Worksheet 9: Nash Bargaining

Assume that two agents are negotiating over how best to divide their quantity of good x, which is normalized to 1. If the players reach an agreement, player 1 receives utility \(u_{1} = x\), and player 2 receives utility \(u_{2} = (1-x)\). If the players do not reach an agreement, player 1 receives a payoff of \(t1 = 0\), and player 2 receives payoff \(t_{2} = a > 0\).

Question 1

Find the Nash bargaining solution to this game.

The Nash bargaining solution is to maximize \(x(1-x-a)\) (i.e., the product of each person’s payoff under agreement less their payoff under disagreement). We can find the solution by differentiating with respect to \(x\) and setting equal to 0, which yields \(1-2x - a = 0\), or \(x=\frac{1}{2}(1-a)\).

Question 2

Explain how this solution varies with \(a\).

As \(a\) increases, \(x\) decreases. Intuitively, this means that as player 2’s outside option improves, less of the total amount \(x\) goes to player 1. In other words, if player 2 has the least to lose in the negotiation, then player 2 will extract a larger share of the joint surplus.