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.

• $X$: the set of possible agreements
• X = {$x_1$, $x_2$)| $x_1 + x_2 = 1$, $x_i \geq 0$}
• $D$: the disagreement outcome
• D = (0,0)
• $u_i$: each player i has preferences, represented by a utility function $u_i$ over $X \cup {D}$

Definition: a bargaining problem is then defined as a pair of $(U,d)$ where $U \in R^2$ and $d \in U$. We assume that

• $U$ is a convex and compact set
• There exists some $v \in U$ such that $v > d$ (i.e., $v_i > d_i$ for some i)

3. Axioms

• Pareto Efficiency
• A bargaining solution $f(U,d)$ is Pareto efficient if there does not exist a $(v_1, v_2) \in U$ such that $v \geq f(U,d)$ and $v_i > f_i(U,d)$ for some $i$. 
• An inefficient outcome is unlikely, since it leaves space for renegotiation.
• Symmetry
• Let $(U,d)$ be such that $(v_1, v_2) \in U$ if and only if $(v_2, v_1) \in U$ and $d_1 = d_2$. Then $f_1(U,d) = f_2 (U,d)$.
• If the players are indistinguishable, the agreement should not discriminate between them.
• Invariance to Equivalent Payoff Representations
• Given a bargaining problem $(U,d)$, consider a different bargaining problem $(U', d')$, for some $\alpha >0, \beta$.
• $U' = \{(\alpha_1 v_1 + \beta_1, \alpha_2 v_2 + \beta_2)| (v_1, v_2 \in U\}$
• $d' = (\alpha_1 d_1 + \beta_1, \alpha_2 d_2 + \beta_2)$
• Then $f_i(U', d') = \alpha_i f_i (U,d) + \beta_i$
• 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 $(U,d)$ and $(U', d)$ be two bargaining problems such that $U' \subset U$, if $f(U,d) \in U'$, then $f(U', d) = f(U,d)$.

4. Nash Bargaining Solution
Definition: We say  that a pair of payoffs $(v^*_1, v^*_2)$ is a Nash bargaining solution if it solves the following optimization problem

• $\max_{v_1, v_2} (v_1 - d_1)(v_2-d_2)$
• subject to 
• $(v_1, v_2) \in U$
• $(v_1, v_2) \geq (d_1, d_2)$

We use $f^N(U,d)$ to denote the Nash Bargaining Solution

Remarks: 

• Existence of an optimal solution: since the set $U$ 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 $f^N(U,d)$ 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