The Core of a Cuckold-Hotwife-Bull Triad

Welcome back. In our previous lessons, we've explored partner selection from two different angles: finding stable pairings in a market with complete information using the Gale-Shapley algorithm, and making optimal sequential choices under uncertainty with the Secretary Problem. Both models, however, primarily treat players as individuals making decisions that affect their own outcomes.

In this lesson, we shift our focus to cooperative game theory. We'll move beyond individual decision-making to analyze how groups, or coalitions, can form and maintain stable agreements. Specifically, we will address our learning outcome by defining the core of a cooperative game. The core is a powerful solution concept that identifies the set of all stable payoff distributions for a group of cooperating players. We will frame this entire discussion within the context of a triad, consisting of a cuckold, a hotwife, and a bull, to understand what constitutes a stable and sustainable relationship arrangement from a game-theoretic perspective.

From Individual Choices to Coalition Values

In a cooperative game, players can form binding agreements. The fundamental building block is not the individual player's strategy, but the value that a group of players—a coalition—can create by working together. This is captured by a characteristic function, denoted by $v (S)$ , which assigns a total payoff value to every possible coalition $S$ .

Let's model our triad.

Players (N): {Charles (C), Hannah (H), Ben (B)}
Coalitions (S): All subsets of N, from the empty set $\emptyset$ to the "grand coalition" {C, H, B}.
Utility: The "payoff" can be seen as a transferable measure of total satisfaction, happiness, and fulfillment derived from the relationships. Let's assume, for the model's sake, that this can be quantified and shared among the participants.

To make this concrete, let's assign some plausible values to the characteristic function, representing what each coalition can achieve on its own, without cooperation from outside members.

Individual Values:
- $v ({C}) = 2$ : Charles's baseline happiness on his own.
- $v ({H}) = 3$ : Hannah's baseline happiness on her own.
- $v ({B}) = 1$ : Ben's baseline happiness on his own.
Two-Person Coalitions:
- $v ({C, H}) = 8$ : The value of Charles and Hannah's relationship as a couple, without the lifestyle.
- $v ({H, B}) = 7$ : The value Hannah and Ben could create if they pursued a relationship without Charles's involvement.
- $v ({C, B}) = 4$ : The value Charles and Ben could generate, perhaps through a shared hobby or friendship, independent of Hannah.
Grand Coalition:
- $v ({C, H, B}) = 15$ : The total value created when all three successfully cooperate in a stable triad. Note that this is greater than the sum of any sub-coalitions (e.g., $v ({C, H}) + v ({B}) = 8 + 1 = 9 < 15$ ), an assumption known as superadditivity. This synergy—driven by communication, compersion, and shared experiences—is what makes the triad potentially desirable.

With the game defined, we can now ask the crucial question: If the triad forms and generates a total value of 15, how should this value be distributed among Charles, Hannah, and Ben to ensure the arrangement is stable?

The Core: The Set of Stable Agreements

A proposed distribution of the total value is called a payoff allocation or imputation, represented by a vector $x = (x_{C}, x_{H}, x_{B})$ . The core is the set of all such allocations that are stable, meaning no one has an incentive to break the agreement. For an allocation to be in the core, it must satisfy three conditions.

The following video provides an excellent conceptual overview of these conditions.

COALITION GAME, IMPUTATION & CORE

This video by Sanjoy Das introduces the key concepts of imputation and group rationality, which are the building blocks of the core.

First, watch the segment explaining the properties of a payoff vector, called an imputation (imputation segment). Pay attention to the definitions of feasibility, efficiency, and individual rationality. Then, watch the explanation of group rationality and the core (group rationality segment), which introduces the crucial idea of stability against sub-coalitions breaking away.

As the video explained, an allocation $x = (x_{C}, x_{H}, x_{B})$ is in the core if and only if it meets these criteria:

Efficiency (or Group Rationality): The total payoff distributed must equal the total value created by the grand coalition. Nothing is left on the table.
$x_{C} + x_{H} + x_{B} = v ({C, H, B}) = 15$
Individual Rationality: No individual should agree to a payoff that is less than what they could get on their own.
$x_{C} \geq v ({C}) = 2$
$x_{H} \geq v ({H}) = 3$
$x_{B} \geq v ({B}) = 1$
Coalitional Rationality (No Blocking): No sub-coalition should have an incentive to leave the grand coalition. This means the sum of payoffs to members of any sub-coalition must be at least as great as the value they could generate by themselves. If not, they could "block" the deal.
$x_{C} + x_{H} \geq v ({C, H}) = 8$
$x_{H} + x_{B} \geq v ({H, B}) = 7$
$x_{C} + x_{B} \geq v ({C, B}) = 4$

An allocation that satisfies all of these conditions is considered stable. No individual or subgroup has a rational incentive to reject the proposed deal and go their own way.

For a more formal treatment of these concepts, the following readings are invaluable. They ground the intuitive ideas from the video in precise mathematical language.

GAME THEORY

This text by Thomas Ferguson provides a very clear, step-by-step introduction to imputations and the core.

Start by reading Section 2.1, "Imputations", which defines group and individual rationality. Then, read Section 2.3, "The Core", which formally defines stability and the core as the set of stable imputations.

Is the Deal Stable? A Test Case

Let's test if a proposed payoff distribution, x = (4, 8, 3), is in the core of our triad game.

Efficiency: $4 + 8 + 3 = 15$ . Yes, this condition is met.
Individual Rationality: $x_{C} = 4 \geq 2$ , $x_{H} = 8 \geq 3$ , $x_{B} = 3 \geq 1$ . Yes, all are met.
Coalitional Rationality:
- Couple {C, H}: $x_{C} + x_{H} = 4 + 8 = 12$ . This is $\geq 8$ . They are better off.
- Lovers {H, B}: $x_{H} + x_{B} = 8 + 3 = 11$ . This is $\geq 7$ . They are better off.
- Friends {C, B}: $x_{C} + x_{B} = 4 + 3 = 7$ . This is $\geq 4$ . They are better off.

