Transcript Sequential Bargaining (Rubinstein Bargaining Model)  Two players divide a cake S  Each in his turn makes an offer, which the other accepts or.

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
1
Sequential Bargaining
(Rubinstein Bargaining Model)
1
denote these stages by 1/2
(x,y) ε S
t=1
2 Y
(x,y)
N
2
t=2
i.e. 1 makes an offer,
2 accepts or rejects
(x,y) ε S
1 Y (δx, δy)
N
1
etc.
(x,y) ε S
2
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
δ
δ
2
Strategies
histories:
2/1
1/2
H n =  h1 , h 2 , ..., h n 
H0 = 
h i  the offer m ade in period i
δ3 2/1
δ4 1/2
3
Sequential Bargaining
(Rubinstein Bargaining Model)
t=1
δ
1/2
Strategies
histories:
t=2
δ2 t = 3
2/1
1/2
H n =  h1 , h 2 , ..., h n 
H0 = 
h i  the offer m ade in period i

 s 1,n  H n - 1   S
s1 = 
s
H

Y
,
N





1,n
n

n odd
n even
4
Sequential Bargaining
(Rubinstein Bargaining Model)
δ
t=1
1/2
t=2
2/1
δ2 t = 3 1/2
payoffs
1 . a g reem en t o n  x , y  in p erio d t
yield s u tilities  δ
t -1
x, δ
t -1
y
2 . If th e p a rties n ever a g ree th ey g et (0 , 0 )
5
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
δ
Nash Equilibria
1 alw ays m akes an offer  x, y   S,
2/1
1 alw ays accepts offers  x.
2 alw ays m akes an offer  x, y   S,
δ2 1/2
δ3 2/1
δ
4
1/2
2 alw ays accepts offers  y.
T h is equ il. is n ot S u bgam e P erfect
e.g.
in t = 1 , player 2 sh ou ld accep t δ y < y
sin ce in t = 2 h e w ill g et y
6
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
Subgame Perfect Equilibria
δ
2/1
δ2 1/2
T h e su b - gam es begin n in g in periods t, t odd,
h ave th e sam e equ il ibria, sim larly for t even
δ3 2/1
δ4 1/2
7
Sequential Bargaining
(Rubinstein Bargaining Model)
Subgame Perfect Equilibria
1/2
L et
δ
2/1

(x, y) is th e payoff of a su bgam e perfect 
P =  (x, y)

equ ilibriu m of a gam e startin g w ith 1/2 

δ2 1/2
δ
3
2/1
δ4 1/2
M = m ax x
(x , y) P
M
odd
1
= m ax x
(x, y) P


 M = su p x 
(x, y) P


m
odd
1
= m in x
(x, y) P
8
Sequential Bargaining
(Rubinstein Bargaining Model)
Subgame Perfect Equilibria
1/2
δ
2/1
Let
δ2 1/2
S =
  x, y  | x +
y  1, x, y  0 
δ3 2/1
δ4 1/2
9
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
Subgame Perfect Equilibria
 1 - δ  1 - δM  , δ  1 - δM 
Can be supported as m
an equilibrium payoff
?
2/1
(δM , 1 - δM )
Can be supported as
an equilibrium payoff
1/2
M
= δ  1 - δM
odd
2
odd
1

= 1 - δ  1 - δM
even
1 - δM = m 2
δM = M
even
1
2 can ensure this payoff
by making this offer
M
M = M
odd
1
10

Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
Subgame Perfect Equilibria
 1 - δ  1 - δM  , δ  1 - δM 
m
M
2/1
odd
2
= δ  1 - δM
= 1 - δ  1 - δM
odd
1
(δM , 1 - δM )

even
1 - δM = m 2
2 will not agree to less
1 cannot take more
1 - δ  1 - δM
1/2
M
>
2
δ M
(1 > δ M )
M = M
odd
1
11

Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
Subgame Perfect Equilibria
 1 - δ  1 - δM  , δ  1 - δM 
M
M 2/1
= 1 -(δM
δ , 11 --δM
δM
)
M =
1-δ
1/2 1 -Mδ
2
=

odd
1
= 1 - δ  1 - δM
even
1 - δM = m 2
1
1+ δ
M = M
odd
1
12

Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
Subgame Perfect Equilibria
 1 - δ  1 - δm  , δ  1 - δm 
m
2/1
(δm , 1 - δm )
odd
1
= 1 - δ  1 - δm 
m = 1 - δ  1 - δm 
m =
1
using similar
1 +arguments
δ
1/2
m
m = m
odd
1
13
Sequential Bargaining
(Rubinstein Bargaining Model)
Subgame Perfect Equilibria
1/2
m = M =
2/1
1
1+ δ
= M
odd
1
= m
odd
1
Similarly the only possible
(SPE) payoff for 2 in 2/1 is
1
1+ δ
1/2
14
Sequential Bargaining
(Rubinstein Bargaining Model)
Subgame Perfect Equilibria
1/2
T he only candidate for S P E :
E ach player dem ands for him self, in his turn
2/1
E ach player agrees to offers 
1/2
1
1+ δ
δ
1+ δ
Check that it is a SPE !!
15
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2/1
Subgame Perfect Equilibria
Graphically
1/2
m in 1 
 m in 2
1/2
16
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
as   1,
Subgame Perfect Equilibria
δ 
 1
,


 1+ δ 1+ δ 
 21 , 21 
Let player i have the discount rate  i
2/1
1/2
Show that there is a unique SPE,
and that it’s payoff is:
 1 - δ 2 δ 2  1 - δ1  
,


1 - δ1 δ 2 
 1 - δ1 δ 2
17
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2
Bargaining with an Outside Option
(a,b)
a+b < 1
δ
δ2
2/1
1/2
2
(a,b)
δ3 2/1
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Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2
δ
δ2
 1 - m ax  b, δ  1 - δm  , m ax b, δ  1 - δm  
(a,b)
1/2
δ3 2/1
m a x  b , δ  1 - δ m 
 δm ,1 - δm 
2/1
2
Bargaining with
an Outside Option
m
(a,b)
m = 1 - m a x  b , δ  1 - δ m 
19
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2
δ
δ2
(a,b)
Bargaining with
an Outside Option
m = 1 - m a x  b , δ  1 - δ m 
2/1
assu m e: b > δ  1 - δ m 
1/2
th en : m = 1 - b
2
(a,b)
check: b > δ  1 - δ  1 - b   ???
δ3 2/1
b>
δ
1+ δ
20
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2
δ
δ2
(a,b)
Bargaining with
an Outside Option
m = 1 - m a x  b , δ  1 - δ m 
n o w assu m e: b < δ  1 - δ m 
2/1
th en : m = 1 - δ  1 - δ m 
1/2
2
δ3 2/1
(a,b)
m =
1
1+ δ
1 

check: b < δ  1 - δ
???

1+ 

δ
b<
1+ δ
21
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2
δ
δ2
(a,b)
Bargaining with
an Outside Option
m = 1 - m a x  b , δ  1 - δ m 
2/1
1/2
2
δ3 2/1
(a,b)
δ
 1
 1 + δ w hen b < 1 + δ

m = 

δ
 1 - b w hen b >
1+ δ

22
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2
δ
δ2
(a,b)
2/1
δ3 2/1
δ
 1
 1 + δ w h en b < 1 + δ

m = 

δ
 1 - b w h en b >
1+ δ

as   1,
1/2
2
Bargaining with
an Outside Option
(a,b)
 21 w hen b < 21

m = 
 1 - b w h en b >

1
2
23
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2
δ
δ2
(a,b)
2/1
Bargaining with
an Outside Option
 21 w hen b < 21

m = 
 1 - b w h en b >

1
2
1/2
2
δ3 2/1
(a,b)
 21

π2 = 1 - m = 
b

w hen
b<
1
2
w h en
b>
1
2
24
Sequential Bargaining
(Rubinstein Bargaining Model)
 21

