Nash Bargaining Problem

Question 1

What is a nash bargaining problem?

Answer 1

Two players try to find an agreement of how to share a resource

Question 2

What happens in a nash bargaining problem if the players do not agree?

Answer 2

They end up at the disagreement point with disagreement utilities (usually 0 for each player)

Question 3

What is the nash bargaining solution defined by?

Answer 3

Players’ utilities instead of the physical distribution

Question 4

What is a physical space / utility space?

Answer 4

A physical space depicts the possible allocations of the physical good, the utility space depicts the possible allocations according to the utilities associated with outcomes

Question 5

What are the steps in going from a physical space to a utility space?

Answer 5

1) find out the utility functions for each of the two players
2) set the restrictions (0 ≤ x1, 0 ≤ x2 and x1 + x2 = 1)
3) replace the final part of the restrictions with the utilities
4) solve the equation

Question 6

What is the nash bargaining solution?

Answer 6

Allocating the goods in such a way that the product of utility gains (nash product) is maximized

Question 7

How is the utility gain calculated for a player?

Answer 7

The difference between the utility of what they achieve and the utility they would get if the bargain failed

πi(x) = ui(xi) − ui(di).

Question 8

What is the formula for the nash product?

Answer 8

N(x) = [u1(x1) − u1(d1)] · [u2(x2) − u2(d2)]

Question 9

What is at each of the extreme points (0,1) or (1, 0)?

Answer 9

The nash product is 0, as one of the players has a gain of 0

Question 10

How do we find a unique maximum?

Answer 10

1) forumlate a constraint maximization problem:
argmax (x1,x2){[u1(x1) − u1(d1)] · [u2(x2) − u2(d2)]} s.t. x1 + x2 = 1

2) Since the disagreement point is (0, 0) and so are the corresponding utilities, we can drop u1(d1) and u2(d2). Furthermore, we use the utility functions

argmax (x1,x2){u1 · u2} s.t. x1 + x2 = 1

3) we insert the constraint and substitue u2
argmax(x1,x2){u1 · 1-u1}

4) We calculate the derivative, equalize it to zero and then solve for x1