Nash Bargaining Problem (Formal) Flashcards
What are the conditions for a set if it contains a nash bargaining solution?
1) it needs to be convex
2) it needs to be closed (contain its borders)
3) it needs to be bounded
What is convex?
Two points in the set need to be able to be connected by a straight line that does not fall outside of the set (no dents in the space of the set) - every point in-between two points of the set is also an element of the set. This makes sure the outcome is unique
What is closed?
It needs to include the points on the boundary of the set. A common example of a set that is not closed is an open interval on the real number line. Consider the set defined by the interval (0,1). This set includes all numbers greater than 0 and less than 1 but does not include the numbers 0 and 1 themselves.
What is bounded?
It needs to be finite. There need to be utility levels that cannot be reached. Why bargain if giving infinite utility is possible?
What is the nash bargaining solution?
a solution/allocation that gives the highest nash product out of all allocations in the set:
What are the properties of the nash bargaining solution?
1) Pareto efficient
2) Symmetric
3) Scale covariant
4) independent of irrelevant alternatives
What is pareto efficiency?
If we cannot make anyone better off without harming another player
(there is no allocation north east)
What is symmetry?
If we can swap players without changing an individual player’s situation. A bargaining solution concept is symmetric if for a symmetric problem it generates a symmetric solution
What is scale covariance?
The way we represent the same preferences does not change the bargainig outcome. If for example player 1 claims double utility for everything it changes the utility space, but not the physical allocation. Same applies if the player adds a const4ant or multiplies the utility function
What is independence of irrelevant alternatives?
The solution does not change if you “cut off” some feasible allocations that are not the solution
