Nash Bargaining for Polycules

Welcome back. In our last lesson, we established that every negotiation is defined by its alternative: the disagreement outcome. You learned to identify the threat points for each person in a vee relationship, understanding that these points—whether they represent a return to the status quo, temporary exclusion, or a full breakup—are the ultimate source of bargaining power.

This lesson takes the next logical step. Now that we know how to define the stakes of a failed negotiation, how can we predict the outcome of a successful one? We will focus on the most celebrated concept in bargaining theory: the Nash Bargaining Solution. Your goal is to apply the Nash bargaining solution to time-allocation disputes between the hinge and partners. This powerful tool will allow you to move beyond simply identifying power imbalances to proposing a specific, mathematically justified "fair" resolution.

From Disagreement to Agreement: The Nash Bargaining Solution

Once we have the set of all possible agreements (the feasible set $S$ ) and the disagreement point $d$ , how do we single out one "solution"? In 1950, John Nash proposed a solution by working backward. Instead of picking a solution and justifying it, he proposed a set of reasonable properties, or axioms, that any "fair" solution ought to satisfy. He then proved that there is only one solution that satisfies them all.

Your background in statistics and formal logic will make the elegance of this axiomatic approach particularly clear. The power of the Nash Bargaining Solution (NBS) comes not from an arbitrary formula, but from its foundation in these compelling principles of fairness and rationality.

The Two Person Bargaining Problem - Game Theory lab

This document provides a concise and formal overview of the Nash bargaining framework. It begins by laying out the axioms that uniquely define the solution.

Please read the section titled "The Axioms of Nash" (from the start of Section 3 to the end of Section 3.5). As you read, focus on the intuition behind each of the five axioms: Strong Efficiency: No value is left on the table. The agreement should be on the Pareto frontier. Individual Rationality: The agreement must be better for everyone than the disagreement point. No one should be forced into a deal that is worse than their outside option. Scale Covariance: The solution should not depend on the units used to measure utility (e.g., whether you measure happiness in "utils" or "mega-utils"). Independence of Irrelevant Alternatives (IIA): If we remove options that wouldn't have been chosen anyway, the final solution shouldn't change. Symmetry: If two players are in identical situations (same utility functions, same disagreement payoffs), they should get the same outcome.

The Nash Product: Finding the Solution

Nash proved that the unique solution satisfying these axioms is the one that maximizes the Nash Product. For a two-player negotiation between Player 1 and Player 2, with utilities $u_{1}$ and $u_{2}$ and a disagreement point $d = (d_{1}, d_{2})$ , the solution is the feasible utility profile $(u_{1}^{*}, u_{2}^{*})$ that maximizes the product of the gains over the disagreement point:

Maximize (u_{1} - d_{1}) (u_{2} - d_{2})

This function elegantly captures the tension and shared interest in a bargain. Each player wants their own term $(u_{i} - d_{i})$ to be as large as possible, but because the potential outcomes are linked (usually, for one person to get more, the other must get less), they must find a balance.

The following document formally states this theorem.

The Two Person Bargaining Problem - Game Theory lab

This reading presents the main theorem that connects the axioms to the maximization problem.

Now, read Section 3.6, "The Nash Bargaining Solution" (the main theorem). This section formally presents the optimization problem that yields the unique solution. Note the term argmax, which you'll recognize from your statistics background as the argument (input) that maximizes the function.

Geometrically, the solution can be visualized as finding the point on the frontier of feasible outcomes that creates the largest possible rectangle when the disagreement point is used as the opposite corner.

This graph illustrates the key components of a bargaining problem. The NBS finds the specific point on the "efficiency line" (the Pareto frontier) that maximizes the area of the rectangle whose corners are the disagreement point and the solution point.

Application: A Time-Allocation Dispute in a Vee

Let's apply this to a core conflict in a vee: a hinge (H) and a partner (P1) are negotiating how to divide a block of discretionary time, say, the 7 evenings of the week. For simplicity, we model this as a two-person bargain.

