Nash Bargaining Solution

Game Theory · Axiom Academy

LESSON Nash Bargaining Solution Unit 6: Cooperative Game Theory - A rigorous approach to fair division and negotiation Nash identified four fundamental axioms that any "fair" bargaining solution should satisfy. These axioms uniquely determine the bargaining outcome. Pareto Efficiency: The solution must lie on the Pareto frontier - no other outcome can make both players better off. Symmetry: If players have identical bargaining positions, they receive equal payoffs. Independence of Irrelevant Alternatives: If the feasible set shrinks but still contains the original solution, the solution remains unchanged. Invariance to Affine Transformations: The solution is independent of how utilities are scaled or shifted. 2. The Nash Bargaining Solution Formula Given a bargaining problem with disagreement point d = (d₁, d₂) and feasible set F , the Nash Bargaining Solution is the unique point that maximizes the product of gains: u, v: Utilities for players 1 and 2 d₁, d₂: Disagreement point utilities (what each player gets if negotiation fails) F: Feasible set of possible agreements The solution maximizes the product of gains from cooperation , not the sum. This ensures both players benefit proportionally from the agreement. The Nash solution has an elegant geometric interpretation: it finds the point in the feasible set where the hyperbola (u - d₁)(v - d₂) = k has the largest possible value of k while still touching the feasible set.

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