π2 = 
b

1/2
2
δ
δ2
(a,b)
2/1
w hen
b<
1
2
w h en
b>
1
2
1
1/2
2
Bargaining with
an Outside Option
(a,b)
1/2
δ3 2/1
1/2
1
b
25
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2
Bargaining with
an Outside Option
Compare this with the Nash
Bargaining Solution of
(a,b)
Δ ,  0, b 
δ
δ2
2/1
disagreement pt.
π2 =
1/2
2
(a,b)
1+ b
2
(1+b)/2
δ3 2/1
b
(1-b)/2
26
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2
δ
δ2
π2 =
(a,b)
Bargaining with
an Outside Option
1+ b
2
2/1
1
1/2
2
Outside Option
(a,b)
1/2
Nash Bargaining Solution
δ3 2/1
1/2
1
b
27
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2
2
δ
δ
2
(a,b)
2/1
Outside Option
1
1/2
Nash Bargaining Solution
1/2
2
δ3 2/1
(a,b)
Bargaining with
an Outside Option
1/2
1
b
• The Nash Bargaining solution
increases with b
• The Outside Option equilibrium
remains constant for small b
28
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2 p
0
(a,b)
Bargaining with random
breakdown of negotiations
1-p
2/1 p
0
(a,b)
1-p
1/2 p
0
(a,b)
1-p
after an offer is rejected,
Nature breaks down the
negotiations with probability p
negotiations continue with
probability 1-p
2/1 p
0
(a,b)
1-p
No need to have a discount rate !!29
Sequential Bargaining
(Rubinstein Bargaining Model)
Bargaining with random
breakdown of negotiations
1/2 p
1 - pb -  1 - p   1 - pa -  1 - p  m  ,
0
(a,b)
(
pb +  1 - p   1 - pa -  1 - p  m 
1-p
2/1 p
0
(a,b)
 pa +  1 - p  m , 1 -
pa -  1 - p  m
)

1-p
1/2 p
0
(a,b)
1-p
2/1 p
0
(a,b)
m
m = 1 - pb -  1 - p   1 - pa -  1 - p  m 
1-p
30
Sequential Bargaining
(Rubinstein Bargaining Model)
Bargaining with random
breakdown of negotiations
1/2 p
0
(a,b) m = 1 - pb -  1 - p   1 - pa -  1 - p  m 


1-p
2/1 p
0
(a,b)
1-p
1/2 p
0
(a,b)
1-p
2/1 p
0
(a,b)
1-p
2

m 1 -  1 - p   = p - pb + p  1 - p  a


m  2 - p = 1 - b + 1 - p  a
m =
1 - b + 1 - p  a
2- p

p 0
1-b+ a
2
31
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2 p
0
(a,b)
1-p
2/1 p
0
(a,b)
1-p
1/2 p
0
(a,b)
m =
Bargaining with random
breakdown of negotiations
1 - b + 1 - p  a

1-b+ a
p 0
2- p
2
The payoff of player 2 :
π2 = 1 - m =
1+ b -a
2
1-p
2/1 p
0
(a,b)
1-p
32
Sequential Bargaining
(Rubinstein Bargaining Model)
Bargaining with random
breakdown of negotiations
1/2 p
0
(a,b)
1+ b -a
π2 =
1-p
This coincides with the
Nash Bargaining Solution of
2/1 p
0
(a,b)
1-p
Δ ,  a, b 
1/2 p
0
(a,b)
1-p
2/1 p
0
(a,b)
1-p
2
 1-b+ a 1+ b -a 
,


2
2


b
a
33
Sequential Bargaining
(Rubinstein Bargaining Model)
1/2 p
0
(a,b)
π2 =
1-p
1-p
2
Δ ,  a, b 
1/2 p
0
(a,b)
1-p
1-p
1+ b -a
This coincides with the
Nash Bargaining Solution of
2/1 p
0
(a,b)
2/1 p
0
(a,b)
Bargaining with random
breakdown of negotiations
 1-b+ a 1+ b -a 
,


2
2


Topics
4
b
a
34