Nash's  Axioms:  The  axiomatic  approach begins  by  assuming that  we are looking for a rule, which will identify a particular outcome. In this way Nash assumes  in  the beginning  that we  are only  interested  in  rules, which identify unique outcomes. Nash then suggests that it would be natural for any such rule to satisfy the following four axioms.

i)  Efficiency

ii)  Individual rationality

iii) Scale covariance

iv)  Independence of irrelevant alternatives.

v)  Symmetry

We will  explain all of them  in  detail after introducing some basic concepts, which will be useful in understanding  the axioms.

There are 2 persons bargaining over  some amount of gain or payoff. We are interested  in  a  solution, which divides  the  gain  in  such  a  way  that  it  is acceptable to both  of  the players.  Any  such bargaining solution not only  depends  on joint payoff  but  also on  the consequences  if  the  bargain  breaks down. In such a situation we define the following:

F: jointly feasible set of payoff. It  is a set feasible vector with two elements in each of the vector. The elements  in  each vector suggest the way the gain  is to be distributed among the two.

Clearly, F⊂ R²  ;  where  R² is the two dimensional Euclidian space. Payoff to the  1^St

We assume that

1)  F  is closed and convex (boundaries of  F are  inside  F  and convex combination of any two points of F lie inside F).

2)  In  the worst  case, when there is disagreement among them the  payoff allocation is given by v =  (v₁,  v₂).

3)    is non-empty and bounded. That  is, there are some common elements between the feasible joint payoff set and the set containing allocations better than the disagreement payoff allocation, but there are not infinitely many.

We  denote the bargaining problem  as (F,  v).  (F, v)  is  said  to be  essential  iff (read as:  if and only  if) ∃y (read as there exists y), where y = ( y₁,  y₂) E  (read as belonging to) F, such that y₁≥ v₁  and y₂ ≥ v₂.  [In a successful solution both the players gain].

We define the bargaining solution function Φ as  follows:

Φ :  (F, v)  +  R²  [such that the solution is in F,  i.e., inside the feasible set]: Φ = (  Φ₁ (F, v), Φ₂(F, 4) The function Φ(F, v) gives solution to a bargaining problem (F,'v).