🤝

A union and a firm are dividing the gains from a deal that is worth more together than apart. Both know that if they walk away, each has a fallback. Those fallbacks, more than any notion of fairness, decide who gets what.

Nash bargaining

Two parties can split a surplus many ways. Nash asked a different question: not how bargaining unfolds, but what a fair, sensible split should satisfy. A few reasonable axioms pin down a single answer, and it turns out to hinge on what each side gets if the talks collapse.

From earlier lessons

Two quick ones before we start.

A risk-neutral player can take 40 for sure or a 50% chance of 100 and otherwise 0. Which has higher expected payoff?

A shared signal that beats the mixed Nash equilibrium of Chicken for both players is a...

Predict, then play

You and another party split a pie worth 1. If you fail to agree, you get your outside option and they get nothing. Drag your share, and watch the Nash product, the thing the fair solution maximizes. Then give yourself a better fallback and see the split move.

Why did that happen?

Take a guess, then reveal the reason

Your outside option is what you can walk away with, so it sets a floor on any deal you would accept. Improve that floor and you can credibly refuse worse offers, which pulls the settlement in your direction. It feels backwards at first, a stronger fallback sounds like less need to bargain, but in fact it is leverage: the other side must give you more to keep you at the table.

The math

Let the pie be 1, your share $x$ , theirs $1 - x$ , and the disagreement point $d = (d_1, d_2)$ , what each gets if talks fail. The Nash bargaining solution maximizes the Nash product

(x - d_1)\,(1 - x - d_2).

Differentiating and setting to zero gives

x^* = \frac{1 + d_1 - d_2}{2}.

With $d = (0,0)$ this is $\tfrac12$ , an even split. With $d_1 = 0.2$ and $d_2 = 0$ it is $0.6$ : your stronger fallback earns you more.

Definition, the four axioms

Nash showed this is the unique split satisfying: efficiency (no waste, the whole pie is divided), symmetry (if the two sides are interchangeable, they split evenly), invariance to affine rescaling of either party’s utility, and independence of irrelevant alternatives (removing unchosen options does not change the outcome). One split meets all four: the one that maximizes the Nash product.

This is an axiomatic answer: it says what a fair split looks like, not how haggling reaches it. The next lesson gives the strategic story.

Work it out

Maximize the Nash product

With $d = (0,0)$ , maximize $x(1 - x)$ . The derivative $1 - 2x = 0$ gives $x = \tfrac12$ : the even split.

Now let your outside option be $d_1 = 0.2$ (theirs still 0). Maximize $(x - 0.2)(1 - x)$ . The derivative is $(1 - x) - (x - 0.2) = 1.2 - 2x$ , zero at $x = 0.6$ . Your improved fallback raised your share from $0.5$ to $0.6$ . The other side’s collapse value, not their need, is what shrank their piece.

Key idea

The Nash solution splits the surplus to maximize the product of each side’s gain over its walk-away value. A better outside option is bargaining power: it wins you a larger share, not a smaller one.

Check yourself

Check yourselfapply

Your outside option improves while the other side's stays at zero. What happens to your Nash-bargaining share?

Check yourselfanalyse

The Nash bargaining solution is called axiomatic. What does that mean here?

Check yourselfbrings back: nash equilibrium

How does the disagreement point here play a role like a benchmark you have seen before?

Sources

Nash, J. F. (1950). The bargaining problem. Econometrica, 18(2), 155-162.