Applications of dynamic games: Bargaining and reputation · 28.10.2014 Daniel Spiro, ECON3200/4200,...

28.10.2014 Daniel Spiro, ECON3200/4200, Lecture 4 1

Applications of dynamic games:

Bargaining and reputation

Lectures in Game Theory

Fall 2014, Lecture 4


Modeling of

negotiations between parties with opposing

interests:

Sequential bargaining games

non-cooperative cooperation between parties

that in the short run want to deviate:

Repeated games


d: default outcome,

disagreement point,

2uBargaining

d

1u

Agreement

creates value, but

how to divide?

A

B

A, B: outcomes

A: efficient outcome if

transfers are possible

Efficient bargaining:

Agree on efficient out-

come and divide surplus.

Applications: Bilateral monopoly

Labor market negotiations

International negotiations


2u

1

The standard

bargaining solution

1u

1

What do the bargaining weights depend on?

First: Normalize.

Then: Divide surplus

according to

bargaining weights, p1

and p2. 1

p

2p


Ultimatum

bargaining

1

2 A

R

1m

,1

m1

1 m

m1 is 1’s

share. 1

u

2u

1

0 ,0

1

Bargaining weight = 1

for the proposer


Two-period

alternating offer

1

2 A

R

1m

,1

m1

1 m

2

1 A

R

2m

,2

m )1(2

m

0 ,0

mt is 1’s

share in

period t.

is the dis-

count factor

1u

2u

1


Three-period

alternating offer

1

2 A

R

1m

,1

m1

1 m

2

1 A

R

2m

,2

m )1(2

m

mt is 1’s

share in

period t.

is the dis-

count factor

1u

2u

2 1

1


1

2 A

R

1m

,1

m1

1 m

2

1 A

R

2m

,2

m )1(2

m

1

Backward induction:

Period 3: 1 suggests m3 =1, 2 will

accept. Payoff is , 0 2

Period 2: 1 indiff between accepting

and rejecting if m2 = . Hence 2 will

suggest m2=, 1 will accept. Payoff is

and (1-m2) = (1- )

2

Period 1: 2 indiff between accepting and rejecting if 1-m1 = (1- ) . Hence, if 1 will suggest m1=1(1) then 2

will accept. Will 1 want to suggest this? Yes since m1 > 2

2

SPNE: s1=[m1=1(1), A if m2, m3 =1]

s2=[A if m1 1(1), m2=, A if m3 0]


Three-period

alternating offer

1

2 A

R

1m

,1

m1

1 m

2

1 A

R

2m

,2

m )1(2

m

mt is 1’s

share in

period t.

is the dis-

count factor

1u

2u

2 1

1


Rubinstein’s

bargaining model

1

2 A

R

1m

,1

m1

1 m

2

1 A

R

2m

,2

m )1(2

m

,3

2m )1(3

2 m

mt is 1’s

share in

period t.

is the dis-

count factor

1u

2u

2 13

2m

Equal bargaining

weight for pl. 1

in periods 1 & 3.

Finding the solution of Rubinstein’s bar-

gaining model with an infinite time horizon

)1(121

mm =

Equal bargaining weight for P1 in periods 1 and 3:

31mm =

Three equations in three unknown. Solution:

In period 1, P1 makes the best proposal that P2

accepts:

3

2

2mm =

In period 2, P2 makes the best proposal that P1

accepts:

==

1

131

mm

=

12

m


Solution of Rubinstein’s bargaining model

with an infinite time horizon

1tm only if and ifaccepts 1

Unique subgame perfect Nash equilibrium:

otherformandherselfform tt

=

=1

111

1

1 suggests 1

=

1

11

1)-(1

tm only if and ifaccepts 2

=

=

=

1

11

1

11

1tt mm suggests 2

1st mover advantage: 1’s share is larger than 2’s.

1’s proposal at time t = 1 is accepted.

If disc. factors are different: It pays to be patient.


Repeated games with an infinite time horizon

D

D

C

C

1 1,

2 2,3 0,

0 3,

Prisoners’

dilemma

2u

1u

1 2 3

1

2

3

Can repetition discipline

the players to cooperate?

Deviating yields a short run gain.

Deviating yields a loss of reputation

that undermines future cooperation.

Yes, if gain

now PV of

future loss


Cooperation in infinitely rep.

.) with(Start .play otherwise

earlier; used been has ifPlay

:strategy"Trigger "

CC

DD

If cooperation breaks down, it will never be restarted.

Prisoners’ Dil.

Subgame perfect Nash equilibrium (NE in all subgames)?

If cooperation has broken down: NE in the subgames.

If cooperation has not broken down:

Short-run gain PV of long-run loss

)12(1

23

21

2

1

2u

1u

1 2 3

1

2

3


“Getting Even”

period. previous inperm"" w/o played

opponent the ifplay to Permitted""

.play to permitted"" unless ,Play

D

D

DC

If 1 deviates, then 1 is punished in the next period.


Short-run gain PV of loss in next period

)02(23 21 2

1

If cooperation has broken down:

)02(01 21 2

1

2u

1u

1 2 3

1

2

3


Repeated games– Economic lessons

Two sources of instability: 1) temptation for deviation in coop stage and

2) temptation to deviate when the player is being punished. With the

numbers in previous example the two factors are equally strong.

Other numbers may change the relative force.

”Getting even” and ”trigger strategy” are ways of upholding

cooperation. In previous example they happen to be equally

stable. Usually they are not.


Repeated games– Economic lessons

The results hinge on the game being infinitelly repeated. Finite PD:

• In last period, DD is only NE no matter of history.

• Therefore payoffs by actions in second to last period is indep of

payoff in last period.

• Hence, in second to last period, DD is only NE

• Similar backward induction leads to DD in all periods.

Further applications


In examples, the will to punish the others was a non-issue. In

many economic settings there is a cost, need to check that

punishment is a NE.

Cooperation for climate control Based on Barrett, Journal of Theoretical Politics 11 (1999), 519–541.

Cooperate: Reduce the emis-

sion of greenhouse gases.

Defect: Do not reduce the

emission of greenh. gases.

( )C n c dn =

In total N countries, where

n countries cooperate.

bnnD

= )(

Payoff function:

Conditions: 0 bdc

( 1) 0b N c dN

Cooperation for climate control (cont.)

An agreement between k participating countries based on:

“Getting Even”: A participating country plays Cooperate

by reducing emission, except if another participating

country has been the sole deviator from “Getting Even”

in the previous period (in which case, the play of Defect in

the sense of no reduction of emission is specified).

Non-participating countries play Defect after any history.

Two observations:

– Not a subgame perfect NE if k is too low.

– Not “renegotiation-proof ” if k is too high.

Subgame perfect Nash equilibrium (NE in all subgames)?

Cooperation for climate control (cont.)


Short-run gain PV of loss in next period

( ) )()()1( cdkcdcdkkb dbd

dck

1

If cooperation has broken down:

( ) )()(0 cdkcdcd d

dck

1

Are the punishing countries better off during punishment (so

they do not want to return to the original equilibrium at once)?

cdkb d

cbk

:1 to close & db =

d

ck

d

c 1

Applications of dynamic games: Bargaining and reputation · 28.10.2014 Daniel Spiro, ECON3200/4200,...

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