1. Define the Bargaining Problem:

Players: Hinge (H), Partner 1 (P1).
Resource: $T = 7$ evenings. Let $t$ be the number of evenings P1 gets, so H gets $7 - t$ .
Utility: People experience diminishing marginal utility from time together. The fifth evening in a week is nice, but not as crucial as the first. We can model this with a square-root function, a standard choice in economics. Let's assume their utilities are $u_{P 1} (t) = t$ and $u_{H} (t) = 7 - t$ .
Disagreement Point (d): As we discussed last lesson, this is key. Let's assume the most basic disagreement: a breakdown in talks means neither H nor P1 gets to use this specific discretionary time for their relationship. Thus, their utility gains are zero. The disagreement point is $d = (d_{P 1}, d_{H}) = (0, 0)$ .

2. Set Up and Maximize the Nash Product:

We need to find the allocation of time $t$ that maximizes the Nash Product:

Maximize (u_{P 1} (t) - d_{P 1}) (u_{H} (t) - d_{H}) = (t - 0) (7 - t - 0) = t (7 - t)

Maximizing $f (t)$ is the same as maximizing $f (t)$ . So we can simplify the problem to maximizing $t (7 - t) = 7 t - t^{2}$ . As you know from calculus, we find the maximum by taking the derivative with respect to $t$ and setting it to zero:

\frac{d}{d t} (7 t - t^{2}) = 7 - 2 t = 0

Solving for $t$ , we get $t = 3.5$ .

The Nash Bargaining Solution suggests an equal split: 3.5 evenings for P1 and 3.5 evenings for H. This result is a direct consequence of the symmetrical setup: the players had identical utility functions and a symmetrical disagreement point.

The Power of Asymmetry: Introducing Outside Options

The real power of the NBS lies in how it handles asymmetries. What if the hinge's disagreement point isn't zero? In our last lesson, we discussed how the hinge's primary source of power is their ability to turn to the other partner (P2).

Let's modify the scenario: If negotiations with P1 break down, H can spend all 7 evenings with P2. Suppose H's relationship with P2 gives H a baseline utility of, say, $u_{H} = 3$ (the utility they'd get from 3 nights, representing a decent but not perfect alternative).

The new disagreement point is $d = (d_{P 1}, d_{H}) = (0, 3)$ .

The Nash Product is now:

Maximize (t - 0) (7 - t - 3)

Solving this is more involved, but the conceptual point is paramount: because $d_{H}$ is now positive, the entire expression shifts. To maximize the product, the term $(7 - t - 3)$ can be smaller, meaning $7 - t$ can be smaller, which in turn means $7 - t$ can be smaller. The solution will now allocate $t < 3.5$ to P1, and $7 - t > 3.5$ to H.

The hinge's better outside option translates directly into a larger share of the negotiated resource. This formalizes the intuition we built in the last lesson. Economists studying household bargaining use this exact logic, where threat points are determined by divorce prospects or non-cooperative behavior within the family.

[PDF] Cooperative Household Models - IZA Institute of Labor Economics

This paper from the IZA Institute of Labor Economics formalizes the application of Nash Bargaining to household economics. It directly addresses the nature of the disagreement point.

Please read the section "Nash Bargaining" (up to the formulation). Notice the explicit discussion of "divorce threat points" and "non-cooperative (Nash equilibrium) behaviour" as the basis for the disagreement utilities. This mirrors our vee scenarios of relationship dissolution or reversion to a less cooperative state. Then, look at Equation (3) (and the surrounding text), which is exactly the Nash Product we have been discussing, where v_i^0 are the threat point utilities.

A Worked Example

Let's solidify this with a concrete problem. Imagine two partners are negotiating an agreement. If they fail, they will play a simple non-cooperative game, and the resulting Nash Equilibrium payoff becomes their disagreement point.

This image presents a non-cooperative game matrix. The problem is to find the Nash Bargaining Solution, assuming the disagreement point d' is the payoff from the Nash Equilibrium of this underlying game.

Step 1: Find the Disagreement Point (The Nash Equilibrium)

First, we must find the Nash Equilibrium (NE) of the game in the image.

