A1 Pareto A2 Symmetry A3 Invariance A4 IIA

There exists a unique satisfying A1- A4 A1 Pareto A2 Symmetry A3 Invariance A4 IIA Theorem: Proof: First, we show that there exists a function satisfying the axioms.

Proof: d For any given bargaining problem define = Does such a point always exist ?? Is it unique ?? Yes !!! =

Proof: does satisfy A1-A4 ?? d Pareto Symmetry = IIA ? Invariance =

Proof: Uniqueness: If satisfies the axioms then: Consider the bargaining problem A2 (Symmetry) By Pareto + Symmetry: By definition: 0 (divide the $) (0,1) (1,0)

Proof: (d1,b) d (a,d2) (d1, d2) For a given bargaining problem = =

Proof: For a given bargaining problem is a degenerate Problem If d

Proof: Consider the bargaining problem (d1,b) (0,1) d (a,d2) (1,0) 0 (d1, d2) For a given (nondegenerate) bargaining problem Find an affine transformation α

Find an affine transformation α Proof: (d1,b) (0,1) d (a,d2) (1,0) 0 (d1, d2)

Proof: (0,1) (1,0) 0 = = (d1,b) d (a,d2) (d1, d2)

Proof: By IIA d

Proof: d end of proof

A2 (nonsymmetric) A Generalization Changing A2 (Symmetry) αmeasures the strength of Player 1 (0,1) B A 0 (1,0)

d With the new A2, define a different For any given bargaining problem define B A

d Following the steps of the previous theorem, is the unique function satisfying the 4 axioms. Does such a point always exist ?? Is it unique ?? B Yes !!! Yes !!! Yes !!! A

A brief mathematical Interlude Consider the (implicit) function Find a tangent at a point (x0,y0) on the curve differentiating y x

A brief mathematical Interlude Find a tangent at a point (x0,y0) on the curve The tangent’s equation: y The intersections with the axis (x=0, y=0) x

A brief mathematical Interlude y (0, y0/β) B (x0 , y0) A x (x0/α,0)

A brief mathematical Interlude Any tangent of the function is split by the tangency point in the ratio y B A x

We find the uniquepoint in S in which the tangent is split in the ratio y x A brief mathematical Interlude end of mathematical Interlude For any convex set S, by maximizing S

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

A1.Without Pareto, satisfies the other axioms. A2.Without Symmetry, satisfies the other axioms. All axioms were used in the proof But are they necessary?

A3.Without Invariance, satisfies the other axioms. (0,1) (1, 0.5) d (0,0) (2,0) All axioms were used in the proof But are they necessary? Nash Bargaining Solution S

A4.Without IIA, the following function satisfies the other axioms. The Kalai Smorodinsky solution All axioms were used in the proof But are they necessary? S d

All axioms were used in the proof But are they necessary? A4.Without IIA, the following function satisfies the other axioms. The Kalai Smorodinsky solution S d

A1 Pareto A2 Symmetry A3 Invariance A4 IIA