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 $t 1 = 0$ , 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 $1 - 2 x - a = 0$ , 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.