an Outside Option

1/2

δ

δ2

δ3


Sequential bargaining rubinstein bargaining model

2/1

(a,b)

(a,b)

1/2

2

2

2/1

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with

an Outside Option

1/2

δ

δ2

δ3


Sequential bargaining rubinstein bargaining model

2/1

(a,b)

(a,b)

1/2

2

2

2/1

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with

an Outside Option

1/2

δ

δ2

δ3


Sequential bargaining rubinstein bargaining model

2/1

(a,b)

(a,b)

1/2

2

2

2/1

1/2

b

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with

an Outside Option

1/2

δ

1

δ2

δ3

1

1/2


Sequential bargaining rubinstein bargaining model

Compare this with the Nash

Bargaining Solution of

2/1

disagreement pt.

(a,b)

(a,b)

1/2

2

2

2/1

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with

an Outside Option

1/2

δ

δ2

(1+b)/2

δ3

b

(1-b)/2


Sequential bargaining rubinstein bargaining model

2/1

(a,b)

(a,b)

1/2

2

2

Outside Option

1

2/1

1/2

b

1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with

an Outside Option

1/2

δ

δ2

Nash Bargaining Solution

δ3


Sequential bargaining rubinstein bargaining model

2/1

(a,b)

(a,b)

1/2

2

2

Outside Option

1

2/1

1/2

b

1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with

an Outside Option

1/2

So where is the disagreement point ??

δ

Nash Bargaining Solution

δ2

  • The Nash Bargaining solution

  • increases with b

  • The Outside Option equilibrium

  • remains constant for small b

δ3


Sequential bargaining rubinstein bargaining model

p

p

p

p

1-p

2/1

1-p

(a,b)

(a,b)

(a,b)

(a,b)

1/2

0

0

0

0

2/1

1-p

1-p

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with random

breakdown of negotiations

1/2

after an offer is rejected,

Nature breaks down the

negotiations with probability p

negotiations continue with

probability 1-p

No need to have a discount rate !!


Sequential bargaining rubinstein bargaining model

p

p

p

p

1-p

2/1

1-p

(a,b)

(a,b)

(a,b)

(a,b)

1/2

0

0

0

0

2/1

1-p

1-p

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with random

breakdown of negotiations

1/2


Sequential bargaining rubinstein bargaining model

p

p

p

p

1-p

2/1

1-p

(a,b)

(a,b)

(a,b)

(a,b)

1/2

0

0

0

0

2/1

1-p

1-p

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with random

breakdown of negotiations

1/2


Sequential bargaining rubinstein bargaining model

p

p

p

p

1-p

2/1

1-p

(a,b)

(a,b)

(a,b)

(a,b)

1/2

0

0

0

0

2/1

1-p

1-p

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with random

breakdown of negotiations

1/2

The payoff of player 2 :


Sequential bargaining rubinstein bargaining model

p

p

p

p

1-p

2/1

This coincides with the

Nash Bargaining Solution of

1-p

(a,b)

(a,b)

(a,b)

(a,b)

1/2

0

0

0

0

2/1

1-p

b

1-p

a

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with random

breakdown of negotiations

1/2


Sequential bargaining rubinstein bargaining model

p

p

p

p

1-p

2/1

This coincides with the

Nash Bargaining Solution of

1-p

(a,b)

(a,b)

(a,b)

(a,b)

1/2

0

0

0

0

2/1

1-p

b

1-p

a

Sequential Bargaining

(Rubinstein Bargaining Model)

Bargaining with random

breakdown of negotiations

1/2

END


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