# Summary of Nash Bargaining

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