Nash bargaining game

The Nash bargaining game is a simple two-player game used to model bargaining interactions. In the Nash Bargaining Game two players demand a portion of some good (usually some amount of money). If the two proposals sum to no more than the total good, then both players get their demand. Otherwise, both get nothing. A Nash bargaining solution is a (Pareto efficient) solution to a Nash bargaining game. According to Walker (2005), Nash's bargaining solution was shown to be the same (by John Harsanyi) as Zeuthen's 1930 solution of the bargaining problem published in the book Problems of Monopoly and Economic Warfare.

Equilibrium analysis
Strategies are represented in the Nash bargaining game by a pair (x, y). x and y are selected from the interval [d, z], where z is the total good. If x + y is equal to or less than z, the first player receives x and the second y. Otherwise both get d. d here represents the disagreement point or the threat of the game; often $$d=0$$.

There are many Nash equilibria in the Nash bargaining game. Any x and y such that x + y = z is a Nash equilibrium. If either player increases their demand, both players receive nothing. If either reduces their demand they will receive less than if they had demanded x or y. There is also a Nash equilibrium where both players demand the entire good. Here both players receive nothing, but neither player can increase their return by unilaterally changing their strategy.

Nash bargaining solution
On the other hand, Nash proposed that a solution should satisfy certain axioms:


 * 1) Invariant to affine transformations or Invariant to equivalent utility representations
 * 2) Pareto optimality
 * 3) Independence of irrelevant alternatives
 * 4) Symmetry

Let us call u the utility function for player 1, v the utility function for player 2. Under these conditions, rational agents will choose what is known as the Nash bargaining solution. Namely, they will seek to maximize $$|u(x)-u(d)||v(y)-v(d)|$$, where $$u(d)$$ and $$v(d)$$, are the status quo utilities (i.e. the utiltity obtained if one decides not to bargain with the other player). The product of the two excess utilities is generally referred to as the Nash product.

Other solution: Monotonicity condition
Independence of Irrelevant Alternatives can be substituted with an appropriate monotonicity condition, thus providing a different solution for the class of bargaining problems. This alternative solution has been introduced by E. Kalai and M. Smorodinsky.

Applications
Recently the Nash bargaining game has been used by some philosophers and economists in order to explain the emergence of human attitudes toward distributive justice (Alexander 2000; Alexander and Skyrms 1999; Binmore 1998, 2005). These authors primarily use evolutionary game theory in order to explain how individuals come to believe that proposing a 50-50 split is the only just solution to the Nash Bargaining Game.