1

A1 ParetoA2 SymmetryA3 InvarianceA4 IIA

A1 ParetoA2 SymmetryA3 InvarianceA4 IIA

First, we show that there exists a function satisfying the axioms.

There exists a unique

satisfying A1- A4

2: f BTheorem:

Proof:

2d

,S,dFor any given bargaining problem

define g S,d

S

=

= Does such a point always exist ??Is it unique ??

Yes !!!

Proof:

3

?

d

g S,d

S

=

=

Proof:

does satisfy A1-A4 ?? g S,d

Pareto

Symmetry

IIA

Invariance

4

(divide the $)

0

Proof:

Consider the bargaining problem ,Δ,0 f S,d g S,d

Uniqueness: If satisfies the axioms then: f S,d

(1,0)

(0,1)

1 12 2f ,0 ,By Pareto + Symmetry:

A2 (Symmetry)

1 2 2 1( , ) ( , )x x x xα

( , ) α α αf S d f S,d

( , ) α α αf 0 f ,0 f ,0

By definition: 1 12 2g ,0 ,

1 12 2f ,0 g ,0 ,

5

Proof:

For a given bargaining problem ,S,d

d

S

g S,d

(a,d2)

(d1,b)

(d1, d2)

=

=

6

Proof:

For a given bargaining problem ,S,d

d

S

g S,d f S,d

If is a degenerate ProblemS,d

7

Proof:

For a given (nondegenerate) bargaining problem ,S,d

0 (1,0)

(0,1)

d

S

g S,d

(a,d2)

(d1,b)

(d1, d2)

Consider the bargaining problem ,Δ,0Find an affine transformation α

α

α

α

8

Proof:

0 (1,0)

(0,1)

d

S

g S,d

(a,d2)

(d1,b)

(d1, d2)

Find an affine transformation α

α

α

α

α x, y = μx + ν, μy + ν

1 2α 0,0 = ν,ν = d ,d 2α 1,0 = μ+ ν ν = a,d, 1α 0,1 = ν, μ ν = d+ ,b

,1 2

1 2

ν,ν

μ, μ

= d ,d

a - d b - d

,1 2a - d b - d > 0,0

9

Proof:

0 (1,0)

(0,1)

d

S

g S,d

(a,d2)(d1, d2)

α

α

α Δ = S

S

αf Δ,0 = f α ,α 0

f S,d 1 12 2α ,

1 12 2,

==

(d1,b)α

g S,d

g S,d f S,d S g S,d S

1 12 2α , 1 1

2 2α 1,0 + 0,1

1 1 1 12 12 2 2 2α 1,0 + α 0,1 a,d d ,b

= g S,d

10

Proof:

d

S

g S,d f S,dS

g S,d f S,d S

By IIA f S,d f S,d

g S,d f S,d

f S,d

11

Proof:

d

S

g S,d f S,d

g S,d f S,d

f S,d

satisfying

g f,

! f A1 - A4

end of proof

120

A Generalization

Changing A2 (Symmetry)

A2 (nonsymmetric)

for some f(Δ,0) = α,1 - α 0 < α < 1.

Δ

(1,0)

(0,1)

A

B

α measures the strength of Player 1

α,1 - α

A

B

1 - α=

α

13d

,S,dFor any given bargaining problem

define g S,d

SA

B

With the new A2, define a different g S,d

A

B

1 - α=

α

14d

g S,dS

Does such a point always exist ??Is it unique ??

Yes !!!

A

B

Following the steps of the previous theorem,

g S,d is the unique function satisfying the 4 axioms.

Yes !!!Yes !!!

A

B

1 - α=

α

15

A brief mathematical InterludeA brief mathematical Interlude

Consider the (implicit) functionα 1-αx y = K

Find a tangent at a point (x0,y0) on the curve

α-1 1-α α -ααx y + 1 - α x y y = 0

0 0αy + 1 - α x y = 0

0

0

αyy = -

1 - α x

differentiating

y

x

16

A brief mathematical InterludeA brief mathematical Interlude

Find a tangent at a point (x0,y0) on the curve

0

0

αy-1 - α x

0

0

y - y=

x - x

x

y

The tangent’s equation:

The intersections with the axis (x=0, y=0)

0 0x y,0 , 0,

α β

17

A

B

A brief mathematical InterludeA brief mathematical Interlude

x

y

0 0

0 0

x yx , y = α ,0 + β 0,

α β

(x0/α,0)

(0, y0/β)

(x0 , y0)

A

B

1 - α=

α

18

A

B

A brief mathematical InterludeA brief mathematical Interlude

x

y

α 1-αx y = KAny tangent of the function

is split by the tangency point in the ratio 1 - α

α

19

A brief mathematical InterludeA brief mathematical Interlude

x

y

α 1-α

(x,y) Smax x y

For any convex set S, by maximizing

S

1-α

α

We find the unique point in S in which the tangent is split

in the ratio

end of mathematical Interludeend of mathematical Interlude

20

α 1-α

1 2(x,y) Smax x - d y - d

To find the Nash Bargaining Solution of a bargaining problem ,S,d

S

d

Nash Bargaining Solution

21

All axioms were used in the proof

But are they necessary?

All axioms were used in the proof

But are they necessary?

A1. Without Pareto,h(S,d) = dsatisfies the other axioms.

A2. Without Symmetry,

satisfies the other axioms.

α 1-α

1 2(x,y) Smax x - d y - d

22

All axioms were used in the proof

But are they necessary?

All axioms were used in the proof

But are they necessary?

A3. Without Invariance,

satisfies the other axioms.

(x, y) S

h(S,d) = max x + y

S

(0,1)

(0,0) (2,0)d

(x, y) Smax xy

h(S,d)

(1, 0.5)

Nash Bargaining Solution

23

All axioms were used in the proof

But are they necessary?

All axioms were used in the proof

But are they necessary?

S

d

A4. Without IIA, the following function satisfies the other axioms.

The Kalai Smorodinsky solution

(S,d)K

1 2max max, (S,d)K

24

All axioms were used in the proof

But are they necessary?

All axioms were used in the proof

But are they necessary?

S

d

A4. Without IIA, the following function satisfies the other axioms.

The Kalai Smorodinsky solution

(S,d)K

1 2max max,(S,d)K

is on the Pareto front of S

1

2

max

max

(S,d) = (x, y)

x=

y

K is on the Pareto front of S

1

2

max

max

(S,d) = (x, y)

x=

y

K