Core Stability in Vee Coalitions

Stability Beyond the Grand Coalition

The core tells us about the stability of the grand coalition against defections by groups. But what if we start with a different structure, like the hinge (H) partnered with P1, and P2 on their own? This structure is written as $\mathcal{C}=\{\{H,P1\},\{P2\}\}$. Is this arrangement stable?

For this structure to be stable, no single individual should have an incentive to change their situation.

• Could H get a better outcome by leaving P1 to be on their own?
• Could P1 get a better outcome by leaving H?
• Could P2 improve their lot by joining H, or P1, or both?

This concept is called the stability of a coalition structure. An arrangement is stable if no player can improve their own payoff by unilaterally moving to another coalition (or forming one of their own). This evaluation requires a method to assign payoffs within any given structure. As we've seen, the Shapley value is an excellent tool for this.

The following reading formally defines this idea.

The definition you just read provides our analytical framework. To check the stability of a given structure, we:

1. Calculate each player's payoff within that structure (e.g., using the Shapley value).
2. Identify all possible unilateral moves for each player.
3. For each potential move, calculate the new payoff the deviating player would receive in the new structure.
4. If no player can find a move that increases their payoff, the original structure is stable.

The Landscape of Vee Stability

The stability of any structure ultimately depends on the value created by different combinations of partners. Theodore Caplow, a sociologist, created a well-known typology of triads based on power dynamics.

Let's use a mapping similar to Caplow's to analyze our vee. We can define a general characteristic function for a vee (H, P1, P2), whom we'll label as players 1, 2, and 3 respectively for simplicity.

• $v(\{1,2,3\})=c$: The total value of the grand coalition.
• $v(\{2,3\})=c_1$: The value of the two non-hinge partners (P1, P2) forming a coalition without the hinge.
• $v(\{1,3\})=c_2$: The value of the Hinge-P2 dyad.
• $v(\{1,2\})=c_3$: The value of the Hinge-P1 dyad.
• $v(\{i\})=0$: The value of any individual acting alone is normalized to zero.

The relationships between $c,c_1,c_2,c_3$ define the game. For instance, a situation where the hinge is critical might have low $c_1$, while a vee with strong independent dyads would have high $c_2$ and $c_3$. The academic paper you just read provides a complete analysis of stability for all possible coalition structures based on these values.

Let's examine two key structures from their findings.

Let's interpret these conditions in the context of a vee relationship:

1. Stability of the Grand Coalition $\{\{H,P1,P2\}\}$
   The table states this structure is stable if (among other conditions) $c_1\ge 0$. This is intuitive: if the two non-hinge partners can create non-negative value on their own ($c_1\ge 0$), it removes one major pressure for them to disrupt the grand coalition. The other, more complex inequalities relate the total value $c$ to the dyadic values, ensuring that no single player is better off leaving the triad.

2. Stability of the Dyad $\{\{H,P1\},\{P2\}\}$
   The table lists several conditions for this structure to be stable, including:

   • $c_1\le 0$: If the two non-hinge partners (P1 and P2) would create negative value by getting together, there is no incentive for P1 to leave the hinge to join P2. This helps stabilize the $\{\{H,P1\},\{P2\}\}$ structure.
   • $c_3\ge c_1$: (This is simplified from one of the conditions). The value of the existing dyad $v(\{H,P1\})$ must be sufficiently high compared to alternatives.
   • $c+c_1-c_3\le 0$: This complex condition essentially means that the benefit of P2 joining the H-P1 dyad (moving to the grand coalition) isn't large enough to entice a change.

By analyzing the values of $c,c_1,c_2,c_3$ for a specific vee, you can use these tables to diagnose which of the five possible coalition structures ($\{\{1,2,3\}\}$, three dyadic variants, and $\{\{1\},\{2\},\{3\}\}$) are stable endpoints. Sometimes only one is stable; sometimes several are. And sometimes, as the paper notes, no structure is stable, predicting a perpetually turbulent dynamic.

Revisiting the Core: The Ultimate Test of the Grand Coalition

How does this new analysis of coalition structures relate to our previous lesson on the core? The two concepts are deeply intertwined.

The core specifically tests the stability of the grand coalition. An empty core signifies that the grand coalition is fundamentally unworkable because at least one dyad can achieve a better per-person outcome by themselves than any plausible division of the grand coalition's total value allows.

A game where the grand coalition is more efficient (per person) than any sub-coalition is called a "totally profitable game." Such games are guaranteed to have a non-empty core. Conversely, if a dyad is "too profitable" compared to the triad, the core may be empty.

Let's translate this into a clear diagnostic for a vee relationship. A game is "not totally profitable"—and thus has an empty core and an unstable grand coalition—if for some coalition $S$, we have:

In our vee, this means the grand coalition is at high risk of collapse if, for example, the Hinge-P1 dyad is exceptionally strong:

This inequality signifies that the per-person value in the dyad is so much higher than in the triad that P2 cannot be offered enough to make it worthwhile for H and P1 to include them. The H-P1 dyad will always be tempted to abandon the grand coalition.

Therefore, the emptiness of the core is the ultimate red flag for the stability of the $\{\{H,P1,P2\}\}$ structure. While the conditions in the first paper we read provide a detailed check, the core provides a powerful, high-level diagnosis of the viability of the full triad.

Conclusion

In this lesson, we expanded our analysis from a binary question (is the grand coalition stable?) to a more sophisticated assessment of the entire landscape of possible relationship structures.

• Key Takeaways:
  • The stability of a coalition structure is determined by whether any individual player has a rational incentive to move to another coalition.
  • We can assess the stability of different structures (e.g., $\{\{H,P1\},\{P2\}\}$) by comparing a player's payoffs (e.g., Shapley value) before and after a potential move.
  • The values of the characteristic function ($c,c_1,c_2,c_3$) determine which structures are stable, allowing for a predictive diagnosis of a relationship's likely configuration.
  • An empty core implies that the per-person value of a sub-coalition is greater than that of the grand coalition. This makes the grand coalition structure inherently unstable and is a strong predictor of its collapse.

This concludes our module on cooperative games and coalition formation. We have built a powerful toolkit to model and analyze the distribution of resources, power, and stability within a vee.

Preview of the next lesson: We will now shift our perspective from the static, rational-choice framework of cooperative game theory to the dynamic world of evolutionary game theory. In the next module, we will begin by modeling the evolution of vee relationship strategies using replicator dynamics, exploring how strategies might spread or die out in a population over time.