Game theory -  A Tutorial

Jean Walrand

Cooperative: Nash Bargaining Equilibrium

Consider two players that want to split $1.00. If P1 gets x, his utility is F(x) and P2’s utility is G(1—x). What is a reasonable value for x? Nash argues that x should be chosen to maximize F(x)G(1—x).

Here is an axiomatic justification illustrated by the figure below.

If the possible rewards are as shown on the left, Nash postulates that (1/2, 1/2) is the fair choice.

If the rewards are as in the middle figure, then Nash says that (1/2, 1/2) is still the fair choice: It was fair before and is still possible, so it should still be the fair choice.

Now, the figure on the right is similar to that in the middle, except that the units of money have been changes, corresponding to a scaling of the axes. Nash argues that fairness should not depend on the units, so that the point (a1/2, a2/2) should be the fair choice.

Accepting these three axioms, we see that the fair choice on the right is such that any deviation from it in the set of feasible rewards results in a negative sum of relative increases of the two rewards.  That is, any deviation decreases the product of the rewards.  Hence, the Nash bargaining equilibrium maximizes the product of the rewards.

See [2] for more details.