For Player 2 (columns), the strategy C is strictly dominated by R (3 > 2 and 0 > -2). So, we can eliminate column C.
The game reduces to:
| | L | R |
|---|---|---|
| T | 3,0 | -1,3 |
| B | 0,-1 | 0,0 |
Now, for Player 2, R is a strictly dominant strategy over L in this reduced game (3 > 0 and 0 > -1).
Since a rational Player 2 will always play R, Player 1 simply needs to choose their best response to R. Comparing T (-1 payoff) and B (0 payoff), Player 1 will choose B.
Therefore, the unique Nash Equilibrium is (B, R), with the payoff vector (0, 0).

This is our disagreement point: $d^{'} = (0, 0)$ .

Step 2: Define the Feasible Set and Frontier

The feasible set of payoffs $S^{'}$ is the convex hull of the payoffs from all pure strategies: (3,0), (1,2), (-1,3), (0,-1), (-2,-2), and (0,0). The "efficiency line" or Pareto frontier is the upper-right boundary of this set. It is formed by the line segments connecting (-1, 3), (1, 2), and (3, 0).

Segment 1 (between (1,2) and (3,0)): The equation is $u_{1} + u_{2} = 3$ .
Segment 2 (between (-1,3) and (1,2)): The equation is $u_{1} + 2 u_{2} = 5$ .

Step 3: Maximize the Nash Product

We need to maximize $(u_{1} - 0) (u_{2} - 0) = u_{1} u_{2}$ subject to $(u_{1}, u_{2})$ being on the Pareto frontier.

On Segment 1 ( $u_{2} = 3 - u_{1}$ ): We maximize $u_{1} (3 - u_{1})$ . The maximum occurs at $u_{1} = 1.5$ , giving the point $(1.5, 1.5)$ . This is a valid point on the segment. The Nash product is $1.5 \times 1.5 = 2.25$ .
On Segment 2 ( $u_{1} = 5 - 2 u_{2}$ ): We maximize $(5 - 2 u_{2}) u_{2}$ . The maximum occurs at $u_{2} = 1.25$ , which gives $u_{1} = 5 - 2 (1.25) = 2.5$ . The point $(2.5, 1.25)$ is not on this segment of the frontier (which ends at $u_{1} = 1$ ). This means the maximum on this segment must be at one of the endpoints. The value of $u_{1} u_{2}$ at (1,2) is 2, and at (-1,3) is -3.

Comparing the candidate values, the maximum Nash Product of 2.25 is achieved at the utility profile (1.5, 1.5).

The Agreement: The Nash Bargaining Solution is the payoff $(1.5, 1.5)$ . This outcome can be achieved if the players agree to a binding contract that uses a correlated strategy (a lottery over pure strategies). Specifically, they can agree to play (T,L) with probability $p$ and (T,C) with probability $1 - p$ . The expected payoff is $p (3, 0) + (1 - p) (1, 2) = (3 p + 1 - p, 2 - 2 p) = (2 p + 1, 2 - 2 p)$ . Setting this equal to (1.5, 1.5), we get $2 p + 1 = 1.5 ⟹ 2 p = 0.5 ⟹ p = 0.25$ . This is consistent for the second payoff: $2 - 2 (0.25) = 1.5$ . So, the agreement is: "We commit to a randomized strategy of playing (T,L) 25% of the time and (T,C) 75% of the time."

Conclusion

In this lesson, you have moved from the "what if" of disagreement to the "what now" of agreement. The Nash Bargaining Solution provides a powerful, axiomatically-grounded method for resolving disputes.

Key Takeaways:

The Nash Bargaining Solution (NBS) finds the unique outcome that is efficient, rational, and abides by principles of symmetry and independence from irrelevant alternatives.
It is found by maximizing the Nash Product: the product of each player's utility gain over their disagreement point.
In a vee context, the NBS can be used to model time-allocation disputes. Symmetrical situations lead to equal splits.
Crucially, asymmetries in outside options (threat points) directly influence the outcome. A partner with a better alternative will secure a larger share of the resource, a principle the NBS quantifies precisely.

You are now able to not only identify sources of power in a negotiation but also calculate a specific, proposed fair outcome. In our next lesson, we will explore a different model of negotiation, the Rubinstein alternating-offers model, which focuses on the process of negotiation over time and how patience itself can be a source of bargaining power.

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