Strategic Network Formation in Polyamory

The Game of Creating Connections

So far, we have mostly treated networks as static structures to be analyzed. The crucial shift in strategic network formation is to view the network itself as the outcome of choices made by self-interested individuals. This perspective, which you'll recognize from your background in game theory and econometrics, emphasizes human agency. Individuals aren't just passive nodes; they are active players shaping the structure around them based on their preferences and expectations.

The following reading from Sanjeev Goyal's lecture notes on network formation introduces this economic approach. It highlights that linking is a deliberate decision with consequences that spill over to others—a concept known as externalities.

As Goyal's notes clarify, to model network formation as a game, we must formally define three core components. Let's break down each one in the context of polyamorous relationships.

1. Players: Who's in the Game?

The players are the individuals who can form or dissolve relationships. In the simplest case of a polyandrous vee, the players are the hinge partner (let's call her H) and her two male partners (P1 and P2). The set of players is simply $N=\{H,P1,P2\}$.

However, the game can be more expansive. We could also include potential new partners, ex-partners who might reconnect, or even the partners of partners in a larger polycule. For our purposes, we will primarily focus on the core individuals within a given relationship structure, but it's important to remember that the set of players defines the boundaries of the system you are analyzing.

2. Action Sets: What Can They Do?

A player's action set defines their available moves. In network formation, the fundamental action is the decision to form or maintain a link.

A crucial distinction, relevant to all relationships, is the rule for link formation:

• Unilateral Linking: One player can create a link to another without reciprocation. Think of "following" someone on social media. One person initiates, and the link exists. This is less common in deep personal relationships but might model unrequited interest or a one-way flow of support.
• Pairwise (or Bilateral) Linking: A link is formed only if both players agree. This is the standard assumption for modeling romantic or sexual relationships, including the connections in a vee. Player A can announce they want to link with Player B, but the link $g_{AB}$ only forms if B also agrees.

Therefore, for any player, their action set consists of proposing, accepting, maintaining, or severing links with every other player in the game. The collection of all links that are successfully formed through these actions creates the network graph, $g$. For example, in a vee, the network consists of the links $g_{HP1}$ and $g_{HP2}$.

3. Payoff Functions: What Do They Get?

The payoff function, or utility function, is the heart of the model. It quantifies what a player gains or loses from a particular network structure. As your work in efficiency analysis and operations research suggests, this is essentially a cost-benefit calculation. The general form is:

$Payof{f}_i(g)=Benefit{s}_i(g)-Cost{s}_i(g)$

where $g$ is the resulting network structure. The benefits and costs depend on the player's position in that network. The paper "Strategic Network Formation with Structural Holes" provides an excellent, concrete model for this.

Let's use the structure from this paper to define a payoff function for a polyamorous vee relationship.

Costs

The cost of a link ($c_{uv}$) represents the time, energy, emotional labor, and material resources required to sustain that connection. These costs are rarely uniform. The cost for a hinge to maintain a relationship with a long-distance partner would be higher than with one who lives nearby. The cost may also reflect emotional difficulty or incompatibility.

For a player $u$, the total cost is the sum of the costs of all links they choose to maintain: $Cost{s}_u(g)={\sum }_{v\in L(u)}{c}_{uv}$, where $L(u)$ is the set of links initiated by $u$.

Benefits

Benefits are twofold, arising from both direct connection and network position.

1. Direct Benefits (${\alpha }_0$): This is the intrinsic value derived from a relationship—companionship, emotional support, sexual intimacy, intellectual connection, etc. In the model, each direct connection provides a fixed benefit. For the hinge (H) in a vee with partners P1 and P2, her direct benefits would be ${\alpha }_{HP1}+{\alpha }_{HP2}$.

2. Network Benefits ($\beta (r)$): This captures the value derived from your position in the wider network, which connects directly to our last lesson on structural holes. In the Kleinberg model, this is specifically the "intermediary benefit" gained from being the middleman on a path of length two between two otherwise unconnected people.

   • In a vee, the hinge H is the unique intermediary between P1 and P2. The path is P1-H-P2. Here, the number of intermediaries is $r=1$. The hinge receives a brokerage benefit of $\beta (1)$.
   • The non-hinge partners, P1 and P2, have no one "under" them to broker, so in this simple model, their network benefit from brokerage is zero. Their payoffs are dominated by the direct benefit of their connection to H, minus its cost.

Assembling the Payoff Function

Using this framework, we can write down the payoff functions for each player in a stable vee:

• Hinge's Payoff:
  $U_H(g_{vee})=({\alpha }_{HP1}+{\alpha }_{HP2})+\beta (1)-(c_{HP1}+c_{HP2})$
  (Direct benefits from two partners + brokerage benefit - costs of two relationships)

• Partner 1's Payoff:
  $U_{P1}(g_{vee})={\alpha }_{P1H}-c_{P1H}$
  (Direct benefit from hinge - cost of one relationship)

• Partner 2's Payoff:
  $U_{P2}(g_{vee})={\alpha }_{P2H}-c_{P2H}$
  (Direct benefit from hinge - cost of one relationship)

This formalization immediately highlights the strategic tensions. The hinge's payoff includes a distinct term, $\beta (1)$, representing the power and opportunity from their brokerage position. Any move by P1 and P2 to form a direct link (turning the vee into a triad) would eliminate this structural hole, and thus this portion of the hinge's payoff. This is the mathematical expression of the strategic incentives we discussed conceptually in the last lesson.

Of course, this model is a simplification. The payoff functions for P1 and P2 could include externalities. For example, P1 might experience compersion from H's relationship with P2, adding a positive term to P1's utility. Conversely, he might experience jealousy, adding a negative term. Your experience in causal analysis is highly relevant here, as defining these functions requires making careful assumptions about what factors drive an individual's utility.

Conclusion

In this lesson, we have laid the formal groundwork for analyzing how relationship networks evolve. We've defined the three essential components of a strategic network formation game and applied them directly to the context of polyamorous relationships.

• Key Takeaways:
  • Players: The individuals with the agency to form or sever connections.
  • Action Sets: The range of linking decisions a player can make, typically requiring mutual consent (pairwise formation) in relationships.
  • Payoff Functions: A utility model that calculates a player's net benefit from a given network structure. This function is typically a trade-off between the costs of maintaining links and the benefits, which include both direct (intrinsic value of a relationship) and network (structural advantage like brokerage) components.

By translating relationship dynamics into players, actions, and payoffs, we have created a formal game that we can now analyze.

Preview of the next lesson:
Now that we have the components of our game, the next logical question is: which networks will actually form? Not all conceivable structures are stable. In the next lesson, we will introduce the crucial concept of pairwise stability. This will allow us to assess whether a given network configuration, like a vee, is a likely equilibrium outcome when every player is strategically managing their relationships based on the payoffs we've just defined.