Bargaining & Markets
Strategic Bargaining
Steady State Market
before: Buyers and Sellers, δtp,
δt(1-p).
 Matching: Seller meets a buyer with
probability α. A buyer meets a seller
with probability β.
 Bargaining: Long term bargainng with
breakdown of negotiations.
 As
1
Strategic Bargaining
Steady State Market
Bargaining & Markets
0
One period in the
life of a matched pair
1/2
1/2
S/B
B/S
0
(1-α)(1-β)
S,B continue
bargaining
α(1- β)
S - newly matched
B -unmatched
β(1-α) αβ
S,B have
new partners
B - newly matched
S -unmatched
2
Bargaining & Markets
Strategic Bargaining
Steady State Market
 VS,
(VB) The expected utility of an
unmatched seller (buyer)
 WS, (WB) The Expected utility of a
matched seller (buyer)
VS = δ αW S +  1 - α VS 
VB = δ  βWB +  1 - β VB 
3
Strategic Bargaining
Steady State Market
Bargaining & Markets
Breakdown of negotiations occurs
with probability q = 1 -  1 - α  1 - β 
The seller’s expected payoff in
the case of breakdown:
US =
UB =
δ αW S + β  1 - α VS 
q
δ  βWB + α  1 - β VB 
q
4
Strategic Bargaining
Steady State Market
Bargaining & Markets
The bargaining between a buyer and a seller is a
sequential game with breakdown:
A
0
1/2
S/B
1/2
B/S
a period
(uS ,uB )
q
q
0
1-q
A
0
(uS ,uB )
1-q
A
5
Bargaining & Markets
Strategic Bargaining
Steady State Market
We seek an equilibrium in which all sellers use the
same strategy, and all buyers use the same
strategy.
We seek an equilibrium in semi-stationary
strategies: Strategies that may depend on the
history of the bargaining within a match but is
independent of who the partner is.
6
Strategic Bargaining
Steady State Market
Bargaining & Markets
x*,  y*  the equilibrium payoff of S at S/B (B/S)
 x* +y*  /2
0
1/2
x*
(uS ,uB )
S/B
1/2
B/S
q
y*
q
0
1-q
A
0
=
quS +  1 - q  δ
x* +y*
2
(uS ,uB )
1-q
A
δ  x* +y*  /2
7
Strategic Bargaining
Steady State Market
Bargaining & Markets
 x* +y* 
x* = 1 - quB   1 - q  δ  1 
2 

x*
(uS ,uB )
 x* +y* 
quB   1 - q  δ  1 
2 

 x* +y* 
δ1 
2


 x* +y*  /2
0
1/2
S/B
1/2
B/S
q
y*
q
0
1-q
0
(uS ,uB )
1-q
A
B
A
δ  x* +y*  /2
8
Bargaining & Markets
Strategic Bargaining
Steady State Market
Alternative calculation
1 - quB -  1 - q  δ  1 - m  + quS +  1 - q  δm 
2
0
1/2
1 - quB   1 - q  δ  1 - m 
(uS ,uB )
quB   1 - q  δ  1 - m 
1- m
S/B
1/2
quS +  1 - q  δm
B/S
q
q
0
1-q
0
(uS ,uB )
1-q
A
B
A
m
9
Strategic Bargaining
Steady State Market
Bargaining & Markets
x* +y*
y* = quS +  1 - q  δ
2
 x* +y* 
1 - x* = quB +  1 - q  δ  1 
2 

VB = δ  βWB +  1 - β VB 
VS = δ αWS +  1 - α VS 
2
B
W = 1x* +y*
2
S
W =
x* +y*
uS =
uB =
δ αW S + β  1 - α VS 
q
δ  βWB + α  1 - β VB 
q
8 equations in 8 variables
x*, y*,VS ,VB ,WS ,WB ,uS ,uB
10
Bargaining & Markets
Strategic Bargaining
Steady State Market
The solution:
x* =
y* =
2  1 - δ  + δα - δ  1 - δ  1 - α  1 - β 
2  1 - δ  + δα + δβ
δα + δ  1 - δ  1 - α  1 - β 
2  1 - δ  + δα + δβ
A seller always demands x* and accepts y* or more.
A buyer always offers y* and accepts 1-x* or more
It can be shown that when all others use these strategies
then this strategy is the best a player can do.
11
Bargaining & Markets
x* =
y* =
Strategic Bargaining
Steady State Market
2  1 - δ  + δα - δ  1 - δ  1 - α  1 - β 
2  1 - δ  + δα + δβ
δα + δ  1 - δ  1 - α  1 - β 
2  1 - δ  + δα + δβ
as δ  1 :
α
x* 
1
α+ β
α
1
y* 
α+ β
12
Bargaining & Markets
Strategic Bargaining
One time entry
and Sellers, p, (1-p). δ=1
 Matching: Each seller meets a buyer.
A buyer meets a seller with probability
S/B. Matching is independent across
periods
 Bargaining: Rejection dissolves the
match.
 Buyers
13
Bargaining & Markets
Strategic Bargaining
One time entry
 Information: Agents
know the time and
they recognize all other agents present.
They have no memory of past events.
(imperfect Recall)
A situation is characterized by :  t, A, j  or  t, A, j, p 
t - time
A - set of agents present
j - identity of the partner
p - the offer just made by the partner
14
Bargaining & Markets
Strategic Bargaining
One time entry
There exists an equilibrium in which
every agent proposes the price 1. A
buyer accepts any price and a seller
accepts price which is at least 1.
prove !!!
15
Bargaining & Markets
Strategic Bargaining
One time entry
There exists an equilibrium in which
every agent proposes the price 1. A
buyer accepts any price and a seller
accepts price which is at least 1.
Although this is not the only equilibrium,
all other equilibria lead to the objects
prove
!!!
being sold at p = 1.
16
Bargaining & Markets
Strategic Bargaining
One time entry
Proof by induction on |S|
Let | S |= 1
B
S
Let Vi (t), V (t) be the expected values
of buyer i and the seller, in some equilibrium.
S
Let m be inf V (t) over
all equilibria and all t.
17
Bargaining & Markets
B
Then :
V
i
B
Strategic Bargaining
One time entry
(t)  1 - m ,
i=1
and for each t there is some buyer
with Vi (t + 1) 
B
1-m
B
The seller is guaranteed to meet this buyer
Why ??? 18
Strategic Bargaining
One time entry
Bargaining & Markets
so if he demands the price
1 - 1-m
B - ε for some 
he will get it.
i.e. m  1 -
1-m
B
-ε
m  1-
B
B-1
hence m  1.
Now assume the proposition
holds for < | S |
19
Bargaining & Markets
S
j
Strategic Bargaining
One time entry
B
Let V (t), Vi (t) be the expected payoffs when
all S and all B players are in the market
S
j
Let m be the inf V (t)
over all equilibria, all t and all j.
,
B
Then :
V
i
B
(t)   1 - m  | S |,
i=1
20
Bargaining & Markets
Strategic Bargaining
One time entry
Hence for any t there a buyer i s.t :
Vi (t + 1)   1 - m  | S | /B
B
if a seller demands the price
1 - (1 - m)
|S|
B
- ε for some 
for as long as all S , B players are in
he will either get it or the market becomes smaller.
21
Strategic Bargaining
One time entry
Bargaining & Markets
hence m  1 - (1 - m)
|S|
B
-ε
or m  1.
In any equilibrium, the sellers obtain the price 1,
as in a competitive equilibrium
22
Bargaining & Markets
Strategic Bargaining
One time entry
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