**1. Desired Properties**

- invariant to coordinate-wise affine transformation
- symmetry-preserving
- efficient
- monotone

**2. Problem Statement**

We consider a two-person bargaining problem formulated as follows

- : to every two-person game we associated a pair , where is a point in the plane and is a subset of the plane.
- The pair has the following intuitive interpretation:
*where is the level of utility that player receives if the two players**do not cooperate*with each other. - Every point
*represents the level of utility for players 1 and 2 that can be reached by an outcome of the game which is feasible for the two players when they**do cooperate*.

**3. ****Assumption**

- Assumption 1: There is at least one point such that . In other words, bargaining may prove
*worthwhile for both players*. - Assumption 2:
*is convex*. This is justified under the assumption if two outcomes of the game give raise to points and in , then randomization of these two outcomes give raise to all convex combinations of and . - Assumption 3: is
*compact*. - Assumption 4:
*for every*. If this is not the case, we can disregard all the points of that fail to satisfy this condition because it is impossible that both players will agree to such a solution.

**4. Axioms**

We let denote the set of pairs that satisfying these four conditions, and we call an element in a bargaining pair.

- Axiom 1:
*Pareto Optimality* - For every there is no such that and .
- Axiom 2:
*Symmetry* - We let be defined by and we require that for every , .
- Axiom 3:
*Invariance*with Respect to*Affine Transformation of Utility* - A is an afine transformation of utility if , , and the maps are of the form for some positive constant and some constant . We require that for such a transformation , .

In addition to the above three axioms, Nash introduced the following

- Axiom of
*Independence of Irrelavant Alternatives* - If and are bargaining pairs such that and , then .
- Interpretation: given a bargaining pair , for every point , consider the product . Then is the unique point in that maximizes this product.
- Many objectives are raised to Nash’s axiom of independence of irrelevant alternatives.

We define some notations,

- Let be a function defined for in the following way
- if is the Pareto of .
- if there is no such .
- thus is the maximum player 2 can get if player 1 get at least x.
- By assumption 1 in the definition of a bargaining pair .
- By the compactness of , and are finite and are attained by points in .
- A pair will be called
*normalized*if and . Clearly every game can be normalized by a unique affine transformation of the utilities.

**Example objective to Nash’s Solution: **consider the following two normalized pair and where

- = convex hull, and
- = convex hull,
- Nash’s solution for is , and for .
- Limitations pf Nash’s solution: Player 2 has good reasons to demand that he get more in the bargaining pair than he does in .

In order to overcome this limitation, Kalai suggests the following alternative axiom.

**Axiom of**: If and are bargaining pairs such that and , then (where .

*Monotonicity*- This axiom states that if, for every utility level that player 1 may demand, the maximum feasible utility level that player 2 can simultaneously reach is increased, then the
*utility level assigned to player 2 according to the solution should also be increased*.

**Theorem**: There is one and only one solution, , satisfying the axioms of monotonicity. The function has the following simple representation. For a pair consider the line joining a to be , . The maximal element (with partial order of ) of on this line is .

**5. How does it work**

- to
*normalize*the utility function of each agent in such a way that it is worth zero at the status quo and one at this agent’s best outcome — given that all others get at least their status quo utility level - to
*sharing equally the benefits*of cooperation. In other words, this solution equalizes the relative benefit from status quo or equivalently the relative frustration until the shadow optimum.

**6. Solution**

- Independence of irrelevant alternatives can be substituted with a monotonicity condition. It is the point which maintains the ratios of maximal gains. In other words, if player 1 could receive a maximum of with player 2’s help (and vice versa for ), then the bargaining solution would yield the
*point*on the Pareto frontier such that .

References

[1] Bargaining problem, wiki

[2] Other solutions to Nash’s bargaining problem, by Ehud Kalai, Meir Smorodinsky, in STOR 1975