# Nash Bargaining Solution

Prior to the introduction of game theory, economic theory had simply two rationality postulates concerning two-person bargaining situations.

- individual rationality: A rational bargainer will not agree to a utility payoff smaller than his conflict payoff (commodity/money payoffs)
- joint rationality: two rational bargainers will not
agree on a utility outcome u = (u
_{1}, u_{2}) if in the feasible set F there is another utility outcome u' = (u'_{1}, u'_{2}) yielding higher payoffs for both of them

In 1930 Danish economist Zeuthen realized the need for a strong bargaining theory predicting a unique agreement point. Von Neumann and Morgenstern sought to solve this problem, however their work did not move beyond the weak bargaining theory of neoclassical economics. But they did introduce von Neumann-Morgenstern utility functions which offered a rigorous and convenient mathematical representation for the various players attitudes toward risk taking—an essential prerequisite for a strong bargaining theory.

John Nash was
the first author to use the tools of game theory to propose a strong bargaining
theory. Firstly, he assumed that the feasible set F of outcomes, defined in
terms of von Neumann-Morgenstern utilities is a compact and convex set and
postulated an agreement point u = (u_{1}, u_{2}) that satisfy the
four axioms:

- Efficiency: similar to the definition of joint rationality
- Symmetry: a symmetric game will have a symmetric
agreement point u = (u
_{1}, u_{2}) with u_{1}= u_{2}. (In a symmetric game the two players have exactly the same strategic possibilities and have exactly the same bargaining power. Therefore, neither player will have any reason to accept an agreement yielding him a lower payoff than his opponent's.) - Linear Invariance: because von Neumann-Morgenstern utility functions are behaviorally equivalent if one can be obtained from the other by an order-preserving linear transformation, if one game G' can be obtained from game G by subjecting player 1's utilities to an order-preserving linear transformation while keeping the other player's utilities unchanged, then a change in the utility outcome will not affect the physical outcome chosen.
- Independence of irrelevant alternatives: let G be a bargaining game with conflict point, with feasible set F, and with agreement point u; and let G' be a game obtained from G by restricting the feasible set to a smaller set F' contained in F in such a way that c and u remain within the new feasible set F', c remaining the conflict point also for G'. Then u will be the agreement point also for this new game G'.

The agreement point of the game is the unique utility vector
u = (u_{1}, u_{2}) that maximizes the product
(u_{1} - c_{1})(u_{2} - c_{2})
[called the Nash product],
subject to (u_{1}, u_{2})
in F and to u_{1} ≥ c_{1} and u_{2} ≥ c_{2} (where
c_{1}, c_{2} are the respective components to the conflict
payoff)

This point is known as the Nash Bargaining Solution.