A new treatment is presented of a classical economic problem, one which occurs in many forms, as bargaining, bilateral monopoly, etc.

— John Nash, The Bargaining Problem (1950)

In 1948, still an undergraduate at the Carnegie Institute of Technology, Nash took a course in international economics, the only economics course of his life. He came out of it with a question: when two people negotiate, how do they share the gains from a deal? Economists thought the answer depended on each side’s psychology, skill and bluffing, and that there was nothing to say in general. Nash showed that, if the deal must satisfy a few reasonable requirements, there is only one possible answer. The paper, written before he arrived at Princeton, was published in 1950 in the journal Econometrica.

The problem

Two people, 1 and 2, can make a deal. Each possible deal gives each of them a gain, measured as utility: $u_1$ and $u_2$. If there is no deal, each keeps what they already had, the disagreement point $(d_1,d_2)$. A deal is only of interest if it gives each person at least what they would have without it.

The four requirements

Nash asked the solution to have four properties:

  1. Efficiency. There must be no other deal that is better for both at the same time.
  2. Symmetry. If the situation is symmetric, with the same possibilities and the same disagreement point for both, both gain the same.
  3. Invariance. The solution does not change if one person’s utility is measured on another scale, as one swaps degrees Celsius for Fahrenheit.
  4. Independence of irrelevant alternatives. If the solution of a problem is still possible in a smaller problem, it is still the solution. Removing options nobody would choose does not change the deal.

Nash’s theorem says that there is a single rule with the four properties: choose the deal that maximises the product of the gains over the disagreement point,

$$\boxed{(u_1-d_1)\cdot(u_2-d_2).}$$

Sharing 10 dollars

Two people are going to share 10 dollars. If there is no deal, nobody gets anything: $d=(0,0)$. Suppose, for simplicity, that utility is the money itself. If person 1 receives $x$, person 2 receives $10-x$, and the product is

$$P(x)=x(10-x)=25-(x-5)^2.$$

The maximum is at $x=5$: each takes 5 dollars. No surprise, in a symmetric case.

Whoever has a good alternative gets more

Now suppose person 1 has an alternative: without a deal, she can get 4 dollars elsewhere, and person 2 still gets nothing. The disagreement point is $(4,0)$. The gains over the disagreement point are $x-4$ and $10-x$, and the product is

$$P(x)=(x-4)(10-x)=9-(x-7)^2,\qquad4\leq x\leq10.$$
The product of the gains over the disagreement point, (x − 4)(10 − x). It is zero when x = 4, because person 1 gains nothing from the deal, and when x = 10, because person 2 gains nothing. The maximum, 9, is at x = 7.
The product of the gains over the disagreement point, (x − 4)(10 − x). It is zero when x = 4, because person 1 gains nothing from the deal, and when x = 10, because person 2 gains nothing. The maximum, 9, is at x = 7.Illustration by Euler’s Eyes

The maximum is at $x=7$: the split is 7 for person 1 and 3 for person 2. Each gains 3 dollars more than they would have without a deal. The Nash solution splits the surplus equally, the value the deal creates beyond what each had already secured. That is why whoever has a good alternative gets more: it is not handed over as a gift, it is the starting point of the negotiation.

A calculation with different utilities

Money is not always worth the same to both. Suppose that, for person 2, who is poorer, the utility of $y$ dollars is $\sqrt y$: the first dollars are worth a lot, the following ones less and less. With disagreement point $(0,0)$, we maximise $x\sqrt{10-x}$. Differentiating,

$$\sqrt{10-x}-\frac{x}{2\sqrt{10-x}}=0\quad\Longrightarrow\quad2(10-x)=x\quad\Longrightarrow\quad x=\frac{20}3\approx6.67.$$

Person 1 takes 6.67 dollars and person 2 takes 3.33. Whoever is more eager for any deal, because each dollar matters more to them, ends up with less. Nash’s mathematics does not say whether this is fair: it says what follows from the four requirements.

Exercise: with a total of 12 and disagreement point (2, 0), what is the Nash split, with utility equal to money?

We maximise $(x-2)(12-x)=-x^2+14x-24=25-(x-7)^2$. The maximum is at $x=7$: the split is (7, 5). The surplus is $12-2=10$, and each gains 5 over the disagreement point.

Exercise: show that, with utilities equal to money, total T and disagreement point (d₁, d₂), person 1 receives d₁ + (T − d₁ − d₂)/2.

The product is $(x-d_1)(T-x-d_2)$. It is a parabola in $x$ with roots at $x=d_1$ and $x=T-d_2$, and the maximum lies halfway between them: $x=\frac{d_1+T-d_2}2=d_1+\frac{T-d_1-d_2}2$. Each receives their disagreement payoff plus half of the surplus $T-d_1-d_2$.

Sources: John F. Nash, “The bargaining problem”, Econometrica 18 (1950) · Ken Binmore, Game Theory: A Very Short Introduction (Oxford, 2007) · 1994 Nobel Prize in Economics, Nobel Foundation.

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