9.2 Nash bargaining solution

Nash Bargaining Solution

The Nash bargaining solution gives a precise answer to a fundamental question in cooperative game theory: when two players can negotiate and make binding agreements, what outcome should they agree on? It does this by identifying the unique point that maximizes both players' gains relative to what they'd get if negotiations broke down, while satisfying a set of four axioms that capture our intuitions about fair and efficient bargaining.

Definition and Properties

The Nash bargaining solution applies to two-player bargaining problems where players can communicate freely and commit to binding agreements. It produces a single, unique outcome.

What makes it special is that it's the only solution satisfying all four of these axioms simultaneously:

• Pareto optimality: No other feasible outcome can make one player better off without making the other worse off. The solution always lands on the efficient frontier.
• Symmetry: If the two players have identical utility functions and face a symmetric feasible set, they receive equal payoffs. The solution doesn't arbitrarily favor one side.
• Scale invariance (invariance to affine transformations): If you rescale or shift a player's utility function (e.g., measuring in different units), the solution outcome doesn't change. This matters because utility functions are only meaningful up to affine transformations.
• Independence of irrelevant alternatives (IIA): If the solution picks outcome $A$ from a large feasible set, and you shrink that set but $A$ is still available, the solution still picks $A$. Removing options that weren't chosen doesn't change the answer.

Mathematical Formulation

The Nash bargaining solution $(x^*, y^*)$ is the point in the feasible set that maximizes the Nash product:

$(x^* - d_1)(y^* - d_2)$

where:

• $d_1$ and $d_2$ are the disagreement payoffs (also called the threat point or status quo point) for players 1 and 2
• The disagreement point $(d_1, d_2)$ represents what each player receives if negotiations fail entirely
• The feasible set $S$ is the set of all achievable payoff pairs $(x, y)$

The solution picks the point on the Pareto frontier of $S$ where the product of both players' gains over disagreement is largest. Geometrically, you're finding the point on the frontier where the hyperbola $(x - d_1)(y - d_2) = k$ is tangent to the boundary of the feasible set, for the highest possible value of $k$.

For a convex feasible set (the standard assumption), this maximizer is unique.

Applying the Nash Bargaining Solution

Two-Player Bargaining Situations

The Nash bargaining solution applies to any setting where two parties negotiate over a surplus: labor-management wage negotiations, splitting profits in a joint venture, international treaty terms, or dividing a shared resource.

To find the solution:

1. Determine the disagreement point $(d_1, d_2)$: what each player gets if no deal is reached.
2. Define the feasible set $S$: all payoff pairs $(x, y)$ that are achievable through some agreement.
3. Maximize the Nash product $(x - d_1)(y - d_2)$ over all $(x, y) \in S$.

Worked Example

Suppose two firms are splitting a joint profit of 100. Firm A's disagreement payoff is 10 (it can earn this independently), and Firm B's is 20.

The feasible set is all pairs $(x, y)$ where $x + y \leq 100$ and $x \geq 10$, $y \geq 20$.

You need to maximize:

$(x - 10)(y - 20)$

subject to $x + y = 100$ (since Pareto optimality pushes us to the boundary).

Substituting $y = 100 - x$:

$(x - 10)(80 - x)$

Take the derivative, set it to zero:

$80 - x - (x - 10) = 0$

$90 - 2x = 0$

$x^* = 45, \quad y^* = 55$

So Firm A gets 45 and Firm B gets 55. Each firm's gain over disagreement is 35, which makes sense: with a linear frontier and equal bargaining weight, the surplus (100 - 10 - 20 = 70) is split equally. But notice that Firm B ends up with a higher total payoff because its disagreement point was higher. The Nash solution splits the surplus equally, not the total pie.

General Step-by-Step Procedure

1. Identify the two players and what they're negotiating over.

2. Determine each player's utility function, mapping outcomes to numerical payoffs.

3. Establish the disagreement point $(d_1, d_2)$.

4. Define the feasible set $S$, including any constraints.

5. Solve for $(x^*, y^*)$ that maximizes $(x - d_1)(y - d_2)$ within $S$.

6. Verify the solution is on the Pareto frontier (no feasible point makes both players strictly better off).

For problems with a linear frontier $x + y = V$, the solution always splits the surplus $V - d_1 - d_2$ equally between the two players.

Efficiency and Fairness of Nash Bargaining

Pareto Efficiency

The Nash bargaining solution always selects an outcome on the Pareto frontier of the feasible set. This is guaranteed directly by the Pareto optimality axiom.

At any interior point (off the frontier), you could increase one player's payoff without decreasing the other's, which would also increase the Nash product. So the maximizer of the Nash product can never be an interior point.

This means the solution never "wastes" available surplus. In a resource allocation problem, for instance, no reallocation could improve one player's outcome without hurting the other.

Fairness Considerations

The symmetry axiom ensures that players who are identical in every relevant respect get equal payoffs. But "fair" and "symmetric" aren't always the same thing in practice.

A few important limitations:

• Disagreement point advantage: A player with a strong outside option (high $d_i$) captures more of the total payoff, even if both players contribute equally to the surplus. The solution rewards having a good fallback position.
• No process considerations: The Nash solution is purely outcome-based. It doesn't model how negotiations actually unfold, who makes offers first, or whether one side has more patience.
• Bargaining power isn't modeled explicitly: The basic Nash solution treats players symmetrically. The generalized Nash bargaining solution addresses this by introducing bargaining power weights $\alpha$ and $1 - \alpha$, maximizing $(x - d_1)^\alpha (y - d_2)^{1-\alpha}$. A higher $\alpha$ shifts the outcome toward player 1.

Nash Bargaining vs Other Solutions

Kalai-Smorodinsky Bargaining Solution

The Kalai-Smorodinsky solution replaces the IIA axiom with individual monotonicity: if the feasible set expands in a way that increases one player's maximum possible payoff, that player should not be made worse off.

The Nash solution can actually violate this. If the feasible set grows in player 1's favor, the Nash solution might paradoxically give player 1 less in certain cases.

The Kalai-Smorodinsky solution works by:

1. Finding each player's maximum feasible gain over disagreement (the "ideal point").
2. Drawing a line from the disagreement point to the ideal point.
3. Selecting the Pareto efficient outcome on that line.

This preserves the ratio of players' maximum possible gains. If player 1 could gain at most 60 and player 2 could gain at most 30, the solution ensures player 1's actual gain is twice player 2's.

Egalitarian and Utilitarian Solutions

• Egalitarian solution: Equalizes the gains from disagreement, so $x^* - d_1 = y^* - d_2$. This prioritizes equality of surplus but may sacrifice total efficiency if the Pareto frontier isn't symmetric.
• Utilitarian solution: Maximizes the sum $x + y$ over the feasible set. This produces the highest total welfare but can give one player almost everything if the frontier is shaped that way.

These represent different tradeoffs. The Nash solution sits between the two: it's more equality-conscious than the utilitarian solution (because the product penalizes extreme imbalances more than the sum does) but less rigid about equality than the egalitarian approach. Choosing among them depends on which axioms you find most compelling for the situation at hand.