Jealousy & Compersion as Network Externalities in a Vee

The Idea of Other-Regarding Preferences

The foundational shift we need to make is to relax the assumption that rational actors care only about their own material or direct payoffs. A more realistic model acknowledges that our feelings, and thus our utility, are often tied to the well-being of others. A paper by Askari et al. introduces a concept called "hyper-rational choice" which provides a straightforward entry point to this idea.

The concept of hyper-rationality suggests that a person's preferences can be divided into two classes: individual preferences (for their own profit/loss) and preferences for others (for others' profit/loss). This allows us to model a player who might derive satisfaction not just from their own outcome, but also from seeing a partner happy (compersion) or, conversely, feel displeasure at a partner's perceived better outcome (jealousy).

In the context of a vee, a non-hinge partner's utility might not only depend on the time they get with the hinge, but also on how the other non-hinge partner is faring. This is the essence of a network-dependent externality.

A Formal Structure for Social Comparison

To move from this general idea to a predictive model, we need a more rigorous mathematical structure. The work of Maccheroni, Marinacci, and Rustichini in "Social Decision Theory" provides a powerful, axiomatic framework for doing exactly this. They propose that an individual's total utility is composed of two parts: their personal, intrinsic utility from an outcome, and a "positional index" that captures the satisfaction or dissatisfaction from social comparison.

The core of their model is presented in Equation (1), which we can write conceptually as:

Let's break this down:

• Intrinsic Utility: This is the standard part of a utility function. It's the happiness you get from your own outcome, isolated from any social context. For example, the enjoyment of a quiet evening alone. This is represented by the first term in their equation: ${\int }_{S}u(f_o(s))dP(s)$.
• Positional Utility: This is the externality term. It captures the feelings that arise from comparing your outcome to the outcomes of others in your reference group (in our case, the other two members of the vee). This is represented by the second term: ${\int }_S\varphi (v(f_o(s)),{\sum }_{i\in I}{\delta }_{v(f_i(s))})dP(s)$.

The function $\varphi$, the "positional index," is where the magic happens. It takes your own outcome's "social value," $v(f_o(s))$, and compares it to the distribution of your partners' outcomes.

The paper makes a subtle but crucial distinction between the private utility function $u$ and the social value function $v$.

• If $u=v$, it models what the authors call a "private emotion." You envy someone for having something because you, personally, would enjoy it. This could model jealousy over a partner getting to spend time with the hinge, because you desire that time for yourself.
• If $u\ne v$, it models a "social emotion." You envy someone for the symbolic or status value of their outcome, even if you wouldn't personally enjoy it. This could model jealousy over a partner attending a high-profile event with the hinge, not because you wanted to go, but because of what it signals about their status within the relationship.

This framework allows us to precisely define how the outcomes of other network members feed into a player's own utility function.

Separating Jealousy (Invidia) from Compersion (Superbia)

The model can be refined even further to distinguish between the feelings you have toward those doing better than you and those doing worse. This is critical for separately modeling jealousy and compersion.

Equation (20) in the paper reformulates the positional utility to depend on three arguments:

1. Your own social value.
2. The distribution of outcomes for peers who are worse off than you.
3. The distribution of outcomes for peers who are better off or equal to you.

This separation gives us a powerful tool for modeling vee dynamics:

• Modeling Jealousy (Invidia): Let's say Partner 1 (P1) has a quiet night in (low utility outcome), while the Hinge (H) and Partner 2 (P2) are on a fancy date (high utility outcome). P1's positional utility function, $\varphi$, is now activated by P2's high outcome appearing in the third argument (peers who are better off). The model specifies that utility is decreasing in this argument, so P1 experiences a direct negative payoff—a "jealousy externality"—as a result of the H-P2 interaction. The structure of the network dictates who is in the "reference group" for this comparison.

• Modeling Compersion (Superbia): Compersion is the joy one feels for a partner's happiness. This can be modeled in a few ways. One interpretation, related to the paper's "pride" (superbia), is feeling good when your outcome is better than others'. A hinge might feel a sense of pride or satisfaction from being able to facilitate positive experiences for both partners, where their own outcome is perceived as highest. More directly, and aligning with the "hyper-rationality" concept, we can define the positional utility function $\varphi$ to be increasing in the outcomes of others, regardless of comparison. If your partner has a high utility outcome, your $\varphi$ is positive. This captures the altruistic component of compersion, where you gain utility simply from their joy.

A Dynamic View: Utility from Reciprocity

The models above focus on comparing static outcomes. However, emotions like jealousy and compersion are often responses to actions perceived as kind, unkind, fair, or unfair. Rabin's model of reciprocity provides a framework for this dynamic aspect.

Rabin's model breaks a player's utility into three parts:

1. Consumption Utility: The direct payoff from the outcome.
2. Kindness Utility: The direct utility (positive or negative) you get from how kindly you perceive another player has acted toward you.
3. Reciprocity Utility: The utility you get from matching the other player's behavior: being kind to those who are kind to you, and unkind to those who are unkind to you.

A player's "kindness" (or fairness) is measured by comparing the payoff they chose to give you with the range of payoffs they could have given you.

Let's apply this to a vee:

• Jealousy as a Response to Unkindness: The hinge has a choice: spend Friday evening with P1 or P2. They choose P2. From P1's perspective, the hinge chose an action that gave P1 a low payoff when a higher payoff (spending the evening together) was possible. This makes the hinge's action appear "unkind" ($f_H<0$). This generates direct disutility for P1. The model also predicts P1 will be motivated to act unkindly in return to gain reciprocity utility. This models the retaliatory cycle that can be triggered by perceived slights.
• Compersion as a Response to Kindness: The hinge is spending the weekend with P2, but before they go, they arrange a special gift or plan a lovely activity for P1. This action is perceived as "kind" ($f_H>0$), generating positive utility for P1. P1 feels good about how they were treated, even though the hinge is with P2. This can foster a desire to be kind in return, creating a positive feedback loop. This is a powerful way to model how thoughtful actions by the hinge can generate compersion and mitigate jealousy.

Conclusion

In this lesson, we have developed a sophisticated toolkit for modeling the emotional externalities of jealousy and compersion within a vee network. We've moved beyond a simple parameter to structured, theory-grounded utility functions.

Key Takeaways:

• Jealousy and compersion can be modeled as network-dependent externalities by incorporating others' outcomes and actions directly into a player's utility function.
• The Social Decision Theory framework separates utility into an intrinsic component and a positional component, allowing us to formally model social comparisons.
• We can distinguish between envy towards those better off (invidia) and pride/satisfaction regarding those worse off (superbia), giving us a nuanced way to model jealousy and some aspects of compersion.
• Rabin's reciprocity model provides a dynamic perspective, where utility is derived from the perceived kindness or unkindness of others' actions, not just their final outcomes.

By combining these approaches, you can construct detailed models that capture the complex interplay of emotions, strategic actions, and network position that defines the dynamics of a vee relationship.

In our next lesson, we will use these enriched utility functions to analyze the overall stability of the system. We will explore what it means for a game on a network to be in equilibrium when players' payoffs are so deeply intertwined.