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6.254 : Game Theory with Engineering Applications
Lecture 14: Nash Bargaining Solution
Asu Ozdaglar
MIT
March 30, 2010
1
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View Full Document Game Theory: Lecture 14
Introduction
Outline
Rubinstein Bargaining Model with Alternating Oﬀers
Nash Bargaining Solution
Relation of Axiomatic and Strategic Model
Reference:
Osborne and Rubinstein, Bargaining and Markets.
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Game Theory: Lecture 14
Strategic Model
Introduction
In this lecture, we discuss an axiomatic approach to the bargaining
problem.
In particular, we introduce the
Nash bargaining solution
and study the
relation between the axiomatic and strategic (noncooperative) models.
As we have seen in the last lecture, the Rubinstein bargaining model
allows two players to oﬀer alternating proposals indeﬁnitely, and it
assumes that future payoﬀs of players 1 and 2 are discounted by
δ
1
,
δ
2
∈
(
0, 1
)
.
3
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View Full Document Game Theory: Lecture 14
Strategic Model
Rubinstein Bargaining Model with Alternating Oﬀers
We showed that the following stationary strategy proﬁle is a subgame
perfect equilibrium for this game.
Player 1 proposes
x
*
1
and accepts oﬀer
y
if, and only if,
y
≥
y
*
1
.
Player 2 proposes
y
*
2
and accepts oﬀer
x
if, and only if,
x
≥
x
*
2
,
where
x
*
1
=
1

δ
2
1

δ
1
δ
2
,
y
*
1
=
δ
1
(
1

δ
2
)
1

δ
1
δ
2
,
x
*
2
=
δ
2
(
1

δ
1
)
1

δ
1
δ
2
,
y
*
2
=
1

δ
1
1

δ
1
δ
2
.
Clearly, an agreement is reached immediately for any values of
δ
1
and
δ
2
.
To gain more insight into the resulting allocation, assume for simplicity that
δ
1
=
δ
2
. Then, we have
If 1 moves ﬁrst, the division will be
(
1
1
+
δ
,
δ
1
+
δ
)
.
If 2 moves ﬁrst, the division will be
(
δ
1
+
δ
,
1
1
+
δ
)
.
4
Game Theory: Lecture 14
Strategic Model
Rubinstein Bargaining Model with Alternating Oﬀers
The ﬁrst mover’s advantage (FMA) is clearly related to the impatience of
the players (i.e., related to the discount factor
δ
):
If
δ
→
1, the FMA disappears and the outcome tends to
(
1
2
,
1
2
)
.
If
δ
→
0, the FMA dominates and the outcome tends to
(
1, 0
)
.
More interestingly, let’s assume the discount factor is derived from some
interest rates
r
1
and
r
2
.
δ
1
=
e

r
1
Δ
t
,
δ
2
=
e

r
2
Δ
t
These equations represent a continuoustime approximation of interest rates.
It is equivalent to interest rates for very small periods of time
Δ
t
:
e

r
i
Δ
t
’
1
1
+
r
i
Δ
t
.
Taking
Δ
t
→
0, we get rid of the ﬁrst mover’s advantage.
lim
Δ
t
→
0
x
*
1
=
lim
Δ
t
→
0
1

δ
2
1

δ
1
δ
2
=
lim
Δ
t
→
0
1

e

r
2
Δ
t
1

e

(
r
1
+
r
2
)
Δ
t
=
r
2
r
1
+
r
2
.
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View Full Document Game Theory: Lecture 14
Strategic Model
Alternative Bargaining Model: Nash’s Axiomatic Model
Bargaining problems represent situations in which:
There is a conﬂict of interest about agreements.
Individuals have the possibility of concluding a mutually beneﬁcial
agreement.
No agreement may be imposed on any individual without his approval.
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This note was uploaded on 05/08/2010 for the course CS 6.254 taught by Professor Asuozdaglar during the Spring '10 term at MIT.
 Spring '10
 AsuOzdaglar

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