Summary of Nash Bargaining

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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

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