1. 1 A1 Pareto A2 Symmetry A3 Invariance A4 IIA A1 Pareto A2 Symmetry A3 Invariance A4 IIA First, we show that there exists a function satisfying the axioms. There exists a unique satisfying A1- A4 Theorem: Proof:

2. 2 d For any given bargaining problem define = = Does such a point always exist ?? Is it unique ?? Yes !!! Proof:

3. 3 ? d = = does satisfy A1-A4 ?? Pareto Symmetry IIA Invariance

4. 4 (divide the $) 0 Proof: Consider the bargaining problem Uniqueness: If satisfies the axioms then: (1,0) (0,1) By Pareto + Symmetry: A2 (Symmetry) By definition:

5. 5 Proof: For a given bargaining problem d (a,d 2 ) (d 1,b) (d 1, d 2 ) = =

6. 6 Proof: For a given bargaining problem d If is a degenerate Problem

7. 7 Proof: For a given (nondegenerate) bargaining problem 0 (1,0) (0,1) d (a,d 2 ) (d 1,b) (d 1, d 2 ) Consider the bargaining problem Find an affine transformation α

8. 8 Proof: 0 (1,0) (0,1) d (a,d 2 ) (d 1,b) (d 1, d 2 ) Find an affine transformation α

9. 9 Proof: 0 (1,0) (0,1) d (a,d 2 ) (d 1, d 2 ) == (d 1,b)

10. 10 Proof: d By IIA

11. 11 Proof: d end of proof

12. 12 0 A Generalization Changing A2 (Symmetry) A2 (nonsymmetric) (1,0) (0,1) A B α measures the strength of Player 1

13. 13 d For any given bargaining problem define A B With the new A2, define a different

14. 14 d Does such a point always exist ?? Is it unique ?? Yes !!! A B Following the steps of the previous theorem, is the unique function satisfying the 4 axioms. Yes !!!

15. 15 A brief mathematical Interlude Consider the (implicit) function Find a tangent at a point (x 0,y 0 ) on the curve differentiating y x

16. 16 A brief mathematical Interlude Find a tangent at a point (x 0,y 0 ) on the curve x y The tangent’s equation: The intersections with the axis (x=0, y=0)

17. 17 A B A brief mathematical Interlude x y (x 0 /α,0) (0, y 0 /β) (x 0, y 0 )

18. 18 A B A brief mathematical Interlude x y Any tangent of the function is split by the tangency point in the ratio

19. 19 A brief mathematical Interlude x y For any convex set S, by maximizing S We find the unique point in S in which the tangent is split in the ratio end of mathematical Interlude

20. 20 To find the Nash Bargaining Solution of a bargaining problem S d Nash Bargaining Solution

21. 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, satisfies the other axioms. A2. Without Symmetry, satisfies the other axioms.

22. 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. S (0,1) (0,0) (2,0) d (1, 0.5) Nash Bargaining Solution

23. 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

24. 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