Now, the question stands of which

point from the bargaining set do we want to choose?

For that, there are several classical approaches.

One of them and

the most famous one is the Nash bargaining solution.

The Nash bargaining solution can be used for n-player games

or for the game with arbitrary number of players.

But in our case, we have two player game but still.

Nash bargaining solution and any other bargaining solution or

cooperative solution in game theory is

defined using the set of axioms.

What is the axiom?

The axiom is a property,

a mathematical property which has a certain physical meaning.

When we define cooperative solution,

we want it to be for example, Pareto-optimal.

So, we want it to has a good outcomes.

We also want to have other properties.

So, we define a set of properties or set of axioms.

Then we derive the solution and then we

try to construct the analytical form

or explicit formula for this solution.

So, for the Nash bargaining solution,

we define four axioms.

The first one is Pareto-optimality.

So, the bargaining solution in our case is

a vector and it is Pareto-optimal

when there is no vector which is better than the chosen

one for any vector value.

So, we say if there is

no vector which is better for any vector value,

then we say that the chosen solution is Pareto-optimal.

Then the second axiom is called symmetry.

We say that the solution satisfies axiom of symmetry if we

will have the same solution when we change the number of players,

or we change the names of the players.

So, solution only depends on some values of payoffs.

So the third axiom is called scale invariance.

It says that if we change

the bargaining set using some linear operator,

for example, we expand the bargaining set

by multiplication or constant or something like that.

Then what we will get is we could

calculate the bargaining solution

using the initial set and the expanded set,

then we can calculate the bargaining solution on

the expanded set as

the bargaining solution with linear operator in the initial set.

That's it. The fourth axiom is called the

independence of irrelevant alternatives.

It says that if we consider the initial bargaining set,

then we define the specific bargaining solution.

Then we can consider the subset of

the initial bargaining set which

includes the corresponding bargaining solution.

Then if we recalculate

some bargaining solution in

the subset of the initial bargaining set,

then it will be

equal to the bargaining solution in the initial bargaining set.

So, it means that the bargaining solution defined using

the initial bargaining set and only depends

on the alternatives or the outcomes which are relevant.

It was proved that there exists

a unique function satisfies the axioms

which were defined on the previous slide

and it can be calculated using the following explicit formula.

The formula seems to be pretty easy to compute,

but as we increase the number of players,

the computation time will be a larger.

Also the solution that satisfies

these four axioms and as it turns

out can be calculated using this formula,

we will call the Nash bargaining solution.

Let's calculate the Nash bargaining solution

for our example of Battle of Sexes.

On the slide, you can see that the Nash bargaining solution is

located right in the middle of the interval a and b.

This is only because

this interval a and b which actually is the set of

Pareto-optimal outcomes is symmetrical

to the disagreement point d. On this slide,

you can see a list of references where

you can study of how we can prove the theory

which corresponds to the Nash bargaining solution and

also you can find there other examples.