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Sequential Bargaining (Rubinstein Bargaining Model)

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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

(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.

histories:

2/1

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

Strategies

δ

δ2

δ3

δ4

histories:

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

Strategies

t = 1

δ

t = 2

δ2

t = 3

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

payoffs

t = 1

δ

t = 2

δ2

t = 3

2/1

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

Nash Equilibria

δ

δ2

δ3

δ4

2/1

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

1/2

Subgame Perfect Equilibria

δ

δ2

δ3

δ4

2/1

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

δ

δ2

δ3

δ4

2/1

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

δ

δ2

δ3

δ4

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

2/1

1/2

2 will not agree to less

1 cannot take more

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

using similar arguments

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

2/1

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

1/2

Check that it is a SPE !!

2/1

1/2

1/2

Sequential Bargaining

(Rubinstein Bargaining Model)

Subgame Perfect Equilibria

Graphically

1/2

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:

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

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

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

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

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

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

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

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

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

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

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

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 !!

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

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

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 :

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

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