1 Bargaining & Markets u As before: Buyers and Sellers, δtp,δtp, δ t (1-p). u Matching: Seller...

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Bargaining & Markets

As 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.

StrategicStrategic BargainingBargainingSteady State MarketSteady State Market

StrategicStrategic BargainingBargainingSteady State MarketSteady State Market

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Bargaining & Markets Strategic BargainingStrategic BargainingSteady State MarketSteady State Market

01/21/2

B/SS/B

0

αβ(1-α)(1-β)β(1-α)

α(1- β) S,B have new partners

B - newly matched S -unmatched

S - newly matched B -unmatched

S,B continue bargaining

One period in the life of a matched pair

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VS, (VB) The expected utility of an

unmatched seller (buyer) WS, (WB) The Expected utility of a

matched seller (buyer)

Strategic BargainingStrategic BargainingSteady State MarketSteady State Market

S S SV = δ αW + 1 - α V

B B BV = δ βW + 1 - β V

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Breakdown of negotiations occurs with probability

Strategic BargainingStrategic BargainingSteady State MarketSteady State Market

q = 1 - 1 - α 1 - β

S SS

δ αW + β 1 - α VU =

q

The seller’s expected payoff in the case of breakdown:

B BB

δ βW + α 1 - β VU =

q

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The bargaining between a buyer and a seller is a sequential game with breakdown:

01/21/2

B/SS/B

q0

1-q

S B(u ,u )q

0S B(u ,u )

1-q

A A

A

a period

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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 are independent of who the partner is.

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01/21/2

B/SS/B

q0

1-q

S B(u ,u )q

0S B(u ,u )

1-q

A A

y*x*

x* +y* /2

δ x* +y* /2

S

x* +y*qu + 1 - q δ

2=

x*, y* the equilibrium payoff of S at S/B (B/S)

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01/21/2

B/SS/B

q0

1-q

S B(u ,u )q

0S B(u ,u )

1-q

A A

y*x*

x* +y* /2

δ x* +y* /2

B

x* +y*x* = 1 - qu 1 - q δ 1 -

2

x* +y*δ 1 -

2

B

x* +y*qu 1 - q δ 1 -

2

B

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01/21/2

B/SS/B

q0

1-q

S B(u ,u )q

0S B(u ,u )

1-q

A A m

Squ + 1 - q δm B1 - qu 1 - q δ 1 - m

Alternative calculationAlternative calculation

Bqu 1 - q δ 1 - m

1 - m B

B S1 - qu - 1 - q δ 1 - m + qu + 1 - q δm

2

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S

B

x* +y*y* = qu + 1 - q δ

2x* +y*

1 - x* = qu + 1 - q δ 1 -2

S SS

B BB

δ αW + β 1 - α Vu =

q

δ βW + α 1 - β Vu =

q

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S B S B S B

equations in 8 variables

x*, y*,V ,V ,W ,W ,u ,u

S S S

B B B

V = δ αW + 1 - α V

V = δ βW + 1 - β V

S

B

x* +y*W =

2x* +y*

W = 1 -2

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2 1 - δ + δα - δ 1 - δ 1 - α 1 - βx* =

2 1 - δ + δα + δβ

δα + δ 1 - δ 1 - α 1 - βy* =

2 1 - δ + δα + δβ

The solution:

It can be shown that when all others use these strategies then this strategy is the best a player can do.

A seller always demands x* and accepts y* or more.A buyer always offers y* and accepts 1-x* or more

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2 1 - δ + δα - δ 1 - δ 1 - α 1 - βx* =

2 1 - δ + δα + δβ

δα + δ 1 - δ 1 - α 1 - βy* =

2 1 - δ + δα + δβ

αx*

α + β

αy

as δ :

*

1

α + β

1

1

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Buyers 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.

StrategicStrategic BargainingBargainingOne time entryOne time entry


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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 : or

- time

- set of agents present

- identity of the partner

- the offer just made by the pa

t

j

p rt

A

ner

t, A, j t, A, j, p

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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 !!!

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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 !!!Although this is not the only equilibrium, all other equilibria lead to the objects being sold at p = 1.

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Proof by induction on |S|


|Let S |= 1B S

iLet be the expected values

of buyer i and the seller, in some eq

V (t)

uili

,

br

V (t)

ium.

inf SLet be over

all equilibria and a

m

l

V (t)

l t.

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Bargaining & Markets StrategicStrategic BargainingBargainingOne time entryOne time entry

B

Bi

i=1

V (t)Then : 1 - m ,

B 1-mi B

and for each there is some bu t

V (t + 1)

yer

with

The seller is guaranteed to meet this buyer

Why ???

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1-mBm 1i.e. - - ε

1-mB1 - -

so if he demands the price

for some

he will ge

ε

t it.

BB-1m 1 -

hence m 1.

assume the proposition

holds for

Now

< | S |

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S Bj iLet be the expected payoffs when

all S and all B players are in t

V (t

he

), V (t)

market

inf SjLet be the

over all equilibria, all t and al

m

l

V (t)

j.

,

B

Bi

i=1

V (t) 1Then : - m | S |,

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Bi

Hence for any t there is a buy i

V (t + 1)

er s.t :

1 - m | S | /B

|S|B

if a seller demands the price

for some

for as long as all S , B players are in

he will either get it or the market becomes s

1 - (1 - m) -

mall

ε

er.

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|S|B m 1 - (1 - m)hence

o

ε

r

-

m 1.

In any equilibrium, the sellers obtain the price 1, as in a competitive equilibrium

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1 Bargaining & Markets u As before: Buyers and Sellers, δtp,δtp, δ t (1-p). u Matching: Seller...

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