Coalition Formation in a Vee: Dyads vs. Grand Coalition

The Central Dilemma: The Grand Coalition vs. Sub-coalitions

As we touched upon previously, the assumption of superadditivity—where merging coalitions is always at least as good as keeping them separate—suggests that the grand coalition $\{H,P_1,P_2\}$ should be the preferred outcome, as it maximizes the total value generated. However, this only looks at the size of the total pie, not how it's sliced.

The fundamental tension in coalition formation is that individuals are not trying to maximize the group's total payoff; they are trying to maximize their own payoff. A smaller, more exclusive coalition might offer a player a larger share of a smaller pie, making it a more attractive option than joining the grand coalition.

This brings us to the two central questions of coalitional game theory.

In a vee, this means we must analyze the incentives for forming the grand coalition versus the three possible dyadic sub-coalitions: $\{H,P_1\}$, $\{H,P_2\}$, and the particularly interesting $\{P_1,P_2\}$.

A Foundational Model: Caplow's Theory of Triads

One of the earliest attempts to formalize the logic of coalition formation came from sociologist Theodore Caplow. He analyzed triads (groups of three) based on the distribution of power, or what game theorists often call Resource Holding Potential (RHP). In a relationship context, RHP can be thought of as a partner's influence, derived from factors like financial resources, time availability, emotional stability, or social connections.

Caplow identified eight distinct types of triads based on the relative power of the three members (A, B, and C).

To understand this classic approach, let's examine the foundational theory.

Let's apply the logic from the reading to our vee context. Imagine a situation where the Hinge (H) is the most powerful individual member, but their power is less than the combined power of the two partners ($P_1$ and $P_2$). This maps directly onto Caplow's Type 6 in the diagram, where $A>B>C$ and $A<(B+C)$.

In this scenario:

• The Hinge (A) is the strongest single player.
• The partners (B and C) are individually weaker.
• Critically, the partners can form an "all-up" coalition $\{P_1,P_2\}$ that is stronger than the Hinge.

Caplow's theory predicts that in such a situation, one of the weaker players will form a coalition with another player to dominate the third. The most likely outcomes are a coalition between A and C, or between B and C. The intuition is that weaker members will seek alliances to improve their position. This brings us to a famous and often counter-intuitive phenomenon.

The Strength-is-Weakness Effect

Caplow's work laid the foundation for what is now known as the "Strength-is-Weakness" effect: the most powerful individual in a group is often excluded from the winning coalition. Why? Because the weaker members realize they can form a "cheapest winning coalition" without the powerful member, thereby securing larger shares of the payoff for themselves.

This is driven by a powerful social heuristic: the equity norm, which states that payoffs should be proportional to the resources contributed. The strongest player, adhering to this norm, will demand the largest share of any coalition's payoff, making them an unattractive partner.

The experimental results from the 5(432) game are striking. Player A has 4 units of power, B has 3, and C has 2. Any two-player coalition is a "winning" one. The findings show:

• The BC coalition ($\{P_1,P_2\}$ in our vee analogy) is formed about 65-67% of the time.
• The strongest player, A (our Hinge), is excluded from the final coalition a majority of the time.
• This happens because A demands a larger share of the payoff (e.g., ~$60 out of $100), making a deal with A less profitable for B or C than a deal with each other.

Coalition Possibilities in the Vee

Synthesizing these concepts, we can analyze the likely coalition structures in a vee relationship $\{H,P_1,P_2\}$:

1. The Grand Coalition $\{H,P_1,P_2\}$: This structure is most likely to be stable when there are significant synergies that only arise from full cooperation. This means $v(\{H,P_1,P_2\})$ must be substantially greater than the value of any other combination of coalitions (e.g., $v(\{H,P_1\})+v(\{P_2\})$). Examples of such synergy include the hinge experiencing profound compersion, or the partners providing a robust mutual support system that simplifies life for all three. Even then, its stability depends on a division of this surplus value that all three members perceive as fair—a topic we'll tackle in the next lesson.

2. The "All-Up" Dyad ($\{P_1,P_2\}$): This dyad forms as a strategic bloc within the vee. This is the Strength-is-Weakness effect in action. $P_1$ and $P_2$ might not have a romantic relationship, but they can form a coalition to increase their collective bargaining power with $H$. For instance, if both feel they are getting insufficient time with $H$, they might agree to present a united front, demanding a new scheduling arrangement. Their coalition is a "cheapest winning coalition" to achieve this specific goal, effectively leveraging their combined influence against the "stronger" Hinge.

3. The "Bridging" or "All-Down" Dyad ($\{H,P_1\}$ or $\{H,P_2\}$): This involves the Hinge and one partner forming a dominant bloc that can sideline or exclude the third member. This might happen if the RHP is distributed such that the Hinge and one partner together are significantly more powerful than any other combination, or if the synergy in one dyad far outweighs the other. For example, if $v(\{H,P_1\})$ is very high, while $v(\{H,P_2\})$ is low, and the $P_1,P_2$ metamour relationship is strained ($v(\{P_1,P_2\})\approx 0$), then the $\{H,P_1\}$ dyad might operate as the core of the relationship, with $P_2$ in a peripheral position. In the extreme, this can lead to the dissolution of the vee and the exclusion of one partner.

The structure that emerges depends critically on the distribution of power (RHP) and the perceived fairness of payoff distributions (the equity norm).

Conclusion

In this lesson, we moved from defining the potential value of coalitions to analyzing which are likely to form. We've seen that while the grand coalition may offer the most collective benefit, individual incentives often lead to the formation of smaller, dyadic sub-coalitions.

• Key Takeaways:
  • The formation of a coalition depends on whether it offers individual members a better outcome than other available options.
  • Caplow's theory of triads provides a classic framework for analyzing coalition formation based on the distribution of power (RHP).
  • The Strength-is-Weakness effect is a powerful dynamic where the strongest player is often excluded because weaker players form a "cheapest winning coalition."
  • This effect is largely driven by the equity norm: strong players demand a proportionally larger share of payoffs, making them less attractive partners.
  • In a vee, this can lead to an "all-up" coalition of partners $\{P_1,P_2\}$ to increase bargaining power, or a "bridging" coalition like $\{H,P_1\}$ that might marginalize $P_2$.

We have now addressed the first major question of cooperative game theory: which coalitions will form? This leaves the second question unanswered: once a coalition forms, how should the value it creates be divided fairly? In our next lesson, we will explore this by introducing the Shapley value, a celebrated method for allocating cooperative payoffs based on each player's marginal contribution.