Modeling Partner Utility and Network Effects

In our previous lesson, we established the fundamental components of a strategic network formation game: the players, their possible actions (forming or severing ties), and the payoff functions that motivate their choices. We conceptualized payoffs as a balance of benefits and costs, with a special focus on the brokerage power derived from occupying a structural hole.

Today, we will build on that foundation to address a more specific task: constructing a simple utility model for a partner considering forming or dissolving a tie, incorporating both direct benefits and network externalities. We will move from the general idea of a payoff function to a concrete, quantifiable model that can be used to analyze a player's decisions. This involves formalizing the intuitive costs and benefits of relationships into a mathematical expression.

From Preferences to Utility

Before we build our model, it's helpful to briefly revisit the concept of utility itself. In game theory and econometrics, a utility function is not an arbitrary assignment of numbers; it is a compact, mathematical representation of an individual's underlying preferences. People don't think in terms of utility points, but they do think about which outcomes they prefer over others. As analysts, we map these preferences to numbers to make them tractable.

The following video from Game Theory 101 offers an excellent refresher on this core principle. It clarifies the crucial relationship between preferences and the utility numbers we use to model them.

Game Theory 101 (#42): Expected Utility Theory

This video discusses the philosophical underpinning of utility theory, explaining that utilities are a tool to represent preferences, not the other way around.

Please watch the segment from the beginning to this point. The key insight is that preferences are the primary concept. The speaker emphasizes that we don't prefer an outcome because its utility is higher; rather, we assign it a higher utility because we prefer it.

With this principle in mind—that our model should reflect underlying preferences—let's construct a utility function that captures the costs and benefits for a person in a polyamorous network.

The Connections Model: A Framework for Utility

A classic and intuitive framework for this task is the connections model, developed by Matthew Jackson and Asher Wolinsky. This model formalizes the idea that we benefit not only from our direct friends (or partners) but also from the "friends of our friends," though to a lesser extent. The benefits decay with social distance.

The survey paper "A Survey of Models of Network Formation" by Matthew Jackson provides a clear description of this model and its components.

A Survey of Models of Network Formation: Stability and ...

This reading details the components of a network game and presents the connections model, a foundational utility model in network theory.

Please read two sections from this paper: First, familiarize yourself with the concepts of Value Functions and Allocation Rules. Start at the heading "Value Functions" and read through to the end of the section on Allocation Rules. This explains how total value is generated by a network and then distributed among its players. Next, read Example 1: The Connections Model. You can find this under the heading "Some Examples". This section provides the specific utility function we will be working with.

Let's break down the utility function from the connections model and apply it to our vee relationship context. The utility of a player $i$ in a given network $g$ is defined as the sum of all benefits they receive, minus the sum of all costs they incur.

U_{i} (g) = Benefits_{i} (g) - Costs_{i} (g)

1. Costs

This is the most straightforward part of the model. Maintaining a relationship requires resources: time, emotional energy, and potentially material assets. The model simplifies this by assigning a cost, $c_{ij}$ , for each direct link that player $i$ maintains. If we assume the cost is the same for all relationships, a player's total cost is simply the number of partners they have, $n_{i} (g)$ , multiplied by the cost per link, $c$ .

Costs_{i} (g) = n_{i} (g) \cdot c

2. Benefits (Direct and Indirect)

The benefit side of the equation is more nuanced. The connections model captures both the value of direct relationships and the value derived from being connected to a wider network—a clear form of network externality.

Direct Benefits: A direct link to another person provides intrinsic value (companionship, support, etc.). The model represents this with a benefit parameter, $δ$ .
Indirect Benefits: You also benefit from your partners' connections. A "partner of a partner" can be a source of information, social opportunities, or support. The model captures this by assuming the benefit decays with the "distance" of the relationship. If a direct link is worth $δ$ , a connection two steps away is worth $δ^{2}$ , a connection three steps away is worth $δ^{3}$ , and so on. We assume $0 < δ < 1$ so that more distant connections are less valuable.

The total benefit for player $i$ is the sum of benefits from every other player $j$ in the network, discounted by their path distance $d (i, j)$ .

