1 Topics in Game Theory SS 2008 Avner Shaked

2

3 n K. n K. Binmore Fun & Games A Text on Game Theory D.C. Heath & Co., 1992

4 n M. n M. Osborne & A. Rubinstein Bargaining and Markets Academic Press, 1990

5 n K. n K. Binmore Fun & Games A Text on Game Theory D.C. Heath & Co., 1992 n M. n M. Osborne & A. Rubinstein Bargaining and Markets Academic Press, 1990

6 A Bargaining Problem S - a feasible set d - a disagreement point Nash Bargaining Theory Nash Verhandlungstheorie John Nash

7 Nash Bargaining Theory u2u2 u1u1 S

8 u2u2 u1u1 bounded closed S

9 Nash Bargaining Theory u2u2 u1u1 A B S

10 Nash Bargaining Theory u2u2 u1u1 d S

11 Nash Bargaining Theory u2u2 u1u1 d S

12 Nash Bargaining Theory d A Nash Bargaining Solution is a function u2u2 u1u1 S

13 Nash Bargaining Theory A Nash Bargaining Solution is a function u2u2 u1u1 S d

14 Axioms A1-A4 A1 (Pareto) A2 (Symmetry) d S

15 Axioms A1-A4 A3 (Invariance to affine transformation) IIA A4 (Independence of Irrelevant Alternatives IIA)

16 Axioms A1-A4 IIA A4 (Independence of Irrelevant Alternatives IIA) u2u2 u1u1 d

17 Axioms A1-A4 IIA A4 (Independence of Irrelevant Alternatives IIA) Gives f(T,d) a flavour of maximum Pasta Fish Meat IIA is violated when

18 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:

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

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

21 (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:

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

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

24 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

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

26 Proof: 0 (1,0) (0,1) d (a,d 2 ) (d 1, d 2 ) == (d 1,b) ??

27 Proof: 0 (1,0) (0,1) d (a,d 2 ) (d 1, d 2 ) == (d 1,b) ??

28 Proof: d By IIA

29 Proof: d end of proof

30 0 A Generalization Changing A2 (Symmetry) A2 (nonsymmetric) (1,0) (0,1) A B α measures the strength of Player 1

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

32 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 !!!

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

34 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)

35 A B A brief mathematical Interlude x y (x 0 /α,0) (0, y 0 /(1-α)) (x 0, y 0 )

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

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

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

39 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.

40 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

41 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

42 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