Lecture 9
Negotiation
14.12 Game Theory
Muhamet Yildiz
1
Examples of Bargaining
•
•
•
•
Buying a car, house, or shopping at a bazaar
Wage Negotiations
International Agreements
Legislative Bargaining
• Litigation
2
Road Map
1. Congressional Bargaining & Agenda
Setting
2. Pretrial Negotiations
3. Bargaining over a dollar
3
Agenda Setting
4
V oting with a fixed agenda
1. 2n+ 1 players
2. Alternatives: XO,x I " , "Xm
3. Each player i has a fixed strict preference
about alternatives: x jO >j Xjl >j .. . >j Xjm
4. There is a fixed binary agenda.
5. Assume: everything above is common
knowledge
5
A binary agenda
Xo
X2
XI
\
A preference profile
1
2
3
Xo
X2
Xl
Xl
Xo
X2
X2
Xl
Xo
X2
XI
Xo
X2
Sophisticated Voters
6
17th Amendment
• Xo =
• Xl =
•
status quo
17th amendment
DePew
Amendment
X2 =
Preference profile
1
2
3
Xo
X2
Xl
X2
Xl
Xo
Xl
Xo
X2
Xo
7
Pretrial Negotiation
8
Model
• Players:
- Plaintiff
- Defendant
• In court Defendant is to pay J to Plaintiff
• Cost of court
- Cp; CD; C = Cp+C D
• Lawyer cost per day:
- Cp ; Cd; C = Cp+C d
Assume: players are risk neutral and no discounting.
9
Timeline - 2n period
T = {1 ,2, ... ,2n-l ,2n,2n+ 1}
1ft is even
1ft < 2n is odd,
Defendant offers settlement st
Plaintiff Accept or Rejects the
offer
If the offer is Accepted,
Plaintiff pays St to the
Defendant and the game ends
Otherwise, we proceed to date
- Plaintiff asks settlement st
- Defendant Accept or Rejects
- If he accepts, Plaintiff pays
St to the Defendant and the
game ends
- Otherwise, we proceed to
date t+ 1
t+i.
At t=2n+ 1, they go to court
10
Backwards Induction
Date
Proposer Settlemeot
20
P
20-1
D
20-2
P
20-3
D
20-4
P
20-5
D
...
2
P
1
D
11
Graphically
-Co-J-2ncd
-Cp+J-2ncp
12
Sequential Bargaining
13
Sequential Bargaining
• N = {1,2}
• X = feasible
expected-utility
pairs (x,y EX)
• U/x,t) = 8t xj O(~
• (0,0) E X
disagreement
payoffs
(.
I
1
14
Timeline - 2n period
T = {l,2, ... ,2n-l,2n}
1ft is odd,
Player I offers some
(xt,Yt),
Player 2 Accept or
Rejects the offer
If the offer is Accepted,
the game ends yielding
8'(x t,Yt),
Otherwise, we proceed
to date t+ 1.
1ft is even
- Player 2 offers some
(xt,Y,),
- Player 1 Accept or Rejects
the offer
- If the offer is Accepted,
the gal):l€ ends yielding
payof{(x"Yt),
- Otherwise, we proceed to
date t+ I , except at t = 2n,
when the game end
yielding (0,0).
15
2
1 ~_+2":"n--,-1,~Y2::;;.n-<I)
2
(0,0)
Reject
Reject
Accept
At t = 2n-l ,
-Accept iff
-Offer
1
2n'Y2n)
( v5:2n-I X2n-I ' 82n-ly 2n-1 )
At t = 2n,
-Accept iff
-Offer
16
The shareofi att+!
=
~~l
~ =
I
SVI
1+1
2
.cr,1
I-VP + I
'
VI =
!
Reject
Reject
Accept
Accept
(Otxt, otyt )
(ot-IXt_l,ot-IYt_l)
17
18
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14.12 Economic Applications of Game Theory
Fall 2012
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