Benefits_{i} (g) = j \neq = i \sum δ^{d (i, j)}

(Note: If there is no path between $i$ and $j$ , $d (i, j) = \infty$ , and $δ^{\infty} = 0$ .)

The Complete Utility Function

Combining these components, the utility for player $i$ in network $g$ under the symmetric connections model is:

U_{i} (g) = j \neq = i \sum δ^{d (i, j)} - n_{i} (g) \cdot c

Applying the Model to a Vee Relationship Decision

Let's use this model to analyze a decision from the perspective of a non-hinge partner, P1, in a vee (P1-H-P2).

Scenario: P1 considers whether to stay in the relationship.

Network $g_{v ee}$ : The vee consists of links P1-H and H-P2.
P1's Connections:
- A direct link to H (distance 1). Benefit: $δ^{1}$ .
- An indirect link to P2 via H (distance 2). Benefit: $δ^{2}$ .
P1's Costs: One direct link (to H). Cost: $c$ .

So, P1's utility in the vee is:

U_{P 1} (g_{v ee}) = (δ^{1} + δ^{2}) - c

To make a decision, P1 compares this utility to their utility from leaving the relationship. If P1 leaves, they are isolated and have no links. Their utility is 0.

Decision Rule: P1 will choose to stay in the vee as long as their utility is positive:

U_{P 1} (g_{v ee}) > 0 ⟹ δ + δ^{2} - c > 0

This simple inequality formally captures the trade-off. The intrinsic benefit from the partner ( $δ$ ) plus the external benefit from being connected to the wider polycule ( $δ^{2}$ ) must outweigh the cost of maintaining the relationship ( $c$ ).

Network Externalities: Compersion and Jealousy

This model gives us a powerful way to formally describe network externalities. An externality occurs when an action by some players affects the utility of another player who wasn't part of the action.

Consider the hinge, H, deciding whether to start a new relationship with P2, while already with P1. This is the decision to form a vee from a dyad.

Before: H is only with P1. P1's utility is $U_{P 1} (g_{d y a d}) = δ - c$ .
After: H forms the vee P1-H-P2. P1's new utility is $U_{P 1} (g_{v ee}) = (δ + δ^{2}) - c$ .

The change in P1's utility due to H's action is $Δ U_{P 1} = U_{P 1} (g_{v ee}) - U_{P 1} (g_{d y a d}) = δ^{2}$ . This is a positive externality. H forming a new relationship with P2 increases P1's utility in this model, because it gives P1 indirect access to a new person. This can be interpreted as a simple model of compersion or the benefits of an expanded social network.

Of course, this model is a simplification. It doesn't capture negative externalities like jealousy or the strain on a hinge's resources. More advanced models, such as the "local spillovers" model mentioned in the LINK reading, could define the benefit $δ$ as a decreasing function of the hinge's number of partners, thereby introducing a negative externality to represent the division of time and attention. Your background in operations research and simulation is perfectly suited for exploring such extensions, where you could define more complex cost and benefit functions based on different assumptions about relationship dynamics.

Conclusion

In this lesson, we have moved from a conceptual outline to a concrete, analytical tool for modeling relationship decisions. By constructing a simple utility model, we can formally represent the trade-offs a person faces when deciding to form, maintain, or dissolve a relationship tie.

Key Takeaways:
- Utility models are formal representations of an individual's preferences, allowing for quantitative analysis of strategic decisions.
- The connections model provides a simple yet powerful utility function: $U_{i} (g) = \sum_{j \neq = i} δ^{d (i, j)} - n_{i} (g) \cdot c$ .
- This model incorporates the direct benefits of a relationship ( $δ$ ), the costs of maintaining it ( $c$ ), and network externalities through indirect benefits from more distant connections ( $δ^{k}$ ).
- By comparing the utility of different network structures, a player can make a rational decision about whether to add or sever a tie.

Preview of the next lesson:
We have now defined how a single player evaluates their options. But a network is formed by the choices of all players. What happens when everyone is simultaneously trying to maximize their own utility? In our next lesson, we will introduce pairwise stability, an equilibrium concept that helps us predict which network structures are likely to emerge and persist when all players act on the incentives defined by their utility models.

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