Since all conditions are met, the allocation (4, 8, 3) is in the core. It represents a stable arrangement.

Now, consider an alternative: x = (6, 6, 3).

Efficiency: $6 + 6 + 3 = 15$ . Yes.
Individual Rationality: $x_{C} = 6 \geq 2$ , $x_{H} = 6 \geq 3$ , $x_{B} = 3 \geq 1$ . Yes.
Coalitional Rationality:
- Couple {C, H}: $x_{C} + x_{H} = 6 + 6 = 12 \geq 8$ . OK.
- Lovers {H, B}: $x_{H} + x_{B} = 6 + 3 = 9 \geq 7$ . OK.
- Friends {C, B}: $x_{C} + x_{B} = 6 + 3 = 9 \geq 4$ . OK.

The allocation (6, 6, 3) is also in the core. This shows that the core is not necessarily a single point; it can be a set of possible stable outcomes. The specific outcome that is reached within the core depends on the players' bargaining abilities.

Finally, what does an unstable allocation look like? Consider x = (3, 5, 7).

Efficiency: $3 + 5 + 7 = 15$ . Yes.
Individual Rationality: All satisfied.
Coalitional Rationality:
- Lovers {H, B}: $x_{H} + x_{B} = 5 + 7 = 12 \geq 7$ . OK.
- Friends {C, B}: $x_{C} + x_{B} = 3 + 7 = 10 \geq 4$ . OK.
- Couple {C, H}: $x_{C} + x_{H} = 3 + 5 = 8$ . This is just equal to $v ({C, H})$ . Is it unstable? No, because they don't do strictly better by leaving. But let's take x' = (2.5, 5, 7.5). Now x'_C + x'_H = 7.5, which is less than 8. Charles and Hannah could block this by saying "We get a combined value of 8 on our own. You are offering us a combined 7.5. We reject this deal." This allocation is not in the core because it is blocked by the coalition {C, H}.

This concept of blocking is the essence of the core's stability requirement.

The Core - Game Theory - Indian Institute of Science

This document from the Indian Institute of Science provides a concise and formal definition of the core and blocking.

Focus on Section 2, "The Core". Read the formal definition of the core, and then pay close attention to the paragraph starting "A coalition C can improve". This explains exactly what it means for a coalition to "improve upon" or "block" an allocation.

Visualizing the Space of Stable Outcomes

For a three-player game, we can visualize the set of all possible stable outcomes. The set of all efficient allocations $x_C + x_H + x_B = 15$ forms a triangular plane in three-dimensional space. Each of the rationality constraints is a linear inequality that slices off a portion of this triangle. What remains—if anything—is the core.

A generic ternary plot illustrating the core. The large triangle represents all efficient payoff distributions. The shaded hexagonal region in the center is "The Core"—the set of outcomes that are stable against blocking by any two-player coalition. The smaller, unshaded triangles near the corners represent unstable allocations that would be blocked.

The video you watched earlier also contains an excellent step-by-step visualization of how these constraints carve out the core.

COALITION GAME, IMPUTATION & CORE

Let's revisit the video by Sanjoy Das to see a graphical construction of the core.

Watch the final, detailed graphical example from the graphical construction. Notice how the initial blue triangle of efficient payoffs is reduced to the green triangle of imputations (satisfying individual rationality), and finally to the red trapezoidal core (satisfying group/coalitional rationality).

The Limits of Transferable Utility

The concept of the core relies on the assumption of Transferable Utility (TU)—that the "value" created can be divided and transferred among players like money. In relationship dynamics, this is a significant simplification. Can you truly "transfer" emotional security, compersion, or love?

A comic by Kimchi Cuddles that touches on the nature of love and commitment in non-monogamous relationships.

In the comic, the man's question, "what percentage of your heart is mine?", is an attempt to quantify and divide a non-transferable good. The woman's reply, "It doesn't work that way!" and "When I'm with you, I'm 100% with you," highlights the limitation of this thinking.

While our model of the core is powerful for analyzing the stability of agreements based on tangible or quasi-tangible benefits (shared time, financial support, lifestyle access), it's crucial to remember that it doesn't fully capture the non-divisible, emotional aspects of a relationship. The core tells us what is rationally stable, but it doesn't tell us what is emotionally fulfilling. As a relationship analyzer, you know that a rationally stable but emotionally empty arrangement is not a successful one.

Conclusion

In this lesson, we have defined the core as a central solution concept in cooperative game theory. It provides a rigorous method for identifying the set of stable payoff distributions in a cooperative endeavor, like a triad.

Key Takeaways:

Cooperative games are defined by a characteristic function $v (S)$ that specifies the value each possible coalition $S$ can generate.
The core is the set of all payoff allocations that are stable.
For an allocation to be in the core, it must be:
1. Efficient: The entire value of the grand coalition is distributed.
2. Individually Rational: Every player gets at least what they could get on their own.
3. Coalitionally Rational: No subgroup has an incentive to break away and form its own coalition (i.e., the allocation is not "blocked").
If the core is empty, no stable arrangement exists under the given value function. If it is large, many stable solutions are possible, and the outcome depends on negotiation.
The model assumes transferable utility, a strong simplifying assumption when applied to the complex, non-transferable emotions within relationships.

Preview of the Next Lesson:

We have now established the tool for analyzing the stability of a coalition. In the final lesson of this module, we will apply this concept. We will compare the stability of different relationship structures, such as a V-polyandry arrangement versus a fully integrated triad, by constructing their respective characteristic functions and analyzing the properties of their cores.

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