Modeling Vee-to-Triad Transitions as a Network Game

The Game of Forming Relationships

So far, we have analyzed games where players choose actions (like how to allocate time) on a fixed network. Now, we consider a game where the primary "action" is choosing the network itself. This is the essence of a strategic network formation game.

The paper "Games on Endogenous Networks" by Sadler and Golub provides a clear, formal structure for this type of game. We will adapt its core components to our polyamorous context.

As you've just read, a network formation game consists of:

• Players (N): In our case, the hinge (H) and her two partners (P1 and P2).
• Actions (S): For our purposes, we'll simplify this. The key strategic decision for each player is whether to consent to forming or maintaining a link with another player.
• Payoffs (ui): The utility each player receives, which depends on the final network graph g. We will continue to use the utility function from our previous lessons: $U_i(g)={\sum }_{j\ne i}{\delta }^{d(i,j)}-n_i(g)\cdot c$.

An outcome of this game is a network structure (a graph g) that is stable. As the reading notes, a strictly pairwise stable outcome is one where no player can unilaterally improve their situation by changing their actions (which we've simplified) and no pair of players has a mutual desire to add a link, nor does any single player wish to sever one.

Modeling the Transition: From Vee to Triad

The most natural evolution from a vee is the formation of a link between the two non-hinge partners, P1 and P2, closing the triangle. This is a classic example of triadic closure.

The principle of triadic closure suggests that if two people have a strong tie to a common third person, they have an increased likelihood of forming a tie themselves. In our game-theoretic model, this "likelihood" is replaced by a formal incentive calculation.

The student blog post "Polyamory as a Network?" discusses this exact phenomenon in the context of real-world polyamorous communities.

Let's model this transition. The vee is unstable and will transition to a triad if both P1 and P2 find it mutually beneficial to form a link. In the vee, the utility for P1 is $U_{P1}(g_{vee})=\delta +{\delta }^2-c$. If they form a link with P2, the graph becomes a triad, and P1's new utility is $U_{P1}(g_{triad})=2\delta -2c$.

P1 and P2 will mutually agree to form the link if their utility increases. For P1 (and symmetrically for P2):

This is precisely the condition under which we previously found the vee to be unstable. In the language of a network formation game, when this condition holds, the vee is not a pairwise stable outcome. Players P1 and P2 have a strategic incentive to add the link {P1, P2}. The resulting triad, where everyone is connected, is stable under this low-cost condition (as no one would want to sever a link). Therefore, the game predicts a transition from the vee to the triad.

Modeling the Transition: From Vee to Dyad or Nothing

What about transitions that involve breaking up? This happens when the cost of maintaining a link becomes too high for a player.

Consider the hinge, H. Her utility in the vee is $U_H(g_{vee})=2\delta -2c$. If she decides the cost c is too high and severs the link with P2, the network becomes a single dyad {H, P1}. Her utility would then be $U_H(g_{dyad})=\delta -c$.

She will make this move if it improves her payoff:

When the cost of maintaining a link exceeds the direct benefit, the hinge is incentivized to start severing connections. The vee is not stable. The network formation game predicts a transition from a vee to a dyad. If $c>\delta$ applies to both relationships, she would sever both links, causing a transition to an empty network.

The blog post you read earlier also touches on a similar idea, linking instability to resource constraints like time or money.

Generalizing the Dynamics: Spillovers and Substitutes

To broaden our understanding, we can use the more general concepts from the "Games on Endogenous Networks" paper. These help describe the underlying forces driving network changes in a way that goes beyond our specific utility formula. Let's imagine players also choose an "action" or "effort" level s (e.g., emotional investment).

1. Positive vs. Negative Spillovers: This describes how one person's action affects their attractiveness as a partner. In relationships, we typically see positive spillovers: if P1 becomes more emotionally invested or a better communicator (higher s_P1), they become a more desirable partner for H.

2. Action-Link Complements vs. Substitutes: This describes how a player's own action level affects their desire for more links.

   • Complements: A person who becomes more invested (s increases) also wants more partners. This describes an expansive, connection-seeking strategy.
   • Substitutes: A person who becomes more invested (s increases) wants fewer partners. This captures the very real constraint of time and energy. Deeper investment in one relationship may make adding another too costly.

Our model of vee dissolution ($c>\delta$) is a classic case of action-link substitutes. The "action" is the maintenance of a link, and its cost c is high. As the hinge adds links, the total cost increases, making her less willing to add or maintain them.

The paper "Games on Endogenous Networks" formally defines these concepts.

The combination of these properties can predict the large-scale structure of stable networks. For example, a combination of positive spillovers (people want to date emotionally invested partners) and action-link substitutes (invested people have less capacity for more partners) tends to produce networks of ordered overlapping cliques. This means people end up in groups with others of a similar investment level, which is a highly intuitive result. The transition from a vee to a triad is the formation of a small clique.

Conversely, a combination of positive spillovers and action-link complements tends to produce nested split graphs—hierarchical networks with a highly connected core. This might describe a "polyamory star" structure where one or two central individuals are connected to many people who are not connected to each other.

Conclusion

In this lesson, we have framed the evolution of relationship structures as a strategic network formation game. This moves our analysis from static snapshots to dynamic transitions.

• Key Takeaways:
  • A strategic network formation game models how rational players choose which relationship links to form or sever based on their utility functions.
  • The transition from a vee to a triad is predicted when the cost of adding the final link is low enough for the two non-hinge partners to mutually benefit ($c<\delta (1-\delta )$). This is a game-theoretic formalization of triadic closure.
  • The transition from a vee to a dyad or an empty network is predicted when the cost of maintaining links is too high for a player, particularly the hinge ($c>\delta$). This models dissolution due to resource constraints.
  • More general properties like action-link substitutes (where more investment makes one want fewer links) can explain the forces driving these transitions and predict what kind of large-scale structures (e.g., cliques) are likely to be stable.

Preview of the next lesson:
Our model of network formation relies on the idea of mutual consent to add a link, a core assumption of pairwise stability. In our next lesson, we will delve deeper into this by examining the role of consent and mutual agreement constraints in these models. We will explore how different rules for forming and dissolving relationships can alter the strategic landscape and the ultimate stability of network structures.