**1. Bargaining problems Scenarios**

Bargaining problems represent situations in which

- There is a
*conflict*of interest about agreements. - Individual have the possibility of concluding a
*mutually beneficial*agreements. - No agreement may be imposed on any individual without his approval

**2. Bargaining problem Definition**

Example: Suppose 2 players must split one unit of good. If no agreement is reached, then players do not receive anything. We define the following notations.

- : the set of possible agreements
- X = {, )| , }
- : the disagreement outcome
- D = (0,0)
- : each player i has preferences, represented by a utility function over

Definition: a bargaining problem is then defined as a pair of where and . We assume that

- is a
*convex*and*compact*set - There exists some such that (i.e.,
*for some i)*

**3. Axioms**

*Pareto Efficiency*- A bargaining solution is Pareto efficient if there does not exist a such that and for some .
- An inefficient outcome is unlikely, since it leaves space for
*renegotiation*. *Symmetry*- Let be such that if and only if and . Then .
- If the players are indistinguishable, the agreement should not discriminate between them.
*Invariance to Equivalent Payoff Representations*- Given a bargaining problem , consider a different bargaining problem , for some .
- Then
- Utility functions are only representation of preferences over outcomes. A transformation of the utility function that maintaining the same ordering over preferences (such as
*linear transformation*) should not alter the outcome of bargaining process. *Independence of Irrelevant Alternatives*- Let and be two bargaining problems such that , if , then .

**4. Nash Bargaining Solution****Definition: **We say that a pair of payoffs is a Nash bargaining solution if it solves the following optimization problem

- subject to

We use to denote the Nash Bargaining Solution

**Remarks**:

*Existence*of an optimal solution: since the set is compact and the objective function of the problem is continuous, there exists an optimal solution for the problem*Uniqueness*of the optimal solution: the objective function of the problem is strictly*quasi-concave*. Therefore, the problem has a unique solution.

**Proposition**: Nash bargaining solution is the unique bargaining solution that satisfies the 4 axioms.

Reference

[1] Game Theory with Engineering Applications: Nash Bargaining Solution, by Asu Ozdaglar, MIT 2010