Consent and Mutual Agreement in Strategic Network Formation

The Implicit Consent Model in Pairwise Stability

Let's begin by formally revisiting the stability concept that has underpinned our recent discussions. The model of pairwise stability is built on a specific, intuitive set of rules about consent.

As you've just read, pairwise stability elegantly captures a common-sense model of consent in many social contexts:

1. Unilateral Consent to Sever: For an existing link ij to be stable, both player i and player j must benefit from it remaining. If either player would be better off severing the link, they can do so unilaterally. This is captured by condition (i): $Y_i(g)\ge Y_i(g-ij)$ AND $Y_j(g)\ge Y_j(g-ij)$.
2. Mutual Consent to Add: For a non-existent link ij to be added, it must benefit both players. If adding the link would make one player worse off, the link is not formed, even if the other player desires it. This is captured by condition (ii): if $Y_i(g+ij)>Y_i(g)$, it must be that $Y_j(g+ij)<Y_j(g)$ for the network g to be stable.

In the context of our vee {H, P1, P2}, this means:

• The hinge H can decide to end her relationship with P2 unilaterally.
• For P1 and P2 to form a new relationship (closing the triad), they must both agree.

This default model seems natural, but it's just one possibility. What happens when we remember that real relationships aren't abstract games, but exist within a rich context of trust, history, and explicit agreements?

Why Context and Agreements Matter

The sterile, context-free scenarios of classic game theory often miss the most important elements of human interaction. The late Nobel laureate Thomas Schelling was a master at highlighting this. In a famous lecture, he deconstructs the Prisoner's Dilemma by asking simple questions about the context that the formal model ignores.

Schelling's point is crucial: the rules of engagement, whether based on social norms (like a "Scout's honor") or explicit negotiation, are not external to the game; they define the game. In polyamorous relationships, these rules are often made explicit through communication and agreements. So, let's treat the consent mechanism not as a fixed assumption, but as a variable we can change.

Changing the Rules: An Experimental View

What if forming a link didn't require mutual consent? Or what if ending one did? A fascinating paper by Gary Charness and Matthew O. Jackson explores exactly this by framing network formation as a game where the "voting rule" for group decisions can be changed.

They map their experiment directly onto the problem of network formation, providing a clear illustration of how consent rules function.

The experimental results are stark: the consent rule is not a minor detail but a primary driver of behavior. People's strategies adapt to the rule structure. Let's apply this insight to our vee model.

Re-evaluating Vee Stability with Different Consent Rules

Our previous analysis used the standard pairwise stability model. Now, let's consider how alternative consent rules would alter the stability of a vee relationship.

Rule for Severing a Link

• Standard Model (Unilateral Sever): The hinge H can break up with partner P2 if her utility from doing so is higher. As we saw, this happens if the cost of the link is greater than its direct benefit ($c>\delta$). This rule grants high individual autonomy.
• Alternative Model (Mutual Consent to Sever): Imagine an agreement where a relationship can only end if both parties consent. For H to break up with P2, P2 would also have to agree. P2's utility from severing the link is negative (they go from receiving benefits to receiving nothing). Therefore, P2 would never agree. Under this rule, links become incredibly "sticky," and the vee would be stable even when it makes the hinge unhappy. This illustrates how such a rule, while seemingly promoting stability, could trap individuals in undesirable situations.

Rule for Adding a Link

• Standard Model (Mutual Consent to Add): For partners P1 and P2 to form a link and create a triad, the move must be beneficial for both. This happens if $c<\delta (1-\delta )$.
• Alternative Model (Unilateral Consent to Add): Imagine a rule where if P1 wants to form a relationship with P2, it happens regardless of P2's wishes. If the condition $c<\delta (1-\delta )$ holds, P1 has an incentive to form the link. Under this rule, P1 could force the triad into existence. This model starkly demonstrates the protective function of the mutual consent norm. It prevents individuals from being forced into connections that do not benefit them, which is a cornerstone of ethical relationship structures.

This analysis shows that the consent mechanism specified in a relationship agreement is a powerful constraint that dictates which structures are stable and which transitions are possible.

The Inherent Tension Between Stability and Group Happiness

We've seen how rules about consent influence stability. This leads to a deeper question you might be familiar with from your Operations Research background: can we design a "perfect" set of rules that ensures the resulting stable network is also the most efficient one (i.e., maximizes the total happiness of the group)?

The surprising answer from network theory is no. There is an inherent tension between individual incentives (stability) and collective well-being (efficiency).

Proposition 6 is a profound result. It states that no allocation rule (a set of rules for distributing value, or utility) can exist that simultaneously satisfies basic fairness conditions (anonymity, component balance) and always guarantees that a stable network is also an efficient one.

The 3-player example in the proof is directly analogous to a polyamorous structure. It shows a situation where making the network stable for each individual requires an allocation of payoffs that is simply impossible—the sum required to keep everyone from breaking or changing links is more than the total value the network generates.

The implication for polyamorous relationships is significant: even with perfectly fair and well-intentioned agreements, there is no system of rules that can guarantee that a stable relationship structure is also the one that maximizes the total well-being of all involved. The goals of individual stability and group optimality can, and often do, conflict.

Conclusion

In this lesson, we have moved consent from an implicit assumption to the central object of our analysis. By evaluating the role of agreement constraints in network formation, we've uncovered several key insights.

• Key Takeaways:
  • Standard models like pairwise stability contain an implicit consent structure: mutual consent to add a link, and unilateral consent to sever one.
  • Altering these consent rules, as demonstrated by experimental evidence, fundamentally changes strategic behavior and resulting network outcomes. The rules define the game.
  • The norm of mutual consent to form relationships serves a critical function: it protects individuals from being forced into connections that are not in their interest.
  • There is a fundamental, mathematically demonstrable tension between the stability of a network (driven by individual incentives) and its efficiency (the collective good). No set of "fair rules" can eliminate this conflict in all situations.

Preview of the next lesson:
This lesson concludes our module on the strategic formation of relationship networks. We've explored how these structures are built and stabilized by the strategic choices of their members. Now, we will shift our focus to what happens within these established networks. In the next module, we will begin exploring the diffusion of information, influence, and even emotions, starting with a lesson that distinguishes between "simple" and "complex" contagion and its relevance for understanding how things spread through a relationship network.